Delay Dependent Local Stabilization Conditions for Time-delay Nonlinear Discrete-time Systems Using Takagi-Sugeno Models

We propose convex conditions for stabilization of nonlinear discrete-time systems with time-varying delay in states through a fuzzy Takagi-Sugeno (T-S) modeling. These conditions are developed from a fuzzy Lyapunov-Krasovskii function and they are formulated in terms of linear matrix inequalities (LMIs). The results can be applied to a class of nonlinear systems that can be exactly represented by T-S fuzzy models inside a specific region called the region of validity. As a consequence, we need to provide an estimate of the set of safe initial conditions called the region of attraction such that the closed-loop trajectories starting in this set are assured to remain in the region of validity and to converge asymptotically to the origin. The estimate of the region of attraction is done with the aid of two sets: one dealing with the current state, and the other concerning the delayed states. Then, we can obtain the feedback fuzzy control law depending on the current state, xk, and the maximum delayed state vector, xk−d̅. It is shown that such a control law can locally stabilize the nonlinear discrete-time system at the origin. We also develop convex optimization procedures for the computation of the fuzzy control gains that maximize the estimates of the region of attraction. We present two examples to demonstrate the efficiency of the developed approach and to compare it with other approaches in the literature.

[1]  Ligang Wu,et al.  Fault Detection for T-S Fuzzy Time-Delay Systems: Delta Operator and Input-Output Methods , 2015, IEEE Transactions on Cybernetics.

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

[3]  Ligang Wu,et al.  Adaptive Fuzzy Control for Nonlinear Networked Control Systems , 2017, IEEE Transactions on Systems, Man, and Cybernetics: Systems.

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

[5]  Fuli Wang,et al.  Robust fast adaptive fault estimation and tolerant control for T-S fuzzy systems with interval time-varying delay , 2017, Int. J. Syst. Sci..

[6]  Antonio Sala,et al.  Polytopic invariant and contractive sets for closed-loop discrete fuzzy systems , 2014, J. Frankl. Inst..

[7]  Eugênio B. Castelan,et al.  Gain-scheduled output control design for a class of discrete-time nonlinear systems with saturating actuators , 2011, Syst. Control. Lett..

[8]  Dong Hwan Lee,et al.  Local stability analysis of T-S fuzzy systems using second-order time derivative of the membership functions , 2017 .

[9]  James Lam,et al.  Stability Analysis and Stabilization for Discrete-Time Fuzzy Systems With Time-Varying Delay , 2009, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics).

[10]  J. Bernussou,et al.  H/sub 2/ and H/sub /spl infin// robust output feedback control for continuous time polytopic systems , 2007 .

[11]  Thierry-Marie Guerra,et al.  Discrete Tagaki-Sugeno models for control: Where are we? , 2009, Annu. Rev. Control..

[12]  Antonio Sala,et al.  Local fuzzy modeling: Maximising the basin of attraction , 2010, International Conference on Fuzzy Systems.

[13]  Yuanqing Xia,et al.  Robust sliding mode control for uncertain discrete-time systems with time delay , 2010 .

[14]  Eugênio B. Castelan,et al.  Fuzzy dynamic output feedback control through nonlinear Takagi-Sugeno models , 2015, Fuzzy Sets Syst..

[15]  Shengyuan Xu,et al.  Robust Controller Design of Uncertain Discrete Time-Delay Systems With Input Saturation and Disturbances , 2012, IEEE Transactions on Automatic Control.

[16]  Young Hoon Joo,et al.  Local stability and local stabilization of discrete-time T–S fuzzy systems with time-delay , 2016 .

[17]  Shouming Zhong,et al.  Stabilization analysis for discrete-time systems with time delay , 2010, Appl. Math. Comput..

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

[19]  Huijun Gao,et al.  Stability analysis and controller synthesis for discrete-time delayed fuzzy systems via small gain theorem , 2013, Inf. Sci..

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

[21]  Thierry-Marie Guerra,et al.  An improved robust stabilization method for discrete-time fuzzy systems with time-varying delays , 2014, J. Frankl. Inst..

[22]  Young Hoon Joo,et al.  On the Generalized Local Stability and Local Stabilization Conditions for Discrete-Time Takagi–Sugeno Fuzzy Systems , 2014, IEEE Transactions on Fuzzy Systems.

[23]  Jian Chu.  Application of a discrete optimal tracking controller to an industrial electric heater with pure delays , 1995 .

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

[25]  Jin Bae Park,et al.  Robust digital implementation of fuzzy control for uncertain systems and its application to active magnetic bearing system , 2012 .

[26]  Luigi Galotto Junior,et al.  Controle ótimo H∞ de sistemas não-lineares com modelos fuzzy takagi-sugeno , 2008 .

[27]  Gang Feng,et al.  Delay-dependent local stabilization of nonlinear discrete-time system using T-S models through convex optimization , 2014, 2014 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE).

[28]  Valter J. S. Leite,et al.  Local Stabilization of Time-Delay Nonlinear Discrete-Time Systems Using Takagi-Sugeno Models and Convex Optimization , 2014 .

[29]  Ligang Wu,et al.  A New Approach to Stability Analysis and Stabilization of Discrete-Time T-S Fuzzy Time-Varying Delay Systems , 2011, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics).

[30]  Yunong Zhang,et al.  Division by zero, pseudo-division by zero, Zhang dynamics method and Zhang-gradient method about control singularity conquering , 2017, Int. J. Syst. Sci..

[31]  Marcio Fantini Miranda,et al.  Robust stabilization of polytopic discrete-time systems with time-varying delay in the states , 2010, 49th IEEE Conference on Decision and Control (CDC).

[32]  S. Niculescu.  Delay Effects on Stability: A Robust Control Approach , 2001 .

[33]  Magdi S. Mahmoud,et al.  Robust stability and stabilization methods for a class of nonlinear discrete-time delay systems , 2010, Appl. Math. Comput..

[34]  B. Barmish.  Stabilization of uncertain systems via linear control , 1983 .