Further Studies on Control Synthesis of Discrete-Time T–S Fuzzy Systems via Useful Matrix Equalities

This paper is concerned with further studies on the control synthesis of discrete-time nonlinear systems in the Takagi-Sugeno (T-S) fuzzy form. To do this, a novel slack variable technique is presented by developing some useful matrix equalities, which are homogenous polynomially parameter-dependent on both the current-time normalized fuzzy weighting functions and the past-time normalized fuzzy weighting functions. Under the framework of homogenous matrix polynomials, the algebraic properties of both the current-time normalized fuzzy weighting functions and the past-time normalized fuzzy weighting functions are collected for the first time into sets of united collection matrices. Consequently, the relaxation quality of control synthesis of discrete-time T-S fuzzy systems is improved, i.e., the convergence of asymptotically necessary and sufficient stabilization conditions is further sped up. Finally, a numerical example is provided to illustrate the effectiveness of the proposed result.

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

[2]  Thierry-Marie Guerra,et al.  Nonquadratic Stabilization Conditions for a Class of Uncertain Nonlinear Discrete Time TS Fuzzy Models: A New Approach , 2008, IEEE Transactions on Automatic Control.

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

[4]  Xiangpeng Xie,et al.  Relaxed Stability Conditions for Continuous-Time T–S Fuzzy-Control Systems Via Augmented Multi-Indexed Matrix Approach , 2011, IEEE Transactions on Fuzzy Systems.

[5]  Jin Bae Park,et al.  Improvement on Nonquadratic Stabilization of Discrete-Time Takagi–Sugeno Fuzzy Systems: Multiple-Parameterization Approach , 2010, IEEE Transactions on Fuzzy Systems.

[6]  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..

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

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

[9]  Jin Bae Park,et al.  Further improvement of periodic control approach for relaxed stabilization condition of discrete-time Takagi-Sugeno fuzzy systems , 2011, Fuzzy Sets Syst..

[10]  Xiao‐Heng Chang.  A Descriptor Representation Approach , 2012 .

[11]  Baocang Ding,et al.  Homogeneous Polynomially Nonquadratic Stabilization of Discrete-Time Takagi–Sugeno Systems via Nonparallel Distributed Compensation Law , 2010, IEEE Transactions on Fuzzy Systems.

[12]  Shumin Fei,et al.  Nonquadratic Stabilization of Continuous T–S Fuzzy Models: LMI Solution for a Local Approach , 2012, IEEE Transactions on Fuzzy Systems.

[13]  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..

[14]  Ricardo C. L. F. Oliveira,et al.  Parameter-Dependent LMIs in Robust Analysis: Characterization of Homogeneous Polynomially Parameter-Dependent Solutions Via LMI Relaxations , 2007, IEEE Transactions on Automatic Control.

[15]  Bing Chen,et al.  Robust Stability for Uncertain Delayed Fuzzy Hopfield Neural Networks With Markovian Jumping Parameters , 2009, IEEE Trans. Syst. Man Cybern. Part B.

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

[17]  Honghai Liu,et al.  Reliable Fuzzy Control for Active Suspension Systems With Actuator Delay and Fault , 2012, IEEE Transactions on Fuzzy Systems.

[18]  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.

[19]  Ricardo C. L. F. Oliveira,et al.  Selective $\hbox{\scr H}_2$ and $\hbox{\scr H}_\infty$ Stabilization of Takagi–Sugeno Fuzzy Systems , 2011, IEEE Transactions on Fuzzy Systems.

[20]  Peng Yang,et al.  Further studies on LMI-based relaxed stabilization conditions for nonlinear systems in Takagi-Sugeno's form , 2006, Autom..

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

[22]  Gang Feng,et al.  H/sub /spl infin// controller synthesis of fuzzy dynamic systems based on piecewise Lyapunov functions and bilinear matrix inequalities , 2005, IEEE Transactions on Fuzzy Systems.

[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]  Thierry-Marie Guerra,et al.  Control Law Proposition for the Stabilization of Discrete Takagi–Sugeno Models , 2009, IEEE Transactions on Fuzzy Systems.

[25]  L X Wang,et al.  Fuzzy basis functions, universal approximation, and orthogonal least-squares learning , 1992, IEEE Trans. Neural Networks.

[26]  Xiangpeng Xie,et al.  Control Synthesis of Discrete-Time T–S Fuzzy Systems Based on a Novel Non-PDC Control Scheme , 2013, IEEE Transactions on Fuzzy Systems.

[27]  Jin Bae Park,et al.  Approaches to extended non-quadratic stability and stabilization conditions for discrete-time Takagi-Sugeno fuzzy systems , 2011, Autom..

[28]  G. Feng,et al.  Controller synthesis of fuzzy dynamic systems based on piecewise Lyapunov functions , 2001, 10th IEEE International Conference on Fuzzy Systems. (Cat. No.01CH37297).