Class-KL estimates and input-to-state stability analysis of impulsive switched systems

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Abstract

In this paper, we investigate input-to-state stability of impulsive switched systems. The goal is to bridge two apparently different, but both useful, stability notions, input-to-state stability and stability in terms of two measures, in the hybrid systems setting. Based on two class-KL function estimates and a comparison theorem for impulsive differential equations, two sets of sufficient Lyapunov-type conditions for input-to-state stability in terms of two measures are obtained for impulsive switched systems. These conditions exploit some nonlinear integral constraints in terms of generalized dwell-time conditions to balance the continuous dynamics and impulsive dynamics so that input-to-state stability is achieved, despite possible instability of individual continuous subsystems or destabilizing impulsive effects. An illustrative example is presented, together with numerical simulations, to demonstrate the main results.

Introduction

Hybrid systems have attracted a lot of attention in recent years due to their numerous applications in various fields of sciences and engineering. Hybrid systems are dynamical systems exhibiting both continuous and discrete dynamic behavior. The interaction of continuous- and discrete-time dynamics in a hybrid system can often lead to rich dynamical behavior and phenomena that are not encountered in purely continuous- or discrete-time systems and hence brings difficulties and challenges to the studies of hybrid systems, such as their stability analysis and control design (see, e.g., [1], [2], [3], [4] and references therein). In this paper, we study stability of hybrid systems under the notions of input-to-state stability [5] and stability in terms of two measures [6].

The notion of input-to-state stability (ISS), originally introduced in [5], has proved very useful in characterizing the effects of external inputs to a control system. The ISS notion has subsequently been extended to discrete-time systems in [7] and to switched systems in [8], [9]. Input-to-state stability properties for hybrid systems are investigated in [10], [11], where the hybrid systems are defined on hybrid time domains. More recently, Chen and Zheng [12] and Hespanha et al. [13] have studied Lyapunov conditions for input-to-state stability of impulsive nonlinear systems with and without time-delays, and Liu et al. [14] have investigated input-to-state properties of impulsive and switching hybrid systems with time-delay.

Stability in terms of two measures, on the other hand, provides a unified notion for Lyapunov stability, partial stability, orbital stability, and stability of an invariant set of nonlinear systems [6], [15], which would otherwise be treated separately. This notion has recently been adopted in the framework of switched systems [14], [16], but not yet exploited for input-to-state stability.

The goal of this paper is to bridge the above two notions of stability, namely input-to-state stability and stability in terms of two measures, in the hybrid systems setting. The clear benefits of doing so include characterization of robustness of hybrid systems affected by noise or disturbances, not only in the Lyapunov stability sense, but also in various other stability notions, such as partial stability, orbital stability, and stability with respect to an invariant set. One additional advantage is that we can choose one of the stability measures to be the output function, the results can also cover input-to-output stability [17].

We focus on two widely studied types of hybrid systems, namely impulsive systems [18] and switched systems [2]. Intuitively, impulsive systems are systems with state jumps and can be used to model real world processes that undergo abrupt changes (impulses) in the state at discrete times [18]. Impulsive dynamical systems can be naturally viewed as a class of hybrid systems [19], [20]. Switched systems, on the other hand, are systems with dynamic switching and can be used to model real systems whose dynamics are chosen from a family of possible choices according to a switching signal [2]. In order to study these two types of hybrid systems in the same framework, we formulate them as impulsive switched systems by introducing integrated signals called impulsive and switching signals [14]. Input-to-state stability properties in terms of two measures are investigated not only under a particular signal, but rather under various classes of signals as in the stability analysis for switched systems [21].

Besides the conceptual unification of two stability notions mentioned above, the results in this paper also include a key improvement, compared with previous results on input-to-state stability of impulsive or switched hybrid systems. Namely, we do not assume that the continuous dynamics of the system either grow exponentially or decay exponentially, which is in contrast with the results in the literature [9], [12], [13], [22]. Note the presence of the nonlinear function ci in both of the main theorems in the current paper. These actually give more natural conditions similar to the Lyapunov characterizations of input/output stability notions of continuous nonlinear systems [23].

The rest of this paper is organized as follows. Basic notation and definitions are given in Section 2, where we formulate an impulsive and switching hybrid system with external input. Particularly, the notion of input-to-state stability in terms of two measures is presented. Section 3 provides two lemmas on class-KL estimates for impulsive switched systems. The main results of this paper, presented in Section 4, give Lyapunov-type sufficient conditions for input-to-state stability of impulsive switched systems in terms of two measures. An example is presented in Section 5 to illustrate the main results. The paper is concluded by Section 6, where the main contributions of this paper are highlighted.

Section snippets

Notation and definitions

Let Z+ denote the set of nonnegative integers, R+ the set of nonnegative real numbers, and Rn the n-dimensional real Euclidean space. For xRn,|x| denotes the Euclidean norm of x. Let C[M;N] denote the set of all continuous functions from MRm to NRn.

Let Ic and Id be two index sets. Consider the following impulsive switched system {x(t)=fik(t,x(t),u(t)),t(tk,tk+1),kZ+,(a)Δx(t)=Ijk(t,x(t),u(t)),t=tk,kZ+{0},(b)x(t0)=x0.(c) where ikIc,jkId,x(t)Rn is the system state, u:R+Rm the system

Class-KL estimates

In this section, we establish two general class-KL estimates for solutions of scalar impulsive switched systems.

Lemma 3.1

Consider the scalar impulsive switched system{y(t)=pik(t)αik(y(t)),t(tk,tk+1),kZ+,(a)y(tk)=g(y(tk)),t=tk,kZ+{0},(b)y(t0)=y0,(c)where ikIc,y00,gK,pi:R+R+ is locally integrable, and αiK is locally Lipschitz for each iIc . If

  • (i)

    τisup{tktk1:kZ+,ik=i}< ; and

  • (ii)

    Niinfq>0g(q)qdsαi(s)>Misupt0tt+τipi(s)ds,

then the system has a unique solution y(t) defined for all tt0 and

Input-to-state stability

In this section, we establish some sufficient conditions for input-to-state stability of impulsive switched systems, as applications of the class-KL estimates we have obtained in the previous section. As mentioned in the introduction, to unify different notions of stability, the input-to-state stability analysis is performed in terms of two measures (h0 and h).

Our first result is concerned with (h0,h)-ISS properties of system (2.1), in the case when all the subsystems governing the continuous

Example

In this section, we present an example to illustrate our main results.

Example 5.1

Consider the following networked hybrid control system {x(t)=Aikx(t)+fik(x(t))+Bikw(t),t(tk,tk+1),(a)y(t)=Cx(t)+v(t),tt0,(b) and {xˆ(t)=Aikxˆ(t)+fˆik(xˆ(t)),t(tk,tk+1),kZ+,(a)yˆ(t)=Cxˆ(t),t(tk,tk+1),kZ+,(b)yˆl(t)={yjk(t),l=jk,yˆl(t),ljk,,t=tk,(c)xˆ(t)=CT(CCT)1yˆ(t),t=tk,(d) where kZ+,l{1,2,,n},x(t)Rn is the system state, w(t)Rm is the disturbance input, CRp×n is the observation matrix (pn), y(t)Rp is the

Conclusions

In this paper, we have investigated the input-to-state stability properties of impulsive switched systems. Sufficient conditions have been established for input-to-state stability in terms of two measures for hybrid systems with both switching and impulse effects. The formulation of hybrid systems is quite general in that it allows both the continuous dynamics and the discrete dynamics to be chosen from a certain family, according to a general impulsive and switching signal. The stability

Acknowledgments

This research was supported in part by the Natural Sciences and Engineering Research Council of Canada, which is gratefully acknowledged. The authors are grateful to the anonymous reviewers for their helpful comments and suggestions.

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