On stability analysis of discrete-time uncertain switched nonlinear time-delay systems

Kermani and Sakly Advances in Difference Equations 2014, 2014:233 http://www.advancesindifferenceequations.com/content/2014/1/233 RESEARCH Open Access On stability analysis of discrete-time uncertain switched nonlinear time-delay systems Marwen Kermani* and Anis Sakly * Correspondence: Research Unit of Industrial Systems Study and Renewable Energy (ESIER), Department of Electrical Engineering, National Engineering School of Monastir (ENIM), Ibn El Jazzar, Skaness, Monastir, 5019, Tunisia Abstract This paper addresses the stability analysis problem for a class of discrete-time switched nonlinear time-delay systems with polytopic uncertainties. These considered systems are characterized by delayed difference nonlinear equations which are given in the state form representation. Then, a transformation under the arrow form is employed. Indeed, by constructing an appropriated common Lyapunov function, and also by resorting to the Kotelyanski lemma and the M-matrix proprieties, new delay-independent stability conditions under arbitrary switching law are deduced. Compared with the existing results of switched systems, those obtained results are formulated in terms of the unknown polytopic uncertain parameters, explicit and easy to apply. Moreover, this method allows us to avoid the search for a common Lyapunov function which is a difficult matter. Finally, a numerical example is presented to illustrate the effectiveness of the proposed approach. Keywords: discrete-time switched nonlinear time-delay systems; polytopic uncertainties; global robust asymptotic stability; M-matrix properties; Kotelyanski lemma; arrow matrix; arbitrary switching 1 Introduction Switched systems are an important class of hybrid systems. Generally speaking, a switched system is composed of a family of subsystems described by differential or difference equations and a switching rule orchestrating the switching between the subsystems that have attracted much attention in control theory and practice during recent decades. Switched systems can be efficiently used to model many practical systems which are inherently multi-model in the sense that several dynamical systems are required to describe their behavior. For example, many physical processes exhibit switched and hybrid nature. Switched systems have strong engineering background in various areas and are often used as a unified modeling tool for a great number of real-world systems, such as power electronics, chemical processes, mechanical systems, automotive industry, aircraft and air traffic control and many other fields [–]. Undoubtedly, stability is the first requirement for a system to work properly; thus, stability analysis of switched systems presents a theoretical challenge, which has attracted growing attention in the literature [–]. However, stability under arbitrary switching is a fundamental issue and an important topic in the design and analysis of these systems. ©2014 Kermani and Sakly; 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. Kermani and Sakly Advances in Difference Equations 2014, 2014:233 http://www.advancesindifferenceequations.com/content/2014/1/233 To solve this matter, many effective methods have been developed [, , , ]. Within this framework, we are required to find conditions that guarantee asymptotic stability under arbitrary switching rules. Indeed, it is well known that the existence of a common Lyapunov function [–] for all subsystems is sufficient to guarantee the stability under arbitrary switching law. However, finding such a function is often difficult even for discretetime switched linear systems []. Consequently, this problem becomes more complicated for discrete-time switched nonlinear systems, and relatively fewer results have been reported in this context. On the other hand, to avoid the problem of the existence of a common Lyapunov function, some attention has been widely given to seeking conditions that guarantee the stability of the switched systems under restricted switching. Although many efficient approaches and important results have been proposed for this alternative, such as the multiple Lyapunov function approach [] and average dwell time method [, ], stability under arbitrary switching, which is considered in this work, remains most preferable for practical systems. Indeed, it offers great flexibility and it allows us to achieve other performances for designing a control law along stability maintained. As is well known, time-delay phenomena are usually confronted in many engineering systems [, –, –, , ], such as chemical engineering systems, hydraulic systems, inferred grinding model, neural network, nuclear reactor, population dynamic model and rolling mill. Recently, stability analysis for discrete-time switched time-delay systems has been investigated [, –, –]. It is noteworthy that there are two divisions in the recent literature addressing the stability analysis of time-delay systems, namely delay-independent criteria and delay-dependent criteria. Therefore, in view of delay-independent criteria, this paper will try to aid the stability analysis under arbitrary switching law. On the other hand, when practical systems are modelled, uncertainties of system parameters are often included. Therefore, most of the systems refer to uncertainties in real applications. Indeed, basically, two kinds of uncertainties are simultaneously encountered in the open literature, widely polytopic uncertainties and norm-bounded. From the practical viewpoint, it is important to investigate switched systems with uncertain parameters [, , , –, , ]. Thus, polytopic uncertainties exist in many real systems, and most of the uncertain systems can be approximated by systems with polytopic uncertainties []. On the other hand, the polytopic uncertain systems are less conservative than systems with norm-bounded uncertainties []. Up to now, discrete-time uncertain switched time-delay systems have received more and more attention. Although many interested and significant results on stability problems for those systems have been established [, , , –], those previous works were mainly focused on several hot topics of Lyapunov stability theory and most of them were interested in the linear case [, –, –]. Thus, due to the complexity of switched nonlinear systems, unfortunately, the available results on the stability of uncertain discretetime switched nonlinear time-delay systems are limited []. Motivated by these mentioned shortcomings for the existing results as well as in the sense of various methods that can be employed, in this paper we aim to establish new stability analysis for a class of discrete-time switched time-delay (...truncated)