Vol.32, No.2ACTA AUTOMATICA SINICAMarch, 2006Robust Exponential Stability of Discrete Time Perturbed Impulsive Switched Systems1)ZONG Guang-DengWU Yu-Qiang(Research Institute of Automation, Qufu Normal University, Qufu273165) (E-mail: zonggdengyahoo.com.cn)AbstractThe robust exponential stability of a class of discrete time impulsive switched systems with structure perturbations is studied. Based on the average dwell time concept and by dividing the total activation time into the time with stable subsystems and the time with unstable subsystems, it is shown that if the average dwell time and the activation time ratio are properly large, the given switched system is robustly exponentially stable with a desired stability degree. Compared with the traditional Lyapunov methods, our layout is more clear and easy to carry out. Simulation resultsvalidate the correctness and effectiveness of the proposed algorithm.Key wordsSwitched systems, robust exponential stability, structure perturbations, average dwell time1Introduction A switched system is an important class of hybrid systems consisting of a family of continuous-time or discrete-time subsystems and a rule that orchestrates the switching among them. Switched systemsare different from the common continuous-time or discrete-time systems, because they have some special properties. It is very important how to choose the rule. Recently, the study on the switched systems analysis and switching control has attracted more and more attentions112. Its main result lies in: 1) many systems encountered in practice exhibit switching phenomena between several subsystems due to the inherent multi-model or various environmental factors3,4; 2) the methods of intelligent controldesign are based on the idea of switching between different controllers3,5; 3) switching controllers can achieve a better performance than the traditional feedback controllers, e.g., Narendra and Balakrishnan improved the system performance by using a set of switched adaptive controllers6. In this paper, under the assumption that the subsystems include stable and unstable subsystems, we study the robust exponential stability of a class of discrete time impulsive switched systems with structure perturbations. 7 considered the exponential stability of a class of continuous time switched systems that only contained stable subsystems, and proved that the switched system is exponentially stable when the dwell time of the switching signal (the time between every two consequent switchings) is large enough. 8 proposed the concept of the average dwell time, and proved that the switched system is still exponentially stable when the average dwell time is large enough.9 studied the switched systems containing stable and unstable subsystems. The total activation time of the switched system was divided into the activation time of the stable subsystems and unstable subsystems. The authors proved that the switched system is exponentially stable when the average dwell time and the total activation time period ratio between the stable subsystems and unstable subsystems are both large enough. Similar to 9, 11 considered the exponential stability of a class of discrete time switched systems. However, there are few results on the robustness of the uncertain switched systems. Meanwhile,for the practical systems, the impulsive effect often occurs when switching exists. Therefore, in this note, we consider the robust exponential stability of the impulsive switched systems with structure perturbations. With the knowledge of the average dwell time and the modular matrix, we prove that the given switched system is robustly exponentially stable with the desired stability degree under the proposed switching signal, if the average dwell time and the total activation time period ratio between the stable subsystems and unstable subsystems are both properly large. Finally, the numerical simulations validate the algorithm in this paper.1) Supported by National Natural Science Foundation of P.R. China (60174042, 60304003, 60574007), Natural Science Foundation of Shandong Province (Y2003G02) and Qufu Normal University Foundation (xj0511) Received February 10, 2004; in revised form December 5, 2005Copyright c? 2006 by Editorial Office of Acta Automatica Sinica. All rights reserved.No.2ZONG Guang-Deng et al.: Robust Exponential Stability of Discrete Time Perturbed 1872Problem formulation and preliminaries Consider the following discrete time perturbed switched systemx(k + 1) = A(k)+ A(k)x(k),x(0) = x0(1)where x(k) Rnis the state, and IN= 1,N is the switching signal which is a piecewise constant function. When (k) = i, i IN, system (1) switches to the i-th subsystem. A(k)describes the structure perturbation of the system mode.Definition 1. |G|m= |gij| is the modular matrix of G = gij, i.e., the elements of |G|mare the absolute values of the corresponding elements of matrix G. Assumption 1. Supp