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Sparsity Methods for Systems and ControlMaximum Hands-off Control
Masaaki Nagahara1
1The University of [email protected]
M. Nagahara (Univ of Kitakyushu) Sparsity Methods 1 / 19
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Table of Contents
1 L0 norm and sparse control
2 A simple example of maximum hands-off control
3 General formulation of maximum hands-off control
4 Conclusion
M. Nagahara (Univ of Kitakyushu) Sparsity Methods 2 / 19
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Table of Contents
1 L0 norm and sparse control
2 A simple example of maximum hands-off control
3 General formulation of maximum hands-off control
4 Conclusion
M. Nagahara (Univ of Kitakyushu) Sparsity Methods 3 / 19
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L0 norm of a function
The support of a function u(t), t ∈ [0, T]:
supp(u) ≜ {t ∈ [0, T] : u(t) , 0}.
The L0 norm of a function u(t):
∥u∥0 ≜ µ(supp(u)
),
µ(S) is the Lebesgue measure (i.e. the length) of a subset S ⊂ [0, T].L0 norm: the total length of time durations on which the signaltakes nonzero values.
M. Nagahara (Univ of Kitakyushu) Sparsity Methods 4 / 19
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L0 norm of a function
The support of a function u(t), t ∈ [0, T]:
supp(u) ≜ {t ∈ [0, T] : u(t) , 0}.
The L0 norm of a function u(t):
∥u∥0 ≜ µ(supp(u)
),
µ(S) is the Lebesgue measure (i.e. the length) of a subset S ⊂ [0, T].L0 norm: the total length of time durations on which the signaltakes nonzero values.
M. Nagahara (Univ of Kitakyushu) Sparsity Methods 4 / 19
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L0 norm of a function
The support of a function u(t), t ∈ [0, T]:
supp(u) ≜ {t ∈ [0, T] : u(t) , 0}.
The L0 norm of a function u(t):
∥u∥0 ≜ µ(supp(u)
),
µ(S) is the Lebesgue measure (i.e. the length) of a subset S ⊂ [0, T].L0 norm: the total length of time durations on which the signaltakes nonzero values.
M. Nagahara (Univ of Kitakyushu) Sparsity Methods 4 / 19
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L0 norm of a function
The support of a function u(t), t ∈ [0, T]:
supp(u) ≜ {t ∈ [0, T] : u(t) , 0}.
The L0 norm of a function u(t):
∥u∥0 ≜ µ(supp(u)
),
µ(S) is the Lebesgue measure (i.e. the length) of a subset S ⊂ [0, T].L0 norm: the total length of time durations on which the signaltakes nonzero values.
M. Nagahara (Univ of Kitakyushu) Sparsity Methods 4 / 19
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Example: L0 norm of a function
The L0 norm
∥u∥0 � µ(supp(u)) � t1 + (T − t2) � T − (t2 − t1).
0
T
u(t) = 0
t
u(t)
t1
t2
If ∥u∥0 is much smaller than the total length T (i.e., ∥u∥0 ≪ T),then the signal is said to be sparse.
M. Nagahara (Univ of Kitakyushu) Sparsity Methods 5 / 19
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Example: L0 norm of a function
The L0 norm
∥u∥0 � µ(supp(u)) � t1 + (T − t2) � T − (t2 − t1).
0
T
u(t) = 0
t
u(t)
t1
t2
If ∥u∥0 is much smaller than the total length T (i.e., ∥u∥0 ≪ T),then the signal is said to be sparse.
M. Nagahara (Univ of Kitakyushu) Sparsity Methods 5 / 19
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Practical benefits of sparsity in control
Let us consider the sparse control signal u(t), t ∈ [0, T].Actuators as electric motors need energy to generate power.If the control u(t) is sparse, we can stop energy supply to theactuator over the time interval [t1 , t2].Such a control is called a hands-off control.
This is also known as coasting.
We can also reduce CO or CO2 emissions, noise, and vibrations.
0
T
u(t) = 0
t
u(t)
t1
t2
M. Nagahara (Univ of Kitakyushu) Sparsity Methods 6 / 19
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Practical benefits of sparsity in control
Let us consider the sparse control signal u(t), t ∈ [0, T].Actuators as electric motors need energy to generate power.If the control u(t) is sparse, we can stop energy supply to theactuator over the time interval [t1 , t2].Such a control is called a hands-off control.
This is also known as coasting.
We can also reduce CO or CO2 emissions, noise, and vibrations.
0
T
u(t) = 0
t
u(t)
t1
t2
M. Nagahara (Univ of Kitakyushu) Sparsity Methods 6 / 19
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Practical benefits of sparsity in control
Let us consider the sparse control signal u(t), t ∈ [0, T].Actuators as electric motors need energy to generate power.If the control u(t) is sparse, we can stop energy supply to theactuator over the time interval [t1 , t2].Such a control is called a hands-off control.
This is also known as coasting.
We can also reduce CO or CO2 emissions, noise, and vibrations.
0
T
u(t) = 0
t
u(t)
t1
t2
M. Nagahara (Univ of Kitakyushu) Sparsity Methods 6 / 19
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Practical benefits of sparsity in control
Let us consider the sparse control signal u(t), t ∈ [0, T].Actuators as electric motors need energy to generate power.If the control u(t) is sparse, we can stop energy supply to theactuator over the time interval [t1 , t2].Such a control is called a hands-off control.
This is also known as coasting.
We can also reduce CO or CO2 emissions, noise, and vibrations.
0
T
u(t) = 0
t
u(t)
t1
t2
M. Nagahara (Univ of Kitakyushu) Sparsity Methods 6 / 19
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Practical benefits of sparsity in control
Let us consider the sparse control signal u(t), t ∈ [0, T].Actuators as electric motors need energy to generate power.If the control u(t) is sparse, we can stop energy supply to theactuator over the time interval [t1 , t2].Such a control is called a hands-off control.
This is also known as coasting.
We can also reduce CO or CO2 emissions, noise, and vibrations.
0
T
u(t) = 0
t
u(t)
t1
t2
M. Nagahara (Univ of Kitakyushu) Sparsity Methods 6 / 19
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Practical benefits of sparsity in control
Let us consider the sparse control signal u(t), t ∈ [0, T].Actuators as electric motors need energy to generate power.If the control u(t) is sparse, we can stop energy supply to theactuator over the time interval [t1 , t2].Such a control is called a hands-off control.
This is also known as coasting.
We can also reduce CO or CO2 emissions, noise, and vibrations.
0
T
u(t) = 0
t
u(t)
t1
t2
M. Nagahara (Univ of Kitakyushu) Sparsity Methods 6 / 19
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Examples of hands-off control
Start-stop system in vehiclesHybrid carsElectric locomotives
1. https://www.carprousa.com/Understanding-Vehicle-StartStop-Systems/a/3
2. https://en.wikipedia.org/wiki/Hybrid_vehicle
3. https://en.wikipedia.org/wiki/Electric_locomotive
M. Nagahara (Univ of Kitakyushu) Sparsity Methods 7 / 19
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Table of Contents
1 L0 norm and sparse control
2 A simple example of maximum hands-off control
3 General formulation of maximum hands-off control
4 Conclusion
M. Nagahara (Univ of Kitakyushu) Sparsity Methods 8 / 19
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A simple example: rocket
r(0) = ξ1 rocket
r(0) = ξ2 r(t), r(t)
r
m F (t)
Control objective: Given T > 0, find F(t), 0 ≤ t ≤ T, such that
r(T) � 0, Ûr(T) � 0.
System model: m Ür(t) � F(t) (Newton’s second law of motion)State variable:
x(t) ≜[r(t)Ûr(t)
]⇒ Ûx(t) �
[Ûr(t)Ür(t)
]�
[0 10 0
]x(t) +
[0
m−1
]u(t).
Control: u(t) � F(t) that we can choose under |u(t)| ≤ Umax.M. Nagahara (Univ of Kitakyushu) Sparsity Methods 9 / 19
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A simple example: rocket
r(0) = ξ1 rocket
r(0) = ξ2 r(t), r(t)
r
m F (t)
Control objective: Given T > 0, find F(t), 0 ≤ t ≤ T, such that
r(T) � 0, Ûr(T) � 0.
System model: m Ür(t) � F(t) (Newton’s second law of motion)State variable:
x(t) ≜[r(t)Ûr(t)
]⇒ Ûx(t) �
[Ûr(t)Ür(t)
]�
[0 10 0
]x(t) +
[0
m−1
]u(t).
Control: u(t) � F(t) that we can choose under |u(t)| ≤ Umax.M. Nagahara (Univ of Kitakyushu) Sparsity Methods 9 / 19
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A simple example: rocket
r(0) = ξ1 rocket
r(0) = ξ2 r(t), r(t)
r
m F (t)
Control objective: Given T > 0, find F(t), 0 ≤ t ≤ T, such that
r(T) � 0, Ûr(T) � 0.
System model: m Ür(t) � F(t) (Newton’s second law of motion)State variable:
x(t) ≜[r(t)Ûr(t)
]⇒ Ûx(t) �
[Ûr(t)Ür(t)
]�
[0 10 0
]x(t) +
[0
m−1
]u(t).
Control: u(t) � F(t) that we can choose under |u(t)| ≤ Umax.M. Nagahara (Univ of Kitakyushu) Sparsity Methods 9 / 19
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A simple example: rocket
r(0) = ξ1 rocket
r(0) = ξ2 r(t), r(t)
r
m F (t)
Control objective: Given T > 0, find F(t), 0 ≤ t ≤ T, such that
r(T) � 0, Ûr(T) � 0.
System model: m Ür(t) � F(t) (Newton’s second law of motion)State variable:
x(t) ≜[r(t)Ûr(t)
]⇒ Ûx(t) �
[Ûr(t)Ür(t)
]�
[0 10 0
]x(t) +
[0
m−1
]u(t).
Control: u(t) � F(t) that we can choose under |u(t)| ≤ Umax.M. Nagahara (Univ of Kitakyushu) Sparsity Methods 9 / 19
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Maximum hands-off control
Control system (rocket)
Ûx(t) �[0 10 0
]x(t) +
[01
]u(t), x(0) �
[11
]For simplicity, we assume m � 1, ξ1 � 1, ξ2 � 1, Umax � 1.
Feasible controlFix T > 0. Find a feasible control u(t), t ∈ [0, T] that drives the statefrom x(0) to x(T) � [0, 0]⊤ that satisfies |u(t)| ≤ 1, for all t ∈ [0, T].
Maximum hands-off control problemFind a feasible control that minimizes the L0 norm of u:
J0(u) � µ(supp(u)) �∫ T
0|u(t)|0dt (the length of the support)
M. Nagahara (Univ of Kitakyushu) Sparsity Methods 10 / 19
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Maximum hands-off control
Control system (rocket)
Ûx(t) �[0 10 0
]x(t) +
[01
]u(t), x(0) �
[11
]For simplicity, we assume m � 1, ξ1 � 1, ξ2 � 1, Umax � 1.
Feasible controlFix T > 0. Find a feasible control u(t), t ∈ [0, T] that drives the statefrom x(0) to x(T) � [0, 0]⊤ that satisfies |u(t)| ≤ 1, for all t ∈ [0, T].
Maximum hands-off control problemFind a feasible control that minimizes the L0 norm of u:
J0(u) � µ(supp(u)) �∫ T
0|u(t)|0dt (the length of the support)
M. Nagahara (Univ of Kitakyushu) Sparsity Methods 10 / 19
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Maximum hands-off control
Control system (rocket)
Ûx(t) �[0 10 0
]x(t) +
[01
]u(t), x(0) �
[11
]For simplicity, we assume m � 1, ξ1 � 1, ξ2 � 1, Umax � 1.
Feasible controlFix T > 0. Find a feasible control u(t), t ∈ [0, T] that drives the statefrom x(0) to x(T) � [0, 0]⊤ that satisfies |u(t)| ≤ 1, for all t ∈ [0, T].
Maximum hands-off control problemFind a feasible control that minimizes the L0 norm of u:
J0(u) � µ(supp(u)) �∫ T
0|u(t)|0dt (the length of the support)
M. Nagahara (Univ of Kitakyushu) Sparsity Methods 10 / 19
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L0 norm and L1 norm
L0 norm:
J0(u) � µ(supp(u)) �∫ T
0|u(t)|0dt
L1 norm
J1(u) �∫ T
0|u(t)|dt ,
−1 0 1
u
|u||u|01
L(u)
M. Nagahara (Univ of Kitakyushu) Sparsity Methods 11 / 19
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L0 norm and L1 norm
L0 norm:
J0(u) � µ(supp(u)) �∫ T
0|u(t)|0dt
L1 norm
J1(u) �∫ T
0|u(t)|dt ,
−1 0 1
u
|u||u|01
L(u)
M. Nagahara (Univ of Kitakyushu) Sparsity Methods 11 / 19
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L1 optimal control
Control system (rocket)
Ûx(t) �[0 10 0
]x(t) +
[01
]u(t), x(0) �
[11
]L1-optimal controlFix T > 0. Find a feasible control u(t), t ∈ [0, T] that drives the statefrom x(0) to x(T) � [0, 0]⊤, that satisfies |u(t)| ≤ 1, ∀ t ∈ [0, T], and thatminimizes the L1 norm of u:
J1(u) �∫ T
0|u(t)|dt .
Also known as fuel optimal control.A convex optimization problem.A closed-form solution is obtained.
M. Nagahara (Univ of Kitakyushu) Sparsity Methods 12 / 19
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L1 optimal control
Control system (rocket)
Ûx(t) �[0 10 0
]x(t) +
[01
]u(t), x(0) �
[11
]L1-optimal controlFix T > 0. Find a feasible control u(t), t ∈ [0, T] that drives the statefrom x(0) to x(T) � [0, 0]⊤, that satisfies |u(t)| ≤ 1, ∀ t ∈ [0, T], and thatminimizes the L1 norm of u:
J1(u) �∫ T
0|u(t)|dt .
Also known as fuel optimal control.A convex optimization problem.A closed-form solution is obtained.
M. Nagahara (Univ of Kitakyushu) Sparsity Methods 12 / 19
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L1 optimal control
Control system (rocket)
Ûx(t) �[0 10 0
]x(t) +
[01
]u(t), x(0) �
[11
]L1-optimal controlFix T > 0. Find a feasible control u(t), t ∈ [0, T] that drives the statefrom x(0) to x(T) � [0, 0]⊤, that satisfies |u(t)| ≤ 1, ∀ t ∈ [0, T], and thatminimizes the L1 norm of u:
J1(u) �∫ T
0|u(t)|dt .
Also known as fuel optimal control.A convex optimization problem.A closed-form solution is obtained.
M. Nagahara (Univ of Kitakyushu) Sparsity Methods 12 / 19
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L1 optimal control
Control system (rocket)
Ûx(t) �[0 10 0
]x(t) +
[01
]u(t), x(0) �
[11
]L1-optimal controlFix T > 0. Find a feasible control u(t), t ∈ [0, T] that drives the statefrom x(0) to x(T) � [0, 0]⊤, that satisfies |u(t)| ≤ 1, ∀ t ∈ [0, T], and thatminimizes the L1 norm of u:
J1(u) �∫ T
0|u(t)|dt .
Also known as fuel optimal control.A convex optimization problem.A closed-form solution is obtained.
M. Nagahara (Univ of Kitakyushu) Sparsity Methods 12 / 19
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A simple example
L1-optimal control u∗(t) and trajectory x∗(t) [Athans and Falb, 1966]
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
time (sec)
u(t
)
Optimal Control
0 0.5 1 1.5
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
x1
x2
state−space trajectory
t=0
t=T
u∗(t) ≡ 0 over [3 −√
10/2, 3 +√
10/2] ≈ [1.4, 4.6]u∗(t) is sparse (∥u∗∥0 � | supp(u∗)| ≈ 1.84 < 5 � T)In fact, it is the sparsest (i.e., maximum hands-off control).
M. Nagahara (Univ of Kitakyushu) Sparsity Methods 13 / 19
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A simple example
L1-optimal control u∗(t) and trajectory x∗(t) [Athans and Falb, 1966]
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
time (sec)
u(t
)
Optimal Control
0 0.5 1 1.5
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
x1
x2
state−space trajectory
t=0
t=T
u∗(t) ≡ 0 over [3 −√
10/2, 3 +√
10/2] ≈ [1.4, 4.6]u∗(t) is sparse (∥u∗∥0 � | supp(u∗)| ≈ 1.84 < 5 � T)In fact, it is the sparsest (i.e., maximum hands-off control).
M. Nagahara (Univ of Kitakyushu) Sparsity Methods 13 / 19
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A simple example
L1-optimal control u∗(t) and trajectory x∗(t) [Athans and Falb, 1966]
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
time (sec)
u(t
)
Optimal Control
0 0.5 1 1.5
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
x1
x2
state−space trajectory
t=0
t=T
u∗(t) ≡ 0 over [3 −√
10/2, 3 +√
10/2] ≈ [1.4, 4.6]u∗(t) is sparse (∥u∗∥0 � | supp(u∗)| ≈ 1.84 < 5 � T)In fact, it is the sparsest (i.e., maximum hands-off control).
M. Nagahara (Univ of Kitakyushu) Sparsity Methods 13 / 19
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Table of Contents
1 L0 norm and sparse control
2 A simple example of maximum hands-off control
3 General formulation of maximum hands-off control
4 Conclusion
M. Nagahara (Univ of Kitakyushu) Sparsity Methods 14 / 19
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L0-optimal control problem
L0-optimal control problemFor the linear time-invariant system
Ûx(t) � Ax(t) + bu(t), t ≥ 0, x(0) � ξ ∈ Rd ,
find a control {u(t) : t ∈ [0, T]} with T > 0 that minimizes
J0(u) � ∥u∥0 �
∫ T
0|u(t)|0dt
subject tox(T) � 0,
and∥u∥∞ ≤ 1.
This is difficult!M. Nagahara (Univ of Kitakyushu) Sparsity Methods 15 / 19
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L0-optimal control problem
L0-optimal control problemFor the linear time-invariant system
Ûx(t) � Ax(t) + bu(t), t ≥ 0, x(0) � ξ ∈ Rd ,
find a control {u(t) : t ∈ [0, T]} with T > 0 that minimizes
J0(u) � ∥u∥0 �
∫ T
0|u(t)|0dt
subject tox(T) � 0,
and∥u∥∞ ≤ 1.
This is difficult!M. Nagahara (Univ of Kitakyushu) Sparsity Methods 15 / 19
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L1-optimal control
L1-optimal control problemFor the linear time-invariant system
Ûx(t) � Ax(t) + bu(t), t ≥ 0, x(0) � ξ ∈ Rd ,
find a control {u(t) : t ∈ [0, T]} with T > 0 that minimizes
J1(u) � ∥u∥1 �
∫ T
0|u(t)|dt
subject tox(T) � 0,
and∥u∥∞ ≤ 1.
This is easy!M. Nagahara (Univ of Kitakyushu) Sparsity Methods 16 / 19
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L1-optimal control
L1-optimal control problemFor the linear time-invariant system
Ûx(t) � Ax(t) + bu(t), t ≥ 0, x(0) � ξ ∈ Rd ,
find a control {u(t) : t ∈ [0, T]} with T > 0 that minimizes
J1(u) � ∥u∥1 �
∫ T
0|u(t)|dt
subject tox(T) � 0,
and∥u∥∞ ≤ 1.
This is easy!M. Nagahara (Univ of Kitakyushu) Sparsity Methods 16 / 19
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Bang-off-bang control
TheoremIf (A, b) is controllable and A is nonsingular, then the L1 optimal control u(t)takes ±1 or 0 for almost all t ∈ [0, T] (if it exists).
A control that takes ±1 or 0 is called a bang-off-bang control.
0
T t
u(t)
t1
t2
0
t
Ttt
1
1
−1
1
M. Nagahara (Univ of Kitakyushu) Sparsity Methods 17 / 19
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Bang-off-bang control
TheoremIf (A, b) is controllable and A is nonsingular, then the L1 optimal control u(t)takes ±1 or 0 for almost all t ∈ [0, T] (if it exists).
A control that takes ±1 or 0 is called a bang-off-bang control.
0
T t
u(t)
t1
t2
0
t
Ttt
1
1
−1
1
M. Nagahara (Univ of Kitakyushu) Sparsity Methods 17 / 19
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Equivalence between L0 and L1 optimal controls
TheoremAssume that there exists an L1-optimal control that is bang-off-bang. Then itis also L0 optimal.
TheoremAssume that there exists at least one L1-optimal control. Assume also that(A, b) is controllable and A is non-singular. Then there exists at least oneL0-optimal control, and the set of L0-optimal controls is equivalent to the set ofL1-optimal controls.
0
T t
u(t)
t1
t2
0
t
Ttt
1
1
−1
1
M. Nagahara (Univ of Kitakyushu) Sparsity Methods 18 / 19
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Equivalence between L0 and L1 optimal controls
TheoremAssume that there exists an L1-optimal control that is bang-off-bang. Then itis also L0 optimal.
TheoremAssume that there exists at least one L1-optimal control. Assume also that(A, b) is controllable and A is non-singular. Then there exists at least oneL0-optimal control, and the set of L0-optimal controls is equivalent to the set ofL1-optimal controls.
0
T t
u(t)
t1
t2
0
t
Ttt
1
1
−1
1
M. Nagahara (Univ of Kitakyushu) Sparsity Methods 18 / 19
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Conclusion
Maximum hands-off control is described as L0-optimal control.Under the assumption of non-singularity, that is, (A, b) iscontrollable and A is nonsingular, L0-optimal control is equivalentto L1-optimal control.Maximum hands-off control is a ternary signal that takes values of±1 and 0. Such a ternary control is called a bang-off-bang control.
M. Nagahara (Univ of Kitakyushu) Sparsity Methods 19 / 19
