Robust stability and stabilization of uncertain switched discrete-time systems
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RESEARCH
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Robust stability and stabilization of uncertain switched discrete-time systems G Rajchakit1* , T Rojsiraphisal2,3 and M Rajchakit1 *
Correspondence: [email protected] 1 Major of Mathematics and Statistics, Faculty of Science, Maejo University, Chiangmai, 50290, Thailand Full list of author information is available at the end of the article
Abstract This paper is concerned with the robust stability and stabilization for a class of switched discrete-time systems with state parameter uncertainty. Firstly, a new matrix inequality considering uncertainties is introduced and proved. By means of it, a novel sufficient condition for robust stability and stabilization of a class of uncertain switched discrete-time systems is presented. Furthermore, based on the result obtained, the switching law is designed and has been performed well, and some sufficient conditions of robust stability and stabilization have been derived for the uncertain switched discrete-time systems using the Lyapunov stability theorem, block matrix method, and inequality technology. Finally, some examples are exploited to illustrate the effectiveness of the proposed schemes. Keywords: switching design; uncertain discrete system; robust stability and stabilization; Lyapunov function; linear matrix inequality
1 Introduction A switched system is a hybrid dynamical system consisting of a finite number of subsystems and a logical rule that manages switching between these subsystems. Switched systems have drawn a great deal of attention in recent years; see [–] and references therein. The motivation for studying switched systems comes partly from the fact that switched systems and switched multicontroller systems have numerous applications in control of mechanical systems, process control, automotive industry, power systems, aircraft and traffic control, and many other fields. An important qualitative property of switched system is stability [–]. The challenge of analyzing the stability of switched system lies partly in the fact that, even if the individual systems are stable, the switched system might be unstable. Using a common quadratic Lyapunov function on all subsystems, the quadratic Lyapunov stability facilitates the analysis and synthesis of switched systems. However, the obtained results within this framework have been recognized to be conservative. In [], various algorithms both for stability and performance analysis of discrete-time piece-wise affine systems were presented. Different classes of Lyapunov functions were considered, and how to compute them through linear matrix inequalities was also shown. Moreover, the tradeoff between the degree of conservativeness and computational requirements was discussed. The problem of stability analysis and control synthesis of switched systems in the discrete-time domain was addressed in []. The approach followed in [] looked at the existence of a switched quadratic Lyapunov function to check asymptotic stability of the switched system under consideration. Two different linear matrix
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