Exponential Stability of Switched Systems with Unstable Subsystems: A Mode-Dependent Average Dwell Time Approach
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Exponential Stability of Switched Systems with Unstable Subsystems: A Mode-Dependent Average Dwell Time Approach Dehua Xie · Hongyu Zhang · Hongbin Zhang · Bo Wang
Received: 5 December 2012 / Revised: 10 April 2013 © Springer Science+Business Media New York 2013
Abstract This article studies the exponential stability of switched systems with unstable subsystems. By using the multiple Lyapunov function (MLF) method combined with mode-dependent average dwell time (MDADT) techniques, less conservative exponential stability conditions are derived in terms of a set of solvable linear matrix inequalities (LMIs). Compared to the existing results, unstable subsystems are considered based on MDADT in this paper. It is revealed that switched systems can be exponentially stable under slow switching schemes and also in the presence of fast switching of unstable subsystems. A numerical example and its simulations are also given to verify the effectiveness of the proposed method. Keywords Switched system · Exponential stability · Mode-dependent average dwell time · Stable and unstable subsystems
1 Introduction Switched systems, typically, composed of a finite number of subsystems and a corresponding switching signal governing the switching law between the subsystems, are a important class of hybrid systems [12]. The motivation for studying switched D. Xie · H. Zhang · H. Zhang () · B. Wang Centre for Nonlinear and Complex Systems, School of Electronic Engineering, University of Electronic Science and Technology of China, Chengdu 611731, P.R. China e-mail: [email protected] D. Xie e-mail: [email protected] H. Zhang e-mail: [email protected] B. Wang e-mail: [email protected]
Circuits Syst Signal Process
systems comes from the fact that many practical plants are inherently multi-model in the sense that several dynamic subsystems are needed to describe their behavior when the system undergoes internal or external abrupt changing of environmental factors [9]. Due to aforementioned characteristics, switched systems have attracted increasing interest in recent years from various fields, such as power electronics and systems, flight control systems, network control systems, and so on; see for example [3, 4, 6, 7, 15, 25], and the references therein. Switched systems have been extensively investigated and have produced many sound and pioneered results [2, 13, 20]. In particular, stability problem and switching law design are focuses for switched systems [21–24, 26]. As is well known, stability, especially exponential stability, is a primary problem in the science and engineering fields. Switched systems, without exception, possess stability analysis problems. In a certain sense, switched systems can be categorized as state-controlled, time-controlled, or a combination of the two [12]. In recent years, much effort has been focused on time-controlled switched systems; for details, see the survey paper [14]. Fortunately, there have been many remarkable results in timecontrolled switched systems. A typical example is the ave
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