New Stability Conditions for Switched Linear Systems: A Reverse-Timer-Dependent Multiple Discontinuous Lyapunov Function Approach

In this article, the stability issues are addressed for switched linear systems (SLSs) with mode-dependent average dwell time (MDADT). By dividing the dwell time into several segments, and constructing a reverse timer which starts timing at the end of each segment, we propose a new reverse-timer-dependent multiple discontinuous Lyapunov function (RTDMDLF), which is more general than the multiple Lyapunov function (MLF) and the multiple discontinuous Lyapunov function (MDLF). With the help of the RTDMDLF approach, several convex and nonconvex stability conditions are derived for SLSs with both stable and unstable subsystems, and the relation of these conditions and existing ones is revealed. Moreover, the stability conditions for SLSs with all stable subsystems are also given. All the results are presented in terms of infinite-dimensional linear matrix inequalities (LMIs), which can be relaxed into computable conditions by using a discretized approach. It is shown that the tighter bound of MDADT can be achieved by the RTDMDLF approach compared with those of the literature. Finally, the advantages of the results are illustrated within three numerical examples.