On stability analysis of linear discrete-time switched systems using quadratic Lyapunov functions

The paper deals with the stability properties of linear discrete-time switched systems with polytopic sets of dynamics. 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 dynamic improves the chance that a quadratic Lyapunov function exists. We prove that the dependence on the dynamic can be actually assumed to be linear, with no prejudice on the effectiveness of the method. Moreover, we show that no gain in the sensibility is obtained if we allow the Lyapunov function to depend on the time as well. We conclude by showing that Lyapunov quadratic stability is a strictly stronger notion that uniform asymptotic stability.