Input-To-State Stability for Nonlinear Model Predictive ... - CiteSeerX

Proceedings of the 45th IEEE Conference on Decision & Control Manchester Grand Hyatt Hotel San Diego, CA, USA, December 13-15, 2006

FrA07.3

Input-to-State Stability for Nonlinear Model Predictive Control L. Magni, D. M. Raimondo and R. Scattolini

Abstract— In this paper 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 ISSLyapunov 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.

I. I NTRODUCTION 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], [3], [4], [5], [6], [7]. Now, several variants of ISS equivalent to the original one have been developed and applied in different contexts (see e.g. [8], [9], [10], [3]). The ISS property has been recently introduced also in the study of nonlinear perturbed discrete-time systems controlled with Model Predictive Control (M P C), see e.g. [11], [12], [13], [14]. 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 openloop MPC formulations with restricted constraints, see [11], [15] or to min-max open-loop and closed-loop formulations, see e.g. [16], [17], [18], [19], [13], [20]. A recent survey on this topic is reported in [12]. This paper is devoted to present some general results related to ISS of nonlinear constrained discrete-time systems; then, these results are used in the framework of MPC. Specifically, the notion of regional ISS is initially introduced to deal with all the cases where the system state and input variables are constrained to remain in some finite sets. Then, the equivalence between the ISS property and The authors acknowledge the financial support of 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, via Ferrata 1, 27100 Pavia, Italy. E-mail: {lalo.magni, davide.raimondo}@unipv.it. R. Scattolini is with Dipartimento di Elettronica e Informazione, Politecnico di Milano, piazza Leonardo da Vinci 32, 20133 Milano, Italy. E-mail:

[email protected]

1-4244-0171-2/06/$20.00 ©2006 IEEE.

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. In the second part of the paper, 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 [11]. The second algorithm resorts to the closed-loop min-max formulation already proposed in [16]. 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. N OTATIONS 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 | · |. The set of signals ψ taking values in some subset Ψ ⊂ Rm is denoted by MΨ , while ψ := maxψ∈Ψ {|ψ|}. The symbol id represents the identity function from R to R, while γ1 ◦γ2 is the composition of two functions γ1 and γ2 from R to R. Given a set A ⊆ Rn , d(ζ, A) := inf {|η − ζ| , η ∈ A} is the point-to-set distance from ζ ∈ Rn to A. The difference between two given sets A ⊆ Rn and B ⊆ Rn with B ⊆ A, is denoted by A\B := {x : x ∈ A, x ∈ / B}. Given a closed set A ⊆ Rn , ∂A denotes the border of A. Definition 1 (K-function): A function γ : R≥0 →R≥0 is of class K (or a ”K-function”) if it is continuous, positive definite and strictly increasing. Definition 2 (K∞ -function): A function γ : R≥0 →R≥0 is of class K∞ if it is a K-function and γ(s) → +∞ as s → +∞. Definition 3 (KL-function): A function β : R≥0 ×Z≥0 → R≥0 is of class KL if, for each fixed t ≥ 0, β(·, t) is of class K, for each fixed s ≥ 0, β(s, ·) is decreasing and β(s, t) → 0 as t → ∞. 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) ∈ Rn is the state, w(k) ∈ Rr is the input (disturbance), limited in a compact set W containing the origin w(k) ∈ W.

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45th IEEE CDC, San Diego, USA, Dec. 13-15, 2006

FrA07.3

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 Ξ ⊆ Rn is a robust positively invariant set for the system (1) if F (x, w) ∈ Ξ, ∀x ∈ Ξ and ∀w ∈ W. Definition 5 (UAG in Ξ): Given a compact set Ξ ⊂ Rn including the origin as an interior point, the system (1) with w ∈ MW satisfies the U AG (Uniform Asymptotic Gain) property in Ξ, if Ξ is robust positively invariant for (1) and if there exists a K-function γ such that for each ε > 0 and ν > 0, ∃T = T (ε, ν) such that |x(k, x ¯, w)| ≤ γ(w) + ε

(2)

for all x ¯ ∈ Ξ with |¯ x| ≤ ν, and all k ≥ T . Definition 6 (LS): The system (1) with w ∈ MW satisfies the LS (Local Stability) property if for each ε > 0, there exists a δ > 0 such that |x(k, x ¯, w)| ≤ ε, ∀k ≥ t

V (x) ≤ α2 (|x|), ∀x ∈ Ω ∆V (x)

∆V (x) ≤ −α4 (V (x)) + σ(|w|), ∀x ∈ Ω, ∀w ∈ W where α4 = α3 ◦ α2−1 . Without loss of generality, assume that (id − α4 ) is a K∞ -function, otherwise take a bigger α2 so that α3 < α2 . Then ¯, w)) ≤ (id − α4 )(V (x(k0 , x ¯, w))) V (x(k0 + 1, x +σ(|w|) ≤ (id − α4 ) ◦ b(||w||) + σ(w) = −(id − ρ) ◦ α4 ◦ b(||w||) +b(||w||) − ρ ◦ α4 ◦ b(||w||) +σ(w).

|x(k, x ¯, w)| ≤ β(|¯ x|, k) + γ(w), ∀k ≥ t, ∀¯ x ∈ Ξ. (4) III. R EGIONAL 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 [8], [10] for the continuous time and [2], [9], [21] for the discrete time case) and assumptions have to be introduced. Lemma 1: The system (1) is ISS in Ξ iff it is U AG in Ξ and LS. Assumption 1: The solution of system (1) is continuous in w = 0, x ¯ = 0, with respect to inputs and initial conditions. Proposition 1: Under Assumption 1, U AG in Ξ implies LS. From Lemma 1 and Proposition 1 the following result is immediately obtained. Theorem 1: The system (1) is ISS in Ξ iff it is U AG in Ξ. The ISS property is now related to the existence of a suitable a-priori non smooth Lyapunov function defined as follows. Definition 8: A function V : Rn → R≥ 0 is called an ISS-Lyapunov function in Ξ for system (1), if Ξ is a compact robust positively invariant set including the origin as an interior point and if there exist compact sets Ω and D, including the origin as an interior point with D ⊂ Ω ⊆ Ξ, some K∞ -functions α1 , α2 , α3 , ρ, some K-functions σ, b = α4−1 ◦ ρ−1 ◦ σ, with α4 = α3 ◦ α2−1 , such that (id − ρ) is a K∞ -function and the following relations hold V (x) ≥ α1 (|x|), ∀x ∈ Ξ

:= V (F (x, w)) − V (x) ≤ −α3 (|x|) + σ(|w|), ∀x ∈ Ξ, ∀w ∈ W

(7) D := {x : d(x, ∂Ω) > c, V (x) ≤ b(w)} ⊂ Ω, c > 0. (8) Then, the following sufficient condition for regional ISS of system (1) can be stated. Theorem 2: If the system (1) admits an ISSLyapunov function in Ξ, then it is ISS in Ξ and limk→∞ d(x(k, x ¯, w), D) = 0. Proof: Let x ¯ ∈ Ξ. First, it is going to be shown that D is robust positively invariant for system (1). To this end, assume that x(k0 , x ¯, w) ∈ D. Then V (x(k0 , x ¯, w)) ≤ b(w); this implies ρ ◦ α4 (V (x(k0 , x ¯, w)) ≤ σ(w). Using (6), (7) can be rewritten as

(3)

for all |¯ x| ≤ δ and all |w(k)| ≤ δ. Definition 7 (ISS in Ξ): Given a compact set Ξ ⊂ Rn including the origin as an interior point, the system (1) with w ∈ MW , is said to be ISS (Input-to-State Stable) in Ξ if Ξ is robust positively invariant for (1) and if there exist a KL-function β and a K-function γ such that

(6)

Considering that ρ ◦ α4 ◦ b(||w||) = σ(w) and (id − ρ) is a K∞ -function, one has ¯, w)) ≤ V (x(k0 + 1, x

−(id − ρ) ◦ α4 ◦ b(||w||) +b(||w||) ≤ b(||w||).

By induction one can show that V (x(k0 +j, x ¯, w)) ≤ b(w) for all j ∈ Z≥0 , that is x(k, x ¯, w) ∈ D for all k ≥ k0 . Hence D is robust positively invariant for system (1). Now it is shown that the state reaches the region Ω in a finite time. For x ∈ Ω\D, ρ ◦ α4 (V (x(k, x ¯, w))) > σ(w) and ρ ◦ α3 ◦ α2−1 (V (x(k, x ¯, w))) > σ(w). By the fact that α2−1 (V (x(k, x ¯, w))) ≤ |x(k, x ¯, w)|, one has ρ ◦ α3 (|x(k, x ¯, w)|) > σ(w). Considering that (id − ρ) is a K∞ -function id(s) > ρ(s), ∀s > 0 then α3 (x(k, x ¯, w)) > ρ◦α3 (|x(k, x ¯, w)|) > σ(w), ∀x ∈ Ω\D which implies

(5)

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−α3 (|x(k, x ¯, w)|) + σ(w) < 0, ∀x ∈ Ω\D.

(9)


FrA07.3

Moreover, in view of (8), ∃¯ c > 0 such that ∀x1 ∈ Ξ\Ω, there exists x2 ∈ Ω\D such that α3 (|x2 |) ≤ α3 (|x1 |) − c¯. Then from (9) it follows that −α3 (|x1 |) + c¯ ≤ −α3 (|x2 |) < −σ(w), ∀x1 ∈ Ξ\Ω. Then ∆V (x(k, x ¯, w)) ≤ −α3 (|x(k, x ¯, w)|) + σ(w) < −¯ c, ∀x ∈ Ξ\Ω so that there exists T1 such that

IV. N ONLINEAR M ODEL P REDICTIVE C ONTROL In this section, the results derived in Theorem 2 are used to analyze the ISS property of open-loop and closed-loop minmax formulations of stabilizing M P C 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 x(k + 1) = f (x(k), u(k)) + w(k), k ≥ t, x(t) = x ¯ (11)

x(T1 , x ¯, w) ∈ Ω. Therefore, starting from Ξ, the state will reach the region Ω in a finite time. If in particular x(T1 , x ¯, w) ∈ D, then the previous result states that the region D is achieved in a finite time. Since D is robust positively invariant, it is true that limk→∞ d(x(k, x ¯, w), D) = 0. Otherwise, if x(T1 , x ¯, w) ∈ / D , ρ ◦ α4 (V (x(T1 , x ¯, w))) > σ(w) and

where x(k) ∈ Rn is the state, u(k) ∈ Rm is the control variable, f (0, 0) = 0, and w(k) ∈ Rn is the additive uncertainty, limited in a compact set W containing the origin

∆V (x(T1 , x ¯, w)) ≤ −α4 (V (x(T1 , x ¯, w))) + σ(w) ¯, w))) = −(id − ρ) ◦ α4 (V (x(T1 , x −ρ ◦ α4 (V (x(T1 , x ¯, w))) + σ(w) ¯, w))) ≤ −(id − ρ) ◦ α4 (V (x(T1 , x ≤ −(id − ρ) ◦ α4 ◦ α1 (|x(T1 , x ¯, w)|)

x(k) ∈ X

(13)

u(k) ∈ U

(14)

where the last step is obtained using (5). Then, ∀ε > 0, ∃T2 (ε ) ≥ T1 such that V (x(T2 , x ¯, w)) ≤ ε + b(w).

(10)

Therefore, starting from Ξ, the state will arrive close to D in a finite time and in D asymptotically. Hence limk→∞ d(x(k, x ¯, w), D) = 0. Now it is going to be shown that the system (1) is U AG in Ξ. Note that if w = 0, then this is true also with ε = 0. But α1 (|x(T2 , x ¯, w)|) ≤ V (x(T2 , x ¯, w)) ≤ ε + b(w) hence

|x(T2 , x ¯, w)| ≤ α1−1 (ε + b(w)).

Noting that, given a K∞ -function θ1 , θ1 (s1 +s2 ) ≤ θ1 (2s1 )+ θ1 (2s2 ), see [13], it follows that |x(T2 , x ¯, w)| ≤ α1−1 (2ε ) + α1−1 (2b(w)). Now, letting ε = α1−1 (2ε ) and γ(w) = α1−1 (2b(w)) the U AG property in Ξ is proven. Finally, in view of Theorem 1, the ISS in Ξ 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 σ, as well as on α2 , α3 and ρ. If w = 0, then σ(||w||) = 0 so that asymptotic stability is guaranteed (for this particular case see [22]). Moreover, since the size of D is directly related to α2 , an accurate estimation of the upper bound α2 is useful in order to compute a smaller region of attraction D.

w(k) ∈ W.

(12)

The state and control variables are required to fulfill the following constraints

where X and U are compact sets of Rn and Rm , respectively containing the origin as an interior point. Assumption 2: Model (11) is locally Lipschitz in x in the domain X × U with Lipschitz constant Lf |f (x1 , u) − f (x2 , u)| ≤ Lf |x1 − x2 |, ∀x1 , x2 ∈ X, ∀u ∈ U. Definition 9 (Robust output admissible set): Given a con¯ ⊆ X is a robust output admissible set trol law u = κ(x), X for the closed-loop system (11) with u(k) = κ(x(k)), if it ¯ and x ¯ implies κ(x(k)) ∈ U, ∀w(k) ∈ is ISS in X ¯ ∈ X W, k ≥ t. A. Open-loop MPC formulation In order to introduce the M P C algorithm formulated according to an open-loop strategy, first let ut1 ,t2 := [u(t1 ) u(t1 +1)... u(t2 )], t2 ≥ t1 . Then, the following finitehorizon optimization problem can be stated. Definition 10 (FHOCP): Given the positive integer N , the stage cost l, the terminal penalty Vf and the terminal set Xf , the Finite Horizon Optimal Control Problem (F HOCP ) consists in minimizing, with respect to ut,t+N −1 , the performance index J(¯ x, ut,t+N −1 , N ) :=

t+N −1

l(x(k), u(k)) + Vf (x(t + N ))

k=t

subject to (i) the nominal state dynamics (11) with w = 0 and x(t) = x ¯; (ii) the constraints (13)-(14), k ∈ [t, t + N − 1]; (iii) the terminal state constraints x(t + N ) ∈ Xf . It is now possible to define a “prototype” of a nonlinear M P C algorithm: at every time instants t, define x ¯ = x(t) and find the optimal control sequence uot,t+N −1 by solving

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the F HOCP . Then, according to the Receding Horizon (RH) strategy, define κM P C (¯ x) = uot,t (¯ x) x) is the first column of uot,t+N −1 , and apply the where uot,t (¯ control law u = κM P C (x). (15) Although the F HOCP has been stated for nominal conditions, under suitable assumptions and by choosing accurately the terminal cost function Vf and the terminal constraint Xf , it is possible to guarantee the ISS property of the closedloop system formed by (11) and (15), subject to constraints (12), (13), (14). Assumption 3: The function l(x, u) is such that l(0, 0) = 0, l(x, u) ≥ αl (|x|) where αl is a K∞ -function. Moreover, l(x, u) is Lipschitz in x, in X × U , with constant Ll , i.e. |l(x1 , u) − l(x2 , u)| ≤ Ll |x1 − x2 |, ∀x1 , x2 ∈ X, ∀u ∈ U. Assumption 4: The design parameters Vf , Xf are such that, given an auxiliary control law κf , 1) Xf ⊆ X, Xf closed, 0 ∈ Xf ; 2) κf (x) ∈ U, ∀x ∈ Xf ; 3) f (x, κf (x)) ∈ Xf , ∀x ∈ Xf ; 4) αVf (|x|) ≤ Vf (x) < βVf (|x|) where αVf and βVf are K∞ -functions; 5) Vf (f (x, κf (x))) − Vf (x) ≤ −l(x, κf (x)), ∀x ∈ Xf ; 6) Vf is Lipschitz in Xf with a Lipschitz constant LVf . Assumption 5: The set W is such that (8) is satisfied, with V (x) := J(x, uot,t+N −1 , N ), α3 = αl , α2 = βVf , Ω = Xf , LN −1 −1

−1 f + Ll L . σ = LJ , where LJ = LVf LN f f −1 Assumption 6: The set X M P C (N ) of states such that a solution of the F HOCP 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 Vf , Xf satisfying Assumption 4 (see e.g [23]). However, with the M P C based on the F HOCP defined above, Assumption 6 is not a-priori satisfied. A way to fulfill it, is shown in [11] by properly restricting the state constraints (ii) and (iii) in the formulation of the F HOCP . The following preliminary result must be introduced first. Lemma 2: Under Assumptions 3-5, (id − α4 ) is a K∞ function. Proof : From Assumption 3 and point 5 of Assumption 4

Vf (f (x, κf (x))) − Vf (x) ≤ −l(x, κf (x)) ≤ −αl (|x|)

V (x, N ) ≥ l(x, κM P C (x)) ≥ αl (|x|), ∀x ∈ X M P C (N ). (16) Moreover, in view of Assumption 4 u ˜t,t+N = [uot,t+N −1 , κf (x(t + N ))] is an admissible, possible suboptimal, control sequence for the F HOCP with horizon N + 1 at time t with cost J(x, u ˜t,t+N , N + 1)

= V (x, N ) − Vf (x(t + N )) +Vf (x(t + N + 1)) +l(x(t + N ), κf (x(t + N ))).

Since u ˜t,t+N is a suboptimal sequence V (x, N + 1) ≤ J(x, u ˜t,t+N , N + 1). But using point 5 of Assumption 4, it follows that J(x, u ˜t,t+N , N + 1)

= V (x, N ) − Vf (x(t + N )) +Vf (x(t + N + 1)) +l(x(t + N ), κf (x(t + N ))) ≤ V (x, N ).

Then V (x, N + 1) ≤ V (x, N ), ∀x ∈ X M P C (N ) with V (x, 0) = Vf (x), ∀x ∈ Xf . Therefore V (x, N ) ≤ V (x, N − 1) ≤ V (x, 0) = Vf (x) < βVf (|x|) , ∀x ∈ Xf .

(17)

Finally, following the proof of Theorem 2 in [11] it is possible to show that ∆V (x, N ) ≤ −αl (|x|)+LJ |w|, ∀x ∈ X M P C (N ), ∀w ∈ W (18) −1 −1 LN N −1 f where LJ = LVf Lf + Ll Lf −1 . Therefore, by (16), (17), (18) and under Assumption 6, the optimal cost J(x, uot,t+N −1 , N ) is an ISS-Lyapunov function for the closed-loop system in X M P C (N ) and hence the closed-loop system (11), (15) is ISS with robust output admissible set X M P C (N ). B. Closed-loop min-max optimization

Then αl (|x|) ≤ Vf (x) − Vf (f (x, κf (x))) ≤ Vf (x) < βVf (|x|) and α3 < α2 , so that the result follows.

Proof : By Theorem 2, if system admits an ISS-Lyapunov function in X M P C (N ), then it is ISS in X M P C (N ). In the following, it will be shown that V (x, N ) := J(x, uot,t+N −1 , N ) is an ISS-Lyapunov function in X M P C (N ). To this end, first note that

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 M P C (N ).

As underlined in Remark 2, the positively invariance of the feasible set X M P C (N ) in a standard open-loop M P C formulation can be achieved through a wise choice of the constraints (ii) and (iii) in the F HOCP . 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. [16], [17], [18], [19], [13]. In the following, it is shown that the ISS

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result of the previous section is also useful to derive the ISS property of min-max M P C. In this framework, at any time instant the controller chooses the input u as 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 κt,t+N −1 = [κ0 (x(t)) κ1 (x(t + 1) . . . κN −1 (x(t + N − 1)] minimizing the cost in the worst disturbance case. Then, the following optimal min-max problem can be stated. Definition 11 (FHCDG): Given the positive integer N , the stage cost l−lw , the terminal penalty Vf and the terminal set Xf , the Finite Horizon Closed-loop Differential Game (F HCDG) problem consists in minimizing, with respect to κt,t+N −1 and maximizing with respect to wt,t+N −1 the cost function J(¯ x, κt,t+N −1 , wt,t+N −1 , N ) t+N −1 {l(x(k), u(k)) − lw (w(k))} + Vf (x(t + N )) := k=t

(19) subject to: (i) the state dynamics (11) with x(t) = x ¯; (ii) the constraints (13)-(14), k ∈ [t, t + N − 1]; (iii) the terminal state constraint x(t + N ) ∈ Xf . o Letting κot,t+N −1 , wt,t+N −1 be the solution of the F HCDG, according to the RH paradigm, the feedback control law u = κM P C (x) is obtained by setting κM P C (x) = κo0 (x) κo0 (x)

κot,t+N −1 .

(11) and (20) subject to constraints (12), (13), (14) is ISS with robust output admissible set X M P C (N ). Proof : The robust positively invariance of X M P C (N ) is easily derived from Assumption 8 by taking κot+1,t+N −1 t + 1 ≤ k ≤ t + N − 1 κ ¯ t+1,t+N = κf (x(t + N )) k =t+N as admissible policy vector at time t + 1 starting from the optimal sequence κot,t+N −1 at time t. Moreover it is possible o to show that V (x, N ) := J(¯ x, κot,t+N −1 , wt,t+N −1 , N ) is an ISS-Lyapunov function for the closed-loop system (11), (20). In fact V (x, N )

≥ ≥

min J(¯ x, κt,t+N −1 , 0, N )

κt,t+N −1

l(x, κf (x)) ≥ αl (|x|), ∀x ∈ X M P C (N ).

(21)

In order to derive the upper bound, consider the following policy vector for the F HCDG with horizon N + 1 κot,t+N −1 t≤k ≤t+N −1 κ ˜ t,t+N = κf (x(t + N )) k =t+N Then J(¯ x, κ ˜ t,t+N , wt,t+N , N + 1) = Vf (x(t + N + 1)) − Vf (x(t + N )) +l(x(t + N ), u(t + N )) − lw (w(t + N )) t+N −1 {l(x(k), u(k)) − lw (w(k))} + Vf (x(t + N )) +

(20)

where is the first element of In order to derive the main stability and performance properties associated to the solution of F HCDG, the following assumptions are introduced. Assumption 7: lw (w) is such that αw (|w|) ≤ lw (w) ≤ βw (|w|) , where αw and βw are K∞ -functions. Assumption 8: The design parameters Vf and Xf are such that, given an auxiliary law κf , 1) Xf ⊆ X, Xf closed, 0 ∈ Xf ; 2) κf (x) ∈ U, ∀x ∈ Xf ; 3) f (x, κf (x)) + w ∈ Xf , ∀x ∈ Xf , ∀w ∈ W; 4) αVf (|x|) ≤ Vf (x) < βVf (|x|) , where αVf and βVf are K∞ -functions; 5) Vf (f (x, κf (x))+w)−Vf (x) ≤ −l(x, κf (x))+lw (w), ∀x ∈ Xf , ∀w ∈ W; 6) Vf is Lipschitz in Xf with Lipschitz constant LV f . Assumption 9: The set W is such that (8) is satisfied, with o V (x) = J(¯ x, κot,t+N −1 , wt,t+N −1 , N ), α3 = αl , α2 = βVf , Ω = Xf , σ = βw . The following preliminary result must be introduced first. Lemma 3: If βVf is chosen such that αl < βVf , then (id− α4 ) is a K∞ -function. The main result can now be stated. Theorem 4: Let X M P C (N ) be the set of states of the system where there exists a solution of the F HCDG. Under Assumptions 1, 3, 7, 8, 9 the closed loop system formed by

o := J(¯ x, κot,t+N −1 , wt,t+N −1 , N )

k=t

so that in view of Assumption 8 J(¯ x, κ ˜ t,t+N , wt,t+N , N + 1) t+N −1 ≤ {l(x(k), u(k)) − lw (w(k))} + Vf (x(t + N )) k=t

which implies V (x, N + 1) ≤ max J(¯ x, κ ˜ t,t+N −1 , wt,t+N −1 , N + 1) w∈MW

≤ max

w∈MW

t+N −1

{l(x(k), u(k)) − lw (w(k))}

k=t

+Vf (x(t + N )) = V (x, N )

(22)

which holds ∀x ∈ X M P C (N ), ∀w ∈ MW . Moreover V (x, N ) ≤ V (x, N − 1) ≤ . . . ≤ V (x, 0) = Vf (x) < βVf (|x|) , ∀x ∈ Xf .

(23)

From the monotonicity property (22) it is easily derived that

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V (f (x, κM P C (x)) + w, N ) − V (x, N ) ≤ −l(x, κM P C (x)) + lw (w) ≤ −αl (|x|) + βw (|w|), ∀x ∈ X M P C (N ), ∀w ∈ W.

(24)


FrA07.3

Then, by (21), (23), (24), the ISS is proven in X M P C (N ). 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 [16], where it is shown how to compute a non linear auxiliary control law based on the solution of a suitable H∞ problem for the linearized system under control. Remark 4: The usual way to derive the upper bound for the value function in X M P C (N ) requires the assumption that the solution uot,t+N −1 of the FHOCP and κot,t+N −1 of the FHCDG are Lipschitz in X M P C (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 Xf , 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 F HOCP and F HCDG respectively, it could be useful to find an upper bound α ¯ 2 of V in a region Ω1 ⊇ Ω. To this regard, define V¯ , 1)α2 α ¯ 2 = max( α2 (r) where V¯ = maxx∈Ω1 (V (x)) and r is such that Br = {x ∈ Rn : |x| ≤ r} ⊆ Ω, as suggested in [13]. This idea can either enlarge or restrict the set W since Ω1 ⊇ Ω but α ¯ 2 ≥ α2 . Remark 5: Following the results reported in [24] for the open-loop and in [16] 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 Xf 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 M P C (N + 1) ⊇ X M P C (N ) ⊇ Xf . V. C ONCLUSIONS Using a suitable, non necessarily continuous, Lyapunov function, regional Input-to-State Stability 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 Model Predictive Control algorithms derived according to openloop 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. R EFERENCES

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