Fuzzy Control System Designs using Redundancy of Descriptor Representation:A Fuzzy Lyapunov Function Approach Kazuo Tanaka, Takashi Nebuya, Hiroshi Ohtake and Hua O. Wang Abstract— This paper presents fuzzy control system designs using redundancy of descriptor representation. A wider class of Takagi-Sugeno fuzzy controllers using the redundancy is employed to derive stabilization conditions for both common Lyapunov functions and fuzzy Lyapunov functions. We show that the fuzzy Lyapunov function approach is less conservative than the common Lyapunov function approach. A design ex- ample also illustrates the utility of the fuzzy Lyapunov function approach using redundancy of descriptor representation. I. I NTRODUCTION Nonlinear control based on the Takagi-Sugeno fuzzy model [1] has received a lot of attention over the last decade (e.g., see [2]- [11]). An advantage of the fuzzy model-based control [12] is to provide a natural, simple and effective design approach although other nonlinear control techniques [13] require special and rather involved knowledge. In addition, it is known that any smooth nonlinear control systems can be approximated by the Takagi-Sugeno fuzzy models (with liner rule consequence) [14]. Recently, piecewise Lyapunov function approaches have received increasing attention as they attempt to relax the conservativeness of stability and stabilization problems. However, stabilization conditions for fuzzy Lyapunov func- tions [15] and piecewise Lyapunov functions [16] become BMIs in general. In [15], the well-known completing square technique was introduced to convert the BMIs into LMIs. The conversion causes conservative results in general. Hence, the converted LMIs do not completely contain the LMIs for the common quadratic Lyapunov function although the fuzzy Lyapunov function contains the common quadratic Lyapunov function as a special case. In this paper, we derive LMI design conditions (that contains the LMIs for the common quadratic Lyapunov function as a special case) using redundancy of descriptor representation. The redundancy also provides us the possibility of designing a fuzzy controller for systems with input nonlinearities. A fuzzy descriptor system design has been already discussed in [17]. The design in [17] did not fully take an advantages of redundancy of descriptor representation. This work was supported in part by a Grant-in-Aid for Scientific Research (C) 15560217 from the Ministry of Education, Science and Culture of Japan. Kazuo Tanaka Takashi Nebuya and Hiroshi Ohtake are with the Department of Mechanical Systems and Intelligent Systems, The University of Electro-Communications, Chofu, Tokyo 182-8585 Japan ktanaka@mce.uec.ac.jp, neb- uya@rc.mce.uec.ac.jp & hohtake@mce.uec.ac.jp Hua O. Wang is with the Department of Aerospace and Mechanical En- gineering, Boston University, Boston, MA 02215 USA wangh@bu.edu This paper is organized as follows. Section II recalls the previous results with respect to fuzzy model and sta- bility conditions. Section III introduces a wider class of Takagi-Sugeno fuzzy controllers and derives stabilization conditions based on common quadratic Lyapunov functions. Section IV presents stability conditions based on fuzzy Lyapunov functions. We show that the fuzzy Lyapunov function approach is less conservative than the common Lyapunov approach. Section V illustrates a design example to demonstrate the utility of the fuzzy Lyapunov function approach using redundancy of descriptor representation. II. FUZZY MODEL AND STABILITY CONDITIONS Consider the following nonlinear systems: ˙ x(t)= f (x(t), u(t)), (1) where x(t)=[x 1 (t) x 2 (t) ··· x n (t)] T is the state vector, u(t) = [u 1 (t) u 2 (t) ··· u m (t)] T is the input vector. Based on the sector nonlinearity concept [12], we can exactly represent (1) with the Takagi-Sugeno fuzzy model (2) (globally or at least semi-globally). Model Rule i: If z 1 (t) is M i1 and ··· and z p (t) is M ip then ˙ x(t)= A i x(t)+ B i u(t) i =1, 2, ··· , r, (2) where z j (t)(j =1, 2, ··· ,p) is the premise variable. The membership function associated with the ith Model Rule and jth premise variable component is denoted by M ij . r denotes the number of Model Rules. Each z j (t) is a measurable time-varying quantity that may be states, inputs, measurable external variables and/or time. It has been tacitly assumed in fuzzy model-based design that each z j (t) does not depend on the inputs u(t). However, the fuzzy control designs using redundancy of descriptor representation permit that each z j (t) depends on the inputs u(t). This is an advantage of fuzzy control designs using redundancy of descriptor representation. The defuzzification process of the model (2) can be represented as ˙ x(t)= r  i=1 h i (z(t)){A i x(t)+ B i u(t)}, (3) where z(t) = [z 1 (t) ··· z p (t)]. From the properties of membership functions, the following relations hold. h i (z(t)) = w i (z(t)) r  i=1 w i (z(t)) ≥ 0, r  i=1 h i (z(t)) = 1, 2005 American Control Conference June 8-10, 2005. Portland, OR, USA 0-7803-9098-9/05/$25.00 ©2005 AACC WeB17.1 1096