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November  2021, 17(6): 3297-3307. doi: 10.3934/jimo.2020119

## Fuzzy event-triggered disturbance rejection control of nonlinear systems

 1 Key laboratory of Advanced Process Control for Light Industry (Ministry of Education), School of Internet of Things Engineering, Jiangnan University, Wuxi, 214122, China 2 School of Electrical Engineering, Computing and Mathematical Sciences, Curtin University, Perth, Western Australia, 6102, Australia 3 School of Electrical Engineering, Computing and Mathematical Sciences, Curtin University, GPO Box U1987, Perth, WA6845, Australia

* Corresponding author: Feng Pan

Received  October 2019 Revised  March 2020 Published  November 2021 Early access  June 2020

The problem of fuzzy based event-triggered disturbance rejection control for nonlinear systems is addressed in this paper. A new fuzzy event based anti rejection controller is designed and a fuzzy reduced disturbance observer is constructed. Sufficient conditions for the closed loop system to be asymptotically stable under an $H_\infty$ performance index are derived. Based on these conditions, the design of a fuzzy event-triggered state feedback controller is formulated and solved. Numerical results are presented to demonstrate the correctness and effectiveness of our theoretical findings.

Citation: Peng Cheng, Yanqing Liu, Yanyan Yin, Song Wang, Feng Pan. Fuzzy event-triggered disturbance rejection control of nonlinear systems. Journal of Industrial and Management Optimization, 2021, 17 (6) : 3297-3307. doi: 10.3934/jimo.2020119
##### References:
 [1] A. Benzaouia and A. E. Hajjaji, Delay-dependent stabilization conditions of controlled positive T-S fuzzy systems with time varying delay, International Journal of Innovative Computing, Information and Control, 7 (2011), 1533-1548. [2] Y.-Y. Cao, Z. L. Lin and Y. Shamash, Set invariance analysis and gain-scheduling control for LPV systems subject to actuator saturation, Systems and Control Letters, 46 (2002), 137-151.  doi: 10.1016/S0167-6911(02)00128-7. [3] X. Chang and G. Yang, Relaxed results on stabilization and state feedback $H_\infty$ control conditions for T-S fuzzy systems, International Journal of Innovative Computing, Information and Control, 7 (2011), 1753-1764. [4] M. Chen and W. Chen, Disturbance observer based robust control for time delay uncertain systems, International Journal of Control, Automation and Systems, 8 (2010), 445-453. [5] T. M. Guerra and L. Vermeiren, LMI-based relaxed nonquadratic stabilization conditions for nonlinear systems in the Takagi-Sugeno's form, Automatica J. IFAC, 40 (2004), 823-829.  doi: 10.1016/j.automatica.2003.12.014. [6] L. Guo and W.-H. Chen, Disturbance attenuation and rejection for systems with nonlinearity via DOBC approach, International Journal of Robust and Nonlinear Control, 15 (2005), 109-125.  doi: 10.1002/rnc.978. [7] L. Guo and S. Y. Cao, Anti-Disturbance Control for Systems with Multiple Disturbances, USA: CRC Press, Boca Raton, FL, 2014.  doi: 10.1201/b15528. [8] L. Guo and S. Y. Cao, Anti-disturbance control theory for systems with multiple disturbances: A survey, ISA Transactions, 53 (2014), 846-849.  doi: 10.1016/j.isatra.2013.10.005. [9] T. Iwasaki, G. Meinsma and M. Y. Fu, Generalized S-procedure and finite frequency KYP lemma, Mathematical Problems in Engineering, 6 (2000), 305-320.  doi: 10.1155/S1024123X00001368. [10] L. L. Lv, S. Y. Tang and L. Zhang, Parametric solutions to generalized periodic Sylvester bimatrix equations, Journal of the Franklin Institute, 357 (2020), 3601-3621.  doi: 10.1016/j.jfranklin.2019.12.031. [11] L. L. Lv and Z. Zhang, Finite iterative solutions to periodic Sylvester matrix equations, Journal of the Franklin Institute, 354 (2017), 2358-2370.  doi: 10.1016/j.jfranklin.2017.01.004. [12] L. L. Lv, Z. Zhang, L. Zhang and X. X. Liu, Gradient based approach for generalized discrete-time periodic coupled Sylvester matrix equations, Journal of the Franklin Institute, 355 (2018), 7691-7705.  doi: 10.1016/j.jfranklin.2018.07.045. [13] X. J. Su, P. Shi, L. Q. Wu and Y.-D. Song, A novel control design on discrete-time Takagi-Sugeno fuzzy systems with time-varying delays, IEEE Trans on Fuzzy Systems, 21 (2013), 655-671.  doi: 10.1109/TFUZZ.2012.2226941. [14] C. Sun, Y. Wang and C. Chang, Switching T-S fuzzy model-based guaranteed cost control for two-wheeled mobile robots, International Journal of Innovative Computing, Information and Control, 8 (2012), 3015-3028. [15] T. Takagi and M. Sugeno, Fuzzy identification of systems and its applications to modeling and control, IEEE Trans on Syst. Man Cybern, 15 (1985), 116-132. [16] K. Tanaka, T. Hori and H. O. Wang, A multiple Lyapunov function approach to stabilization of fuzzy control systems, IEEE Trans on Fuzzy Syst., 11 (2003), 582-589.  doi: 10.1109/TFUZZ.2003.814861. [17] L. Wu, X. Su, P. Shi and J. Qiu, Model approximation for discrete-time state-delay systems in the T-S fuzzy framework, IEEE Trans on Fuzzy Systems, 19 (2011), 366-378. [18] X. M. Yao and L. Guo, Composite anti-disturbance control for Markovian jump nonlinear systems via disturbance observer, Automatica J. IFAC, 49 (2013), 2538-2545.  doi: 10.1016/j.automatica.2013.05.002. [19] Y. Y. Yin, X. Chen and F. Liu, Disturbance rejection control for Markov jump systems with nonhomogeneous processes, The 27th Chinese Control and Decision Conference (2015 CCDC), Qingdao, China, (2015), 15340479. doi: 10.1109/CCDC.2015.7162007. [20] Y. Y. Yin, Z. L. Lin, Y. Q. Liu and K. L. Teo, Event-triggered constrained control of positive systems with input saturation, International Journal of Robust and Nonlinear Control, 28 (2018), 3532-3542.  doi: 10.1002/rnc.4097. [21] Y. Y. Yin, Y. Q. Liu, K. L. Teo and S. Wang, Event-triggered probabilistic robust control of linear systems with input constrains: By scenario optimization approach, International Journal of Robust and Nonlinear Control, 28 (2018), 144-153.  doi: 10.1002/rnc.3858. [22] Y. Y. Yin, L. J. Zhu, F. Liu, K. L. Teo and S. Wang, Asynchronous $H_\infty$ control for nonhomogeneous higher-level Markov jump systems, Journal of the Franklin Institute, 357 (2020), 4697–4708. doi: 10.1016/j.jfranklin.2020.02.010. [23] L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-353.  doi: 10.1016/S0019-9958(65)90241-X. [24] H. B. Zeng, K. L. Teo, Y. He and W. Wang, Sampled-data-based dissipative control of T-S fuzzy systems, Applied Mathematical Modelling, 65 (2019), 415-427.  doi: 10.1016/j.apm.2018.08.012. [25] H.-B. Zeng, K. L. Teo, Y. He and W. Wang, Sampled-data stabilization of chaotic systems based on a T-S fuzzy model, Information Sciences, 483 (2019), 262-272.  doi: 10.1016/j.ins.2019.01.046.

show all references

##### References:
 [1] A. Benzaouia and A. E. Hajjaji, Delay-dependent stabilization conditions of controlled positive T-S fuzzy systems with time varying delay, International Journal of Innovative Computing, Information and Control, 7 (2011), 1533-1548. [2] Y.-Y. Cao, Z. L. Lin and Y. Shamash, Set invariance analysis and gain-scheduling control for LPV systems subject to actuator saturation, Systems and Control Letters, 46 (2002), 137-151.  doi: 10.1016/S0167-6911(02)00128-7. [3] X. Chang and G. Yang, Relaxed results on stabilization and state feedback $H_\infty$ control conditions for T-S fuzzy systems, International Journal of Innovative Computing, Information and Control, 7 (2011), 1753-1764. [4] M. Chen and W. Chen, Disturbance observer based robust control for time delay uncertain systems, International Journal of Control, Automation and Systems, 8 (2010), 445-453. [5] T. M. Guerra and L. Vermeiren, LMI-based relaxed nonquadratic stabilization conditions for nonlinear systems in the Takagi-Sugeno's form, Automatica J. IFAC, 40 (2004), 823-829.  doi: 10.1016/j.automatica.2003.12.014. [6] L. Guo and W.-H. Chen, Disturbance attenuation and rejection for systems with nonlinearity via DOBC approach, International Journal of Robust and Nonlinear Control, 15 (2005), 109-125.  doi: 10.1002/rnc.978. [7] L. Guo and S. Y. Cao, Anti-Disturbance Control for Systems with Multiple Disturbances, USA: CRC Press, Boca Raton, FL, 2014.  doi: 10.1201/b15528. [8] L. Guo and S. Y. Cao, Anti-disturbance control theory for systems with multiple disturbances: A survey, ISA Transactions, 53 (2014), 846-849.  doi: 10.1016/j.isatra.2013.10.005. [9] T. Iwasaki, G. Meinsma and M. Y. Fu, Generalized S-procedure and finite frequency KYP lemma, Mathematical Problems in Engineering, 6 (2000), 305-320.  doi: 10.1155/S1024123X00001368. [10] L. L. Lv, S. Y. Tang and L. Zhang, Parametric solutions to generalized periodic Sylvester bimatrix equations, Journal of the Franklin Institute, 357 (2020), 3601-3621.  doi: 10.1016/j.jfranklin.2019.12.031. [11] L. L. Lv and Z. Zhang, Finite iterative solutions to periodic Sylvester matrix equations, Journal of the Franklin Institute, 354 (2017), 2358-2370.  doi: 10.1016/j.jfranklin.2017.01.004. [12] L. L. Lv, Z. Zhang, L. Zhang and X. X. Liu, Gradient based approach for generalized discrete-time periodic coupled Sylvester matrix equations, Journal of the Franklin Institute, 355 (2018), 7691-7705.  doi: 10.1016/j.jfranklin.2018.07.045. [13] X. J. Su, P. Shi, L. Q. Wu and Y.-D. Song, A novel control design on discrete-time Takagi-Sugeno fuzzy systems with time-varying delays, IEEE Trans on Fuzzy Systems, 21 (2013), 655-671.  doi: 10.1109/TFUZZ.2012.2226941. [14] C. Sun, Y. Wang and C. Chang, Switching T-S fuzzy model-based guaranteed cost control for two-wheeled mobile robots, International Journal of Innovative Computing, Information and Control, 8 (2012), 3015-3028. [15] T. Takagi and M. Sugeno, Fuzzy identification of systems and its applications to modeling and control, IEEE Trans on Syst. Man Cybern, 15 (1985), 116-132. [16] K. Tanaka, T. Hori and H. O. Wang, A multiple Lyapunov function approach to stabilization of fuzzy control systems, IEEE Trans on Fuzzy Syst., 11 (2003), 582-589.  doi: 10.1109/TFUZZ.2003.814861. [17] L. Wu, X. Su, P. Shi and J. Qiu, Model approximation for discrete-time state-delay systems in the T-S fuzzy framework, IEEE Trans on Fuzzy Systems, 19 (2011), 366-378. [18] X. M. Yao and L. Guo, Composite anti-disturbance control for Markovian jump nonlinear systems via disturbance observer, Automatica J. IFAC, 49 (2013), 2538-2545.  doi: 10.1016/j.automatica.2013.05.002. [19] Y. Y. Yin, X. Chen and F. Liu, Disturbance rejection control for Markov jump systems with nonhomogeneous processes, The 27th Chinese Control and Decision Conference (2015 CCDC), Qingdao, China, (2015), 15340479. doi: 10.1109/CCDC.2015.7162007. [20] Y. Y. Yin, Z. L. Lin, Y. Q. Liu and K. L. Teo, Event-triggered constrained control of positive systems with input saturation, International Journal of Robust and Nonlinear Control, 28 (2018), 3532-3542.  doi: 10.1002/rnc.4097. [21] Y. Y. Yin, Y. Q. Liu, K. L. Teo and S. Wang, Event-triggered probabilistic robust control of linear systems with input constrains: By scenario optimization approach, International Journal of Robust and Nonlinear Control, 28 (2018), 144-153.  doi: 10.1002/rnc.3858. [22] Y. Y. Yin, L. J. Zhu, F. Liu, K. L. Teo and S. Wang, Asynchronous $H_\infty$ control for nonhomogeneous higher-level Markov jump systems, Journal of the Franklin Institute, 357 (2020), 4697–4708. doi: 10.1016/j.jfranklin.2020.02.010. [23] L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-353.  doi: 10.1016/S0019-9958(65)90241-X. [24] H. B. Zeng, K. L. Teo, Y. He and W. Wang, Sampled-data-based dissipative control of T-S fuzzy systems, Applied Mathematical Modelling, 65 (2019), 415-427.  doi: 10.1016/j.apm.2018.08.012. [25] H.-B. Zeng, K. L. Teo, Y. He and W. Wang, Sampled-data stabilization of chaotic systems based on a T-S fuzzy model, Information Sciences, 483 (2019), 262-272.  doi: 10.1016/j.ins.2019.01.046.
System trajectories under disturbance rejection controller
Estimation of disturbance
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