Class- estimates and input-to-state stability analysis of impulsive switched systems☆
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 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- 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 denote the set of nonnegative integers, the set of nonnegative real numbers, and the -dimensional real Euclidean space. For denotes the Euclidean norm of . Let denote the set of all continuous functions from to .
Let and be two index sets. Consider the following impulsive switched system where is the system state, the system
Class- estimates
In this section, we establish two general class- estimates for solutions of scalar impulsive switched systems. Lemma 3.1 Consider the scalar impulsive switched systemwhere is locally integrable, and is locally Lipschitz for each . If ; and ,
then the system has a unique solution defined for all 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- 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 ( and ).
Our first result is concerned with -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 and where is the system state, is the disturbance input, is the observation matrix (), 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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2020, Nonlinear Analysis: Hybrid SystemsCitation Excerpt :The basis of the mathematical theory of impulsive systems as well as fundamental results on the existence and local stability of solutions are summarized in the monographs by Samoilenko and Perestyuk [1], Lakshmikantham et al. [2], Samoilenko and Perestyuk [7]. A more refined technique for the ISS and global stability analysis of impulsive control systems that relies on the concept of a candidate Lyapunov function with nonlinear rate functions has been employed in Liu et al., Dashkovskiy and Mironchenko, Feketa and Bajcinca, Mancilla-Aguilar and Haimovich [20, 23, 24, 25, 26, 27]. The nonlinear rate functions are used to bound the evolution of the candidate Lyapunov function along the nonlinear flows and jumps of the impulsive system more precisely than what is possible by means of an exponential-type Lyapunov function [9,10].
Input-to-state stability for impulsive switched systems with incommensurate impulsive switching signals
2020, Communications in Nonlinear Science and Numerical SimulationCitation Excerpt :Thus, a more comprehensive model named impulsive switched system was constituted, which cannot be well described by purely continuous or purely discrete models. A fundamental analysis of the properties of impulsive switched system can be found in [21], and there are some latest research results, see [22–26]. When investigating the dynamics of a nonlinear system, it is necessary to characterize the effects of external inputs.
Stability conditions of switched nonlinear systems with unstable subsystems and destabilizing switching behaviors
2019, Applied Mathematics and Computation
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Research supported in part by NSERC Canada.