Fuzzy Sets and Systems 160 (2009) 2776 – 2795 www.elsevier.com/locate/fss A membership-function-dependent approach for stability analysis and controller synthesis of Takagi–Sugeno models  Miguel Bernal a , 1 , Thierry Marie Guerra b, ∗ , Alexandre Kruszewski c a National Research System, Mexico b LAMIH UMR CNRS 8530, University of Valenciennes Hainaut-Cambrésis, France c LAGIS UMR CNRS 8146, Ecole Centrale de Lille, France Received 28 May 2008; received in revised form 5 February 2009; accepted 11 February 2009 Available online 23 February 2009 Abstract This paper presents a new approach for stability analysis and controller design of Takagi–Sugeno (TS) models. The analysis considers information derived from existing or induced order relations among the membership functions. Partitioning of the state- space and the use of piecewise Lyapunov functions (PWLF) arise naturally as a consequence of induced order relations. Conditions under the novel approach can be expressed as linear matrix inequalities (LMIs) facilitating the inclusion of performance design. Examples are provided to show the advantages over the classical quadratic approach. © 2009 Elsevier B.V. All rights reserved. Keywords: Linear matrix inequalities (LMI); Stabilization; Takagi–Sugeno models 1. Introduction In recent years, Takagi–Sugeno (TS) models [21] have been the subject of an intensive research by virtue of their approximation capabilities. They can represent exactly a nonlinear model in a compact set of the domain of the state variables [24]. TS models consist in a set of linear models blended together with nonlinear functions holding the convex-sum property [22]. The stabilization problem is usually addressed via the so-called PDC (parallel distributed compensation) control law [27]. It consists in a set of linear-state feedbacks blended together using the same nonlinear functions as the TS model. Stability and stabilization of TS models are usually investigated through the direct Lyapunov method. An LMI (linear matrix inequality) formulation [3] of these problems is preferred, since LMIs can be easily solved by convex optimization techniques. This formulation is directly achieved by quadratic Lyapunov functions and many results concerning robustness and performance under this approach have been developed [22,24,27]. Nevertheless, quadratic-  The present research work has been supported by International Campus on Safety and Intermodality in Transportation, the Nord-Pas-de-Calais Region, the European Community, the Regional Delegation for Research and Technology, the Ministry of Higher Education and Research, and the National Center for Scientific Research. The authors gratefully acknowledge the support of these institutions. ∗ Corresponding author. Tel.: +33 32751 1350. E-mail addresses: galadali_2002@yahoo.com.mx (M. Bernal), Guerra@univ-valenciennes.fr (T.M. Guerra). 1 During this work, he was post-doc at the LAMIH. 0165-0114/$-see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2009.02.005