LMI approach for exponential stabilization of continuous-time delayed Takagi-Sugeno systems N. Daraoui * , M. Nachidi ** , F. Tadeo * and A. Benzaouia *** * Dpto. de Ingenier´ ıa de Sistemas y Autom´atica, Universidad de Valladolid, 47005, Spain. E-mail: ndaraoui@gmail.com ** Department of Modeling, Information and Systems, University of Picardie-Jules Verne, 7 Rue du Moulin Neuf - 80000, Amiens, France *** Research Unit: Constrained and Robust Regulation, Department of Physics, Faculty of Science Semlalia, P.B 2390, Marrakech, Morocco. Key-words: Nonlinear systems, Takagi-Sugeno models, fuzzy Lyapunov functions, decay rate, LMI. Abstract: This paper deals with the analysis and design of controllers for a class of continuous- time delayed Takagi-Sugeno fuzzy systems. Sufficient conditions for the exponential stability are derived based on a fuzzy Lyapunov function. These conditions are expressed in the form of linear matrix inequalities, that depend on the maximum bound of the time delay and the desired decay rate of the system. A Non parallel distributed compensation controller is considered. The controller design problem is presented as an optimization under linear matrix inequalities constraints. This fuzzy controller provides a maximum domain of constraints determined by the time derivative of the membership functions, while guaranteeing the desired decay rate.The effectiveness of the proposed approaches is shown through a numerical example. 1. INTRODUCTION Fuzzy systems in the form of the Takagi-Sugeno (T-S) models have attracted much attention in the last decade. These fuzzy dynamic models were proposed by (Takagi and Sugeno, 1985) to represent nonlinear dynamical sys- tems. It has been shown that the T-S models give an effective way to represent complex nonlinear systems by a piecewise interpolation of several linear models through membership functions. These fuzzy models are described by a family of fuzzy IF-THEN rules that represent local linear input-output relations of the system. The overall fuzzy model of the system is achieved by the fuzzy mix- ing of these local linear models. Thus, controller design methodologies can be developed to achieve stability and performance for the nonlinear system. Therefore, Takagi- Sugeno models are considered a powerful solution to bridge the gap between linear control and fuzzy logic control tar- geting complex nonlinear systems (Takagi, 1995), (Wang et al., 1996). The stabilization problem for systems in T-S fuzzy model has been studied extensively and numerous methods have been proposed (Kim et al., 2000) , (Nachidi et al., 2010). For instance, the paper of (Feng, 2006) mainly surveys the stability of fuzzy control based on T-S models, where three types of Lyapunov functions are considered: a com- mon quadratic Lyapunov function, a piecewise quadratic Lyapunov function and a fuzzy Lyapunov function. The control design is carried out based on a fuzzy model via the so-called parallel distributed compensation (PDC) scheme (Wang et al., 1996). The idea is that a linear feedback control is designed for each local linear model, and the resulting overall controller is a fuzzy blending of each individual linear controller, so it is nonlinear in general. The greatest difficulty in using the continuous-time PDC approach with a fuzzy Lyapunov function is that sufficient conditions are usually in the form of bilinear matrix in- equalities (BMIs). To address this problem, in Tanaka et al. (2003), a non PDC was further adopted, and results based on this idea were obtained by Nachidi et al. (2009). However, dealing with the stability of T-S systems, few of these papers considered the effect of the time delays that occur in many dynamical systems such as biolog- ical systems, chemical systems, nuclear reactions, elec- trical networks, etc. Nonlinear systems with time delay constitute basic mathematical models of real phenom- ena. The stability and synthesis of time delay systems is an important issue addressed by many authors and for which surveys can be found in several monographs, for example, Gu (2003) and Kolmanovskii and Richard (1999). In the context of T-S systems with time delay, Cao and Frank (2001) and Guan and Chen (2004) can be mentioned, where delay-independent and delay-dependent stability conditions were obtained. However, the results considered common quadratic Lyapunov functions, which leads to much conservativeness in the stability tests. To avoid this conservatism, fuzzy Lyapunov functions that include the membership functions have recently been used to establish stability and stabilization conditions for T- S systems (Tanaka et al., 2007). Inspired by the work of (Nachidi, 2009) in which stability conditions in LMIs form, for continuous-time fuzzy systems have been derived in the Lyapunov sense without requiring a priori knowledge of bounds on the time-derivative of each membership func- Preprints of the 18th IFAC World Congress Milano (Italy) August 28 - September 2, 2011 Copyright by the International Federation of Automatic Control (IFAC) 3873