A systematic approach to improve multiple Lyapunov function stability and stabilization conditions for fuzzy systems

This paper presents a systematic approach for decreasing conservativeness in stability analysis and control design for Takagi-Sugeno (TS) systems. This approach is based on the idea of multiple Lyapunov functions together with simple techniques for introducing slack matrices. Unlike some previous approaches based on multiple Lyapunov functions, both the stability and the stabilization conditions are written as linear matrix inequality (LMI) problems. The proposed approach reduces the number of inequalities and guarantees extra degrees of freedom to the LMI problems. Numeric examples illustrate the effectiveness of this method.

[1]  Kazuo Tanaka,et al.  A Descriptor System Approach to Fuzzy Control System Design via Fuzzy Lyapunov Functions , 2007, IEEE Transactions on Fuzzy Systems.

[2]  J. Ragot,et al.  Nonquadratic stability analysis of Takagi-Sugeno models , 2002, Proceedings of the 41st IEEE Conference on Decision and Control, 2002..

[3]  Tao Li,et al.  A new approach to robust and non-fragile Hinfinity control for uncertain fuzzy systems , 2007, Inf. Sci..

[4]  Guang-Hong Yang,et al.  State feedback control of continuous-time T-S fuzzy systems via switched fuzzy controllers , 2008, Inf. Sci..

[5]  PooGyeon Park,et al.  H∞ state-feedback controller design for discrete-time fuzzy systems using fuzzy weighting-dependent Lyapunov functions , 2003, IEEE Trans. Fuzzy Syst..

[6]  Chein-Chung Sun,et al.  Discrete $H_{2}/H_{\infty}$ Nonlinear Controller Design Based on Fuzzy Region Concept and Takagi–Sugeno Fuzzy Framework , 2006, IEEE Transactions on Circuits and Systems I: Regular Papers.

[7]  Johan Löfberg,et al.  YALMIP : a toolbox for modeling and optimization in MATLAB , 2004 .

[8]  Kazuo Tanaka,et al.  An approach to fuzzy control of nonlinear systems: stability and design issues , 1996, IEEE Trans. Fuzzy Syst..

[9]  Kazuo Tanaka,et al.  Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach , 2008 .

[10]  Peng Shi,et al.  Robust Hinfinity output feedback control design for fuzzy dynamic systems with quadratic D stability constraints: An LMI approach , 2006, Inf. Sci..

[11]  Huijun Gao,et al.  Improved Hinfinite control of discrete-time fuzzy systems: a cone complementarity linearization approach , 2005, Inf. Sci..

[12]  S. Bittanti,et al.  Affine Parameter-Dependent Lyapunov Functions and Real Parametric Uncertainty , 1996 .

[13]  Gang Feng,et al.  Analysis and design for a class of complex control systems part II: Fuzzy controller design , 1997, Autom..

[14]  P. Borne,et al.  Fuzzy systems and controllers: Lyapunov tools for a regionwise approach , 2005 .

[15]  H. C. Pietrobom,et al.  On relaxed LMI-based designs for fuzzy regulators and fuzzy observers , 2001, ECC.

[16]  Petr Ekel,et al.  Improved asymptotic stability analysis for uncertain delayed state neural networks , 2009 .

[17]  James Lam,et al.  Control Design for Fuzzy Systems Based on Relaxed Nonquadratic Stability and $H_{\infty}$ Performance Conditions , 2007, IEEE Transactions on Fuzzy Systems.

[18]  Peng Shi,et al.  Robust Hinfinity fuzzy filter design for uncertain nonlinear singularly perturbed systems with Markovian jumps: An LMI approach , 2007, Inf. Sci..

[19]  Sangchul Won,et al.  A new fuzzy Lyapunov function approach for a Takagi-Sugeno fuzzy control system design , 2006, Fuzzy Sets Syst..

[20]  Euntai Kim,et al.  New approaches to relaxed quadratic stability condition of fuzzy control systems , 2000, IEEE Trans. Fuzzy Syst..

[21]  Bor-Sen Chen,et al.  Mixed H2/H∞ fuzzy output feedback control design for nonlinear dynamic systems: an LMI approach , 2000, IEEE Trans. Fuzzy Syst..

[22]  Gang Feng,et al.  H/sub /spl infin// controller synthesis of fuzzy dynamic systems based on piecewise Lyapunov functions and bilinear matrix inequalities , 2005, IEEE Transactions on Fuzzy Systems.

[23]  Yung-Sheng Liu,et al.  A new LMI-based approach to relaxed quadratic stabilization of T-S fuzzy control systems , 2003, SMC'03 Conference Proceedings. 2003 IEEE International Conference on Systems, Man and Cybernetics. Conference Theme - System Security and Assurance (Cat. No.03CH37483).

[24]  Kazuo Tanaka,et al.  A multiple Lyapunov function approach to stabilization of fuzzy control systems , 2003, IEEE Trans. Fuzzy Syst..

[25]  Sing Kiong Nguang,et al.  Static output feedback controller design for fuzzy systems: an ILMI approach , 2006 .

[26]  Kazuo Tanaka,et al.  Fuzzy regulators and fuzzy observers: relaxed stability conditions and LMI-based designs , 1998, IEEE Trans. Fuzzy Syst..

[27]  Reza Langari,et al.  An LMI-based H fuzzy control system design with TS framework , 2000, Inf. Sci..

[28]  Lotfi A. Zadeh,et al.  Is there a need for fuzzy logic? , 2008, NAFIPS 2008 - 2008 Annual Meeting of the North American Fuzzy Information Processing Society.

[29]  M. C. D. Oliveiraa,et al.  A new discrete-time robust stability condition ( , 1999 .

[30]  Jin Bae Park,et al.  Robust fuzzy control of nonlinear systems with parametric uncertainties , 2001, IEEE Trans. Fuzzy Syst..

[31]  Huai-Ning Wu,et al.  Reliable ${H}_{\infty}$ Fuzzy Control for a Class of Discrete-Time Nonlinear Systems Using Multiple Fuzzy Lyapunov Functions , 2007, IEEE Transactions on Circuits and Systems II: Express Briefs.

[32]  Fernando de Oliveira Souza,et al.  Robust Hinfinity Control for Master-Slave Synchronization of Lur'e Systems with Time-Delay Feedback Control , 2008, Int. J. Bifurc. Chaos.

[33]  Vilma Alves de Oliveira,et al.  Robust $H_{\infty}$ Fuzzy Control Approach for a Class of Markovian Jump Nonlinear Systems , 2006, IEEE Transactions on Fuzzy Systems.

[34]  Fernando de Oliveira Souza,et al.  Further Results on Master-Slave Synchronization of General Lur'e Systems with Time-Varying Delay , 2008, Int. J. Bifurc. Chaos.

[35]  G. Feng,et al.  A Survey on Analysis and Design of Model-Based Fuzzy Control Systems , 2006, IEEE Transactions on Fuzzy Systems.

[36]  Zhi-Ren Tsai,et al.  An LMI-based stable T-S fuzzy model with parametric uncertainties using multiple Lyapunov function approach , 2004, IEEE Conference on Cybernetics and Intelligent Systems, 2004..

[37]  Karl-Erik Årzén,et al.  Piecewise quadratic stability of fuzzy systems , 1999, IEEE Trans. Fuzzy Syst..

[38]  Peng Yang,et al.  Further studies on LMI-based relaxed stabilization conditions for nonlinear systems in Takagi-Sugeno's form , 2006, Autom..

[39]  J. Geromel,et al.  A new discrete-time robust stability condition , 1999 .

[40]  Shengyuan Xu,et al.  Robust Hinfinite control for discrete-time fuzzy systems via basis-dependent Lyapunov functions , 2005, Inf. Sci..

[41]  Petr Ekel,et al.  Asymptotic stability analysis in uncertain multi-delayed state neural networks via Lyapunov-Krasovskii theory , 2007, Math. Comput. Model..

[42]  G. Feng,et al.  Controller synthesis of fuzzy dynamic systems based on piecewise Lyapunov functions , 2001, 10th IEEE International Conference on Fuzzy Systems. (Cat. No.01CH37297).

[43]  Kazuo Tanaka,et al.  Stability analysis and design of fuzzy control systems , 1992 .

[44]  Fernando de Oliveira Souza,et al.  Improved robust ℋ ∞ control for neutral systems via discretised Lyapunov-Krasovskii functional , 2008, Int. J. Control.

[45]  Bor-Sen Chen,et al.  Mixed Fuzzy Output Feedback Control Design for Nonlinear Dynamic Systems: An LMI Approach , 2000 .

[46]  Leonardo A. B. Tôrres,et al.  Chaotic Synchronization and Information Transmission Experiments: A Fuzzy Relaxed H∞ Control Approach , 2007 .

[47]  Michio Sugeno,et al.  Fuzzy identification of systems and its applications to modeling and control , 1985, IEEE Transactions on Systems, Man, and Cybernetics.

[48]  Kazuo Tanaka,et al.  Fuzzy control systems design and analysis , 2001 .