IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 15, NO. 3, JUNE 2007 333 A Descriptor System Approach to Fuzzy Control System Design via Fuzzy Lyapunov Functions Kazuo Tanaka, Member, IEEE, Hiroshi Ohtake, Member, IEEE, and Hua O. Wang, Senior Member, IEEE Abstract—There has been a flurry of research activities in the analysis and design of fuzzy control systems based on linear matrix inequalities (LMIs). This paper presents a descriptor system ap- proach to fuzzy control system design using fuzzy Lyapunov func- tions. The design conditions are still cast in terms of LMIs but the proposed approach takes advantage of the redundancy of de- scriptor systems to reduce the number of LMI conditions which leads to less computational requirement. To obtain relaxed LMI conditions, new types of fuzzy controller and fuzzy Lyapunov func- tion are proposed. A salient feature of the LMI conditions derived in this paper is to relate the feasibility of the LMIs to the switching speed of each linear subsystem (to be exact, to the lower bounds of time derivatives of membership functions). To illustrate the va- lidity and applicability of the proposed approach, two design ex- amples are provided. The first example shows that the LMI condi- tions based on the fuzzy Lyapunov function are less conservative than those based on a common (standard) Lyapunov function. The second example illustrates the utility of the fuzzy Lyapunov func- tion approach in comparison with a piecewise Lyapunov function approach. Index Terms—Descriptor representation, fuzzy control, fuzzy Lyapunov function, redundancy. I. INTRODUCTION N ONLINEAR control systems based on the Takagi–Sugeno (T–S) fuzzy model [1] have received a great deal of atten- tion over the last decade (e.g., see [2]–[15]). The main advantage of such fuzzy model-based control methodology [16] is that it provides a natural, simple and effective design approach to com- plement other nonlinear control techniques (e.g., [17]) that re- quire special and rather involved knowledge. Moreover, there is no loss of generality in adopting the T–S fuzzy model based control design framework as it has been established that any smooth nonlinear control systems can be approximated by the T–S fuzzy models (with linear rule consequence) [18]. Within the general framework of T–S fuzzy model-based control sys- tems, there has been, in particular, a flurry of research activities in the analysis and design of fuzzy control systems based on linear matrix inequalities (LMIs) (e.g., [16]). Manuscript received March 22, 2005; revised November 6, 2005 and January 10, 2006. This work was supported in part by a Grant-in-Aid for Scientific Re- search (C) 15560217 from the Ministry of Education, Science, and Culture of Japan. K. Tanaka and H. Ohtake are with the Department of Mechanical Engineering and Intelligent Systems, The University of Electro-Communications, Tokyo 182-8585, Japan (e-mail: ktanaka@mce.uec.ac.jp; hohtake@mce.uec.ac.jp). H. O. Wang is with the Department of Aerospace and Mechanical Engi- neering, Boston University, Boston, MA, 02215 USA (e-mail: wangh@bu.edu). Digital Object Identifier 10.1109/TFUZZ.2006.880005 In this paper, we present a descriptor system approach to fuzzy control system design via fuzzy Lyapunov functions. The design conditions are still cast in terms of LMIs but the proposed approach takes advantage of the redundancy of descriptor sys- tems to reduce the number of LMI conditions which leads to less computational requirement. It is well known that linear de- scriptor systems [19] describe a larger class of systems than con- ventional linear state-space models. A descriptor system is also much tighter than a state-space expression for representing in- dependent parametric perturbations. Analysis and design of the linear descriptor systems have been extensively discussed in the literature. A fuzzy descriptor representation that is a kind of non- linear descriptor system was first stated in [20], [21]. A number of papers (e.g., [22]) extended the results in [20] and [21]. How- ever, these papers did not fully take advantage of the redundancy of descriptor representation in control system design. Further- more, these papers only addressed the common Lyapunov func- tion approach which typically led to conservative results. In this paper, we propose an approach with new types of fuzzy con- troller and fuzzy Lyapunov function to take full advantage of the redundancy of fuzzy descriptor systems to reduce the number of LMI conditions and to render less conservative stability and sta- bilization results. With regard to relax the conservativeness of stability and stabilization problems using Lyapunov approach, recently piecewise or switched Lyapunov function approaches [23]–[26] have received increasing attention. However, stabilization con- ditions for fuzzy Lyapunov functions [27] and piecewise Lyapunov functions [28], [29] are in terms of bilinear matrix inequalities (BMIs) in general. In [27], BMI conditions have been converted into LMI conditions by way of the well-known completing square technique. In general such a conversion leads to conservative results. To overcome the conservative- ness, in this paper, we directly obtain LMI design conditions for stabilizing fuzzy controllers (without using the completing square technique) by introducing new types of fuzzy controller and fuzzy Lyapunov function. A salient feature of the LMI conditions derived in this paper is to relate the feasibility of the LMIs to the switching speed of each linear subsystems (exactly speaking, to the lower bounds of time derivatives of membership functions). The details are illustrated through the design examples in Section V-B. The rest of the paper is organized as follows. Section II recalls the T–S fuzzy model and control. Section III presents a design method based on common Lyapunov functions within fuzzy de- scriptor systems. Section IV introduces new types of fuzzy con- troller and fuzzy Lyapunov functions to take advantage of the redundancy of descriptor systems. Section V entails two design 1063-6706/$25.00 © 2007 IEEE