Quasi‐Time‐Dependent Controller for Discrete‐Time Switched Linear Systems With Mode‐Dependent Average Dwell‐Time

Abstract

This paper is concerned with the stability and stabilization problem of a class of discrete-time switched systems with mode-dependent average dwell time (MDADT). A novel Lyapunov function, which is both mode-dependent (MD) and quasi-time-dependent (QTD), is established. The new established Lyapunov function is allowed to increase at some certain time instants. A QTD H ∞ controller is designed such that the system is globally uniformly asymptotically stable (GUAS) and has a guaranteed H ∞ performance index. The new QTD robust controller designed in this paper is less conservative than the mode independent one which is frequently considered in literatures. Finally, a numerical example and a practical example are provided to illustrate the effectiveness of the developed results.Definition 2. For a switching signal (t) and any T ≥ t ≥ 0, let N p (T, t) be the switching numbers and the p th subsystem is activated over the interval [t, T]. T p (T, t) denotes the total running time of the p th subsystem over the interval [t, T] with p ∈ . We say that (t) has a mode-dependent average dwell time p if there exists a positive number N 0p (We call N 0p the mode-dependent chatter bounds here) and p satisfyDefinition 3 ([26]). For > 0, system (1)-(2) is said to be GUAS with an l 2 -gain, if under zero initial condition, system (1)-(2) is GUAS and ‖z‖ l 2 [0,T] ≤ ‖ ‖ l 2 [0,T] holds for all nonzero ∈ l 2 [0, T], 0 ≤ T < ∞.Definition 4. For > 0 and let 0 < < 1, the switched system (1)-(2) is said to be GUAS with an l 2 -gain, if under zero initial condition, system (1)-(2) is GUAS and ∑ ∞ k=k 0(1 − )z T (k)z(k) ≤ ∑ ∞ k=k 0

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