This paper investigates input-to-state stability (ISS) for a class of discrete time-delay systems with switching and impulsive signals. By the Lyapunov–Krasovskii technique, a dwell-time bound and delay bound are clearly presented to contribute to the ISS for discrete time-delay systems. A significant subsequence method of the switched and impulsive sequence will firstly be applied to study the ISS of discrete systems. Based on this method, some new conditions guaranteeing ISS are presented. Compared with the existing results on related problems, the obtained stability criteria are less conservative as it is only required for the specially designed Lyapunov function to be non-increasing along each of the defined subsequences of the switched and impulsive time. Examples of network control systems are presented to illustrate the main results at the end.

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.

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