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 & Optimization, 2020, 10 (3) : 345-354. doi: 10.3934/naco.2020006
References:
[1]

AbdennebiNizar 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.   Google Scholar

[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.   Google Scholar

[3]

T. S. ChiangC. 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.   Google Scholar

[4]

Q. GaoH. 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.   Google Scholar

[5]

S. GuoF. Zhu and L. Xu, Unknown input observer design for Takagi-Sugeno fuzzy stochastic system, International Journal of Control Automation & Systems, 13 (2015), 1003-1009.   Google Scholar

[6]

C. HanG. 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.  Google Scholar

[7]

M. KchaouH. 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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[13]

L. RenS. 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.   Google Scholar

[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.  Google Scholar

[15]

A. Si-AmmourS. 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.  Google Scholar

[16]

H. WangB. 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.  Google Scholar

[17]

Y. WangY. 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.  Google Scholar

[18]

J. Wu, Robust stabilization for uncertain T-S fuzzy singular system, International Journal of Machine Learning & Cybernetics, 7 (2016), 699-706.   Google Scholar

[19]

Y. XiaH. YangM. 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.  Google Scholar

[20]

L. XiaoH. 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.  Google Scholar

[21]

X. G. YanS. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[25]

Y. ZhaoJ. 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.  Google Scholar

show all references

References:
[1]

AbdennebiNizar 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.   Google Scholar

[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.   Google Scholar

[3]

T. S. ChiangC. 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.   Google Scholar

[4]

Q. GaoH. 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.   Google Scholar

[5]

S. GuoF. Zhu and L. Xu, Unknown input observer design for Takagi-Sugeno fuzzy stochastic system, International Journal of Control Automation & Systems, 13 (2015), 1003-1009.   Google Scholar

[6]

C. HanG. 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.  Google Scholar

[7]

M. KchaouH. 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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[13]

L. RenS. 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.   Google Scholar

[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.  Google Scholar

[15]

A. Si-AmmourS. 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.  Google Scholar

[16]

H. WangB. 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.  Google Scholar

[17]

Y. WangY. 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.  Google Scholar

[18]

J. Wu, Robust stabilization for uncertain T-S fuzzy singular system, International Journal of Machine Learning & Cybernetics, 7 (2016), 699-706.   Google Scholar

[19]

Y. XiaH. YangM. 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.  Google Scholar

[20]

L. XiaoH. 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.  Google Scholar

[21]

X. G. YanS. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[25]

Y. ZhaoJ. 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.  Google Scholar

Figure 1.  Trajectory of state $ x_{1}(t) $ before adding controller
Figure 2.  Trajectory of state $ x_{2}(t) $ before adding controller
Figure 3.  control input signal
Figure 4.  Trajectory of state $ x_{1}(t) $ after adding controller
Figure 5.  Trajectory of state $ x_{2}(t) $ after adding controller
Figure 6.  Trajectory of sliding mode variable
[1]

Cecilia Cavaterra, Denis Enăchescu, Gabriela Marinoschi. Sliding mode control of the Hodgkin–Huxley mathematical model. Evolution Equations & Control Theory, 2019, 8 (4) : 883-902. doi: 10.3934/eect.2019043

[2]

James P. Nelson, Mark J. Balas. Direct model reference adaptive control of linear systems with input/output delays. Numerical Algebra, Control & Optimization, 2013, 3 (3) : 445-462. doi: 10.3934/naco.2013.3.445

[3]

Hao Sun, Shihua Li, Xuming Wang. Output feedback based sliding mode control for fuel quantity actuator system using a reduced-order GPIO. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020375

[4]

Tayel Dabbous. Adaptive control of nonlinear systems using fuzzy systems. Journal of Industrial & Management Optimization, 2010, 6 (4) : 861-880. doi: 10.3934/jimo.2010.6.861

[5]

Magdi S. Mahmoud. Output feedback overlapping control design of interconnected systems with input saturation. Numerical Algebra, Control & Optimization, 2016, 6 (2) : 127-151. doi: 10.3934/naco.2016004

[6]

Nasim Ullah, Ahmad Aziz Al-Ahmadi. A triple mode robust sliding mode controller for a nonlinear system with measurement noise and uncertainty. Mathematical Foundations of Computing, 2020, 3 (2) : 81-99. doi: 10.3934/mfc.2020007

[7]

Peng Cheng, Yanqing Liu, Yanyan Yin, Song Wang, Feng Pan. Fuzzy event-triggered disturbance rejection control of nonlinear systems. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020119

[8]

Hongbiao Fan, Jun-E Feng, Min Meng. Piecewise observers of rectangular discrete fuzzy descriptor systems with multiple time-varying delays. Journal of Industrial & Management Optimization, 2016, 12 (4) : 1535-1556. doi: 10.3934/jimo.2016.12.1535

[9]

Hooton Edward, Balanov Zalman, Krawcewicz Wieslaw, Rachinskii Dmitrii. Sliding Hopf bifurcation in interval systems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3545-3566. doi: 10.3934/dcds.2017152

[10]

Xiu-Fang Liu, Gen-Qi Xu. Exponential stabilization of Timoshenko beam with input and output delays. Mathematical Control & Related Fields, 2016, 6 (2) : 271-292. doi: 10.3934/mcrf.2016004

[11]

Yaping Wu, Qian Xu. The existence and structure of large spiky steady states for S-K-T competition systems with cross-diffusion. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 367-385. doi: 10.3934/dcds.2011.29.367

[12]

K. Aruna Sakthi, A. Vinodkumar. Stabilization on input time-varying delay for linear switched systems with truncated predictor control. Numerical Algebra, Control & Optimization, 2020, 10 (2) : 237-247. doi: 10.3934/naco.2019050

[13]

Xingyue Liang, Jianwei Xia, Guoliang Chen, Huasheng Zhang, Zhen Wang. $ \mathcal{H}_{\infty} $ control for fuzzy markovian jump systems based on sampled-data control method. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020368

[14]

Zhaohua Gong, Chongyang Liu, Yujing Wang. Optimal control of switched systems with multiple time-delays and a cost on changing control. Journal of Industrial & Management Optimization, 2018, 14 (1) : 183-198. doi: 10.3934/jimo.2017042

[15]

Ying Wu, Zhaohui Yuan, Yanpeng Wu. Optimal tracking control for networked control systems with random time delays and packet dropouts. Journal of Industrial & Management Optimization, 2015, 11 (4) : 1343-1354. doi: 10.3934/jimo.2015.11.1343

[16]

Hernán Cendra, María Etchechoury, Sebastián J. Ferraro. Impulsive control of a symmetric ball rolling without sliding or spinning. Journal of Geometric Mechanics, 2010, 2 (4) : 321-342. doi: 10.3934/jgm.2010.2.321

[17]

Shu Zhang, Yuan Yuan. The Filippov equilibrium and sliding motion in an internet congestion control model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 1189-1206. doi: 10.3934/dcdsb.2017058

[18]

Giuseppe Buttazzo, Lorenzo Freddi. Optimal control problems with weakly converging input operators. Discrete & Continuous Dynamical Systems - A, 1995, 1 (3) : 401-420. doi: 10.3934/dcds.1995.1.401

[19]

Anna Chiara Lai, Paola Loreti. Self-similar control systems and applications to zygodactyl bird's foot. Networks & Heterogeneous Media, 2015, 10 (2) : 401-419. doi: 10.3934/nhm.2015.10.401

[20]

D. J. W. Simpson, R. Kuske. Stochastically perturbed sliding motion in piecewise-smooth systems. Discrete & Continuous Dynamical Systems - B, 2014, 19 (9) : 2889-2913. doi: 10.3934/dcdsb.2014.19.2889

 Impact Factor: 

Metrics

  • PDF downloads (50)
  • HTML views (247)
  • Cited by (0)

Other articles
by authors

[Back to Top]