IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 19, NO. 3, JUNE 2011 505 LMI-Based Stability Analysis for Fuzzy-Model-Based Control Systems Using Artificial T–S Fuzzy Model H. K. Lam, Senior Member, IEEE Abstract—This paper investigates the stability of fuzzy-model- based (FMB) control systems. An alternative stability-analysis ap- proach using an artificial fuzzy system based on the Lyapunov stability theory is proposed. To facilitate the stability analysis, the continuous membership functions of the Takagi–Sugeno (T–S) fuzzy model are represented by the staircase ones. With the nice property of the staircase membership functions, it turns the set of infinite number of linear-matrix-inequality (LMI) based stability conditions into a finite one. Furthermore, the staircase member- ship functions carrying system information can be brought to the stability conditions to relax the stability conditions. The stability of the original FMB control systems is guaranteed by the satisfaction of the LMI-based stability conditions. The proposed stability anal- ysis is applied to the FMB control systems of which the T–S fuzzy model and fuzzy controller do not share the same premise mem- bership functions and, thus, is able to enhance the design flexibility of the fuzzy controller. A simulation example is given to illustrate the merits of the proposed approach. Index Terms—Fuzzy control, linear-matrix inequality (LMI), stability analysis, staircase membership functions, Takagi–Sugeno (T–S) fuzzy model. I. INTRODUCTION F UZZY-MODEL-BASED (FMB) control [1] offers a sys- tematic approach to deal with nonlinear control problems. With the Takagi–Sugeno (T–S) fuzzy model [2], [3], it offers a general framework to represent the nonlinear system as an aver- age weighted sum of linear subsystems. The T–S fuzzy model exhibits a nice property to facilitate the stability analysis and control synthesis. A fuzzy controller was proposed in [4] and [5] to close the feedback loop of the nonlinear plant represented by the T–S fuzzy model to form an FMB control system. Stability is an essential issue for the FMB control systems. It has drawn the attention of the researchers in the fuzzy-control community for the past decades. The Lyapunov stability theory is the most-popular analysis tool to investigate the stability of the FMB control systems. Basic stability conditions in the form of Lyapunov inequalities were reported in [4] and [5]. If there exists a solution to the stability conditions, the FMB control Manuscript received November 9, 2009; revised April 27, 2010 and October 25, 2010; accepted January 5, 2011. Date of publication February 17, 2011; date of current version June 6, 2011. This work was supported by the King’s College London and by the Engineering and Physical Sciences Research Council under Project EP/E05627X/1. The author is with the Division of Engineering, King’s College London, London, WC2R 2LS, U.K. (e-mail: hak-keung.lam@kcl.ac.uk). Digital Object Identifier 10.1109/TFUZZ.2011.2116027 system is guaranteed to be asymptotically stable. As the mem- bership functions are not considered, the stability conditions achieved are very conservative. However, it allows the member- ship functions to be chosen freely for the fuzzy controller, and thus, enhances the controller-design flexibility. In [6]–[8], the same authors and other researchers proposed some approaches to bring the boundary information of the membership functions, such as the lower and upper bounds of the membership func- tions and/or their multiplications, to the stability analysis for this class of FMB control systems. As more information of the system nonlinearity is considered, more relaxed stability con- ditions can be obtained, compared with the work given in [4] and [5]. The parallel-distribution-compensation (PDC) technique was proposed in [9] for the design of fuzzy controller. It is required that the fuzzy controller share the same premise membership functions as those of the T–S fuzzy model. As some cross terms of membership functions of the FMB control system are in common, they can be grouped in the stability analysis, thereby leading to more relaxed stability conditions. Further relaxed stability conditions were reported in [10]–[17] by utilizing the property of the fuzzy summations at different levels. As the de- sign flexibility of the fuzzy controller under the PDC vanishes (i.e., the membership functions of the fuzzy controller are not allowed to be chosen freely but determined by the T–S fuzzy model), the implementation cost (i.e., the cost of implementing the fuzzy controller physically) will increase if the membership functions of the T–S fuzzy model are complicated. The stabil- ity conditions and the design of the fuzzy controller were for- mulated as a linear–matrix-inequality (LMI) problem [4]–[17]. The solution to the LMI-based stability conditions can be found numerically using convex-programming techniques [18]. By in- corporating the boundary information of the membership func- tions [19], [20], more relaxed stability conditions for the FMB control systems under the PDC can be obtained. It should be noted that the stability conditions in [9]–[17] contain no information about the membership functions of the T–S fuzzy model and the fuzzy controller. Once a fea- sible solution is found, the stability of the FMB control system is guaranteed for any shapes of membership functions. Practically, a fuzzy controller should be designed for a specified nonlinear plant but not for a family. By considering the nonlin- ear plant to be controlled, the membership functions are already known, and it is a waste if we do not consider this information in the stability analysis. In this paper, we propose an alterna- tive stability-analysis approach to investigate the stability of the 1063-6706/$26.00 © 2011 IEEE