Regional Input-to-State Stability for Nonlinear Model Predictive Control

1548 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 51, NO. 9, SEPTEMBER 2006 Regional Input-to-State Stability for Nonlinear Model Predictive Control L. Magni, D. M. Raimondo, and R. Scattolini Abstract In this note, regional input-to-state stability (ISS) is introduced and studied in order to analyze the domain of attraction of nonlinear constrained systems with disturbances. ISS is derived by means of a non smooth ISS-Lyapunov function with an upper bound guaranteed only in a sub-region of the domain of attraction. These results are used to study the ISS properties of nonlinear model predictive control (MPC) algorithms. Index Terms Input-to-state stability, nonlinear model predictive control, nonlinear systems, robustness. I. INTRODUCTION Input-to-state stability (ISS) is one of the most important tools to study the dependence of state trajectories of nonlinear continuous and discrete time systems on the magnitude of inputs, which can represent control variables or disturbances. The concept of ISS was first introduced in [1] and then further exploited by many authors in view of its equivalent characterization in terms of robust stability, dissipativity and input-output stability, see e.g., [2] [6]. Now, several variants of ISS equivalent to the original one have been developed and applied in different contexts (see, e.g., [7] [9], and [3]). The ISS property has been recently introduced also in the study of nonlinear perturbed discrete-time systems controlled with model predictive control (MPC); see, e.g., [10] [13]. In fact, the development of MPC synthesis methods with enhanced robustness characteristics is motivated by the widespread success of MPC and by the availability of many MPC algorithms for nonlinear systems guaranteeing stability in nominal conditions and under state and control constraints. Different approaches have been followed so far to derive robust MPC algorithms, either by resorting to open-loop MPC formulations with restricted constraints, see [10] and [14] or to min-max open-loop and closed-loop formulations; see, e.g., [15] [19] and [12]. A recent survey on this topic is reported in [11]. In order to apply the ISS property to MPC, global results are in general not useful in view of the presence of state and input constraints. On the other hand, local results, see, e.g., [2], [20], do not allow to analyze the properties of the predictive control law in terms of its region of attraction., in this note the notion of regional ISS is initially introduced, see also [6], and the equivalence between the ISS property and the existence of a suitable Lyapunov function is established. Notably, this Lyapunov function is not required to be smooth nor to be upper bounded in the whole region of attraction. An estimation of the region where the state of the system converges asymptotically is also given. Manuscript received December 22, 2005; revised March 17, 2006 and May 25, 2006. Recommended by Associate Editor S. Celikovsky. This work was supported by the MIUR projects Advanced Methodologies for Control of Hybrid Systems and Identification and Adaptive Control of Industrial Systems. L. Magni and D. M. Raimondo are with Dipartimento di Informatica e Sistemistica, Universita di Pavia, 27100 Pavia, Italy (e-mail: lalo.magni@unipv.it; davide.raimondo@unipv.it). R. Scattolini is with Dipartimento di Elettronica e Informazione, Politecnico di Milano, 20133 Milan, Italy (e-mail: riccardo.scattolini@elet.polimi.it). Digital Object Identifier 10.1109/TAC.2006.880808 In the second part of this note, the achieved results are used to derive the ISS properties of two families of MPC algorithms for nonlinear systems. The first one relies on an open-loop formulation for the nominal system, where state and terminal constraints are modified to improve robustness, see also [10]. The second algorithm resorts to the closed-loop min max formulation already proposed in [15]. No continuity assumptions are required on the value function or on the resulting MPC control law, which indeed are difficult and not immediately verifiable hypothesis. II. NOTATIONS AND BASIC DEFINITIONS Let R; R 0 ; Z, and Z 0 denote the real, the non-negative real, the integer and the non-negative integer numbers, respectively. Euclidean norm is denoted as j1j. The set of signals taking values in some subset 9 R m is denoted by M 9, while k k := max 29fj jg. The symbol id represents the identity function from R to R, while 1 2 is the composition of two functions 1 and 2 from R to R. Given a set A R n ;d(; A):=inffj 0 j; 2 Ag is the point-to-set distance from 2R n to A. The difference between two given sets A R n and B R n with B A, is denoted by AnB := fx : x 2 A; x 2 Bg. Given a closed set A R n ;@Adenotes the border of A. Definition 1 (K-Function): A function : R 0!R 0 is of class K (or a K-function ) if it is continuous, positive definite and strictly increasing. Definition 2(K 1 -Function): A function : R 0!R 0 is of class K 1 if it is a K-function and (s)! +1 as s! +1. Definition 3(KL-Function): A function : R 0 2Z!R 0 is of class KL if, for each fixed t 0;( 1;t) is of class K, for each fixed s 0;(s; 1 ) is decreasing and (s; t)! 0 as t!1. Consider the following nonlinear discrete-time dynamic system: x(k +1)=F (x(k);w(k)); k t; x(t) =x (1) where F (0; 0) = 0;x(k) 2R n is the state, w(k) 2R r is the input (disturbance), limited in a compact set W containing the origin w(k) 2W: The transient of the system (1) with initial state x and input w is denoted by x(k; x; w);k t. Definition 4 (Robust Positively Invariant Set): A set 4 R n is a robust positively invariant set for the system (1) if F (x; w) 2 4; 8x 2 4 and 8w 2W. Definition 5 (UAG in 4): Given a compact set 4 R n including the origin as an interior point, the system (1) with w 2M W satisfies the UAG (Uniform Asymptotic Gain) property in 4,if 4 is robust positively invariant for (1) and if there exists a K-function such that for each ">0 and >0; 9T = T ("; ) such that jx(k; x; w)j(kwk)+" (2) for all x 2 4 with jxj, and all k T. Definition 6 (LS): The system (1) with w 2M W satisfies the LS (Local Stability) property if for each ">0, there exists a >0 such that for all jxj and all jw(k)j. jx(k; x; w)j" 8k t (3) 0018-9286/$20.00 2006 IEEE

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 51, NO. 9, SEPTEMBER 2006 1549 Definition 7 (ISS in 4): Given a compact set 4 R n including the origin as an interior point, the system (1) with w 2MW, is said to be ISS (Input-to-State Stable) in 4 if 4 is robust positively invariant for (1) and if there exist a KL-function and a K-function such that jx(k; x; w)j (jxj; k)+(kwk) 8k t; 8x 2 4: (4) III. REGIONAL ISS This section introduces the notion of regional ISS, useful to consider systems with constrained states and inputs, and establishes its equivalence with the existence of a suitable Lyapunov function. To this end, some preliminary results (see [7], [9] for the continuous-time and [2], [8], [20] for the discrete-time case) and assumptions have to be introduced. Lemma 1: System (1) is ISS in 4 iff it is UAG in 4 and LS. Assumption 1: The solution of (1) is continuous in w =0; x =0, with respect to inputs and initial conditions. Proposition 1: Under Assumption 1, UAG in 4 implies LS. From Lemma 1 and Proposition 1 the following result is immediately obtained. Theorem 1: System (1) is ISS in 4 iff it is UAG in 4. The ISS property is now related to the existence of a suitable a-priori nonsmooth Lyapunov function defined as follows. Definition 8: A function V : R n!r0is called an ISS-Lyapunov function in 4 for system (1), if 4 is a compact robust positively invariant set including the origin as an interior point and if there exist compact sets and D, including the origin as an interior point with D 4, some K1-functions 1; 2; 3;, some K-functions ; b = 01 4 01, with 4 = 3 01 2, such that (id 0 ) is a K1-function and the following relations hold: V (x) 1 (jxj) 8x 2 4 (5) V (x) 2(jxj) 8x 2 (6) 1V (x) :=V (F (x; w)) 0 V (x) 0 3 (jxj) +(jwj) 8x 2 4; 8w 2W (7) D := fx : d(x; @) >c V (x) b(kwk)g; c > 0: (8), the following sufficient condition for regional ISS of system (1) can be stated. Theorem 2: If (1) admits an ISS-Lyapunov function in 4, then it is ISS in 4 and limk!1 d(x(k; x; w);d)=0. Proof: Let x 2 4. First, it is going to be shown that D is robust positively invariant for system (1). To this end, assume that x(k 0 ; x; w) 2 D., V (x(k 0 ; x; w)) b(kwk); this implies 4 (V (x(k 0 ; x; w)) (kwk). Using (6) and (7) can be rewritten as 1V (x) 0 4(V (x)) + (jwj) 8x 2 ; 8w 2W where 4 = 3 01 2. Without loss of generality, assume that (id 0 4)is a K1-function, otherwise take a bigger 2 so that 3 < 2. V (x(k 0 +1; x; w)) (id 0 4 )(V (x(k 0 ; x; w))) + (jwj) (id 0 4 ) b(kwk)+(kwk) = 0(id 0 ) 4 b(kwk)+b(kwk) 0 4 b(kwk)+(kwk): Considering that 4b(kwk) =(kwk) and (id0) is a K1-function, one has V (x(k 0 +1; x; w)) 0(id 0 ) 4 b(kwk)+b(kwk) b(kwk): By induction one can show that V (x(k 0 + j; x; w)) b(kwk) for all j 2Z0, that is x(k; x; w) 2 D for all k k 0. Hence D is robust positively invariant for system (1). Now it is shown that the state reaches the region in a finite time. For x 2 nd; 4 (V (x(k; x; w))) >(kwk) and 3 01 2 (V (x(k; x; w))) >(kwk): By the fact that 01 2 (V (x(k; x; w))) jx(k; x; w)j; one has 3 (jx(k; x; w)j) >(kwk): Considering that (id 0 ) is a K1-function then id(s) >(s) 8s >0 3 (x(k; x; w)) > 3 (jx(k; x; w)j) >(kwk) which implies 8x 2 nd 0 3 (jx(k; x; w)j)+(kwk) < 0 8x 2 nd: (9) Moreover, in view of (8), 9c >0 such that 8x 1 2 4n, there exists x 2 2 nd such that 3 (jx 2 j) 3 (jx 1 j)0c. from (9) it follows that 0 3(jx 1j) +c 0 3(jx 2j) < 0(kwk) 8x 1 2 4n: 1V (x(k; x; w)) 0 3 (jx(k; x; w)j)+(kwk) < 0c 8x 2 4n so that there exists T 1 such that x(t 1; x; w) 2 : Therefore, starting from 4, the state will reach the region in a finite time. If in particular x(t 1 ; x; w) 2 D, then the previous result states that the region D is achieved in a finite time. Since D is robust positively

1550 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 51, NO. 9, SEPTEMBER 2006 invariant, it is true that lim k!1 d(x(k; x; w);d)=0. Otherwise, if x(t 1 ; x; w) =2 D; 4 (V (x(t 1 ; x; w))) >(kwk) and 1V (x(t 1 ; x; w)) 0 4 (V (x(t 1 ; x; w))) + (kwk) = 0(id 0 ) 4(V (x(t 1; x; w))) 0 4 (V (x(t 1 ; x; w))) + (kwk) 0(id 0 ) 4 (V (x(t 1 ; x; w))) 0(id 0 ) 4 1 (jx(t 1 ; x; w)j) where x(k) 2R n is the state, u(k) 2R m is the control variable, f (0; 0) = 0, and w(k) 2R n is the additive uncertainty, limited in a compact set W containing the origin w(k) 2W: (12) The state and control variables are required to fulfill the following constraints: where the last step is obtained using (5)., 8" 0 > 0; T 1 such that 9T 2 (" 0 ) x(k) 2 X (13) u(k) 2 U (14) V (x(t 2 ; x; w)) " 0 + b(kwk): (10) Therefore, starting from 4, the state will arrive close to D in a finite time and in D asymptotically. Hence lim k!1 d(x(k; x; w);d)=0. Now it is going to be shown that the system (1) is UAG in 4. Note that if kwk 6= 0; then this is true also with " 0 =0. However where X and U are compact sets of R n and R m, respectively containing the origin as an interior point. Assumption 2: Model (11) is locally Lipschitz in x in the domain X 2 U with Lipschitz constant L f jf (x 1 ;u) 0 f (x 2 ;u)j L f jx 1 0 x 2 j 8x 1 ;x 2 2 X; 8u 2 U: hence 1 (jx(t 2 ; x; w)j) V (x(t 2 ; x; w)) " 0 + b(kwk) Definition (Robust Output Admissible Set): Given a control law u = (x); X X is a robust output admissible set for the closed-loop system (11) with u(k) =(x(k)), if it is ISS in X and x 2 X implies (x(k)) 2 U; 8w(k) 2W; k t. jx(t 2 ; x; w)j 01 1 (" 0 + b(kwk)): Noting that, given a K 1 -function 1 ; 1 (s 1 +s 2 ) 1 (2s 1 )+ 1 (2s 2 ), see [12], it follows that jx(t 2 ; x; w)j 01 1 (2" 0 )+ 01 1 (2b(kwk)): Now, letting " = 01 1 (2"0 ) and (kwk) = 01 1 (2b(kwk)) the UAG property in 4 is proven. Finally, in view of Theorem 1, the ISS in 4 is proven. Remark 1: Theorem 2 gives an estimation of the region D where the state of the system converges asymptotically. This region depends on the bound on w through, as well as on 2; 3 and. Ifw = 0, then (kwk) = 0 so that asymptotic stability is guaranteed (for this particular case see [21]). Moreover, since the size of D is directly related to 2, an accurate estimation of the upper bound 2 is useful in order to compute a smaller region of attraction D. IV. NONLINEAR MODEL PREDICTIVE CONTROL In this section, the results derived in Theorem 2 are used to analyze the ISS property of open-loop and closed-loop min-max formulations of stabilizing MPC for nonlinear systems. Notably, in the following it is not necessary to assume the regularity of the value function and of the resulting control law. Consider the nonlinear perturbed discrete-time dynamic system A. Open-Loop MPC Formulation In order to introduce the MPC algorithm formulated according to an open-loop strategy, first let u t ;t := [u(t 1 ) u(t 1 + 1);...;u(t 2)]; t 2 t 1., the following finite-horizon optimization problem can be stated. Definition 10 (FHOCP): Given the positive integer N, the stage cost l, the terminal penalty V f and the terminal set X f, the finite horizon optimal control problem (FHOCP) consists in minimizing, with respect to u t;, the performance index J(x; u t; ;N):= l(x(k);u(k)) + V f (x(t + N )) subject to i) the nominal state dynamics (11) with w =0and x(t) =x; ii) the constraints (13) (14), k 2 [t; t + N 0 1]; iii) the terminal state constraints x(t + N ) 2 X f. It is now possible to define a prototype of a nonlinear MPC algorithm: At every time instants t, define x = x(t) and find the optimal control sequence u o t; by solving the FHOCP., according to the Receding Horizon (RH) strategy, define MPC (x) =u o t;t(x) where u o t;t(x) is the first column of u o t;, and apply the control law x(k +1)=f (x(k);u(k)) + w(k); k t; x(t) =x (11) u = MPC (x): (15)

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 51, NO. 9, SEPTEMBER 2006 1551 Although the FHOCP has been stated for nominal conditions, under suitable assumptions and by choosing accurately the terminal cost function V f and the terminal constraint X f, it is possible to guarantee the ISS property of the closed-loop system formed by (11) and (15), subject to constraints (12) (14). Assumption 3: The function l(x; u) is such that l(0; 0) = 0; l(x; u) l (jxj) where l is a K1-function. Moreover, l(x; u) is Lipschitz in x, inx 2 U, with constant L l, i.e., Moreover, in view of Assumption 4 ~u t;t+n =[u o t;; f (x(t + N ))] is an admissible, possible suboptimal, control sequence for the FHOCP with horizon N +1at time t with cost jl(x 1;u) 0 l(x 2;u)j L l jx 1 0 x 2j 8x 1;x 2 2 X 8u 2 U: Assumption 4: The design parameters V f ;X f are such that, given an auxiliary control law f 1) X f X; X f closed, 0 2 X f ; 2) f (x) 2 U; 8x 2 X f ; 3) f (x; f (x)) 2 X f ; 8x 2 X f ; 4) V (jxj) V f (x) < V (jxj) where V and V are K1-functions; 5) V f (f(x; f (x))) 0 V f (x) 0l(x; f (x)); 8x 2 X f ; 6) V f is Lipschitz in X f with a Lipschitz constant L V. Assumption 5: The set W is such that (8) is satisfied, with V (x) := J(x; u o t;;n); 3 = l ; 2 = V ; =X f ; = L J, where L J = L V L N01 f + L l (L N01 f 0 1=L f 0 1). Assumption 6: The set X MPC (N ) of states such that a solution of the FHOCP exists is a robust positively invariant set for the closed-loop system (11), (15). Remark 2: Many methods have been proposed in the literature to compute V f ;X f satisfying Assumption 4 (see, e.g., [22]). However, with the MPC based on the FHOCP defined previously, Assumption 6 is not a-priori satisfied. A way to fulfill it, is shown in [10] by properly restricting the state constraints ii) and iii) in the formulation of the FHOCP. The following preliminary result must be introduced first. Lemma 3: Under Assumptions 3 5, (id 0 4 ) is a K1-function. Proof: From Assumption 3 and point 5 of Assumption 4 V f (f(x; f (x))) 0 V f (x) 0l(x; f (x)) 0 l (jxj) l (jxj) V f (x) 0 V f (f(x; f (x))) V f (x) < V (jxj) and 3 < 2, so that the result follows. The main result can now be stated. Theorem 3: Under Assumptions 1 6 the closed-loop system formed by (11) and (15) subject to constraints (12), (13), (14) is ISS with robust output admissible set X MPC (N). Proof: By Theorem 2, if system admits an ISS-Lyapunov function in X MPC (N), then it is ISS in X MPC (N). In the following, it will be shown that V (x; N ) := J(x; u o t;;n) is an ISS-Lyapunov function in X MPC (N). To this end, first note that V (x; N ) l(x; MPC (x)) l (jxj) 8x 2 X MPC (N): (16) J(x; ~u t;t+n ;N +1)=V (x; N ) 0 V f (x(t + N )) + V f (x(t + N + 1)) + l(x(t + N ); f (x(t + N ))): Since ~u t;t+n is a suboptimal sequence V (x; N +1) J(x; ~u t;t+n ;N +1): However, using point 5 of Assumption 4, it follows that J(x; ~u t;t+n ;N +1) = V (x; N ) 0 V f (x(t + N )) + V f (x(t + N + 1)) + l(x(t + N ); f (x(t + N ))) V (x; N ): V (x; N +1) V (x; N ) with V (x; 0) = V f (x); 8x 2 X f. Therefore 8x 2 X MPC (N) V (x; N ) V (x; N 0 1) V (x; 0) = V f (x) < V (jxj) 8x 2 X f : (17) Finally, following the proof of [10, Th. 2] it is possible to show that 1V (x; N ) 0 l (jxj) +L Jjwj 8x 2 X MPC (N) 8w 2W (18) where L J = L V L N01 f + L l (L N01 f 0 1=L f 0 1). Therefore, by (16) (18) and under Assumption 6, the optimal cost J(x; ut;;n) o is an ISS-Lyapunov function for the closed-loop system in X MPC (N) and, hence, the closed-loop system (11), (15) is ISS with robust output admissible set X MPC (N). B. Closed-Loop Min Max Optimization As underlined in Remark 2, the positively invariance of the feasible set X MPC (N) in a standard open-loop MPC formulation can be achieved through a wise choice of the constraints ii) and iii) in the FHOCP. However, this solution can be extremely conservative and can provide a small robust output admissible set, so that a less stringent approach explicitly accounting for the intrinsic feedback nature of any RH implementation has been proposed, see, e.g., [15] [18], [12]. In the following, it is shown that the ISS result of the previous section is also useful to derive the ISS property of min max MPC. In this framework, at any time instant the controller chooses the input u as

1552 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 51, NO. 9, SEPTEMBER 2006 a function of the current state x, so as to guarantee that the effect of the disturbance w is compensated for any choice made by the nature. Hence, instead of optimizing with respect to a control sequence, at any time t the controller has to choose a vector of feedback control policies t; =[ 0 (x(t)); 1 (x(t + 1))... N01(x(t + N 0 1))] minimizing the cost in the worst disturbance case., the following optimal min max problem can be stated. Definition 11 [FHCDG]: Given the positive integer N, the stage cost l 0 l w, the terminal penalty V f and the terminal set X f, the finite horizon closed-loop differential game (FHCDG) problem consists in minimizing, with respect to t; and maximizing with respect to w t; the cost function J(x; t;;w t;;n): = fl(x(k);u(k)) 0 l w (w(k))g + V f (x(t + N )) (19) subject to i) the state dynamics (11) with x(t) =x; ii) the constraints (12) (14), k 2 [t; t + N 0 1]; iii) the terminal state constraint x(t + N ) 2 X f. Letting o t;;w o t; be the solution of the FHCDG, according to the RH paradigm, the feedback control law u = MPC (x) is obtained by setting MPC (x) = o 0(x) (20) where o 0(x) is the first element of o t;. In order to derive the main stability and performance properties associated to the solution of FHCDG, the following assumptions are introduced. Assumption 7: l w(w) is such that w(jwj) l w(w) w(jwj), where w and w are K1-functions. Assumption 8: The design parameters V f and X f are such that, given an auxiliary law f 1) X f X; X f closed, 0 2 X f ; 2) f (x) 2 U; 8x 2 X f ; 3) f (x; f (x)) + w 2 X f ; 8x 2 X f ; 8w 2W; 4) V (jxj) V f (x) < V (jxj), where V and V are K1-functions; 5) V f (f(x; f (x)) + w) 0 V f (x) 0l(x; f (x)) + l w (w);8x 2 X f ; 8w 2W; 6) V f is Lipschitz in X f with Lipschitz constant L Vf. Assumption 9: The set W is such that (8) is satisfied, with V (x) = J(x; o t;;w o t;;n); 3 = l ; 2 = V ; = X f ; = w. The following preliminary result must be introduced first. Lemma 3: If V is chosen such that l < V, then (id 0 4 ) is a K1-function. The main result can now be stated. Theorem 4: Let X MPC (N) be the set of states of the system where there exists a solution of the FHCDG. Under Assumptions 1, 3, 7, 8, and 9, the closed-loop system formed by (11) and (20) subject to constraints (12), (13), (14) is ISS with robust output admissible set X MPC (N). Proof: The robust positively invariance of X MPC (N) is easily derived from Assumption 8 by taking t+1;t+n = o t+1; t +1 k t + N 0 1 f (x(t + N )) k = t + N as admissible policy vector at time t +1starting from the optimal sequence o t; at time t. Moreover, it is possible to show that V (x; N ):=J(x; o t;;w o t;;n) is an ISS-Lyapunov function for the closed-loop system (11), (20). In fact V (x; N ):=J x; o t;;w o t;;n min l(x; 0 (x)) l (jxj) J(x; t;; 0;N) 8x 2 X MPC (N): (21) In order to derive the upper bound, consider the following policy vector for the FHCDG with horizon: N +1 ~ t;t+n = o t; t k t + N 0 1 f (x(t + N )) k = t + N: J(x; ~ t;t+n ;w t;t+n ;N +1) = V f (x(t + N + 1)) 0 V f (x(t + N )) + l(x(t + N );u(t + N )) 0 l w (w(t + N )) + fl(x(k);u(k)) 0 l w(w(k))g + V f (x(t + N )) so that in view of Assumption 8 which implies J(x; ~ t;t+n ;w t;t+n ;N +1) fl(x(k);u(k)) 0 l w (w(k))g + V f (x(t + N )) V (x; N +1) max w2m J(x; ~ t;;w t;;n +1) max fl(x(k);u(k)) 0 l w (w(k))g w2m + V f (x(t + N )) = V (x; N ) (22) which holds 8x 2 X MPC (N); 8w 2MW. Moreover V (x; N ) V (x; N 0 1)... V (x; 0) = V f (x) < V (jxj) 8x 2 X f : (23) From the monotonicity property (22), it is easily derived that V (f(x; MPC (x)) + w; N) 0 V (x; N ) 0l(x; MPC (x)) + l w(w) 0 l (jxj) + w (jwj) 8x 2 X MPC (N); 8w 2W: (24), by (21), (23), and (24), the ISS is proven in X MPC (N).

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 51, NO. 9, SEPTEMBER 2006 1553 Remark 3: The computation of the auxiliary control law, of the terminal penalty and of the terminal inequality constraint satisfying Assumption 8, is not trivial at all. In this regard, a solution has been proposed for affine system in [15], where it is shown how to compute a non linear auxiliary control law based on the solution of a suitable H1 problem for the linearized system under control. Remark 4: The usual way to derive the upper bound for the value function in X MPC (N ) requires the assumption that the solution u o t; of the FHOCP and o t; of the FHCDG are Lipschitz in X MPC (N). On the contrary, Theorem 2 gives the possibility to find the upper bound for the ISS Lyapunov function only in a subset of the robust output admissible set. This can be derived in X f, without assuming any regularity of the control strategies, by using the monotonicity properties (17) and (23) respectively. However, in order to enlarge the set W that satisfies Assumptions 5 and 9 for the FHOCP and FHCDG, respectively, it could be useful to find an upper bound 2 of V in a region 1. To this regard, define 2 =max V 2(r) ; 1 2 where V = max x2 (V (x)) and r is such that B r = fx 2R n : jxj rg, as suggested in [12]. This idea can either enlarge or restrict the set W since 1 but 2 2. Remark 5: Following the results reported in [23] for the open-loop and in [15] for the closed-loop min-max MPC formulations, it is easy to show that the robust output admissible sets guaranteed by the NMPC control law include the terminal region X f used in the optimization problem. Moreover the robust output admissible set guaranteed with a longer optimization horizon includes the one obtained with a shorter horizon, i.e., X MPC (N +1) X MPC (N) X f. V. CONCLUSION Using a suitable, non-necessarily continuous, Lyapunov function, regional ISS for discrete-time nonlinear constrained systems has been established. Moreover an estimation of the region where the state of the system converges asymptotically is given. This result has been used to study the robustness characteristics of MPC algorithms derived according to open-loop and closed-loop min-max formulations. No continuity assumptions on the optimal Receding Horizon control law have been required. It is believed that these results can be used to improve the analysis of existing MPC algorithms as well as to develop new synthesis methods with enhanced robustness properties. REFERENCES [7] E. D. Sontag and Y. Wang, New characterizations of input-to statestability, IEEE Trans. Autom. Control, vol. 41, no. 9, pp. 1283 1294, Sep. 1996. [8] K. Gao and Y. Lin, On equivalent notions of input-to-state stability for nonlinear discrete time systems, in Proc. IASTED Int. Conf. Control and Applications, Cancun, Mexico, May 2000, pp. 81 86. [9] E. D. Sontag and Y. Wang, On characterizations of the input-to-state stability property, Syst. 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