Existence of Similarity Transformation Converting BMIs to LMIs

Under a mild condition, we investigate the existence of a similarity transformation regarding robust Hinfin stability analysis and observer-controller synthesis problems. The observer-based controller is capable of disturbance-rejection in the presence of unknown but bounded disturbance. We present results, which illustrate plainly the role of the similarity transformation in converting a BMI problem into an LMI problem in a unified fashion applicable to both continuous- and discrete-time problems with or without uncertainty. Finally the validity and applicability of the approach are illuminated by examples.

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

[2]  G. Feng,et al.  Quadratic stabilization of uncertain discrete-time fuzzy dynamic systems , 2001 .

[3]  J. Buckley.  Sugeno type controllers are universal controllers , 1993 .

[4]  Ali Jadbabaie,et al.  Fuzzy observer-based control of nonlinear systems , 1997, Proceedings of the 36th IEEE Conference on Decision and Control.

[5]  Yong-Yan Cao,et al.  Robust H∞ disturbance attenuation for a class of uncertain discrete-time fuzzy systems , 2000, IEEE Trans. Fuzzy Syst..

[6]  Anthony Tzes,et al.  Quadratic stability analysis of the Takagi-Sugeno fuzzy model , 1998, Fuzzy Sets Syst..

[7]  P. Gahinet,et al.  A linear matrix inequality approach to H∞ control , 1994 .

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

[9]  Gang Feng,et al.  Analysis and design of fuzzy control systems using dynamic fuzzy-state space models , 1999, IEEE Trans. Fuzzy Syst..

[10]  C. Scherer,et al.  Multiobjective output-feedback control via LMI optimization , 1997, IEEE Trans. Autom. Control..

[11]  Wilfrid Perruquetti,et al.  Non-quadratic stabilisation of discrete Takagi Sugeno fuzzy models , 2001, 10th IEEE International Conference on Fuzzy Systems. (Cat. No.01CH37297).

[12]  Joongseon Joh,et al.  On the stability issues of linear Takagi-Sugeno fuzzy models , 1998, IEEE Trans. Fuzzy Syst..

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

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

[15]  Kiriakos Kiriakidis,et al.  Robust stabilization of the Takagi-Sugeno fuzzy model via bilinear matrix inequalities , 2001, IEEE Trans. Fuzzy Syst..

[16]  Kazuo Tanaka,et al.  Robust stabilization of a class of uncertain nonlinear systems via fuzzy control: quadratic stabilizability, H∞ control theory, and linear matrix inequalities , 1996, IEEE Trans. Fuzzy Syst..

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

[18]  Anders Rantzer,et al.  Computation of piecewise quadratic Lyapunov functions for hybrid systems , 1997, 1997 European Control Conference (ECC).

[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]  Zengqi Sun,et al.  Analysis and design of fuzzy controller and fuzzy observer , 1998, IEEE Trans. Fuzzy Syst..

[21]  Hao Ying,et al.  Sufficient conditions on uniform approximation of multivariate functions by general Takagi-Sugeno fuzzy systems with linear rule consequent , 1998, IEEE Trans. Syst. Man Cybern. Part A.

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

[23]  Gang Feng,et al.  H∞ control of uncertain fuzzy continuous-time systems , 2000, Fuzzy Sets Syst..

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

[25]  G. Feng.  Approaches to quadratic stabilization of uncertain fuzzy dynamic systems , 2001 .

[26]  Euntai Kim,et al.  Stability analysis and synthesis for an affine fuzzy control system via LMI and ILMI: a continuous case , 2002, IEEE Trans. Fuzzy Syst..

[27]  Eun Tae Jeung,et al.  Robust fuzzy H[infin] control for uncertain nonlinear systems via state feedback: an LMI approach , 2001, Fuzzy Sets Syst..

[28]  M. Mattei,et al.  A note on the solution of a class of BMIs for H/sub /spl infin// problems , 2000, Proceedings of the 39th IEEE Conference on Decision and Control (Cat. No.00CH37187).

[29]  Gang Feng,et al.  Analysis and design of fuzzy control systems using dynamic fuzzy global models , 1995, Fuzzy Sets Syst..

[30]  B. Anderson,et al.  A game theoretic approach to H ∞ control for time-varying systems , 1992 .

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

[32]  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..

[33]  Pierre Apkarian,et al.  Robust pole placement in LMI regions , 1999, IEEE Trans. Autom. Control..

[34]  Ji-Chang Lo,et al.  Robust H/sub /spl infin// control for fuzzy systems with Frobenius norm-bounded uncertainties , 2006, IEEE Transactions on Fuzzy Systems.

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

[36]  Pierre Apkarian,et al.  New fuzzy control model and dynamic output feedback parallel distributed compensation , 2004, IEEE Transactions on Fuzzy Systems.

[37]  P. Borne,et al.  Relaxed stability conditions for Takagi-Sugeno's fuzzy models , 2000, Ninth IEEE International Conference on Fuzzy Systems. FUZZ- IEEE 2000 (Cat. No.00CH37063).

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

[39]  Mohammed Chadli,et al.  Relaxed stability conditions for Takagi-Sugeno fuzzy systems , 2000, Smc 2000 conference proceedings. 2000 ieee international conference on systems, man and cybernetics. 'cybernetics evolving to systems, humans, organizations, and their complex interactions' (cat. no.0.

[40]  Kazuo Tanaka,et al.  T-S fuzzy model with linear rule consequence and PDC controller: a universal framework for nonlinear control systems , 2000, Ninth IEEE International Conference on Fuzzy Systems. FUZZ- IEEE 2000 (Cat. No.00CH37063).

[41]  Euntai Kim,et al.  Stability analysis and synthesis for an affine fuzzy system via LMI and ILMI: discrete case , 2001, IEEE Trans. Syst. Man Cybern. Part B.

[42]  Marcelo C. M. Teixeira,et al.  Stabilizing controller design for uncertain nonlinear systems using fuzzy models , 1999, IEEE Trans. Fuzzy Syst..

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

[44]  Akira Ichikawa,et al.  Output stabilization of Takagi-Sugeno fuzzy systems , 2000, Fuzzy Sets Syst..

[45]  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.

[46]  Shaocheng Tong,et al.  Observer-based robust fuzzy control of nonlinear systems with parametric uncertainties , 2002, Fuzzy Sets Syst..

[47]  Ji-Chang Lo,et al.  Robust H/sub /spl infin// nonlinear control via fuzzy static output feedback , 2003 .

[48]  M. Sugeno,et al.  Structure identification of fuzzy model , 1988 .

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

[50]  Jong Hae Kim,et al.  Output feedback robust H∞ control of uncertain fuzzy dynamic systems with time-varying delay , 2000, IEEE Trans. Fuzzy Syst..

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

[52]  Euntai Kim,et al.  A new sliding-mode control with fuzzy boundary layer , 2001, Fuzzy Sets Syst..

[53]  Ji-Chang Lo,et al.  Robust H[infin] nonlinear modeling and control via uncertain fuzzy systems , 2004, Fuzzy Sets Syst..