Enlarged terminal sets guaranteeing stability of receding horizon control

Transcription

1 Enlarged terminal sets guaranteeing stability of receding horizon control J.A. De Doná a, M.M. Seron a D.Q. Mayne b G.C. Goodwin a a School of Electrical Engineering and Computer Science, The University of Newcastle, Callaghan 2308, NSW, Australia b Department of Electrical and Electronic Engineering, Imperial College of Science, Technology and Medicine, London, UK Abstract The purpose of this paper is to relax the terminal conditions typically used to ensure stability in model predictive control, thereby enlarging the domain of attraction for a given prediction horizon. Using some recent results, we present novel conditions that employ, as the terminal cost, the finite-horizon cost resulting from a nonlinear controller u = sat(kx) and, as the terminal constraint set, the set in which this controller is optimal for the finite-horizon constrained optimal control problem. It is shown that this solution provides a considerably larger terminal constraint set than is usually employed in stability proofs for model predictive control. Key words: constrained control, receding horizon, stability, terminal conditions Revised paper: SCL Corresponding author. addresses: eejose@ee.newcastle.edu.au (J.A. De Doná), marimar@eie.fceia.unr.edu.ar (M.M. Seron), d.mayne@ic.ac.uk (D.Q. Mayne), eegcg@ec .newcastle.edu.au (G.C. Goodwin). Preprint submitted to Systems & Control Letters 13 April 2002

2 1 Introduction Model predictive control is a form of control in which the current control is obtained by solving, at each sampling instant, a finite-horizon open-loop optimal control problem and applying the first element of the optimal control sequence so obtained. The method is particularly appealing when it is desirable that the system satisfy certain constraints. Here we consider regulation with a single input which is required to satisfy amplitude constraints. Several ingredients of the online optimal control problem directly affect closed-loop stability; these are: the terminal cost F ( ), the terminal constraint set X f (both of which are employed in the optimal control problem solved online), and the local controller κ f ( ) that allows one to establish existence of feasible solutions to the optimal control problem (see, e.g., [1,7,9]). An ideal choice for the terminal cost F ( ) would be the infinite-horizon value function V ( ) 0 (for the constrained optimal control problem), in which case, the finite-horizon value function is simply VN( ) 0 =V ( ). 0 With this choice, the advantages of an infinite-horizon problem automatically accrue and stability is easy to establish. However, constraints generally render this approach intractable. Usually, then, X f is chosen to be an appropriate neighborhood of the origin in which V ( ) 0 is exactly (or approximately) known, and F ( ) is set equal to V ( ) 0 or its approximation. When the system being controlled is linear, F ( ) is often chosen (see [10,11]) to be the value function of the infinite-horizon unconstrained optimal control problem, κ f ( ) is chosen to be the optimal controller (κ f (x) = Kx) for this problem, and X f the maximal output admissible set O (defined by (4.2) below) for the closed-loop system using the local controller κ f ( ). In this case, F (x) =x T Px for all x X f (P and K are obtained from the solution of an algebraic Riccati equation, see (3.7) and (3.8) below.) The purpose of this paper is to provide new terminal ingredients for model 2

3 predictive control of input constrained linear systems. The ingredients are an improvement over those previously used in that the terminal constraints set X f is strictly larger than O, thus increasing, for given horizon length N, the domain of attraction of model predictive control. To obtain the improved terminal conditions, we employ recent results [3,4] that show that the nonlinear controller κ nl (x) := sat(kx) is optimal in a region X f that includes the maximal output admissible set O. The proposed terminal cost function F ( ), while no longer quadratic, is convex. The new terminal constraint set X f is, however, no longer convex, so that the online optimal control problem P N is not easily solved. We overcome this difficulty by pre-computing a horizon length N such that, at all states x encountered, the solution of P N (x) isalso the solution of another optimal control problem PN(x) that has no terminal constraint but is otherwise identical to P N (x). Problem PN(x) is aconvex optimal control problem (because the terminal cost F ( ) and the path cost l( ) are convex and the system linear) that is easily solved. This paper therefore extends a previous result [8] that dealt with the case of stable plants and is valid for arbitrary linear plants with a constrained single input. 2 Definitions and notation The system considered is x(k +1)=Ax(k)+Bu(k) (2.1) or, more concisely, x + = f(x, u) :=Ax + Bu, wherex IR n and u IR are, respectively, the current state and control and x + is the successor state. The pair (A, B) is assumed controllable. The control is required to satisfy the constraint u(k) Ω for all k, where, by way of illustration, we take Ω:=[ 1, 1]. The following notation will be employed. The solution of (2.1) at time k, when 3

4 the initial state is x at time i and the control sequence is u, isx u (k; x, i); to simplify notation, x u (k; x) :=x u (k; x, 0), i.e. the initial time is dropped when it is zero. For all ɛ>0, B ɛ := {x x ɛ}. For any set X in, say, IR n, X c denotes the complement of X (in IR n ). IR + := {x IR x>0} and for each IR +, the function sat ( ) isdefinedbysat (u) :=sign(u) min{ u, }. The function sat( )isdefinedtobesat 1 ( ). In the sequel denotes concatenation, i.e. (a b)(x) :=a(b(x)), a 0 (x) :=x and, for all i =1, 2,..., a i (x) :=(a i 1 a)(x) =(a a i 1 )(x). 3 Model predictive control In model predictive control, a finite-horizon optimal control problem P N (x) defined below is repeatedly solved. Because of time invariance, the initial time in the optimal control problem may be taken to be zero. Thus P N (x) isdefined by P N (x) : VN(x) 0 =min u V N (x, u) (3.1) subject to the control constraint u Ω N and the terminal constraint x(n) X f,whereu := {u(0),u(1),...,u(n 1)} is a sequence of N control actions. We consider a quadratic incremental cost with terminal weighting as: N 1 V N (x, u):= l(x(k),u(k)) + F (x(n)) (3.2) k=0 l(x, u):= x 2 Q + u 2 R (3.3) and x(k) :=x u (k; x), k =0, 1,...,N. We assume that Q and R are positive definite, and denote x 2 Q := x T Qx and u 2 R := u T Ru. In (3.2), F ( ) is the terminal cost. At event (x, k) (at state x, timek), problem P N (x) is solved yielding the optimal control sequence u 0 (x) ={u 0 (0; x),u 0 (1; x),...,u 0 (N 1; x)}, the optimal state sequence x 0 (x) ={x 0 (0; x),x 0 (1; x),...,x 0 (N; x)} (where x 0 (0; x) =x, the initial state) and the value function V 0 N(x) =V N (x, u 0 (x)). The first control u 0 (0; x) is applied to the plant so that the (implicit) model 4

5 predictive control law is u = κ N (x) :=u 0 (0; x), (3.4) and the procedure is repeated as a new state becomes available. 3.1 Closed loop stability If F ( ) andx f are chosen appropriately (see, for example, [9]), the receding horizon control law (3.4) can be shown to be stabilizing subject to certain feasibility assumptions. For any function θ : IR n IR n,let θ( ) bedefinedby θ(x, u) :=θ(f(x, u)) θ(x) (3.5) where f(x, u) =Ax + Bu. We recall the following stability pro-forma: Theorem 1 ([9]) Suppose the terminal cost function F : X f IR, the terminal constraint set X f and the local control law κ f : X f IR satisfy: A1: X f is closed and 0 X f, A2: κ f (x) Ω, x X f (control constraint satisfied in X f ), A3: X f is positively invariant for the system, x + = f(x, κ f (x)), A4: [ F + l](x, κ f (x)) 0, x X f (F ( ) is a local Lyapunov function). Then [ VN 0 + l](x, κ N (x)) 0 for all x X N, the (compact) set of states steerable to X f by an admissible control in time N or less. Also X N is positively invariant for the closed-loop system x + = f(x, κ N (x)) where κ N ( ) is the model predictive control law. Corollary 1 ([9]) Suppose Q>0 and R>0, that (F ( ), X f,κ f ( )) satisfy A1 A4 and that, in addition, there exists a finite c such that F (x) c x 2 for all x X f. Then the origin is exponentially stable for the closed-loop system x + = f(x, κ N (x)) with a region of attraction X N. 5

6 3.2 Regional characterization of the value function We consider in this section an optimization problem P M(x) which is defined as problem P N (x) (3.1) with horizon M, terminal cost F (x) =x T Px and with the terminal constraint removed, i.e., X f = IR n. Let u (x) = {u (0; x),u (1; x),...,u (M 1; x)} denote the solution of P M(x), x (x) = {x (0; x),x (1; x),...,x (M; x)} the corresponding state trajectory (x (0; x) = x) andv M(x) =V M (x, u (x)) the optimal value function. We review some recent results [3,4] that show that the nonlinear controller κ nl ( ) definedby κ nl (x) :=sat( Kx)= sat(kx) (3.6) is optimal for the problem P M(x) in a non-trivial region of the state-space (non-trivial in the sense that the region includes the output admissible set, which is defined in the sequel). In (3.6), the vector K is the optimal gain for the unconstrained infinite-horizon problem, which is computed from the (unique positive semi-definite) solution of the algebraic Riccati equation P = A T PA+ Q K T RK, (3.7) where K := R 1 B T PA, R := R + B T PB. (3.8) In the sequel we consider both a linear controller u = Kx and a nonlinear controller u = sat(kx). The closed-loop satisfies x + = φ l (x) when the linear controller is used and x + = φ nl (x) when the nonlinear controller is employed, where the mappings φ l : IR n IR n and, φ nl : IR n IR n are defined by φ l (x):=a K x, A K := A BK, (3.9) φ nl (x):=ax Bsat(Kx) (3.10) For all i {1, 2,...,M} (M 1 an integer) the function δ i ( ) isdefinedby δ i (x) :=Kx sat i (Kx) (3.11) 6

7 where the saturation bounds i are defined by 1 := 1, i := 1 + i 2 j=0 KA j B, i = 2, 3,...,M. The set X 0 := IR n, and for each i {1, 2,...,M 1}, the set X i IR n is defined by X i := {x δ i (A i 1 A K x)=0} = {x K i x i } (3.12) where K i := KA i 1 A K.WealsorequirethesetsY i (i {1, 2,...,M}) and Z M defined by Y i := X 0 X 1... X i 1, (3.13) Z 1 := Y 1 = IR n, (3.14) Z M := {x φ k nl(x) Y M k,k =0, 1,...,M 2}, for M 2. (3.15) We can now state the main results of [3,4]: Lemma 1 ([3,4]) For any i {1, 2,...,M 1} δ i (A i 1 φ nl (x)) 2 = δ i+1 (A i x) 2, for all x X i. (3.16) where φ nl ( ), δ i ( ) and X i are defined, respectively, in (3.10), (3.11) and (3.12). Theorem 2 ([3,4]) For any integer M>0, the optimal value function V M(x) and optimal control law κ M(x) for problem P M(x) satisfy V M(x) =x T Px+ R M δ k (A k 1 x) 2 (3.17) and κ M(x) =κ nl (x) = sat(kx) (3.18) k=1 for all x Z M, the set defined in (3.15). Since the functions x δ k (A k 1 x) 2 are convex, so is the value function V M( ). 7

8 4 Terminal conditions 4.1 Standard specification of (F ( ), X f,κ f ( )) A triple (F ( ), X f,κ f ( )) satisfying A1 A4 of Theorem 1 and F (x) c x 2 for all x X f ensures exponential stability as shown in 3.1above.Ausefulchoice of terminal conditions [10,11] for the problem considered is to choose F ( )to be the value function Vuc( ) 0 for the unconstrained infinite-horizon optimal control problem P uc (x) for the same system (2.1), defined as P uc (x) : V 0 uc(x) =min u V uc (x, u) (4.1) with cost V uc (x, u) := k=0 l(x(k),u(k)), where l(x, u) = x 2 Q + u 2 R as before. (Note that P uc ( ) does not have either a terminal cost nor a terminal constraint; both are irrelevant since, if a solution to the problem exists, x 0 (k; x) 0ask.) Thus, in the constrained optimization problem P N ( ) (3.1), solved at each time instant in model predictive control, the terminal cost function F ( ) normally used is F (x) :=Vuc(x) 0 =x T Px where P > 0 is the (unique positive semidefinite) solution of the algebraic Riccati equation (3.7) (3.8). In this context, the local controller is defined by κ f (x) := Kx where K is computed from (3.8), and is, therefore, the optimal controller for the unconstrained infinite-horizon problem P uc ( ). The set X f is usually taken to be the maximal output admissible set O defined in [5], i.e. O := {x KA j Kx Ω, j=0, 1,...}. (4.2) An interesting consequence of this choice for (F ( ), X f,κ f ( )) is that V (x) 0 = F (x) for all x in X f and that VN(x) 0 =V (x) 0 for all x X N such that the terminal constraint is not active (i.e. x 0 (N; x) lies in the interior of X f ); if N is so chosen, the terminal constraint may be omitted from P N ( ). 8

9 4.2 New specification of (F ( ), X f,κ f ( )) It is the purpose of this paper to propose a larger terminal constraint set X f, thus increasing the provable domain of attraction of model predictive control (or reducing N in those variants that omit the terminal constraint from the optimal control problem P N but increase the horizon N until this constraint is satisfied). To this end we employ the results described in 3.2; namely, the regional characterization of the value function V M( ) (3.17) when the optimal control law κ M( ) =κ nl ( ) (3.18) is employed for states x in a region Z M of the state space. We show in this section that the new triple (F ( ), X f,κ f ), obtained using these elements, satisfies conditions A1 A4 of Theorem 1 and constitutes an improvement over previous results (cf. 4.1). It can be readily seen from (3.18) that the control law κ nl ( ) satisfies A2. Our problem then reduces to finding a set X f that satisfies A1, is positively invariant under the control law κ nl ( ) and in which V M( ) is a local Lyapunov function. Definition 1 Define the sets X M, ȲM, Z M IR n, for any M 1, by X M := {x δ M (A M 1 φ nl (x)) = 0} = {x KA M 1 φ nl (x) M }, (4.3) Ȳ M := Y M X M D S = X 0...X M 1 X M D S, (4.4) and Z M := {x φ k nl(x) ȲM,k =0, 1, 2,...}, (4.5) where the set Y M is as in (3.13) and D S is a design set, used to ensure compactness of ȲM. In the case when Y M X M is compact, D S can be chosen to be D S = IR n ; otherwise, D S is chosen to be an arbitrarily large compact set such that O D S. (4.6) 9

10 The value function V M( ) is a candidate for the terminal constraint F ( ), the set Z M is a candidate for the terminal constraint set X f, and the control law κ nl ( ) is a candidate for the local control κ f ( ). To establish that the triple (V M( ), Z M,κ nl ( )) satisfies A1 A4 of Theorem 1 is our next task and requires some preliminary results. Proposition 1 (i) The set Z M is the maximal positively invariant set in ȲM for the system x + = φ nl (x). (ii) Z M ȲM. (iii) Z M is compact and contains the origin in its interior. Proof: (i) This follows from the definition (4.5) of Z M if Z M is not empty. Let c>0 be such that the level set L := {x x T Px c} O ȲM; since both O [5] and ȲM contain the origin in their interiors and since P > 0, such a c exists. Since κ nl (x) = Kx and φ nl (x) =A K x in O, and since L is positively invariant for x + = A K x = φ nl (x), it follows that L Z M.But L contains the origin in its interior (since P>0). (ii) From definition (4.5), x Z M implies x ȲM. (iii) Since ȲM is compact and φ nl ( ) is continuous, each set {x φ k nl(x) ȲM}, k =0, 1,..., is compact. Hence Z M is compact. That Z M contains the origin in its interior follows from the facts that Z M L and L contains the origin in its interior. Proposition 2 Z M Z M. Proof: Notice first, from (3.13) and (4.4), that ȲM Y i, i =1, 2,...,M. It follows from definition (4.5) that x Z M implies φ k nl(x) ȲM, for all k. Hence, φ k nl(x) Y i, for all k and i =2, 3,...,M, and we conclude from (3.15) that x Z M. Proposition 3 For all x Z M : [ V M + l](x, κ nl (x)) = 0. (4.7) 10

11 Proof: Making use of (3.5), (3.7), (3.8), (3.11), (3.16), (3.17), (4.3), (4.4), and the fact that Z M ȲM (Proposition 1) and Z M Z M (Proposition 2), we obtain, for x Z M, [ V M + l](x, κ nl (x)) = V M(φ nl (x)) + l(x, κ nl (x)) V M(x) = V M(x)+ R δ M (A M 1 φ nl (x)) 2 V M(x) =0. For all j =1, 2,...,letW j be defined by W j := {x φ i nl(x) ȲM for i =0, 1,...,j 1andφ j nl (x) L} (4.8) where L is defined in the proof of Proposition 1. Proposition 4 There exists an integer i such that Z M = W i ; i.e., ZM is finitely determined. Proof: (i) Let max{v M(x) x ȲM} = c 1 <. From (4.7) we have that V M(x, κ nl (x)) = l(x, κ nl (x)) x 2 Q c 2 x 2, x Z M. There exists a c 3 (0, ) such that V M(x, κ nl (x)) c 3, for all x Z M closure(l c ). Hence for all x Z M there exists an integer i c 1 /c 3 such that φ i nl(x) L. Hence x Z M implies x W i. (ii) Suppose x W i so that φ i nl(x) ȲM for i =0, 1,...,i 1andφ i nl L. Since L is positively invariant for x + = φ nl (x), φ j nl (x) L ȲM for j = i,i +1,i +2,... Hence x Z M. From the definition (4.5) it is clear (as shown in Proposition 1) that Z M is the maximal positively invariant set in ȲM for the closed-loop system x + = φ nl (x) and, hence, X f := Z M satisfies A3. We now establish that O is a subset of Z M. 11

12 Proposition 5 O Z M. Proof: By definition Z M is the maximal positively invariant set in ȲM for the closed-loop system x + = φ nl (x). The set O is also a positively invariant set for x + = φ nl (x) (since φ nl ( ) =φ l ( ) ino ). It suffices, therefore, to establish that O ȲM; i.e., that O X i for i =1, 2,...,M 1andO X M, since O D S by design (cf. (4.6)). Assume, therefore, that x O,sothat KA j Kx 1, j =0, 1,... (4.9) For any i {1, 2,...,M} A i K =(A BK)A i 1 K which implies = A(A BK)A i 2 K = AA i 1 K BKA i 1 K BKA i 1 K KA i 1 A K x = KA i Kx + i 2 = = A i 1 A K i 2 j=0 From (4.9) and (4.10), we obtain the inequality j=0 A j BKA i 1 j K KA j BKA i 1 j K x (4.10) i 2 KA i 1 A K x KA i Kx + i 2 j=0 KA j B KA i 1 j K x (4.11) 1+ KA j B = i (4.12) j=0 This implies x X i for i =1, 2,...,M 1 (cf. (3.12)). To show that O X M notice, from (4.3), that X M can be written as the union of three sets: X M = V 1 V 2 V 3,whereV 1 := {x Kx 1} {x KA M 1 A K x M }, V 2 := {x Kx < 1} {x KA M 1 (Ax + B) M } and V 3 := {x Kx > 1} {x KA M 1 (Ax B) M }. Then, it follows from (4.9) and (4.12) that x O implies x V 1 and, hence, x X M. Example 1 In this example, the relative size of the sets O, ȲM and Z M is illustrated. Consider the system x + = Ax + Bu, y = Cx with A = 12

13 [1 0 ; 0.4 1], B = [0.4; 0.08] and C = [0 1], which is the zero-order hold discretization, with a sampling period of 0.4 sec., of the double integrator ẋ 1 = u, ẋ 2 = x 1, y = x 2. In Eqs. (3.2) (3.3) we take: Q = I 2 2 and R =0.25. The matrix P and the gain K were computed from (3.7) (3.8). In Figure 1 we show the set ȲM, the maximal output admissible set O and 8 6 Ȳ M 4 2 Z M x2 0 O x 1 Fig. 1. Set boundaries for the example. (In the case of ZM an estimate obtained numerically is shown. Note that O Z M ȲM.) an estimate of the set Z M. In this example Y M is compact and Y M X M. Therefore, we take D S = IR n in (4.4) and obtain ȲM = Y M = M 1 i=0 X i. ThevalueofM used was M =10. However, in this example, the set ȲM (and, hence, ZM ) does not change with M for M 7. The estimate of ZM was obtained numerically by gridding the set ȲM and checking the condition in (4.5) for each point of the grid. In the implementation of the MPC algorithm, the test x 0 (N; x) X f is not performed explicitly (see 4.3 below) and, thus, an explicit characterization of X f := Z M is not required. We have included an estimate of ZM in the figure for illustration purposes. As proven in Proposition 4, the set Z M is finitely determined and, in this example, the test φ k nl(x) ȲM (cf. (4.5)) is stopped for k such that φ k nl(x) O.Asthe figure shows, the new terminal constraint set Z M is considerably larger than 13

14 the usually employed O. We now state the main result of this paper. Theorem 3 The triple (F ( ), X f,κ f ( )) where F ( ) :=V M( ), X f := Z M and κ f ( ) :=κ nl ( ), for any integer M 1, satisfies conditions A1 A4 of Theorem 1. Proof: (i) It is established in Proposition 1 above that X f := Z M is closed and contains the origin in its interior, thus satisfying A1. (ii) κ f ( ) :=κ nl ( ) satisfies A2 by definition (cf. (3.18)). (iii) That X f := Z M is positively invariant for the system x + = f(x, κ f (x)) = φ nl (x) is established in Proposition 1 above. (iv) We have shown in Proposition 3 (cf. (4.7)) that F ( ) :=V M( ) satisfies [ F + l](x, κ nl (x)) = 0 for all x X f, thus satisfying A Implementation of the MPC algorithm Since Z M is not necessarily convex (as Figure 1 illustrates) and F ( ) =V M( )is not quadratic, P N ( ) is not a quadratic program, nor even a convex program. However, because Z M is finitely specified, it is easy to determine whether or not a given state lies in Z M. This makes it possible to achieve stabilizing model predictive control by solving online, for a suitably chosen horizon N, theconvex optimization problem PN(x) defined as problem P N (x) but without the (non-convex) terminal constraint x(n) X f = Z M.Let u (x) = {u (0; x),u (1; x),...,u (N 1; x)} denote the solution of PN(x) and x (x) ={x (0; x),x (1; x),...,x (N; x)} the corresponding state trajectory (x (0; x) =x). 14

15 The set D s, used in (4.4), is chosen to be φ M nl (O ); with this choice, the solution u (x) ={u 0(x),u 1(x),...,u M 1(x)} of P M(x) steers any x Z M into O in M steps or less so that V M(x) =V (x). Problem P N ( ) is now similar to the problem considered in [2] with the optimal value function F (x) =x T Px replaced by the optimal value function F (x) =V M(x), the terminal constraint set X f = O replaced by X f = Z M and the locally optimal control law κ f (x) = Kx replaced by κ f (x) = sat(kx); in each case, F : X f IR is the value function for an infinite horizon optimal control problem, and the terminal constraint set X f is positively invariant for x + = φ nl (x). Thus, as shown in Theorem 1 of [2], for each x such that V (x) 0 <, there exists a N such that x (N; x) Z M,sothatu (x) =u 0 (x) andv N(x) =V 0 N(x) =V 0 (x). The actual trajectory of the plant is therefore the same as the projected trajectory, so that the value of N chosen for the initial state x serves for all subsequent states. Exponential stability of the origin with a domain of attraction the domain of V 0 ( ) follows using V 0 ( ) as a Lyapunov function. 5 Conclusions We have shown how to obtain new terminal ingredients F ( ), X f, κ f ( ) (for the optimal control problem employed in model predictive control) that ensure closed-loop stability. The ingredients provide a larger terminal constraint set than that provided by previous approaches, thus potentially increasing the domain of attraction of model predictive control. Examples show that the new constraint set X f is larger than the output admissible set O conventionally employed. References [1] C.C. Chen and L. Shaw, On receding horizon feedback control, Automatica 18 (1982)

16 [2] D. Chmielewsky and V. Manousiouthakis, On constrained infinite-time linear quadratic optimal control, Systems and Control Letters 29 (1996) [3] J.A. De Doná and G.C. Goodwin, Ellucidation of the state-space regions wherein model predictive control and anti-windup strategies achieve identical control policies, in: Proc. American Control Conference, Chicago [4] J.A. De Doná and G.C. Goodwin, Characterisation of regions in which model predictive control policies have a finite dimensional parameterisation, Technical Report EE99044, School of Electrical Engineering and Computer Science, The University of Newcastle, Australia. [5] E.G. Gilbert and K.T. Tan, Linear systems with state and control constraints: The theory and application of maximal output admissible sets, IEEE Transactions on Automatic Control 36 (1991) [6] A. Jadbabaie, J. Yu and J. Hauser, Unconstrained receding horizon control of nonlinear systems, IEEE Transactions on Automatic Control 46 (2001) [7] S.S. Keerthi and E.G. Gilbert, Optimal, infinite horizon feedback laws for a general class of constrained discrete time systems: Stability and moving-horizon approximations, Journal of Optimization Theory and Applications 57 (1988) [8] D.Q. Mayne, J.A. De Doná and G.C. Goodwin, Improved stabilising conditions for model predictive control, in: Proc. 39th IEEE Conference on Decision and Control, Sydney 2000, pp [9] D.Q. Mayne, J.B. Rawlings, C.V. Rao and P.O.M. Scokaert, Constrained model predictive control: Stability and optimality, Automatica 36 (2000) [10] P.O.M. Scokaert and J.B. Rawlings, Constrained linear quadratic regulation, IEEE Transactions on Automatic Control 43 (1998) [11] M. Sznaier and M.J. Damborg, Suboptimal control of linear systems with state and control inequality constraints, in: Proc. 26th IEEE Conference on Decision and Control, Los Angeles 1987, pp