M -matrix based stability analysis switched nonlinear time-varying delay systems
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M-matrix based stability analysis switched nonlinear time-varying delay systems Marwen Kermani1,2
· Anis Sakly1
Received: 26 February 2020 / Revised: 4 October 2020 / Accepted: 15 October 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract This paper addresses the stability analysis problem for a class of switched nonlinear time-varying delays systems. In particular, the considered time-varying delays depend on the subsystem number. Based on a common Lyapunov function, the M-matrix proprieties, and through the l1,∞ induced norms notion, new algebraic stability criteria under arbitrary switching are derived. Then, by employing the aggregation techniques and the Borne–Gentina criterion, the obtained results are extended to a class of switched nonlinear systems modeled by difference equations. By some comparisons, it is shown that the proposed approach allows us to avoid searching a common Lyapunov function, considered a hard task even for simple cases. Lastly, numerical simulations are included to illustrate the effectiveness of the obtained results. Keywords Discrete-time switched nonlinear system · Time-varying delays · Global asymptotic stability · Borne–Gentina criterion · Common Lyapunov function · Arrow form state matrix · Arbitrary switching
1 Introduction Switched systems are a special class of hybrid systems, including a collection of subsystems modeled by differential or difference equations, equipped with a switching law that coordinates the switching among them. Recently, switched systems have been widely applied in various disciplines of science and engineering such as power systems [1], automotive industry, automated highways [2], network control systems [3], and air traffic control systems [4]. Stability analysis of switched systems presents a theoretical challenge, which has attracted interest from many scientists [5–32]. According to the switching mechanism, two main problems have been addressed in the literature which are respectively, arbitrary switching and specific
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Marwen Kermani [email protected] Anis Sakly [email protected]
1
Laboratory of Automation, Electrical Systems and Environment (LAESE), National Engineering School of Monastir (ENIM), University of Monastir, Ibn El Jazzar, Skaness, 5019 Monastir, Tunisia
2
National School of Advanced Science and Technology of Borj Cedria, University of Carthage, BP 122 Hammam-Chott, 1164 Tunis, Tunisia
switching. Indeed, stability under arbitrary switching is a fundamental issue and an important topic in the design and analysis of such systems. This kind of switching strategy is considered in the cases where the switching mechanism is unknown or randomly generated. The most important result of this issue is based on a common Lyapunov function (CLF) for all the subsystems [6–8]. However, the existence of a CLF remains very difficult even for a family of linear stationary switched systems [9]. On the other hand, although it exists many results of studying switched systems under restricted switching [10, 11]
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