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114

Tracking Control for NeuromuscularElectrical Stimulation

1Michael Malisoff Roy P Daniels Professor of Mathematics

Louisiana State University

1JOINT WITH MARCIO DE QUEIROZ (LSU) IASSON KARAFYLLIS (NTUA)MIROSLAV KRSTIC (UCSD) AND RUZHOU YANG (LSU)

SPONSORED BY NSFECCSEPCN PROGRAM

1Applied and Computational Mathematics SeminarGeorgia Tech ndash Skiles 005 ndash April 20 2015

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

214

Problem and Our Solution

NMES artificially stimulates skeletal muscles to restore functionin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)

It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion

Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes

Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)

Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

214

Problem and Our Solution

NMES artificially stimulates skeletal muscles to restore functionin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)

It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion

Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes

Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)

Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

214

Problem and Our Solution

NMES artificially stimulates skeletal muscles to restore functionin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)

It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion

Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes

Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)

Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

214

Problem and Our Solution

NMES artificially stimulates skeletal muscles to restore functionin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)

It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion

Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes

Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)

Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

214

Problem and Our Solution

NMES artificially stimulates skeletal muscles to restore functionin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)

It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion

Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes

Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)

Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

214

Problem and Our Solution

NMES artificially stimulates skeletal muscles to restore functionin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)

It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion

Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes

Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)

Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn

We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function u

The functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty

D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ)

Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded

γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t)

s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state

sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn)

andL2

ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed system

E2 Find bounds on the allowable delays τ such thatY prime(t) = F

(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

NMES on Leg Extension Machine

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Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

TVSpotwmv
Media File (videox-ms-wmv)

514

NMES on Leg Extension Machine

(Loading Video)

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

AN14NMESVidmov
Media File (videoquicktime)

514

NMES on Leg Extension Machine

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i

where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi

The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Potential Controller (continued)

Euler iterations used for control

zk+1 = Ω(Ti + khi hi zk v) for k = 0 Ni minus 1 where

z0 =

(tan (q(Ti))minus tan (qd (Ti))

q(Ti )cos2(q(Ti ))

minus qd (Ti )cos2(qd (Ti ))

) hi = τ

Ni

and Ω [0+infin)2 times R2 rarr R2 is defined by

Ω(T h x v) =

[Ω1(T h x v)Ω2(T h x v)

](8)

and the formulas

Ω1(T h x v) = x1 + hx2 and

Ω2(T h x v) = x2 + ζ2d (T ) +int T+h

T g1(ζd (s)+x)ds

+int T+h

T g2 (ζd (s)+x) v(s minus τ)ds minus ζ2d (T +h)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1214

NMES Theorem (IK MM MK Ruzhou IJRNC)

For each pair (τ r) isin (0infin)2 there exist a locally boundedfunction N a constant ω isin (0 micro2) and a locally Lipschitzfunction C satisfying C(0) = 0 such that For all sample timesTi in [0infin) such that supige0 (Ti+1 minus Ti) le r and each initialcondition the solution (q(t) q(t) v(t)) with

Ni = N(∣∣∣(tan(q(Ti)) q(Ti )

cos2(q(Ti ))

)minus ζd (Ti)

∣∣∣+||v minus vd ||[Timinusτ Ti ]

) (9)

satisfies|q(t)minus qd (t)|+ |q(t)minus qd (t)|+ ||v minus vd ||[tminusτ t]

le eminusωtC(|q(0)minusqd (0)|+|q(0)minusqd (0)|

cos2(q(0)) + ||v0 minus vd ||[minusτ 0])

for all t ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

214

Problem and Our Solution

NMES artificially stimulates skeletal muscles to restore functionin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)

It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion

Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes

Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)

Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

214

Problem and Our Solution

NMES artificially stimulates skeletal muscles to restore functionin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)

It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion

Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes

Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)

Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

214

Problem and Our Solution

NMES artificially stimulates skeletal muscles to restore functionin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)

It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion

Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes

Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)

Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

214

Problem and Our Solution

NMES artificially stimulates skeletal muscles to restore functionin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)

It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion

Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes

Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)

Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

214

Problem and Our Solution

NMES artificially stimulates skeletal muscles to restore functionin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)

It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion

Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes

Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)

Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

214

Problem and Our Solution

NMES artificially stimulates skeletal muscles to restore functionin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)

It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion

Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes

Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)

Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn

We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function u

The functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty

D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ)

Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded

γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t)

s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state

sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn)

andL2

ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed system

E2 Find bounds on the allowable delays τ such thatY prime(t) = F

(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

NMES on Leg Extension Machine

(Loading Video)

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

TVSpotwmv
Media File (videox-ms-wmv)

514

NMES on Leg Extension Machine

(Loading Video)

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

AN14NMESVidmov
Media File (videoquicktime)

514

NMES on Leg Extension Machine

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i

where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi

The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Potential Controller (continued)

Euler iterations used for control

zk+1 = Ω(Ti + khi hi zk v) for k = 0 Ni minus 1 where

z0 =

(tan (q(Ti))minus tan (qd (Ti))

q(Ti )cos2(q(Ti ))

minus qd (Ti )cos2(qd (Ti ))

) hi = τ

Ni

and Ω [0+infin)2 times R2 rarr R2 is defined by

Ω(T h x v) =

[Ω1(T h x v)Ω2(T h x v)

](8)

and the formulas

Ω1(T h x v) = x1 + hx2 and

Ω2(T h x v) = x2 + ζ2d (T ) +int T+h

T g1(ζd (s)+x)ds

+int T+h

T g2 (ζd (s)+x) v(s minus τ)ds minus ζ2d (T +h)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1214

NMES Theorem (IK MM MK Ruzhou IJRNC)

For each pair (τ r) isin (0infin)2 there exist a locally boundedfunction N a constant ω isin (0 micro2) and a locally Lipschitzfunction C satisfying C(0) = 0 such that For all sample timesTi in [0infin) such that supige0 (Ti+1 minus Ti) le r and each initialcondition the solution (q(t) q(t) v(t)) with

Ni = N(∣∣∣(tan(q(Ti)) q(Ti )

cos2(q(Ti ))

)minus ζd (Ti)

∣∣∣+||v minus vd ||[Timinusτ Ti ]

) (9)

satisfies|q(t)minus qd (t)|+ |q(t)minus qd (t)|+ ||v minus vd ||[tminusτ t]

le eminusωtC(|q(0)minusqd (0)|+|q(0)minusqd (0)|

cos2(q(0)) + ||v0 minus vd ||[minusτ 0])

for all t ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

214

Problem and Our Solution

NMES artificially stimulates skeletal muscles to restore functionin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)

It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion

Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes

Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)

Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

214

Problem and Our Solution

NMES artificially stimulates skeletal muscles to restore functionin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)

It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion

Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes

Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)

Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

214

Problem and Our Solution

NMES artificially stimulates skeletal muscles to restore functionin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)

It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion

Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes

Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)

Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

214

Problem and Our Solution

NMES artificially stimulates skeletal muscles to restore functionin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)

It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion

Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes

Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)

Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

214

Problem and Our Solution

NMES artificially stimulates skeletal muscles to restore functionin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)

It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion

Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes

Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)

Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn

We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function u

The functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty

D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ)

Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded

γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t)

s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state

sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn)

andL2

ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed system

E2 Find bounds on the allowable delays τ such thatY prime(t) = F

(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

NMES on Leg Extension Machine

(Loading Video)

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

TVSpotwmv
Media File (videox-ms-wmv)

514

NMES on Leg Extension Machine

(Loading Video)

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

AN14NMESVidmov
Media File (videoquicktime)

514

NMES on Leg Extension Machine

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i

where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi

The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Potential Controller (continued)

Euler iterations used for control

zk+1 = Ω(Ti + khi hi zk v) for k = 0 Ni minus 1 where

z0 =

(tan (q(Ti))minus tan (qd (Ti))

q(Ti )cos2(q(Ti ))

minus qd (Ti )cos2(qd (Ti ))

) hi = τ

Ni

and Ω [0+infin)2 times R2 rarr R2 is defined by

Ω(T h x v) =

[Ω1(T h x v)Ω2(T h x v)

](8)

and the formulas

Ω1(T h x v) = x1 + hx2 and

Ω2(T h x v) = x2 + ζ2d (T ) +int T+h

T g1(ζd (s)+x)ds

+int T+h

T g2 (ζd (s)+x) v(s minus τ)ds minus ζ2d (T +h)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1214

NMES Theorem (IK MM MK Ruzhou IJRNC)

For each pair (τ r) isin (0infin)2 there exist a locally boundedfunction N a constant ω isin (0 micro2) and a locally Lipschitzfunction C satisfying C(0) = 0 such that For all sample timesTi in [0infin) such that supige0 (Ti+1 minus Ti) le r and each initialcondition the solution (q(t) q(t) v(t)) with

Ni = N(∣∣∣(tan(q(Ti)) q(Ti )

cos2(q(Ti ))

)minus ζd (Ti)

∣∣∣+||v minus vd ||[Timinusτ Ti ]

) (9)

satisfies|q(t)minus qd (t)|+ |q(t)minus qd (t)|+ ||v minus vd ||[tminusτ t]

le eminusωtC(|q(0)minusqd (0)|+|q(0)minusqd (0)|

cos2(q(0)) + ||v0 minus vd ||[minusτ 0])

for all t ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

214

Problem and Our Solution

NMES artificially stimulates skeletal muscles to restore functionin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)

It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion

Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes

Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)

Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

214

Problem and Our Solution

NMES artificially stimulates skeletal muscles to restore functionin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)

It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion

Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes

Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)

Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

214

Problem and Our Solution

NMES artificially stimulates skeletal muscles to restore functionin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)

It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion

Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes

Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)

Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

214

Problem and Our Solution

NMES artificially stimulates skeletal muscles to restore functionin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)

It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion

Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes

Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)

Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn

We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function u

The functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty

D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ)

Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded

γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t)

s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state

sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn)

andL2

ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed system

E2 Find bounds on the allowable delays τ such thatY prime(t) = F

(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

NMES on Leg Extension Machine

(Loading Video)

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

TVSpotwmv
Media File (videox-ms-wmv)

514

NMES on Leg Extension Machine

(Loading Video)

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

AN14NMESVidmov
Media File (videoquicktime)

514

NMES on Leg Extension Machine

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i

where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi

The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Potential Controller (continued)

Euler iterations used for control

zk+1 = Ω(Ti + khi hi zk v) for k = 0 Ni minus 1 where

z0 =

(tan (q(Ti))minus tan (qd (Ti))

q(Ti )cos2(q(Ti ))

minus qd (Ti )cos2(qd (Ti ))

) hi = τ

Ni

and Ω [0+infin)2 times R2 rarr R2 is defined by

Ω(T h x v) =

[Ω1(T h x v)Ω2(T h x v)

](8)

and the formulas

Ω1(T h x v) = x1 + hx2 and

Ω2(T h x v) = x2 + ζ2d (T ) +int T+h

T g1(ζd (s)+x)ds

+int T+h

T g2 (ζd (s)+x) v(s minus τ)ds minus ζ2d (T +h)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1214

NMES Theorem (IK MM MK Ruzhou IJRNC)

For each pair (τ r) isin (0infin)2 there exist a locally boundedfunction N a constant ω isin (0 micro2) and a locally Lipschitzfunction C satisfying C(0) = 0 such that For all sample timesTi in [0infin) such that supige0 (Ti+1 minus Ti) le r and each initialcondition the solution (q(t) q(t) v(t)) with

Ni = N(∣∣∣(tan(q(Ti)) q(Ti )

cos2(q(Ti ))

)minus ζd (Ti)

∣∣∣+||v minus vd ||[Timinusτ Ti ]

) (9)

satisfies|q(t)minus qd (t)|+ |q(t)minus qd (t)|+ ||v minus vd ||[tminusτ t]

le eminusωtC(|q(0)minusqd (0)|+|q(0)minusqd (0)|

cos2(q(0)) + ||v0 minus vd ||[minusτ 0])

for all t ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

214

Problem and Our Solution

NMES artificially stimulates skeletal muscles to restore functionin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)

It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion

Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes

Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)

Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

214

Problem and Our Solution

NMES artificially stimulates skeletal muscles to restore functionin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)

It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion

Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes

Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)

Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

214

Problem and Our Solution

NMES artificially stimulates skeletal muscles to restore functionin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)

It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion

Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes

Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)

Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn

We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function u

The functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty

D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ)

Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded

γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t)

s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state

sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn)

andL2

ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed system

E2 Find bounds on the allowable delays τ such thatY prime(t) = F

(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

NMES on Leg Extension Machine

(Loading Video)

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

TVSpotwmv
Media File (videox-ms-wmv)

514

NMES on Leg Extension Machine

(Loading Video)

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

AN14NMESVidmov
Media File (videoquicktime)

514

NMES on Leg Extension Machine

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i

where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi

The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Potential Controller (continued)

Euler iterations used for control

zk+1 = Ω(Ti + khi hi zk v) for k = 0 Ni minus 1 where

z0 =

(tan (q(Ti))minus tan (qd (Ti))

q(Ti )cos2(q(Ti ))

minus qd (Ti )cos2(qd (Ti ))

) hi = τ

Ni

and Ω [0+infin)2 times R2 rarr R2 is defined by

Ω(T h x v) =

[Ω1(T h x v)Ω2(T h x v)

](8)

and the formulas

Ω1(T h x v) = x1 + hx2 and

Ω2(T h x v) = x2 + ζ2d (T ) +int T+h

T g1(ζd (s)+x)ds

+int T+h

T g2 (ζd (s)+x) v(s minus τ)ds minus ζ2d (T +h)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1214

NMES Theorem (IK MM MK Ruzhou IJRNC)

For each pair (τ r) isin (0infin)2 there exist a locally boundedfunction N a constant ω isin (0 micro2) and a locally Lipschitzfunction C satisfying C(0) = 0 such that For all sample timesTi in [0infin) such that supige0 (Ti+1 minus Ti) le r and each initialcondition the solution (q(t) q(t) v(t)) with

Ni = N(∣∣∣(tan(q(Ti)) q(Ti )

cos2(q(Ti ))

)minus ζd (Ti)

∣∣∣+||v minus vd ||[Timinusτ Ti ]

) (9)

satisfies|q(t)minus qd (t)|+ |q(t)minus qd (t)|+ ||v minus vd ||[tminusτ t]

le eminusωtC(|q(0)minusqd (0)|+|q(0)minusqd (0)|

cos2(q(0)) + ||v0 minus vd ||[minusτ 0])

for all t ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

214

Problem and Our Solution

NMES artificially stimulates skeletal muscles to restore functionin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)

It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion

Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes

Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)

Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

214

Problem and Our Solution

NMES artificially stimulates skeletal muscles to restore functionin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)

It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion

Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes

Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)

Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn

We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function u

The functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty

D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ)

Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded

γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t)

s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state

sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn)

andL2

ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed system

E2 Find bounds on the allowable delays τ such thatY prime(t) = F

(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

NMES on Leg Extension Machine

(Loading Video)

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

TVSpotwmv
Media File (videox-ms-wmv)

514

NMES on Leg Extension Machine

(Loading Video)

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

AN14NMESVidmov
Media File (videoquicktime)

514

NMES on Leg Extension Machine

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i

where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi

The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Potential Controller (continued)

Euler iterations used for control

zk+1 = Ω(Ti + khi hi zk v) for k = 0 Ni minus 1 where

z0 =

(tan (q(Ti))minus tan (qd (Ti))

q(Ti )cos2(q(Ti ))

minus qd (Ti )cos2(qd (Ti ))

) hi = τ

Ni

and Ω [0+infin)2 times R2 rarr R2 is defined by

Ω(T h x v) =

[Ω1(T h x v)Ω2(T h x v)

](8)

and the formulas

Ω1(T h x v) = x1 + hx2 and

Ω2(T h x v) = x2 + ζ2d (T ) +int T+h

T g1(ζd (s)+x)ds

+int T+h

T g2 (ζd (s)+x) v(s minus τ)ds minus ζ2d (T +h)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1214

NMES Theorem (IK MM MK Ruzhou IJRNC)

For each pair (τ r) isin (0infin)2 there exist a locally boundedfunction N a constant ω isin (0 micro2) and a locally Lipschitzfunction C satisfying C(0) = 0 such that For all sample timesTi in [0infin) such that supige0 (Ti+1 minus Ti) le r and each initialcondition the solution (q(t) q(t) v(t)) with

Ni = N(∣∣∣(tan(q(Ti)) q(Ti )

cos2(q(Ti ))

)minus ζd (Ti)

∣∣∣+||v minus vd ||[Timinusτ Ti ]

) (9)

satisfies|q(t)minus qd (t)|+ |q(t)minus qd (t)|+ ||v minus vd ||[tminusτ t]

le eminusωtC(|q(0)minusqd (0)|+|q(0)minusqd (0)|

cos2(q(0)) + ||v0 minus vd ||[minusτ 0])

for all t ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

214

Problem and Our Solution

NMES artificially stimulates skeletal muscles to restore functionin human limbs (CragoJezernikKoo-LeonessaLevy-Mizrahi)

It entails voltage excitation of skin or implanted electrodes toproduce muscle contraction joint torque and motion

Delays in muscle response come from finite propagation ofchemical ions synaptic transmission delays and other causes

Delay compensating controllers have realized some trackingobjectives including use on humans (Dixon Sharma 2011)

Our new control only needs sampled observations allows anydelay and tracks position and velocity under a state constraint

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn

We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function u

The functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty

D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ)

Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded

γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t)

s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state

sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn)

andL2

ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed system

E2 Find bounds on the allowable delays τ such thatY prime(t) = F

(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

NMES on Leg Extension Machine

(Loading Video)

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

TVSpotwmv
Media File (videox-ms-wmv)

514

NMES on Leg Extension Machine

(Loading Video)

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

AN14NMESVidmov
Media File (videoquicktime)

514

NMES on Leg Extension Machine

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i

where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi

The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Potential Controller (continued)

Euler iterations used for control

zk+1 = Ω(Ti + khi hi zk v) for k = 0 Ni minus 1 where

z0 =

(tan (q(Ti))minus tan (qd (Ti))

q(Ti )cos2(q(Ti ))

minus qd (Ti )cos2(qd (Ti ))

) hi = τ

Ni

and Ω [0+infin)2 times R2 rarr R2 is defined by

Ω(T h x v) =

[Ω1(T h x v)Ω2(T h x v)

](8)

and the formulas

Ω1(T h x v) = x1 + hx2 and

Ω2(T h x v) = x2 + ζ2d (T ) +int T+h

T g1(ζd (s)+x)ds

+int T+h

T g2 (ζd (s)+x) v(s minus τ)ds minus ζ2d (T +h)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1214

NMES Theorem (IK MM MK Ruzhou IJRNC)

For each pair (τ r) isin (0infin)2 there exist a locally boundedfunction N a constant ω isin (0 micro2) and a locally Lipschitzfunction C satisfying C(0) = 0 such that For all sample timesTi in [0infin) such that supige0 (Ti+1 minus Ti) le r and each initialcondition the solution (q(t) q(t) v(t)) with

Ni = N(∣∣∣(tan(q(Ti)) q(Ti )

cos2(q(Ti ))

)minus ζd (Ti)

∣∣∣+||v minus vd ||[Timinusτ Ti ]

) (9)

satisfies|q(t)minus qd (t)|+ |q(t)minus qd (t)|+ ||v minus vd ||[tminusτ t]

le eminusωtC(|q(0)minusqd (0)|+|q(0)minusqd (0)|

cos2(q(0)) + ||v0 minus vd ||[minusτ 0])

for all t ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn

We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function u

The functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty

D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ)

Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded

γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t)

s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state

sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn)

andL2

ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed system

E2 Find bounds on the allowable delays τ such thatY prime(t) = F

(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

NMES on Leg Extension Machine

(Loading Video)

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

TVSpotwmv
Media File (videox-ms-wmv)

514

NMES on Leg Extension Machine

(Loading Video)

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

AN14NMESVidmov
Media File (videoquicktime)

514

NMES on Leg Extension Machine

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i

where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi

The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Potential Controller (continued)

Euler iterations used for control

zk+1 = Ω(Ti + khi hi zk v) for k = 0 Ni minus 1 where

z0 =

(tan (q(Ti))minus tan (qd (Ti))

q(Ti )cos2(q(Ti ))

minus qd (Ti )cos2(qd (Ti ))

) hi = τ

Ni

and Ω [0+infin)2 times R2 rarr R2 is defined by

Ω(T h x v) =

[Ω1(T h x v)Ω2(T h x v)

](8)

and the formulas

Ω1(T h x v) = x1 + hx2 and

Ω2(T h x v) = x2 + ζ2d (T ) +int T+h

T g1(ζd (s)+x)ds

+int T+h

T g2 (ζd (s)+x) v(s minus τ)ds minus ζ2d (T +h)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1214

NMES Theorem (IK MM MK Ruzhou IJRNC)

For each pair (τ r) isin (0infin)2 there exist a locally boundedfunction N a constant ω isin (0 micro2) and a locally Lipschitzfunction C satisfying C(0) = 0 such that For all sample timesTi in [0infin) such that supige0 (Ti+1 minus Ti) le r and each initialcondition the solution (q(t) q(t) v(t)) with

Ni = N(∣∣∣(tan(q(Ti)) q(Ti )

cos2(q(Ti ))

)minus ζd (Ti)

∣∣∣+||v minus vd ||[Timinusτ Ti ]

) (9)

satisfies|q(t)minus qd (t)|+ |q(t)minus qd (t)|+ ||v minus vd ||[tminusτ t]

le eminusωtC(|q(0)minusqd (0)|+|q(0)minusqd (0)|

cos2(q(0)) + ||v0 minus vd ||[minusτ 0])

for all t ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn

We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function u

The functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty

D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ)

Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded

γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t)

s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state

sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn)

andL2

ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed system

E2 Find bounds on the allowable delays τ such thatY prime(t) = F

(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

NMES on Leg Extension Machine

(Loading Video)

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

TVSpotwmv
Media File (videox-ms-wmv)

514

NMES on Leg Extension Machine

(Loading Video)

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

AN14NMESVidmov
Media File (videoquicktime)

514

NMES on Leg Extension Machine

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i

where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi

The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Potential Controller (continued)

Euler iterations used for control

zk+1 = Ω(Ti + khi hi zk v) for k = 0 Ni minus 1 where

z0 =

(tan (q(Ti))minus tan (qd (Ti))

q(Ti )cos2(q(Ti ))

minus qd (Ti )cos2(qd (Ti ))

) hi = τ

Ni

and Ω [0+infin)2 times R2 rarr R2 is defined by

Ω(T h x v) =

[Ω1(T h x v)Ω2(T h x v)

](8)

and the formulas

Ω1(T h x v) = x1 + hx2 and

Ω2(T h x v) = x2 + ζ2d (T ) +int T+h

T g1(ζd (s)+x)ds

+int T+h

T g2 (ζd (s)+x) v(s minus τ)ds minus ζ2d (T +h)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1214

NMES Theorem (IK MM MK Ruzhou IJRNC)

For each pair (τ r) isin (0infin)2 there exist a locally boundedfunction N a constant ω isin (0 micro2) and a locally Lipschitzfunction C satisfying C(0) = 0 such that For all sample timesTi in [0infin) such that supige0 (Ti+1 minus Ti) le r and each initialcondition the solution (q(t) q(t) v(t)) with

Ni = N(∣∣∣(tan(q(Ti)) q(Ti )

cos2(q(Ti ))

)minus ζd (Ti)

∣∣∣+||v minus vd ||[Timinusτ Ti ]

) (9)

satisfies|q(t)minus qd (t)|+ |q(t)minus qd (t)|+ ||v minus vd ||[tminusτ t]

le eminusωtC(|q(0)minusqd (0)|+|q(0)minusqd (0)|

cos2(q(0)) + ||v0 minus vd ||[minusτ 0])

for all t ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn

We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function u

The functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty

D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ)

Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded

γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t)

s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state

sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn)

andL2

ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed system

E2 Find bounds on the allowable delays τ such thatY prime(t) = F

(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

NMES on Leg Extension Machine

(Loading Video)

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

TVSpotwmv
Media File (videox-ms-wmv)

514

NMES on Leg Extension Machine

(Loading Video)

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

AN14NMESVidmov
Media File (videoquicktime)

514

NMES on Leg Extension Machine

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i

where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi

The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Potential Controller (continued)

Euler iterations used for control

zk+1 = Ω(Ti + khi hi zk v) for k = 0 Ni minus 1 where

z0 =

(tan (q(Ti))minus tan (qd (Ti))

q(Ti )cos2(q(Ti ))

minus qd (Ti )cos2(qd (Ti ))

) hi = τ

Ni

and Ω [0+infin)2 times R2 rarr R2 is defined by

Ω(T h x v) =

[Ω1(T h x v)Ω2(T h x v)

](8)

and the formulas

Ω1(T h x v) = x1 + hx2 and

Ω2(T h x v) = x2 + ζ2d (T ) +int T+h

T g1(ζd (s)+x)ds

+int T+h

T g2 (ζd (s)+x) v(s minus τ)ds minus ζ2d (T +h)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1214

NMES Theorem (IK MM MK Ruzhou IJRNC)

For each pair (τ r) isin (0infin)2 there exist a locally boundedfunction N a constant ω isin (0 micro2) and a locally Lipschitzfunction C satisfying C(0) = 0 such that For all sample timesTi in [0infin) such that supige0 (Ti+1 minus Ti) le r and each initialcondition the solution (q(t) q(t) v(t)) with

Ni = N(∣∣∣(tan(q(Ti)) q(Ti )

cos2(q(Ti ))

)minus ζd (Ti)

∣∣∣+||v minus vd ||[Timinusτ Ti ]

) (9)

satisfies|q(t)minus qd (t)|+ |q(t)minus qd (t)|+ ||v minus vd ||[tminusτ t]

le eminusωtC(|q(0)minusqd (0)|+|q(0)minusqd (0)|

cos2(q(0)) + ||v0 minus vd ||[minusτ 0])

for all t ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function u

The functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty

D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ)

Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded

γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t)

s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state

sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn)

andL2

ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed system

E2 Find bounds on the allowable delays τ such thatY prime(t) = F

(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

NMES on Leg Extension Machine

(Loading Video)

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

TVSpotwmv
Media File (videox-ms-wmv)

514

NMES on Leg Extension Machine

(Loading Video)

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

AN14NMESVidmov
Media File (videoquicktime)

514

NMES on Leg Extension Machine

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i

where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi

The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Potential Controller (continued)

Euler iterations used for control

zk+1 = Ω(Ti + khi hi zk v) for k = 0 Ni minus 1 where

z0 =

(tan (q(Ti))minus tan (qd (Ti))

q(Ti )cos2(q(Ti ))

minus qd (Ti )cos2(qd (Ti ))

) hi = τ

Ni

and Ω [0+infin)2 times R2 rarr R2 is defined by

Ω(T h x v) =

[Ω1(T h x v)Ω2(T h x v)

](8)

and the formulas

Ω1(T h x v) = x1 + hx2 and

Ω2(T h x v) = x2 + ζ2d (T ) +int T+h

T g1(ζd (s)+x)ds

+int T+h

T g2 (ζd (s)+x) v(s minus τ)ds minus ζ2d (T +h)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1214

NMES Theorem (IK MM MK Ruzhou IJRNC)

For each pair (τ r) isin (0infin)2 there exist a locally boundedfunction N a constant ω isin (0 micro2) and a locally Lipschitzfunction C satisfying C(0) = 0 such that For all sample timesTi in [0infin) such that supige0 (Ti+1 minus Ti) le r and each initialcondition the solution (q(t) q(t) v(t)) with

Ni = N(∣∣∣(tan(q(Ti)) q(Ti )

cos2(q(Ti ))

)minus ζd (Ti)

∣∣∣+||v minus vd ||[Timinusτ Ti ]

) (9)

satisfies|q(t)minus qd (t)|+ |q(t)minus qd (t)|+ ||v minus vd ||[tminusτ t]

le eminusωtC(|q(0)minusqd (0)|+|q(0)minusqd (0)|

cos2(q(0)) + ||v0 minus vd ||[minusτ 0])

for all t ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty

D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ)

Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded

γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t)

s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state

sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn)

andL2

ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed system

E2 Find bounds on the allowable delays τ such thatY prime(t) = F

(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

NMES on Leg Extension Machine

(Loading Video)

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

TVSpotwmv
Media File (videox-ms-wmv)

514

NMES on Leg Extension Machine

(Loading Video)

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

AN14NMESVidmov
Media File (videoquicktime)

514

NMES on Leg Extension Machine

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i

where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi

The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Potential Controller (continued)

Euler iterations used for control

zk+1 = Ω(Ti + khi hi zk v) for k = 0 Ni minus 1 where

z0 =

(tan (q(Ti))minus tan (qd (Ti))

q(Ti )cos2(q(Ti ))

minus qd (Ti )cos2(qd (Ti ))

) hi = τ

Ni

and Ω [0+infin)2 times R2 rarr R2 is defined by

Ω(T h x v) =

[Ω1(T h x v)Ω2(T h x v)

](8)

and the formulas

Ω1(T h x v) = x1 + hx2 and

Ω2(T h x v) = x2 + ζ2d (T ) +int T+h

T g1(ζd (s)+x)ds

+int T+h

T g2 (ζd (s)+x) v(s minus τ)ds minus ζ2d (T +h)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1214

NMES Theorem (IK MM MK Ruzhou IJRNC)

For each pair (τ r) isin (0infin)2 there exist a locally boundedfunction N a constant ω isin (0 micro2) and a locally Lipschitzfunction C satisfying C(0) = 0 such that For all sample timesTi in [0infin) such that supige0 (Ti+1 minus Ti) le r and each initialcondition the solution (q(t) q(t) v(t)) with

Ni = N(∣∣∣(tan(q(Ti)) q(Ti )

cos2(q(Ti ))

)minus ζd (Ti)

∣∣∣+||v minus vd ||[Timinusτ Ti ]

) (9)

satisfies|q(t)minus qd (t)|+ |q(t)minus qd (t)|+ ||v minus vd ||[tminusτ t]

le eminusωtC(|q(0)minusqd (0)|+|q(0)minusqd (0)|

cos2(q(0)) + ||v0 minus vd ||[minusτ 0])

for all t ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ)

Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded

γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t)

s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state

sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn)

andL2

ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed system

E2 Find bounds on the allowable delays τ such thatY prime(t) = F

(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

NMES on Leg Extension Machine

(Loading Video)

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

TVSpotwmv
Media File (videox-ms-wmv)

514

NMES on Leg Extension Machine

(Loading Video)

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

AN14NMESVidmov
Media File (videoquicktime)

514

NMES on Leg Extension Machine

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i

where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi

The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Potential Controller (continued)

Euler iterations used for control

zk+1 = Ω(Ti + khi hi zk v) for k = 0 Ni minus 1 where

z0 =

(tan (q(Ti))minus tan (qd (Ti))

q(Ti )cos2(q(Ti ))

minus qd (Ti )cos2(qd (Ti ))

) hi = τ

Ni

and Ω [0+infin)2 times R2 rarr R2 is defined by

Ω(T h x v) =

[Ω1(T h x v)Ω2(T h x v)

](8)

and the formulas

Ω1(T h x v) = x1 + hx2 and

Ω2(T h x v) = x2 + ζ2d (T ) +int T+h

T g1(ζd (s)+x)ds

+int T+h

T g2 (ζd (s)+x) v(s minus τ)ds minus ζ2d (T +h)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1214

NMES Theorem (IK MM MK Ruzhou IJRNC)

For each pair (τ r) isin (0infin)2 there exist a locally boundedfunction N a constant ω isin (0 micro2) and a locally Lipschitzfunction C satisfying C(0) = 0 such that For all sample timesTi in [0infin) such that supige0 (Ti+1 minus Ti) le r and each initialcondition the solution (q(t) q(t) v(t)) with

Ni = N(∣∣∣(tan(q(Ti)) q(Ti )

cos2(q(Ti ))

)minus ζd (Ti)

∣∣∣+||v minus vd ||[Timinusτ Ti ]

) (9)

satisfies|q(t)minus qd (t)|+ |q(t)minus qd (t)|+ ||v minus vd ||[tminusτ t]

le eminusωtC(|q(0)minusqd (0)|+|q(0)minusqd (0)|

cos2(q(0)) + ||v0 minus vd ||[minusτ 0])

for all t ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ)

Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded

γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t)

s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state

sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn)

andL2

ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed system

E2 Find bounds on the allowable delays τ such thatY prime(t) = F

(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

NMES on Leg Extension Machine

(Loading Video)

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

TVSpotwmv
Media File (videox-ms-wmv)

514

NMES on Leg Extension Machine

(Loading Video)

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

AN14NMESVidmov
Media File (videoquicktime)

514

NMES on Leg Extension Machine

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i

where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi

The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Potential Controller (continued)

Euler iterations used for control

zk+1 = Ω(Ti + khi hi zk v) for k = 0 Ni minus 1 where

z0 =

(tan (q(Ti))minus tan (qd (Ti))

q(Ti )cos2(q(Ti ))

minus qd (Ti )cos2(qd (Ti ))

) hi = τ

Ni

and Ω [0+infin)2 times R2 rarr R2 is defined by

Ω(T h x v) =

[Ω1(T h x v)Ω2(T h x v)

](8)

and the formulas

Ω1(T h x v) = x1 + hx2 and

Ω2(T h x v) = x2 + ζ2d (T ) +int T+h

T g1(ζd (s)+x)ds

+int T+h

T g2 (ζd (s)+x) v(s minus τ)ds minus ζ2d (T +h)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1214

NMES Theorem (IK MM MK Ruzhou IJRNC)

For each pair (τ r) isin (0infin)2 there exist a locally boundedfunction N a constant ω isin (0 micro2) and a locally Lipschitzfunction C satisfying C(0) = 0 such that For all sample timesTi in [0infin) such that supige0 (Ti+1 minus Ti) le r and each initialcondition the solution (q(t) q(t) v(t)) with

Ni = N(∣∣∣(tan(q(Ti)) q(Ti )

cos2(q(Ti ))

)minus ζd (Ti)

∣∣∣+||v minus vd ||[Timinusτ Ti ]

) (9)

satisfies|q(t)minus qd (t)|+ |q(t)minus qd (t)|+ ||v minus vd ||[tminusτ t]

le eminusωtC(|q(0)minusqd (0)|+|q(0)minusqd (0)|

cos2(q(0)) + ||v0 minus vd ||[minusτ 0])

for all t ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded

γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t)

s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state

sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn)

andL2

ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed system

E2 Find bounds on the allowable delays τ such thatY prime(t) = F

(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

NMES on Leg Extension Machine

(Loading Video)

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

TVSpotwmv
Media File (videox-ms-wmv)

514

NMES on Leg Extension Machine

(Loading Video)

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

AN14NMESVidmov
Media File (videoquicktime)

514

NMES on Leg Extension Machine

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i

where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi

The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Potential Controller (continued)

Euler iterations used for control

zk+1 = Ω(Ti + khi hi zk v) for k = 0 Ni minus 1 where

z0 =

(tan (q(Ti))minus tan (qd (Ti))

q(Ti )cos2(q(Ti ))

minus qd (Ti )cos2(qd (Ti ))

) hi = τ

Ni

and Ω [0+infin)2 times R2 rarr R2 is defined by

Ω(T h x v) =

[Ω1(T h x v)Ω2(T h x v)

](8)

and the formulas

Ω1(T h x v) = x1 + hx2 and

Ω2(T h x v) = x2 + ζ2d (T ) +int T+h

T g1(ζd (s)+x)ds

+int T+h

T g2 (ζd (s)+x) v(s minus τ)ds minus ζ2d (T +h)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1214

NMES Theorem (IK MM MK Ruzhou IJRNC)

For each pair (τ r) isin (0infin)2 there exist a locally boundedfunction N a constant ω isin (0 micro2) and a locally Lipschitzfunction C satisfying C(0) = 0 such that For all sample timesTi in [0infin) such that supige0 (Ti+1 minus Ti) le r and each initialcondition the solution (q(t) q(t) v(t)) with

Ni = N(∣∣∣(tan(q(Ti)) q(Ti )

cos2(q(Ti ))

)minus ζd (Ti)

∣∣∣+||v minus vd ||[Timinusτ Ti ]

) (9)

satisfies|q(t)minus qd (t)|+ |q(t)minus qd (t)|+ ||v minus vd ||[tminusτ t]

le eminusωtC(|q(0)minusqd (0)|+|q(0)minusqd (0)|

cos2(q(0)) + ||v0 minus vd ||[minusτ 0])

for all t ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

314

What are Delayed Control Systems

These are doubly parameterized families of ODEs of the form

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

) Y (t) isin Y (1)

Y sube Rn We have freedom to choose the control function uThe functions δ [0infin)rarr D represent uncertainty D sube Rm

Yt (θ) = Y (t + θ) Specify u to get a singly parameterized family

Y prime(t) = G(t Yt δ(t)) Y (t) isin Y (2)

where G(t Yt d) = F(t Y (t)u(t Y (t minus τ))d)

Typically we construct u such that all trajectories of (2) for allpossible choices of δ satisfy some control objective

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded

γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t)

s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state

sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn)

andL2

ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed system

E2 Find bounds on the allowable delays τ such thatY prime(t) = F

(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

NMES on Leg Extension Machine

(Loading Video)

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

TVSpotwmv
Media File (videox-ms-wmv)

514

NMES on Leg Extension Machine

(Loading Video)

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

AN14NMESVidmov
Media File (videoquicktime)

514

NMES on Leg Extension Machine

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i

where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi

The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Potential Controller (continued)

Euler iterations used for control

zk+1 = Ω(Ti + khi hi zk v) for k = 0 Ni minus 1 where

z0 =

(tan (q(Ti))minus tan (qd (Ti))

q(Ti )cos2(q(Ti ))

minus qd (Ti )cos2(qd (Ti ))

) hi = τ

Ni

and Ω [0+infin)2 times R2 rarr R2 is defined by

Ω(T h x v) =

[Ω1(T h x v)Ω2(T h x v)

](8)

and the formulas

Ω1(T h x v) = x1 + hx2 and

Ω2(T h x v) = x2 + ζ2d (T ) +int T+h

T g1(ζd (s)+x)ds

+int T+h

T g2 (ζd (s)+x) v(s minus τ)ds minus ζ2d (T +h)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1214

NMES Theorem (IK MM MK Ruzhou IJRNC)

For each pair (τ r) isin (0infin)2 there exist a locally boundedfunction N a constant ω isin (0 micro2) and a locally Lipschitzfunction C satisfying C(0) = 0 such that For all sample timesTi in [0infin) such that supige0 (Ti+1 minus Ti) le r and each initialcondition the solution (q(t) q(t) v(t)) with

Ni = N(∣∣∣(tan(q(Ti)) q(Ti )

cos2(q(Ti ))

)minus ζd (Ti)

∣∣∣+||v minus vd ||[Timinusτ Ti ]

) (9)

satisfies|q(t)minus qd (t)|+ |q(t)minus qd (t)|+ ||v minus vd ||[tminusτ t]

le eminusωtC(|q(0)minusqd (0)|+|q(0)minusqd (0)|

cos2(q(0)) + ||v0 minus vd ||[minusτ 0])

for all t ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded

γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t)

s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state

sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn)

andL2

ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed system

E2 Find bounds on the allowable delays τ such thatY prime(t) = F

(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

NMES on Leg Extension Machine

(Loading Video)

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

TVSpotwmv
Media File (videox-ms-wmv)

514

NMES on Leg Extension Machine

(Loading Video)

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

AN14NMESVidmov
Media File (videoquicktime)

514

NMES on Leg Extension Machine

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i

where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi

The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Potential Controller (continued)

Euler iterations used for control

zk+1 = Ω(Ti + khi hi zk v) for k = 0 Ni minus 1 where

z0 =

(tan (q(Ti))minus tan (qd (Ti))

q(Ti )cos2(q(Ti ))

minus qd (Ti )cos2(qd (Ti ))

) hi = τ

Ni

and Ω [0+infin)2 times R2 rarr R2 is defined by

Ω(T h x v) =

[Ω1(T h x v)Ω2(T h x v)

](8)

and the formulas

Ω1(T h x v) = x1 + hx2 and

Ω2(T h x v) = x2 + ζ2d (T ) +int T+h

T g1(ζd (s)+x)ds

+int T+h

T g2 (ζd (s)+x) v(s minus τ)ds minus ζ2d (T +h)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1214

NMES Theorem (IK MM MK Ruzhou IJRNC)

For each pair (τ r) isin (0infin)2 there exist a locally boundedfunction N a constant ω isin (0 micro2) and a locally Lipschitzfunction C satisfying C(0) = 0 such that For all sample timesTi in [0infin) such that supige0 (Ti+1 minus Ti) le r and each initialcondition the solution (q(t) q(t) v(t)) with

Ni = N(∣∣∣(tan(q(Ti)) q(Ti )

cos2(q(Ti ))

)minus ζd (Ti)

∣∣∣+||v minus vd ||[Timinusτ Ti ]

) (9)

satisfies|q(t)minus qd (t)|+ |q(t)minus qd (t)|+ ||v minus vd ||[tminusτ t]

le eminusωtC(|q(0)minusqd (0)|+|q(0)minusqd (0)|

cos2(q(0)) + ||v0 minus vd ||[minusτ 0])

for all t ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

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Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded

γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t)

s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state

sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn)

andL2

ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed system

E2 Find bounds on the allowable delays τ such thatY prime(t) = F

(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

NMES on Leg Extension Machine

(Loading Video)

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

TVSpotwmv
Media File (videox-ms-wmv)

514

NMES on Leg Extension Machine

(Loading Video)

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

AN14NMESVidmov
Media File (videoquicktime)

514

NMES on Leg Extension Machine

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i

where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi

The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Potential Controller (continued)

Euler iterations used for control

zk+1 = Ω(Ti + khi hi zk v) for k = 0 Ni minus 1 where

z0 =

(tan (q(Ti))minus tan (qd (Ti))

q(Ti )cos2(q(Ti ))

minus qd (Ti )cos2(qd (Ti ))

) hi = τ

Ni

and Ω [0+infin)2 times R2 rarr R2 is defined by

Ω(T h x v) =

[Ω1(T h x v)Ω2(T h x v)

](8)

and the formulas

Ω1(T h x v) = x1 + hx2 and

Ω2(T h x v) = x2 + ζ2d (T ) +int T+h

T g1(ζd (s)+x)ds

+int T+h

T g2 (ζd (s)+x) v(s minus τ)ds minus ζ2d (T +h)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1214

NMES Theorem (IK MM MK Ruzhou IJRNC)

For each pair (τ r) isin (0infin)2 there exist a locally boundedfunction N a constant ω isin (0 micro2) and a locally Lipschitzfunction C satisfying C(0) = 0 such that For all sample timesTi in [0infin) such that supige0 (Ti+1 minus Ti) le r and each initialcondition the solution (q(t) q(t) v(t)) with

Ni = N(∣∣∣(tan(q(Ti)) q(Ti )

cos2(q(Ti ))

)minus ζd (Ti)

∣∣∣+||v minus vd ||[Timinusτ Ti ]

) (9)

satisfies|q(t)minus qd (t)|+ |q(t)minus qd (t)|+ ||v minus vd ||[tminusτ t]

le eminusωtC(|q(0)minusqd (0)|+|q(0)minusqd (0)|

cos2(q(0)) + ||v0 minus vd ||[minusτ 0])

for all t ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded

γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t)

s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state

sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn)

andL2

ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed system

E2 Find bounds on the allowable delays τ such thatY prime(t) = F

(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

NMES on Leg Extension Machine

(Loading Video)

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

TVSpotwmv
Media File (videox-ms-wmv)

514

NMES on Leg Extension Machine

(Loading Video)

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

AN14NMESVidmov
Media File (videoquicktime)

514

NMES on Leg Extension Machine

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i

where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi

The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Potential Controller (continued)

Euler iterations used for control

zk+1 = Ω(Ti + khi hi zk v) for k = 0 Ni minus 1 where

z0 =

(tan (q(Ti))minus tan (qd (Ti))

q(Ti )cos2(q(Ti ))

minus qd (Ti )cos2(qd (Ti ))

) hi = τ

Ni

and Ω [0+infin)2 times R2 rarr R2 is defined by

Ω(T h x v) =

[Ω1(T h x v)Ω2(T h x v)

](8)

and the formulas

Ω1(T h x v) = x1 + hx2 and

Ω2(T h x v) = x2 + ζ2d (T ) +int T+h

T g1(ζd (s)+x)ds

+int T+h

T g2 (ζd (s)+x) v(s minus τ)ds minus ζ2d (T +h)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1214

NMES Theorem (IK MM MK Ruzhou IJRNC)

For each pair (τ r) isin (0infin)2 there exist a locally boundedfunction N a constant ω isin (0 micro2) and a locally Lipschitzfunction C satisfying C(0) = 0 such that For all sample timesTi in [0infin) such that supige0 (Ti+1 minus Ti) le r and each initialcondition the solution (q(t) q(t) v(t)) with

Ni = N(∣∣∣(tan(q(Ti)) q(Ti )

cos2(q(Ti ))

)minus ζd (Ti)

∣∣∣+||v minus vd ||[Timinusτ Ti ]

) (9)

satisfies|q(t)minus qd (t)|+ |q(t)minus qd (t)|+ ||v minus vd ||[tminusτ t]

le eminusωtC(|q(0)minusqd (0)|+|q(0)minusqd (0)|

cos2(q(0)) + ||v0 minus vd ||[minusτ 0])

for all t ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded

γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t)

s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state

sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn)

andL2

ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed system

E2 Find bounds on the allowable delays τ such thatY prime(t) = F

(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

NMES on Leg Extension Machine

(Loading Video)

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

TVSpotwmv
Media File (videox-ms-wmv)

514

NMES on Leg Extension Machine

(Loading Video)

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

AN14NMESVidmov
Media File (videoquicktime)

514

NMES on Leg Extension Machine

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i

where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi

The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Potential Controller (continued)

Euler iterations used for control

zk+1 = Ω(Ti + khi hi zk v) for k = 0 Ni minus 1 where

z0 =

(tan (q(Ti))minus tan (qd (Ti))

q(Ti )cos2(q(Ti ))

minus qd (Ti )cos2(qd (Ti ))

) hi = τ

Ni

and Ω [0+infin)2 times R2 rarr R2 is defined by

Ω(T h x v) =

[Ω1(T h x v)Ω2(T h x v)

](8)

and the formulas

Ω1(T h x v) = x1 + hx2 and

Ω2(T h x v) = x2 + ζ2d (T ) +int T+h

T g1(ζd (s)+x)ds

+int T+h

T g2 (ζd (s)+x) v(s minus τ)ds minus ζ2d (T +h)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1214

NMES Theorem (IK MM MK Ruzhou IJRNC)

For each pair (τ r) isin (0infin)2 there exist a locally boundedfunction N a constant ω isin (0 micro2) and a locally Lipschitzfunction C satisfying C(0) = 0 such that For all sample timesTi in [0infin) such that supige0 (Ti+1 minus Ti) le r and each initialcondition the solution (q(t) q(t) v(t)) with

Ni = N(∣∣∣(tan(q(Ti)) q(Ti )

cos2(q(Ti ))

)minus ζd (Ti)

∣∣∣+||v minus vd ||[Timinusτ Ti ]

) (9)

satisfies|q(t)minus qd (t)|+ |q(t)minus qd (t)|+ ||v minus vd ||[tminusτ t]

le eminusωtC(|q(0)minusqd (0)|+|q(0)minusqd (0)|

cos2(q(0)) + ||v0 minus vd ||[minusτ 0])

for all t ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded

γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t)

s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state

sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn)

andL2

ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed system

E2 Find bounds on the allowable delays τ such thatY prime(t) = F

(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

NMES on Leg Extension Machine

(Loading Video)

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

TVSpotwmv
Media File (videox-ms-wmv)

514

NMES on Leg Extension Machine

(Loading Video)

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

AN14NMESVidmov
Media File (videoquicktime)

514

NMES on Leg Extension Machine

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i

where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi

The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Potential Controller (continued)

Euler iterations used for control

zk+1 = Ω(Ti + khi hi zk v) for k = 0 Ni minus 1 where

z0 =

(tan (q(Ti))minus tan (qd (Ti))

q(Ti )cos2(q(Ti ))

minus qd (Ti )cos2(qd (Ti ))

) hi = τ

Ni

and Ω [0+infin)2 times R2 rarr R2 is defined by

Ω(T h x v) =

[Ω1(T h x v)Ω2(T h x v)

](8)

and the formulas

Ω1(T h x v) = x1 + hx2 and

Ω2(T h x v) = x2 + ζ2d (T ) +int T+h

T g1(ζd (s)+x)ds

+int T+h

T g2 (ζd (s)+x) v(s minus τ)ds minus ζ2d (T +h)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1214

NMES Theorem (IK MM MK Ruzhou IJRNC)

For each pair (τ r) isin (0infin)2 there exist a locally boundedfunction N a constant ω isin (0 micro2) and a locally Lipschitzfunction C satisfying C(0) = 0 such that For all sample timesTi in [0infin) such that supige0 (Ti+1 minus Ti) le r and each initialcondition the solution (q(t) q(t) v(t)) with

Ni = N(∣∣∣(tan(q(Ti)) q(Ti )

cos2(q(Ti ))

)minus ζd (Ti)

∣∣∣+||v minus vd ||[Timinusτ Ti ]

) (9)

satisfies|q(t)minus qd (t)|+ |q(t)minus qd (t)|+ ||v minus vd ||[tminusτ t]

le eminusωtC(|q(0)minusqd (0)|+|q(0)minusqd (0)|

cos2(q(0)) + ||v0 minus vd ||[minusτ 0])

for all t ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t)

s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state

sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn)

andL2

ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed system

E2 Find bounds on the allowable delays τ such thatY prime(t) = F

(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

NMES on Leg Extension Machine

(Loading Video)

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

TVSpotwmv
Media File (videox-ms-wmv)

514

NMES on Leg Extension Machine

(Loading Video)

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

AN14NMESVidmov
Media File (videoquicktime)

514

NMES on Leg Extension Machine

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i

where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi

The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Potential Controller (continued)

Euler iterations used for control

zk+1 = Ω(Ti + khi hi zk v) for k = 0 Ni minus 1 where

z0 =

(tan (q(Ti))minus tan (qd (Ti))

q(Ti )cos2(q(Ti ))

minus qd (Ti )cos2(qd (Ti ))

) hi = τ

Ni

and Ω [0+infin)2 times R2 rarr R2 is defined by

Ω(T h x v) =

[Ω1(T h x v)Ω2(T h x v)

](8)

and the formulas

Ω1(T h x v) = x1 + hx2 and

Ω2(T h x v) = x2 + ζ2d (T ) +int T+h

T g1(ζd (s)+x)ds

+int T+h

T g2 (ζd (s)+x) v(s minus τ)ds minus ζ2d (T +h)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1214

NMES Theorem (IK MM MK Ruzhou IJRNC)

For each pair (τ r) isin (0infin)2 there exist a locally boundedfunction N a constant ω isin (0 micro2) and a locally Lipschitzfunction C satisfying C(0) = 0 such that For all sample timesTi in [0infin) such that supige0 (Ti+1 minus Ti) le r and each initialcondition the solution (q(t) q(t) v(t)) with

Ni = N(∣∣∣(tan(q(Ti)) q(Ti )

cos2(q(Ti ))

)minus ζd (Ti)

∣∣∣+||v minus vd ||[Timinusτ Ti ]

) (9)

satisfies|q(t)minus qd (t)|+ |q(t)minus qd (t)|+ ||v minus vd ||[tminusτ t]

le eminusωtC(|q(0)minusqd (0)|+|q(0)minusqd (0)|

cos2(q(0)) + ||v0 minus vd ||[minusτ 0])

for all t ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t)

s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state

sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn)

andL2

ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed system

E2 Find bounds on the allowable delays τ such thatY prime(t) = F

(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

NMES on Leg Extension Machine

(Loading Video)

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

TVSpotwmv
Media File (videox-ms-wmv)

514

NMES on Leg Extension Machine

(Loading Video)

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

AN14NMESVidmov
Media File (videoquicktime)

514

NMES on Leg Extension Machine

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i

where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi

The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Potential Controller (continued)

Euler iterations used for control

zk+1 = Ω(Ti + khi hi zk v) for k = 0 Ni minus 1 where

z0 =

(tan (q(Ti))minus tan (qd (Ti))

q(Ti )cos2(q(Ti ))

minus qd (Ti )cos2(qd (Ti ))

) hi = τ

Ni

and Ω [0+infin)2 times R2 rarr R2 is defined by

Ω(T h x v) =

[Ω1(T h x v)Ω2(T h x v)

](8)

and the formulas

Ω1(T h x v) = x1 + hx2 and

Ω2(T h x v) = x2 + ζ2d (T ) +int T+h

T g1(ζd (s)+x)ds

+int T+h

T g2 (ζd (s)+x) v(s minus τ)ds minus ζ2d (T +h)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1214

NMES Theorem (IK MM MK Ruzhou IJRNC)

For each pair (τ r) isin (0infin)2 there exist a locally boundedfunction N a constant ω isin (0 micro2) and a locally Lipschitzfunction C satisfying C(0) = 0 such that For all sample timesTi in [0infin) such that supige0 (Ti+1 minus Ti) le r and each initialcondition the solution (q(t) q(t) v(t)) with

Ni = N(∣∣∣(tan(q(Ti)) q(Ti )

cos2(q(Ti ))

)minus ζd (Ti)

∣∣∣+||v minus vd ||[Timinusτ Ti ]

) (9)

satisfies|q(t)minus qd (t)|+ |q(t)minus qd (t)|+ ||v minus vd ||[tminusτ t]

le eminusωtC(|q(0)minusqd (0)|+|q(0)minusqd (0)|

cos2(q(0)) + ||v0 minus vd ||[minusτ 0])

for all t ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state

sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn)

andL2

ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed system

E2 Find bounds on the allowable delays τ such thatY prime(t) = F

(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

NMES on Leg Extension Machine

(Loading Video)

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

TVSpotwmv
Media File (videox-ms-wmv)

514

NMES on Leg Extension Machine

(Loading Video)

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

AN14NMESVidmov
Media File (videoquicktime)

514

NMES on Leg Extension Machine

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i

where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi

The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Potential Controller (continued)

Euler iterations used for control

zk+1 = Ω(Ti + khi hi zk v) for k = 0 Ni minus 1 where

z0 =

(tan (q(Ti))minus tan (qd (Ti))

q(Ti )cos2(q(Ti ))

minus qd (Ti )cos2(qd (Ti ))

) hi = τ

Ni

and Ω [0+infin)2 times R2 rarr R2 is defined by

Ω(T h x v) =

[Ω1(T h x v)Ω2(T h x v)

](8)

and the formulas

Ω1(T h x v) = x1 + hx2 and

Ω2(T h x v) = x2 + ζ2d (T ) +int T+h

T g1(ζd (s)+x)ds

+int T+h

T g2 (ζd (s)+x) v(s minus τ)ds minus ζ2d (T +h)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1214

NMES Theorem (IK MM MK Ruzhou IJRNC)

For each pair (τ r) isin (0infin)2 there exist a locally boundedfunction N a constant ω isin (0 micro2) and a locally Lipschitzfunction C satisfying C(0) = 0 such that For all sample timesTi in [0infin) such that supige0 (Ti+1 minus Ti) le r and each initialcondition the solution (q(t) q(t) v(t)) with

Ni = N(∣∣∣(tan(q(Ti)) q(Ti )

cos2(q(Ti ))

)minus ζd (Ti)

∣∣∣+||v minus vd ||[Timinusτ Ti ]

) (9)

satisfies|q(t)minus qd (t)|+ |q(t)minus qd (t)|+ ||v minus vd ||[tminusτ t]

le eminusωtC(|q(0)minusqd (0)|+|q(0)minusqd (0)|

cos2(q(0)) + ||v0 minus vd ||[minusτ 0])

for all t ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn)

andL2

ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed system

E2 Find bounds on the allowable delays τ such thatY prime(t) = F

(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

NMES on Leg Extension Machine

(Loading Video)

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

TVSpotwmv
Media File (videox-ms-wmv)

514

NMES on Leg Extension Machine

(Loading Video)

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

AN14NMESVidmov
Media File (videoquicktime)

514

NMES on Leg Extension Machine

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i

where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi

The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Potential Controller (continued)

Euler iterations used for control

zk+1 = Ω(Ti + khi hi zk v) for k = 0 Ni minus 1 where

z0 =

(tan (q(Ti))minus tan (qd (Ti))

q(Ti )cos2(q(Ti ))

minus qd (Ti )cos2(qd (Ti ))

) hi = τ

Ni

and Ω [0+infin)2 times R2 rarr R2 is defined by

Ω(T h x v) =

[Ω1(T h x v)Ω2(T h x v)

](8)

and the formulas

Ω1(T h x v) = x1 + hx2 and

Ω2(T h x v) = x2 + ζ2d (T ) +int T+h

T g1(ζd (s)+x)ds

+int T+h

T g2 (ζd (s)+x) v(s minus τ)ds minus ζ2d (T +h)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1214

NMES Theorem (IK MM MK Ruzhou IJRNC)

For each pair (τ r) isin (0infin)2 there exist a locally boundedfunction N a constant ω isin (0 micro2) and a locally Lipschitzfunction C satisfying C(0) = 0 such that For all sample timesTi in [0infin) such that supige0 (Ti+1 minus Ti) le r and each initialcondition the solution (q(t) q(t) v(t)) with

Ni = N(∣∣∣(tan(q(Ti)) q(Ti )

cos2(q(Ti ))

)minus ζd (Ti)

∣∣∣+||v minus vd ||[Timinusτ Ti ]

) (9)

satisfies|q(t)minus qd (t)|+ |q(t)minus qd (t)|+ ||v minus vd ||[tminusτ t]

le eminusωtC(|q(0)minusqd (0)|+|q(0)minusqd (0)|

cos2(q(0)) + ||v0 minus vd ||[minusτ 0])

for all t ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn)

andL2

ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed system

E2 Find bounds on the allowable delays τ such thatY prime(t) = F

(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

NMES on Leg Extension Machine

(Loading Video)

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

TVSpotwmv
Media File (videox-ms-wmv)

514

NMES on Leg Extension Machine

(Loading Video)

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

AN14NMESVidmov
Media File (videoquicktime)

514

NMES on Leg Extension Machine

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i

where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi

The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Potential Controller (continued)

Euler iterations used for control

zk+1 = Ω(Ti + khi hi zk v) for k = 0 Ni minus 1 where

z0 =

(tan (q(Ti))minus tan (qd (Ti))

q(Ti )cos2(q(Ti ))

minus qd (Ti )cos2(qd (Ti ))

) hi = τ

Ni

and Ω [0+infin)2 times R2 rarr R2 is defined by

Ω(T h x v) =

[Ω1(T h x v)Ω2(T h x v)

](8)

and the formulas

Ω1(T h x v) = x1 + hx2 and

Ω2(T h x v) = x2 + ζ2d (T ) +int T+h

T g1(ζd (s)+x)ds

+int T+h

T g2 (ζd (s)+x) v(s minus τ)ds minus ζ2d (T +h)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1214

NMES Theorem (IK MM MK Ruzhou IJRNC)

For each pair (τ r) isin (0infin)2 there exist a locally boundedfunction N a constant ω isin (0 micro2) and a locally Lipschitzfunction C satisfying C(0) = 0 such that For all sample timesTi in [0infin) such that supige0 (Ti+1 minus Ti) le r and each initialcondition the solution (q(t) q(t) v(t)) with

Ni = N(∣∣∣(tan(q(Ti)) q(Ti )

cos2(q(Ti ))

)minus ζd (Ti)

∣∣∣+||v minus vd ||[Timinusτ Ti ]

) (9)

satisfies|q(t)minus qd (t)|+ |q(t)minus qd (t)|+ ||v minus vd ||[tminusτ t]

le eminusωtC(|q(0)minusqd (0)|+|q(0)minusqd (0)|

cos2(q(0)) + ||v0 minus vd ||[minusτ 0])

for all t ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn)

andL2

ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed system

E2 Find bounds on the allowable delays τ such thatY prime(t) = F

(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

NMES on Leg Extension Machine

(Loading Video)

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

TVSpotwmv
Media File (videox-ms-wmv)

514

NMES on Leg Extension Machine

(Loading Video)

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

AN14NMESVidmov
Media File (videoquicktime)

514

NMES on Leg Extension Machine

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i

where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi

The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Potential Controller (continued)

Euler iterations used for control

zk+1 = Ω(Ti + khi hi zk v) for k = 0 Ni minus 1 where

z0 =

(tan (q(Ti))minus tan (qd (Ti))

q(Ti )cos2(q(Ti ))

minus qd (Ti )cos2(qd (Ti ))

) hi = τ

Ni

and Ω [0+infin)2 times R2 rarr R2 is defined by

Ω(T h x v) =

[Ω1(T h x v)Ω2(T h x v)

](8)

and the formulas

Ω1(T h x v) = x1 + hx2 and

Ω2(T h x v) = x2 + ζ2d (T ) +int T+h

T g1(ζd (s)+x)ds

+int T+h

T g2 (ζd (s)+x) v(s minus τ)ds minus ζ2d (T +h)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1214

NMES Theorem (IK MM MK Ruzhou IJRNC)

For each pair (τ r) isin (0infin)2 there exist a locally boundedfunction N a constant ω isin (0 micro2) and a locally Lipschitzfunction C satisfying C(0) = 0 such that For all sample timesTi in [0infin) such that supige0 (Ti+1 minus Ti) le r and each initialcondition the solution (q(t) q(t) v(t)) with

Ni = N(∣∣∣(tan(q(Ti)) q(Ti )

cos2(q(Ti ))

)minus ζd (Ti)

∣∣∣+||v minus vd ||[Timinusτ Ti ]

) (9)

satisfies|q(t)minus qd (t)|+ |q(t)minus qd (t)|+ ||v minus vd ||[tminusτ t]

le eminusωtC(|q(0)minusqd (0)|+|q(0)minusqd (0)|

cos2(q(0)) + ||v0 minus vd ||[minusτ 0])

for all t ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

414

What is One Possible Control Objective

Input-to-state stability generalizes global asymptotic stability

Y prime(t) = G(t Yt ) Y (t) isin Y (Σ)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)(UGAS)

Our γi rsquos are 0 at 0 strictly increasing and unbounded γi isin Kinfin

Tracking ErrorY (t) = s(t)minus sr (t) s(t) = state sr (t) = reference signal

Y prime(t) = G(t Yt δ(t)

) Y (t) isin Y (Σpert)

|Y (t)| le γ1(et0minustγ2(|Yt0 |[minusτ 0])

)+ γ2(|δ|[t0t]) (ISS)

Find γi rsquos by building certain LKFs for Y prime(t) = G(t Yt 0)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn)

andL2

ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed system

E2 Find bounds on the allowable delays τ such thatY prime(t) = F

(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

NMES on Leg Extension Machine

(Loading Video)

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

TVSpotwmv
Media File (videox-ms-wmv)

514

NMES on Leg Extension Machine

(Loading Video)

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

AN14NMESVidmov
Media File (videoquicktime)

514

NMES on Leg Extension Machine

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i

where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi

The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Potential Controller (continued)

Euler iterations used for control

zk+1 = Ω(Ti + khi hi zk v) for k = 0 Ni minus 1 where

z0 =

(tan (q(Ti))minus tan (qd (Ti))

q(Ti )cos2(q(Ti ))

minus qd (Ti )cos2(qd (Ti ))

) hi = τ

Ni

and Ω [0+infin)2 times R2 rarr R2 is defined by

Ω(T h x v) =

[Ω1(T h x v)Ω2(T h x v)

](8)

and the formulas

Ω1(T h x v) = x1 + hx2 and

Ω2(T h x v) = x2 + ζ2d (T ) +int T+h

T g1(ζd (s)+x)ds

+int T+h

T g2 (ζd (s)+x) v(s minus τ)ds minus ζ2d (T +h)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1214

NMES Theorem (IK MM MK Ruzhou IJRNC)

For each pair (τ r) isin (0infin)2 there exist a locally boundedfunction N a constant ω isin (0 micro2) and a locally Lipschitzfunction C satisfying C(0) = 0 such that For all sample timesTi in [0infin) such that supige0 (Ti+1 minus Ti) le r and each initialcondition the solution (q(t) q(t) v(t)) with

Ni = N(∣∣∣(tan(q(Ti)) q(Ti )

cos2(q(Ti ))

)minus ζd (Ti)

∣∣∣+||v minus vd ||[Timinusτ Ti ]

) (9)

satisfies|q(t)minus qd (t)|+ |q(t)minus qd (t)|+ ||v minus vd ||[tminusτ t]

le eminusωtC(|q(0)minusqd (0)|+|q(0)minusqd (0)|

cos2(q(0)) + ||v0 minus vd ||[minusτ 0])

for all t ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn)

andL2

ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed system

E2 Find bounds on the allowable delays τ such thatY prime(t) = F

(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

NMES on Leg Extension Machine

(Loading Video)

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

TVSpotwmv
Media File (videox-ms-wmv)

514

NMES on Leg Extension Machine

(Loading Video)

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

AN14NMESVidmov
Media File (videoquicktime)

514

NMES on Leg Extension Machine

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i

where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi

The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Potential Controller (continued)

Euler iterations used for control

zk+1 = Ω(Ti + khi hi zk v) for k = 0 Ni minus 1 where

z0 =

(tan (q(Ti))minus tan (qd (Ti))

q(Ti )cos2(q(Ti ))

minus qd (Ti )cos2(qd (Ti ))

) hi = τ

Ni

and Ω [0+infin)2 times R2 rarr R2 is defined by

Ω(T h x v) =

[Ω1(T h x v)Ω2(T h x v)

](8)

and the formulas

Ω1(T h x v) = x1 + hx2 and

Ω2(T h x v) = x2 + ζ2d (T ) +int T+h

T g1(ζd (s)+x)ds

+int T+h

T g2 (ζd (s)+x) v(s minus τ)ds minus ζ2d (T +h)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1214

NMES Theorem (IK MM MK Ruzhou IJRNC)

For each pair (τ r) isin (0infin)2 there exist a locally boundedfunction N a constant ω isin (0 micro2) and a locally Lipschitzfunction C satisfying C(0) = 0 such that For all sample timesTi in [0infin) such that supige0 (Ti+1 minus Ti) le r and each initialcondition the solution (q(t) q(t) v(t)) with

Ni = N(∣∣∣(tan(q(Ti)) q(Ti )

cos2(q(Ti ))

)minus ζd (Ti)

∣∣∣+||v minus vd ||[Timinusτ Ti ]

) (9)

satisfies|q(t)minus qd (t)|+ |q(t)minus qd (t)|+ ||v minus vd ||[tminusτ t]

le eminusωtC(|q(0)minusqd (0)|+|q(0)minusqd (0)|

cos2(q(0)) + ||v0 minus vd ||[minusτ 0])

for all t ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn)

andL2

ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed system

E2 Find bounds on the allowable delays τ such thatY prime(t) = F

(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

NMES on Leg Extension Machine

(Loading Video)

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

TVSpotwmv
Media File (videox-ms-wmv)

514

NMES on Leg Extension Machine

(Loading Video)

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

AN14NMESVidmov
Media File (videoquicktime)

514

NMES on Leg Extension Machine

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i

where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi

The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Potential Controller (continued)

Euler iterations used for control

zk+1 = Ω(Ti + khi hi zk v) for k = 0 Ni minus 1 where

z0 =

(tan (q(Ti))minus tan (qd (Ti))

q(Ti )cos2(q(Ti ))

minus qd (Ti )cos2(qd (Ti ))

) hi = τ

Ni

and Ω [0+infin)2 times R2 rarr R2 is defined by

Ω(T h x v) =

[Ω1(T h x v)Ω2(T h x v)

](8)

and the formulas

Ω1(T h x v) = x1 + hx2 and

Ω2(T h x v) = x2 + ζ2d (T ) +int T+h

T g1(ζd (s)+x)ds

+int T+h

T g2 (ζd (s)+x) v(s minus τ)ds minus ζ2d (T +h)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1214

NMES Theorem (IK MM MK Ruzhou IJRNC)

For each pair (τ r) isin (0infin)2 there exist a locally boundedfunction N a constant ω isin (0 micro2) and a locally Lipschitzfunction C satisfying C(0) = 0 such that For all sample timesTi in [0infin) such that supige0 (Ti+1 minus Ti) le r and each initialcondition the solution (q(t) q(t) v(t)) with

Ni = N(∣∣∣(tan(q(Ti)) q(Ti )

cos2(q(Ti ))

)minus ζd (Ti)

∣∣∣+||v minus vd ||[Timinusτ Ti ]

) (9)

satisfies|q(t)minus qd (t)|+ |q(t)minus qd (t)|+ ||v minus vd ||[tminusτ t]

le eminusωtC(|q(0)minusqd (0)|+|q(0)minusqd (0)|

cos2(q(0)) + ||v0 minus vd ||[minusτ 0])

for all t ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn)

andL2

ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed system

E2 Find bounds on the allowable delays τ such thatY prime(t) = F

(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

NMES on Leg Extension Machine

(Loading Video)

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

TVSpotwmv
Media File (videox-ms-wmv)

514

NMES on Leg Extension Machine

(Loading Video)

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

AN14NMESVidmov
Media File (videoquicktime)

514

NMES on Leg Extension Machine

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i

where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi

The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Potential Controller (continued)

Euler iterations used for control

zk+1 = Ω(Ti + khi hi zk v) for k = 0 Ni minus 1 where

z0 =

(tan (q(Ti))minus tan (qd (Ti))

q(Ti )cos2(q(Ti ))

minus qd (Ti )cos2(qd (Ti ))

) hi = τ

Ni

and Ω [0+infin)2 times R2 rarr R2 is defined by

Ω(T h x v) =

[Ω1(T h x v)Ω2(T h x v)

](8)

and the formulas

Ω1(T h x v) = x1 + hx2 and

Ω2(T h x v) = x2 + ζ2d (T ) +int T+h

T g1(ζd (s)+x)ds

+int T+h

T g2 (ζd (s)+x) v(s minus τ)ds minus ζ2d (T +h)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1214

NMES Theorem (IK MM MK Ruzhou IJRNC)

For each pair (τ r) isin (0infin)2 there exist a locally boundedfunction N a constant ω isin (0 micro2) and a locally Lipschitzfunction C satisfying C(0) = 0 such that For all sample timesTi in [0infin) such that supige0 (Ti+1 minus Ti) le r and each initialcondition the solution (q(t) q(t) v(t)) with

Ni = N(∣∣∣(tan(q(Ti)) q(Ti )

cos2(q(Ti ))

)minus ζd (Ti)

∣∣∣+||v minus vd ||[Timinusτ Ti ]

) (9)

satisfies|q(t)minus qd (t)|+ |q(t)minus qd (t)|+ ||v minus vd ||[tminusτ t]

le eminusωtC(|q(0)minusqd (0)|+|q(0)minusqd (0)|

cos2(q(0)) + ||v0 minus vd ||[minusτ 0])

for all t ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed system

E2 Find bounds on the allowable delays τ such thatY prime(t) = F

(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

NMES on Leg Extension Machine

(Loading Video)

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

TVSpotwmv
Media File (videox-ms-wmv)

514

NMES on Leg Extension Machine

(Loading Video)

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

AN14NMESVidmov
Media File (videoquicktime)

514

NMES on Leg Extension Machine

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i

where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi

The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Potential Controller (continued)

Euler iterations used for control

zk+1 = Ω(Ti + khi hi zk v) for k = 0 Ni minus 1 where

z0 =

(tan (q(Ti))minus tan (qd (Ti))

q(Ti )cos2(q(Ti ))

minus qd (Ti )cos2(qd (Ti ))

) hi = τ

Ni

and Ω [0+infin)2 times R2 rarr R2 is defined by

Ω(T h x v) =

[Ω1(T h x v)Ω2(T h x v)

](8)

and the formulas

Ω1(T h x v) = x1 + hx2 and

Ω2(T h x v) = x2 + ζ2d (T ) +int T+h

T g1(ζd (s)+x)ds

+int T+h

T g2 (ζd (s)+x) v(s minus τ)ds minus ζ2d (T +h)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1214

NMES Theorem (IK MM MK Ruzhou IJRNC)

For each pair (τ r) isin (0infin)2 there exist a locally boundedfunction N a constant ω isin (0 micro2) and a locally Lipschitzfunction C satisfying C(0) = 0 such that For all sample timesTi in [0infin) such that supige0 (Ti+1 minus Ti) le r and each initialcondition the solution (q(t) q(t) v(t)) with

Ni = N(∣∣∣(tan(q(Ti)) q(Ti )

cos2(q(Ti ))

)minus ζd (Ti)

∣∣∣+||v minus vd ||[Timinusτ Ti ]

) (9)

satisfies|q(t)minus qd (t)|+ |q(t)minus qd (t)|+ ||v minus vd ||[tminusτ t]

le eminusωtC(|q(0)minusqd (0)|+|q(0)minusqd (0)|

cos2(q(0)) + ||v0 minus vd ||[minusτ 0])

for all t ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed system

E2 Find bounds on the allowable delays τ such thatY prime(t) = F

(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

NMES on Leg Extension Machine

(Loading Video)

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

TVSpotwmv
Media File (videox-ms-wmv)

514

NMES on Leg Extension Machine

(Loading Video)

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

AN14NMESVidmov
Media File (videoquicktime)

514

NMES on Leg Extension Machine

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i

where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi

The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Potential Controller (continued)

Euler iterations used for control

zk+1 = Ω(Ti + khi hi zk v) for k = 0 Ni minus 1 where

z0 =

(tan (q(Ti))minus tan (qd (Ti))

q(Ti )cos2(q(Ti ))

minus qd (Ti )cos2(qd (Ti ))

) hi = τ

Ni

and Ω [0+infin)2 times R2 rarr R2 is defined by

Ω(T h x v) =

[Ω1(T h x v)Ω2(T h x v)

](8)

and the formulas

Ω1(T h x v) = x1 + hx2 and

Ω2(T h x v) = x2 + ζ2d (T ) +int T+h

T g1(ζd (s)+x)ds

+int T+h

T g2 (ζd (s)+x) v(s minus τ)ds minus ζ2d (T +h)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1214

NMES Theorem (IK MM MK Ruzhou IJRNC)

For each pair (τ r) isin (0infin)2 there exist a locally boundedfunction N a constant ω isin (0 micro2) and a locally Lipschitzfunction C satisfying C(0) = 0 such that For all sample timesTi in [0infin) such that supige0 (Ti+1 minus Ti) le r and each initialcondition the solution (q(t) q(t) v(t)) with

Ni = N(∣∣∣(tan(q(Ti)) q(Ti )

cos2(q(Ti ))

)minus ζd (Ti)

∣∣∣+||v minus vd ||[Timinusτ Ti ]

) (9)

satisfies|q(t)minus qd (t)|+ |q(t)minus qd (t)|+ ||v minus vd ||[tminusτ t]

le eminusωtC(|q(0)minusqd (0)|+|q(0)minusqd (0)|

cos2(q(0)) + ||v0 minus vd ||[minusτ 0])

for all t ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed system

E2 Find bounds on the allowable delays τ such thatY prime(t) = F

(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

NMES on Leg Extension Machine

(Loading Video)

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

TVSpotwmv
Media File (videox-ms-wmv)

514

NMES on Leg Extension Machine

(Loading Video)

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

AN14NMESVidmov
Media File (videoquicktime)

514

NMES on Leg Extension Machine

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i

where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi

The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Potential Controller (continued)

Euler iterations used for control

zk+1 = Ω(Ti + khi hi zk v) for k = 0 Ni minus 1 where

z0 =

(tan (q(Ti))minus tan (qd (Ti))

q(Ti )cos2(q(Ti ))

minus qd (Ti )cos2(qd (Ti ))

) hi = τ

Ni

and Ω [0+infin)2 times R2 rarr R2 is defined by

Ω(T h x v) =

[Ω1(T h x v)Ω2(T h x v)

](8)

and the formulas

Ω1(T h x v) = x1 + hx2 and

Ω2(T h x v) = x2 + ζ2d (T ) +int T+h

T g1(ζd (s)+x)ds

+int T+h

T g2 (ζd (s)+x) v(s minus τ)ds minus ζ2d (T +h)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1214

NMES Theorem (IK MM MK Ruzhou IJRNC)

For each pair (τ r) isin (0infin)2 there exist a locally boundedfunction N a constant ω isin (0 micro2) and a locally Lipschitzfunction C satisfying C(0) = 0 such that For all sample timesTi in [0infin) such that supige0 (Ti+1 minus Ti) le r and each initialcondition the solution (q(t) q(t) v(t)) with

Ni = N(∣∣∣(tan(q(Ti)) q(Ti )

cos2(q(Ti ))

)minus ζd (Ti)

∣∣∣+||v minus vd ||[Timinusτ Ti ]

) (9)

satisfies|q(t)minus qd (t)|+ |q(t)minus qd (t)|+ ||v minus vd ||[tminusτ t]

le eminusωtC(|q(0)minusqd (0)|+|q(0)minusqd (0)|

cos2(q(0)) + ||v0 minus vd ||[minusτ 0])

for all t ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

NMES on Leg Extension Machine

(Loading Video)

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

TVSpotwmv
Media File (videox-ms-wmv)

514

NMES on Leg Extension Machine

(Loading Video)

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

AN14NMESVidmov
Media File (videoquicktime)

514

NMES on Leg Extension Machine

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i

where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi

The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Potential Controller (continued)

Euler iterations used for control

zk+1 = Ω(Ti + khi hi zk v) for k = 0 Ni minus 1 where

z0 =

(tan (q(Ti))minus tan (qd (Ti))

q(Ti )cos2(q(Ti ))

minus qd (Ti )cos2(qd (Ti ))

) hi = τ

Ni

and Ω [0+infin)2 times R2 rarr R2 is defined by

Ω(T h x v) =

[Ω1(T h x v)Ω2(T h x v)

](8)

and the formulas

Ω1(T h x v) = x1 + hx2 and

Ω2(T h x v) = x2 + ζ2d (T ) +int T+h

T g1(ζd (s)+x)ds

+int T+h

T g2 (ζd (s)+x) v(s minus τ)ds minus ζ2d (T +h)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1214

NMES Theorem (IK MM MK Ruzhou IJRNC)

For each pair (τ r) isin (0infin)2 there exist a locally boundedfunction N a constant ω isin (0 micro2) and a locally Lipschitzfunction C satisfying C(0) = 0 such that For all sample timesTi in [0infin) such that supige0 (Ti+1 minus Ti) le r and each initialcondition the solution (q(t) q(t) v(t)) with

Ni = N(∣∣∣(tan(q(Ti)) q(Ti )

cos2(q(Ti ))

)minus ζd (Ti)

∣∣∣+||v minus vd ||[Timinusτ Ti ]

) (9)

satisfies|q(t)minus qd (t)|+ |q(t)minus qd (t)|+ ||v minus vd ||[tminusτ t]

le eminusωtC(|q(0)minusqd (0)|+|q(0)minusqd (0)|

cos2(q(0)) + ||v0 minus vd ||[minusτ 0])

for all t ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

What is a Lyapunov-Krasovskii Functional (LKF)

Definition We call V ] an ISS-LKF for Y prime(t) = G(t Yt δ(t))provided there exist functions γi isin Kinfin such that

L1 γ1(|φ(0)|) le V ](t φ) le γ2(|φ|[minusτ 0])for all (t φ) isin [0infin)times C([minusτ 0]Rn) and

L2ddt

[V ](t Yt )

]le minusγ3(V ](t Yt )) + γ4(|δ(t)|)

along all trajectories of the system

Emulation Approach for Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)

E1 Find u such that Y prime(t) = F(t Y (t)u(t Y (t))0

)satisfies

UGAS and a LF V for this undelayed unperturbed systemE2 Find bounds on the allowable delays τ such that

Y prime(t) = F(t Y (t)u(t Y (t minus τ)) δ(t)

)satisfies ISS

E3 Step E2 is often done by converting V into an ISS-LKF V ]

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

NMES on Leg Extension Machine

(Loading Video)

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

TVSpotwmv
Media File (videox-ms-wmv)

514

NMES on Leg Extension Machine

(Loading Video)

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

AN14NMESVidmov
Media File (videoquicktime)

514

NMES on Leg Extension Machine

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i

where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi

The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Potential Controller (continued)

Euler iterations used for control

zk+1 = Ω(Ti + khi hi zk v) for k = 0 Ni minus 1 where

z0 =

(tan (q(Ti))minus tan (qd (Ti))

q(Ti )cos2(q(Ti ))

minus qd (Ti )cos2(qd (Ti ))

) hi = τ

Ni

and Ω [0+infin)2 times R2 rarr R2 is defined by

Ω(T h x v) =

[Ω1(T h x v)Ω2(T h x v)

](8)

and the formulas

Ω1(T h x v) = x1 + hx2 and

Ω2(T h x v) = x2 + ζ2d (T ) +int T+h

T g1(ζd (s)+x)ds

+int T+h

T g2 (ζd (s)+x) v(s minus τ)ds minus ζ2d (T +h)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1214

NMES Theorem (IK MM MK Ruzhou IJRNC)

For each pair (τ r) isin (0infin)2 there exist a locally boundedfunction N a constant ω isin (0 micro2) and a locally Lipschitzfunction C satisfying C(0) = 0 such that For all sample timesTi in [0infin) such that supige0 (Ti+1 minus Ti) le r and each initialcondition the solution (q(t) q(t) v(t)) with

Ni = N(∣∣∣(tan(q(Ti)) q(Ti )

cos2(q(Ti ))

)minus ζd (Ti)

∣∣∣+||v minus vd ||[Timinusτ Ti ]

) (9)

satisfies|q(t)minus qd (t)|+ |q(t)minus qd (t)|+ ||v minus vd ||[tminusτ t]

le eminusωtC(|q(0)minusqd (0)|+|q(0)minusqd (0)|

cos2(q(0)) + ||v0 minus vd ||[minusτ 0])

for all t ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

NMES on Leg Extension Machine

(Loading Video)

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

TVSpotwmv
Media File (videox-ms-wmv)

514

NMES on Leg Extension Machine

(Loading Video)

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

AN14NMESVidmov
Media File (videoquicktime)

514

NMES on Leg Extension Machine

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i

where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi

The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Potential Controller (continued)

Euler iterations used for control

zk+1 = Ω(Ti + khi hi zk v) for k = 0 Ni minus 1 where

z0 =

(tan (q(Ti))minus tan (qd (Ti))

q(Ti )cos2(q(Ti ))

minus qd (Ti )cos2(qd (Ti ))

) hi = τ

Ni

and Ω [0+infin)2 times R2 rarr R2 is defined by

Ω(T h x v) =

[Ω1(T h x v)Ω2(T h x v)

](8)

and the formulas

Ω1(T h x v) = x1 + hx2 and

Ω2(T h x v) = x2 + ζ2d (T ) +int T+h

T g1(ζd (s)+x)ds

+int T+h

T g2 (ζd (s)+x) v(s minus τ)ds minus ζ2d (T +h)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1214

NMES Theorem (IK MM MK Ruzhou IJRNC)

For each pair (τ r) isin (0infin)2 there exist a locally boundedfunction N a constant ω isin (0 micro2) and a locally Lipschitzfunction C satisfying C(0) = 0 such that For all sample timesTi in [0infin) such that supige0 (Ti+1 minus Ti) le r and each initialcondition the solution (q(t) q(t) v(t)) with

Ni = N(∣∣∣(tan(q(Ti)) q(Ti )

cos2(q(Ti ))

)minus ζd (Ti)

∣∣∣+||v minus vd ||[Timinusτ Ti ]

) (9)

satisfies|q(t)minus qd (t)|+ |q(t)minus qd (t)|+ ||v minus vd ||[tminusτ t]

le eminusωtC(|q(0)minusqd (0)|+|q(0)minusqd (0)|

cos2(q(0)) + ||v0 minus vd ||[minusτ 0])

for all t ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

NMES on Leg Extension Machine

(Loading Video)

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

AN14NMESVidmov
Media File (videoquicktime)

514

NMES on Leg Extension Machine

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i

where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi

The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Potential Controller (continued)

Euler iterations used for control

zk+1 = Ω(Ti + khi hi zk v) for k = 0 Ni minus 1 where

z0 =

(tan (q(Ti))minus tan (qd (Ti))

q(Ti )cos2(q(Ti ))

minus qd (Ti )cos2(qd (Ti ))

) hi = τ

Ni

and Ω [0+infin)2 times R2 rarr R2 is defined by

Ω(T h x v) =

[Ω1(T h x v)Ω2(T h x v)

](8)

and the formulas

Ω1(T h x v) = x1 + hx2 and

Ω2(T h x v) = x2 + ζ2d (T ) +int T+h

T g1(ζd (s)+x)ds

+int T+h

T g2 (ζd (s)+x) v(s minus τ)ds minus ζ2d (T +h)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1214

NMES Theorem (IK MM MK Ruzhou IJRNC)

For each pair (τ r) isin (0infin)2 there exist a locally boundedfunction N a constant ω isin (0 micro2) and a locally Lipschitzfunction C satisfying C(0) = 0 such that For all sample timesTi in [0infin) such that supige0 (Ti+1 minus Ti) le r and each initialcondition the solution (q(t) q(t) v(t)) with

Ni = N(∣∣∣(tan(q(Ti)) q(Ti )

cos2(q(Ti ))

)minus ζd (Ti)

∣∣∣+||v minus vd ||[Timinusτ Ti ]

) (9)

satisfies|q(t)minus qd (t)|+ |q(t)minus qd (t)|+ ||v minus vd ||[tminusτ t]

le eminusωtC(|q(0)minusqd (0)|+|q(0)minusqd (0)|

cos2(q(0)) + ||v0 minus vd ||[minusτ 0])

for all t ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

514

NMES on Leg Extension Machine

Leg extension machine at Warren Dixonrsquos NCR Lab at U of FL

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i

where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi

The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Potential Controller (continued)

Euler iterations used for control

zk+1 = Ω(Ti + khi hi zk v) for k = 0 Ni minus 1 where

z0 =

(tan (q(Ti))minus tan (qd (Ti))

q(Ti )cos2(q(Ti ))

minus qd (Ti )cos2(qd (Ti ))

) hi = τ

Ni

and Ω [0+infin)2 times R2 rarr R2 is defined by

Ω(T h x v) =

[Ω1(T h x v)Ω2(T h x v)

](8)

and the formulas

Ω1(T h x v) = x1 + hx2 and

Ω2(T h x v) = x2 + ζ2d (T ) +int T+h

T g1(ζd (s)+x)ds

+int T+h

T g2 (ζd (s)+x) v(s minus τ)ds minus ζ2d (T +h)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1214

NMES Theorem (IK MM MK Ruzhou IJRNC)

For each pair (τ r) isin (0infin)2 there exist a locally boundedfunction N a constant ω isin (0 micro2) and a locally Lipschitzfunction C satisfying C(0) = 0 such that For all sample timesTi in [0infin) such that supige0 (Ti+1 minus Ti) le r and each initialcondition the solution (q(t) q(t) v(t)) with

Ni = N(∣∣∣(tan(q(Ti)) q(Ti )

cos2(q(Ti ))

)minus ζd (Ti)

∣∣∣+||v minus vd ||[Timinusτ Ti ]

) (9)

satisfies|q(t)minus qd (t)|+ |q(t)minus qd (t)|+ ||v minus vd ||[tminusτ t]

le eminusωtC(|q(0)minusqd (0)|+|q(0)minusqd (0)|

cos2(q(0)) + ||v0 minus vd ||[minusτ 0])

for all t ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i

where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi

The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Potential Controller (continued)

Euler iterations used for control

zk+1 = Ω(Ti + khi hi zk v) for k = 0 Ni minus 1 where

z0 =

(tan (q(Ti))minus tan (qd (Ti))

q(Ti )cos2(q(Ti ))

minus qd (Ti )cos2(qd (Ti ))

) hi = τ

Ni

and Ω [0+infin)2 times R2 rarr R2 is defined by

Ω(T h x v) =

[Ω1(T h x v)Ω2(T h x v)

](8)

and the formulas

Ω1(T h x v) = x1 + hx2 and

Ω2(T h x v) = x2 + ζ2d (T ) +int T+h

T g1(ζd (s)+x)ds

+int T+h

T g2 (ζd (s)+x) v(s minus τ)ds minus ζ2d (T +h)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1214

NMES Theorem (IK MM MK Ruzhou IJRNC)

For each pair (τ r) isin (0infin)2 there exist a locally boundedfunction N a constant ω isin (0 micro2) and a locally Lipschitzfunction C satisfying C(0) = 0 such that For all sample timesTi in [0infin) such that supige0 (Ti+1 minus Ti) le r and each initialcondition the solution (q(t) q(t) v(t)) with

Ni = N(∣∣∣(tan(q(Ti)) q(Ti )

cos2(q(Ti ))

)minus ζd (Ti)

∣∣∣+||v minus vd ||[Timinusτ Ti ]

) (9)

satisfies|q(t)minus qd (t)|+ |q(t)minus qd (t)|+ ||v minus vd ||[tminusτ t]

le eminusωtC(|q(0)minusqd (0)|+|q(0)minusqd (0)|

cos2(q(0)) + ||v0 minus vd ||[minusτ 0])

for all t ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i

where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi

The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Potential Controller (continued)

Euler iterations used for control

zk+1 = Ω(Ti + khi hi zk v) for k = 0 Ni minus 1 where

z0 =

(tan (q(Ti))minus tan (qd (Ti))

q(Ti )cos2(q(Ti ))

minus qd (Ti )cos2(qd (Ti ))

) hi = τ

Ni

and Ω [0+infin)2 times R2 rarr R2 is defined by

Ω(T h x v) =

[Ω1(T h x v)Ω2(T h x v)

](8)

and the formulas

Ω1(T h x v) = x1 + hx2 and

Ω2(T h x v) = x2 + ζ2d (T ) +int T+h

T g1(ζd (s)+x)ds

+int T+h

T g2 (ζd (s)+x) v(s minus τ)ds minus ζ2d (T +h)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1214

NMES Theorem (IK MM MK Ruzhou IJRNC)

For each pair (τ r) isin (0infin)2 there exist a locally boundedfunction N a constant ω isin (0 micro2) and a locally Lipschitzfunction C satisfying C(0) = 0 such that For all sample timesTi in [0infin) such that supige0 (Ti+1 minus Ti) le r and each initialcondition the solution (q(t) q(t) v(t)) with

Ni = N(∣∣∣(tan(q(Ti)) q(Ti )

cos2(q(Ti ))

)minus ζd (Ti)

∣∣∣+||v minus vd ||[Timinusτ Ti ]

) (9)

satisfies|q(t)minus qd (t)|+ |q(t)minus qd (t)|+ ||v minus vd ||[tminusτ t]

le eminusωtC(|q(0)minusqd (0)|+|q(0)minusqd (0)|

cos2(q(0)) + ||v0 minus vd ||[minusτ 0])

for all t ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i

where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi

The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Potential Controller (continued)

Euler iterations used for control

zk+1 = Ω(Ti + khi hi zk v) for k = 0 Ni minus 1 where

z0 =

(tan (q(Ti))minus tan (qd (Ti))

q(Ti )cos2(q(Ti ))

minus qd (Ti )cos2(qd (Ti ))

) hi = τ

Ni

and Ω [0+infin)2 times R2 rarr R2 is defined by

Ω(T h x v) =

[Ω1(T h x v)Ω2(T h x v)

](8)

and the formulas

Ω1(T h x v) = x1 + hx2 and

Ω2(T h x v) = x2 + ζ2d (T ) +int T+h

T g1(ζd (s)+x)ds

+int T+h

T g2 (ζd (s)+x) v(s minus τ)ds minus ζ2d (T +h)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1214

NMES Theorem (IK MM MK Ruzhou IJRNC)

For each pair (τ r) isin (0infin)2 there exist a locally boundedfunction N a constant ω isin (0 micro2) and a locally Lipschitzfunction C satisfying C(0) = 0 such that For all sample timesTi in [0infin) such that supige0 (Ti+1 minus Ti) le r and each initialcondition the solution (q(t) q(t) v(t)) with

Ni = N(∣∣∣(tan(q(Ti)) q(Ti )

cos2(q(Ti ))

)minus ζd (Ti)

∣∣∣+||v minus vd ||[Timinusτ Ti ]

) (9)

satisfies|q(t)minus qd (t)|+ |q(t)minus qd (t)|+ ||v minus vd ||[tminusτ t]

le eminusωtC(|q(0)minusqd (0)|+|q(0)minusqd (0)|

cos2(q(0)) + ||v0 minus vd ||[minusτ 0])

for all t ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

614

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

MI (q) = Jq inertial effects of shank-foot complex about theknee-joint J = inertia of the combined shank and foot

Mv (q) = b1q + b2 tanh (b3q) viscous effects due to damping inthe musculotendon complex with constants bi gt 0

Me(q) = k1qeminusk2q + k3 tan (q) elastic effects due to jointstiffness with constants ki gt 0 We introduce the tan term toaccommodate our state constraint q isin (minusπ2 π2)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i

where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi

The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Potential Controller (continued)

Euler iterations used for control

zk+1 = Ω(Ti + khi hi zk v) for k = 0 Ni minus 1 where

z0 =

(tan (q(Ti))minus tan (qd (Ti))

q(Ti )cos2(q(Ti ))

minus qd (Ti )cos2(qd (Ti ))

) hi = τ

Ni

and Ω [0+infin)2 times R2 rarr R2 is defined by

Ω(T h x v) =

[Ω1(T h x v)Ω2(T h x v)

](8)

and the formulas

Ω1(T h x v) = x1 + hx2 and

Ω2(T h x v) = x2 + ζ2d (T ) +int T+h

T g1(ζd (s)+x)ds

+int T+h

T g2 (ζd (s)+x) v(s minus τ)ds minus ζ2d (T +h)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1214

NMES Theorem (IK MM MK Ruzhou IJRNC)

For each pair (τ r) isin (0infin)2 there exist a locally boundedfunction N a constant ω isin (0 micro2) and a locally Lipschitzfunction C satisfying C(0) = 0 such that For all sample timesTi in [0infin) such that supige0 (Ti+1 minus Ti) le r and each initialcondition the solution (q(t) q(t) v(t)) with

Ni = N(∣∣∣(tan(q(Ti)) q(Ti )

cos2(q(Ti ))

)minus ζd (Ti)

∣∣∣+||v minus vd ||[Timinusτ Ti ]

) (9)

satisfies|q(t)minus qd (t)|+ |q(t)minus qd (t)|+ ||v minus vd ||[tminusτ t]

le eminusωtC(|q(0)minusqd (0)|+|q(0)minusqd (0)|

cos2(q(0)) + ||v0 minus vd ||[minusτ 0])

for all t ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i

where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi

The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Potential Controller (continued)

Euler iterations used for control

zk+1 = Ω(Ti + khi hi zk v) for k = 0 Ni minus 1 where

z0 =

(tan (q(Ti))minus tan (qd (Ti))

q(Ti )cos2(q(Ti ))

minus qd (Ti )cos2(qd (Ti ))

) hi = τ

Ni

and Ω [0+infin)2 times R2 rarr R2 is defined by

Ω(T h x v) =

[Ω1(T h x v)Ω2(T h x v)

](8)

and the formulas

Ω1(T h x v) = x1 + hx2 and

Ω2(T h x v) = x2 + ζ2d (T ) +int T+h

T g1(ζd (s)+x)ds

+int T+h

T g2 (ζd (s)+x) v(s minus τ)ds minus ζ2d (T +h)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1214

NMES Theorem (IK MM MK Ruzhou IJRNC)

For each pair (τ r) isin (0infin)2 there exist a locally boundedfunction N a constant ω isin (0 micro2) and a locally Lipschitzfunction C satisfying C(0) = 0 such that For all sample timesTi in [0infin) such that supige0 (Ti+1 minus Ti) le r and each initialcondition the solution (q(t) q(t) v(t)) with

Ni = N(∣∣∣(tan(q(Ti)) q(Ti )

cos2(q(Ti ))

)minus ζd (Ti)

∣∣∣+||v minus vd ||[Timinusτ Ti ]

) (9)

satisfies|q(t)minus qd (t)|+ |q(t)minus qd (t)|+ ||v minus vd ||[tminusτ t]

le eminusωtC(|q(0)minusqd (0)|+|q(0)minusqd (0)|

cos2(q(0)) + ||v0 minus vd ||[minusτ 0])

for all t ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i

where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi

The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Potential Controller (continued)

Euler iterations used for control

zk+1 = Ω(Ti + khi hi zk v) for k = 0 Ni minus 1 where

z0 =

(tan (q(Ti))minus tan (qd (Ti))

q(Ti )cos2(q(Ti ))

minus qd (Ti )cos2(qd (Ti ))

) hi = τ

Ni

and Ω [0+infin)2 times R2 rarr R2 is defined by

Ω(T h x v) =

[Ω1(T h x v)Ω2(T h x v)

](8)

and the formulas

Ω1(T h x v) = x1 + hx2 and

Ω2(T h x v) = x2 + ζ2d (T ) +int T+h

T g1(ζd (s)+x)ds

+int T+h

T g2 (ζd (s)+x) v(s minus τ)ds minus ζ2d (T +h)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1214

NMES Theorem (IK MM MK Ruzhou IJRNC)

For each pair (τ r) isin (0infin)2 there exist a locally boundedfunction N a constant ω isin (0 micro2) and a locally Lipschitzfunction C satisfying C(0) = 0 such that For all sample timesTi in [0infin) such that supige0 (Ti+1 minus Ti) le r and each initialcondition the solution (q(t) q(t) v(t)) with

Ni = N(∣∣∣(tan(q(Ti)) q(Ti )

cos2(q(Ti ))

)minus ζd (Ti)

∣∣∣+||v minus vd ||[Timinusτ Ti ]

) (9)

satisfies|q(t)minus qd (t)|+ |q(t)minus qd (t)|+ ||v minus vd ||[tminusτ t]

le eminusωtC(|q(0)minusqd (0)|+|q(0)minusqd (0)|

cos2(q(0)) + ||v0 minus vd ||[minusτ 0])

for all t ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

714

Knee Joint Dynamics (Sharma et al 2009)

MI (q) + Mv (q) + Me (q) + Mg (q) = microc (KD)

Mg (q) =Mgl sin(q) gravitational component M = mass ofshank and foot g = gravitational acceleration l = distancebetween knee-joint and lumped center of mass of shank-foot

microc = ζ(q)F knee torque F = total muscle force at tendonζ(q) = positive valued moment arm

F = ξ(q q)v(t minus τ) v = voltage across quadricepsτ = latency between applying voltage and force production

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i

where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi

The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Potential Controller (continued)

Euler iterations used for control

zk+1 = Ω(Ti + khi hi zk v) for k = 0 Ni minus 1 where

z0 =

(tan (q(Ti))minus tan (qd (Ti))

q(Ti )cos2(q(Ti ))

minus qd (Ti )cos2(qd (Ti ))

) hi = τ

Ni

and Ω [0+infin)2 times R2 rarr R2 is defined by

Ω(T h x v) =

[Ω1(T h x v)Ω2(T h x v)

](8)

and the formulas

Ω1(T h x v) = x1 + hx2 and

Ω2(T h x v) = x2 + ζ2d (T ) +int T+h

T g1(ζd (s)+x)ds

+int T+h

T g2 (ζd (s)+x) v(s minus τ)ds minus ζ2d (T +h)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1214

NMES Theorem (IK MM MK Ruzhou IJRNC)

For each pair (τ r) isin (0infin)2 there exist a locally boundedfunction N a constant ω isin (0 micro2) and a locally Lipschitzfunction C satisfying C(0) = 0 such that For all sample timesTi in [0infin) such that supige0 (Ti+1 minus Ti) le r and each initialcondition the solution (q(t) q(t) v(t)) with

Ni = N(∣∣∣(tan(q(Ti)) q(Ti )

cos2(q(Ti ))

)minus ζd (Ti)

∣∣∣+||v minus vd ||[Timinusτ Ti ]

) (9)

satisfies|q(t)minus qd (t)|+ |q(t)minus qd (t)|+ ||v minus vd ||[tminusτ t]

le eminusωtC(|q(0)minusqd (0)|+|q(0)minusqd (0)|

cos2(q(0)) + ||v0 minus vd ||[minusτ 0])

for all t ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i

where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi

The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Potential Controller (continued)

Euler iterations used for control

zk+1 = Ω(Ti + khi hi zk v) for k = 0 Ni minus 1 where

z0 =

(tan (q(Ti))minus tan (qd (Ti))

q(Ti )cos2(q(Ti ))

minus qd (Ti )cos2(qd (Ti ))

) hi = τ

Ni

and Ω [0+infin)2 times R2 rarr R2 is defined by

Ω(T h x v) =

[Ω1(T h x v)Ω2(T h x v)

](8)

and the formulas

Ω1(T h x v) = x1 + hx2 and

Ω2(T h x v) = x2 + ζ2d (T ) +int T+h

T g1(ζd (s)+x)ds

+int T+h

T g2 (ζd (s)+x) v(s minus τ)ds minus ζ2d (T +h)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1214

NMES Theorem (IK MM MK Ruzhou IJRNC)

For each pair (τ r) isin (0infin)2 there exist a locally boundedfunction N a constant ω isin (0 micro2) and a locally Lipschitzfunction C satisfying C(0) = 0 such that For all sample timesTi in [0infin) such that supige0 (Ti+1 minus Ti) le r and each initialcondition the solution (q(t) q(t) v(t)) with

Ni = N(∣∣∣(tan(q(Ti)) q(Ti )

cos2(q(Ti ))

)minus ζd (Ti)

∣∣∣+||v minus vd ||[Timinusτ Ti ]

) (9)

satisfies|q(t)minus qd (t)|+ |q(t)minus qd (t)|+ ||v minus vd ||[tminusτ t]

le eminusωtC(|q(0)minusqd (0)|+|q(0)minusqd (0)|

cos2(q(0)) + ||v0 minus vd ||[minusτ 0])

for all t ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i

where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi

The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Potential Controller (continued)

Euler iterations used for control

zk+1 = Ω(Ti + khi hi zk v) for k = 0 Ni minus 1 where

z0 =

(tan (q(Ti))minus tan (qd (Ti))

q(Ti )cos2(q(Ti ))

minus qd (Ti )cos2(qd (Ti ))

) hi = τ

Ni

and Ω [0+infin)2 times R2 rarr R2 is defined by

Ω(T h x v) =

[Ω1(T h x v)Ω2(T h x v)

](8)

and the formulas

Ω1(T h x v) = x1 + hx2 and

Ω2(T h x v) = x2 + ζ2d (T ) +int T+h

T g1(ζd (s)+x)ds

+int T+h

T g2 (ζd (s)+x) v(s minus τ)ds minus ζ2d (T +h)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1214

NMES Theorem (IK MM MK Ruzhou IJRNC)

For each pair (τ r) isin (0infin)2 there exist a locally boundedfunction N a constant ω isin (0 micro2) and a locally Lipschitzfunction C satisfying C(0) = 0 such that For all sample timesTi in [0infin) such that supige0 (Ti+1 minus Ti) le r and each initialcondition the solution (q(t) q(t) v(t)) with

Ni = N(∣∣∣(tan(q(Ti)) q(Ti )

cos2(q(Ti ))

)minus ζd (Ti)

∣∣∣+||v minus vd ||[Timinusτ Ti ]

) (9)

satisfies|q(t)minus qd (t)|+ |q(t)minus qd (t)|+ ||v minus vd ||[tminusτ t]

le eminusωtC(|q(0)minusqd (0)|+|q(0)minusqd (0)|

cos2(q(0)) + ||v0 minus vd ||[minusτ 0])

for all t ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i

where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi

The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Potential Controller (continued)

Euler iterations used for control

zk+1 = Ω(Ti + khi hi zk v) for k = 0 Ni minus 1 where

z0 =

(tan (q(Ti))minus tan (qd (Ti))

q(Ti )cos2(q(Ti ))

minus qd (Ti )cos2(qd (Ti ))

) hi = τ

Ni

and Ω [0+infin)2 times R2 rarr R2 is defined by

Ω(T h x v) =

[Ω1(T h x v)Ω2(T h x v)

](8)

and the formulas

Ω1(T h x v) = x1 + hx2 and

Ω2(T h x v) = x2 + ζ2d (T ) +int T+h

T g1(ζd (s)+x)ds

+int T+h

T g2 (ζd (s)+x) v(s minus τ)ds minus ζ2d (T +h)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1214

NMES Theorem (IK MM MK Ruzhou IJRNC)

For each pair (τ r) isin (0infin)2 there exist a locally boundedfunction N a constant ω isin (0 micro2) and a locally Lipschitzfunction C satisfying C(0) = 0 such that For all sample timesTi in [0infin) such that supige0 (Ti+1 minus Ti) le r and each initialcondition the solution (q(t) q(t) v(t)) with

Ni = N(∣∣∣(tan(q(Ti)) q(Ti )

cos2(q(Ti ))

)minus ζd (Ti)

∣∣∣+||v minus vd ||[Timinusτ Ti ]

) (9)

satisfies|q(t)minus qd (t)|+ |q(t)minus qd (t)|+ ||v minus vd ||[tminusτ t]

le eminusωtC(|q(0)minusqd (0)|+|q(0)minusqd (0)|

cos2(q(0)) + ||v0 minus vd ||[minusτ 0])

for all t ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

814

Knee Joint Dynamics (Sharma et al 2009)

MI (q)︷︸︸︷Jq +

Mv (q)︷ ︸︸ ︷b1q + b2 tanh (b3q) +

Me(q)︷ ︸︸ ︷k1qeminusk2q + k3 tan (q)

+Mgl sin (q)︸ ︷︷ ︸Mg (q)

= A (q q) v (t minus τ) q isin (minusπ2

π2 )

(3)

A = scaled moment arm v = voltage control

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

Our RequirementsF (minusπ2 π2)rarr [0infin) is C2 and limqrarrplusmnπ2 F (q) =infin

G (minusπ2 π2)times Rrarr (0infin) is C1 and boundedH Rrarr R is C1 and infxisinR xH(x) ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i

where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi

The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Potential Controller (continued)

Euler iterations used for control

zk+1 = Ω(Ti + khi hi zk v) for k = 0 Ni minus 1 where

z0 =

(tan (q(Ti))minus tan (qd (Ti))

q(Ti )cos2(q(Ti ))

minus qd (Ti )cos2(qd (Ti ))

) hi = τ

Ni

and Ω [0+infin)2 times R2 rarr R2 is defined by

Ω(T h x v) =

[Ω1(T h x v)Ω2(T h x v)

](8)

and the formulas

Ω1(T h x v) = x1 + hx2 and

Ω2(T h x v) = x2 + ζ2d (T ) +int T+h

T g1(ζd (s)+x)ds

+int T+h

T g2 (ζd (s)+x) v(s minus τ)ds minus ζ2d (T +h)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1214

NMES Theorem (IK MM MK Ruzhou IJRNC)

For each pair (τ r) isin (0infin)2 there exist a locally boundedfunction N a constant ω isin (0 micro2) and a locally Lipschitzfunction C satisfying C(0) = 0 such that For all sample timesTi in [0infin) such that supige0 (Ti+1 minus Ti) le r and each initialcondition the solution (q(t) q(t) v(t)) with

Ni = N(∣∣∣(tan(q(Ti)) q(Ti )

cos2(q(Ti ))

)minus ζd (Ti)

∣∣∣+||v minus vd ||[Timinusτ Ti ]

) (9)

satisfies|q(t)minus qd (t)|+ |q(t)minus qd (t)|+ ||v minus vd ||[tminusτ t]

le eminusωtC(|q(0)minusqd (0)|+|q(0)minusqd (0)|

cos2(q(0)) + ||v0 minus vd ||[minusτ 0])

for all t ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i

where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi

The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Potential Controller (continued)

Euler iterations used for control

zk+1 = Ω(Ti + khi hi zk v) for k = 0 Ni minus 1 where

z0 =

(tan (q(Ti))minus tan (qd (Ti))

q(Ti )cos2(q(Ti ))

minus qd (Ti )cos2(qd (Ti ))

) hi = τ

Ni

and Ω [0+infin)2 times R2 rarr R2 is defined by

Ω(T h x v) =

[Ω1(T h x v)Ω2(T h x v)

](8)

and the formulas

Ω1(T h x v) = x1 + hx2 and

Ω2(T h x v) = x2 + ζ2d (T ) +int T+h

T g1(ζd (s)+x)ds

+int T+h

T g2 (ζd (s)+x) v(s minus τ)ds minus ζ2d (T +h)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1214

NMES Theorem (IK MM MK Ruzhou IJRNC)

For each pair (τ r) isin (0infin)2 there exist a locally boundedfunction N a constant ω isin (0 micro2) and a locally Lipschitzfunction C satisfying C(0) = 0 such that For all sample timesTi in [0infin) such that supige0 (Ti+1 minus Ti) le r and each initialcondition the solution (q(t) q(t) v(t)) with

Ni = N(∣∣∣(tan(q(Ti)) q(Ti )

cos2(q(Ti ))

)minus ζd (Ti)

∣∣∣+||v minus vd ||[Timinusτ Ti ]

) (9)

satisfies|q(t)minus qd (t)|+ |q(t)minus qd (t)|+ ||v minus vd ||[tminusτ t]

le eminusωtC(|q(0)minusqd (0)|+|q(0)minusqd (0)|

cos2(q(0)) + ||v0 minus vd ||[minusτ 0])

for all t ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i

where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi

The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Potential Controller (continued)

Euler iterations used for control

zk+1 = Ω(Ti + khi hi zk v) for k = 0 Ni minus 1 where

z0 =

(tan (q(Ti))minus tan (qd (Ti))

q(Ti )cos2(q(Ti ))

minus qd (Ti )cos2(qd (Ti ))

) hi = τ

Ni

and Ω [0+infin)2 times R2 rarr R2 is defined by

Ω(T h x v) =

[Ω1(T h x v)Ω2(T h x v)

](8)

and the formulas

Ω1(T h x v) = x1 + hx2 and

Ω2(T h x v) = x2 + ζ2d (T ) +int T+h

T g1(ζd (s)+x)ds

+int T+h

T g2 (ζd (s)+x) v(s minus τ)ds minus ζ2d (T +h)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1214

NMES Theorem (IK MM MK Ruzhou IJRNC)

For each pair (τ r) isin (0infin)2 there exist a locally boundedfunction N a constant ω isin (0 micro2) and a locally Lipschitzfunction C satisfying C(0) = 0 such that For all sample timesTi in [0infin) such that supige0 (Ti+1 minus Ti) le r and each initialcondition the solution (q(t) q(t) v(t)) with

Ni = N(∣∣∣(tan(q(Ti)) q(Ti )

cos2(q(Ti ))

)minus ζd (Ti)

∣∣∣+||v minus vd ||[Timinusτ Ti ]

) (9)

satisfies|q(t)minus qd (t)|+ |q(t)minus qd (t)|+ ||v minus vd ||[tminusτ t]

le eminusωtC(|q(0)minusqd (0)|+|q(0)minusqd (0)|

cos2(q(0)) + ||v0 minus vd ||[minusτ 0])

for all t ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i

where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi

The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Potential Controller (continued)

Euler iterations used for control

zk+1 = Ω(Ti + khi hi zk v) for k = 0 Ni minus 1 where

z0 =

(tan (q(Ti))minus tan (qd (Ti))

q(Ti )cos2(q(Ti ))

minus qd (Ti )cos2(qd (Ti ))

) hi = τ

Ni

and Ω [0+infin)2 times R2 rarr R2 is defined by

Ω(T h x v) =

[Ω1(T h x v)Ω2(T h x v)

](8)

and the formulas

Ω1(T h x v) = x1 + hx2 and

Ω2(T h x v) = x2 + ζ2d (T ) +int T+h

T g1(ζd (s)+x)ds

+int T+h

T g2 (ζd (s)+x) v(s minus τ)ds minus ζ2d (T +h)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1214

NMES Theorem (IK MM MK Ruzhou IJRNC)

For each pair (τ r) isin (0infin)2 there exist a locally boundedfunction N a constant ω isin (0 micro2) and a locally Lipschitzfunction C satisfying C(0) = 0 such that For all sample timesTi in [0infin) such that supige0 (Ti+1 minus Ti) le r and each initialcondition the solution (q(t) q(t) v(t)) with

Ni = N(∣∣∣(tan(q(Ti)) q(Ti )

cos2(q(Ti ))

)minus ζd (Ti)

∣∣∣+||v minus vd ||[Timinusτ Ti ]

) (9)

satisfies|q(t)minus qd (t)|+ |q(t)minus qd (t)|+ ||v minus vd ||[tminusτ t]

le eminusωtC(|q(0)minusqd (0)|+|q(0)minusqd (0)|

cos2(q(0)) + ||v0 minus vd ||[minusτ 0])

for all t ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

914

Tracking Problem

q(t) = minusdFdq (q(t))minus H(q(t)) + G(q(t) q(t))v(t minus τ) (4)

F (q) = k1 exp(minusk2q)Jk2

2

(exp(k2q)minus 1minus k2q

)+mgl

J

(1minus cos(q)

)+ k3

J ln(

1cos(q)

)

G(q q) = 1JA(q q) and

H(q) = b2J tanh(b3q) + b1

J q

(5)

qd (t) = minusdFdq (qd (t))minus H(qd (t)) + G(qd (t) qd (t))vd (t minus τ) (6)

max||qd ||infin ||vd ||infin ||vd ||infin ltinfin and ||qd ||infin lt π2 (7)

We want (q minus qd q minus qd )rarr 0 in a UGAS exponential way

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i

where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi

The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Potential Controller (continued)

Euler iterations used for control

zk+1 = Ω(Ti + khi hi zk v) for k = 0 Ni minus 1 where

z0 =

(tan (q(Ti))minus tan (qd (Ti))

q(Ti )cos2(q(Ti ))

minus qd (Ti )cos2(qd (Ti ))

) hi = τ

Ni

and Ω [0+infin)2 times R2 rarr R2 is defined by

Ω(T h x v) =

[Ω1(T h x v)Ω2(T h x v)

](8)

and the formulas

Ω1(T h x v) = x1 + hx2 and

Ω2(T h x v) = x2 + ζ2d (T ) +int T+h

T g1(ζd (s)+x)ds

+int T+h

T g2 (ζd (s)+x) v(s minus τ)ds minus ζ2d (T +h)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1214

NMES Theorem (IK MM MK Ruzhou IJRNC)

For each pair (τ r) isin (0infin)2 there exist a locally boundedfunction N a constant ω isin (0 micro2) and a locally Lipschitzfunction C satisfying C(0) = 0 such that For all sample timesTi in [0infin) such that supige0 (Ti+1 minus Ti) le r and each initialcondition the solution (q(t) q(t) v(t)) with

Ni = N(∣∣∣(tan(q(Ti)) q(Ti )

cos2(q(Ti ))

)minus ζd (Ti)

∣∣∣+||v minus vd ||[Timinusτ Ti ]

) (9)

satisfies|q(t)minus qd (t)|+ |q(t)minus qd (t)|+ ||v minus vd ||[tminusτ t]

le eminusωtC(|q(0)minusqd (0)|+|q(0)minusqd (0)|

cos2(q(0)) + ||v0 minus vd ||[minusτ 0])

for all t ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i

where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi

The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Potential Controller (continued)

Euler iterations used for control

zk+1 = Ω(Ti + khi hi zk v) for k = 0 Ni minus 1 where

z0 =

(tan (q(Ti))minus tan (qd (Ti))

q(Ti )cos2(q(Ti ))

minus qd (Ti )cos2(qd (Ti ))

) hi = τ

Ni

and Ω [0+infin)2 times R2 rarr R2 is defined by

Ω(T h x v) =

[Ω1(T h x v)Ω2(T h x v)

](8)

and the formulas

Ω1(T h x v) = x1 + hx2 and

Ω2(T h x v) = x2 + ζ2d (T ) +int T+h

T g1(ζd (s)+x)ds

+int T+h

T g2 (ζd (s)+x) v(s minus τ)ds minus ζ2d (T +h)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1214

NMES Theorem (IK MM MK Ruzhou IJRNC)

For each pair (τ r) isin (0infin)2 there exist a locally boundedfunction N a constant ω isin (0 micro2) and a locally Lipschitzfunction C satisfying C(0) = 0 such that For all sample timesTi in [0infin) such that supige0 (Ti+1 minus Ti) le r and each initialcondition the solution (q(t) q(t) v(t)) with

Ni = N(∣∣∣(tan(q(Ti)) q(Ti )

cos2(q(Ti ))

)minus ζd (Ti)

∣∣∣+||v minus vd ||[Timinusτ Ti ]

) (9)

satisfies|q(t)minus qd (t)|+ |q(t)minus qd (t)|+ ||v minus vd ||[tminusτ t]

le eminusωtC(|q(0)minusqd (0)|+|q(0)minusqd (0)|

cos2(q(0)) + ||v0 minus vd ||[minusτ 0])

for all t ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i

where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi

The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Potential Controller (continued)

Euler iterations used for control

zk+1 = Ω(Ti + khi hi zk v) for k = 0 Ni minus 1 where

z0 =

(tan (q(Ti))minus tan (qd (Ti))

q(Ti )cos2(q(Ti ))

minus qd (Ti )cos2(qd (Ti ))

) hi = τ

Ni

and Ω [0+infin)2 times R2 rarr R2 is defined by

Ω(T h x v) =

[Ω1(T h x v)Ω2(T h x v)

](8)

and the formulas

Ω1(T h x v) = x1 + hx2 and

Ω2(T h x v) = x2 + ζ2d (T ) +int T+h

T g1(ζd (s)+x)ds

+int T+h

T g2 (ζd (s)+x) v(s minus τ)ds minus ζ2d (T +h)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1214

NMES Theorem (IK MM MK Ruzhou IJRNC)

For each pair (τ r) isin (0infin)2 there exist a locally boundedfunction N a constant ω isin (0 micro2) and a locally Lipschitzfunction C satisfying C(0) = 0 such that For all sample timesTi in [0infin) such that supige0 (Ti+1 minus Ti) le r and each initialcondition the solution (q(t) q(t) v(t)) with

Ni = N(∣∣∣(tan(q(Ti)) q(Ti )

cos2(q(Ti ))

)minus ζd (Ti)

∣∣∣+||v minus vd ||[Timinusτ Ti ]

) (9)

satisfies|q(t)minus qd (t)|+ |q(t)minus qd (t)|+ ||v minus vd ||[tminusτ t]

le eminusωtC(|q(0)minusqd (0)|+|q(0)minusqd (0)|

cos2(q(0)) + ||v0 minus vd ||[minusτ 0])

for all t ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i

where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi

The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Potential Controller (continued)

Euler iterations used for control

zk+1 = Ω(Ti + khi hi zk v) for k = 0 Ni minus 1 where

z0 =

(tan (q(Ti))minus tan (qd (Ti))

q(Ti )cos2(q(Ti ))

minus qd (Ti )cos2(qd (Ti ))

) hi = τ

Ni

and Ω [0+infin)2 times R2 rarr R2 is defined by

Ω(T h x v) =

[Ω1(T h x v)Ω2(T h x v)

](8)

and the formulas

Ω1(T h x v) = x1 + hx2 and

Ω2(T h x v) = x2 + ζ2d (T ) +int T+h

T g1(ζd (s)+x)ds

+int T+h

T g2 (ζd (s)+x) v(s minus τ)ds minus ζ2d (T +h)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1214

NMES Theorem (IK MM MK Ruzhou IJRNC)

For each pair (τ r) isin (0infin)2 there exist a locally boundedfunction N a constant ω isin (0 micro2) and a locally Lipschitzfunction C satisfying C(0) = 0 such that For all sample timesTi in [0infin) such that supige0 (Ti+1 minus Ti) le r and each initialcondition the solution (q(t) q(t) v(t)) with

Ni = N(∣∣∣(tan(q(Ti)) q(Ti )

cos2(q(Ti ))

)minus ζd (Ti)

∣∣∣+||v minus vd ||[Timinusτ Ti ]

) (9)

satisfies|q(t)minus qd (t)|+ |q(t)minus qd (t)|+ ||v minus vd ||[tminusτ t]

le eminusωtC(|q(0)minusqd (0)|+|q(0)minusqd (0)|

cos2(q(0)) + ||v0 minus vd ||[minusτ 0])

for all t ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i

where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi

The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Potential Controller (continued)

Euler iterations used for control

zk+1 = Ω(Ti + khi hi zk v) for k = 0 Ni minus 1 where

z0 =

(tan (q(Ti))minus tan (qd (Ti))

q(Ti )cos2(q(Ti ))

minus qd (Ti )cos2(qd (Ti ))

) hi = τ

Ni

and Ω [0+infin)2 times R2 rarr R2 is defined by

Ω(T h x v) =

[Ω1(T h x v)Ω2(T h x v)

](8)

and the formulas

Ω1(T h x v) = x1 + hx2 and

Ω2(T h x v) = x2 + ζ2d (T ) +int T+h

T g1(ζd (s)+x)ds

+int T+h

T g2 (ζd (s)+x) v(s minus τ)ds minus ζ2d (T +h)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1214

NMES Theorem (IK MM MK Ruzhou IJRNC)

For each pair (τ r) isin (0infin)2 there exist a locally boundedfunction N a constant ω isin (0 micro2) and a locally Lipschitzfunction C satisfying C(0) = 0 such that For all sample timesTi in [0infin) such that supige0 (Ti+1 minus Ti) le r and each initialcondition the solution (q(t) q(t) v(t)) with

Ni = N(∣∣∣(tan(q(Ti)) q(Ti )

cos2(q(Ti ))

)minus ζd (Ti)

∣∣∣+||v minus vd ||[Timinusτ Ti ]

) (9)

satisfies|q(t)minus qd (t)|+ |q(t)minus qd (t)|+ ||v minus vd ||[tminusτ t]

le eminusωtC(|q(0)minusqd (0)|+|q(0)minusqd (0)|

cos2(q(0)) + ||v0 minus vd ||[minusτ 0])

for all t ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i

where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi

The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Potential Controller (continued)

Euler iterations used for control

zk+1 = Ω(Ti + khi hi zk v) for k = 0 Ni minus 1 where

z0 =

(tan (q(Ti))minus tan (qd (Ti))

q(Ti )cos2(q(Ti ))

minus qd (Ti )cos2(qd (Ti ))

) hi = τ

Ni

and Ω [0+infin)2 times R2 rarr R2 is defined by

Ω(T h x v) =

[Ω1(T h x v)Ω2(T h x v)

](8)

and the formulas

Ω1(T h x v) = x1 + hx2 and

Ω2(T h x v) = x2 + ζ2d (T ) +int T+h

T g1(ζd (s)+x)ds

+int T+h

T g2 (ζd (s)+x) v(s minus τ)ds minus ζ2d (T +h)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1214

NMES Theorem (IK MM MK Ruzhou IJRNC)

For each pair (τ r) isin (0infin)2 there exist a locally boundedfunction N a constant ω isin (0 micro2) and a locally Lipschitzfunction C satisfying C(0) = 0 such that For all sample timesTi in [0infin) such that supige0 (Ti+1 minus Ti) le r and each initialcondition the solution (q(t) q(t) v(t)) with

Ni = N(∣∣∣(tan(q(Ti)) q(Ti )

cos2(q(Ti ))

)minus ζd (Ti)

∣∣∣+||v minus vd ||[Timinusτ Ti ]

) (9)

satisfies|q(t)minus qd (t)|+ |q(t)minus qd (t)|+ ||v minus vd ||[tminusτ t]

le eminusωtC(|q(0)minusqd (0)|+|q(0)minusqd (0)|

cos2(q(0)) + ||v0 minus vd ||[minusτ 0])

for all t ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1014

Voltage Controller

Error variablesx1 = tan(q)minus tan(qd ) and x2 = q

cos2(q) minusqd

cos2(qd )

Three parts of the control scheme assuming t0 = 0

A numerical prediction ξ(Ti) = zNi of the error variables at timeTi + τ using (q(Ti) q(Ti)) isin (minusπ2 π2)times R

An intersample prediction ξ = (ξ1 ξ2) of the error variables forthe time interval between two consecutive measurements

Applying the predictor feedback v(t) ie the nominal controlwith the state variables replaced by their predicted values

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i

where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi

The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Potential Controller (continued)

Euler iterations used for control

zk+1 = Ω(Ti + khi hi zk v) for k = 0 Ni minus 1 where

z0 =

(tan (q(Ti))minus tan (qd (Ti))

q(Ti )cos2(q(Ti ))

minus qd (Ti )cos2(qd (Ti ))

) hi = τ

Ni

and Ω [0+infin)2 times R2 rarr R2 is defined by

Ω(T h x v) =

[Ω1(T h x v)Ω2(T h x v)

](8)

and the formulas

Ω1(T h x v) = x1 + hx2 and

Ω2(T h x v) = x2 + ζ2d (T ) +int T+h

T g1(ζd (s)+x)ds

+int T+h

T g2 (ζd (s)+x) v(s minus τ)ds minus ζ2d (T +h)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1214

NMES Theorem (IK MM MK Ruzhou IJRNC)

For each pair (τ r) isin (0infin)2 there exist a locally boundedfunction N a constant ω isin (0 micro2) and a locally Lipschitzfunction C satisfying C(0) = 0 such that For all sample timesTi in [0infin) such that supige0 (Ti+1 minus Ti) le r and each initialcondition the solution (q(t) q(t) v(t)) with

Ni = N(∣∣∣(tan(q(Ti)) q(Ti )

cos2(q(Ti ))

)minus ζd (Ti)

∣∣∣+||v minus vd ||[Timinusτ Ti ]

) (9)

satisfies|q(t)minus qd (t)|+ |q(t)minus qd (t)|+ ||v minus vd ||[tminusτ t]

le eminusωtC(|q(0)minusqd (0)|+|q(0)minusqd (0)|

cos2(q(0)) + ||v0 minus vd ||[minusτ 0])

for all t ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i

where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi

The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Potential Controller (continued)

Euler iterations used for control

zk+1 = Ω(Ti + khi hi zk v) for k = 0 Ni minus 1 where

z0 =

(tan (q(Ti))minus tan (qd (Ti))

q(Ti )cos2(q(Ti ))

minus qd (Ti )cos2(qd (Ti ))

) hi = τ

Ni

and Ω [0+infin)2 times R2 rarr R2 is defined by

Ω(T h x v) =

[Ω1(T h x v)Ω2(T h x v)

](8)

and the formulas

Ω1(T h x v) = x1 + hx2 and

Ω2(T h x v) = x2 + ζ2d (T ) +int T+h

T g1(ζd (s)+x)ds

+int T+h

T g2 (ζd (s)+x) v(s minus τ)ds minus ζ2d (T +h)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1214

NMES Theorem (IK MM MK Ruzhou IJRNC)

For each pair (τ r) isin (0infin)2 there exist a locally boundedfunction N a constant ω isin (0 micro2) and a locally Lipschitzfunction C satisfying C(0) = 0 such that For all sample timesTi in [0infin) such that supige0 (Ti+1 minus Ti) le r and each initialcondition the solution (q(t) q(t) v(t)) with

Ni = N(∣∣∣(tan(q(Ti)) q(Ti )

cos2(q(Ti ))

)minus ζd (Ti)

∣∣∣+||v minus vd ||[Timinusτ Ti ]

) (9)

satisfies|q(t)minus qd (t)|+ |q(t)minus qd (t)|+ ||v minus vd ||[tminusτ t]

le eminusωtC(|q(0)minusqd (0)|+|q(0)minusqd (0)|

cos2(q(0)) + ||v0 minus vd ||[minusτ 0])

for all t ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi

The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Potential Controller (continued)

Euler iterations used for control

zk+1 = Ω(Ti + khi hi zk v) for k = 0 Ni minus 1 where

z0 =

(tan (q(Ti))minus tan (qd (Ti))

q(Ti )cos2(q(Ti ))

minus qd (Ti )cos2(qd (Ti ))

) hi = τ

Ni

and Ω [0+infin)2 times R2 rarr R2 is defined by

Ω(T h x v) =

[Ω1(T h x v)Ω2(T h x v)

](8)

and the formulas

Ω1(T h x v) = x1 + hx2 and

Ω2(T h x v) = x2 + ζ2d (T ) +int T+h

T g1(ζd (s)+x)ds

+int T+h

T g2 (ζd (s)+x) v(s minus τ)ds minus ζ2d (T +h)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1214

NMES Theorem (IK MM MK Ruzhou IJRNC)

For each pair (τ r) isin (0infin)2 there exist a locally boundedfunction N a constant ω isin (0 micro2) and a locally Lipschitzfunction C satisfying C(0) = 0 such that For all sample timesTi in [0infin) such that supige0 (Ti+1 minus Ti) le r and each initialcondition the solution (q(t) q(t) v(t)) with

Ni = N(∣∣∣(tan(q(Ti)) q(Ti )

cos2(q(Ti ))

)minus ζd (Ti)

∣∣∣+||v minus vd ||[Timinusτ Ti ]

) (9)

satisfies|q(t)minus qd (t)|+ |q(t)minus qd (t)|+ ||v minus vd ||[tminusτ t]

le eminusωtC(|q(0)minusqd (0)|+|q(0)minusqd (0)|

cos2(q(0)) + ||v0 minus vd ||[minusτ 0])

for all t ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Controller

v(t) = g2(ζd (t+τ))vd (t)minusg1(ζd (t+τ)+ξ(t))+g1(ζd (t+τ))minus(1+micro2)ξ1(t)minus2microξ2(t)g2(ζd (t+τ)+ξ(t))

for all t isin [Ti Ti+1) and each i where

g1(x) = minus(1 + x21 )dF

dq (tanminus1(x1)) +2x1x2

21+x2

1minus (1 + x2

1 )H(

x21+x2

1

)

g2(x) = (1 + x21 )G

(tanminus1(x1) x2

1+x21

)

ζd (t) = (ζ1d (t) ζ2d (t)) =(

tan(qd (t)) qd (t)cos2(qd (t))

)

ξ1(t) = eminusmicro(tminusTi )

(ξ2(Ti) + microξ1(Ti)) sin(t minus Ti)

+ ξ1(Ti) cos(t minus Ti)

ξ2(t) = eminusmicro(tminusTi )minus(microξ2(Ti) + (1 + micro2)ξ1(Ti)

)sin(t minus Ti)

+ ξ2(Ti) cos(t minus Ti)

and ξ(Ti) = zNi The time-varying Euler iterations zk at eachtime Ti use measurements (q(Ti) q(Ti))

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Potential Controller (continued)

Euler iterations used for control

zk+1 = Ω(Ti + khi hi zk v) for k = 0 Ni minus 1 where

z0 =

(tan (q(Ti))minus tan (qd (Ti))

q(Ti )cos2(q(Ti ))

minus qd (Ti )cos2(qd (Ti ))

) hi = τ

Ni

and Ω [0+infin)2 times R2 rarr R2 is defined by

Ω(T h x v) =

[Ω1(T h x v)Ω2(T h x v)

](8)

and the formulas

Ω1(T h x v) = x1 + hx2 and

Ω2(T h x v) = x2 + ζ2d (T ) +int T+h

T g1(ζd (s)+x)ds

+int T+h

T g2 (ζd (s)+x) v(s minus τ)ds minus ζ2d (T +h)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1214

NMES Theorem (IK MM MK Ruzhou IJRNC)

For each pair (τ r) isin (0infin)2 there exist a locally boundedfunction N a constant ω isin (0 micro2) and a locally Lipschitzfunction C satisfying C(0) = 0 such that For all sample timesTi in [0infin) such that supige0 (Ti+1 minus Ti) le r and each initialcondition the solution (q(t) q(t) v(t)) with

Ni = N(∣∣∣(tan(q(Ti)) q(Ti )

cos2(q(Ti ))

)minus ζd (Ti)

∣∣∣+||v minus vd ||[Timinusτ Ti ]

) (9)

satisfies|q(t)minus qd (t)|+ |q(t)minus qd (t)|+ ||v minus vd ||[tminusτ t]

le eminusωtC(|q(0)minusqd (0)|+|q(0)minusqd (0)|

cos2(q(0)) + ||v0 minus vd ||[minusτ 0])

for all t ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1114

Voltage Potential Controller (continued)

Euler iterations used for control

zk+1 = Ω(Ti + khi hi zk v) for k = 0 Ni minus 1 where

z0 =

(tan (q(Ti))minus tan (qd (Ti))

q(Ti )cos2(q(Ti ))

minus qd (Ti )cos2(qd (Ti ))

) hi = τ

Ni

and Ω [0+infin)2 times R2 rarr R2 is defined by

Ω(T h x v) =

[Ω1(T h x v)Ω2(T h x v)

](8)

and the formulas

Ω1(T h x v) = x1 + hx2 and

Ω2(T h x v) = x2 + ζ2d (T ) +int T+h

T g1(ζd (s)+x)ds

+int T+h

T g2 (ζd (s)+x) v(s minus τ)ds minus ζ2d (T +h)

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1214

NMES Theorem (IK MM MK Ruzhou IJRNC)

For each pair (τ r) isin (0infin)2 there exist a locally boundedfunction N a constant ω isin (0 micro2) and a locally Lipschitzfunction C satisfying C(0) = 0 such that For all sample timesTi in [0infin) such that supige0 (Ti+1 minus Ti) le r and each initialcondition the solution (q(t) q(t) v(t)) with

Ni = N(∣∣∣(tan(q(Ti)) q(Ti )

cos2(q(Ti ))

)minus ζd (Ti)

∣∣∣+||v minus vd ||[Timinusτ Ti ]

) (9)

satisfies|q(t)minus qd (t)|+ |q(t)minus qd (t)|+ ||v minus vd ||[tminusτ t]

le eminusωtC(|q(0)minusqd (0)|+|q(0)minusqd (0)|

cos2(q(0)) + ||v0 minus vd ||[minusτ 0])

for all t ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1214

NMES Theorem (IK MM MK Ruzhou IJRNC)

For each pair (τ r) isin (0infin)2 there exist a locally boundedfunction N a constant ω isin (0 micro2) and a locally Lipschitzfunction C satisfying C(0) = 0 such that For all sample timesTi in [0infin) such that supige0 (Ti+1 minus Ti) le r and each initialcondition the solution (q(t) q(t) v(t)) with

Ni = N(∣∣∣(tan(q(Ti)) q(Ti )

cos2(q(Ti ))

)minus ζd (Ti)

∣∣∣+||v minus vd ||[Timinusτ Ti ]

) (9)

satisfies|q(t)minus qd (t)|+ |q(t)minus qd (t)|+ ||v minus vd ||[tminusτ t]

le eminusωtC(|q(0)minusqd (0)|+|q(0)minusqd (0)|

cos2(q(0)) + ||v0 minus vd ||[minusτ 0])

for all t ge 0

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Jq + b1q + b2 tanh (b3q) + k1qeminusk2q + k3 tan (q)

+Mgl sin (q) = A (q q) v (t minus τ) q isin (minusπ2

π2 )

(10)

τ = 007s A(q q) = aeminus2q2sin(q) + b

J = 039 kg-m2rad b1 = 06 kg-m2(rad-s) a = 0058b2 = 01 kg-m2(rad-s) b3 = 50 srad b = 00284k1 = 79 kg-m2(rad-s2) k2 = 1681radk3 = 117 kg-m2(rad-s2) M = 438 kg l = 0248 m

(11)

qd (t) =π

8sin(t) (1minus exp (minus8t)) rad (12)

q(0) = 05 rad q(0) = 0 rads v(t) = 0 on [minus0070)Ni = N = 10 and Ti+1 minus Ti = 0014s and micro = 2

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1314

MATLAB Simulation

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

1414

Summary of NMES Research

NMES is an important emerging technology that can helprehabilitate patients with motor neuron disorders

It produces difficult tracking control problems that containdelays state constraints and uncertainties

Our new sampled predictive control design overcame thesechallenges and can track a large class of reference trajectories

By incorporating the state constraint on the knee position ourcontrol can help ensure patient safety for any input delay value

Our control used a new numerical solution approximationmethod that covers many other time-varying models

In future work we hope to apply input-to-state stability to betterunderstand the effects of uncertainties under state constraints

Karafyllis (NTUA) Krstic (UCSD) Malisoff (LSU) et al Tracking Control for Neuromuscular Electrical Stimulation

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