ISS small-gain theorem for networked discrete-time switching systems

In this paper it is proved that a networked discrete-time switching system, equipped with a given switches digraph, is input-to-state stable, provided that there exist multiple Lyapunov functions (one for each mode) for each subsystem in the network, satisfying suitable standard inequalities, and provided that a set of suitable vector small-gain conditions are satisfied. The small-gain theorem here provided for the input-to-state stability takes into account the switches digraph. That is, the less is the number of edges in the switches digraph, the less is the number of involved Lyapunov inequalities and small-gain conditions which, if satisfied, guarantee the input-to-state stability of the entire switching system under study. The multiple Lyapunov functions for the entire system, guaranteeing the input-to-state stability, are determined by the multiple Lyapunov functions for each subsystem in the family. To the author's best knowledge, this is the first paper in the literature concerning small-gain theorems for the input-to-state stability of nonlinear discrete-time switching systems with given switches digraphs.