# American Institute of Mathematical Sciences

September  2020, 10(3): 345-354. doi: 10.3934/naco.2020006

## Sliding mode control for uncertain T-S fuzzy systems with input and state delays

 School of science, Shenyang University of Technology, Shenyang, Liaoning 110870, China

* Corresponding author: Ruxia Zhang

Received  May 2019 Revised  August 2019 Published  February 2020

Fund Project: The first author is supported by is supported by National Nature Science Foundation under grant NO.61673099 and Provincial Education Department Key Project (LZGD2017039)

In this paper, the problem of sliding mode control (SMC) for uncertain T-S (Tagaki-Sugeno) fuzzy systems with input and state delays is investigated, in which the nonlinear uncertain terms are unknown, and also unmatched. For the T-S fuzzy model of the controlled object, a method based on sliding mode compensator is designed, and the system is controlled by sliding mode. Based on solving linear matrix inequalities (LMI), we obtain the design method of sliding mode and controller. The sufficient conditions for the asymptotical stability of the sliding mode dynamics are given by using LMI technique and the Lyapunov stability theory, and it has been shown that the state trajectories can be driven onto the sliding surface in a finite time. Finally, a numerical example is provided to illustrate the effectiveness of the proposed theories.

Citation: Yuan Li, Ruxia Zhang, Yi Zhang, Bo Yang. Sliding mode control for uncertain T-S fuzzy systems with input and state delays. Numerical Algebra, Control and Optimization, 2020, 10 (3) : 345-354. doi: 10.3934/naco.2020006
##### References:
 [1] Abdennebi, Nizar and Mansour, A new sliding function for discrete predictive sliding mode control of time delay systems, International Journal of Automation and Computing, 10 (2013), 288-295. [2] P. Balasubramaniam and T. Senthilkumar, Delay - Dependent robust stabilization and H$\infty$ control for uncertain stochastic T-S fuzzy systems with discrete interval and distributed time-varying delays, International Journal of Automation & Computing, 10 (2013), 18-31. [3] T. S. Chiang, C. S. Chiu and P. Liu, Adaptive TS-FNN control for a class of uncertain multi-time-delay systems: The exponentially stable sliding mode- based approach, International Journal of Adaptive Control & Signal Processing, 23 (2010), 378-399. [4] Q. Gao, H. Huang and G. Feng, Robust H$\infty$ stabilization of uncertain T-S fuzzy systems via dynamic integral sliding mode control, Ifac Proceedings Volumes, 46 (2013), 485-490. [5] S. Guo, F. Zhu and L. 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Zhang, Robust H$_\infty$ sliding mode observer design for a class of Takagi-Sugeno fuzzy descriptor systems with time-varying delay, Applied Mathematics & Computation, 337 (2018), 158-178.  doi: 10.1016/j.amc.2018.05.008. [10] Z. Liu and C. Gao, A new result on robust H$\infty$ control for uncertain time-delay singular systems via sliding mode control, Complexity, 21 (2016), 165-177.  doi: 10.1002/cplx.21793. [11] R. M. Nagarale and B. M. Patre, Exponential function based fuzzy sliding mode control of uncertain nonlinear systems, International Journal of Dynamics & Control, 4 (2016), 67-75.  doi: 10.1007/s40435-014-0117-2. [12] T. Niknam and M. H. Khooban, Fuzzy sliding mode control scheme for a class of non-linear uncertain chaotic systems, Iet Science Measurement & Technology, 7 (2013), 249-255. [13] L. Ren, S. Xie and Z. G. Miao, Fuzzy robust sliding mode control of a class of uncertain systems, Journal of Central South University, 23 (2016), 2296-2304. [14] D. B. Salem and W. Saad, Integral sliding mode control for systems with time-varying input and state delays, International Conference on Engineering & Mis, (2017), 978–982. [15] A. Si-Ammour, S. Djennoune and M. Bettayeb, A sliding mode control for linear fractional systems with input and state delays, Communications in Nonlinear Science & Numerical Simulation, 14 (2009), 2310-2318.  doi: 10.1016/j.cnsns.2008.05.011. [16] H. Wang, B. Zhou and R. Lu, New stability and stabilization criteria for a class of fuzzy singular systems with time-varying delay, Journal of the Franklin Institute, 351 (2014), 3766-3781.  doi: 10.1016/j.jfranklin.2013.02.030. [17] Y. Wang, Y. Xia and H. Li, A new integral sliding mode design method for nonlinear stochastic systems, Automatica, 90 (2018), 304-309.  doi: 10.1016/j.automatica.2017.11.029. [18] J. Wu, Robust stabilization for uncertain T-S fuzzy singular system, International Journal of Machine Learning & Cybernetics, 7 (2016), 699-706. [19] Y. Xia, H. Yang, M. Fu and P. Shi, Sliding mode control for linear systems with time-varying input and state delays, Circuits Systems, and Signal Processing, 30 (2011), 629-641.  doi: 10.1007/s00034-010-9237-x. [20] L. Xiao, H. Su and J. Chu, Sliding mode prediction tracking control design for uncertain systems, Asian Journal of Control, 9 (2010), 317-325.  doi: 10.1111/j.1934-6093.2007.tb00417.x. [21] X. G. Yan, S. Spurgeon and Y. Korlov, Output feedback control synthesis for non-linear time- delay systems using a sliding-mode observer, IMA Journal of Mathematical Control and Information, 31 (2014), 501-518.  doi: 10.1093/imamci/dnt028. [22] S. Y. Yoon and Z. Lin, Robust output regulation of linear time-delay systems: A state predictor approach, International Journal of Robust & Nonlinear Control, 26 (2016), 1686-1740.  doi: 10.1002/rnc.3374. [23] J. Yu and Z. Yi, Stability analysis and fuzzy control for uncertain delayed T-S nonlinear systems, International Journal of Fuzzy Systems, 18 (2016), 1-8.  doi: 10.1007/s40815-016-0203-z. [24] Y. Zhang, Robust stability and H$\infty$ control of discrete-time uncertain impulsive systems with time-varying delay, Circuits Systems & Signal Processing, 35 (2016), 3882-3912. [25] Y. Zhao, J. Shen and D. Chen, New stability criterion for discrete-time genetic regulatory networks with time-varying delays and stochastic disturbances, Mathematical Problems in Engineering, 2016 (2016), 1-13.  doi: 10.1155/2016/7634680.

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##### References:
 [1] Abdennebi, Nizar and Mansour, A new sliding function for discrete predictive sliding mode control of time delay systems, International Journal of Automation and Computing, 10 (2013), 288-295. [2] P. Balasubramaniam and T. Senthilkumar, Delay - Dependent robust stabilization and H$\infty$ control for uncertain stochastic T-S fuzzy systems with discrete interval and distributed time-varying delays, International Journal of Automation & Computing, 10 (2013), 18-31. [3] T. S. Chiang, C. S. Chiu and P. Liu, Adaptive TS-FNN control for a class of uncertain multi-time-delay systems: The exponentially stable sliding mode- based approach, International Journal of Adaptive Control & Signal Processing, 23 (2010), 378-399. [4] Q. Gao, H. Huang and G. Feng, Robust H$\infty$ stabilization of uncertain T-S fuzzy systems via dynamic integral sliding mode control, Ifac Proceedings Volumes, 46 (2013), 485-490. [5] S. Guo, F. Zhu and L. Xu, Unknown input observer design for Takagi-Sugeno fuzzy stochastic system, International Journal of Control Automation & Systems, 13 (2015), 1003-1009. [6] C. Han, G. Zhang and L. Wu, Sliding mode control of T-S fuzzy descriptor systems with time-delay, Journal of the Franklin Institute, 349 (2012), 1430-1444.  doi: 10.1016/j.jfranklin.2011.07.001. [7] M. Kchaou, H. Gassara and A. E. Hajjaji, Dissipativity-based integral sliding mode control for a class of Takagi-Sugeno fuzzy singular systems with time-varying delay, Control Theory & Applications Iet, 8 (2014), 2045-2054.  doi: 10.1049/iet-cta.2014.0101. [8] Y. M. Li and Y. Y. Li, Fuzzy control for nonlinear uncertain T-S fuzzy systems with time-varying delays, Applied Mechanics and Materials, 6 (2013), 341-342. [9] R. Li and Q. Zhang, Robust H$_\infty$ sliding mode observer design for a class of Takagi-Sugeno fuzzy descriptor systems with time-varying delay, Applied Mathematics & Computation, 337 (2018), 158-178.  doi: 10.1016/j.amc.2018.05.008. [10] Z. Liu and C. Gao, A new result on robust H$\infty$ control for uncertain time-delay singular systems via sliding mode control, Complexity, 21 (2016), 165-177.  doi: 10.1002/cplx.21793. [11] R. M. Nagarale and B. M. Patre, Exponential function based fuzzy sliding mode control of uncertain nonlinear systems, International Journal of Dynamics & Control, 4 (2016), 67-75.  doi: 10.1007/s40435-014-0117-2. [12] T. Niknam and M. H. Khooban, Fuzzy sliding mode control scheme for a class of non-linear uncertain chaotic systems, Iet Science Measurement & Technology, 7 (2013), 249-255. [13] L. Ren, S. Xie and Z. G. Miao, Fuzzy robust sliding mode control of a class of uncertain systems, Journal of Central South University, 23 (2016), 2296-2304. [14] D. B. Salem and W. Saad, Integral sliding mode control for systems with time-varying input and state delays, International Conference on Engineering & Mis, (2017), 978–982. [15] A. Si-Ammour, S. Djennoune and M. Bettayeb, A sliding mode control for linear fractional systems with input and state delays, Communications in Nonlinear Science & Numerical Simulation, 14 (2009), 2310-2318.  doi: 10.1016/j.cnsns.2008.05.011. [16] H. Wang, B. Zhou and R. Lu, New stability and stabilization criteria for a class of fuzzy singular systems with time-varying delay, Journal of the Franklin Institute, 351 (2014), 3766-3781.  doi: 10.1016/j.jfranklin.2013.02.030. [17] Y. Wang, Y. Xia and H. Li, A new integral sliding mode design method for nonlinear stochastic systems, Automatica, 90 (2018), 304-309.  doi: 10.1016/j.automatica.2017.11.029. [18] J. Wu, Robust stabilization for uncertain T-S fuzzy singular system, International Journal of Machine Learning & Cybernetics, 7 (2016), 699-706. [19] Y. Xia, H. Yang, M. Fu and P. Shi, Sliding mode control for linear systems with time-varying input and state delays, Circuits Systems, and Signal Processing, 30 (2011), 629-641.  doi: 10.1007/s00034-010-9237-x. [20] L. Xiao, H. Su and J. Chu, Sliding mode prediction tracking control design for uncertain systems, Asian Journal of Control, 9 (2010), 317-325.  doi: 10.1111/j.1934-6093.2007.tb00417.x. [21] X. G. Yan, S. Spurgeon and Y. Korlov, Output feedback control synthesis for non-linear time- delay systems using a sliding-mode observer, IMA Journal of Mathematical Control and Information, 31 (2014), 501-518.  doi: 10.1093/imamci/dnt028. [22] S. Y. Yoon and Z. Lin, Robust output regulation of linear time-delay systems: A state predictor approach, International Journal of Robust & Nonlinear Control, 26 (2016), 1686-1740.  doi: 10.1002/rnc.3374. [23] J. Yu and Z. Yi, Stability analysis and fuzzy control for uncertain delayed T-S nonlinear systems, International Journal of Fuzzy Systems, 18 (2016), 1-8.  doi: 10.1007/s40815-016-0203-z. [24] Y. Zhang, Robust stability and H$\infty$ control of discrete-time uncertain impulsive systems with time-varying delay, Circuits Systems & Signal Processing, 35 (2016), 3882-3912. [25] Y. Zhao, J. Shen and D. Chen, New stability criterion for discrete-time genetic regulatory networks with time-varying delays and stochastic disturbances, Mathematical Problems in Engineering, 2016 (2016), 1-13.  doi: 10.1155/2016/7634680.
Trajectory of state $x_{1}(t)$ before adding controller
Trajectory of state $x_{2}(t)$ before adding controller
control input signal
Trajectory of state $x_{1}(t)$ after adding controller
Trajectory of state $x_{2}(t)$ after adding controller
Trajectory of sliding mode variable
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