Asymptotic Stability Analysis for Switched Stochastic Nonlinear Systems Using Mode-dependent Uniformly Stable Functions
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ISSN:1598-6446 eISSN:2005-4092 http://www.springer.com/12555
Asymptotic Stability Analysis for Switched Stochastic Nonlinear Systems Using Mode-dependent Uniformly Stable Functions Dianfeng Zhang, Yong-Feng Gao, and Sheng-Li Du* Abstract: In this paper, we intend to investigate uniform global asymptotic stability in probability (UGAS-P) for a class of time-varying switched stochastic nonlinear systems. Conventional criteria on stability for switched stochastic systems are based on the negativity of the infinitesimal generator of Lyapunov functions, it is demonstrated that these criteria are conservative. Taking this fact into account, the infinitesimal generator for each active subsystem acting on Lyapunov functions is relaxed to be indefinite with the help of uniformly stable function (USF). Subsequently, improved criteria on asymptotic stability are proposed by applying the weakened condition and modedependent average dwell time (MDADT) technique. In addition, numerical examples are presented to verify the effectiveness of the obtained results. Keywords: Asymptotic stability, mode-dependent average dwell time, switched stochastic nonlinear systems, uniformly stable function.
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INTRODUCTION
Switched systems consisting of a family of continuous (or discrete) time subsystems orchestrated by a switching rule have been studied extensively due to a wide range of engineering applications and increasingly complexity of engineering systems in recent decades (see, for example, [1–5] and references therein). As a fundamental and challenging issue, stability analysis of switched systems has attracted a lot of attention. Various approaches have been developed to tackle the stability of switched systems under different switching mechanisms, see [6–10] and [5] for a good survey. In particular, uniform asymptotic stability (UAS) for time-varying systems has received more and more attention because of its inherent robustness [11, 12]. Recently, many interesting results have been presented to analyze the UAS of the switched systems, for instance, the extension of LaSalle’s invariance principle [13], the limiting systems [14, 15]. However, the existence of switching behaviors and the time-varying characteristic of the systems makes it quite challenging in using LaSalle’s invariance principle and guaranteeing the common limiting systems. This motivates us to continue the investigation of this direction. It is known that switching signals play a crucial role in stability analysis of switched systems, which can be
roughly divided into two major types: arbitrary switching and restricted switching. Common Lyapunov function approach is frequently used to develop the stability conditions of switched systems under arbitrary switching. It describes that if a common Lyapunov function has a uniformly negative definite derivative along any subsystem, then the switched system is uniformly global asymptotic stable [8, 9]. In fact, however, it is quite difficult to find such a common Lyapunov function [15, 16]. Thus, considerable attention
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