Input-to-state stability of switched systems with counter-dependent multiple discontinuous Lyapunov function

Abstract

This paper investigates the input-to-state stability (ISS) of switched systems under the mode-dependent average dwell time (MDADT) switching. A new non-equidistant partitioning (NEP) method for maximum and actual dwell time intervals is proposed, enhancing the flexibility in constructing the Lyapunov function. Building upon the proposed NEP method, a novel counter-dependent multiple discontinuous Lyapunov function (CDMDLF) is constructed by incorporating a counter, where the value of the counter is the partitioning number of the current mode-running time interval. Furthermore, the evolution of Lyapunov functions at adjacent partitioning points dynamically depends on the previous partitioning point rather than a fixed constant, which is more flexible and practical. Utilizing the developed CDMDLF, some less conservative stability criteria are established for nonlinear and linear switched systems. It is revealed that our proposed method covers the multiple discontinuous Lyapunov function (MDLF) and the switching-signal-based MDLF (SMDLF) methods as exceptional cases while providing tighter bounds on the MDADT. In the end, two numerical examples are given to illustrate the advantages and effectiveness of the presented approach.