Design of polynomial fuzzy observer–controller for nonlinear systems with state delay: sum of squares approach

ABSTRACT This paper investigates the problem of observer-based control for two classes of polynomial fuzzy systems with time-varying delay. The first class concerns a special case where the polynomial matrices do not depend on the estimated state variables. The second one is the general case where the polynomial matrices could depend on unmeasurable system states that will be estimated. For the last case, two design procedures are proposed. The first one gives the polynomial fuzzy controller and observer gains in two steps. In the second procedure, the designed gains are obtained using a single-step approach to overcome the drawback of a two-step procedure. The obtained conditions are presented in terms of sum of squares (SOS) which can be solved via the SOSTOOLS and a semi-definite program solver. Illustrative examples show the validity and applicability of the proposed results.

[1]  Ahmed El Hajjaji,et al.  Control of Time Delay Polynomial Fuzzy Model Subject to Actuator Saturation , 2016, Int. J. Fuzzy Syst..

[2]  Yong Zhang,et al.  Weighted Fuzzy Observer-Based Fault Detection Approach for Discrete-Time Nonlinear Systems via Piecewise-Fuzzy Lyapunov Functions , 2016, IEEE Transactions on Fuzzy Systems.

[3]  Arkadi Nemirovski,et al.  Lmi Control Toolbox For Use With Matlab , 2014 .

[4]  Witold Pedrycz,et al.  Analysis of stability and robust stability of polynomial fuzzy model-based control systems using a sum-of-squares approach , 2014, Soft Comput..

[5]  W. P. M. H. Heemels,et al.  Stability analysis of networked control systems: A sum of squares approach , 2010, CDC.

[6]  Guanghong Yang,et al.  Fault‐tolerant control synthesis for a class of nonlinear systems: Sum of squares optimization approach , 2009 .

[7]  Jos F. Sturm,et al.  A Matlab toolbox for optimization over symmetric cones , 1999 .

[8]  Jianbin Qiu,et al.  Real-Time Fault Detection Approach for Nonlinear Systems and its Asynchronous T–S Fuzzy Observer-Based Implementation , 2017, IEEE Transactions on Cybernetics.

[9]  Jianbin Qiu,et al.  Approaches to T–S Fuzzy-Affine-Model-Based Reliable Output Feedback Control for Nonlinear Itô Stochastic Systems , 2017, IEEE Transactions on Fuzzy Systems.

[10]  Graziano Chesi,et al.  LMI Techniques for Optimization Over Polynomials in Control: A Survey , 2010, IEEE Transactions on Automatic Control.

[11]  Shu-Guang Cao,et al.  Design of fuzzy control systems with guaranteed stability , 1997, Fuzzy Sets Syst..

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

[13]  Kazuo Tanaka,et al.  Polynomial fuzzy observer design: A sum of squares approach , 2009, Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference.

[14]  Kazuo Tanaka,et al.  An SOS-based observer design for polynomial fuzzy systems , 2011, Proceedings of the 2011 American Control Conference.

[15]  Ahmed El Hajjaji,et al.  Observer-Based Robust $H_{\infty }$ Reliable Control for Uncertain T–S Fuzzy Systems With State Time Delay , 2010, IEEE Transactions on Fuzzy Systems.

[16]  Kazuo Tanaka,et al.  A polynomial observer design for a wider class of polynomial fuzzy systems , 2011, 2011 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE 2011).

[17]  Peter J Seiler,et al.  Quantitative local analysis of nonlinear systems using sum-of-squares decompositions (T-1) , 2009 .

[18]  Mourad Kchaou,et al.  Control of Time Delay Fuzzy Descriptor Systems with Actuator Saturation , 2014, Circuits Syst. Signal Process..

[19]  Ji-Chang Lo,et al.  SOS-based fuzzy stability analysis via homogeneous Lyapunov functions , 2014, 2014 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE).

[20]  Wang Jicheng,et al.  A ROBUSTNESS ANALYSIS FOR NONLINEAR SYSTEMS , 1990 .

[21]  James Lam,et al.  α-Dissipativity analysis of singular time-delay systems , 2011, Autom..

[22]  Bing Chen,et al.  Fuzzy guaranteed cost control for nonlinear systems with time-varying delay , 2005, IEEE Transactions on Fuzzy Systems.

[23]  Mourad Kchaou,et al.  Robust observer‐based control design for uncertain singular systems with time‐delay , 2014 .

[24]  Hamid Reza Karimi,et al.  New results on H∞ dynamic output feedback control for Markovian jump systems with time‐varying delay and defective mode information , 2014 .

[25]  Honghai Liu,et al.  Stability Analysis of Polynomial-Fuzzy-Model-Based Control Systems Using Switching Polynomial Lyapunov Function , 2013, IEEE Transactions on Fuzzy Systems.

[26]  Ahmed El Hajjaji,et al.  Comment on "Observer-based robust fuzzy control of nonlinear systems with parametric uncertainties" , 2006, Fuzzy Sets Syst..

[27]  A. El-Hajjaji,et al.  On stabilizability of stochastic fuzzy systems , 2006, 2006 American Control Conference.

[28]  Giuseppe Franzè,et al.  A robust fault detection filter for polynomial nonlinear systems via sum-of-squares decompositions , 2012, Syst. Control. Lett..

[29]  Kevin Guelton,et al.  Sum-of-Squares Stability Analysis of Takagi-Sugeno Systems Based on Multiple Polynomial Lyapunov Functions , 2013 .

[30]  W. P. M. H. Heemels,et al.  Stability analysis of networked control systems: A sum of squares approach , 2010, 49th IEEE Conference on Decision and Control (CDC).

[31]  H. Gassara,et al.  Robust control of T‐S fuzzy systems with time‐varying delay using new approach , 2010 .

[32]  Ahmed El Hajjaji,et al.  Stability analysis and stabilization of polynomial fuzzy systems with time-delay via a Sum Of Squares (SOS) approach , 2015, 2015 American Control Conference (ACC).

[33]  Kazuo Tanaka,et al.  Polynomial Fuzzy Observer Designs: A Sum-of-Squares Approach , 2012, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics).

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

[35]  B. Chen,et al.  Delay-Dependent Robust H∞ Control for T-S Fuzzy Systems With Time Delay , 2005, IEEE Trans. Fuzzy Syst..

[36]  Jianbin Qiu,et al.  Fuzzy-Model-Based Reliable Static Output Feedback $\mathscr{H}_{\infty }$ Control of Nonlinear Hyperbolic PDE Systems , 2016, IEEE Transactions on Fuzzy Systems.

[37]  Yong-Yan Cao,et al.  Analysis and synthesis of nonlinear time-delay systems via fuzzy control approach , 2000, IEEE Trans. Fuzzy Syst..

[38]  H. Gassara,et al.  Local stabilization of Polynomial Fuzzy Model with time delay: SOS approach , 2017 .

[39]  Jianbin Qiu,et al.  Mode-dependent nonrational output feedback control for continuous-time semi-Markovian jump systems with time-varying delay , 2015 .

[40]  Jianbin Qiu,et al.  A New Design of $H$ -Infinity Piecewise Filtering for Discrete-Time Nonlinear Time-Varying Delay Systems via T–S Fuzzy Affine Models , 2017, IEEE Transactions on Systems, Man, and Cybernetics: Systems.