Delay-independent stability criteria under arbitrary switching of a class of switched nonlinear time-delay systems

Kermani and Sakly Advances in Difference Equations (2015) 2015:225 DOI 10.1186/s13662-015-0560-1 RESEARCH Open Access Delay-independent stability criteria under arbitrary switching of a class of switched nonlinear time-delay systems Marwen Kermani* and Anis Sakly * Correspondence: Research Unit of Industrial Systems Study and Renewable Energy (ESIER), National Engineering School of Monastir, Ibn El Jazzar, Skaness, Monastir, 5019, Tunisia Abstract This paper addresses the stability problem of a class of switched nonlinear time-delay systems modeled by delay differential equations. Indeed, by transforming the system representation under the arrow form, using a constructed Lyapunov function, the aggregation techniques, the Borne-Gentina practical stability criterion associated with the M-matrix properties, new delay-independent conditions to test the global asymptotic stability of the considered systems are established. In addition, these stability conditions are extended to be generalized for switched nonlinear systems with multiple delays. Note that the results obtained are explicit, they are simple to use, and they allow us to avoid the problem of searching a common Lyapunov function. Finally, an example is provided, with numerical simulations, to demonstrate the effectiveness of the proposed method. Keywords: continuous switched nonlinear systems time delays; global asymptotic stability; Borne-Gentina criterion; common Lyapunov function; arrow form state matrix; arbitrary switching 1 Introduction Switched systems are a class of important hybrid systems which consist of a finite number of subsystems that are governed by differential or difference equations and a switching law which defines a specific subsystem being activated during a certain interval of time. Due to the physical properties or various environmental factors, many real-world systems can be modeled as switched systems such as computer science, autonomous transmission systems, computer disc drivers, control systems, electrical engineering and technology, automotive industry, air traffic management, chemical systems, power systems and communication networks, and other applications [–]. On the other hand, considerable efforts have been made as regards the analysis and the design of switched systems. There are still many open and challenging issues remaining to be tackled, despite great successes reported during the past several decades. Among those research topics, stability analysis and stabilization have attracted most attention [–, –]. Hence, several methods have been proposed for these matters. It is commonly recognized that there are mainly three basic types of problems considering the stability and the stabilization issues of switched systems [–]: (i) guaranteeing of asymptotical stability of the switched system with arbitrary switching; (ii) identification of the limited but useful class of stabilizing switching © 2015 Kermani and Sakly. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Kermani and Sakly Advances in Difference Equations (2015) 2015:225 Page 2 of 20 laws; and (iii) construction of asymptotically stabilizing switching signals. Specifically, the stability analysis under arbitrary switching problem (i) which will be focused on in this work deals with the case that all subsystems are stable. This problem seems trivial, but it is fundamental and important [, –], since we can find many examples where all subsystems are stable but inappropriate switching rules can make the whole system unstable. In addition, stability under arbitrary switching is a desirable property of switched systems due to its practical importance and also it allows us to consider higher control specifications for the system. For this problem, it is well known that the existence of a common Lyapunov function for individual systems guarantees stability of the switched system under arbitrary switching [, ]. Therefore, this method is usually very difficult to apply even for continuous-time switched linear systems [, ]; however, it becomes more complicated for switched nonlinear systems. Yet, some attempts are presented to construct a common Lyapunov function for nonlinear switched systems [, ]. On the other hand, time delay is a common phenomenon encountered in various practical and engineering systems [, ] such as chemical processes, nuclear reactors, models of lasers, electrical systems, aircraft stabilization, biological systems, and systems with lossless transmission lines; and most of them appear in the form of time-varying delay. It is a well-known fact that the presence of delays is an inherent feature of many physical processes, the big sources of instability and poor performances in switched systems. Thus, it is important to investigate the stability analysis problem for switched delay systems [–, , , –]. It is noted that current methods of the analysis and design for time-delay systems can be classified into two categories: delay-independent criteria and delay-dependent ones. In this work, in view of a delay-independent analysis, we expect to aid in studying stability analysis of switched systems under an arbitrary switching law. Presently, the most important consideration in the analysis of switched systems is their stability. Recently, many researchers focused on switched time-delay systems. Indeed, the stability analysis problem of switched time-delay systems has attracted a lot of attention from many researchers [, –]. However, the presence of delays makes this problem much more complicated. Thus, the main approach for stability analysis under arbitrary switching relies on the use of a Lyapunov-Krasovskii functional and the LMI approach for constructing a common Lyapunov function []. In fact, getting such a function becomes more complicated even for switched linear systems. Consequently, few results have been obtained for continuous-time switched nonlinear time-delay systems []. Motivated by these mentioned shortcomings for the existing results in this framework as well in the sense of various methods that can be employed in this paper, we address this challenging problem. Indeed, based on the construction of a common Lyapunov function as well as the use of the Borne-Gentina practical stability criterion [–, –] associated with the M-matrix properties [, ], new delay-independent sufficient stability conditions for continuous-time switched nonlinear time-delay systems under arbitrary switching are established. Subsequently, these obtained results are extended to be generalized for continuous-time switched nonlinear s (...truncated)