Lyapunov functions for discrete-time arbitrary switching linear systems- Joint setting for mode dependent and independent constructions

The paper aims to present and prove a general result regarding the usage of Lyapunov functions in testing the global exponential stability of equilibrium {0} for discrete-time arbitrary switching linear systems. The generality consists, on the one hand, in the possibility of working with both mode-dependent and mode-independent Lyapunov functions, and on the other hand, in the diversity of available concrete expressions (weighted vector norms). The flexible mathematical formulation allows the retrieval of several stability criteria that had been reported by separate articles, as various particular cases included in the generality of our new result. From this general result, individual criteria are derived by considering extended state-space representations for the description of systems dynamics, as well as several types of absolute norms in the definition of Lyapunov functions.