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

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.