Gipsa-lab - Optimal & Predictive Controlmazen.alamir/files/... · 2009. 12. 14. · 1 Control ofmulti-variablecoupled dynamical systems 2 Handlingconstraintson the state and on the

Optimal & Predictive Control

Optimal & Predictive Controland some related topics . . .

Mazen Alamir

French National Research Center (CNRS)Gipsa-lab, Control Systems Department.

University of GrenobleFrance

Mazen Alamir French National Research Center (CNRS) Gipsa-lab, Control Systems Department. University of Grenoble France

Optimal & Predictive Control 1/125


Table of contents

Temporary download location:

http://www.mazenalamir.fr/files/predictive_ense3.pdf




Illustrative example

Illustrative Example (1)

System equations:

x = f(x, u, w)yc = hm(x, u, w)yr = hr(x, u, w)

x state vector

yc constrained output vector

yr vector of regulated variables

u manipulated variable (control input)

w vector of non measured disturbances

Boeing 747






System equations:


Control objective

Keep the regulated output yr as close as pos-sible to some desired value ydr while respect-ing the constraints. Boeing 747






System equations:


Operational constraints

yminc ≤ yc(t) ≤ ymaxc

umin ≤ u(t) ≤ umax

δmin ≤ u(t+ τ)− u(t)) ≤ δmaxBoeing 747





Illustrative Example

0 2 4 6 8 10 12-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

time (sec)

State Evolution

0 2 4 6 8 10 12-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

time (sec)

Regulated output Evolution

0 2 4 6 8 10 12-6

-4

-2

0

2

4

6

time (sec)

Control Input Evolution

0 2 4 6 8 10 12-3

-2

-1

0

1

2

3

time (sec)

Control Input Slope

yminc = −10,−10 ; y

maxc = +10,+10

umin

= −4,−4 ; umax

= +4,+4δ

min= −50,−50 ; δ

max= +50,+50 ; no disturbance w = 0

Boeing 747






0 2 4 6 8 10 12-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

time (sec)

State Evolution

0 2 4 6 8 10 12-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

time (sec)


0 2 4 6 8 10 12-6

-4

-2

0

2

4

6

time (sec)


0 2 4 6 8 10 12-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

time (sec)

Control Input Slope

yminc = −0.5,−10 ; y

maxc = +10,+10

umin

= −4,−4 ; umax

= +4,+4δ

min= −50,−50 ; δ = +50,+50 ; no disturbance w = 0

Boeing 747






0 2 4 6 8 10 12-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

time (sec)

State Evolution

0 2 4 6 8 10 12-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

time (sec)


0 2 4 6 8 10 12-6

-4

-2

0

2

4

6

time (sec)


0 2 4 6 8 10 12-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

time (sec)

Control Input Slope

yminc = −0.5,−10 ; y

maxc = +10,+10

umin

= −4,−4 ; umax

= +4,+4δ

min= −0.5,−0.5 ; δ = +1,+1 ; no disturbance w = 0

Boeing 747






0 2 4 6 8 10 12-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

time (sec)

State Evolution

0 2 4 6 8 10 12-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

time (sec)


0 2 4 6 8 10 12-6

-4

-2

0

2

4

6

time (sec)


0 2 4 6 8 10 12-3

-2

-1

0

1

2

3

4

time (sec)

Control Input Slope

yminc = −10,−10 ; y

maxc = +10,+10

umin

= −4,−4 ; umax

= +4,+4δ

min= −50,−50 ; δ

max= +50,+50 ; disturbance w = (0.2, 0.2)

Boeing 747






0 2 4 6 8 10 12-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

time (sec)

State Evolution

0 2 4 6 8 10 12-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

time (sec)


0 2 4 6 8 10 12-6

-4

-2

0

2

4

6

time (sec)


0 2 4 6 8 10 12-2

-1

0

1

2

3

4

time (sec)

Control Input Slope

yminc = −0.5,−10 ; y

maxc = +10,+10

umin

= −4,−4 ; umax

= +4,+4δ

min= −50,−50 ; δ = +50,+50 ; disturbance w = (0.2, 0.2)

Boeing 747






0 2 4 6 8 10 12-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

time (sec)

State Evolution

0 2 4 6 8 10 12-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

time (sec)


0 2 4 6 8 10 12-6

-4

-2

0

2

4

6

time (sec)


0 2 4 6 8 10 12-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

time (sec)

Control Input Slope

yminc = −0.5,−10 ; y

maxc = +10,+10

umin

= −4,−4 ; umax

= +4,+4δ

min= −0.5,−0.5 ; δ = +1,+1 ; disturbance w = (0.2, 0.2)

Boeing 747






0 2 4 6 8 10 12-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

time (sec)

State Evolution

0 2 4 6 8 10 12-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

time (sec)


0 2 4 6 8 10 12-6

-4

-2

0

2

4

6

time (sec)


0 2 4 6 8 10 12-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

time (sec)

Control Input Slope

yminc = −0.5,−10 ; y

maxc = +10,+10

umin

= −4,−4 ; umax

= +4,+4δ

min= −0.5,−0.5 ; δ = +1,+1 ; disturbance w = (1, 1)

Boeing 747





Advantages of Model Predictive Control

1 Control of multi-variable coupled dynamical systems

2 Handling constraints on the state and on the control input

3 Express optimality concerns

4 Conceptually easy handling of nonlinearities in the system model.

5 Systematic design










5 Systematic design










5 Systematic design










5 Systematic design










5 Systematic design




Model Predictive Control: An intuitive Presentation

What is model predictive control ?

Feedback implementation of optimalcontrol using:

Finite prediction horizon

on-line computation

> 5800 successful industrialapplications in

Petrochemical Industry

Food industry

Aerospace, car industryR. Findeisen: NMPC for Fast Nonlinear Systems (Workshop CDC 2006, San Diego)

Non-exhaustive MPC Vendor List

• ABB• ACT• Adaptics• Adaptive Resources• Adersa Home Page• Aspen Technology• Aurel Systems Inc.• Batch CAD• Bonner and Moore• Brainwave• C.F. Picou and Associates• Chemstations• Comdale Technologies• Control Arts Inc.• Control Consulting Inc.• Control Dynamics Homepage• Controlsoft Incorporated• Cybosoft• DOT Products• Trieber Controls• Yokogawa APC• US Process Control L.L.C.• Eldridge Engineering Inc.

• Elsag Bailey• Envision Systems Inc.• Gensym• Enterprise Control Technologies• Fantoft Process Group• MATHWORKS• Honeywell• Hyprotech• Inferential Control Company• IntellOpt• Knowledgescape• MDC Technology• Neuralware• Nexus Engineering• Objectspace• Optimal Control Research• Pavilion Technologies• Predictive Control Ltd.• Process System Consultants• RSI• Simulation and Advanced Controls Inc.• Simtech• Texas Controls Inc.

Typically linear applications and tools








on-line computation



Food industry













on-line computation



Food industry

Aerospace, car industry

R. Findeisen: NMPC for Fast Nonlinear Systems (Workshop CDC 2006, San Diego)












on-line computation



Food industry











This commercial products mean that

1 MPC is useful

2 MPC is generic

3 MPC is not that straightforward

Let us try to understand the basic idea of MPC . . .







1 MPC is useful

2 MPC is generic









1 MPC is useful

2 MPC is generic









1 MPC is useful

2 MPC is generic









1 MPC is useful

2 MPC is generic







Control keywords

System model (State, control, measurement, disturbance)

Constraints (on state, control)

performance index (Operational cost, consumption, performance)

Stability

Robustness





Control keywords




Stability

Robustness





Control keywords




Stability

Robustness





Control keywords




Stability

Robustness





Control keywords




Stability

Robustness





Control keywords




Stability

Robustness





NMPC: An intuitive Strategy

Current state

Desired state






Current state

Desired state

State constraint






Current state

Desired state

Performance index :

=

Length of the steering path






Current state

Desired state






Current state

Desired state






Current state

Desired state






Current state

Desired state






Current state

Desired state






Current state

Desired state






Current state

Desired state






Current state

Desired state






Desired state

Initially computed trajectory

Closed loop trajectory





A simple feedback principle (informal)

At each decision instant, evaluate the situation

Based on the evaluation, compute the best strategy

Apply the beginning of the strategy until the next decision instant

Re-evaluate the situation

Recompute the best strategy

Apply the first part until the next decision instant

Keep doing












Keep doing












Keep doing












Keep doing












Keep doing












Keep doing












Keep doing




Formal Definition

Formal Definition: The System Model

We consider a discrete-time dynamical system given by:

x(k + 1) = f(x(k), u(k))

x ∈ Rn is the state vector

u ∈ Rm is the vector of manipulated variables (input vector)

x(k) is a short notation for x(kτ) where τ is some sampling period

u(k) is a constant control applied during [kτ, (k + 1)τ ]A sequence of future control inputs on [k, k +Nτ ]:

u(k) =(u(k) . . . u(k +N − 1)

)T ; u(k + i) ∈ Rm

The short notation k+ is sometimes used to denote k + 1Similarly, x+(k) is sometimes used to denote x(k + 1) = x(k+)




Formal Definition



x(k + 1) = f(x(k), u(k))





u(k) =(u(k) . . . u(k +N − 1)

)T ; u(k + i) ∈ Rm





Formal Definition



x(k + 1) = f(x(k), u(k))





u(k) =(u(k) . . . u(k +N − 1)

)T ; u(k + i) ∈ Rm





Formal Definition



x(k + 1) = f(x(k), u(k))




u(k) is a constant control applied during [kτ, (k + 1)τ ]

A sequence of future control inputs on [k, k +Nτ ]:

u(k) =(u(k) . . . u(k +N − 1)

)T ; u(k + i) ∈ Rm





Formal Definition



x(k + 1) = f(x(k), u(k))





u(k) =(u(k) . . . u(k +N − 1)

)T ; u(k + i) ∈ Rm





Formal Definition



x(k + 1) = f(x(k), u(k))





u(k) =(u(k) . . . u(k +N − 1)

)T ; u(k + i) ∈ Rm

The short notation k+ is sometimes used to denote k + 1

Similarly, x+(k) is sometimes used to denote x(k + 1) = x(k+)




Formal Definition



x(k + 1) = f(x(k), u(k))





u(k) =(u(k) . . . u(k +N − 1)

)T ; u(k + i) ∈ Rm





Formal Definition

A Simple Feedback Principle (Formal Definition)

At decision instant k, measure the state x(k)

Based on x(k), compute the best sequence of actions :

u0(x(k)) :=(u0(k; x(k)) u0(k + 1; x(k)) . . . u0(k + i; x(k)) . . .

)Apply the control u0(k;x(k)) on the sampling period [k, k + 1]At decision instant k + 1, measure the state x(k + 1)Based on x(k + 1), compute the best sequence of actions :

u0(x(k + 1)) :=(u0(k + 1;x(k + 1)) u0(k + 2;x(k + 1)) . . .

)Apply the control u0(k + 1;x(k + 1)) on the sampling period[k + 1, k + 2]. . .




Formal Definition


At decision instant k, measure the state x(k)Based on x(k), compute the best sequence of actions :


)

Apply the control u0(k;x(k)) on the sampling period [k, k + 1]At decision instant k + 1, measure the state x(k + 1)Based on x(k + 1), compute the best sequence of actions :

u0(x(k + 1)) :=(u0(k + 1;x(k + 1)) u0(k + 2;x(k + 1)) . . .





Formal Definition




)Apply the control u0(k;x(k)) on the sampling period [k, k + 1]

At decision instant k + 1, measure the state x(k + 1)Based on x(k + 1), compute the best sequence of actions :

u0(x(k + 1)) :=(u0(k + 1;x(k + 1)) u0(k + 2;x(k + 1)) . . .





Formal Definition




)Apply the control u0(k;x(k)) on the sampling period [k, k + 1]At decision instant k + 1, measure the state x(k + 1)

Based on x(k + 1), compute the best sequence of actions :

u0(x(k + 1)) :=(u0(k + 1;x(k + 1)) u0(k + 2;x(k + 1)) . . .





Formal Definition





u0(x(k + 1)) :=(u0(k + 1;x(k + 1)) u0(k + 2;x(k + 1)) . . .

)

Apply the control u0(k + 1;x(k + 1)) on the sampling period[k + 1, k + 2]. . .




Formal Definition





u0(x(k + 1)) :=(u0(k + 1;x(k + 1)) u0(k + 2;x(k + 1)) . . .

)Apply the control u0(k + 1;x(k + 1)) on the sampling period[k + 1, k + 2]

. . .




Formal Definition





u0(x(k + 1)) :=(u0(k + 1;x(k + 1)) u0(k + 2;x(k + 1)) . . .





Formal Definition





u0(x(k + 1)) :=(u0(k + 1;x(k + 1)) u0(k + 2;x(k + 1)) . . .





Formal Definition

A Sampled State Feedback




A state feedback

We have defined a sampled state feedback

u(k) = u0(k;x(k))




Formal Definition

A Key Task: optimization

At decision instant k, measure the state x(k)

Based on x(k), compute the best sequence of actions :



u0(x(k + 1)) :=(u0(k + 1;x(k + 1)) u0(k + 2;x(k + 1)) . . .





Formal Definition




)

Apply the control u0(k;x(k)) on the sampling period [k, k + 1]At decision instant k + 1, measure the state x(k + 1)Based on x(k + 1), compute the best sequence of actions :

u0(x(k + 1)) :=(u0(k + 1;x(k + 1)) u0(k + 2;x(k + 1)) . . .





Formal Definition




)Apply the control u0(k;x(k)) on the sampling period [k, k + 1]

At decision instant k + 1, measure the state x(k + 1)Based on x(k + 1), compute the best sequence of actions :

u0(x(k + 1)) :=(u0(k + 1;x(k + 1)) u0(k + 2;x(k + 1)) . . .





Formal Definition





Based on x(k + 1), compute the best sequence of actions :

u0(x(k + 1)) :=(u0(k + 1;x(k + 1)) u0(k + 2;x(k + 1)) . . .





Formal Definition





u0(x(k + 1)) :=(u0(k + 1;x(k + 1)) u0(k + 2;x(k + 1)) . . .

)

Apply the control u0(k + 1;x(k + 1)) on the sampling period[k + 1, k + 2]. . .




Formal Definition





u0(x(k + 1)) :=(u0(k + 1;x(k + 1)) u0(k + 2;x(k + 1)) . . .

)Apply the control u0(k + 1;x(k + 1)) on the sampling period[k + 1, k + 2]

. . .




Formal Definition





u0(x(k + 1)) :=(u0(k + 1;x(k + 1)) u0(k + 2;x(k + 1)) . . .





Formal Definition




)Apply the control u0(k;x(k)) on the sampling period

We need an optimization problem (time invariant systems)

P(x(k)) : minu

V (x(k),u) | u ∈ U(x(k))

u0(x(k)) is A solution of P(x(k))




Formal Definition

Brief recalls on optimization

An optimization, problem is defined by the following elements:

A cost function to be minimized V (u)A vector of decision variable u (degrees of freedom)

A set of admissible values of u, say U

The formal expression of an optimization problem is given by:

minu∈U

V (u)

which has to be interpreted as follows:

Minimize the cost function V (u) in the decision variable uunder the constraint u ∈ U




Formal Definition



A cost function to be minimized V (u)

A vector of decision variable u (degrees of freedom)



minu∈U

V (u)






Formal Definition






minu∈U

V (u)






Formal Definition






minu∈U

V (u)






Formal Definition






minu∈U

V (u)






Formal Definition






minu∈U

V (u)






Formal Definition

A simple example (1)

Let us take the oversimplified example:

x+ = x+ u

and apply the Model Predictive Control (MPC) design to regulate xaround xd = 1. Moreover assume:

A given prediction horizon Npu(0) = u(1) = · · · = u(Np − 1) = p ∈ [umin, umax]




Formal Definition



x+ = x+ u


A given prediction horizon Np

u(0) = u(1) = · · · = u(Np − 1) = p ∈ [umin, umax]




Formal Definition



x+ = x+ u



U =u | u(j) = p ∀j ∈ 0, . . . , Np−1




Formal Definition



x+ = x+ u



U =u | u(j) = p ∀j ∈ 0, . . . , Np−1




Formal Definition



x+ = x+ u



U =u | u(j) = p ∀j ∈ 0, . . . , Np−1




Formal Definition



x+ = x+ u



U =u | u(j) = p ∀j ∈ 0, . . . , Np−1




Formal Definition



x+ = x+ u



V (x(k), p) =Np∑i=1

|x(k + i, p)− 1|2




Formal Definition



x+ = x+ u



V (x(k), p) =Np∑i=1

|[x(k)− 1] + i · p|2




Formal Definition



x+ = x+ u



V (x(k), p) = ‖

x(k)− 1x(k)− 1

...x(k)− 1

+

12...

Np

· p‖2




Formal Definition


V (x(k), p) = ‖

x(k)− 1x(k)− 1

...x(k)− 1

︸︷︷︸

ψ0(x(k))

+

12...Np

︸︷︷︸ψ1

·p‖2

Therefore,

V (x(k), p) =[ψT1 ψ1

]· p2 +

[2ψT1 ψ0(x(k))

]· p+ ψ2

0(x(k))

The unconstrained optimal solution is therefore given by:

punc(x(k)) = −ψT1 ψ0(x(k))ψT1 ψ1




Formal Definition


V (x(k), p) = ‖

x(k)− 1x(k)− 1

...x(k)− 1

︸︷︷︸

ψ0(x(k))

+

12...Np

︸︷︷︸ψ1

·p‖2

Therefore,

V (x(k), p) =[ψT1 ψ1

]· p2 +

[2ψT1 ψ0(x(k))

]· p+ ψ2

0(x(k))






Formal Definition


V (x(k), p) = ‖

x(k)− 1x(k)− 1

...x(k)− 1

︸︷︷︸

ψ0(x(k))

+

12...Np

︸︷︷︸ψ1

·p‖2

Therefore,

V (x(k), p) =[ψT1 ψ1

]· p2 +

[2ψT1 ψ0(x(k))

]· p+ ψ2

0(x(k))






Formal Definition


p0(x(k)) =

umin if punc(x(k)) < uminumax if punc(x(k)) > umaxpunc(x(k)) otherwise

Therefore,

V (x(k), p) =[ψT1 ψ1

]· p2 +

[2ψT1 ψ0(x(k))

]· p+ ψ2

0(x(k))






Formal Definition


This gives the state feedback law:

u = p0(x(k))

where

p0(x(k)) =

umin if punc(x(k)) < uminumax if punc(x(k)) > umaxpunc(x(k)) otherwise

in which





Formal Definition

Simulation of the closed-loop evolutionNsim=20;Np=2;lesx=zeros(Nsim,1);lesx(1)=0;lesu=zeros(Nsim,1);umin=-0.2;umax=+0.2;for i=1:Nsim-1,

lesu(i)=Kdex(lesx(i),Np,umin,umax)lesx(i+1)=lesx(i)+lesu(i);

endlesu(Nsim)=Kdex(lesx(Nsim),Np,umin,umax);subplot(211),plot(lesx,’m-o’,’LineWidth’,2);grid on;xlim([1 Nsim]);xlabel(’sampling period’,’FontSize’,16);hold on;title(’State Evolution’,’FontSize’,16);subplot(212),stairs(lesu,’m-o’,’LineWidth’,2);grid on;xlim([1 Nsim]);xlabel(’sampling period’,’FontSize’,17);hold on;title(’State Evolution’,’FontSize’,16);figure(gcf)




Formal Definition

Simulation of the closed-loop evolution%--------------------------------------------function u=Kdex(x,Np,umin,umax)

inter=p_unc(x,Np);u=max(umin,min(umax,inter));

return%--------------------------------------------function f=p_unc(x,Np)

inter1=psi1(Np);inter0=psi0(x,Np);f=-inter1’*inter0/(inter1’*inter1);

return%--------------------------------------------function f=psi1(Np)

f=cumsum(ones(Np,1));return%-------------------------------------------function f=psi0(x,Np)

f=ones(Np,1)*(x-1);return%-------------------------------------------




Formal Definition

Simulation of the closed-loop evolution

2 4 6 8 10 12 14 16 18 200

0.2

0.4

0.6

0.8

1

sampling period

State Evolution

2 4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

sampling period

Control Evolution

Unconstrained case umin = −∞, umax = +∞




Formal Definition


2 4 6 8 10 12 14 16 18 200

0.2

0.4

0.6

0.8

1

sampling period

State Evolution

2 4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

sampling period

Control Evolution

Constrained case umin = −0.4, umax = +0.4




Formal Definition


2 4 6 8 10 12 14 16 18 200

0.2

0.4

0.6

0.8

1

sampling period

State Evolution

2 4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

sampling period

Control Evolution

Constrained case umin = −0.2, umax = +0.2




Formal Definition

Closed-loop versus Open-Loop

Remember the introductory example:

Desired state

Initially computed trajectory

Closed loop trajectory




Formal Definition


Remember that the admissible open-loop control profile for productionsatisfies:

u(0) = u(1) = · · · = u(Np − 1) = p ∈ [umin, umax]

while the closed-loop was such that:




Formal Definition


Remember that the admissible open-loop control profile for productionsatisfies:

u(0) = u(1) = · · · = u(Np − 1) = p ∈ [umin, umax]

while the closed-loop was such that:




Formal Definition

Crucial property of MPC

Property of MPC

Simple open-loop profiles may still lead to Very rich resulting closed-looptime profiles of control input




Formal Definition

Back to the example

Note that in the unconstrained case:

u(k) = punc(x(k)) = −[ ψT1ψT1 ψ1

]· ψ0(x(k))

where

ψ0(x) =

x(k)− 1...

x(k)− 1

and ψ1(x) =

1...Np

which can clearly be put in the following form:

u(k) = −[K(Np)

]· x(k) + v(Np)

which nothing but a linear state feedback with a feed-forward term




Formal Definition

Crucial property of MPC

Property 1 of MPC

Simple open-loop profiles may still lead to Very rich resulting closed-looptime profiles of control input

Property 2 of MPC

In the linear unconstrained case, MPC is just a systematic way to define alinear state feedback.




Formal Definition

Back to the example

In the example, we assumed that the sequence of future control satisfies:

u(k) = u(k + 1) = · · · = u(k +Np − 1) = p

This can also be written as follows:

u(k + i, p) = p ∈ R

This means that the degree of freedom is p, and once p is known, thewhole control sequence if known.

This is called a control parameterization.




Formal Definition

Trivial control parameterization

In the general case where u ∈ Rm, when the following choice is done:

p =

u(k)

u(k + 1)...

u(k +Np − 1)

∈ RNp·m

The corresponding parameterization is called the trivial parameterization.Namely, all the unknowns are taken as degrees of freedom.




Formal Definition

Back to the example

Remember that the cost function was given by:

V (x(k), p) =Np∑i=1

|x(k + i, p)− 1|2

A particular feature of the example lies in the fact that the future statetrajectory is given by

x(k + i, p) = x(k) + i · p (Affine in the decision variable p)

The fact that V is quadratic in x(k + i, p) induced a cost function thatwas quadratic in the decision variable p. This highly simplified theoptimization.

Question: To what extent is this feature general ?




Formal Definition

Back to the example


V (x(k), p) =Np∑i=1

|[x(k)− 1] + i · p|2








Formal Definition

Back to the example


V (x(k), p) =Np∑i=1

|[x(k)− 1] + i · p|2








Formal Definition

Back to the example


V (x(k), p) =Np∑i=1

|[x(k)− 1] + i · p|2








MPC for Unconstrained LTI Systems

Case of time invariant linear system

Consider the linear time invariant (LTI) system given by:

x(k + 1) = Ax(k) +Bu(k)

According to the system dynamics, one has:

x(k + 2) = Ax(k + 1) +Bu(k + 1)

= A[Ax(k) +Bu(k)

]+Bu(k + 1)

= A2x(k) + [AB,B](

u(k)u(k + 1)

)and more generally:

x(k + i) = Aix(k) + [Ai−1B, . . . , AB,B]

u(k)

...u(k + i− 2)u(k + i− 1)







x(k + 1) = Ax(k) +Bu(k)


x(k + 2) = Ax(k + 1) +Bu(k + 1)

= A[Ax(k) +Bu(k)

]+Bu(k + 1)

= A2x(k) + [AB,B](

u(k)u(k + 1)

)

and more generally:

x(k + i) = Aix(k) + [Ai−1B, . . . , AB,B]

u(k)

...u(k + i− 2)u(k + i− 1)







x(k + 1) = Ax(k) +Bu(k)


x(k + 2) = Ax(k + 1) +Bu(k + 1)

= A[Ax(k) +Bu(k)

]+Bu(k + 1)

= A2x(k) + [AB,B](

u(k)u(k + 1)

)and more generally:

x(k + i) = Aix(k) + [Ai−1B, . . . , AB,B]

u(k)

...u(k + i− 2)u(k + i− 1)






x(k + i) = Aix(k) + [Ai−1B, . . . , AB,B]

u(k)

...u(k + i− 2)u(k + i− 1)

(1)

and using the notation of the trivial parameterization, namely:

p =

u(k)

u(k + 1)...

u(k +Np − 1)

∈ RNp·m

Equation (2) becomes:

x(k + i) = Aix(k) +(

[Ai−1B, . . . , AB,B] ·Π(m,Np)1→i

)p





State prediction of LTI systems

For a LTI system given by x+ = Ax + Bu, the prediction of the futurestate trajectory under the trivial parameterization is given by:

x(k + i, p) =[Φi]x(k) +

[Ψi

]p

Φi = Ai

Ψi = [Ai−1B, . . . , AB,B] ·Π(m,Np)1→i

Π(m,Np)1→i :=

(I(i·m) Oi·m×(Np−i)·m

)

This generalizes the example case where:

x(k + i, p) = x(k) + i · p = [1]x(k) + [i]p





State prediction of LTI systems

For a LTI system given by x+ = Ax + Bu, the prediction of the futurestate trajectory under the trivial parameterization is given by:

x(k + i, p) =[Φi]x(k) +

[Ψi

]p

Φi = Ai

Ψi = [Ai−1B, . . . , AB,B] ·Π(m,Np)1→i

Π(m,Np)1→i :=


)This generalizes the example case where:

x(k + i, p) = x(k) + i · p = [1]x(k) + [i]p





Output prediction of LTI systems

For a LTI system given by x+ = Ax + Bu with the output y = Cx, theprediction of the future output trajectory under the trivial parameterizationis given by:

y(k + i, p) =[CΦi

]x(k) +

[CΨi

]p

Φi = Ai

Ψi = [Ai−1B, . . . , AB,B] ·Π(m,Np)1→i

Π(m,Np)1→i :=


)





Prediction using outputs

Since the prediction of the state can be obtained from output predictionby using:

C = I

We shall work exclusively with output prediction.





Back to the example

Remember that in order to regulate at xr = 1, the cost functionV (x(k), p) has been defined by:

V (x(k), p) =Np∑i=1

|x(k + i, p)− xr|2

Now, more generally, assume some output reference trajectory yr(k + i),we need to compute the quadratic cost:





Back to the example


V (x(k), p) =Np∑i=1

|x(k + i, p)− xr|2


V (x(k), p) =Np∑i=1

‖y(k + i, p)− yr(k + i)‖2Q





Back to the example


V (x(k), p) =Np∑i=1

|x(k + i, p)− xr|2


V (x(k), p) =Np∑i=1

∥∥[CΦi]x(k) +

[CΨi

]p− yr(k + i)

∥∥2

Q





Back to the example


V (x(k), p) =Np∑i=1

|x(k + i, p)− xr|2


V (x(k), p) =Np∑i=1

∥∥[CΨi

]p+

[CΦi

]x(k)− yr(k + i)

∥∥2

Q





Back to the example


V (x(k), p) =Np∑i=1

|x(k + i, p)− xr|2


V (x(k), p) =Np∑i=1

∥∥∥∥∥∥∥[CΨi

]︸︷︷︸Ωi,1

p+[CΦi

]x(k)− yr(k + i)︸︷︷︸

Ωi,0

∥∥∥∥∥∥∥2

Q





Back to the example


V (x(k), p) =Np∑i=1

|x(k + i, p)− xr|2


V (x(k), p) =Np∑i=1

∥∥[Ωi,1]p+[Ωi,0(x(k), yr(k + i))

]∥∥2

Q





Back to the example


V (x(k), p) =Np∑i=1

|x(k + i, p)− xr|2


V (x(k), p) =Np∑i=1

pT[ΩTi,1QΩi,1

]p+ 2

[ΩTi,0QΩi,1

]p+

[ΩTi,0QΩi,0

]





Back to the example


V (x(k), p) =Np∑i=1

|x(k + i, p)− xr|2


V (x(k), p) = pT

[Np∑i=1

[ΩTi,1QΩi,1

]]p+2

[Np∑i=1

[ΩTi,0QΩi,1

]]p+

[Np∑i=1

[ΩTi,0QΩi,0

]]





The Quadratic Cost Expression

Given a LTI system x+ = Ax+Bu and the trivial parameterization definedby p ∈ RmNp , the cost function:

V (x(k), p) =Np∑i=1

‖y(k + i, p)− yr(k + i)‖2Q

where y = Cx, can be put in the following quadratic form:

V (x(k), p) = pT[H2

]p+ 2

[h1(x(k), yr(k + ·))

]p+

[h0(x(k), yr(k + ·))

]where

H2 =∑Np

i=1

[ΩTi,1QΩi,1

]where Ωi,1 = CΨi

h1 =∑Np

i=1

[ΩTi,0QΩi,1

]where Ωi,0 = [CΦi]x(k)− yr(k + i)

h0 =∑Np

i=1

[ΩTi,0QΩi,0

]Mazen Alamir French National Research Center (CNRS) Gipsa-lab, Control Systems Department. University of Grenoble France





Given a LTI system x+ = Ax+Bu and the trivial parametrization definedby p ∈ RmNp , the cost function:

V (x(k), p) =Np∑i=1

‖y(k + i, p)− yr(k + i)‖2Q

where y = Cx, can be put in the following quadratic form:

V (x(k), p) = pT[H2

]p+ 2

[h1(x(k), yr(k + ·))

]p+

[h0(x(k), yr(k + ·))

]Unique solution if H2 > 0 which gives the unconstrained state feedback:

u(k) = Π(m,Np)1→1 · punc

(x(k), yr(k + ·)

)avec

punc(x(k), yr(k + ·)

)= −H−1

2 · hT1(x(k), yr(k + ·)

)Mazen Alamir French National Research Center (CNRS) Gipsa-lab, Control Systems Department. University of Grenoble France





In order to obtain H2 > 0, penalties are added on the control such that

V (x(k), p) =Np∑i=1

[‖y(k + i, p)− yr(k + i)‖2Q + . . .

+‖Π(m,Np)i→i · p︸︷︷︸u(k+i)

‖2R

which leads to a slight modification on H2 that becomes:

H2 =Np∑i=1

[ΩTi,1QΩi,1

]+[Π(m,Np)i→i

]TR[Π(m,Np)i→i

]





Unconstrained State Feedback (LTI systems)


(x(k), yr(k + ·)

)

which takes the general form:

u(k) = −[K(Np)

]· x(k) + v(Np, yr(k + ·))

with

K(Np) = Π(m,Np)1→1 ·H−1

2 ·Np∑i=1

[ΩTi,1QCΦi

]v(Np, yr(k + ·)) := Π(m,Np)

1→1 ·H−12 ·

Np∑i=1

[ΩTi,1Q · yr(k + i)

]







(x(k), yr(k + ·)

)︸︷︷︸−H−1

2 ·hT1

(x(k),yr(k+·)

)


u(k) = −[K(Np)

]· x(k) + v(Np, yr(k + ·))

with

K(Np) = Π(m,Np)1→1 ·H−1

2 ·Np∑i=1

[ΩTi,1QCΦi

]v(Np, yr(k + ·)) := Π(m,Np)

1→1 ·H−12 ·

Np∑i=1


]






u(k) = −Π(m,Np)1→1 ·H−1

2 · hT1(x(k), yr(k + ·)

)


u(k) = −[K(Np)

]· x(k) + v(Np, yr(k + ·))

with

K(Np) = Π(m,Np)1→1 ·H−1

2 ·Np∑i=1

[ΩTi,1QCΦi

]v(Np, yr(k + ·)) := Π(m,Np)

1→1 ·H−12 ·

Np∑i=1


]






u(k) = −Π(m,Np)1→1 ·H−1

2 · hT1

(x(k), yr(k + ·)︸︷︷︸∑Np

i=1

[ΩT

i,1QΩi,0(x(k),yr(k+i)]


u(k) = −[K(Np)

]· x(k) + v(Np, yr(k + ·))

with

K(Np) = Π(m,Np)1→1 ·H−1

2 ·Np∑i=1

[ΩTi,1QCΦi

]v(Np, yr(k + ·)) := Π(m,Np)

1→1 ·H−12 ·

Np∑i=1


]






u(k) = −Π(m,Np)1→1 ·H−1

2 ·Np∑i=1

[ΩTi,1QΩi,0(x(k), yr(k + i)

]


u(k) = −[K(Np)

]· x(k) + v(Np, yr(k + ·))

with

K(Np) = Π(m,Np)1→1 ·H−1

2 ·Np∑i=1

[ΩTi,1QCΦi

]v(Np, yr(k + ·)) := Π(m,Np)

1→1 ·H−12 ·

Np∑i=1


]






u(k) = −Π(m,Np)1→1 ·H−1

2 ·Np∑i=1

[ΩTi,1QΩi,0(x(k), yr(k + i)︸︷︷︸[

CΦi

]x(k)−yr(k+i)

]


u(k) = −[K(Np)

]· x(k) + v(Np, yr(k + ·))

with

K(Np) = Π(m,Np)1→1 ·H−1

2 ·Np∑i=1

[ΩTi,1QCΦi

]v(Np, yr(k + ·)) := Π(m,Np)

1→1 ·H−12 ·

Np∑i=1


]






u(k) = −Π(m,Np)1→1 ·H−1

2 ·Np∑i=1

[ΩTi,1Q · [

[CΦi

]x(k)− yr(k + i)]

]


u(k) = −[K(Np)

]· x(k) + v(Np, yr(k + ·))

with

K(Np) = Π(m,Np)1→1 ·H−1

2 ·Np∑i=1

[ΩTi,1QCΦi

]v(Np, yr(k + ·)) := Π(m,Np)

1→1 ·H−12 ·

Np∑i=1


]






u(k) = −Π(m,Np)1→1 ·H−1

2 ·Np∑i=1

[ΩTi,1Q · [

[CΦi

]x(k)− yr(k + i)]

]which takes the general form:

u(k) = −[K(Np)

]· x(k) + v(Np, yr(k + ·))

with

K(Np) = Π(m,Np)1→1 ·H−1

2 ·Np∑i=1

[ΩTi,1QCΦi

]v(Np, yr(k + ·)) := Π(m,Np)

1→1 ·H−12 ·

Np∑i=1







u(k) = −Π(m,Np)1→1 ·H−1

2 ·Np∑i=1

[ΩTi,1Q · [

[CΦi

]x(k)− yr(k + i)]

]which takes the general form:

u(k) = −[K(Np)

]· x(k) + v(Np, yr(k + ·))

The feedback gain K(Np) depends only on Np

The feed-forward term v(Np, yr(k + ·)) depends on Np and thereference trajectory yr(·)

The stability of the closed-loop system depends on eigenvalues ofthe closed-loop matrix:

Acl(Np) := A−B ·K(Np)





Deeper in the unconstrained feed-forward term

Recall that the feed-forward term is given by:

v(Np, yr(k + ·)) := Π(m,Np)1→1 ·H−1

2 ·Np∑i=1


]This can be put in the following form:

v(Np, yr(k + ·)) :=

Π(m,Np)1→1 ·H−1

2 ·(ΩT1,1Q, . . . ,Ω

TNp,1

Q)·

yr(k + 1)...

yr(k +Np)





Deeper in the unconstrained feed-forward term

Recall that the feed-forward term is given by:

v(Np, yr(k + ·)) := Π(m,Np)1→1 ·H−1

2 ·Np∑i=1


]This can be put in the following form:

v(Np, yr(k + ·)) :=

Π(m,Np)1→1 ·H−1

2 ·(ΩT1,1Q, . . . ,Ω

TNp,1

Q)︸︷︷︸

G(Np)

·

yr(k + 1)...

yr(k +Np)





Unconstrained State Feedback (LTI Systems)

The predictive control law for a linear time invariant system with noconstraints:

u(k) = −[K(Np)

]· x(k) +

[G(Np)

] yr(k + 1)...

yr(k +Np)

with

K(Np) := Π(m,Np)1→1 ·H−1

2 ·Np∑i=1

[ΩTi,1QCΦi

]G(Np) := Π(m,Np)

1→1 ·H−12 ·

(ΩT1,1Q, . . . ,Ω

TNp,1

Q)





Particular case: Constant Set-point

In the case where yr(k + ·) = yr is constant, one obtains:

u(k) = −[K(Np)

]· x(k) +

[Gc(Np)

]· yr

with

K(Np) := Π(m,Np)1→1 ·H−1

2 ·Np∑i=1

[ΩTi,1QCΦi

]Gc(Np) := Π(m,Np)

1→1 ·H−12 ·

Np∑i=1

ΩTi,1Q





%==========================================================================% This function computes the feedback gains for a given prediction% horizon Np and given weighting matrices Q and R%==========================================================================function [K,G,Gc]=K_Np(A,B,C,Np,Q,R)

[n,m]=size(B);[ny,n]=size(C);H=H2(A,B,C,Np,Q,R);inter=zeros(Np*m,n);interG=zeros(Np*m,Np*ny);interGc=zeros(Np*m,ny);for i=1:Np,

O1i=Omega_i_1(A,B,C,Np,i);Phii=Phi_i(A,i);inter=inter+O1i’*Q*C*Phii;interG(:,(i-1)*ny+1:i*ny)=O1i’*Q;interGc=interGc+O1i’*Q;

endK=Pi_select(m,Np,1,1)*(H\inter);G=Pi_select(m,Np,1,1)*(H\interG);Gc=Pi_select(m,Np,1,1)*(H\interGc);

return%==========================================================================





Example: The Triple Integrator

Example: The triple integrator

Let us take the triple integrator system

x1 = x2

x2 = x3

x3 = u

y = x1

Obtain the discrete-time representation

Compute K(Np) for Np = 1, . . . , 15plot the stability indicator:

S(Np) := maxi∈1,2,3

|λi(A−B ·K(Np)|

using different Q’s, R = 1 and C = (1, 0, 0) and τ = 0.1.






%==========================================================================% Example: Triple integrator.%==========================================================================A=[0 1 0;0 0 1;0 0 0];B=[0;0;1];C=[1 0 0];Q=5;R=1;tau=0.1;[Ad,Bd]=c2d(A,B,tau);S=zeros(15,1);Np_max=35;Np=(1:1:Np_max)’;n_Np=length(Np);for Np=1:Np_max,

[K,G,Gc]=K_Np(Ad,Bd,C,Np,Q,R);S(Np)=max(abs(eig(Ad-Bd*K)));

endplot(S,’g-o’,’LineWidth’,2);grid on;xlabel(’Prediction Horizon N_p’,’FontSize’,16);title(’Stability indicator S(N_p)’,’FontSize’,16);xlim([1 Np_max]);figure(gcf);hold on;plot(ones(size(S)),’r’,’LineWidth’,2);set(gca,’FontSize’,16)%==========================================================================






5 10 15 20 25 30 35 40 45 500.92

0.94

0.96

0.98

1

1.02

1.04

Prediction Horizon Np

Stability indicator S(Np)

Q=1Q=2Q=5Q=10stability bound

(R = 1,τ = 0.1)






Stability vs the prediction horizon & the cost function

Consequently, under the conditions:

C = (1, 0, 0)R = 1

the stability conditions are given by:

Weighting matrix Q stability condition1 Np ≥ 202 Np ≥ 175 Np ≥ 14

10 Np ≥ 13





Penalizing The State Trajectory

Penalizing the state trajectory

Suppose that we define the cost function

V (x(k), p) =Np∑i=1

[‖x(k + i, p)‖2Q + ‖u(k + i)‖2R

]This can be done using the same algorithm provided that:

C = I3×3

%=================================% Example: Triple integrator.%=================================A=[0 1 0;0 0 1;0 0 0];B=[0;0;1];C=eye(3);Q=eye(3);R=1;......






5 10 15 20 25 30 35 40 45 500.92

0.94

0.96

0.98

1

1.02

1.04


Stability indicator S(Np)

Q=1Q=2Q=5Q=10C=eye(3), Q=eye(3)stability bound

(R = 1,τ = 0.1)







Stability is obtained for all prediction horizon !This is a generic property of Model Predictive Control:

Penalizing the whole state vector increases the stability.

Observe that when Np →∞, one obtains the infinite horizon LQregulator that minimizes the cost function:

V (x(k), p) =+∞∑i=1

[‖x(k + i, p)‖2Q + ‖u(k + i)‖2R

]that can be obtained using the Matlab command:

[K,S,E] = DLQR(A,B,Q,R,N) calculates the optimal gain matrix Ksuch that the state-feedback law u[n] = -Kx[n] minimizes thecost function

J = Sum x’Qx + u’Ru + 2*x’Nu

subject to the state dynamics x[n+1] = Ax[n] + Bu[n].










V (x(k), p) =+∞∑i=1

[‖x(k + i, p)‖2Q + ‖u(k + i)‖2R

]

that can be obtained using the Matlab command:













V (x(k), p) =+∞∑i=1

[‖x(k + i, p)‖2Q + ‖u(k + i)‖2R

]that can be obtained using the Matlab command:









10 20 30 40 50 60 70 80 90 1000

0.5

1

1.5

2

2.5


Components of K(Np) (*) / components of K

LQR




Unconstrained Tracking of Exogenous Signal


Remember that the unconstrained MPC state feedback is given by:

u(k) = −[K(Np)

]· x(k) +

[G(Np)

] yr(k + 1)...

yr(k +Np)

Let us try this formulaes to force the output of the triple integrator totrack the following reference signal:

yr(t) = sin(2π5t)





%--------------------------------------------------------------------------x0=[1;0;0];Np=20;tsim=20;lest=(0:tau:tsim)’;nt=size(lest,1);yref=sin(2*pi/5*(0:tau:2*tsim)’);Q=diag(10000);R=1;[K,G,Gc]=K_Np(Ad,Bd,C,Np,Q,R);lesx=zeros(nt,3);lesy=zeros(nt,1);lesu=zeros(nt,1);%--------------------------------------------------------------------------yref=sin(2*pi/5*(0:tau:2*tsim)’);lesx(1,:)=x0’;lesy(1)=C*lesx(1,:)’;for i=1:nt-1,

yref_pred=yref(i+1:i+Np);u=-K*lesx(i,:)’+G*yref_pred;lesu(i,:)=u’;xplus=Ad*lesx(i,:)’+Bd*u;lesx(i+1,:)=xplus’;lesy(i+1)=C*xplus;

end





Unconstrained Tracking of Exogenous Signal: Results (1)

0 2 4 6 8 10 12 14 16 18 20-1

-0.5

0

0.5

1

time

Tracking results with Q=100

0 2 4 6 8 10 12 14 16 18 20-2

-1

0

1

2

time

Control Evolution with Q=100






0 2 4 6 8 10 12 14 16 18 20-1

-0.5

0

0.5

1

time


0 2 4 6 8 10 12 14 16 18 20-6

-4

-2

0

2

4

time







0 2 4 6 8 10 12 14 16 18 20-1

-0.5

0

0.5

1

time


0 2 4 6 8 10 12 14 16 18 20-30

-20

-10

0

10

20

time






Unconstrained Tracking a Square Reference

0 2 4 6 8 10 12 14 16 18 20-0.5

0

0.5

1

1.5

time


0 2 4 6 8 10 12 14 16 18 20-10

-5

0

5

time







0 2 4 6 8 10 12 14 16 18 20-0.5

0

0.5

1

1.5

time


0 2 4 6 8 10 12 14 16 18 20-100

-50

0

50

time







0 2 4 6 8 10 12 14 16 18 20-0.5

0

0.5

1

1.5

time


0 2 4 6 8 10 12 14 16 18 20-2000

-1000

0

1000

2000

3000

time







0 2 4 6 8 10 12 14 16 18 20-0.5

0

0.5

1

1.5

time


0 2 4 6 8 10 12 14 16 18 20-5000

0

5000

time






What about actuator saturations . . . ?

The last plots shows that handling the actuator saturations is mandatory.

Imagine that the control input u is constrained, namely:

u ∈ [umin,+umax] ; umax = −umin = 1000

What happens if we brutally saturate the computed feedback, that is:

u(k) = minumax,max

umin,KMPC(x(k))

(2)

=: Sat(u(k), umin, umax

)(3)





Constrained Tracking a Square Reference

0 2 4 6 8 10 12 14 16 18 20-0.5

0

0.5

1

1.5

time


0 2 4 6 8 10 12 14 16 18 20-5000

0

5000

time







0 2 4 6 8 10 12 14 16 18 20-0.5

0

0.5

1

1.5

time

Tracking results with Q=100000000 with saturation at umax

=1000

0 2 4 6 8 10 12 14 16 18 20

-1000

-500

0

500

1000

time


=1000






0 2 4 6 8 10 12 14 16 18 20-10

-5

0

5

10

time


=1000

0 2 4 6 8 10 12 14 16 18 20

-1000

-500

0

500

1000

time


=1000






0 2 4 6 8 10 12 14 16 18 20-1000

-500

0

500

1000

time


=1000

0 2 4 6 8 10 12 14 16 18 20

-1000

-500

0

500

1000

time


=1000






0 2 4 6 8 10 12 14 16 18 20-1

-0.5

0

0.5

1x 10

5

time


=1000

0 2 4 6 8 10 12 14 16 18 20

-1000

-500

0

500

1000

time


=1000




Handling Constraint in MPC formulation

Constraints need to be handled Explicitly . . . !

There is no actuator without constraints

Many state variables (or combinaison of state variables) have to beconstrained:

Either for safety reasons,

Or in order to meet the specification requirements,

etc.

High performance requirement generally steers the system to theboundary of the admissible operating domain and therefore leads toactive constraints.










etc.











etc.











etc.











etc.











etc.






General Form of Constraints

Three kinds of constraints will be considered for LTI systems:

1 Trajectory constraints:

CT · x(k + i) +DT · u(k + i− 1) ≤ bT for all i ∈ 1, . . . , Np

2 Final constraints:

CF · x(k +Np) +DF · u(k +Np − 1) = bF

3 Rate constraints on control inputs

u(k+ i) ∈ u(k+ i−1) + [δmin, δmax] for all i ∈ 0, . . . , Np − 1










CF · x(k +Np) +DF · u(k +Np − 1) = bF












CF · x(k +Np) +DF · u(k +Np − 1) = bF












CF · x(k +Np) +DF · u(k +Np − 1) = bF












CF · x(k +Np) +DF · u(k +Np − 1) = bF







Trajectory Constraints

Trajectory constraints

CT · x(k + i) +DT · u(k + i− 1) ≤ bT

+Cc−CcOO

x(k + i) +

OO+I−I

u(k + i− 1) ≤

+ymaxc

−yminc

+umax

−umin

yc(k + i) = Cc · x(k + i) ≤ ymaxc

yc(k + i) = Cc · x(k + i) ≥ yminc

u(k + i− 1) ≤ umax

u(k + i− 1) ≥ umin





Trajectory Constraints

Trajectory constraints

CT · x(k + i) +DT · u(k + i− 1) ≤ bT

where

CT :=

+Cc−CcOO

; DT :=

OO+I−I

; bT :=

+ymaxc

−yminc

+umax

−umin





Trajectory Constraints Re-writing

CT · x(k + i) +DT · u(k + i− 1) ≤ bT

Consequently, the trajectory constraints given by:

CT · x(k + i) +DT · u(k + i− 1) ≤ bT

is equivalent to the following linear system of inequalities:[Γ(i)T

]· p ≤ Θ(i)

T where

Γ(i)T := CTΨi +DT ·Π

(m,Np)i→i and

Θ(i)T := bT −

[CTΦi

]x(k)






CT · x(k + i, p) +DT ·Π(m,Np)i→i · p ≤ bT


CT · x(k + i) +DT · u(k + i− 1) ≤ bT


]· p ≤ Θ(i)

T where


(m,Np)i→i and

Θ(i)T := bT −

[CTΦi

]x(k)






CT · x(k + i, p)︸︷︷︸[CT Φi

]x(k)+

[CT Ψi

]p

+DT ·Π(m,Np)i→i · p ≤ bT


CT · x(k + i) +DT · u(k + i− 1) ≤ bT


]· p ≤ Θ(i)

T where


(m,Np)i→i and

Θ(i)T := bT −

[CTΦi

]x(k)






[CTΦi

]x(k) +

[CTΨi +DT ·Π

(m,Np)i→i

]p ≤ bT


CT · x(k + i) +DT · u(k + i− 1) ≤ bT


]· p ≤ Θ(i)

T where


(m,Np)i→i and

Θ(i)T := bT −

[CTΦi

]x(k)






[CTΨi +DT ·Π

(m,Np)i→i

]p ≤ bT −

[CTΦi

]x(k)


CT · x(k + i) +DT · u(k + i− 1) ≤ bT


]· p ≤ Θ(i)

T where


(m,Np)i→i and

Θ(i)T := bT −

[CTΦi

]x(k)






[CTΨi +DT ·Π

(m,Np)i→i

]p ≤ bT −

[CTΦi

]x(k)


CT · x(k + i) +DT · u(k + i− 1) ≤ bT


]· p ≤ Θ(i)

T where


(m,Np)i→i and

Θ(i)T := bT −

[CTΦi

]x(k)






Therefore, by concatenating the constraints for i = 1, . . . , Np:Γ(1)T...

Γ(Np)T

· p ≤

Θ(1)T...

Θ(Np)T

that can be written in the following condensed form:

[ΓT]· p ≤

[ΘT

]=[ΘT

]1

+[ΘT

]2· x(k)






Therefore, by concatenating the constraints for i = 1, . . . , Np:Γ(1)T...

Γ(Np)T

︸︷︷︸

ΓT

·p ≤

Θ(1)T...

Θ(Np)T

︸︷︷︸

ΘT

that can be written in the following condensed form:

[ΓT]· p ≤

[ΘT

]=[ΘT

]1

+[ΘT

]2· x(k)






where

[ΘT

]1

=

bT...bT

∈ RNp·nyc

[ΘT

]2

= −

CTΦ1

...CTΦNp

∈ R(Np·nyc )×n






%--------------------------------------------------------------------------% Function that computes the matrices Gamma_T, Theta_T_1 and Theta_T_2% enabling trajectory constraints handling.%--------------------------------------------------------------------------function [Gamma_T,Theta_T_1,Theta_T_2]=...

traj_constr(A,B,Cc,Np,yc_min,yc_max,u_min,u_max)[n,m]=size(B);nyc=size(Cc,1);C_T=[Cc;-Cc;zeros(m,n);zeros(m,n)];D_T=[zeros(2*nyc,m);eye(m);-eye(m)];b_T=[yc_max;-yc_min;u_max;-u_min];Gamma_T=zeros(Np*nyc,Np*m);Theta_T_1=zeros(Np*size(b_T,1),1);Theta_T_2=zeros(Np*size(b_T,1),n);for i=1:Np,

ind=((i-1)*size(b_T,1)+1:i*size(b_T,1))’;Phi=Phi_i(A,i);Psi=Psi_i(A,B,Np,i);Gamma_i=C_T*Psi+D_T*Pi_select(m,Np,i,i);Theta_T_1(ind,:)=b_T;Theta_i_2(ind,:)=-C_T*Phi;Gamma_T(ind,:)=Gamma_i;

endreturn%--------------------------------------------------------------------------





Final Equality Constraints


CF · x(k +Np) +DF · u(k +Np − 1) = bF

(Cfinal

O

)x(k +Np) +

(OI

)u(k +Np − 1) =

(yfinalufinal

)

Cfinal · x(k +Np) = yfinal

u(k +Np − 1) = ufinal







CF · x(k +Np) +DF · u(k +Np − 1) = bF

where

CF =(CfinalOm×n

); DF =

(Onyfinal

×mIm×m

); bF =

(yfinalufinal

)







CF · x(k +Np) +DF · u(k +Np − 1) ≤ bF

Using the same development as for the trajectory constraints, the abovecondition becomes:[

ΓF]· p =

[ΘF

]=[ΘF

]1

+[ΘF

]2· x(k) where

ΓF := CFΨNp+DF ·Π

(m,Np)Np→Np

and

[ΘF ]1 := bF

[ΘF ]2 = −[CFΦNp






%--------------------------------------------------------------------------% Function that computes the matrices Gamma_F, Theta_F_1 and Theta_F_2% enabling final (terminal) constraints handling.%--------------------------------------------------------------------------function [Gamma_F,Theta_F_1,Theta_F_2]=final_constr(A,B,Cfinal,Np,yfinal,ufinal)

[n,m]=size(B);nyfinal=size(Cfinal,1);C_F=[Cfinal;zeros(m,nyfinal)];D_F=[zeros(nyfinal,m);eye(m)];b_F=[yfinal;ufinal];Gamma_F=C_F*Psi_i(A,B,Np,Np)+D_F*Pi_select(m,Np,Np,Np);Theta_F_1=b_F;Theta_F_2=-C_F*Phi_i(A,Np);

return%--------------------------------------------------------------------------





Rate constraints on Control Inputs


u(k + i) ∈ u(k + i− 1) + [δmin, δmax] for all i ∈ 0, . . . , Np − 1

This can be written as follows:









δmin ≤ u(k)− u(k − 1) ≤ δmaxδmin ≤ u(k + 1)− u(k) ≤ δmax...

......

δmin ≤ u(k +Np − 1)− u(k +Np − 2) ≤ δmax









δmin ≤[Π(m,Np)

1→1

]p− u(k − 1) ≤ δmax

δmin ≤ u(k + 1)− u(k) ≤ δmax...

......

δmin ≤ u(k +Np − 1)− u(k −+Np − 2) ≤ δmax









δmin ≤[Π(m,Np)

1→1

]p− u(k − 1) ≤ δmax

δmin ≤[Π(m,Np)

2→2 −Π(m,Np)1→1

]p ≤ δmax

......

...

δmin ≤ u(k +Np − 1)− u(k −+Np − 2) ≤ δmax









δmin ≤[Π(m,Np)

1→1

]p− u(k − 1) ≤ δmax

δmin ≤[Π(m,Np)

2→2 −Π(m,Np)1→1

]p ≤ δmax

......

...

δmin ≤[Π(m,Np)Np→Np

−Π(m,Np)

(Np−1)→(Np−1)

]p ≤ δmax









δminδmin

...δmin

≤

Π(m,Np)1→1

Π(m,Np)2→2 −Π(m,Np)

1→1...

Π(m,Np)Np→Np

−Π(m,Np)

(Np−1)→(Np−1)

p−

u(k − 1)Om×1

...Om×1

≤δmaxδmax

...δmax









δminδmin

...δmin

︸︷︷︸δmin

≤

Π(m,Np)

1→1

Π(m,Np)2→2 −Π(m,Np)

1→1...

Π(m,Np)Np→Np

−Π(m,Np)

(Np−1)→(Np−1)

︸︷︷︸

∆(m,Np)

p−

u(k − 1)Om×1

...Om×1

︸︷︷︸T ·u(k−1)

≤

δmaxδmax

...δmax

︸︷︷︸δmax









(+∆(m,Np)

−∆(m,Np)

)p ≤

(+δmax−δmin

)+(

+T−T

)· u(k − 1)

where

δmin ∈ Rm·Np ; δmax ∈ Rm·Np ; ∆(m,Np) ∈ R(m·Np)×(m·Np)






%--------------------------------------------------------------------------% Function that computes the matrices Delta_bar, delta_bar and T_bar% enabling rate constraints to be handled.%--------------------------------------------------------------------------function [Delta_bar,delta_bar,Tbar]=rate_constr(Np,delta_min,delta_max)

m=size(delta_min);delta_bar_min=zeros(Np*m,1);delta_bar_max=zeros(Np*m,1);Delta=zeros(m*Np,Np*m);T=zeros(Np*m,m);T(1:m,1:m)=eye(m);for i=1:Np,

ind=((i-1)*m+1:i*m)’;delta_bar_min(ind)=delta_min;delta_bar_max(ind)=delta_max;if (i==1),

Delta(ind,:)=Pi_select(m,Np,1,1);else

Delta(ind,:)=Pi_select(m,Np,i,i)-Pi_select(m,Np,i-1,i-1);end

endDelta_bar=[Delta;-Delta];delta_bar=[delta_bar_max;-delta_bar_min];Tbar=[T;-T];

return%--------------------------------------------------------------------------





Summary of The Constraints

Summary of the Constraints1 Trajectory constraints:[

ΓT]· p ≤

[ΘT

]1

+[ΘT

]2· x(k)

2 Rate constraints on control inputs(+∆(m,Np)

−∆(m,Np)

)p ≤

(+δmax−δmin

)+(

+T−T

)· u(k − 1)

3 Final constraints:[ΓF]· p =

[ΘF

]1

+[ΘF

]2· x(k)






Summary of the Constraints

The whole set of constraints can be split into Inequality and Equalityconstraints as follows:[

Aineq]· p ≤

[Bineq

(x(k), u(k − 1)

)][Aeq

]· p =

[Beq

(x(k)

)]withAineq = R(2Np·(nyc +2m))×(Np·m)

Aeq ∈ R(n+m)×(Np·m)

n is the state vector dimension

m is the control vector dimension

Np is the prediction horizon length

nycis the constrained output vector dimension








Aineq]· p ≤

[Bineq

(x(k), u(k − 1)

)][Aeq

]· p =

[Beq

(x(k)

)]with

Aeq := ΓF ; Beq := [ΘF ]1 + [ΘF ]2 · x(k)

Aineq :=

ΓT+∆(m,Np)

−∆(m,Np)

Bineq :=

[ΘT ]1+δmax−δmin

+

[ΘT ]2OO

· x(k) +

O+T−T

u(k − 1)








Aineq]· p ≤

[Bineq

(x(k), u(k − 1)

)][Aeq

]· p =

[Beq

(x(k)

)]with

Aeq := ΓF ; Beq := [ΘF ]1 + [ΘF ]2 · x(k)

Aineq :=

ΓT+∆(m,Np)

−∆(m,Np)

Bineq :=

[ΘT ]1+δmax−δmin

+

[ΘT ]2OO

· x(k) +

O+T−T

u(k − 1)





The Constrained Optimization Problem


We have the following optimization problem to solve at the decisioninstant k:

minp

[pT[H2

]p+ 2

[h1(x(k), yr(k + ·))

]p]

under the constraints[Aineq

]· p ≤

[Bineq

(x(k), u(k − 1)

)];[Aeq

]· p =

[Beq

(x(k)

)]

There are quite efficient solvers for this problem known as a QuadraticProgramming (QP) problem





Matlab help for QuadProg Solver

>> help quadprog

QUADPROG Quadratic programming.

X=QUADPROG(H,f,A,b) attempts to solve the quadratic programming problem:

min 0.5*x’*H*x + f’*x subject to: A*x <= bx

X=QUADPROG(H,f,A,b,Aeq,beq) solves the problem above while additionallysatisfying the equality constraints Aeq*x = beq.





Expression of the feedback law


The solution of the Quadratic Programming Problem is denoted by:

p(x(k), u(k − 1))

and according by receding-horizon principle, the feedback law is:

u(k) = Π(m,Np)1→1 · p

(x(k), u(k − 1)

)

Note that there is no more explicit expression of the state feedback.

This is no more an affine-in-the-state feedback.







We have the following optimization problem to solve at the decisioninstant k:

minp

[pT[H2

]p+ 2

[h1(x(k), yr(k + ·))

]p]


]· p ≤

[Bineq

(x(k), u(k − 1)

)];[Aeq

]· p =

[Beq

(x(k)

)]

We need to examine more closely how to compute h1.







u(k) = −Π(m,Np)1→1 ·H−1

2 · hT1

(x(k), yr(k + ·)

)︸︷︷︸∑Np

i=1

[ΩT

i,1QΩi,0(x(k),yr(k+i)]

Therefore, we can write:

avec

Yr(k) :=

yr(k + 1)...

yr(k +Np)

∈ Rnyr ·Np







u(k) = −Π(m,Np)1→1 ·H−1

2 ·Np∑i=1

[ΩTi,1QΩi,0(x(k), yr(k + i))

]︸︷︷︸

hT1 (x(k),yr(k+·))


avec

Yr(k) :=

yr(k + 1)...

yr(k +Np)

∈ Rnyr ·Np







u(k) = −Π(m,Np)1→1 ·H−1

2 ·Np∑i=1

[ΩTi,1QΩi,0(x(k), yr(k + i)︸︷︷︸[

CΦi

]x(k)−yr(k+i)

]︸︷︷︸

hT1 (x(k),yr(k+·))


avec

Yr(k) :=

yr(k + 1)...

yr(k +Np)

∈ Rnyr ·Np







u(k) = −Π(m,Np)1→1 ·H−1

2 ·Np∑i=1

[ΩTi,1Q ·

[CΦi

]x(k)− yr(k + i)]

]︸︷︷︸

hT1 (x(k),yr(k+·))


avec

Yr(k) :=

yr(k + 1)...

yr(k +Np)

∈ Rnyr ·Np







u(k) = −Π(m,Np)1→1 ·H−1

2 ·Np∑i=1

[ΩTi,1Q ·

[CΦi

]x(k)− yr(k + i)]

]︸︷︷︸

hT1 (x(k),yr(k+·))Therefore, we can write:

hT1 =

[Np∑i=1

[ΩTi,1Q ·

[CΦi

]]]· x(k)−

Np∑i=1


]

avec

Yr(k) :=

yr(k + 1)...

yr(k +Np)

∈ Rnyr ·Np







u(k) = −Π(m,Np)1→1 ·H−1

2 ·Np∑i=1

[ΩTi,1Q ·

[CΦi

]x(k)− yr(k + i)]

]︸︷︷︸

hT1 (x(k),yr(k+·))Therefore, we can write:

hT1 =

[Np∑i=1

[ΩTi,1Q ·

[CΦi

]]]︸︷︷︸

F1

·x(k)−[ Np∑i=1

[ΩTi,1Q ·Π

(nyr ,Np)i→i

]]︸︷︷︸

F2

·Yr(k)

avec

Yr(k) :=

yr(k + 1)...

yr(k +Np)

∈ Rnyr ·Np






To summarize:

hT1 =[F1

]· x(k) +

[F2

]· Yr(k)

where

F1 :=∑Np

i=1

[ΩTi,1Q ·

[CΦi

]]F2 := −

∑Np

i=1

[ΩTi,1Q ·Π

(nyr ,Np)i→i

]

Yr(k) :=

yr(k + 1)...

yr(k +Np)

∈ Rnyr ·Np






%==========================================================================% This function computes the matrices needed to define the quadratic cost%==========================================================================function [H,F1,F2]=quad_pb_mat(A,B,C,Np,Q,R)

[n,m]=size(B);[ny,n]=size(C);F1=zeros(Np*m,n);F2=zeros(Np*m,Np*ny);H=H2(A,B,C,Np,Q,R);for i=1:Np,

O1i=Omega_i_1(A,B,C,Np,i);Phii=Phi_i(A,i);F1=F1+O1i’*Q*C*Phii;F2=F2-O1i’*Q*Pi_select(ny,Np,i,i);

endreturn%==========================================================================





Back to the triple integrator example


0 2 4 6 8 10 12 14 16 18 20-0.5

0

0.5

1

1.5

time


=1000

0 2 4 6 8 10 12 14 16 18 20

-1000

-500

0

500

1000

time


=1000






Constrained tracking of a triple integrator (1)Let us see how constrained MPC performs ?

%--------------------------------------------------------------------------% Constrained Tracking for a triple integrator%--------------------------------------------------------------------------clear all;clfA=[0 1 0;0 0 1;0 0 0];B=[0;0;1];C=[1 0 0];tau=0.1;[n,m]=size(B);[Ad,Bd]=c2d(A,B,tau);%--------------------------------------------------------------------------






Constrained tracking of a triple integrator (1)

%--------------------------------------------------------------------------x0=[1;0;0];Np=20;tsim=20;lest=(0:tau:tsim)’;nt=size(lest,1);T=5;r=0.5;Q=diag(1e8);R=1;Cc=[1 0 0;0 1 0];Cfinal=[0 0 0];yfinal=0;ufinal=0;yc_min=[-inf;-inf];yc_max=[+inf;+inf];u_min=-1000;u_max=+1000;delta_min=[-inf];delta_max=[+inf];%---------------------------------







%--------------------------------------------------------------------------[A_ineq,B_ineq_1,B_ineq_2,B_ineq_3,A_eq,B_eq_1,B_eq_2]=...

mat_constr(Ad,Bd,Cc,Cfinal,Np,yc_min,yc_max,yfinal,...u_min,u_max,ufinal,delta_min,delta_max);

%--------------------------------------------------------------------------[H,F1,F2]=quad_pb_mat(Ad,Bd,C,Np,Q,R);%--------------------------------------------------------------------------

Remember the quadprog syntaxe:

>> help quadprog











%--------------------------------------------------------------------------[A_ineq,B_ineq_1,B_ineq_2,B_ineq_3,A_eq,B_eq_1,B_eq_2]=...


%--------------------------------------------------------------------------[H,F1,F2]=quad_pb_mat(Ad,Bd,C,Np,Q,R);%--------------------------------------------------------------------------

Remember the quadprog syntaxe:

>> help quadprog











%--------------------------------------------------------------------------lesx=zeros(nt,3);lesy=zeros(nt,1);lesu=zeros(nt,1);yref=yref_de_t((0:tau:2*tsim)’,T,r);lesx(1,:)=x0’;lesy(1)=C*lesx(1,:)’;u_=0;for i=1:nt-1,

disp(i);yref_pred=yref(i+1:i+Np);h1=(F1*lesx(i,:)’+F2*yref_pred)’;B_ineq=B_ineq_1+B_ineq_2*lesx(i,:)’+B_ineq_3*u_;B_eq=B_eq_1+B_eq_2*lesx(i,:)’;u_sol=quadprog(H,h1,A_ineq,B_ineq,A_eq,B_eq);u=u_sol(1:m);lesu(i,:)=u;u_=u;xplus=Ad*lesx(i,:)’+Bd*u;lesx(i+1,:)=xplus’;lesy(i+1)=C*xplus;

end%--------------------------------------------------------------------------







%--------------------------------------------------------------------------subplot(311),plot(lest,yref(1:nt),’r-.’,lest,lesy,’m’,’LineWidth’,2);xlabel(’time’,’FontSize’,20,’FontName’,’garamond’);title([’Q=’,num2str(Q),’ / u_max= ’,...

num2str(u_max),’ / \delta_max=’,num2str(delta_max)],...’FontSize’,20,’FontName’,’garamond’);

set(gca,’FontSize’,16);grid on;figure(gcf);hold on;%--------------------------------------------------------------------------subplot(312),stairs(lest,lesu,’m’,’LineWidth’,2);xlabel(’time’,’FontSize’,20,’FontName’,’garamond’);title([’Control Evolution’],’FontSize’,20,’FontName’,’garamond’);set(gca,’FontSize’,16);grid on;figure(gcf);hold on;ylim([1.2*u_min,1.2*u_max])%--------------------------------------------------------------------------les_du=[diff(lesu);0];subplot(313),stairs(lest,les_du,’m’,’LineWidth’,2);xlabel(’time’,’FontSize’,20,’FontName’,’garamond’);title([’Evolution of \Delta u(k)=u(k)-u(k-1)’],...

’FontSize’,20,’FontName’,’garamond’);set(gca,’FontSize’,16);grid on;figure(gcf);hold on;ylim([1.2*delta_min,1.2*delta_max])%--------------------------------------------------------------------------






0 2 4 6 8 10 12 14 16 18 20-2

0

2

time

Q=10000000000 / umax= Inf / δ

max=Inf

0 2 4 6 8 10 12 14 16 18 20-4000-2000

020004000

time

Control Evolution

0 2 4 6 8 10 12 14 16 18 20

-5000

0

5000

time

Evolution of ∆ u(k)=u(k)-u(k-1)






0 2 4 6 8 10 12 14 16 18 20-2

0

2

time

Q=10000000000 / umax= 1000 / δ

max=Inf

0 2 4 6 8 10 12 14 16 18 20-1000

0

1000

time

Control Evolution

0 2 4 6 8 10 12 14 16 18 20

-1000

0

1000

time







0 2 4 6 8 10 12 14 16 18 20-2

0

2

time

Q=10000000000 / umax= 1000 / δ

max=100

0 2 4 6 8 10 12 14 16 18 20-1000

0

1000

time

Control Evolution

0 2 4 6 8 10 12 14 16 18 20-100

0

100

time







0 2 4 6 8 10 12 14 16 18 20-0.5

00.5

11.5

time

Q=10000000000 / umax

= 1000 / δmax

=100 / yc

max= [0.9 Inf]

0 2 4 6 8 10 12 14 16 18 20-1000

0

1000

time

Control Evolution

0 2 4 6 8 10 12 14 16 18 20-100

0

100

time







0 2 4 6 8 10 12 14 16 18 20-0.5

00.5

11.5

time

Q=10000000000 / umax= 1000 / δ

max=100 / y

c

max= [Inf 1]

0 2 4 6 8 10 12 14 16 18 20-1000

0

1000

time

Control Evolution

0 2 4 6 8 10 12 14 16 18 20-100

0

100

time







%--------------------------------------------------------------------------x0=[0;0;0];Np=20;tsim=20;lest=(0:tau:tsim)’;nt=size(lest,1);T=5;r=0.5;Q=diag(1e10);R=1;Cc=[1 0 0;0 1 0];Cfinal=[0 0 0];yfinal=0;ufinal=0;yc_min=[-inf;-inf];yc_max=[+inf;+1];u_min=-1000;u_max=+1000;delta_min=[-100];delta_max=[+100];%--------------------------------------------------------------------------






%--------------------------------------------------------------------------x0=[0;0;0];Np=20;tsim=20;lest=(0:tau:tsim)’;nt=size(lest,1);T=5;r=0.5;Q=diag(1e10);R=1;Cc=[1 0 0;0 1 0];Cfinal=[0 0 0];yfinal=0;ufinal=0;yc_min=[-inf;-1];yc_max=[+inf;+1];u_min=-1000;u_max=+1000;delta_min=[-100];delta_max=[+100];%--------------------------------------------------------------------------






0 2 4 6 8 10 12 14 16 18 20-0.5

00.5

11.5

time

Q=10000000000 / umax= 1000 / δ

max=100 / y

c

max= [Inf 1] / y

c

min= [-Inf -1]

0 2 4 6 8 10 12 14 16 18 20-1000

0

1000

time

Control Evolution

0 2 4 6 8 10 12 14 16 18 20-100

0

100

time







0 2 4 6 8 10 12 14 16 18 20-0.5

00.5

11.5

time

Q=10000000000 / umax= 1000 / δ

max=100 / y

c

max= [Inf 2] / y

c

min= [-Inf -2]

0 2 4 6 8 10 12 14 16 18 20-1000

0

1000

time

Control Evolution

0 2 4 6 8 10 12 14 16 18 20-100

0

100

time







0 2 4 6 8 10 12 14 16 18 20-0.5

00.5

11.5

time

Q=10000000000 / umax= 1000 / δ

max=100 / y

c

max= [Inf 3] / y

c

min= [-Inf -3]

0 2 4 6 8 10 12 14 16 18 20-1000

0

1000

time

Control Evolution

0 2 4 6 8 10 12 14 16 18 20-100

0

100

time







0 2 4 6 8 10 12 14 16 18 20

-1

0

1

time

Q=10000000000 / umax= 1000 / δ

max=100 / y

c

max= [Inf 3] / y

c

min= [-Inf -3]

0 2 4 6 8 10 12 14 16 18 20-100

0

100

time

Control Evolution

0 2 4 6 8 10 12 14 16 18 20-100

0

100

time







0 2 4 6 8 10 12 14 16 18 20

-1

0

1

time

Q=10000000000 / umax= 1000 / δ

max=10 / y

c

max= [Inf 3] / y

c

min= [-Inf -3]

0 2 4 6 8 10 12 14 16 18 20-50

0

50

time

Control Evolution

0 2 4 6 8 10 12 14 16 18 20-10

0

10

time







0 2 4 6 8 10 12 14 16 18 20

-1

0

1

time

Q=10000000000 / umax= 1000 / δ

max=1 / y

c

max= [Inf 3] / y

c

min= [-Inf -3]

0 2 4 6 8 10 12 14 16 18 20-10

0

10

time

Control Evolution

0 2 4 6 8 10 12 14 16 18 20-1

0

1

time







0 2 4 6 8 10 12 14 16 18 20-2

0

2

time

Q=10000000000 / umax

= 1000 / δmax

=0.5 / yc

max= [Inf 3] / y

c

min= [-Inf -3] / N

p= 20

0 2 4 6 8 10 12 14 16 18 20-10

0

10

time

Control Evolution

0 2 4 6 8 10 12 14 16 18 20-0.5

0

0.5

time







0 2 4 6 8 10 12 14 16 18 20-2

0

2

time

Q=10000000000 / umax

= 1000 / δmax

=0.5 / yc

max= [Inf 3] / y

c

min= [-Inf -3] / N

p= 50

0 2 4 6 8 10 12 14 16 18 20-5

0

5

time

Control Evolution

0 2 4 6 8 10 12 14 16 18 20-0.5

0

0.5

time







0 10 20 30 40 50 60 70 80 90 100-2

0

2

time

Q=10000000000 / umax

= 1000 / δmax

=0.5 / yc

max= [Inf 3] / y

c

min= [-Inf -3] / N

p= 50

0 10 20 30 40 50 60 70 80 90 100-5

0

5

time

Control Evolution

0 10 20 30 40 50 60 70 80 90 100-0.5

0

0.5

time







0 2 4 6 8 10 12 14 16 18 20-2

0

2

time

Q=10000000000 / umax

= 10 / δmax

=10 / yc

max= [Inf 0.8] / y

c

min= [-Inf -0.8] / N

p= 50

0 2 4 6 8 10 12 14 16 18 20-10

0

10

time

Control Evolution

0 2 4 6 8 10 12 14 16 18 20-10

0

10

time





Control Profile Parametrization

Need for control profile parametrization

Why to parameterize control profile ?

In the formulation used so far, the dimension of the decision variable (thenumber of degrees of freedom) is given by

np = Np ·m

where

m is the number of control variable [u ∈ Rm]

Np is the length of the prediction horizon

Example:

Tp = 1 s , τ = 10 ms m = 2 −→ np = 200 !!






Why to parameterize control profile ?

In the formulation used so far, the dimension of the decision variable (thenumber of degrees of freedom) is given by

np = Np ·m

where

m is the number of control variable [u ∈ Rm]

Np is the length of the prediction horizon

Example:

Tp = 1 s , τ = 10 ms m = 2 −→ np = 200 !!






First example of control profile parametrization

Assume that the degrees of freedom are represented by:

u(k) ; u(k +N1) and u(k +N2)

while the other u(k + i) are obtained by a linear interpolation, namely:






First example of control profile parametrization

u(k + i) :=

u(k) + i

N1

[u(k +N1)− u(k)

]if i ≤ N1

u(k +N1) +i−N1

N2−N1

[u(k +N2)− u(k +N1)

]if i ∈ [N1, N2]

u(k +N2) otherwise






First example of control profile parametrizationbut the expression

u(k + i) :=

u(k) + i

N1

[u(k +N1)− u(k)

]if i ≤ N1

u(k +N1) +i−N1

N2−N1

[u(k +N2)− u(k +N1)

]if i ∈ [N1, N2]

u(k +N2) otherwise

can be written equivalently as follows:

u(k + i) = Ri ·

u(k)u(k +N1)u(k +N2)

= Ri · p where p ∈ R3m

provided that Ri is defined by:

Ri =

(

(1− iN1

)I iN1

I O)

if i ≤ N1(O (1− i−N1

N2−N1)I i−N1

N2−N1I)

if i ∈ [N1, N2](O O I

)otherwise






First example of control profile parametrizationbut the expression

u(k + i) :=

u(k) + i

N1

[u(k +N1)− u(k)

]if i ≤ N1

u(k +N1) +i−N1

N2−N1

[u(k +N2)− u(k +N1)

]if i ∈ [N1, N2]

u(k +N2) otherwise

can be written equivalently as follows:

u(k + i) = Ri ·

pr︷︸︸︷ u(k)u(k +N1)u(k +N2)

= Ri · pr where pr ∈ R3m

provided that Ri is defined by:

Ri =

(

(1− iN1

)I iN1

I O)

if i ≤ N1(O (1− i−N1

N2−N1)I i−N1

N2−N1I)

if i ∈ [N1, N2](O O I

)otherwise






Therefore,

p =

u(k)

u(k + 1)...

u(k +Np)

p =

u(k)

u(k + 1)...

u(k +Np)

=

R0

R1

...RNp−1

︸︷︷︸

ΠR(Np)∈R(Np·m)×(3m)

·pr

To summarize, the reduced parametrization is given by:

p =[ΠR(Np)

]· pr

The number of degrees of freedom is 3m for any prediction horizon Np.






Therefore,

p =

u(k)

u(k + 1)...

u(k +Np)

p =

u(k)

u(k + 1)...

u(k +Np)

=

R0

R1

...RNp−1

︸︷︷︸


·pr


p =[ΠR(Np)

]· pr







Therefore,

p =

u(k)

u(k + 1)...

u(k +Np)

p =

u(k)

u(k + 1)...

u(k +Np)

=

R0

R1

...RNp−1

︸︷︷︸


·pr


p =[ΠR(Np)

]· pr







More generally:

u(k + i) is obtained by linear interpolation between u(k +Nji) andu(k +Nji+1) where i ∈ [Nji , Nji+1[






u(k + i) = u(k +Nji) +i−Nji

Nji+1 −Nji

[u(k +Nji+1)− u(k +Nji)

]where

ji := maxj=1,...,nr

j such that Nj ≤ i







Nji+1 −Nji


]where

ji := maxj=1,...,nr


This leads to

u(k + i) = Ri ·

u(k)

u(k +N1)

.

.

.u(k +Nnr )

︸︷︷︸

pr

= Li · pr







Nji+1 −Nji


]where

ji := maxj=1,...,nr


This leads to

u(k + i) = Ri ·

u(k)

u(k +N1)

.

.

.u(k +Nnr )

︸︷︷︸

pr

= Li · pr

with Nj0 = 0 and for all i ∈ 1, . . . , Nnr− 1,

Ri =

O . . . (1−i−Nji

Nji+1 −Nji

)︸︷︷︸jith block

Im

i−NjiNji+1−Nji

Im O . . .O







Nji+1 −Nji


]where

ji := maxj=1,...,nr


This leads to

u(k + i) = Ri ·

u(k)

u(k +N1)

.

.

.u(k +Nnr )

︸︷︷︸

pr

= Li · pr

and for all i ≥ Nnr ,

Ri :=(O . . . O Im

)






Using the above notation, the previous parametrization can be given by:

p =[ΠR(Np)

]pr

where

[ΠR(Np)

]:=

R0

R1

...RNp−1

∈ R(Np·m)×((nr+1)·m)

Here, one has

nr = 2 ⇒ pr ∈ R3m






Using the above notation, the previous parametrization can be given by:

p =[ΠR(Np)

]pr

where

[ΠR(Np)

]:=

R0

R1

...RNp−1

∈ R(Np·m)×((nr+1)·m)

Here, one has

nr = 2 ⇒ pr ∈ R3m






%--------------------------------------------------------------------------% This function compute the parametrization matrix R used in the% expression p=R*p_r where p is the original piece-wise control profile% while p_r is the vector of reduced parametrization.%--------------------------------------------------------------------------function Pi_R=compute_Pi_R(lesN,Np,m)

lesN=[1;lesN];nr=length(lesN);Pi_R=zeros(Np*m,nr*m);for i=1:Np,

if (i==1),Pi_R(1:m,1:m)=eye(m);

elseif (i>=lesN(nr)),Pi_R((i-1)*m+1:i*m,(nr-1)*m+1:nr*m)=eye(m);

elseji=max(find(lesN<=i));Pi_R((i-1)*m+1:i*m,m*(ji-1)+1:m*ji)=...

(1-(i-lesN(ji))/(lesN(ji+1)-lesN(ji)))*eye(m);Pi_R((i-1)*m+1:i*m,m*ji+1:m*(ji+1))=...

(i-lesN(ji))/(lesN(ji+1)-lesN(ji))*eye(m);end

endreturn%--------------------------------------------------------------------------






%--------------------------------------------------------------------------

% test compute_R

%--------------------------------------------------------------------------

clf;

m=1;

Np=10;

lesN=[5;8];

pr=[1;-1;0];

Pi_R=compute_Pi_R(lesN,Np,m);

plot((0:1:Np-1)’,R*pr,’b-’,’LineWidth’,2,’marker’,’o’);

grid on;hold on;

xlim([0 Np-1]);set(gca,’FontSize’,18);

ylim([-1.2 1.2])

xlabel(’i’,’FontSize’,18);ylabel(’u(k+i)’,’FontSize’,18)

plot([1;lesN]-1,pr,’ro’,’LineWidth’,12);

figure(gcf);

%--------------------------------------------------------------------------

-1 0 1 2 3 4 5 6 7 8 9

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

i

u(k+

i)

3 unknowns rather than 10






%--------------------------------------------------------------------------

% test compute_R

%--------------------------------------------------------------------------

clf;

m=1;

Np=51;

lesN=[5;8;20];

pr=[1;-0.5;0.5;0];

Pi_R=compute_R(lesN,Np,m);

plot((0:1:Np-1)’,R*pr,’b-’,’LineWidth’,2,’marker’,’o’);

grid on;hold on;

xlim([-1 Np-1]);set(gca,’FontSize’,18);

ylim([-1.2 1.2])


plot([1;lesN]-1,pr,’ro’,’LineWidth’,12);

figure(gcf);

%--------------------------------------------------------------------------

0 5 10 15 20 25 30 35 40 45 50

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

i

u(k+

i)







%--------------------------------------------------------------------------

% test compute_R

%--------------------------------------------------------------------------

clf;

m=2;

Np=51;

lesN=[5;8;20];

pr=[[1;-0.8];[-0.5;0.8];[0.5;-1];[0;0.2]];

Pi_R=compute_R(lesN,Np,m);

lesu=(reshape(R*pr,2,Np))’;

plot((0:1:Np-1)’,lesu(:,1),’b-’,(0:1:Np-1)’,lesu(:,2),’k-’,...

’LineWidth’,2,’marker’,’o’);

grid on;hold on;

xlim([-1 Np-1]);set(gca,’FontSize’,18);

ylim([-1.2 1.2])


plot([1;lesN]-1,(reshape(pr,2,length(pr)/2))’,’ro’,’LineWidth’,12);

figure(gcf);

%--------------------------------------------------------------------------

0 5 10 15 20 25 30 35 40 45 50

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

i

u(k+

i)






Modified Optimization Problem

There are different (many) other possible reduced parametrizationthat we do not discussed here.

How does the constrained MPC problem change when a reduceddimensional parametrization:

p = ΠR · pr

is used ?






There are different (many) other possible reduced parametrizationthat we do not discussed here.

How does the constrained MPC problem change when a reduceddimensional parametrization:

p = ΠR · pr

is used ?







Remember that we had to solve the following constrained optimizationproblem:

minp

[pT[H2

]p+ 2

[F1x(k) + F2Yr(k)

]Tp]


]· p ≤

[Bineq

(x(k), u(k − 1)

)][Aeq

]· p =

[Beq

(x(k)

)]

Now, we have the additional constraint

p = ΠR · pr








minp

[pT[H2

]p+ 2

[F1x(k) + F2Yr(k)

]Tp]


]· p ≤

[Bineq

(x(k), u(k − 1)

)][Aeq

]· p =

[Beq

(x(k)

)]


p = ΠR · pr








minpr

[pTr[ΠTR ·H2 ·ΠR

]· pr + 2

[F1x(k) + F2Yr(k)

]TΠR · pr]

under the constraints[Aineq ·ΠR

]· pr ≤

[Bineq

(x(k), u(k − 1)

)][Aeq ·ΠR

]· pr =

[Beq

(x(k)

)]


p = ΠR · pr






The new optimization problem is given by:

minpr

[pTr[Hr

2 ] · pr + 2[hr1(x(k), Yr(k))

]· pr]

under the constraints:[Arineq

]pr ≤ Bineq

(x(k), u(k − 1)

)[Areq

]pr = Beq

(x(k)

)where:

Hr2 := ΠT

R ·H2 ·ΠR

hr1

(x(k), Yr(k)

):=

[F1x(k) + F2Yr(k)

]TΠR

Arineq := Aineq ·ΠR

Areq := Aeq ·ΠR





validation of the reduced parametrization

Reconsider the tracking of a sinusoidal signal by the triple integrator.

First parametrization

Classical trivial piece-wiseparametrization.

Np = 7 → 7 degrees offreedom.

second parametrization

Np = 20nr = 5N1 = 2, N2 = 3, N3 = 4,N4 = 5, N5 = 155 degrees of freedom.






Classical parametrization with 7 degrees of freedom

0 2 4 6 8 10 12 14 16 18 20-2

0

2

time

Q=10000000000 / umax

= Inf / δmax

=10 / ycmax= [Inf Inf] / y

cmin= [-Inf -Inf] / N

p= 7

0 2 4 6 8 10 12 14 16 18 20-100

0

100

time

Control Evolution

0 2 4 6 8 10 12 14 16 18 20-10

0

10

time

Evolution of Δ u(k)=u(k)-u(k-1)







0 2 4 6 8 10 12 14 16 18 20-10

0

10

time

Q=10000000000 / umax

= Inf / δmax



p= 7

0 2 4 6 8 10 12 14 16 18 20-100

0

100

time

Control Evolution

0 2 4 6 8 10 12 14 16 18 20-10

0

10

time








0 2 4 6 8 10 12 14 16 18 20-2000

0

2000

time

Q=10000000000 / umax

= Inf / δmax



p= 7

0 2 4 6 8 10 12 14 16 18 20-100

0

100

time

Control Evolution

0 2 4 6 8 10 12 14 16 18 20-10

0

10

time







Np=20;%--------------------------------------------------------------------------% Computing the reduced parametrization%--------------------------------------------------------------------------lesN=[2;3;4;5;15];%lesN=(2:1:Np)’;Pi_R=compute_R(lesN,Np,m);%--------------------------------------------------------------------------[A_ineq,B_ineq_1,B_ineq_2,B_ineq_3,A_eq,B_eq_1,B_eq_2]=...


A_ineq_r=A_ineq*Pi_R;A_eq_r=A_eq*Pi_R;%--------------------------------------------------------------------------[H,F1,F2]=quad_pb_mat(Ad,Bd,C,Np,Q,R);Hr=Pi_R’*H*Pi_R;%--------------------------------------------------------------------------






for i=1:nt-1,

disp(i);

yref_pred=yref(i+1:i+Np);

h1=(F1*lesx(i,:)’+F2*yref_pred)’*Pi_R;

B_ineq=B_ineq_1+B_ineq_2*lesx(i,:)’+B_ineq_3*u_;

B_eq=B_eq_1+B_eq_2*lesx(i,:)’;

pr_sol=quadprog(Hr,h1,A_ineq_r,B_ineq,A_eq_r,B_eq);

u_sol=Pi_R*pr_sol;

u=u_sol(1:m);lesu(i,:)=u;u_=u;

xplus=Ad*lesx(i,:)’+Bd*u;

lesx(i+1,:)=xplus’;

lesy(i+1)=C*xplus;

end






Reduced parametrization with 5 degrees of freedom

0 2 4 6 8 10 12 14 16 18 20-2

0

2

time

Q=10000000000 / umax

= Inf / δmax



p= 20

0 2 4 6 8 10 12 14 16 18 20

-10

0

10

time

Control Evolution

0 2 4 6 8 10 12 14 16 18 20-10

0

10

time







Advantage of reduced dimensional parametrization

The reduced parametrization enables to conciliate:

A low number of degrees of freedom

A long prediction horizon that may be necessary for stability




Disturbance handling

Disturbance Handling

Assume that the dynamic system is affected by a disturbance vector w asfollows:

x(k + 1) = Adx(k) +Bdu(k) +Gdw(k)y(k) = Cx(k)

where

x is the state vector

u is the manipulated input vector

y is the measurement vector

w is the disturbance vector





Different configurations

The disturbance vector can be

Measured with known dynamics

Measured with unknown dynamics

Unmeasured with known dynamics

Unmeasured with unknown dynamics

In case where the disturbance is not modeled, predictive control need touse some assumption on the future behavior of w during the futureprediction horizon.




























































presentpast future

time





presentpast future

time





presentpast future

time

Measured or estimated using observer





presentpast future

time

Has to be computed in order to have an optimal behavior





presentpast future

time

If not known, assumptionshave to be done





Standard assumptions on future disturbance behavior

In case where the dynamics of the uncertainty is not known:

presentpast future

time

constant behavior







presentpast future

time

Vanishing behavior







presentpast future

time

Vanishing behavior

How to estimate the Disturbance ?





Recall on observer design

Remember that for a discrete time linear system:

x+ = Ax+Bu

y = Cx

The observation paradigm is to know how to estimate the state x(k)using the past measurements:(

y(k), y(k − 1), . . . , y(k −NO + 1))

When is this possible ?

if this is possible, how to design the corresponding algorithm ?







x+ = Ax+Bu

y = Cx


y(k), y(k − 1), . . . , y(k −NO + 1))









x+ = Ax+Bu

y = Cx


y(k), y(k − 1), . . . , y(k −NO + 1))









x+ = Ax+Bu

y = Cx


This is possible if the the following condition is satisfied:

rank(

CCA

...CAn−1

) = n (dimension of the state x)







x+ = Ax+Bu

y = Cx

how to design the corresponding algorithm ?

The dynamic observer equation is given by:

x+ = (A− LC)x+Bu+ Ly

where L ∈ Rn×ny is computed such that:

maxi∈1,...,n

|λi(A− LC)| < 1

[for instance, use L = dlqr(AT , CT , Q,R,N)]





Disturbance Estimation

x+ = Ax+Bu+Gw

y = Cx

Generally, it is assumed that the disturbance has a constant dynamic:

x+ = Ax+Bu+Gw

w+ = w

y = Cx

This can be written equivalently as follows(xw

)+

=(

A 0nw×1

0nw×n Inw

)(xw

)+(

B0nw×nu

)u

y =(C 0ny×nw

)(xw

)Mazen Alamir French National Research Center (CNRS) Gipsa-lab, Control Systems Department. University of Grenoble France




Disturbance Estimation

(xw

)+

=(

A 0nw×1

0nw×n Inw

)(xw

)+(

B0nw×nu

)u

y =(C 0ny×nw

)(xw

)that can be written shortly as follows:

x+ = Ax+ Bu

y = Cx

The same analysis and design can be done on this extended system:

rank(

CCA

...CAn−1

) = n+ nw



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Gipsa-lab - Optimal & Predictive Controlmazen.alamir/files/... · 2009. 12. 14. · 1 Control ofmulti-variablecoupled dynamical systems 2 Handlingconstraintson the state and on the