Decentralized and distributed controlsisdin.unipv.it/lab/personale/pers_hp/ferrari/EECI_DEDICO/6a-DMPC_DeDiCo.pdfDecentralized and distributed control Centralized control for constrained

Decentralized and distributed controlCentralized control for constrained discrete-time systems

M. Farina1 G. Ferrari Trecate2

1Dipartimento di Elettronica, Informazione e Bioingegneria (DEIB)Politecnico di Milano, Italy

[email protected]

2Dipartimento di Ingegneria Industriale e dell’Informazione (DIII)Universita degli Studi di Pavia, [email protected]

EECI-HYCON2 Graduate School on Control 2015Supelec, France

Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School 2015 1 / 46

[email protected]

[email protected]

Outline

1 Classical MPC solutions for nominal system

2 Robust MPC

3 Remarks

4 Conclusions

5 Suggested readings


Outline


2 Robust MPC

3 Remarks

4 Conclusions



Classical MPC solutions for nominal system

MPC is an on-line optimization-based control approach which

allows to account for operational constraints,

allows to account for multi-variable systems,

allows to account for non linear systems,

can be extended to deal with continuous and discrete decision variablesand to include logic relations,

has been recently used for control of large-scale systems.


Classical MPC solutions for nominal systemIngredients

The discrete-time model of the system

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

where x ∈ Rn, u ∈ Rm.

The constraints {x ∈ X⊆ Rn

u ∈ U⊆ Rm

where X and U are convex neighborhoods of the origin.

The auxiliary control law

u(k) = Kx(k)

(its properties will be later specified)



Terminal setThe positively invariant terminal set Xf ⊆ X defined in such a way that,if x(k) ∈ Xf , then {

x(k + i) ∈ Xf

Kx(k + i) ∈ U

for all i ≥ 0, if the state is controlled with the auxiliary control law

x(k +1) = (A+BK)x(k)



Remark: the constraint sets X and U, as well as the terminal set Xf , can be,e.g.,

polytopic, i.e., described as an intersection of a finite number of halfspaces (the sets are convex), e.g., given scalars ci and vectors fi

Xf = {x ∈ Rn : fTi x≤ ci , for all i}

ellipsoidal, i.e., described by, for a given scalar c > 0:

Xf = {x ∈ Rn : xT Hx≤ c}

where H = HT ≥ 0.



For example, an ellipsoidal positively invariant terminal set for thesystem (controlled with a stabilizing auxiliary control law)

x(k +1) = (A+BK)x(k)

isXf = {x ∈ Rn : xT Px≤ c}

where P is such that

(A+BK)T P(A+BK)−P < 0

V(x) = xT Px is a Lyapunov function for the system and, if x(k) ∈ Xf

V(x(k+1))=x(k)T (A+BK)T P(A+BK)x(k)T<x(k)T Px(k)T =V(x(k))≤c

and then x(k +1) ∈ Xf

Methods for computing polytopic invariant sets have also beendeveloped.



The cost functionV(x(t : t +N),u(t : t +N−1)) :=

∑t+N−1k=t

12{‖x(k)‖2Q +‖u(k)‖2R}︸︷︷︸+ Vf (x(t +N))︸︷︷︸

stage cost arrival cost

where‖x‖2H = xT Hx

Q > 0, R > 0, Vf is a positive definite function, (i.e., Vf (0) = 0 andVf (x)> 0 if x 6= 0). Furthermore

x(t : t +N),u(t : t +N−1)

denote the sequences {x(t), . . . ,x(t +N)} and {u(t), . . . ,u(t +N−1)},respectively.



The MPC optimization problemThe MPC problem consists in the following optimization, at each timestep t

V∗(x(t)) = minu(t :t+N−1)

V(x(t : t +N),u(t : t +N−1))

subject tox(k +1) = Ax(k)+Bu(k)x(k) ∈ X, for k = t , . . . , t +N−1u(k) ∈ U, for k = t , . . . , t +N−1x(t +N) ∈ Xf



Result of the MPC optimization problemThe result of the MPC problem (solved at each time step t) is theoptimal input sequence

u(t : t +N−1|t) = u(t |t), . . . ,u(t +N−1|t)

According to the receding horizon criterion, at instant t only the firstelement u(t |t) is applied. This implicitly defines a time-invariant MPCcontrol law

u(t) = u(t |t) = KMPC(x(t))



Remark: MPC is a close-loop control method, i.e.,

1. at time step t x(t) (or itsestimate, obtained throughan observer from themeasurement y(t)) isevaluated;

2. the MPC optimizationproblem is solved on-line;

3. the control input u(t/t) iscomputed and applied attime t ;

4. t +1→ t , and go to step 1.


Classical MPC solutions for nominal systemAssumptions

Results of MPC-controlled systems can be established. Two possiblesolutions can be adopted.

I) Zero terminal constraint

auxiliary control law: u(k) = 0,terminal constraint: Xf = {0} positively invariant under

the auxiliary control law, i.e.,,x(t +1) = Ax(t)

arrival cost: Vf ≡ 0



II) Stabilizing auxiliary control law

auxiliary control law: u(k) = Kx(k) such that A+BK is as. stable,terminal constraint: Xf = {x : ‖x‖2P ≤ α}, (positively invariant under

the auxiliary control law)arrival cost: Vf (x) = 1

2‖x‖2P

Matrix P is selected in such a way that

‖x(k +1)‖2P ≤ ‖x(k)‖2P− (‖x(k)‖2Q +‖u(k)‖2R)

under the auxiliary control law, i.e. u(k) = Kx(k) and

x(k +1) = (A+BK)x(k)



For computing P (method II)):a typical choice is to set K =−(R+BT PB)−1BT PA (LQ control),where P solves the algebraic Riccati equation

P = AT PA+Q−AT PB(R+BT PB)−1BT PA

the simplest choice is, first to find K with alternative methods (e.g.,eigenvalue assignment), and then to let P be the solution of thediscrete-time Lyapunov equation (remark that K is given)

(A+BK)T P(A+BK)−P =−(Q+KT RK)

if A is stable a simple solution is to set K = 0, and to let P be thesolution of the discrete-time Lyapunov equation

AT PA−P =−Q


Classical MPC solutions for nominal systemMain results

Main resultsUnder the stated assumptions, it is possible to prove:

(i) recursive feasibility, i.e., if the MPC problem has a solution attime t , then has a solution at time t +1;

(ii) convergence to the origin, i.e., x(t)→ 0 as t →+∞.

For simplicity, we now consider only the case II) (stabilizing auxiliarycontrol law).


Classical MPC solutions for nominal systemMain results - recursive feasibility

We briefly sketch the proof of the recursive feasibility.

Assume that, at step t , the MPC problem is feasible:I there exists an optimal trajectory u(t : t +N−1|t),I denote with x(t +1 : t +N|t) the trajectory computed with model

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

with x(t) as initial condition and with u(t : t +N−1|t) as input sequence,

in view of the feasibility at time t :a) x(k |t) ∈ X for all k = t , . . . , t +N−1,b) u(k |t) ∈ U for all k = t , . . . , t +N−1,c) x(t +N|t) ∈ Xf ⊆ X,

in view of a), and of the fact that Xf is invariant with respect to the auxiliary control law:I u(t +N|t) = Kx(t +N|t) ∈ U,I x(t +N +1|t) = (A+BK)x(t +N|t) ∈ Xf

This proves that u(t +1 : t +N|t) is a feasible (although not optimal) trajectory for the MPCproblem at time t +1 (where the state of the system is x(t +1) = x(t +1|t)).





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









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









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






Classical MPC solutions for nominal systemMain results - convergence

A sketch of the proof of convergence is the following.

As we have already proved, u(t +1 : t +N|t) is a feasible (non-optimal) solution to theMPC problem at time t +1, where

I u(t +1 : t +N−1|t) is given by the solution of the MPC problem at time t ,I u(t +N|t) = Kx(t +N|t),I furthermore x(t +N +1|t) = (A+BK)x(t +N|t),

we compute the value of V with respect to such a feasible non-optimal solution:

V (x(t +1 : t +N +1|t),u(t +1 : t +N|t)) == ∑

t+Nk=t+1

12{‖x(k |t)‖

2Q +‖u(k |t)‖2R}+Vf (x(t +N +1|t))

= ∑t+N−1k=t

12{‖x(k |t)‖

2Q +‖u(k |t)‖2R}+Vf (x(t +N|t))+

− 12{‖x(t |t)‖

2Q +‖u(t |t)‖2R}+

12{‖x(t +N|t)‖2Q +‖u(t +N|t)‖2R}+

−Vf (x(t +N|t))+Vf (x(t +N +1|t))

= V(x(t : t +N|t),x(t : t +N−1|t))+− 1

2{‖x(t |t)‖2Q +‖u(t |t)‖2R}+

12{‖x(t +N|t)‖2Q +‖Kx(t +N|t)‖2R}+

−Vf (x(t +N|t))+Vf (x(t +N +1|t))







V (x(t +1 : t +N +1|t),u(t +1 : t +N|t)) == ∑

t+Nk=t+1

12{‖x(k |t)‖

2Q +‖u(k |t)‖2R}+Vf (x(t +N +1|t))

= ∑t+N−1k=t

12{‖x(k |t)‖

2Q +‖u(k |t)‖2R}+Vf (x(t +N|t))+

− 12{‖x(t |t)‖

2Q +‖u(t |t)‖2R}+

12{‖x(t +N|t)‖2Q +‖u(t +N|t)‖2R}+

−Vf (x(t +N|t))+Vf (x(t +N +1|t))

= V(x(t : t +N|t),x(t : t +N−1|t))+− 1

2{‖x(t |t)‖2Q +‖u(t |t)‖2R}+

12{‖x(t +N|t)‖2Q +‖Kx(t +N|t)‖2R}+

−Vf (x(t +N|t))+Vf (x(t +N +1|t))







V (x(t +1 : t +N +1|t),u(t +1 : t +N|t)) == ∑

t+Nk=t+1

12{‖x(k |t)‖

2Q +‖u(k |t)‖2R}+Vf (x(t +N +1|t))

= ∑t+N−1k=t

12{‖x(k |t)‖

2Q +‖u(k |t)‖2R}+Vf (x(t +N|t))+

− 12{‖x(t |t)‖

2Q +‖u(t |t)‖2R}+

12{‖x(t +N|t)‖2Q +‖u(t +N|t)‖2R}+

−Vf (x(t +N|t))+Vf (x(t +N +1|t))

= V(x(t : t +N|t),x(t : t +N−1|t))+− 1

2{‖x(t |t)‖2Q +‖u(t |t)‖2R}+

12{‖x(t +N|t)‖2Q +‖Kx(t +N|t)‖2R}+

−Vf (x(t +N|t))+Vf (x(t +N +1|t))







V (x(t +1 : t +N +1|t),u(t +1 : t +N|t)) == ∑

t+Nk=t+1

12{‖x(k |t)‖

2Q +‖u(k |t)‖2R}+Vf (x(t +N +1|t))

= ∑t+N−1k=t

12{‖x(k |t)‖

2Q +‖u(k |t)‖2R}+Vf (x(t +N|t))+

− 12{‖x(t |t)‖

2Q +‖u(t |t)‖2R}+

12{‖x(t +N|t)‖2Q +‖u(t +N|t)‖2R}+

−Vf (x(t +N|t))+Vf (x(t +N +1|t))

= V(x(t : t +N|t),x(t : t +N−1|t))+− 1

2{‖x(t |t)‖2Q +‖u(t |t)‖2R}+

12{‖x(t +N|t)‖2Q +‖Kx(t +N|t)‖2R}+

−Vf (x(t +N|t))+Vf (x(t +N +1|t))



in view of the assumption on the terminal constraint:

Vf (x(t +N +1|t))≤ Vf (x(t +N|t))− 12{‖x(t +N|t)‖2Q +‖Kx(t +N|t)‖2R}

Therefore

V (x(t +1 : t +N +1|t),u(t +1 : t +N|t))≤ V(x(t : t +N|t),u(t : t +N−1|t))+− 1

2{‖x(t |t)‖2Q +‖u(t |t)‖2R}

Recalling thatV(x(t : t +N|t),u(t : t +N−1|t)) = V∗(x(t))

and that, in view of the sub-optimality of x(t +1 : t +N +1|t) and u(t +1 : t +N|t)

V∗(x(t +1))≤ V(x(t +1 : t +N +1|t),u(t +1 : t +N|t))

we obtain thatV∗(x(t +1))≤ V∗(x(t))− 1

2{‖x(t |t)‖2Q +‖u(t |t)‖2R}

from the latter it follows that V∗(x(t +1)) is decreasing, which implies that

‖x(t |t)‖2Q→ 0

as t →+∞ which, in view of the positive-definiteness of Q, implies that x(t)→ 0 as t →+∞.



in view of the assumption on the terminal constraint:

Vf (x(t +N +1|t))≤ Vf (x(t +N|t))− 12{‖x(t +N|t)‖2Q +‖Kx(t +N|t)‖2R}

Therefore

V (x(t +1 : t +N +1|t),u(t +1 : t +N|t))≤ V(x(t : t +N|t),u(t : t +N−1|t))+− 1

2{‖x(t |t)‖2Q +‖u(t |t)‖2R}

Recalling thatV(x(t : t +N|t),u(t : t +N−1|t)) = V∗(x(t))

and that, in view of the sub-optimality of x(t +1 : t +N +1|t) and u(t +1 : t +N|t)

V∗(x(t +1))≤ V(x(t +1 : t +N +1|t),u(t +1 : t +N|t))

we obtain thatV∗(x(t +1))≤ V∗(x(t))− 1

2{‖x(t |t)‖2Q +‖u(t |t)‖2R}

from the latter it follows that V∗(x(t +1)) is decreasing, which implies that

‖x(t |t)‖2Q→ 0

as t →+∞ which, in view of the positive-definiteness of Q, implies that x(t)→ 0 as t →+∞.



An extensionThe assumption Q > 0 can be relaxed to Q≥ 0 provided that the pair(A,Q) is detectable.


Outline


2 Robust MPC

3 Remarks

4 Conclusions



Robust ”tube based” MPC

Perturbed systems

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

where w(k) is a bounded disturbance, i.e., w(k) ∈W, where W iscompact and contains the origin.

ProblemTo devise an MPC controller that provides convergence, (worst-case)optimality, and constraint satisfaction for all possible realizations of thebounded disturbance w(k).

Two main approaches:min-max approach, which leads to burdensome optimizationproblems;tube-based approach.


Robust ”tube based” MPCPerturbed systems

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

where w(k) is a bounded disturbance, i.e., w(k) ∈W, where W is compact andcontains the origin.

Nominal model

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

Robust control law

u(k) = u(k)+K(x(k)− x(k))

Denote z(k) = x(k)− x(k). The variable z(k) evolves according to

z(k +1) = (A+BK)z(k)+w(k)

irrespective of u(k).Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School 2015 23 / 46

Robust ”tube based” MPCRobust positively invariant (RPI) set

z(k +1) = (A+BK)z(k)+w(k)

If A+BK is as. stable, then there exists a robust positively invariant (RPI) set Z suchthat, if z(t) ∈ Z and w(k) ∈W for all k ≥ t , then

z(t + i) ∈ Z

for all i ≥ 0, i.e.,(A+BK)Z⊕W⊆ Z



Minkowski sum:

C = A⊕B = {c = a+b : a ∈ A,b ∈ B}

Minkowski difference:

C = AB = {c : c⊕B ⊆ A}



Minkowski sum:

C = A⊕B = {c = a+b : a ∈ A,b ∈ B}

Minkowski difference:

C = AB = {c : c⊕B ⊆ A}



Since z(k) = x(k)− x(k) ∈ Z, in order to meet the state and inputconstraints {

x ∈ X⊆ Rn

u ∈ U⊆ Rm

where X and U are convex neighborhoods of the origin, it is sufficientto satisfy the following tightened constraints{

x ∈ Xu ∈ U

where X= XZ and U= UKZ.



Auxiliary control law

u(k) = Kx(k)

where A+BK is asymptotically stable.

Positively invariant terminal set

It is the invariant set Xf ⊆ X for the nominal model such that, ifx(k) ∈ Xf , then {

x(k + i) ∈ Xf

Kx(k + i) ∈ U

for all i ≥ 0, if the nominal state evolves according to

x(k +1) = (A+BK)x(k)



The cost functionV(x(t : t +N), u(t : t +N−1)) :=

∑t+N−1k=t

12{‖x(k)‖2Q +‖u(k)‖2R}︸︷︷︸+ Vf (x(t +N))︸︷︷︸

stage cost arrival cost

where Q > 0, R > 0, and Vf is a positive definite function.



The tube-based robust MPC optimization problemThe tube-based robust MPC problem consists in the followingoptimization, at time t

V∗(x(t)) = minu(t :t+N−1)

V(x(t : t +N), u(t : t +N−1))




Result of the MPC optimization problemThe result of the MPC problem (solved at each time step t) is optimal nominal input sequence

u(t : t +N−1|t) = u(t/t), . . . , u(t +N−1/t)

According to the receding horizon criterion at instant t only the first element u(t/t) isapplied to the nominal model

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

in such a way that x(t +1) is computed;

the robust MPC control input for the perturbed system is, at time t

u(t) = u(t/t)+K(x(t)− x(t))

the nominal state trajectory x(t) is independent of the perturbed state trajectory x(t), butthe invariance property guarantees that z(t) = x(t)− x(t) remains bounded, i.e.,

x(t) = x(t)⊕Z



Main idea:

at time t = 0, x(0) = x0 and setx(0) such that

x(0) ∈ x(0)⊕Z

at time step t1. solve the nominal MPC

problem with tightenedconstraints;

2. the nominal input u(t/t) iscomputed and applied tothe nominal model:x(t +1) is computed;

3. the robust input u(t) =u(t/t)+K(x(t)− x(t)) isapplied to the real system;

4. t +1→ t and go to step 1.



Remarks:

Since the MPC method isactually applied to the nominalsystem, it is apparent that

x(t)→ 0 as t → ∞

in view of the invarianceproperty

x(t) ∈ x(t)⊕Z

for all t ;

it follows that

x(t)→ Z as t → ∞

for better performance: find Z assmall as possible!


Robust ”tube based” MPC”Improved” approach

In the previously discussed tubebased approach, the evolution of thenominal system is not affected by theevolution of the real system.



In the previously discussed tubebased approach, the evolution of thenominal system is not affected by theevolution of the real system.

ProblemHow to introduce a feedback from thereal system to the nominal controller?



The optimization problem is reformulated.

The improved tube-based MPC problemThe tube-based improved robust MPC problem consists in the following optimization,at time t

V∗(x(t)) = minx(t),u(t :t+N−1)

V(x(t : t +N), u(t : t +N−1))


with the additional constraintx(t)− x(t) ∈ Z

Note that a degree of freedom has been added: x(t) is now an argument of theoptimization problem.



Result of the MPC optimization problemThe result of the MPC problem (solved at each time step t) is the optimal nominalinput sequence

u(t : t +N−1|t) = u(t/t), . . . , u(t +N−1/t)

and the nominal state (at instant t):x(t/t)

the robust MPC control input for the perturbed system is, at time t

u(t) = u(t/t)+K(x(t)− x(t))

the nominal state trajectory x(t) now depends on the perturbed state trajectoryx(t),in view of the additional constraint

x(t) = x(t)⊕Z



Main idea:

at time t = 0, x(0) = x0;

at time step t1. solve the nominal MPC

problem with tightenedconstraints and with theadditional constraint onthe initial value x(t);

2. the robust input u(t) =u(t/t)+K(x(t)− x(t)) isapplied to the real system;

3. t +1→ t and go to step 1.



PropertiesAlso for this approach, recursive feasibility and stability propertiescan be established.


Outline


2 Robust MPC

3 Remarks

4 Conclusions



Remarks

MPC is computationally demanding, i.e.,on-line optimization;many applications involve non-linear systems;many applications require small sampling time;many application involve large-scale systems - large scaleoptimization problems;even explicit methods (off-line computation of the MPC control lawKMPC(x(t)) or an approximation of it) are demanding - memoryand computational power.

There is a strong need to develop distributed and/or decentralizedMPC methods to cope with large-scale systems.


Remarks

Main issues:scalability: as the order of the system grows, the main goal is todivide the problem into small-scale subproblems and to keep

I the computational/memory burden, andI the transmission/communication load

as limited as possible;reliability and robustness: large-scale systems involve relevantmodel uncertainties and disturbances, and the possibility thatparts or subsystems are removed, added, or replaced:adaptivity to structural changes: large-scale plants require thatparts or subsystems are removed, added, or replaced, without thenecessity to re-design the overall control system and architecture.


Outline


2 Robust MPC

3 Remarks

4 Conclusions



ConclusionsTake-home messages:

MPC is a relatively simple feedback control algorithm,MPC methods are available for coping with disturbances, withequivalent computational burden,its use can become prohibitive on large-scale systems (it is on-lineoptimization-based);


ConclusionsKey concepts and references

centralized nominal MPC [1,2];centralized robust tube-based MPC [3];centralized robust ”improved” tube-based MPC [4];


Outline


2 Robust MPC

3 Remarks

4 Conclusions



Suggested readings

Books1. J. B. Rawlings and D. Q. Mayne. Model Predictive Control: Theory

and Design. Nob Hill Publishing, Madison, WI, 2009.


Suggested readings

Papers2. D. Q. Mayne, J. B. Rawlings, C. V. Rao and P. O. M. Scokaert,

Constrained model predictive control: stability and optimality,Automatica, 36(6):789–814, 2000

3. W. Langson, I. Chrissochoos, S.V. Rakovic, and Mayne D. Q.Robust model predictive control using tubes. Automatica, 40(1),2004.

4. D.Q. Mayne, M.M. Seron, and V. Rakovic. Robust modelpredictive control of constrained linear systems with boundeddisturbances. Automatica, 41:219–224, 2005.


Documents

Decentralized and distributed controlsisdin.unipv.it/lab/personale/pers_hp/ferrari/EECI_DEDICO/6a-DMPC_DeDiCo.pdfDecentralized and distributed control Centralized control for constrained