Asymptotically necessary and sufficient conditions for stability and performance in fuzzy control: Applications of Polya's theorem

Stability and performance requirements in fuzzy control of Takagi-Sugeno systems are usually stated as fuzzy summations, i.e., sums of terms, related to Lyapunov functions, which are weighted by membership functions or products of them. This paper presents an application to fuzzy control of Polya's theorems on positive forms on the standard simplex. The result provides a set of progressively less conservative sufficient conditions for proving positivity of fuzzy summations; such conditions are less and less conservative as a complexity parameter, n, increases. Particular cases of such conditions are those in [C.-H. Fang, Y.-S. Liu, S.-W. Kau, L. Hong, C.-H. Lee, A new LMI-based approach to relaxed quadratic stabilization of T-S fuzzy control systems, IEEE Trans. Fuzzy Systems 14 (2006) 286-397; X. Liu, Q. Zhang, New approaches to H"~ controller designs based on fuzzy observers for T-S fuzzy systems via LMI, Automatica 39 (9) (2003) 1571-1582], with n=2 and 3, respectively. The proposed conditions are asymptotically exact, i.e., necessary and sufficient when n tends to infinity or, equivalently, when a tolerance parameter tends to zero.

[1]  Robert Babuska,et al.  Perspectives of fuzzy systems and control , 2005, Fuzzy Sets Syst..

[2]  Stephen P. Boyd,et al.  Linear Matrix Inequalities in Systems and Control Theory , 1994 .

[3]  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..

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

[5]  E. Yaz.  Linear Matrix Inequalities In System And Control Theory , 1998, Proceedings of the IEEE.

[6]  Antonio Sala,et al.  Relaxed Stability and Performance Conditions for Takagi–Sugeno Fuzzy Systems With Knowledge on Membership Function Overlap , 2007, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics).

[7]  Graziano Chesi,et al.  Polynomially parameter-dependent Lyapunov functions for robust stability of polytopic systems: an LMI approach , 2005, IEEE Transactions on Automatic Control.

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

[9]  B. Reznick,et al.  A new bound for Pólya's Theorem with applications to polynomials positive on polyhedra , 2001 .

[10]  Marcelo C. M. Teixeira,et al.  On relaxed LMI-based designs for fuzzy regulators and fuzzy observers , 2001, 2001 European Control Conference (ECC).

[11]  Kazuo Tanaka,et al.  A fuzzy Lyapunov approach to fuzzy control system design , 2001, Proceedings of the 2001 American Control Conference. (Cat. No.01CH37148).

[12]  Pierre Apkarian,et al.  Parameterized linear matrix inequality techniques in fuzzy control system design , 2001, IEEE Trans. Fuzzy Syst..

[13]  C. W. Scherer,et al.  Relaxations for Robust Linear Matrix Inequality Problems with Verifications for Exactness , 2005, SIAM J. Matrix Anal. Appl..

[14]  Jesús A. De Loera,et al.  Erratum to: “An effective version of Pólya's theorem on positive definite forms” [J. Pure Appl. Algebra 108 (1996) 231–240]☆ , 2001 .

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

[16]  Jesús A. De Loera,et al.  An effective version of Pólya's theorem on positive definite forms , 1996 .

[17]  Antonio Sala,et al.  Relaxed LMI conditions for closed-loop fuzzy systems with tensor-product structure , 2007, Eng. Appl. Artif. Intell..

[18]  Sheng-Luen Chung,et al.  Robust static output-feedback stabilization for nonlinear discrete-time systems with time delay via fuzzy control approach , 2005, IEEE Trans. Fuzzy Syst..

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

[20]  Kazuo Tanaka,et al.  Parallel distributed compensation for Takagi-Sugeno fuzzy models: multiobjective controller design , 1999, Proceedings of the 1999 American Control Conference (Cat. No. 99CH36251).

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

[22]  Pierre-Alexandre Bliman,et al.  A Convex Approach to Robust Stability for Linear Systems with Uncertain Scalar Parameters , 2003, SIAM J. Control. Optim..

[23]  L. Xiaodong,et al.  New approaches to H∞ controller designs based on fuzzy observers for T-S fuzzy systems via LMI , 2003, Autom..

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

[25]  Tong-heng Lee,et al.  Technical communique: Improvement on observer-based H∞ control for T-S fuzzy systems , 2005 .