A Delay-dependent Condition for the Exponential Stability of Switched Linear Systems with Time-varying Delay

Kreangkri Ratchagit

doi:10.5539/jmr.v2n3p120

Abstract

We use a Lyapunov-Krasovskii functional approach to establish the exponential stability of linear systems with timevaryingdelay. Our delay-dependent condition allows to compute simultaneously the two bounds that characterize theexponential stability rate of the solution. A simple procedure for constructing switching rule is also presented.

Highlights

As an important class of hybrid systems, switched system is a family of differential equations together with rules to switch between them
A switched system can be described by a differential equation of the form x = fα(t, x), where { fα(.) : α ∈ Ω}, is a family of functions that is parameterized by some index set Ω, and α(·) ∈ Ω depending on the system state in each time is a switching rule/signal
2006), studying a switching system composed of a finite number of linear delay differential equations, it was shown that the asymptotic stability of this kind of switching systems may be achieved by using a common Lyapunov function method switching rule

Summary

Introduction

As an important class of hybrid systems, switched system is a family of differential equations together with rules to switch between them. Many important results have been obtained for switched linear systems, there are few results concerning the stability of the systems with time delay. 2003) studied the asymptotic stability for switched linear systems with time delay, but the result was limited to symmetric systems. 2006), studying a switching system composed of a finite number of linear delay differential equations, it was shown that the asymptotic stability of this kind of switching systems may be achieved by using a common Lyapunov function method switching rule. There are some other results concerning asymptotic stability for switched linear systems with time delay, but we do not find any result on exponential stability even for the switched systems without delay except We study the exponential stability of a class of switched linear systems with time-varying delay.

Problem formulation

Findings

Conclusions