Thai Journal of Mathematics Special Issue (Annual Meeting in Mathematics, 2008) : 1-13 www.math.science.cmu.ac.th/thaijournal Online ISSN 1686-0209 Stability and Robust Stability of Discrete-Time Switched Systems with Delays** J. Thipcha and P. Niamsup * Abstract : In this paper, we study stability and roust stability of the zero solution of discrete-time switched systems with delays. And we give sufficient conditions which guarantee that the zero solution of discrete-time switched systems with delays is asymptotically stable and robustly stable. Numerical simulations are also given to illustrate the results. Keywords : Switched system, Asymptotically stable, Robustly stable, Lyapunov function, Multiple Delay. 2000 Mathematics Subject Classification : Applied Mathematics. 1 Introduction An important class of hybrid dynamical systems is a class of switched sys- tems with delays, which compose of a family of continuous-time or discrete-time subsystems and a rule orchestrating the switching between these subsystems. A discrete-time switched system can be described by a difference equation of the form x k+1 = f i k (x k ,x k-h ), k ∈ Z + = {0, 1, 2, ...} where h ≥ 1 is the state delay, {f i k (·): i k ∈ M} is a family of functions from R 2n to R n that is parameterized by the index set M, and i k ∈M is a switching signal. The set M is typically a finite set. Some examples of such switched systems: Automobile with a manual gearbox,see [5]. The motion of a car that travels along a fixed path can be characterised by two continuous states: velocity v and position s. The system has two input: the throttle angle (u) and the engaged gear (g). It is evident that the manner in which the velocity of the car responds to the throttle input depends on the engaged gear. Recently, there have been many studies of *Corresponding author: **This paper is a part of Jenjira’s thesis from Chiang Mai University. c  2008 by the Mathematical Association of Thailand All rights reserve.