Analysis of switched normal discrete-time systems

In this paper, we study stability and L₂ gain properties for a class of switched systems which are composed of normal discrete-time subsystems. When all subsystems are Schur stable, we show that a common quadratic Lyapunov function exists for all subsystems and that the switched normal system is exponentially stable under arbitrary switching. For L₂ gain analysis, we introduce an expanded matrix including each subsystem´s coefficient matrices. Then, we show that if the expanded matrix is normal and Schur stable so that each subsystem is Schur stable and has unity L₂ gain, then the switched normal system also has unity L₂ gain under arbitrary twitching. The key point is to establish a common quadratic Lyapunov function for all subsystems in the sense of unity L₂ gain.