Discretized Lyapunov Function Approach for Switched Linear Systems under Dwell Time Constraint

Abstract

This paper is concerned with the stability and disturbance attenuation properties of switched linear system with dwell time constraint. A novel time-scheduled Lyapunov function is introduced to deal with the problems studied in this paper. To numerically check the existence of such time-scheduled Lyapunov function, the discretized Lyapunov function technique usually used in time-delay system is developed in the context of switched system in continuous-time cases. Based on discretized Lyapunov function, sufficient conditions ensuring dwell-time constrained switched system global uniformly asymptotically stable are established, then the disturbance attenuation properties in the sense ofL2gain are studied. The main advantage of discretized Lyapunov function approach is that the derived sufficient conditions are convex in subsystem matrices, which makes the analysis results easily used and generalized. Thus, theH∞control synthesis problem is considered. On the basis of analysis results in hand, the control synthesis procedures including both controller and switching law design are unified into one-step method which explicitly facilitates the control synthesis process. Several numerical examples are provided to illustrate the results within our paper.

Unified dwell time–based stability and stabilization criteria for switched linear stochastic systems and their application to intermittent control

“…Remark Since switched systems involve switching signals at variable times, they are indeed a class of time‐varying systems. Therefore, some analysis methods of time‐varying systems have been extended to switched systems in recent years, such as other works . The quadratic time‐dependent Lyapunov function V ( t , x )= x T P σ( t ) ( t ) x is more desirable because it can obtain verifiable stability criteria in terms of algebraic equations or matrix inequalities.…”

“…The dwell time–based stability or stabilization results had been obtained for switched linear deterministic systems , where the work of Allerhand and Shaked only establishing the minimum dwell time–based stability. In the following, we will compare the conservatives of our results with those literatures from the constant dwell time, the minimum dwell time, and the ranged dwell time.…”

“…According to the dynamic behaviors of the subsystems, the stability analysis of the switched systems can be roughly classified into 3 categories. In the first category, assuming that the switched systems, consisting of all stable subsystems, arbitrary switching or minimum dwell time switching problems, are studied. By using the comparison principle and the multiple Lyapunov functions, some average dwell‐time stability criteria were given in the work of Chatterjee and Liberzon .…”

“…In the work of Allerhand and Shaked, a discretized Lyapunov function approach was proposed for establishing a minimum dwell‐time stability criterion of linear switched systems. Later, Shaked and Gershon extended the approach to solve the control problem for switched stochastic systems, and the results in the work of Zhao et al are the special cases in the aforementioned work . Recently, Briat has proposed the clock‐dependent Lyapunov function method to obtain minimum and mode‐dependent dwell‐time stabilization conditions for switched systems.…”