MPC for tracking of piece-wise constant referente for constrained linear systems

MPC FOR TRACKING OF PIECE-WISE CONSTANT REFERENCES FOR CONSTRAINED LINEAR SYSTEMS D. Limon, I. Alvarado, T. Alamo, and E.F. Camacho Departamento de Ingenier´ ıa de Sistemas y Autom´ atica, Universidad de Sevilla. Avda. Camino de los Descubrimientos s/n. 41092 Sevilla, SPAIN. Telephone: +34 954487488 Fax: +34 954487340, e-mail: {limon,alvarado,alamo,eduardo}@cartuja.us.es Abstract: Model predictive control (MPC) is one of the few techniques which is able to handle with constraints on both state and input of the plant. The admissible evolution and asymptotically convergence of the closed loop system is ensured by means of a suitable choice of the terminal cost and terminal contraint. However, most of the existing results on MPC are designed for a regulation problem. If the desired steady state changes, the MPC controller must be redesigned to guarantee the feasibility of the optimization problem, the admissible evolution as well as the asymptotic stability. In this paper a novel formulation of the MPC is proposed to track varying references. This controller ensures the feasibility of the optimization problem, constraint satisfaction and asymptotic evolution of the system to any admissible steady-state. Hence, the proposed MPC controller ensures the offset free tracking of any sequence of piece-wise constant admissible set points. Moreover this controller requires the solution of a single QP at each sample time, it is not a switching controller and improves the performance of the closed loop system. Copyright©2005 IFAC. Keywords: Predictive control, Tracking, Constrained linear systems, Invariants, Asymptotic stability. 1. INTRODUCTION Model predictive control (MPC) is one of the few control techniques which is able to consider constraints (on both state and inputs of the system) in the design of the control law. This is achieved by predicting the evolution of the system and computing the admissible sequence of control inputs which makes the system evolve satisfying the constraints. This problem can be posed as an optimization problem. To obtain a feedback policy, the obtained sequence of control inputs 1The authors would like to acknowledge MCYT-Spain (contract DPI2002-04375-c03-01) for funding this work. is applied in a receding horizon manner, solving the optimization problem at each sample time. Nowadays the theoretical foundation of MPC is wellknown and under some assumptions, asymptotic stability is guaranteed. This is achieved by means of a suitable penalization of the terminal state and an additional terminal constraint (Mayne, Rawlings, Rao & Scokaert 2000). Most of the results on MPC consider the regulation problem, i.e. steering the system to the desired steady state which is assumed to be the origin. The purpose of this paper is to design an MPC algorithm to track a sequence of admissible set points. It is clear that for Copyright (c) 2005 IFAC. All rights reserved 16th Triennial World Congress, Prague, Czech Republic 135 a given non-zero set point, a suitable choice of the steady state can be chosen and the problem can be posed as a regulation problem translating the state and input of the system (Muske & Rawlings 1993). However, since the stabilizing choice of the terminal cost and constraints depends on the desired steady state, a changing reference requires to re-design the MPC at each change of the reference. The computational amount that the design of a stabilizing MPC requires may make this approach not viable. Some results exist to solve the tracking problem for constrained linear system. The first remarkable approach is the so-called command governor (Gilbert, Kolmanovsky & Tan 1994), where a nonlinear lowpass filter of the reference is added to guarantee the admissible evolution of the system to the reference. This can be seen as adding an artificial reference (the output of the filter). This is computed at each sampling time to ensure the admissible evolution of the system, converging to the desired reference. In (Bemporad, Casavola & Mosca 1997) a command governor is designed to minimize a performance index of the predicted evolution of the system. In (Blanchini & Miani 2000) it is proved that any control invariant set for the constrained system is a tracking domain of attraction and an interpolation-based control law is proposed. In (Bemporad et al. 1997, Rossiter & Kouvaritakis 1998) is shown that there exists similarities between predictive controllers and command governors: both compute the control action to guarantee the constraint satisfaction and the convergence to the reference. The main difference is how this control action is considered in the associated optimization problem. In (Chisci & Zappa 2003) a dual-mode strategy for tracking based on MPC is presented: if the MPC is not feasible, the controller switches to a feasibility recovery mode, which steers the system to the feasibility region of the MPC. In this paper a novel MPC strategy for tracking is presented. In a similar way to the command governors, an artificial reference is considered as decision variable of the MPC. The feasibility is ensured by considering an invariant set for tracking as terminal constraint and the offset free control is ensured by penalizing the deviation of the artificial reference from the desired reference. The obtained optimization problem is a standard QP and the MPC control law is able to track any admissible reference, and hence, any piece-wise constant sequence of references. Notation: vector (x,t,r)denotes [xT,tT,rT]T; for a given λ,λX={λ·x:x∈X};int(X)denotes the interior of set X; a matrix Tdefinite positive is denoted as T>0 and T>Pdenotes that T−P>0. Consider a∈IRna,b∈IRnb, and set Γ⊂IRna+nb, then projection operation is defined as Pro ja(Γ) = {a∈IRna: ∃b∈IRnb,(a,b)∈Γ}. 2. PROBLEM DESCRIPTION Let a discrete-time linear system be described by: x+=Ax +Bu (1) y=Cx +Du where x∈IRnis the current state of the system, u∈ IRmis the current input, y∈IRpis the current otput and x+is the successor state. The state of the system and the control input applied at sampling time kare denoted as x(k)and u(k)respectively. The system is subject to hard constraints on state and control: x(k)∈X,u(k)∈U for all k≥0. The sets Xand Uare compact convex polyhedra containing the origin in their interior. They are given by X={x∈IRn:Ax·x≤bx}(2) U={u∈IRm:Au·u≤bu}(3) The problem we consider is to track a piece-wise constant sequence of set points or references s(k) in such a way that the constraints are satisfied for all the time. For this purpose we propose an MPC formulation which allows one to reach any admissible set point swith offset-free. 3. CALCULATION OF THE ADMISSIBLE STEADY STATES Consider a set-point tand a steady state of the system (xs,us)associated to this set-point satisfying: A−I B C D ·xs us=0 t(4) Denote E=A−I B C D F=A−I B 0 C D Ip where Ipis the identity matrix of order p. It is assumed that the rank of Eis equal to the rank of F. This assures that the steady state equation (4) has a solution for any set-point t. Thus, any solution of (4) can be posed as zs=xs us=M·t+N·r=Mx Mu·t+Nx Nu·r(5) where ris an auxiliary variable and its size depends on the rank of E; notice that if Eis full column rank, then the solution is given by zs=M·t. If rank of Eis less than n+m, then vector rcan be thought as free variables in the selection of the steady state and input for a desired set point t. 136 The set of all admissible steady states of the system, is denoted as Xs, i.e. Xs={xs∈X:∃us∈ Usuch that (A−I)·xs+B·us=0}. Analogously, the set of admissible steady inputs is denoted as Us. The set of all admissible set-points is denoted as Si.e. S={s=C·xs+D·us: xs∈X,us∈Uand (A−I)·xs+ B·us=0}. 4. CALCULATION OF AN INVARIANT SET FOR TRACKING Assume that the following controller given by u=K·(x−xs)+us,(6) where xsand usis the steady state we want to reach, asymptotically stabilizes the closed loop system. It is well known that if the controller gain Kis such that A+BK has all its eigenvalues inside the unit circle, then the system is steered to the desired steady state. Since the system is constrained, this controller leads to an admissible evolution of the system only in a neighborhood of the origin. Substituting (5) in (6), matrices Ltand Lrcan be found such that u=K·x+Lt·t+Lr·r(7) Consider the extended state w= (x,t,r), then the closed loop system can be posed as   x t r  + =  A+BK BLtBLr 0I0 0 0 I ·  x t r (8) that is, w+=Aw·w. Because of reasons that will be clearer later, define set Wλ={w= (x,t,r):u=K·x+Lt·t+Lr·r∈U,x∈ X,xs=Mx·t+Nx·r∈λXand us=Mu·t+Nu·r∈λU}; this set is a polyhedron given by     Ax0 0 AuK Au·LtAu·Lr 0Ax·MxAx·Nx 0Au·MuAu·Nu    ·  x t r ≤    bx bu λ·bx λ·bu    (9) It is clear that the set of constraints for system (8) is W=W1. We say that a set Ωwis an admissible invariant set for tracking, for system (8) constrained to W, if for all w∈Ωw, then Aw·w∈Ωwand Ωw⊆W. The maximal admissible invariant set for tracking is given by (Gilbert & Tan 1991): Ow ∞={w:Ai ww∈W,∀i≥0} This set might be not finitely determined by a finite set of constraints. Consider the maximal admissible invariant set for tracking considering Wλas constraint set, which is given by Ow ∞,λ={w:Ai ww∈Wλ,∀i≥0} Following similar arguments to (Gilbert et al. 1994), it can be shown that for any λ∈(0,1),Ow ∞,λis finitely determined and λOw ∞⊂Ow ∞,λ⊂Ow ∞. Notice that since λcan be chosen arbitrarily close to 1, the obtained invariant set is arbitrarily close to the real maximal invariant set Ow ∞. In what follows, superscript wdenotes that set Ωw is defined in the extended state, while no superscript denotes that set Ωis defined in the state vector space x, i.e. Ω=Pro jx(Ωw). 5. MPC FOR TRACKING Assume that a stabilizing gain controller Kfor the system (1) is computed and an admissible invariant set for tracking Xw fis obtained. Based on these, we present an MPC formulation which guarantees offsetfree tracking to any admissible steady state ˆ xscontained in Xf. In a similar way to the command governors, the proposed MPC considers an artificial reference given by (xs,us)as decision variable of the associated cost. Moreover, the deviation between the artificial steady state xsand the desired steady state ˆ xsis penalized. If ˆ xsis an admissible steady state, then this penalization guarantees offset-free tracking; however if ˆ xsis not admissible, this penalization makes the system evolve to an admissible steady state such that its deviation with the desired (although unreachable) steady state is minimized. The proposed cost is VN(x,s,u,t,r) = N−1 ∑ i=0kx(i)−xsk2 Q+ku(i)−usk2 R +kx(N)−xsk2 P+kxs−ˆ xsk2 T where xis the current state, sis desired set point to be tracked, uis a sequence of Nfuture control inputs, x(i) is the predicted state of the system at time igiven by x(i+1) = A·x(i)+ B·u(i), with x(0) = x,xs=Mx·t+ Nx·rand us=Mu·t+Nu·r. For a given desired set point s, an associated steady state ˆ xsis obtained by means of a linear mapping ˆ xs=Hx·sin such a way that some performance index is minimized. Matrices Q,R,Pand Tare assumed to be definite positive. Therefore, u,t, and rare the decision variables and x and sare the parameters of the proposed cost function. Note that this cost can be posed as a quadratic function of the decision variables. The MPC optimization problem PN(x,s)is given by 137 V∗ N(x,s) = min u,t,rVN(x,s,u,t,r) s.t.x(0) = x. x(j+1) = A·x(j)+B·u(j), u(j)∈U, x(j)∈X,j=0,···,N−1. (x(N),t,r)∈Xw f. where Xw fis an invariant set for tracking in the extended space (x,t,r). Since this region is a polyhedron, the optimization problem is a standard quadratic problem, that can be efficiently computed. 6. STABILITY ANALYSIS In this section we provide sufficient conditions to guarantee asymptotic stability of the proposed controller, in such a way that the closed loop system asymptotically reaches any desired admissible steady state s∈S. Before stating the main result of the paper, some technical lemmas must be proved. All the proofs can be found in the appendix. We denote hereafter O∞(xs)as the maximal invariant set of states that can be steered to xsin an admissible way by the control law (6). Lemma 1. Let ˆ xsbe an admissible steady state and let xsand usbe a steady state and input for system (1) such that xs∈int(X)and u∈int(U). Let Kbe a stabilizing linear controller with a Lyapunov matrix P. Then there exists λ∈[0,1]and ¯ xs=λxs+ (1−λ)ˆ xs such that: (1) xs∈O∞(¯ xs). (2) For all T>Pand xs6=ˆ xsthen kxs−¯ xsk2 P+k¯ xs−ˆ xsk2 T<kxs−ˆ xsk2 T Next, a (standard) lemma is presented: Lemma 2. Consider system (1) subject to contraints (2) and (3). Let u=K·xbe a stabilizing controller with an associated Lyapunov matrix Psuch that (A+BK)TP(A+BK)−P=−(Q+KTRK) where Q∈IRn×nand R∈IRm×mare definite positive matrices. Consider any x∈Xf=Pro jx(Xw f). Consider an admissible steady state xssuch that x∈O∞(xs). Then we have that V∗ N(x,s)≤kx−xsk2 P+kxs−ˆ xsk2 T where ˆ xsis an steady state associated to s. Based on these lemmas, the following one can be proved: Lemma 3. Consider a given reference sand the selected associated steady state ˆ xs; assume that for a given state xthe optimal solution of PN(x,s)is such that kx−x∗ skQ=0 (i.e. x=x∗ s), then kx−ˆ xskQ=0. Now the main result of the paper is presented: Theorem 4. Consider a system (1) subject to contraints (2) and (3). Let Q∈IRn×nand R∈IRm×mbe definite positive matrices. Let u=K·xbe a stabilizing controller with an associated Lyapunov matrix Psuch that (A+BK)TP(A+BK)−P=−(Q+KTRK) Consider a matrix T>P. Assume that Xw f=Ow ∞,λ, computed for a given λ∈(0,1), and for the gain matrix K(see section 4). Then for any feasible initial state x0∈XNand for any admissible set point s∈λS, the proposed MPC controller steers asymptotically the system to sin an admissible way. Remark 5. The set of the admissible steady states that can be tracked without offset is λXs. Since λXs⊂O∞ and since the evolution of the system remains in XN, the system can be steered to any admissible reference. Then, a sequence of piecewise admissible references can be tracked without offset. If the desired reference s(and hence the associated steady state ˆ xs) is not admissible, then the controller steers the system to an admissible steady state xsin such a way that the distance kxs−ˆ xskTis minimized. Remark 6. The stability theorem can be extended to consider Xw fas any admissible invariant set for tracking. In this case, the set of references that can be tracked without offset is ˆ S={s∈IRp:∃(x,s,r)∈ Xw f,x=Mx·s+Nx·r}. Remark 7. The proposed controller with s=0, i.e the origin as the desired steady state, provides a larger domain of attraction and a better performance (lower optimal cost) than the MPC formulated for regulation. 7. EXAMPLE Consider a LTI system given by: A=1 1 0 1 ,B=0.5 0.0 0.0 1.0,and C=1 0 . The system is constrained to kxk∞≤5 and kuk∞≤3. The steady state and input are characterized by the matrices M=1 0 0 0 T,N=0 0.4472 −0.8944 0 T. The weighting matrices have been chosen as Q= 0.01·I2and R=I2. The local controller gain and the 138 −5 −4 −3 −2 −1 0 1 2 3 4 5 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 x1 x 2 s1 s2 s3 Fig. 1. State portrait of the system. 0 10 20 30 40 50 60 70 80 90 −5 −3 −1 1 3 5 x 1 0 10 20 30 40 50 60 70 80 90 −2 −1 0 1 2 3 samples x 2 Fig. 2. Evolution of the states of the system Lyapunov matrix Phas been computed using an LQR. The computed maximal invariant set for tracking is shown in figure 1 in dashed line. The system is controlled by the proposed MPC with a control horizon N=10 and a deviation penalization matrix T=1.1·P. The domain of attraction of the proposed controller is depicted in figure 1 in solid line. To illustrate the proposed controller, a piece-wise constant sequence of references has been tracked. The first value of the set point is s1=4.5, the second s2=−4.5 and the third s3=0. The evolution of the state is shown in figure 1. In figure 2 the evolution of the states is plotted. As it can be seen, the system evolution is admissible for all time and for drastic changes of the set point. Notice also that the controller steers the system to the desired set points. In this figure the artificial reference x∗ s(k)of each state is drawn in dashed line. Appendix A. PROOFS Proof of Lemma 1 Fact (1): Let Pbe a Lyapunov matrix of the closed loop system x+= (A+BK)·x. Since xs∈int(X), there exists α>0 such that {x:kx−xskP≤α}⊆X. Analogously, there exists an β>0 such that {u:ku− uskP≤β}⊆U. Consider a positive constant ε=min(α 2,β 2kKkP ) where kKkPdenotes the induced weighted euclidean norm of matrix K. Consider λ1∈[0,1]such that (1− λ1)kxs−ˆ xskP≤ε. Consider λ2∈[0,1]such that (1− λ2)kus−ˆ uskP≤β/2. Consider λ≥max(λ1,λ2)and denote ¯ xs=λxs+(1−λ)ˆ xsand ¯ us=λus+(1−λ)ˆ us. Then: •For all {x:kx−¯ xskP≤ε}, we have that u= K(x−¯ xs)+ ¯ usis admissible. ku−uskP=kK(x−¯ xs)+ ¯ us−uskP ≤kKkP·kx−¯ xskP+k¯ us−uskP We have that ¯ us−us= (1−λ)(ˆ us−us)and hence ku−uskP≤kKkP·kx−¯ xskP+(1−λ)kˆ us−uskP ≤kKkP·ε+(1−λ)kˆ us−uskP ≤β/2+(1−λ)kˆ us−uskP≤β and then u∈U. • {x:kx−¯ xskP≤ε}⊆X. In effect kx−xskP≤ kx−¯ xskP+k¯ xs−xskP= kx−¯ xskP+ (1−λ)kˆ xs−xskP≤2ε≤α, and hence is contained in X. From these two facts and the property of Lyapunov matrix Pwe derive that {x:kx−¯ xskP≤ε}is an admissible invariant set, and hence it is contained in O∞(¯ xs). Since kxs−¯ xskP= (1−λ)kxs−ˆ xskP≤ε, we have that xs∈{x:kx−¯ xskP≤ε}and hence it is contained in O∞(¯ xs). Fact (2): In virtue of the previous fact we have that there exists a λ∈[0,1]such that for ¯ xs=λxs+ (1−λ)ˆ xswe have that xs∈O∞(¯ xs). Considering the expression of ¯ xsit is easy to see that xs−¯ xs= (1− λ)(xs−ˆ xs)and ¯ xs−ˆ xs=λ(xs−ˆ xs). Then we get that kxs−¯ xskP= (1−λ)kxs−ˆ xskPand k¯ xs−ˆ xskT= λkxs−ˆ xskT. From this result and the assumption that T>P, we can state that kxs−¯ xskP+k¯ xs−ˆ xskT= (1−λ)kxs−ˆ xskP+λkxs−ˆ xskT< (1−λ)kxs−ˆ xskT+λkxs−ˆ xskT=kxs−ˆ xskT Since a+b<cimplies that a2+b2<c2for all a,b,c>0, we have that kxs−¯ xsk2 P+k¯ xs−ˆ xsk2 T< kxs−ˆ xsk2 T, which proves the fact. Proof of lemma 2 It suffices to note that the sequence of control inputs obtained from the control law u= 139 K(x−xs)+us, denoted as us, is a feasible solution for the MPC optimization problem since x∈O∞(xs). Consider that AK=A+BK and Q∗=Q+KT·R·K, then from Lyapunov equation P−AT K·P·AK=Q∗we derive that kx(i)−xsk2 P−kx(i+1)−xsk2 P=kx(i)− xsk2 Q∗for all x(i). Summing this terms we have that N−1 ∑ i=0kx(i)−xsk2 Q∗=kx−xsk2 P−kx(N)−xsk2 P Therefore, V∗ N(x,s)≤ N−1 ∑ i=0 kx(i)−xsk2 Q∗ z}| { kx(i)−xsk2 Q+kK·(x(i)−xs)k2 R +kx(N)−xsk2 P+kxs−ˆ xsk2 T =kx−xsk2 P+kxs−ˆ xsk2 T Proof of lemma 3 It is proved by contradiction. Assume that x=x∗ sand x6=ˆ xs. Since x=x∗ sis a steady state of the system, the control sequence given by the steady input is the optimal solution of PN(x∗ s,s)and hence V∗ N(x∗ s,s) = kx∗ s−ˆ xsk2 T. Since x∗ s6=ˆ xs, in virtue of lemma 1 it is inferred that there exists a steady state ¯ xs(and an input ¯ us), described by ¯ tand ¯r, such that x∗ s∈O∞(¯ xs). Then the sequence ¯ uderived from the control law u=K(x− ¯ xs) + ¯ usis admissible and hence, from lemma 2 we have that VN(x∗ s,s,¯ u,¯ t,¯r)≤kx∗ s−¯ xsk2 P+k¯ xs−ˆ xsk2 T In virtue of lemma 1 we have that VN(x∗ s,s,¯ u,¯ t,¯r)<kx∗ s−ˆ xsk2 T=V∗ N(x∗ s,s) which contradicts the fact of the optimality ofV∗ N(x∗ s,s), and then x=x∗ s=ˆ xs. Proof of theorem 4 In what follows we denote the optimal solution to the optimization problem by the superscript ∗. Thus, (u∗(k),t∗(k),r∗(k)) denotes the optimal solution obtained in the optimization problem solved at sampling time k. Moreover, x∗ s(k)and u∗ s(k) denote the optimal steady state and input associated to t∗(k)and r∗(k).x∗(i;k)denotes the optimal predicted evolution of the system. Feasibility: Assume that the state at the current state k,xkis such that xk∈XNand assume that the optimal solution is (u∗(k),t∗(k),r∗(k)) with an optimal cost V∗ N(xk,s). Let xk+1be the state at the next sampling time. Consider t(k+1) = t∗(k),r(k+ 1) = r∗(k)and a control sequence u(k+1) = {u∗(1;k),···,u∗(N−1;k), K(x∗(N;k)−x∗ s(k))+u∗ s(k)} Then, it is easy to see that(u(k+1),t(k+1),r(k+ 1)) is feasible due to the feasibility of the optimal solution at kand the positively invariance of Xw f. Consequently, xk+1∈XN. Convergence: Consider the feasible solution at time k+1 previously presented. Following standard steps in the stability proofs of MPC (Mayne et al. 2000), we get that V∗ N(xk+1,s)≤VN(xk+1,s,u(k+1),t(k+1),r(k+1)) ≤V∗ N(xk,s)−kxk−x∗ s(k)k2 Q Due to the definite positiveness of the optimal cost and its non-increasing evolution, we infer that lim k→∞kxk−x∗ s(k)kQ=0 and in virtue of lemma 3 we have that lim k→∞kxk−ˆ xskQ=0 Consequently, the system output evolves to s. REFERENCES Bemporad, A., Casavola, A. & Mosca, E. (1997), ‘Nonlinear control of constrained linear systems via predictive reference management.’, IEEE Transactions on Automatic Control 42, 340–349. Blanchini, F. & Miani, S. 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