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 switching. The key point is establishing a common quadratic Lyapunov function for all subsystems in the sense of unity L₂ gain.