Decentralized and distributed control

Transcription

1 Decentralized and distributed control Centralized control for constrained discrete-time systems M. Farina 1 G. Ferrari Trecate 2 1 Dipartimento di Elettronica, Informazione e Bioingegneria (DEIB) Politecnico di Milano, Italy marcello.farina@polimi.it 2 Dipartimento di Ingegneria Industriale e dell Informazione (DIII) Università degli Studi di Pavia, Italy giancarlo.ferrari@unipv.it EECI-HYCON2 Graduate School on Control 2015 Supélec, France Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 46

2 Outline 1 Classical MPC solutions for nominal system 2 Robust MPC 3 Remarks 4 Conclusions 5 Suggested readings Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 46

3 Outline 1 Classical MPC solutions for nominal system 2 Robust MPC 3 Remarks 4 Conclusions 5 Suggested readings Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 46

4 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 variables and to include logic relations, has been recently used for control of large-scale systems. Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 46

5 Classical MPC solutions for nominal system Ingredients The discrete-time model of the system where x R n, u R m. x(k + 1) = Ax(k) + Bu(k) The constraints { x X R n u U R m where X and U are convex neighborhoods of the origin. The auxiliary control law u(k) = Kx(k) (its properties will be later specified) Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 46

6 Classical MPC solutions for nominal system Ingredients Terminal set The positively invariant terminal set X f X defined in such a way that, if x(k) X f, then { x(k + i) X f 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) Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 46

7 Classical MPC solutions for nominal system Ingredients Remark: the constraint sets X and U, as well as the terminal set X f, can be, e.g., polytopic, i.e., described as an intersection of a finite number of half spaces (the sets are convex), e.g., given scalars c i and vectors f i X f = {x R n : f T i x c i, for all i} ellipsoidal, i.e., described by, for a given scalar c > 0: where H = H T 0. X f = {x R n : x T Hx c} Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 46

8 Classical MPC solutions for nominal system Ingredients For example, an ellipsoidal positively invariant terminal set for the system (controlled with a stabilizing auxiliary control law) x(k + 1) = (A + BK)x(k) is where P is such that X f = {x R n : x T Px c} (A + BK) T P(A + BK) P < 0 V(x) = x T Px is a Lyapunov function for the system and, if x(k) X f 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) X f Methods for computing polytopic invariant sets have also been developed. Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 46

9 Classical MPC solutions for nominal system Ingredients The cost function V(x(t : t + N),u(t : t + N 1)) := t+n 1 k=t 1 2 { x(k) 2 Q + u(k) 2 R } + }{{} V f (x(t + N)) }{{} stage cost arrival cost where x 2 H = xt Hx Q > 0, R > 0, V f is a positive definite function, (i.e., V f (0) = 0 and V f (x) > 0 if x 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. Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 46

10 Classical MPC solutions for nominal system Ingredients The MPC optimization problem The MPC problem consists in the following optimization, at each time step t V (x(t)) = min V(x(t : t + N),u(t : t + N 1)) subject to u(t:t+n 1) x(k + 1) = Ax(k) + Bu(k) x(k) X, for k = t,...,t + N 1 u(k) U, for k = t,...,t + N 1 x(t + N) X f Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 46

11 Classical MPC solutions for nominal system Ingredients Result of the MPC optimization problem The result of the MPC problem (solved at each time step t) is the optimal 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) is applied. This implicitly defines a time-invariant MPC control law u(t) = u(t t) = K MPC (x(t)) Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 46

12 Classical MPC solutions for nominal system Ingredients Remark: MPC is a close-loop control method, i.e., 1. at time step t x(t) (or its estimate, obtained through an observer from the measurement y(t)) is evaluated; 2. the MPC optimization problem is solved on-line; 3. the control input u(t/t) is computed and applied at time t; 4. t + 1 t, and go to step 1. Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 46

13 Classical MPC solutions for nominal system Assumptions Results of MPC-controlled systems can be established. Two possible solutions can be adopted. I) Zero terminal constraint auxiliary control law: u(k) = 0, terminal constraint: X f = {0} positively invariant under the auxiliary control law, i.e.,, x(t + 1) = Ax(t) arrival cost: V f 0 Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 46

14 Classical MPC solutions for nominal system Assumptions II) Stabilizing auxiliary control law auxiliary control law: terminal constraint: arrival cost: u(k) = Kx(k) such that A + BK is as. stable, X f = {x : x 2 P α}, (positively invariant under the auxiliary control law) V f (x) = 1 2 x 2 P Matrix P is selected in such a way that x(k + 1) 2 P x(k) 2 P ( x(k) 2 Q + u(k) 2 R ) under the auxiliary control law, i.e. u(k) = Kx(k) and x(k + 1) = (A + BK)x(k) Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 46

15 Classical MPC solutions for nominal system Assumptions For computing P (method II)): a typical choice is to set K = (R + B T PB) 1 B T PA (LQ control), where P solves the algebraic Riccati equation P = A T PA + Q A T PB(R + B T PB) 1 B T 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 the discrete-time Lyapunov equation (remark that K is given) (A + BK) T P(A + BK) P = (Q + K T RK) if A is stable a simple solution is to set K = 0, and to let P be the solution of the discrete-time Lyapunov equation A T PA P = Q Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 46

16 Classical MPC solutions for nominal system Main results Main results Under the stated assumptions, it is possible to prove: (i) recursive feasibility, i.e., if the MPC problem has a solution at time 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 auxiliary control law). Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 46

17 Classical MPC solutions for nominal system Main results - recursive feasibility We briefly sketch the proof of the recursive feasibility. Assume that, at step t, the MPC problem is feasible: there exists an optimal trajectory u(t : t + N 1 t), 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, Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 46

18 Classical MPC solutions for nominal system Main results - recursive feasibility We briefly sketch the proof of the recursive feasibility. Assume that, at step t, the MPC problem is feasible: there exists an optimal trajectory u(t : t + N 1 t), 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) X f X, Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 46

19 Classical MPC solutions for nominal system Main results - recursive feasibility We briefly sketch the proof of the recursive feasibility. Assume that, at step t, the MPC problem is feasible: there exists an optimal trajectory u(t : t + N 1 t), 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) X f X, in view of a), and of the fact that X f is invariant with respect to the auxiliary control law: u(t + N t) = Kx(t + N t) U, x(t + N + 1 t) = (A + BK)x(t + N t) X f Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 46

20 Classical MPC solutions for nominal system Main results - recursive feasibility We briefly sketch the proof of the recursive feasibility. Assume that, at step t, the MPC problem is feasible: there exists an optimal trajectory u(t : t + N 1 t), 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) X f X, in view of a), and of the fact that X f is invariant with respect to the auxiliary control law: u(t + N t) = Kx(t + N t) U, x(t + N + 1 t) = (A + BK)x(t + N t) X f This proves that u(t + 1 : t + N t) is a feasible (although not optimal) trajectory for the MPC problem at time t + 1 (where the state of the system is x(t + 1) = x(t + 1 t)). Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 46

21 Classical MPC solutions for nominal system Main 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 the MPC problem at time t + 1, where u(t + 1 : t + N 1 t) is given by the solution of the MPC problem at time t, u(t + N t) = Kx(t + N t), furthermore x(t + N + 1 t) = (A + BK)x(t + N t), Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 46

22 Classical MPC solutions for nominal system Main 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 the MPC problem at time t + 1, where u(t + 1 : t + N 1 t) is given by the solution of the MPC problem at time t, u(t + N t) = Kx(t + N t), 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+n 1 k=t+1 2 { x(k t) 2 Q + u(k t) 2 R } + Vf (x(t + N + 1 t)) Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 46

23 Classical MPC solutions for nominal system Main 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 the MPC problem at time t + 1, where u(t + 1 : t + N 1 t) is given by the solution of the MPC problem at time t, u(t + N t) = Kx(t + N t), 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+n 1 k=t+1 2 { x(k t) 2 Q + u(k t) 2 R } + Vf (x(t + N + 1 t)) = t+n 1 1 k=t 2 { x(k t) 2 Q + u(k t) 2 R } + Vf (x(t + N t))+ 2 1 { x(t t) 2 Q + u(t t) 2 R } { x(t + N t) 2 Q + u(t + N t) 2 R }+ V f (x(t + N t)) + V f (x(t + N + 1 t)) Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 46

24 Classical MPC solutions for nominal system Main 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 the MPC problem at time t + 1, where u(t + 1 : t + N 1 t) is given by the solution of the MPC problem at time t, u(t + N t) = Kx(t + N t), 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+n 1 k=t+1 2 { x(k t) 2 Q + u(k t) 2 R } + Vf (x(t + N + 1 t)) = t+n 1 1 k=t 2 { x(k t) 2 Q + u(k t) 2 R } + Vf (x(t + N t))+ 2 1 { x(t t) 2 Q + u(t t) 2 R } { x(t + N t) 2 Q + u(t + N t) 2 R }+ V f (x(t + N t)) + V f (x(t + N + 1 t)) = V(x(t : t + N t),x(t : t + N 1 t))+ 1 2 { x(t t) 2 Q + u(t t) 2 R } { x(t + N t) 2 Q + Kx(t + N t) 2 R }+ V f (x(t + N t)) + V f (x(t + N + 1 t)) Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 46

25 Classical MPC solutions for nominal system Main results - convergence in view of the assumption on the terminal constraint: Therefore V Recalling that V f (x(t + N + 1 t)) V f (x(t + N t)) 1 2 { x(t + N t) 2 Q + Kx(t + N t) 2 R } (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) 2 Q + u(t t) 2 R } V(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 that V (x(t + 1)) V (x(t)) 1 2 { x(t t) 2 Q + u(t t) 2 R } Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 46

26 Classical MPC solutions for nominal system Main results - convergence in view of the assumption on the terminal constraint: Therefore V Recalling that V f (x(t + N + 1 t)) V f (x(t + N t)) 1 2 { x(t + N t) 2 Q + Kx(t + N t) 2 R } (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) 2 Q + u(t t) 2 R } V(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 that V (x(t + 1)) V (x(t)) 1 2 { x(t t) 2 Q + u(t t) 2 R } from the latter it follows that V (x(t + 1)) is decreasing, which implies that x(t t) 2 Q 0 as t + which, in view of the positive-definiteness of Q, implies that x(t) 0 as t +. Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 46

27 Classical MPC solutions for nominal system Main results - convergence An extension The assumption Q > 0 can be relaxed to Q 0 provided that the pair (A, Q) is detectable. Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 46

28 Outline 1 Classical MPC solutions for nominal system 2 Robust MPC 3 Remarks 4 Conclusions 5 Suggested readings Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 46

29 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 is compact and contains the origin. Problem To devise an MPC controller that provides convergence, (worst-case) optimality, and constraint satisfaction for all possible realizations of the bounded disturbance w(k). Two main approaches: min-max approach, which leads to burdensome optimization problems; tube-based approach. Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 46

30 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 is compact and contains the origin. Nominal model ^x(k + 1) = A^x(k) + B^u(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 / 46

31 Robust tube based MPC Robust 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 such that, 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 Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 46

32 Robust tube based MPC Minkowski sum: C = A B = {c = a + b : a A,b B} Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 46

33 Robust tube based MPC Minkowski sum: C = A B = {c = a + b : a A,b B} Minkowski difference: C = A B = {c : c B A} Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 46

34 Robust tube based MPC Since z(k) = x(k) ^x(k) Z, in order to meet the state and input constraints { x X R n u U R m where X and U are convex neighborhoods of the origin, it is sufficient to satisfy the following tightened constraints where ^X = X Z and ^U = U KZ. { ^x ^X ^u ^U Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 46

35 Robust tube based MPC Auxiliary control law ^u(k) = K^x(k) where A + BK is asymptotically stable. Positively invariant terminal set It is the invariant set ^X f ^X for the nominal model such that, if ^x(k) ^X f, then { ^x(k + i) ^X f K^x(k + i) ^U for all i 0, if the nominal state evolves according to ^x(k + 1) = (A + BK)^x(k) Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 46

36 Robust tube based MPC The cost function V(^x(t : t + N), ^u(t : t + N 1)) := t+n 1 k=t 1 2 { ^x(k) 2 Q + ^u(k) 2 R } + }{{} V f (^x(t + N)) }{{} stage cost arrival cost where Q > 0, R > 0, and V f is a positive definite function. Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 46

37 Robust tube based MPC The tube-based robust MPC optimization problem The tube-based robust MPC problem consists in the following optimization, at time t V (^x(t)) = min V(^x(t : t + N), ^u(t : t + N 1)) ^u(t:t+n 1) subject to ^x(k + 1) = A^x(k) + B^u(k) ^x(k) ^X, for k = t,...,t + N 1 ^u(k) ^U, for k = t,...,t + N 1 ^x(t + N) ^X f Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 46

38 Robust tube based MPC Result of the MPC optimization problem The 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) is applied to the nominal model in such a way that ^x(t + 1) is computed; ^x(t + 1) = A^x(t) + B^u(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) is independent of the perturbed state trajectory x(t), but the invariance property guarantees that z(t) = x(t) ^x(t) remains bounded, i.e., x(t) = ^x(t) Z Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 46

39 Robust tube based MPC Main idea: at time t = 0, x(0) = x 0 and set ^x(0) such that x(0) ^x(0) Z at time step t 1. solve the nominal MPC problem with tightened constraints; 2. the nominal input ^u(t/t) is computed and applied to the nominal model: ^x(t + 1) is computed; 3. the robust input u(t) = ^u(t/t) + K(x(t) ^x(t)) is applied to the real system; 4. t + 1 t and go to step 1. Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 46

40 Robust tube based MPC Remarks: Since the MPC method is actually applied to the nominal system, it is apparent that ^x(t) 0 as t in view of the invariance property for all t; it follows that x(t) ^x(t) Z x(t) Z as t for better performance: find Z as small as possible! Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 46

41 Robust tube based MPC Improved approach In the previously discussed tube based approach, the evolution of the nominal system is not affected by the evolution of the real system. Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 46

42 Robust tube based MPC Improved approach In the previously discussed tube based approach, the evolution of the nominal system is not affected by the evolution of the real system. Problem How to introduce a feedback from the real system to the nominal controller? Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 46

43 Robust tube based MPC Improved approach The optimization problem is reformulated. The improved tube-based MPC problem The tube-based improved robust MPC problem consists in the following optimization, at time t V (x(t)) = min V(^x(t : t + N), ^u(t : t + N 1)) ^x(t),^u(t:t+n 1) subject to with the additional constraint ^x(k + 1) = A^x(k) + B^u(k) ^x(k) ^X, for k = t,...,t + N 1 ^u(k) ^U, for k = t,...,t + N 1 ^x(t + N) ^X f x(t) ^x(t) Z Note that a degree of freedom has been added: ^x(t) is now an argument of the optimization problem. Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 46

44 Robust tube based MPC Improved approach Result of the MPC optimization problem The result of the MPC problem (solved at each time step t) is the optimal nominal input 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 trajectory x(t),in view of the additional constraint x(t) = ^x(t) Z Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 46

45 Robust tube based MPC Improved approach Main idea: at time t = 0, x(0) = x 0 ; at time step t 1. solve the nominal MPC problem with tightened constraints and with the additional constraint on the initial value ^x(t); 2. the robust input u(t) = ^u(t/t) + K(x(t) ^x(t)) is applied to the real system; 3. t + 1 t and go to step 1. Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 46

46 Robust tube based MPC Improved approach Properties Also for this approach, recursive feasibility and stability properties can be established. Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 46

47 Outline 1 Classical MPC solutions for nominal system 2 Robust MPC 3 Remarks 4 Conclusions 5 Suggested readings Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 46

48 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 scale optimization problems; even explicit methods (off-line computation of the MPC control law K MPC (x(t)) or an approximation of it) are demanding - memory and computational power. There is a strong need to develop distributed and/or decentralized MPC methods to cope with large-scale systems. Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 46

49 Remarks Main issues: scalability: as the order of the system grows, the main goal is to divide the problem into small-scale subproblems and to keep the computational/memory burden, and the transmission/communication load as limited as possible; reliability and robustness: large-scale systems involve relevant model uncertainties and disturbances, and the possibility that parts or subsystems are removed, added, or replaced: adaptivity to structural changes: large-scale plants require that parts or subsystems are removed, added, or replaced, without the necessity to re-design the overall control system and architecture. Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 46

50 Outline 1 Classical MPC solutions for nominal system 2 Robust MPC 3 Remarks 4 Conclusions 5 Suggested readings Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 46

51 Conclusions Take-home messages: MPC is a relatively simple feedback control algorithm, MPC methods are available for coping with disturbances, with equivalent computational burden, its use can become prohibitive on large-scale systems (it is on-line optimization-based); Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 46

52 Conclusions Key concepts and references centralized nominal MPC [1,2]; centralized robust tube-based MPC [3]; centralized robust improved tube-based MPC [4]; Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 46

53 Outline 1 Classical MPC solutions for nominal system 2 Robust MPC 3 Remarks 4 Conclusions 5 Suggested readings Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 46

54 Suggested readings Books 1. J. B. Rawlings and D. Q. Mayne. Model Predictive Control: Theory and Design. Nob Hill Publishing, Madison, WI, Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 46

55 Suggested readings Papers 2. D. Q. Mayne, J. B. Rawlings, C. V. Rao and P. O. M. Scokaert, Constrained model predictive control: stability and optimality, Automatica, 36(6): , W. Langson, I. Chrissochoos, S.V. Rakovic, and Mayne D. Q. Robust model predictive control using tubes. Automatica, 40(1), D.Q. Mayne, M.M. Seron, and V. Rakovic. Robust model predictive control of constrained linear systems with bounded disturbances. Automatica, 41: , Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School / 46