Qualitative analysis of discrete-time switched systems

In this paper, we investigate some qualitative properties for time-controlled switched systems consisting of several linear discrete-time subsystems. First, we study exponential stability of the switched system with commutation property, stable combination and average dwell time. When all subsystem matrices are commutative pairwise and there exists a stable combination of unstable subsystem matrices, we propose a class of stabilizing switching laws where Schur stable subsystems (if exist) are activated arbitrarily while unstable ones are activated in sequence with their duration time periods satisfying a specified ratio. For more general switched system whose subsystem matrices are not commutative pairwise, we show that the switched system is exponentially stable if the average dwell time is chosen sufficiently large and the total activation time ratio between Schur stable and unstable subsystems is not smaller than a specified constant. Secondly, we use an average dwell time approach incorporated with a piecewise Lyapunov function to study the L₂ gain of the switched system. We show that when all subsystems are Schur stable and achieve an L₂ gain smaller than a positive scalar γ0, (1) if all subsystems have a common Lyapunov function in the sense of L₂ gain, then the switched system achieves the same L₂ gain γ0 under arbitrary switching; (2) if there does not exist a common Lyapunov function, then the switched system under an average dwell time scheme achieves a weighted L₂ gain γ0, and the weighted L₂ gain approaches normal L₂ gain if the average dwell time is chosen sufficiently large.