Feedback stabilization of switched systems under arbitrary switching: A convex characterization

In this paper, we study stabilizability of discrete-time switched linear systems where the switching signal is considered as an arbitrary external input (and not a control variable). We characterize feedback stabilization via a hierarchy of necessary and sufficient linear matrix inequalities (LMIs) conditions based on novel graph structures. We analyze both the cases in which the controller has (or has not) access to the current switching mode, the so-called mode-dependent and mode-independent settings, providing specular results. Moreover, our approach provides explicit piecewise-linear and memory-dependent linear controllers, highlighting the connections with existing stabilization approaches. The effectiveness of the proposed technique is finally illustrated with the help of some numerical examples.