Trajectory‐based small gain theorems for ISpS and ISS of large‐scale networks of switched systems with arbitrary switchings

Abstract

The authors consider a large-scale switched non-linear system which is composed as an interconnection of N non-linear switched systems. Having assumed that every subsystem is stable uniformly with respect to the unobservable switching signals, it can happen that the entire interconnection is unstable. This study provides a sufficient condition for the uniform stability of the entire large-scale interconnection. The framework of input-to-state stability (ISS) is used for this purpose and the stability condition is of the small-gain type which is given in terms of trajectory-based ISS gains. To prove this result we extend the well-known result by Sontag and Wang on characterisations of the ISS property in terms of simpler stability notions to the case of switched systems with arbitrary switchings.