Comparison between classes of state-quadratic Lyapunov functions for discrete-time linear polytopic and switched systems

The paper deals with the stability properties of linear discrete-time switched systems with polytopic sets of modes. The most classical way of studying the uniform asymptotic stability of such a system is to check for the existence of a quadratic Lyapunov function. It is known from the literature that letting the Lyapunov function depend on the time-varying switching parameter improves the chance that a quadratic Lyapunov function exists. Our objective is to compare different notions of quadratic stability. The contribution of this paper is twofold. In the first part we consider switching systems satisfying a certain non-degeneracy assumption and we prove that, for such systems, no gain in the stability analysis is obtained if we allow the Lyapunov function to depend explicitly also on time. In the second part we consider the case where the non-degeneracy assumption is violated. We prove that in this case allowing the Lyapunov function to depend on time is less conservative. We also show that new LMI conditions can be used in order to characterize the existence of a time-dependent quadratic Lyapunov function. Moreover in the paper we discuss the case where the variation of the switching parameter is bounded by a prescribed constant between two subsequent times.