Necessary and sufficient condition for stabilizability of discrete-time linear switched systems: a set-theory approach*

In this paper, the stabilizability of discrete-time linear switched systems is considered. Several sufficient conditions for stabilizability are proposed in the literature, but not a necessary and sufficient one. The main contribution is a computation-oriented necessary and sufficient condition for stabilizability based on set-theory. Based on such a condition, an algorithm for computing the Lyapunov functions and a procedure to design the stabilizing switching control law are provided. The generic algorithm is based on the invariance of unions of compact, convex sets containing the origin and is applied to numerical examples using ellipsoids and polytopes. It will be shown in particular that no assumption is made on the existence of a Schur convex combination of the matrices, assumption on which the Lyapunov-Metzler inequalities approach is based. Further discussions with respect to the literature and concluding remarks are also proposed.