Stability Analysisfor Uncertain Delayed TS Fuzzy Systems

This paper studies the global exponential stability for a Takagi-Sugeno (T-S) fuzzy system with bounded uncertain delays. Most existing T-S method represent global nonlinear systems by connecting local linear systems with linguistic description. However, many complex systems cannot be represented by linear systems. In this paper, a class of local nonlinear systems having nice dynamic properties is employed to represent some global complex systems. Moreover, the delays are any uncertain bounded continuous functions. Sufficient conditions for global exponential stability of these delayed global complex systems are derived. Criteria for design of nonlinear fuzzy controllers to feedback control the stability of global nonlinear fuzzy systems are given. Keywords-Fuzzy systems; Global exponential stability; Delays I. I NTRODUCTION Since Tanaka and Sugeno [1] proposed Takagi-Sugeno (TS) fuzzy model in 1985, a great number of results have been reported for T-S systems [2], [3], [4], [5]. The T-S model gives an effective method to combine some simple local systems with their linguistic description to represent complex nonlinear dynamic systems. Stability of T-S model fuzzy systems is quite important for practical applications. It has been widely studied by many authors, see, for examples, [6], [7], [8]. Time delays in dynamic systems has been studied for many years. T-S model of fuzzy systems with delays was first introduced in [9]. It is well known that delays can affect dynamics of some nonlinear systems, a stable system may become unstable by introducing some delays [10]. In recent years, some authors have paid their attention to control of nonlinear systems with delays by using T-S fuzzy models. There exist two kinds of delays, one is continuous, see, for examples [11], [12], [13], [14]. The other is discrete, see, for examples, [15], [16]. In control engineering, delays are difficult to be known exactly, so stability for systems with uncertain delays are quite interesting [17]. In this paper, the delays are assumed to be any uncertain bounded continuous functions. Stability conditions will be derived, some of them will be represented by simple algebraic inequalities and are easy to check. As we know, in most reported stability results of T-S model, simple linear systems are used to form global nonlinear fuzzy systems [18], [19], [20]. However, there are many complex nonlinear fuzzy systems cannot be connected by using local linear systems. In this paper, unlike using local linear systems in previous study, a class of nonlinear systems with delays having nice dynamical properties [21] will be used as local systems to form some global complex nonlinear fuzzy systems by T-S method. Since global exponential stability (GES) is more interesting than global asymptotic stability, Our stability conditions will guarantee the global exponential stability of the global complex nonlinear fuzzy delayed systems. This paper is organized as follows. In Section II, some preliminaries for delayed fuzzy control systems will be given. In Section III, conditions for global exponential stability of fuzzy systems with delays will be proposed and proved. In Section IV, simulations will be given. This paper will be concluded in Section V. II. PRELIMINARIES Consider a T-S fuzzy time-delay model which is composed of r plant rules. For eachs = 1, · · · , r, the sth plant rule can be represented as follows: Plant Rule s: IF α1(t) is M1s AND · · · AND αp(t) is Mps THEN ẋ(t) = −x(t) + Wsg(x(t)) + Jsg ( x(t− τs(t)) ) (1) for t ≥ 0, wherex(t) = (x1(t), · · · , xn(t)) is the state vector, r is the number of IF-THEN rules, α1(t), · · · , αp(t) are the premise variables and each Mis(i = 1, · · · , p) is the fuzzy set corresponding to αi(t) and plant rules. u(t) is the control input vector,τs(t) is the time delay which satisfies 0 ≤ τs(t) ≤ τ . For any x ∈ R, g(x) = (gx1 ) , · · · , gxn) )T , and the function g is defined as follows: g(s) = |s + 1| − |s− 1| 2 , s ∈ R. The function g is continuous but non-differentiable. So the local system is nonlinear which is the main feature of this paper different from others. The system of (1) can be described as follows: ẋi(t) = −xi(t) + n ∑ j=1 [ W s ijg ( xj(t) ) + J ijg ( xj(t− τs(t)) )] (2) International Conference on Automatic Control Theory and Application (ACTA 2014) © 2014. The authors Published by Atlantis Press 71 for t ≥ 0 and i = 1, · · · , n. WhereWs = ( W s ij ) n×n, Js = ( J ij ) n×n is the constant matrix. Let Mis(αi(t)) be the membership function of the fuzzy setMis at the positionαi(t) and denote