Relaxed Stabilization Criterion for T–S Fuzzy Systems by Minimum-Type Piecewise-Lyapunov-Function-Based Switching Fuzzy Controller

This paper proposes the minimum-type piecewise-Lyapunov-function-based switching fuzzy controller that switches accompanying the piecewise Lyapunov function. By applying this switching fuzzy controller with the minimum-type piecewise Lyapunov function, the relaxed stabilization criterion is obtained for continuous Takagi-Sugeno (T-S) fuzzy systems. Some conditions of the relaxed stabilization criterion are represented by bilinear matrix inequalities (BMIs), which contain some bilinear terms as the product of a full matrix and a scalar. According to the literature, the path-following method is very effective for this kind of BMI problem; hence, it is utilized to obtain solutions of the criterion. Xie et al. in 1997 chose two types (i.e., minimum type and maximum type) of piecewise Lyapunov functions as the Lyapunov function candidates. The reasons for why this study only chooses the minimum-type piecewise Lyapunov function as the Lyapunov function candidate are illustrated. Moreover, the numerical example shows the relaxation of the proposed criterion.

[1]  M. Fu,et al.  Piecewise Lyapunov functions for robust stability of linear time-varying systems , 1995, Proceedings of 1995 34th IEEE Conference on Decision and Control.

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

[3]  Stephen P. Boyd,et al.  A path-following method for solving BMI problems in control , 1999, Proceedings of the 1999 American Control Conference (Cat. No. 99CH36251).

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

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

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

[7]  Bor-Sen Chen,et al.  Fuzzy tracking control design for nonlinear dynamic systems via T-S fuzzy model , 2001, IEEE Trans. Fuzzy Syst..

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

[9]  Chein-Chung Sun,et al.  Constrained fuzzy controller design of discrete Takagi-Sugeno fuzzy models , 2003, Fuzzy Sets Syst..

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

[11]  Xiaodong Liu,et al.  Approaches to quadratic stability conditions and H∞ control designs for T-S fuzzy systems , 2003, IEEE Trans. Fuzzy Syst..

[12]  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..

[13]  Thierry-Marie Guerra,et al.  LMI-based relaxed nonquadratic stabilization conditions for nonlinear systems in the Takagi-Sugeno's form , 2004, Autom..

[14]  Péter Baranyi,et al.  TP model transformation as a way to LMI-based controller design , 2004, IEEE Transactions on Industrial Electronics.

[15]  Ji-Chang Lo,et al.  Observer-based robust H/sub /spl infin// control for fuzzy systems using two-step procedure , 2004, IEEE Transactions on Fuzzy Systems.

[16]  Michio Sugeno,et al.  Stabilization of nonlinear systems based on piecewise Lyapunov functions , 2004, 2004 IEEE International Conference on Fuzzy Systems (IEEE Cat. No.04CH37542).

[17]  G. Feng,et al.  Piecewise H∞ controller design of discrete time fuzzy systems , 2004, IEEE Trans. Syst. Man Cybern. Part B.

[18]  Wen-June Wang,et al.  A relaxed stability criterion for T-S fuzzy discrete systems , 2004, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics).

[19]  Chung-Hsun Sun,et al.  A relaxed stability criterion for T-S fuzzy discrete system , 2004, IEEE International Conference on Networking, Sensing and Control, 2004.

[20]  Wen-June Wang,et al.  Relaxed stability and stabilization conditions for a T-S fuzzy discrete system , 2005, Fuzzy Sets Syst..

[21]  Tingshu Hu,et al.  Conjugate Lyapunov functions for saturated linear systems , 2005, Autom..

[22]  Chun-Hsiung Fang,et al.  A new LMI-based approach to relaxed quadratic stabilization of T-S fuzzy control systems , 2006, IEEE Trans. Fuzzy Syst..

[23]  Luca Zaccarian,et al.  Stability and Performance for Saturated Systems via Quadratic and Nonquadratic Lyapunov Functions , 2006, IEEE Transactions on Automatic Control.

[24]  Franco Blanchini,et al.  Stabilizability of switched linear systems does not imply the existence of convex Lyapunov functions , 2006, CDC.

[25]  Tingshu Hu,et al.  Conjugate convex Lyapunov functions for dual linear differential inclusions , 2006, IEEE Transactions on Automatic Control.

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

[27]  Tingshu Hu,et al.  Nonlinear control design for linear differential inclusions via convex hull of quadratics , 2007, Autom..

[28]  Y.-J. Chen,et al.  Relaxed Stabilization Criteria for Discrete-Time T–S Fuzzy Control Systems Based on a Switching Fuzzy Model and Piecewise Lyapunov Function , 2007, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics).

[29]  Hak-Keung Lam,et al.  BMI-Based Stability and Performance Design for Fuzzy-Model-Based Control Systems Subject to Parameter Uncertainties , 2007, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics).

[30]  Antonio Sala,et al.  Asymptotically necessary and sufficient conditions for stability and performance in fuzzy control: Applications of Polya's theorem , 2007, Fuzzy Sets Syst..

[31]  Thierry-Marie Guerra,et al.  Continuous Takagi–Sugeno's Models: Reduction of the Number of LMI Conditions in Various Fuzzy Control Design Technics , 2007, IEEE Transactions on Fuzzy Systems.

[32]  Tingshu Hu,et al.  Stabilization of Switched Systems via Composite Quadratic Functions , 2008, IEEE Transactions on Automatic Control.

[33]  Ricardo C. L. F. Oliveira,et al.  Convergent LMI Relaxations for Quadratic Stabilizability and ${{\mathscr H}}_{\infty}$ Control of Takagi–Sugeno Fuzzy Systems , 2009, IEEE Transactions on Fuzzy Systems.

[34]  Zehui Mao,et al.  $H_\infty$-Filter Design for a Class of Networked Control Systems Via T–S Fuzzy-Model Approach , 2010, IEEE Transactions on Fuzzy Systems.

[35]  Yufei Xu,et al.  H∞ filter design for a class of networked control systems via T-S fuzzy model approach , 2010, International Conference on Fuzzy Systems.

[36]  Baocang Ding,et al.  Stabilization of Takagi–Sugeno Model via Nonparallel Distributed Compensation Law , 2008, IEEE Transactions on Fuzzy Systems.

[37]  Renfa Li,et al.  New Results on a Delay-Derivative-Dependent Fuzzy H $^\infty$ Filter Design for T–S Fuzzy Systems , 2011, IEEE Transactions on Fuzzy Systems.

[38]  Zhiyu Xi,et al.  Piecewise Sliding-Mode Control for T–S Fuzzy Systems , 2011, IEEE Transactions on Fuzzy Systems.

[39]  Jin Bae Park,et al.  A New Fuzzy Lyapunov Function for Relaxed Stability Condition of Continuous-Time Takagi–Sugeno Fuzzy Systems , 2011, IEEE Transactions on Fuzzy Systems.