Input-to-state stability for hybrid delayed systems with admissible edge-dependent switching signals

Abstract

This study examines the input-to-state stability (ISS) for a class of hybrid time-delay systems with impulsive and switching signals. Compared with the existing results, the Lyapunov-like function is allowed to decrease and increase and have decreasing and increasing conditions alternately during the operation time of activated subsystems. Moreover, some sufficient criteria for ISS of impulsive and switched hybrid systems are obtained by using the method of admissible edge-dependent average dwell time (AED-ADT) together with the action of multiple Lyapunov-like function. When all of the subsystems are ISS, an impulsive switched hybrid system is ISS if AED-ADT satisfies a lower bound even if the impulses are destabilising. When all of the subsystems are unstable, a new definition of fast AED-ADT on the basis of the definition of AED-ADT is proposed. The new concept posits that the impulsive switched hybrid system is ISS if AED-ADT satisfies an upper bound coupled with stabilising impulses. For a special case in which the Lyapunov-like function decreases and increases alternately during the running time of some subsystems, a novel transformation technique of the extended Lyapunov-like functions is built to deal with the difficulty invoked by the external disturbance. A relationship is established amongst the AED-ADT scheme, impulses, the interval length of increasing of Lyapunov function and the scale between decreasing and increasing of Lyapunov function such that the impulsive switched hybrid system is ISS. The acquired stability results can be used to a large class of hybrid delayed systems because the transformation technique builds the bridge from the special case to the general case. We also provide an example to illustrate the validity of the results.