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Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference 2005 Seville, Spain, December 12-15, 2005

WeB16.3

Results on input-to-state stability for hybrid systems

Chaohong Cai, Andrew R. Teel

Abstract— We show that, like continuous-time systems, zeroinput locally asymptotically stable hybrid systems are locally input-to-state-stable (LISS). We demonstrate by examples that, unlike continuous-time systems, zero-input locally exponentially stable hybrid systems may not be LISS with linear gain, inputto-state stable (ISS) hybrid systems may not admit any ISS Lyapunov function, and nonuniform ISS hybrid systems may not be (uniformly) ISS. We then provide a strengthened ISS condition as an equivalence to the existence of an ISS Lyapunov function for hybrid systems. This strengthened condition reduces to standard ISS for continuous-time and discrete-time systems. Finally under some other assumptions we establish the equivalence among ISS, several asymptotic characterizations of ISS, and the existence of an ISS Lyapunov function for hybrid systems.

I. I NTRODUCTION Input-to-state stability (ISS), introduced in [15], is a useful stability notion for studying the robustness of nonlinear control systems affected by noise or disturbances [16], [11], [8]. Some key results related to ISS for continuous-time systems are: ? zero-input local asymptotic stability (0-LAS) implies local input-to-state stability (LISS) [18, Lemma I.2]; ? ISS is equivalent to the existence of an ISS Lyapunov function [17]; ? ISS has asymptotic characterizations [18, Theorem 1]; ? zero-input local exponential stability (0-LES) and local Lipschitz property imply LISS with linear gain [3]. In this paper, we investigate similar statements for hybrid systems. Hybrid systems are those whose trajectories can ?ow in continuous time and also jump at discrete instants. The system variables can be dynamical processes (states) and logical processes (modes). In this paper, we will mainly consider a hybrid system that is a combination of a differential equation on a ?ow set and a difference equation on a jump set. To study (robust) stability of hybrid systems, we will use the solution de?ned in [6], [7]. This solution notion has been used to establish sequential compactness of solutions and the “upper semicontinuous” dependence of solutions with respect to (w.r.t.) initial conditions [7], and hence to make LaSalle’s invariance principle [14] and smooth converse Lyapunov theorems [4] available for hybrid systems. In turn, these results enable the results of the current paper. For starters, we show the implication from 0-LAS to LISS for hybrid systems (see Theorem 1 in Section IV) by

Research partially supported by the ARO under Grant DAAD19-03-10144, the NSF under Grants CCR-0311084 and ECS-0324679, and the AFOSR under Grant F49620-03-1-0203. Authors are with Department of Electrical & Computer Engineering, University of California, Santa Barbara. Email: cai, teel@http://www.wendangku.net/doc/514ae408b52acfc789ebc993.html.

using a result in [7]. However, the combination of the four components of the hybrid system — differential equation, ?ow set, difference equation, and jump set — may also exhibit complex dynamical behaviors and hence yield different behaviors from those of continuous-time systems. For hybrid systems, we will demonstrate by examples that ISS may not imply the existence of an ISS Lyapunov function, that 0-LES and local Lipschitz property may not imply LISS with linear gain, and that nonuniform ISS (i.e. the combination of the asymptotic gain property and global stability) may not imply (uniform) ISS. It has been shown that ISS is equivalent to the existence of an ISS Lyapunov function for continuous-time systems [17], discrete-time systems [9], and switched systems with arbitrary switching signals [13]. For hybrid systems, it is not hard to show the implication from ISS Lyapunov function to ISS by using a hybrid comparison lemma; hence, the more technical work is to show the converse: under what additional conditions does ISS imply the existence of an ISS Lyapunov function? The answers to this question are stated as Theorem 2 and Theorem 3 in Section IV. We provide in Theorem 2 a strengthened ISS condition as an equivalence to the existence of an ISS Lyapunov function for hybrid systems. This strengthened condition reduces to standard ISS for continuous-time systems and discrete-time systems. We present in Theorem 3 some other assumptions to make asymptotic characterizations of ISS available and hence to establish the equivalence between them and the existence of an ISS Lyapunov function for hybrid systems. The rest of the paper is organized as follows. Section II provides a description of hybrid systems, solutions, and stability concepts. Section III gives the three aforementioned counterexamples. Section IV presents main theorems, whose (sketches of) proofs are provided in appendices so as to make this paper self-contained to some extent. Section V gives conclusions. Finally, we list the basic de?nitions and notation:

? ? ?

? ?

?

R+ = [0, +∞) and N+ = {0, 1, 2, ...}. B represents the open unit ball in Euclidean space. Given a vector v = [v1 , v2 , · · · , vn ] ∈ Rn , v denotes its transpose, and |v| denotes its Euclidean norm, i.e. 1/n n |v| = ( i=1 |vi |n ) . Given a set A ? Rn , the sets A and coA stand for the closure and the closed convex hull, respectively, of A. Given a compact set A ? Rn , a point x ∈ Rn , and a constant c > 0, denote |x|A := miny∈A |x ? y| and A[c] := {ξ ∈ Rn : |ξ|A ≤ c}. A function α : R≥0 → R≥0 is said to belong to

0-7803-9568-9/05/$20.00 ?2005 IEEE

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? ?

class-K (α ∈ K) if it is continuous, zero at zero, and strictly increasing. It is said to belong to class-K∞ if, in addition, it is unbounded. Denote by α?1 the inverse function of α ∈ K. A function β : R+ × R+ → R+ is said to belong to class-KL (β ∈ KL) if it satis?es: (i) for each t ≥ 0, β(·, t) is nondecreasing and lim β(s, t) = 0, and (ii) for each s ≥ 0, β(s, ·) is nonincreasing and lim β(s, t) = 0. t→∞ A function γ : R+ × R+ × R+ → R+ is said to belong to class-KLL (γ ∈ KLL) if, for each r ≥ 0, γ(·, ·, r) ∈ KL and γ(·, r, ·) ∈ KL.

s→0+

?

II. H YBRID SYSTEMS AND STABILITY DEFINITIONS A. Hybrid systems Consider hybrid systems Hu with state x and input u Hu := x = f (x, u) for x ∈ C, ˙ x+ = g(x, u) for x ∈ D, (1)

where x ∈ Rn , u ∈ Rm , f : Rn ×Rm → Rn , g : Rn ×Rm → Rn , and C, D ? Rn . For simplicity of notation, we use the data (f, g, C, D) to represent hybrid system Hu . The solutions to Hu are de?ned on hybrid time domains, as used in [6], [7], and [5]. We call a subset E ? R+ × N+ J a compact hybrid time domain if E = j=0 ([tj , tj+1 ], j) for some ?nite sequence of times 0 = t0 ≤ t1 ≤ t2 ≤ ... ≤ tJ+1 . We say E is a hybrid time domain if for all (T, J) ∈ E, E ∩ ([0, T ] × {0, 1, ...J}) is a compact hybrid time domain. On each hybrid time domain there is a natural ordering of points: (t, j) (s, k) if t + j ≤ s + k. Equivalently, this can be characterized by t ≤ s and j ≤ k. A hybrid signal is a function de?ned on a hybrid time domain. Speci?cally, hybrid signal u : dom u → Rm is called a hybrid input in this paper. A hybrid signal x : dom x → Rn is called a hybrid arc if x(·, j) is locally absolutely continuous for each j. A hybrid arc x : dom x → Rn and a hybrid input u : dom u → Rm is a solution pair (x,u) to Hu if (S0) dom x = dom u; (S1) for all j ∈ N+ and almost all t such that (t, j) ∈ dom x, x(t, j) ∈ C, x(t, j) = f (x(t, j), u(t, j)); ˙ (2)

where Γ(u) denotes the set of all (s, k) ∈ dom u such that (s, k + 1) ∈ dom u. When t + j → ∞, u (t,j) is denoted by u ∞ . We denote by Lm the set of hybrid inputs (in Rm ) ∞ that have ?nite · ∞ . Throughout this paper, we assume u ∈ Lm for (1). ∞ A solution pair to Hu is maximal if it cannot be extended, and it is complete if its hybrid time domain is unbounded. Denote by Su (ξ) the set of all maximal solution pairs (x, u) to Hu with x(0, 0) = ξ ∈ Rn . By slight abuse of notation, we will use x(t, j, ξ, u) to denote x(·, ·) evaluated at (t, j) ∈ dom x, where (x, u) ∈ Su (ξ). When u ≡ 0 and (x, 0) ∈ Su (ξ), we simply say x ∈ S0 (ξ) and call x a maximal solution starting from ξ ∈ Rn . The hybrid system Hu is forward complete if, for each ξ ∈ Rn , each (x, u) ∈ Su (ξ) is complete. We impose the following basic conditions for Hu : Standing Assumption 1: For Hu = (f, g, C, D), ? f and g are continuous, and f is locally Lipschitz in x uniformly on any compact subset of Rm 1 ; n n ? C and D are closed in R , and C ∪ D = R . The solution results in [2], [5], [7] tell us that the existence of solutions with u ∈ Lm to the hybrid system (1) is ∞ guaranteed by Standing Assumption 1. B. Stability Consider a hybrid system Hu in (1) and let A be a compact subset of Rn (throughout this subsection). The set A is 0(locally) stable (0-LS) if for each ε > 0 there exists δ > 0 such that for each ξ ∈ A[δ] , each solution x ∈ S0 (ξ) is complete and satis?es |x(t, j, ξ, 0)|A ≤ ε for all (t, j) ∈ dom x; it is 0-attractive if there exists μ > 0 such that for each ξ ∈ A[μ] , each solution x ∈ S0 (ξ) is complete and |x(t, j, ξ, 0)|A = 0; and it is 0satis?es lim

(t,j)∈dom x, t+j→∞

input locally asymptotically stable (0-LAS) if it is both 0stable and 0-attractive. The set of points ξ ∈ Rn such that all solutions in S0 (ξ) are complete and converge to A is called 0 the 0-input basin of attraction for A and is denoted BA . From 0 Proposition 6.1(i) in [7], we know that BA is open (since C ∪ D = Rn ). The set A is 0-input globally asymptotically 0 stable (0-GAS) if A is 0-LAS and BA = Rn . The set A is 0-input locally exponentially stable (0-LES) if there exist r > 0, λ > 0, and c > 0 such that, for each ξ ∈ A[r] , each solution x ∈ S0 (ξ) satis?es |x(t, j, ξ, 0)|A ≤ c|ξ|A e?λ(t+j) ?(t, j) ∈ dom x.

(S2) for all (t, j) ∈ dom x such that (t, j + 1) ∈ dom x, x(t, j) ∈ D, x(t, j + 1) = g(x(t, j), u(t, j)). (3)

We emphasize from the de?nition of solution pair that the jump set D (respectively, the ?ow set C) enables jumps (respectively, ?ows). Given any hybrid input, de?ne its supremum norm from (0, 0) to (t, j) ∈ dom u as ? ? ? ? u (t,j):=max ess.sup |u(s, k)|, sup |u(s, k)| , ? ? (s,k)∈Γ(u), (s,k)∈dom u\Γ(u),

(s,k) (t,j) (s,k) (t,j)

De?nition 1: System Hu is (uniformly) input-to-state stable (ISS) w.r.t. A if there exist γ ∈ KLL and κ ∈ K such that, for each ξ ∈ Rn , each solution pair (x, u) ∈ Su (ξ) satis?es |x(t, j, ξ, u)|A ≤ max γ(|ξ|A , t, j), κ for each (t, j) ∈ dom x.

1 For each compact U ? Rm and each compact K ? Rn , there exists some constant L > 0 such that |f (x, u) ? f (z, u)| ≤ L|x ? z| for all x, z ∈ K and all u ∈ U .

u

(t,j)

(4)

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De?nition 2: System Hu is locally input-to-state stable (LISS) w.r.t. A if there exist r > 0, γ ∈ KLL, and κ ∈ K such that, for each ξ ∈ A[r] , each solution pair (x, u) ∈ Su (ξ) with u ∞ ≤ r satis?es (4) for each (t, j) ∈ dom x. System Hu is LISS w.r.t. A with ?nite gain if κ in De?nition 2 is a linear function. Like continuous-time and discrete-time systems, we can have Lyapunov characterizations of ISS for hybrid systems (cf. [17, Theorem 1] and [9, Theorem 4]). De?nition 3: A smooth function V : Rn → R+ is called an ISS-Lyapunov function w.r.t. A for system (1) if there exist α1 , α2 , α3 ∈ K∞ and ρ ∈ K such that α1 (|ξ|A ) ≤ V (ξ) ≤ α2 (|ξ|A ) ?ξ ∈ Rn (5) and, for all ξ ∈ Rn and all u ∈ Rm satisfying |ξ|A ≥ ρ(|u|), (6) ?V (ξ) · f (ξ, u) ≤ ?α3 (|ξ|A ) ?ξ ∈ C, V (g(ξ, u)) ? V (ξ) ≤ ?α3 (|ξ|A ) ?ξ ∈ D. (7) Inspired by [8, Lemma 10.4.2], we can have an equivalent de?nition of ISS-Lyapunov function for (1). Proposition 1: For system (1), a smooth function V : Rn → R+ , satisfying (5) with α1 , α2 ∈ K∞ , is an ISSLyapunov function w.r.t. A if and only if there exist α3 ∈ ? ? K∞ and ρ ∈ K such that, for all ξ ∈ Rn and all u ∈ Rm , ? ?V (ξ) · f (ξ, u) ≤ ?? 3 (|ξ|A ) + ρ(|u|) ?ξ ∈ C, (8) α V (g(ξ, u)) ? V (ξ) ≤ ?? 3 (|ξ|A ) + ρ(|u|) ?ξ ∈ D. (9) α ? The next proposition, whose converse does not generally hold (see Example 2 in Subsection III-B), is a corollary of Theorem 2 in Section IV. Proposition 2: If system (1) has an ISS-Lyapunov function w.r.t. A, then Hu is ISS w.r.t. A. Like continuous-time and discrete-time systems, we can also have asymptotic characterizations of ISS for hybrid systems (cf. [18, Theorem 1] and [9, Theorem 4]). De?nition 4: System Hu has the asymptotic gain (AG) property w.r.t. A if there exists κ ∈ K such that, for each ξ ∈ Rn , each solution pair (x, u) ∈ Su (ξ) satis?es De?nition 5: System Hu has global stability (GS) w.r.t. A if there exists α, κ ∈ K such that, for each ξ ∈ Rn , each solution pair (x, u) ∈ Su (ξ) satis?es De?nition 6: System Hu is nonuniform ISS w.r.t. A if it has the AG property and GS w.r.t. A. Proposition 3: For system (1), if Hu is ISS w.r.t. A, then it is nonuniform ISS w.r.t. A. Proposition 3 is straightforward, but its converse does not generally hold (see Example 3 in Subsection III-C). A weaker concept than the AG property is the following. De?nition 7: System Hu has the limit property w.r.t. A if there exists κ ∈ K such that, for each ξ ∈ Rn , each solution pair (x, u) ∈ Su (ξ) satis?es

(t,j)∈dom x (t,j)∈dom x (t,j)∈dom x,t+j→∞

x2

3 8

15 64

1 3

?

1 2

?

0

1 4

1 3

1 2

x1

?

8 27

Fig. 1. Example 1 (the lined area is not plotted)

III. E XAMPLES A. Example 1: 0-LES + Lipschitz ? LISS with ?nite gain In this subsection, we provide a planar example of hybrid systems, where 0-LES and the local Lipschitz property of f do not imply LISS with ?nite gain. Let x = [x1 , x2 ] ∈ R2 and u ∈ R. De?ne f (x, u) := [x2 + u, g(x, u) := 0 , C := R2 \ D , where D+ :={x : x1 ≥ 0, x2 ≥ 0}

∞

?x1 + u] ,

D := D+ ∪ D? ∪ {0} ,

x : |x| ≥

3 8 ,

lim sup

|x(t, j, ξ, u)|A ≤ κ( u

∞ ). n=1

x:

1 1 1 ? ≤ |x| ≤ 2n + 2 (2n + 2)3 2n + 1

D? :={x : x1 ≤ 0, x2 ≤ 0}

∞

sup

|x(t, j, ξ, u)|A ≤ max {α(|ξ|A ), κ( u

∞ )} .

x:

n=1

1 1 1 ? ≤ |x| ≤ 3 2n + 1 (2n + 1) 2n

.

Note that f is locally Lipschitz. As shown in Fig. 1, inside the disk with radius 1/2, the jump set D is partitioned alternatively between the ?rst quadrant and the third quadrant 1 with overlap n3 (for example, see grey area in Fig. 1). De?ne the hybrid system Hu := (f, g, C, D) and let A := {0}. One can verify Hu satis?es Standing Assumption 1. When u = 0, since no circle in R2 is a subset of C, we conclude, for each ξ ∈ R2 , each x ∈ S0 (ξ) satis?es |x(t, j, ξ, 0)| ≤ e2π |ξ|e?(t+j) ?(t, j) ∈ dom x,

inf

|x(t, j, ξ, u)|A ≤ κ( u

∞ ).

which means the origin is 0-LES (in a global sense). Now, 1 pick any initial condition ξ ∈ C with |ξ| = n . There

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exists some solution x(·, ·, ξ, u), where (x, u) ∈ Su (ξ) and 1 |u(·, 0)| ∝ n3 , ?owing on C and rotating with a radius of 1 approximately n (say n = 3, then there exists some solution x(·, ·, ξ, u) that can rotate with a radius of approximately 1 3 8 in the ?rst, second, and fourth quadrant but exactly 27 in the third quadrant). Namely,

1 1 ∝ ( u (t,j) ) 3 , ?(t, j) ∈ dom x, n which means that Hu can not be LISS with ?nite gain.

x2

D C 0 1 2 3 C 4

x1

|x(t, j, ξ, u)| ≈

Fig. 2. Example 3 (the dark area is not plotted)

B. Example 2: ISS ? existence of ISS Lyapunov functions The following planar example shows the converse of Proposition 2 fails for hybrid systems. Let x = [x1 , x2 ] ∈ R2 and u = [u1 , u2 ] ∈ R2 . De?ne f (x, u) := [|u1 ? u2 | ? 1, g(x, u) := 0 , D := R2 . One can verify that the hybrid system Hu := (f, g, C, D) satis?es Standing Assumption 1 and is ISS w.r.t. A := {0}. Nevertheless, Hu does not admit an ISS Lyapunov function. Otherwise, one could pick u = [2, 0] and υ = [0, 2] ; then for any ξ ∈ C satisfying |ξ| ≥ max{ρ(|u|), ρ(|υ|)}, where ρ ∈ K comes from De?nition 3, using (6) we have, ?V (ξ) · (f (ξ, u) + f (ξ, υ)) ≤ ?2α3 (V (ξ)). (10) u 1 ? u2 ] ,

C := {x ∈ R2 : x1 ≥ 0, x2 = 0} ,

Let A := {0} and de?ne the hybrid system Hu := (f, g, C, D), which indeed satis?es Standing Assumption 1. Note that f (x, u) = 0 and cos(θ(x, u)) ∈ [0, 1] for all x and u and that cos(θ(x, u)) = 0 only for sin(u/2) = ±1 and x1 = 2n, where n is an arbitrary integer. These imply that |x1 (·, 0, ξ, u)| increases as long as x(·, 0, ξ, u) ?ows on C. Since C is de?ned as a union of isolated trapezoids (for example, see the grey area), each x(·, ·, ξ, u) with ξ ∈ C will ?ow to the boundary of C in ?nite time and then jump to the origin. Therefore, we conclude that, for each ξ ∈ R2 , each (x, u) ∈ Su (ξ) satis?es |x(t, j, ξ, u)| ≤ 2|ξ| for |x(t, j, ξ, u)| = 0, all (t, j) ∈ dom x and lim which establish GS and the AG property for Hu . Nevertheless, Hu is not ISS. Suppose there exist γ ∈ KLL and κ ∈ K such that (4) holds. Then pick two positive integers n and such that n > κ(π) and γ(2n, , 0) ≤ 1. Consider 1 ξ = 2n ? 2 , 0 and pick u(·, 0) ∈ {?π, π} in such a way to assure |x2 (·, 0, ξ, u)| ≤ 2 1 . Consequently, we +1 have x1 = |x1 ? 2n|3 . De?ne z := 2n ? x1 gives the ˙ 1 differential equation z = ?z 3 , which takes time t? = 2 + 2 ˙ 1 1 ? to decrease from z(0) = 2 to z(t ) = 2 +1 . In particular, |x(t? , 0, ξ, u)| > max{κ(π), 1}, which contradicts (4). IV. M AIN R ESULTS A. 0-LAS implies LISS Inspired by a result on LISS for continuous-time systems [18, Lemma I.2], we propose the following implication from 0-LAS to LISS. Theorem 1: If the compact set A ? Rn is 0-LAS for Hu , then Hu is LISS w.r.t. A. Proof: See Appendix I. Remark 1: The proof of Theorem 1 does not require the Lipschitz condition but only the continuity of f . B. Existence of ISS Lyapunov function As Example 2 in Subsection III-B shows, ISS does not imply the existence of an ISS Lyapunov function for hybrid systems. The main reason behind this is that the solutions to the differential inclusion x ∈ f (x, εB) may not be dense any ˙ more in the solutions to x ∈ cof (x, εB) on the ?ow set C, ˙ which, unfortunately, may not be Rn for hybrid systems (cf. the Relaxation Theorem for differential inclusions, Theorem 10.4.4 in [1]). In order to achieve ISS characterizations, one may require nice behaviors on the ?ow and jump set boundaries (like Theorem 2), or one may require f (x, u) to

(t,j)∈dom x, t+j→∞

On the other hand, de?ne the function f : R2 → R2 by letting f (ξ) := f (ξ, u) + f (ξ, υ) = [2, 0] for each ξ ∈ R2 and de?ne a new hybrid system H := (f , 0, C, D) (without inputs), and then we can have some solution to H ?owing on C and blowing up. This contradicts the combination of (10) and (5). C. Example 3: nonuniform ISS ? ISS The following planar example shows the converse of Proposition 3 fails for hybrid systems. Let x = [x1 , x2 ] ∈ R2 and u ∈ R. De?ne a periodic function ψ : R → R by letting ψ(x1 ) := |x1 ? 2n|3 for each integer n and each x1 ∈ [2n ? 1, 2n + 1]. De?ne f (x, u) := [cos(θ(x, u)), g(x, u) := 0 ,

n n C := ∪∞ (C+ ∩ C? ) , n=1

sin(θ(x, u))] ,

D := R2 \ C , where θ(x, u) := ?

∞

u π + arcsin ψ(x1 ) sin , 2 2 n C+ := {x : x1 ≥ 2n ? 1 and x1 ? 2n ≤ x2 ≤ 2n ? x1 } , :=

=1

n C?

2n ?

1 1 , 2n ? × R. 2 2 +1

Note that f is locally Lipschitz and that f (x, u) has no convex property w.r.t. u.

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have a convex property w.r.t. u (like Theorem 3 in the next subsection). Theorem 2: Let A ? Rn be compact. For (1), Hu admits an ISS-Lyapunov function w.r.t. A if and only if there exists a continuous function σ : Rn → R+ such that A = {ξ : σ(ξ) = 0} and the hybrid system Huσ := (f, g, Cσ , Dσ ) is forward complete and ISS w.r.t. A, where Cσ := {x ∈ Rn : (x + σ(x)B) ∩ C = ?}, Dσ := {x ∈ Rn : (x + σ(x)B) ∩ D = ?}. Sketch of proof: See Appendix II. Remark 2: If C = Rn and D = ?, then Cσ = C and hence Theorem 2 for A = {0} becomes the equivalence between ISS and ISS Lyapunov function for continuoustime systems (see the equivalence 1?2 of Theorem 1 in [17]). Similarly, if D = Rn and C = ?, then Theorem 2 for A = {0} becomes the one for discrete-time systems (see the equivalence 1?4 of Theorem 1 in [9]). Remark 3: The property of forward completeness does not necessarily carry over from Hu to Huσ . Consider Example 2 by rede?ning f (x, u) := x3 |u1 ? u2 |, u1 ? u2 . 1 Clearly, Hu is forward complete, but Huσ is not: one can ?nd x(·, 0) to ?ow in Cσ and blow up (in the x1 coordinate) in ?nite time, where (x, u) is a maximal solution pair starting from Cσ and u(·, 0) is chosen appropriately. C. Asymptotic characterizations of ISS For continuous-time systems, nonuniform ISS is equivalent to ISS even without assuming that f (x, u) has a convex property w.r.t. u (see [18, Theorem 1]), but there is no such equivalence for hybrid systems (see Example 3 in Subsection III-C). If f (x, u) is assumed with a convex property w.r.t. u, then asymptotic characterizations of ISS will carry over from continuous-time systems to hybrid systems. The following theorem provides a hybrid version of [18, Theorem 1] and [9, Theorem 1]. Theorem 3: Let A ? Rn be compact. For (1), assume that Hu is forward complete and that f (x, εB) = cof (x, εB) for each x ∈ Rn and each ε > 0. Then the following statements are equivalent: 1) Hu is ISS w.r.t. A; 2) Hu is nonuniform ISS w.r.t. A; 3) Hu has the AG property w.r.t. A and the set A is 0-LS for Hu ; 4) Hu satis?es the limit property w.r.t. A and the set A is 0-LS for Hu ; 5) Hu satis?es the AG property and is LISS w.r.t. A; 6) Hu admits an ISS-Lyapunov function w.r.t. A. Sketch of proof: See Appendix III. V. C ONCLUSIONS We have demonstrated similarities and differences between ISS results for continuous-time systems and hybrid systems. We have investigated conditions to guarantee Lyapunov and asymptotic characterizations of ISS for hybrid systems. These characterizations parallel what has been developed previously for continuous-time and discrete-time systems.

A PPENDIX I P ROOF OF T HEOREM 1 Suppose Hu is 0-LAS with the 0-input basin of attraction 0 BA . Then Theorem 6.2 in [7] implies the existence of c > 0, 0 a proper indicator ω (see de?nition in [4]) for A on BA , and a continuous γ ∈ KLL such that ω(η) = |η|A for all η ∈ A[c] 0 and, for each ξ ∈ BA , each x ∈ S0 (ξ) satis?es ω(x(t, j, ξ, 0)) ≤ γ(ω(ξ), t, j) ?(t, j) ∈ dom x.

Furthermore, using Theorem 6.2 in [7], we have the following claim. 0 Claim 1: For each ε > 0 and each compact set K ? BA , there exists δ > 0 such that the solutions xδ to the hybrid inclusion Hδ := (fδ , gδ , C, D) with initial condition ξ ∈ K satisfy, for all (t, j) ∈ dom xδ , ω(xδ (t, j, ξ)) ≤ γ(ω(ξ), t, j) + ε, where fδ (ξ) := co{v ∈ Rn : v = f (ξ, u), u ∈ δB}, gδ (ξ) := {v ∈ Rn : v = g(ξ, u), u ∈ δB}. Given any ρ ∈ K∞ satisfying ρ(r) ≥ γ(r, 0, 0) ≥ r for each r ≥ 0, let K = A[ρ?1 (c)] and, without loss of generality, let α ∈ K be such that Claim 1 holds with δ = α(ε). De?ne r := min{ρ?1 (c), supε>0 α(ε)}. Then κ := α?1 is a classK function on [0, r). Using Claim 1, we have r > 0, γ ∈ KLL, and κ ∈ K for De?nition 2. A PPENDIX II S KETCH OF PROOF OF T HEOREM 2 Given any α ∈ K∞ and any continuous function σ : Rn → R+ , de?ne set-valued mappings Fo , G, F , Fσ , Gσ : Rn ? Rn as follows: Fo (x) := {v ∈ Rn : v = f (x, u), u ∈ α(|x|A )B}, G(x) := {v ∈ Rn : v = g(x, u), u ∈ α(|x|A )B}, F (x) := coFo (x), Fσ (x) := coF ((x + σ(x)B) ∩ C) + σ(x)B, Gσ (x) := {v : v ∈ g + σ(g)B, g ∈ G((x + σ(x)B) ∩ D)}. A. Necessity Let V : Rn → R+ and ρ ∈ K come from De?nition 3. Pick ρ ∈ K∞ to majorize ρ. De?ne the function α := ρ?1 . First, we use the ISS Lyapunov function V to show that A is globally asymptotically stable for the hybrid inclusion H := (F, G, C, D). Then the combination of Corollary 2 and Theorem 6 in [4] implies the existence of a continuous function σ : Rn → R+ and a smooth function V : Rn → R+ such that A = {x : σ(x) = 0} and V is a smooth Lyapunov function (see [4, De?nition 2]) for the perturbed hybrid inclusion HΣ := (Fσ , Gσ , Cσ , Dσ ). Now consider the hybrid system Huσ := (f, g, Cσ , Dσ ), and note the solution relationship between Huσ and HΣ . With the properties of V and the following lemma, which is a hybrid version of Comparison Principle (cf. [12, Lemma 4.3] and [10, Lemma 4.4]), we can follow similar arguments

5407

in the proof of [17, Lemma 2.14] to establish the necessity in Theorem 2. Lemma 1 (Hybrid Comparison Principle): For each α ∈ K, there exists γα ∈ KLL with the following properties: if ˙ a hybrid arc z : dom z → R+ satis?es z(t, j) ≤ ?α(z(t, j)) and z(t, j + 1) ? z(t, j) ≤ ?α(z(t, j)), then z(t, j) ≤ γα (z(0, 0), t, j) ≤ z(0, 0) for each (t, j) ∈ dom z. B. Suf?ciency Let γ ∈ KLL and κ ∈ K come from De?nition 1. Pick s κ ∈ K∞ to majorize κ. De?ne α(s) := κ?1 ( 2 ) for all s ≥ 0. ˉ ˉ Consider Hoσ := (Fo , G, Cσ , Dσ ). Then the ISS assumption of Huσ implies each maximal solution x to Hoσ starting from any ξ ∈ Rn satisfy, for each (t, j) ∈ dom x, |x(t, j, ξ)|A ≤max γ(|ξ|A , t, j), 1 sup |x(s, k, ξ)|A , 2 (s,k)∈dom x,

(s,k) (t,j)

Finally, we follow similar arguments in [18] to show 4?5. Without loss of generality, let κ? ∈ K∞ come from De?nition 7. If u ∞ = 0, then combining the 0-LS of A and the limit property gives 0-LAS of A for Hu , and then Theorem 1 gives LISS w.r.t. A for Hu . If r := max{|ξ|A , κ? ( u ∞ )} > 0, then de?ne θ(r) := sup{|x(t, j, ξ, u)|A : ξ ∈ A[2r] , (x, u) ∈ Su (ξ), (t, j) ∈ dom x, u

∞

≤ κ?1 (r)}. ?

which immediately gives the uniform stability of A for Hoσ . Furthermore, with the routine small-gain arguments, one can use the inequality above to establish the uniform attractivity of A for Hoσ . Then using Proposition 1 in [4] we conclude that A is globally asymptotically stable for Hoσ , which makes the following lemma applicable. Lemma 2: If there exists a continuous function σ : X → R+ such that A = {x : σ(x) = 0} is globally asymptotically stable for Hoσ , then A is also globally asymptotically stable for the hybrid system H := (F, G, C, D). Now, using Theorem 1 in [4] we have a smooth Lyapunov function w.r.t. A for H. De?ning the function ρ := α?1 for De?nition 3, we establish the suf?ciency in Theorem 2. A PPENDIX III S KETCH OF PROOF OF T HEOREM 3 The implications 1?2, 2?3, and 3?4 are obvious. The implication 6?1 comes from Proposition 2. Next we show 5?6. Let r > 0, γ ∈ KLL, and κ1 ∈ K come from De?nition 2. Let κ2 ∈ K come from De?nition 4. Pick κ ∈ K∞ to majorize κ1 and κ2 . De?ne ˉ s α(s) := κ?1 ( 2 ) for all s ≥ 0. Consider the hybrid system ˉ Ho := (Fo , G, C, D), where the set-valued mappings Fo and G are de?ned in Appendix II. Note from assumptions that Fo (x) is convex for each ξ ∈ Rn . Using the AG property w.r.t. A, following similar arguments to the proof of [18, Lemma II.1], and using the properties of κ and α, we can show that, each maximal ˉ solution x to Ho starting from any ξ ∈ Rn satis?es

(t,j)∈dom x, t+j→∞

The limit property assumption gives the existence of (s, k) ∈ dom x such that x(s, k, ξ, u) ∈ A[3r/2] . Then we can use the following lemma to conclude that θ(r) < ∞ and hence choose θ : R+ → R+ as a nondecreasing function. Lemma 3: Let U be a compact subset of Rm . Let K1 and and K2 be compact subsets of Rn such that K1 + B ? K2 , where > 0. Assume, for each ξ ∈ K2 and each (x, u) ∈ Su (ξ) with u(·, ·) ∈ U , there exists (s, k) ∈ dom x such that x(s, k, ξ, u) ∈ K1 . Then the (in?nite horizon) reachable set starting from K2 is bounded. Combining the property of θ and the LISS w.r.t. A gives GS w.r.t. A, say with α, κ ∈ K for De?nition 5. Then ? ? de?ning the function κ := max{? ? κ? , κ} for De?nition 4 α ? and following similar arguments to the proof of [18, Lemma I.4], we can obtain the AG property w.r.t. A for Hu and hence establish 4?5. R EFERENCES

[1] J.-P. Aubin and H. Frankowska. Set-valued Analysis. Birkh¨ user, 1990. a [2] J.-P. Aubin, J. Lygeros, M. Quincampoix, S. Sastry S, and N. Seube. Impulse differential inclusions: a viability approach to hybrid systems. IEEE Trans. Auto. Cont., 47:2–20, 2002. [3] C.I. Byrnes, A. Isidori, and L. Praly. On the asymptotic properties of a system arising in non-equilibrium theory of output regulation. Mittag Lef?er Institute, Stockholm, Sweden, 2003. [4] C. Cai, A. R. Teel, and R. Goebel. Converse Lyapunov theorems and robust asymptotic stability for hybrid systems. In Proc. 24th ACC, pages 12–17, 2005. [5] P. Collins. A trajectory-space approach to hybrid systems. In 16th Int. Symp. Math. Theory Networks & Systems, 2004. [6] R. Goebel, J. Hespanha, A.R. Teel, C. Cai, and R. Sanfelice. Hybrid systems: generalized solutions and robust stability. In IFAC Symposium on Nonlinear Control Systems, Stuttgart, Germany, 2004. [7] R. Goebel and A. R. Teel. Results on solution sets to hybrid systems with applications to stability theory. In Proc. 24th ACC, pages 557– 562, 2005. [8] A. Isidori. Nonlinear Control Systems II. Springer, 1999. [9] Z. Jiang and Y. Wang. Input-to-state stability for discrete-time nonlinear systems. Automatica, 37:857–869, 2001. [10] Z. Jiang and Y. Wang. A converse Lyapunov theorem for discrete-time systems with disturbances. Sys. & Cont. Lett., 45:49–58, 2002. [11] H.K. Khalil. Nonlinear Systems. Prentice Hall, third edition, 2002. [12] Y. Lin, E.D. Sontag, and Y. Wang. A smooth converse Lyapunov theorem for robust stability. SIAM J. Cont. & Opt., 34:124–, 1996. [13] J. L. Mancilla-Aguilar and R. A. Garcia. On converse Lyapunov theorems for ISS and iISS switched systems. Sys. & Cont. Lett., 42:47–53, 2001. [14] R. G. Sanfelice, R. Goebel, and A. R. Teel. Results on convergence in hybrid systems via detectability and an invariance principle. In Proc. 24th ACC, pages 551–556, 2005. [15] E. Sontag. Smooth stabilization implies coprime factorization. IEEE Trans. Auto. Cont., 34:435–443, 1989. [16] E.D. Sontag. Input to state stability: basic concepts and results. Lecture Notes in Mathematics (CIME Course, Cetraro), June 2004. [17] E.D. Sontag and Y. Wang. On characterizations of the input-to-state stability property. Sys. & Cont. Lett., 24:351–359, 1995. [18] E.D. Sontag and Y. Wang. New characterizations of input to state stability. IEEE Trans. Auto. Cont., 41:1283–1294, 1996.

lim sup |x(t, j, ξ)|A ≤

1 lim sup |x(t, j, ξ)|A , 2 (t,j)∈dom x,

t+j→∞

which gives the global attractivity of A for Ho . Using the LISS property of Hu , we can establish the local stability of A for Ho . Using [7, Proposition 6.1(iii)], we conclude that A is (uniformly) globally asymptotically stable for Ho . Using Theorem 1 in [4] we have a smooth Lyapunov function w.r.t. A for Ho . De?ning the function ρ := α?1 for De?nition 3 establishes the implication 5?6.

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