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New sufficient conditions for the asymptotic stability of discrete time-delay systems

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8 pages
This paper is concerned with asymptotic stability of switched discrete time-delay systems. The system to be considered is subject to interval time-varying delays, which allows the delay to be a fast time-varying function and the lower bound is not restricted to zero. Based on the discrete Lyapunov functional, a switching rule for the asymptotic stability for the system is designed via linear matrix inequalities. Numerical example is included to illustrate the effectiveness of the result.
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SangapateAdvances in Difference Equations2012,2012:28 http://www.advancesindifferenceequations.com/content/2012/1/28
R E S E A R C HOpen Access New sufficient conditions for the asymptotic stability of discrete timedelay systems P Sangapate
Correspondence: prisayarat@mju.ac. th Department of Mathematics, Maejo University, Chiangmai, 50290, Thailand
Abstract This paper is concerned with asymptotic stability of switched discrete timedelay systems. The system to be considered is subject to interval timevarying delays, which allows the delay to be a fast timevarying function and the lower bound is not restricted to zero. Based on the discrete Lyapunov functional, a switching rule for the asymptotic stability for the system is designed via linear matrix inequalities. Numerical example is included to illustrate the effectiveness of the result. Keywords:Switching design, discrete system, asymptotic stability, Lyapunov func tion, linear matrix inequality
Introduction As an important class of hybrid systems, switched systems arise in many practical pro cesses that cannot be described by exclusively continuous or exclusively discrete mod els, such as manufacturing, communication networks, automotive engineering control and chemical processes (see, e.g., [13] and the references therein). On the other hand, timedelay phenomena are very common in practical systems. A switched system with timedelay individual subsystems is called a switched timedelay system; in particular, when the subsystems are linear, it is then called a switched timedelay linear system. During the last decades, the stability analysis of switched linear continuous/discrete timedelay systems has attracted a lot of attention [418]. The main approach for stabi lity analysis relies on the use of LyapunovKrasovskii functionals and linear matrix ine qulity (LMI) approach for constructing a common Lyapunov function [1924]. Although many important results have been obtained for switched linear continuous time systems, there are few results concerning the stability of switched linear discrete systems with timevarying delays. It was shown in [5,7,11] that when all subsystems are asymptotically stable, the switching system is asymptotically stable under an arbi trary switching rule. The asymptotic stability for switching linear discrete timedelay systems has been studied in [10], but the result was limited to constant delays. In [11], a class of switching signals has been identified for the considered switched discrete time delay systems to be stable under the average dwell time scheme. This paper studies asymptotic stability problem for switched linear discrete systems with interval timevarying delays. Specifically, our goal is to develop a constructive way to design switching rule to asymptotically stabilize the system. By using improved Lya punovKrasovskii functionals combined with LMIs technique, we propose new criteria
© 2012 Sangapate; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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