A New Relaxed Stability Condition for Takagi-Sugeno Fuzzy Control

1. Introduction
The concept of fuzzy sets was introduced by Zadeh in 1965. Since then, fuzzy set theory has been deployed in a large variety of applications, e.g., fuzzy control [1], approximate reasoning [2], pattern recognition [3], etc. Takagi-Sugeno (T-S) fuzzy model, proposed by Takagi and Sugeno in 1984 [4], has become a landmark in fuzzy control theory. A T-S fuzzy model can efficiently represent a nonlinear plant by a weighted sum of linear plants. In order to stabilize a fuzzy model based (FMB) control system, the well-known parallel distribution compensaCorresponding Author: Kairui Cao is with the National Key Laboratory of Tunable Laser Technology, Harbin Institute of Technology, Harbin, P. R. China. E-mail: kcaohit@gmail.com Manuscript received 1 Nov. 2013; revised 5 May 2014; accepted 5 June 2014.
© 2014 TFSA
328
International Journal of Fuzzy Systems, Vol. 16, No. 3, September 2014
bership functions. Inspired by the work covered in [25, 26, 29-34], we consider a possibility of obtaining more relaxed results by incorporating both the QFLFs and SMFs in the stability analysis of FBM control systems. According to the Lyapunov theory, we will see that the stability of a FMB control system is guaranteed by the negative definiteness of several fuzzy summations. Then SMFs are applied to bring the information of membership functions into stability analysis. In the following analysis, we will see that the proposed method delivers less conservative results than some of existing stability conditions. The rest of this paper is organized as follows. Section 2 introduces the T-S fuzzy model, non-PDC fuzzy controller, and SMFs. Based on QFLFs and SMFs, the stability analysis of FMB control systems is investigated in Section 3. In Section 4, two numerical simulation examples are provided to demonstrate the efficiency of the proposed method. Finally, some remarks and conclusions are drawn in Section 5.
Kairui Cao, X. Z. Gao, H. K. Lam, A. V. Vasilakos, and Witold Pedrycz Abstract1
This paper presents a new relaxed stability condition for Takagi-Sugeno (T-S) fuzzy control systems. Using quadratic fuzzy Lyapunov functions (QFLFs), the stability of closed-loop control system is guaranteed by the negative definiteness of several fuzzy summations. However, since the membership functions are continuous, the negative definiteness of these fuzzy summations implies an infinite number of linear matrix inequalities (LMIs), which cannot be solved directly by conventional convex optimization methods. To handle this problem, Staircase Membership Functions (SMFs) are employed to convert the infinite number of LMIs into a finite one. At the same time, the information of membership functions is brought into stability analysis, which substantially relaxes the proposed stability condition. The efficiency of the presented approach is demonstrated by using two simulation examples. Keywords: Takagi-Sugeno (T-S) fuzzy models, fuzzy control systems, quadratic fuzzy Lyapunov functions (QFLFs), staircase membership functions (SMFs), stability analysis. tion (PDC) scheme was developed [5]. FMB control is an efficient method to deal with nonlinear control problems, such as stability analysis [5-10], robust filtering [11], sampled-data control [12], time-delayed control [13], and DC-DC converter control [14]. Stability analysis for FMB control systems is a basis of real world applications of fuzzy controllers [15]. Recent research works have focused on how to derive more relaxed stability conditions. Based on the commonly used quadratic Lyapunov functions and PDC fuzzy controllers, a series of stability conditions have been obtained [16]. By arranging the interactions of fuzzy logic rules into a single matrix, a more relaxed stability condition has been proposed [17]. Furthermore, some further relaxed stability conditions have been presented as well [18-24]. In the abovementioned works, the information residing with the membership functions was not considered. To incorporate the information of membership functions to stability analysis, the staircase membership functions (SMFs) were implemented to approximate the membership functions of fuzzy models and fuzzy controllers in [25, 26]. Then the fuzzy summation inequality, which delivers a sufficient condition to guarantee the stability of closed-loop control systems, could be converted to a finite number of linear matrix inequalities (LMIs). The advantage of that is the information of membership functions can be used to carry out stability analysis. Therefore, the derived stability conditions in [25, 26] led to less conservative results. This stresses a fact that the information of membership functions plays an important role in reducing the conservativeness of stability analysis for FMB control systems. On the other hand, another effective way of reducing the conservativeness of stability conditions is to adopt complex Lyapunov functions in stability analysis. Piecewise quadratic Lyapunov functions were used for the stability analysis of FMB control systems in [27, 28]. In [29-34], the fuzzy Lyapunov functions (FLFs) and quadratic fuzzy Lyapunov functions (QFLFs) were proposed. Correspondingly, the PDC controller has also been evolved to a non-PFra Baidu bibliotekC form, which provides greater design flexibility compared to the former one. However, the non-PDC controller exhibits a more complicated structure. Usually, it requires the information of time derivatives of mem-
International Journal of Fuzzy Systems, Vol. 16, No. 3, September 2014
327
A New Relaxed Stability Condition for Takagi-Sugeno Fuzzy Control Systems Using Quadratic Fuzzy Lyapunov Functions and Staircase Membership Functions