A Delay-dependent Stability Criteria for T-s Fuzzy System with Time-delays

This paper addresses the criterion of delay-dependent stability for nonlinear systems with time-delays which can be represented by Takagi–Sugeno (T–S) fuzzy model with time-delays. Based on Leibniz-Newton formula, the e¤ect of states and the delay states to the stability is considered to construct a Lyapunov functional and to form some special equations, which facilitate proposing a new delay-dependent stability criterion with less conservatism compared with the existing methods, which breaked through the limit to the varying ratio of timedelay. An illustrative example is presented to demonstrate the validity of the proposed stability criterion. Copyright 2005 IFAC Keywords: delay-dependent stability; Takagi–Sugeno (T–S) fuzzy system; Linear Matrix Inequality(LMI); Leibniz-Newton formula 1. INTRODUCTION During the past several years, Takagi-Sugeno (TS) fuzzy systems have been concerned(K. Tanaka, et al., 1996, J. Joh et al., 1998, G. Feng, 2001). The advantage of fuzzy modes, compared with conventional mathematical models, is which can represent complex nonlinear systems by fuzzy sets and fuzzy reasoning. The T-S fuzzy systems are to combine some simple local linear dynamic systems with their linguistic description to represent highly nonlinear dynamic systems. This method is feasible, since in many situations, human experts can provide linguistic descriptions of local systems in terms of IF-THEN rules. The method is quite interesting because it gives a way to smoothly connect local linear systems to form global nonlinear 1 Partially supported by NSFC (NO. 60274023).Corresponding Author: Dr. Xinping Guan. [email protected] systems by fuzzy membership functions. T-S fuzzy model can provide an e¤ective representation of complex nonlinear systems in terms of fuzzy sets and fuzzy reasoning applied to a set of linear input-output submodels. On the other side, the problem of stability of time-delay systems has received considerable attention in the last two decades. It can be classi…ed into two categories: delay-dependent and delayindependent criteria(E. Fridman, 2001). Since delay-dependent criteria make use of information on the length of delays, they are less conservative than delay-independent ones, especially, when the size of delay is small. To reduce the conservatism, many researchers have made great e¤orts and put forward lots of approaches. The main approaches consist of model transformation of the prototype system, the construction of new LyapunovKrasovskii functional and the development of new bounding techniques for the inner product of involved cross-terms. It has been proved that the transformed system is not equivalent to the original one, through the …rst-order transformation. This is because additional eigenvalues are introduced into the system characteristic equation. And the neutral transformation also yields a unequivalent system to the original one. A descriptor model transformation can transform the original system to an equivalent descriptor form representation and will not introduce additional dynamics(E. Fridman et al., 2001). In order to reduce the conservatism arisen from the inequality a b aXa + bX b; a; b 2 R; X > 0; used to determine the stability of the system, Park(Park, et al., 1998) introduced a free matrix M , and obtain a less conservative inequality, which is the well-known Park Inequality. And Moon etc. (Y. S. Moon et al., 2001(a) ; Y. S. Moon et al., 2001(b)) extended it to a more general form, the Moon inequality. In their analysis, they applied the Leibniz-Newton formula, but they just replaced some of the term x (t ) with x (t) R t t _ x (s) ds and do not consider the e¤ect of x (t) ; x (t ) and _ x (t) to the stability result. Wu etc. (M. Wu, et al., 2004) proposed a new analysis approach applying the Leibniz-Newton formula. However they only consider the linear systems, and they don’t break through the limit to the varying ratio of the time-delay, so their result is somehow conservative. Delayed fuzzy systems were …rst introduced and studied in(Y. Cao, et al., 2000) with developing the T-S fuzzy model. Generally speaking, the dynamic behaviors of systems with delay are more complicated than those without any delays. Fuzzy delayed systems are combined with fuzzy linguistic descriptions to achieve nonlinearity. The stability of T-S model fuzzy systems without delays has been widely studied in the recent years, and many results have been reported. However as for T–S fuzzy systems with time-delays, there are relatively seldom literatures that consider the delaydependent stability(Y. Cao, et al., 2001; Cao, Y. Y., 2001 ;X. P. Guan, et al., 2001 ).We extend the method of Min Wu to T-S fuzzy system, and get new criteria of asymptotical stability. In many cases, it is di¢ cult to know the delays exactly, but we can estimate the bounds of the delays in advance. In this paper, the upper bound of the varying ratio of time-delay is not necessary to be known. 2. PROBLEM FORMULATION In this section, we …rst consider a nonlinear timedelay system represented by the T-S fuzzy model with time-delay : Rule i : If M1 (t) is Fi1 and M2 (t) is Fi2 and and Mg (t) is Fig Then <: _ x (t) = Aix (t) +Adix (t d (t)) ; x (t) = (t) ; t 0; 0 d (t) ; _ d (t) : (1) where x 2 R denotes the state vector,M1 (t) ;M2 (t) ; :::;Mg (t) are the premise variables, and are positive constant upper bound for delay time and its derivative respectively, Fij is the fuzzy set, i = 1; :::; n; j = 1; :::; g: n is the number of If-Then rules. (t) is an initial value, Ai and Adi represent constant matrices with appropriate dimensions, respectively. Using center-average defuzzier, product inference and singleton fuzzier, the dynamic fuzzy model (1) can be expressed as the following global model: