On the conservativeness of fuzzy and fuzzy-polynomial control of nonlinear systems

A fairly general class of nonlinear plants can be modeled as fuzzy systems, i.e., as a time-varying convex combination of “vertex” linear systems. As many linear LMI control results naturally generalize to such fuzzy systems, LMI formulations for fuzzy control became the tool of choice in the 1990s. Important results have since been obtained in the fuzzy arena, although significant sources of conservativeness remain. This paper reviews some of the sources of conservativeness of fuzzy control designs based on the linear vertex models instead of the original nonlinear equations. Then, ideas that may overcome some of the conservativeness issues (but increasing computational requirements) are discussed. Recently, the sum of squares paradigm extended some linear results to polynomial systems; this idea can be used for the so-called fuzzy polynomial systems that are also discussed in this work.

[1]  Kevin Guelton,et al.  LMI BASED H∞ CONTROLLER DESIGN FOR UNCERTAIN TAKAGI-SUGENO DESCRIPTORS SUBJECT TO EXTERNAL DISTURBANCES , 2007 .

[2]  Graziano Chesi,et al.  Homogeneous Lyapunov functions for systems with structured uncertainties , 2003, Autom..

[3]  Kazuo Tanaka,et al.  Stabilization of Polynomial Fuzzy Systems via a Sum of Squares Approach , 2007, 2007 IEEE 22nd International Symposium on Intelligent Control.

[4]  Pedro Albertos,et al.  Iterative Identification and Control: Advances in Theory and Applications , 2002 .

[5]  Vladimir A. Yakubovich,et al.  Linear Matrix Inequalities in System and Control Theory (S. Boyd, L. E. Ghaoui, E. Feron, and V. Balakrishnan) , 1995, SIAM Rev..

[6]  Bart Kosko,et al.  Fuzzy Systems as Universal Approximators , 1994, IEEE Trans. Computers.

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

[8]  Shuzhi Sam Ge,et al.  A noniterative neuro-fuzzy based identification method for Hammerstein processes , 2005 .

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

[10]  Florin Gheorghe Filip,et al.  Decision support and control for large-scale complex systems , 2008, Annu. Rev. Control..

[11]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

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

[13]  Thierry-Marie Guerra,et al.  Conditions of output stabilization for nonlinear models in the Takagi-Sugeno's form , 2006, Fuzzy Sets Syst..

[14]  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..

[15]  Antonio Sala,et al.  Iterative identification and control : advances in theory and applications , 2002 .

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

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

[18]  Hak-Keung Lam,et al.  Stability analysis of fuzzy control systems subject to uncertain grades of membership , 2005, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics).

[19]  Michael Athans,et al.  Guaranteed properties of gain scheduled control for linear parameter-varying plants , 1991, Autom..

[20]  Salim Labiod,et al.  Adaptive fuzzy control of a class of SISO nonaffine nonlinear systems , 2007, Fuzzy Sets Syst..

[21]  Joseba Jokin Quevedo Casín,et al.  Identification and control : the gap between theory and practice , 2007 .

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

[23]  José Luis Navarro,et al.  Target-shaped possibilistic clustering applied to local-model identification , 2006, Eng. Appl. Artif. Intell..

[24]  Robert Babuska,et al.  Neuro-fuzzy methods for nonlinear system identification , 2003, Annu. Rev. Control..

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

[26]  Carsten W. Scherer,et al.  LMI Relaxations in Robust Control , 2006, Eur. J. Control.

[27]  Katta G. Murty,et al.  Some NP-complete problems in quadratic and nonlinear programming , 1987, Math. Program..

[28]  Antonio Sala,et al.  Extensions to “Stability Analysis of Fuzzy Control Systems Subject to Uncertain Grades of Membership” , 2008, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics).

[29]  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 .

[30]  Tor Arne Johansen,et al.  Computation of Lyapunov functions for smooth nonlinear systems using convex optimization , 2000, Autom..

[31]  Edoardo Mosca,et al.  Constrained predictive control of nonlinear plants via polytopic linear system embedding , 2000 .

[32]  R. Freeman,et al.  Robust Nonlinear Control Design: State-Space and Lyapunov Techniques , 1996 .

[33]  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..

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

[35]  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.

[36]  Tao Zhang,et al.  Stable Adaptive Neural Network Control , 2001, The Springer International Series on Asian Studies in Computer and Information Science.

[37]  James C. Bezdek,et al.  Pattern Recognition with Fuzzy Objective Function Algorithms , 1981, Advanced Applications in Pattern Recognition.

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

[39]  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).

[40]  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).

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

[42]  G. Chesi.  Characterizing the positive polynomials which are not SOS , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[43]  A. Kruszewski,et al.  Lois de commande pour une classe de modèles non linéaires sous la forme Takagi-Sugeno : Mise sous forme LMI , 2006 .

[44]  Antonio Sala,et al.  Some remarks on polynomial fuzzy control , 2009, 2009 IEEE Control Applications, (CCA) & Intelligent Control, (ISIC).

[45]  A. Papachristodoulou,et al.  Nonlinear control synthesis by sum of squares optimization: a Lyapunov-based approach , 2004, 2004 5th Asian Control Conference (IEEE Cat. No.04EX904).

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

[47]  Salim Labiod,et al.  Stabilization of Takagi-Sugeno discrete models: towards an unification of the results , 2007, 2007 IEEE International Fuzzy Systems Conference.

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

[49]  Ferenc Szeifert,et al.  Modified Gath-Geva fuzzy clustering for identification of Takagi-Sugeno fuzzy models , 2002, IEEE Trans. Syst. Man Cybern. Part B.

[50]  Thierry-Marie Guerra,et al.  A membership-function-dependent H∞ controller design for Takagi-Sugeno models , 2008, 2008 IEEE International Conference on Fuzzy Systems (IEEE World Congress on Computational Intelligence).

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

[52]  H.K. Lam,et al.  Fuzzy controller with stability and performance rules for nonlinear systems , 2007, Fuzzy Sets Syst..

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

[54]  A. Kruszewski,et al.  New Approaches for the Stabilization of Discrete Takagi-Sugeno Fuzzy Models , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

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

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

[57]  Christopher J. Harris,et al.  Piecewise Lyapunov stability conditions of fuzzy systems , 2001, IEEE Trans. Syst. Man Cybern. Part B.

[58]  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..

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

[60]  Han-Xiong Li,et al.  Conventional fuzzy control and its enhancement , 1996, IEEE Trans. Syst. Man Cybern. Part B.

[61]  Antonio Sala,et al.  Relaxed Stability and Performance LMI Conditions for Takagi--Sugeno Fuzzy Systems With Polynomial Constraints on Membership Function Shapes , 2008, IEEE Transactions on Fuzzy Systems.

[62]  A. Isidori.  Nonlinear Control Systems , 1985 .

[63]  Antonio Sala,et al.  Encoding fuzzy possibilistic diagnostics as a constrained optimization problem , 2008, Inf. Sci..

[64]  Antonio Sala.  Introducing shape-dependent relaxed conditions in fuzzy control of nonlinear systems in Takagi-Sugeno form , 2008, 2008 IEEE International Conference on Fuzzy Systems (IEEE World Congress on Computational Intelligence).

[65]  Chris J. Harris,et al.  Neurofuzzy state estimators and their applications , 1999 .

[66]  Karl-Erik Årzén,et al.  AI in the Feedback Loop: A Survey of Alternative Approaches , 1996 .

[67]  Antonio Sala,et al.  A Triangulation Approach to Asymptotically Exact Conditions for Fuzzy Summations , 2009, IEEE Transactions on Fuzzy Systems.