Input‐to‐state stability of discrete‐time switched delayed systems with delay‐dependent impulses: Admissible edge‐dependent average impulsive interval

Abstract

This article addresses the input‐to‐state stability (ISS) problem for general discrete‐time impulsive and switched delayed systems coupled by delay‐dependent impulses. In our studied systems, the impulsive signal has a large degree of freedom, that is, the impulse effect can occur simultaneously with or different from the switching effect. A new Lyapunov–Krasovskii functional is proposed to deal with the delay effect of switching signal and impulsive jump, which is divided into the delay‐independent part and delay‐dependent part. A new concept named admissible edge‐dependent average impulsive interval (AED‐AII) is set up to characterize the impulse sequence relative to the admissible edge‐dependent average dwell time (AED‐ADT) concept of the switching signal. Based on AED‐ADT approach and AED‐AII technique, some improved sufficient outcomes guaranteeing ISS of discrete‐time impulsive and switched delayed systems are presented, respectively, for stabilizing and destabilizing delay‐dependent impulses. Meanwhile, the relationship among AED‐AII, AED‐ADT, the jump amplitude of switching signal and impulse signal at their respective discrete points, the decay rate of Lyapunov functional for system dynamics is connected to pursue ISS of the whole system. Compared with previous works, our setups enjoy the following characteristics: it thoroughly digs the influence of time delay existing in impulses and system dynamics on the function part of L‐K functional, which are less conservative relative to other works; the proposed AED‐ADT switching signals and AED‐AII jump signals cover a large set of hybrid systems. Two numerical examples are given to illustrate the effectiveness and less conservativeness of the obtained results.

Stability analysis for stochastic time‐delayed systems allowing admissible edge‐dependent switching signals

“…The characteristic of MDAII technique only focused on the currently activated impulse. In References 24 and 25, AED‐AII method was proposed according to properties of edges. AED‐AII requires that the type of impulses to be activated relies on the subsystems being activated, which leads to some limitations in application.…”

“…It is worth noting that MDADT method is only relevant to the currently active subsystem. Because of the characteristics of edges, a connection is built between the current subsystem and the next subsystem to be activated, which was defined as edge dependence in References 24 and 25. In References 24 and 25, the gap between two successive switchings was described by admissible edge‐dependent average dwell time ($$$ \mathrm{AED}\hbox{-} \mathrm{ADT} $$$).…”

“…Because of the characteristics of edges, a connection is built between the current subsystem and the next subsystem to be activated, which was defined as edge dependence in References 24 and 25. In References 24 and 25, the gap between two successive switchings was described by admissible edge‐dependent average dwell time ($$$ \mathrm{AED}\hbox{-} \mathrm{ADT} $$$). Besides, for impulsive interval, mode‐dependent average impulsive interval ($$$ \mathrm{MDAII} $$$) was introduced in Reference 26.…”

“…Besides, for impulsive interval, mode‐dependent average impulsive interval ($$$ \mathrm{MDAII} $$$) was introduced in Reference 26. To declare the relationship between the current impulse and the next impulse to be activated, a method named admissible edge‐dependent average impulsive interval ($$$ \mathrm{AED}\hbox{-} \mathrm{AII} $$$) was proposed 24,25 . However, because of the impulses depending on the subsystems, AED‐AII technique needs to know which subsystems is activated, which exists some limitations.…”

“…The main contributions contain: A new approach named admissible amputated edge‐dependent average impulsive interval ($$$ \mathrm{AAED}\hbox{-} \mathrm{AII} $$$) is proposed according to properties of edges. Compared with AED‐AII in References 24 and 25 and MDAII, the characteristic of AAED‐AII is that impulsive generation is independent of subsystem, that is, the type of impulses is unpredictable according to activated subsystems. Concurrently, for AAED‐AII, a relation is established between the current impulse and the next impulse. The subsystem switching sequence in this article is allowed to be different from the impulsive jump sequence.…”