Necessary and Sufficient Conditions for Stability of Discrete-Time Switched Linear Systems with Ranged Dwell Time

This letter deals with the stability analysis problem of discrete-time switched linear systems with ranged dwell time. A novel concept called $L$ -switching-cycle is proposed, which contains sequences of multiple activation cycles satisfying the prescribed ranged dwell time constraint. Based on $L$ -switching-cycle, two sufficient conditions are proposed to ensure the global uniform asymptotic stability of discrete-time switched linear systems. It is noted that two conditions are equivalent in stability analysis with the same $L$ -switching-cycle. These two sufficient conditions can be viewed as generalizations of the clock-dependent Lyapunov and multiple Lyapunov function methods, respectively. Furthermore, it has been proven that the proposed $L$ -switching-cycle can eventually achieve the nonconservativeness in stability analysis as long as a sufficiently long $L$ -switching-cycle is adopted. A numerical example is provided to illustrate our theoretical results.