Stability and robust stabilization of discrete-time switched systems with time-delays: LMI approach

• Published in: Applied mathematics and computation Vol. 206; no. 2; pp. 570 - 578
• Main Author: Ibrir, Salim
• Format: Journal Article Conference Proceeding
• Language: English
• Published: Amsterdam Elsevier Inc 15.12.2008; Elsevier
• Subjects: Algebra; Algebraic geometry; Calculus of variations and optimal control; Convex optimization; Discrete-time systems; Exact sciences and technology; Global analysis, analysis on manifolds; Mathematical analysis; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Numerical linear algebra; Sciences and techniques of general use; Switched systems; System theory; Time-delay systems; Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds; Switched systems; Discrete-time systems; Time-delay systems; Convex optimization; System theory; Numerical linear algebra; Optimization method; Linear time; Discrete system; Delay system; Systems theory; Direct method; Asymptotic stability; Numerical analysis; Sufficient condition; Linear system; Matrix inversion; Applied mathematics; Discrete time; Numerical stability; Lyapunov function; Mathematical programming
• ISSN: 0096-3003; 1873-5649
• DOI: 10.1016/j.amc.2008.05.149

We present sufficient linear matrix inequality conditions for asymptotic stability and stabilizability of switched discrete-time linear systems subject to time-delays and norm bounded uncertainties. Namely, if these LMIs are solvable then, the switched system is exponentially stable for arbitrary switching. In fact, we show that any family of switched time-delay systems satisfying these conditions possesses a quadratic common Lyapunov function. We also discuss the implication of this result on the stabilizability of this class of systems by switching controllers that use common Lyapunov functions. We show that even the analysis is carried out through a common Lyapunov function, the applied controller is mode dependent.