Lyapunov–Krasovskii functionals for input-to-state stability of switched non-linear systems with time-varying input delay

This study investigates the input-to-state stability issue and the Lyapunov-Krasovskii functional construction for a class of switched non-linear systems with time-varying input delay under mode-dependent average dwell-time scheme. The authors assume that an exponential stabilising state feedback controller has been predesigned for the nominal system and the Lyapunov function for the nominal closed-loop system is available. When input delay and disturbance are involved, the closed-loop system may be unstable under the average dwell time for the nominal system. In allusion to this instance, by introducing a positive definite relaxation matrix and constructing a novel piecewise Lyapunov-Krasovskii functional based on the known Lyapunov function, they provide sufficient conditions and the upper bound of the input delay such that the closed-loop system is input-to-state stable with respect to the disturbance. Finally, numerical examples are given to illustrate the effective of the proposed method.