A Triangulation Approach to Asymptotically Exact Conditions for Fuzzy Summations

Many Takagi-Sugeno (T-S) fuzzy control-synthesis problems in the literature are expressed as the problem of finding decision variables in a double convex sum (fuzzy summation) of positive definite matrices. Matricespsila coefficients in the summation take values in the standard simplex. This paper presents a triangulation approach to the problem of generating simplicial partitions of the standard simplex in order to set up a family of sufficient conditions and, in parallel, another family of necessary ones for fuzzy summations. The conditions proposed in this paper are asymptotically exact as the size of the involved simplices decreases; its conservativeness vanishes for a sufficiently fine partition (sufficiently dense mesh of vertex points). The set of conditions is in the form of linear matrix inequalities (LMIs), for which efficient software is available.

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

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

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

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

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

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

[7]  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..

[8]  David P. Dobkin,et al.  The quickhull algorithm for convex hulls , 1996, TOMS.

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

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

[11]  P. Apkarian,et al.  Mixed H2/H∞ multi-channel linear parameter-varying control in discrete time , 2000 .

[12]  Chun-Hsiung Fang,et al.  Robust H∞ fuzzy static output feedback control of T-S fuzzy systems with parametric uncertainties , 2007, Fuzzy Sets Syst..

[13]  Gang Feng.  Controller synthesis of fuzzy dynamic systems based on piecewise Lyapunov functions , 2003, IEEE Trans. Fuzzy Syst..

[14]  Kazuo Tanaka,et al.  Model construction, rule reduction, and robust compensation for generalized form of Takagi-Sugeno fuzzy systems , 2001, IEEE Trans. Fuzzy Syst..

[15]  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..

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

[17]  C. Scherer,et al.  Linear Matrix Inequalities in Control , 2011 .

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

[19]  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).

[20]  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).

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

[22]  Guang-Hong Yang,et al.  Robust static output feedback control for linear discrete-time systems with time-varying uncertainties , 2008, Syst. Control. Lett..

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

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

[25]  Antonio Sala,et al.  On the conservativeness of fuzzy and fuzzy-polynomial control of nonlinear systems , 2009, Annu. Rev. Control..