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 to establish a common quadratic Lyapunov function for all subsystems in the sense of unity ℒ2 gain.
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