Equivalence 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 and viable 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. The contribution of this paper is twofold. We first prove that under a non-degeneracy assumption the dependence on the switching function can be actually assumed to be linear with no prejudice on the effectiveness of the method. Moreover, we show that no gain is obtained even if we allow the Lyapunov function to depend on the time. Second, we introduce the notion of eventual accessible sets and we show that, in the degenerate case, it leads to a relaxation of the LMI conditions to check stability of switched linear systems. As a consequence, equivalence between different notions of quadratic stability can still be established under an additional assumption but, in general, allowing the Lyapunov function to depend on time leads to less conservative LMI conditions, as we explicitly show through an example. We also discuss the case where the variation of the switching parameter is bounded by a prescribed constant between two subsequent times.