Stability and ℒ2 gain analysis for a class of switched symmetric systems

In this paper, we study stability and L₂ gain properties for a class of switched systems which are composed of a finite number of linear time-invariant symmetric sub-systems. We focus our attention mainly on discrete-time systems. When all subsystems are Schur stable, we show that the switched system is exponentially stable under arbitrary switching. Furthermore, we show that when all subsystems are Schur stable and have L₂ gains smaller than a positive scalar γ, the switched system is exponentially stable and has an L₂ gain smaller than the same γ under arbitrary switching. The key idea for both stability and L₂ gain analysis in this paper is to establish a common Lyapunov function for all subsystems in the switched system.