Stability analysis and feedback control of T-S fuzzy hyperbolic delay model for a class of nonlinear systems with time-varying delay

Document Type: Research Paper

Authors

School of Mathematics and Statistics, Xidian University, Xi'an, 710071, China

Abstract

In this paper, a new T-S fuzzy hyperbolic delay model for a class of nonlinear systems with time-varying delay, is presented to address the problems of stability analysis and feedback control. Fuzzy controller is designed based on the parallel distributed compensation (PDC), and with a new Lyapunov function, delay dependent asymptotic stability conditions of the closed-loop system are derived via linear matrix inequalities (LMIs). Besides, considering the differences between the model and the real system, we extent the model to uncertain T-S fuzzy hyperbolic delay model. Based on the uncertain model, a robust $H_{\infty}$ fuzzy controller is obtained and stability conditions are developed in terms of LMIs. The main advantage of the control based on T-S fuzzy hyperbolic delay model is that it can achieve small control amplitude via ``soft'' constraint approach. Finally, a numerical example and the Van de Vusse example are given to validate the advantages of the proposed method.

Keywords


[1] P. Balasubramaniam and V. M. Revathi, H1 Filtering for Markovian switching system with
mode-dependent time-varying delays, Circuits Systems and Signal Processing, 33(2) (2014),
347{369.
[2] P. Balasubramaniam and T. Senthilkumar, Delay-dependent robust stabilization and H1
control for uncertain stochastic T-S fuzzy systems with multiple time delays, Iranian Journal
of Fuzzy Systems, 9(2) (2012), 89{111.
[3] Y. Y. Cao and P. M. Frank, Stability analysis and synthesis of nonlinear time-delay systems
via linear Takagi-Sugeno fuzzy models, Fuzzy sets and systems, 124(2) (2001), 213{229.
[4] M. L. Chen and J. M. Li, Modeling and control of T-S fuzzy hyperbolic model for a class of
nonlinear systems, Proceedings of International Conference on Modelling, Identi cation and
Control, (2012), 57{62.
[5] M. L. Chen and J. M. Li, Non-fragile guaranteed cost control for Takagi-Sugeno fuzzy hyper-
bolic systems, International Journal of Systems Science, 46(9) (2015), 1614{1627.
[6] T. H. Chen, C. C. Kung and K. H. Su, The piecewise Lyapunov functions based the delay-
independent H1 controller design for a class of time-delay T-S fuzzy system, IEEE Interna-
tional Conference on Systems, Man and Cybernetics, (2007), 121{126.
[7] C. S. Chiu, W. T. Yang and T. S. Chiang, Robust output feedback control of T-S fuzzy time-
delay systems, IEEE Symposium on Computational Intelligence in Control and Automation,
(2013), 45{50.
[8] G. Feng, A survey on analysis and design of model-based fuzzy control systems, IEEE Trans-
actions on Fuzzy Systems, 14(5) (2006), 676{697.
[9] Daniel W. C. Ho and Y. Niu, Robust fuzzy design for nonlinear uncertain stochastic systems
via sliding-mode control, IEEE Transactions on Fuzzy Systems, 15(3) (2007), 350{358.
[10] Z. Hong and Z. F. Li, Stabilization for a class of T-S uncertain nonlinear systems with
Time-Delay, Chinese Control and Decision Conference, (2012), 375{380.
[11] M. Y. Hsiao, C. H. Liu, S. H. Tsai and et al, A Takagi-Sugeno fuzzy-model-based modeling
method, IEEE International Conference on Fuzzy Systems, (2010), 1{6.
[12] J. M. Li, G. Zhang, Non-fragile guaranteed cost control of T-S fuzzy time varying delay
systems with local bilinear models, Iranian Journal of Fuzzy Systems, 9(2) (2012), 43{62.
[13] Y. M. Li and S. C. Tong, Prescribed performance adaptive fuzzy output-feedback dynamic
surface control for nonlinear large-scale systems with time delays, Information Sciences, 292
(2015), 125{142.
[14] C. H. Lien and K. W. Yu, Robust control for Takagi-Sugeno fuzzy systems with time-varying
state and input delays, Chaos, Solitons and Fractals, 35(5) (2008), 1003{1008.
[15] C. Lin, Q. G. Wang and T. H. Lee, Delay-dependent LMI conditions for stability and stabi-
lization of T-S fuzzy systems with bounded time-delay, Fuzzy Sets Systems, 157(9) (2006),
1229{1247.
[16] C. Lin, Q. G. Wang, T. H. Lee and Y. He, Fuzzy weighting-dependent approach to H1 lter
design for time-delay fuzzy systems, IEEE Transactions on Signal Processing, 55(6) (2007),
2746{2751.
[17] T. Takagi and M. Sugeno, Fuzzy identi cation of systems and its applications to modelling
and control, IEEE Transactions on Systems, Man and Cybernetics, 15(1) (1985), 116{132.

[18] K. Tanaka, T. Ikeda and H. O. Wang, Fuzzy regulators and fuzzy observers: relaxed stability
conditions and LMI-based designs, IEEE Transactions on Fuzzy Systems, 6(2) (1998), 250{
265.
[19] K. Tanaka and H. O. Wang, Fuzzy control systems design and analysis: A linear matrix
inequality approach, John Wiley and Sons, 2002.
[20] S. H. Tsai and C. J. Fang, A novel relaxed stabilization condition for a class of T-S time-delay
fuzzy systems, IEEE International Conference on Fuzzy Systems, (2014), 2294{2299.
[21] C. S. Tseng, B. S. Chen and H. J. Uang, Fuzzy tracking control design for nonlinear dynamic
systems via T-S fuzzy model, IEEE Transactions on Fuzzy Systems, 9(3)(2001), 381{392.
[22] G. Wang, Y. Wang and D. S. Yang, New sucient conditions for delay-dependent robust
H1 control of uncertain nonlinear system based on fuzzy hyperbolic model with time-varying
delays, Chinese Control and Decision Conference, (2012), 1138{1143.
[23] S. B. Wang, Y. Y. Wang and L. K. Zhang, Time-delay dependent state feedback fuzzy-
predictive control of time-delay T-S fuzzy model, Fifth International Conference on Fuzzy
Systems and Knowledge Discovery, (2008), 129{133.
[24] T. T. Wang, H. C. Yan, H. B. Shi and H. Zhang, Event-triggered H1 control for networked
T-S fuzzy systems with time delay, IEEE International Conference on Information and Au-
tomation, (2014), 194{199.
[25] Y. Y. Wang, H. G. Zhang, J. Y. Zhang and et al, An sos-based observer design for discrete-
time polynomial fuzzy systems, International Journal of Fuzzy Systems, 17(1) (2015), 94{104.
[26] G. L. Wei, G. Feng and Z. D. Wang, Robust H1 control for discrete-time fuzzy systems with
in nite-distributed delays, IEEE Transactions on Fuzzy Systems, 17(1) (2009), 224{232.
[27] H. N. Wu and H. X. Li, New approach to delay dependent stability analysis and stabilization
for continuous-time fuzzy systems with time-varying delay, IEEE Transactions on Fuzzy
Systems, 15(3) (2007), 482{493.
[28] H. G. Zhang, Fuzzy hyperbolic model - modeling, control and application, Beijing: Science
Press, 2009.
[29] H. G. Zhang, Q. X. Gong and Y. C. Wang, Delay-dependent robust H1 control for uncertain
fuzzy hyperbolic systems with multiple delays, Progress in Natural Science, 18(1) (2008),
97{104.
[30] H. G. Zhang, X. R. Liu, Q. X. Gong and et al, New sucient conditions for robust H1 fuzzy
hyperbolic tangent control of uncertain nonlinear systems with time-varying delay, Fuzzy Sets
and Systems, 161(15) (2010), 1993{2011.
[31] H. G. Zhang, S. X. Lun and D. R. Liu, Fuzzy H1 lter design for a class of nonlinear discrete-
time systems with multiple time delays, IEEE Transactions on Fuzzy Systems, 15(3) (2007),
453{469.
[32] H. G. Zhang and Y. B. Quan, Modeling, identication and control of a class of nonlinear
system, IEEE Transactions on Fuzzy Systems, 9(2) (2001), 349{354.
[33] H. G. Zhang and X. P. Xie, Relaxed Stability Conditions for Continuous-Time T-S Fuzzy-
Control Systems Via Augmented Multi-Indexed Matrix Approach, IEEE Transactions on
Fuzzy Systems, 19(3) (2011), 478{492.
[34] H. G. Zhang, J. L. Zhang, G. H. Yang and et al, Leader-based optimal coordination con-
trol for the consensus problem of multiagent di erential games via fuzzy adaptive dynamic
programming, IEEE Transactions on Fuzzy Systems, 23(1) (2015), 152{163.
[35] J. H. Zhang, P. Shi and J. Q. Qiu, Non-fragile guaranteed cost control for uncertain stochastic
nonlinear time-delay systems, Journal of the Franklin Institute, 346(7) (2009), 676{690.
[36] Z. Y. Zhang, C. Lin and B. Chen, New stability and stabilization conditions for T-S fuzzy
systems with time delay, Fuzzy Sets and Systems, 263(C) (2015), 82{91.
[37] Y. Zhao and H. J. Gao, Fuzzy-model-based control of an overhead crane with input delay and
actuator saturation, IEEE Transactions on Fuzzy Systems, 20(1) (2012), 181{186.