# American Institute of Mathematical Sciences

July  2022, 15(7): 1713-1731. doi: 10.3934/dcdss.2021168

## Switching mechanism-based event-triggered fuzzy adaptive control with prescribed performance for MIMO nonlinear systems

 1 College of Electrical Engineering, Liaoning University of Technology, Jinzhou, China 2 College of Science, Liaoning University of Technology, Jinzhou, China 3 Northeastern University, Shenyang, China 4 College of Science, Liaoning University of Technology, Jinzhou, China

* Corresponding author: Yongming Li

Received  September 2021 Revised  October 2021 Published  July 2022 Early access  December 2021

Fund Project: This work is supported by National Natural Science Foundation (NNSF) of China under Grant 62173172 and Grant 61822307

This paper investigates the switching mechanism-based event-trig-gered fuzzy adaptive control issue of multi-input and multi-output (MIMO) nonlinear systems with prescribed performance (PP). Utilizing fuzzy logic systems (FLSs) to approximate unknown nonlinear functions. By using the switching threshold strategy, the system has more flexibility in strategy selection. The proposed control scheme can better solve the communication resource limitation. On account of the Lyapunov stability theory, the stability of the controlled system is proved. And all signals of the controlled system are bounded. Moreover, the tracking errors are controlled in a diminutive realm of the origin within the PP bounded. Simultaneously, the Zeno behavior is avoided. Finally, illustrate the effectiveness of the control scheme that has been proposed by demonstrating some simulation consequences.

Citation: Ruitong Wu, Yongming Li, Jun Hu, Wei Liu, Shaocheng Tong. Switching mechanism-based event-triggered fuzzy adaptive control with prescribed performance for MIMO nonlinear systems. Discrete and Continuous Dynamical Systems - S, 2022, 15 (7) : 1713-1731. doi: 10.3934/dcdss.2021168
##### References:
 [1] W. W. Bai, T. S. Li and S. C. Tong, NN reinforcement learning adaptive control for a class of nonstrict-feedback discrete-time systems, IEEE Transactions on Cybernetics, 50 (2020), 4573-4584.  doi: 10.1109/TCYB.2020.2963849. [2] C. P. Bechlioulis and G. A. Rovithakis, A low-complexity global approximation-free control scheme with prescribed performance for unknown pure feedback systems, Automatica, 50 (2014), 1217-1226.  doi: 10.1016/j.automatica.2014.02.020. [3] C. Deng, C. Wen, J. Huang, X. M. Zhang and Y. Zou, Distributed observer-based cooperative control approach for uncertain nonlinear MASs under event-triggered communication, IEEE Transactions on Automatic Control, (2021), 1–1. doi: 10.1109/TAC.2021.3090739. [4] K. W. Li and Y. M. Li, Adaptive fuzzy finite-time dynamic surface control for high-order nonlinear system with output constraints, International Journal of Control, Automation and Systems, 19 (2021), 112-123.  doi: 10.1007/s12555-019-0986-4. [5] T. S. Li, W. W. Bai, Q. Liu, Y. Long and C. L. Philip Chen, Distributed fault-tolerant containment control protocols for the discrete-time multi-agent systems via reinforcement learning method, IEEE Transactions on Neural Networks and Learning Systems, (2021), 1–13. doi: 10.1109/TNNLS.2021.3121403. [6] X. D. Li and P. Li, Stability of time-delay systems with impulsive control involving stabilizing delays, Automatica, 124 (2021), 109336.  doi: 10.1016/j.automatica.2020.109336. [7] X. D. Li and X. Y. Yang, Lyapunov stability analysis for nonlinear systems with state-dependent state delay, Automatica, 112 (2020), 108674.  doi: 10.1016/j.automatica.2019.108674. [8] Y. M. Li, K. W. Li and S. C. Tong, Finite-time adaptive fuzzy output feedback dynamic surface control for MIMO non-strict feedback systems, IEEE Transactions on Fuzzy Systems, 27 (2019), 96-110. [9] Y. M. Li, Y. J. Liu and S. C. Tong, Observer-based neuro-adaptive optimized control for a class of strict-feedback nonlinear systems with state constraints, IEEE Transactions on Neural Networks and Learning Systems, (2021), 1–15. doi: 10.1109/TNNLS.2021.3051030. [10] Y. M. Li and S. C. Tong, Fuzzy adaptive control design strategy of nonlinear switched large-scale systems, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 48 (2018), 2209-2218.  doi: 10.1109/TSMC.2017.2703127. [11] L. Liu, D. Wang, Z. H. Peng and Q. L. Han, Distributed path following of multiple under-actuated autonomous surface vehicles based on data-driven neural predictors via integral concurrent learning, IEEE Transactions on Neural Networks and Learning Systems, 32 (2021), 5334-5344.  doi: 10.1109/TNNLS.2021.3100147. [12] M. Liu, L. X. Zhang, P. Shi and Y. X. Zhao, Fault estimation sliding-mode observer with digital communication constraints, IEEE Trans. Automat. Control, 63 (2018), 3434-3441.  doi: 10.1109/TAC.2018.2794826. [13] Z. Liu, F. Wang, Y. Zhang, X. Chen and C. L. P. Chen, Adaptive tracking control for a class of nonlinear systems with a fuzzy dead-zone input, IEEE Transactions on Fuzzy Systems, 23 (2015), 193-204. [14] W. Qian, W. W. Xing and S. M. Fei, $H_{\infty}$ state estimation for neural networks with general activation function and mixed time-varying delays, IEEE Trans. Automat. Control, 32 (2021), 3909-3918.  doi: 10.1109/TNNLS.2020.3016120. [15] J. Qiu, K. Sun, T. Wang and H. Gao, Observer-based fuzzy adaptive event-triggered control for pure-feedback nonlinear systems with prescribed performance, IEEE Transactions on Fuzzy Systems, 27 (2019), 2152-2162.  doi: 10.1109/TFUZZ.2019.2895560. [16] Z. W. Ruan, Q. M. Yang, S. Z. S. Ge and Y. X. Sun, Adaptive fuzzy fault tolerant control of uncertain MIMO nonlinear systems with output constraints and unknown control directions, IEEE Transactions on Fuzzy Systems, (2021), 1–1. doi: 10.1109/TFUZZ.2021.3055336. [17] X. F. Shao and D. Ye, Fuzzy adaptive event-triggered secure control for stochastic nonlinear high-order MASs subject to DoS attacks and actuator faults, IEEE Transactions on Fuzzy Systems, 29 (2021), 3812-3821.  doi: 10.1109/TFUZZ.2020.3028657. [18] W. Sun, S. F. Su, J. W. Xia and Y. Q. Wu, Adaptive tracking control of wheeled inverted pendulums with periodic disturbances, IEEE Transactions on Cybernetics, 50 (2020), 1867-1876. [19] S. C. Tong, X. Min and Y. X. Li, Observer-based adaptive fuzzy tracking control for strict-feedback nonlinear systems with unknown control gain functions, IEEE Transactions on Cybernetics, 50 (2020), 3903-3913.  doi: 10.1109/TCYB.2020.2977175. [20] S. C. Tong, K. K. Sun and S. Sui, Observer-based adaptive fuzzy decentralized optimal control design for strict-feedback nonlinear large-scale systems, IEEE Transactions on Fuzzy Systems, 26 (2018), 569-584.  doi: 10.1109/TFUZZ.2017.2686373. [21] J. H. Wang, Z. Liu, C. L. Philip Chen and Y. Zhang, Event-triggered neural adaptive failure compensation control for stochastic systems with dead-zone output, Nonlinear Dynamics, 96 (2019), 2179-2196.  doi: 10.1007/s11071-019-04916-8. [22] T. Wang, Y. F. Zhang, J. B. Qiu and H. J. Gao, Adaptive fuzzy backstepping control for a class of nonlinear systems with sampled and delayed measurements, IEEE Transactions on Fuzzy Systems, 23 (2015), 302-312. [23] W. Wang and S. Tong, Observer-based adaptive fuzzy containment control for multiple uncertain nonlinear systems, IEEE Transactions on Fuzzy Systems, 27 (2019), 2079-2089.  doi: 10.1109/TFUZZ.2019.2893339. [24] L. B. Wu, J. H. Park, X. P. Xie, C. Gao and N. N. Zhao, Fuzzy adaptive event-triggered control for a class of uncertain nonaffine nonlinear systems with full state constraints, IEEE Transactions on Fuzzy Systems, 29 (2021), 904-916. [25] L. Xing, C. Wen, Z. Liu, H. Su and J. Cai, Adaptive compensation for actuator failures with event-triggered input, Automatica, 85 (2017), 129-136.  doi: 10.1016/j.automatica.2017.07.061. [26] L. Xing, C. Wen, Z. Liu, H. Su and J. Cai, Event-triggered adaptive control for a class of uncertain nonlinear systems, IEEE Trans. Automat. Control, 62 (2017), 2071-2076.  doi: 10.1109/TAC.2016.2594204. [27] W. Q. Xu, X. P. Liu, H. Q. Wang and Y. C. Zhou, Event-triggered adaptive NN tracking control for MIMO nonlinear discrete-time systems, IEEE Transactions on Neural Networks and Learning Systems, (2021), 1–11. doi: 10.1109/TNNLS.2021.3084965. [28] J. P. Yu, P. Shi and L. Zhao, Finite-time command filtered backstepping control for a class of nonlinear systems, Automatica, 92 (2018), 173-180.  doi: 10.1016/j.automatica.2018.03.033. [29] S. Zeghlache, L. Benyettou, A. Djerioui and M. Z. Ghellab, Twin rotor MIMO system experimental validation of robust adaptive fuzzy control against wind effects, IEEE Systems Journal, (2020), 1–11. doi: 10.1109/JSYST.2020.3034993. [30] Y. Zhang, X. H. Su, Z. Liu and C. L. P. Chen, Event-triggered adaptive fuzzy tracking control with guaranteed transient performance for MIMO nonlinear uncertain systems, IEEE Transactions on Cybernetics, 51 (2021), 736-749.  doi: 10.1109/TCYB.2019.2894343.

show all references

##### References:
 [1] W. W. Bai, T. S. Li and S. C. Tong, NN reinforcement learning adaptive control for a class of nonstrict-feedback discrete-time systems, IEEE Transactions on Cybernetics, 50 (2020), 4573-4584.  doi: 10.1109/TCYB.2020.2963849. [2] C. P. Bechlioulis and G. A. Rovithakis, A low-complexity global approximation-free control scheme with prescribed performance for unknown pure feedback systems, Automatica, 50 (2014), 1217-1226.  doi: 10.1016/j.automatica.2014.02.020. [3] C. Deng, C. Wen, J. Huang, X. M. Zhang and Y. Zou, Distributed observer-based cooperative control approach for uncertain nonlinear MASs under event-triggered communication, IEEE Transactions on Automatic Control, (2021), 1–1. doi: 10.1109/TAC.2021.3090739. [4] K. W. Li and Y. M. Li, Adaptive fuzzy finite-time dynamic surface control for high-order nonlinear system with output constraints, International Journal of Control, Automation and Systems, 19 (2021), 112-123.  doi: 10.1007/s12555-019-0986-4. [5] T. S. Li, W. W. Bai, Q. Liu, Y. Long and C. L. Philip Chen, Distributed fault-tolerant containment control protocols for the discrete-time multi-agent systems via reinforcement learning method, IEEE Transactions on Neural Networks and Learning Systems, (2021), 1–13. doi: 10.1109/TNNLS.2021.3121403. [6] X. D. Li and P. Li, Stability of time-delay systems with impulsive control involving stabilizing delays, Automatica, 124 (2021), 109336.  doi: 10.1016/j.automatica.2020.109336. [7] X. D. Li and X. Y. Yang, Lyapunov stability analysis for nonlinear systems with state-dependent state delay, Automatica, 112 (2020), 108674.  doi: 10.1016/j.automatica.2019.108674. [8] Y. M. Li, K. W. Li and S. C. Tong, Finite-time adaptive fuzzy output feedback dynamic surface control for MIMO non-strict feedback systems, IEEE Transactions on Fuzzy Systems, 27 (2019), 96-110. [9] Y. M. Li, Y. J. Liu and S. C. Tong, Observer-based neuro-adaptive optimized control for a class of strict-feedback nonlinear systems with state constraints, IEEE Transactions on Neural Networks and Learning Systems, (2021), 1–15. doi: 10.1109/TNNLS.2021.3051030. [10] Y. M. Li and S. C. Tong, Fuzzy adaptive control design strategy of nonlinear switched large-scale systems, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 48 (2018), 2209-2218.  doi: 10.1109/TSMC.2017.2703127. [11] L. Liu, D. Wang, Z. H. Peng and Q. L. Han, Distributed path following of multiple under-actuated autonomous surface vehicles based on data-driven neural predictors via integral concurrent learning, IEEE Transactions on Neural Networks and Learning Systems, 32 (2021), 5334-5344.  doi: 10.1109/TNNLS.2021.3100147. [12] M. Liu, L. X. Zhang, P. Shi and Y. X. Zhao, Fault estimation sliding-mode observer with digital communication constraints, IEEE Trans. Automat. Control, 63 (2018), 3434-3441.  doi: 10.1109/TAC.2018.2794826. [13] Z. Liu, F. Wang, Y. Zhang, X. Chen and C. L. P. Chen, Adaptive tracking control for a class of nonlinear systems with a fuzzy dead-zone input, IEEE Transactions on Fuzzy Systems, 23 (2015), 193-204. [14] W. Qian, W. W. Xing and S. M. Fei, $H_{\infty}$ state estimation for neural networks with general activation function and mixed time-varying delays, IEEE Trans. Automat. Control, 32 (2021), 3909-3918.  doi: 10.1109/TNNLS.2020.3016120. [15] J. Qiu, K. Sun, T. Wang and H. Gao, Observer-based fuzzy adaptive event-triggered control for pure-feedback nonlinear systems with prescribed performance, IEEE Transactions on Fuzzy Systems, 27 (2019), 2152-2162.  doi: 10.1109/TFUZZ.2019.2895560. [16] Z. W. Ruan, Q. M. Yang, S. Z. S. Ge and Y. X. Sun, Adaptive fuzzy fault tolerant control of uncertain MIMO nonlinear systems with output constraints and unknown control directions, IEEE Transactions on Fuzzy Systems, (2021), 1–1. doi: 10.1109/TFUZZ.2021.3055336. [17] X. F. Shao and D. Ye, Fuzzy adaptive event-triggered secure control for stochastic nonlinear high-order MASs subject to DoS attacks and actuator faults, IEEE Transactions on Fuzzy Systems, 29 (2021), 3812-3821.  doi: 10.1109/TFUZZ.2020.3028657. [18] W. Sun, S. F. Su, J. W. Xia and Y. Q. Wu, Adaptive tracking control of wheeled inverted pendulums with periodic disturbances, IEEE Transactions on Cybernetics, 50 (2020), 1867-1876. [19] S. C. Tong, X. Min and Y. X. Li, Observer-based adaptive fuzzy tracking control for strict-feedback nonlinear systems with unknown control gain functions, IEEE Transactions on Cybernetics, 50 (2020), 3903-3913.  doi: 10.1109/TCYB.2020.2977175. [20] S. C. Tong, K. K. Sun and S. Sui, Observer-based adaptive fuzzy decentralized optimal control design for strict-feedback nonlinear large-scale systems, IEEE Transactions on Fuzzy Systems, 26 (2018), 569-584.  doi: 10.1109/TFUZZ.2017.2686373. [21] J. H. Wang, Z. Liu, C. L. Philip Chen and Y. Zhang, Event-triggered neural adaptive failure compensation control for stochastic systems with dead-zone output, Nonlinear Dynamics, 96 (2019), 2179-2196.  doi: 10.1007/s11071-019-04916-8. [22] T. Wang, Y. F. Zhang, J. B. Qiu and H. J. Gao, Adaptive fuzzy backstepping control for a class of nonlinear systems with sampled and delayed measurements, IEEE Transactions on Fuzzy Systems, 23 (2015), 302-312. [23] W. Wang and S. Tong, Observer-based adaptive fuzzy containment control for multiple uncertain nonlinear systems, IEEE Transactions on Fuzzy Systems, 27 (2019), 2079-2089.  doi: 10.1109/TFUZZ.2019.2893339. [24] L. B. Wu, J. H. Park, X. P. Xie, C. Gao and N. N. Zhao, Fuzzy adaptive event-triggered control for a class of uncertain nonaffine nonlinear systems with full state constraints, IEEE Transactions on Fuzzy Systems, 29 (2021), 904-916. [25] L. Xing, C. Wen, Z. Liu, H. Su and J. Cai, Adaptive compensation for actuator failures with event-triggered input, Automatica, 85 (2017), 129-136.  doi: 10.1016/j.automatica.2017.07.061. [26] L. Xing, C. Wen, Z. Liu, H. Su and J. Cai, Event-triggered adaptive control for a class of uncertain nonlinear systems, IEEE Trans. Automat. Control, 62 (2017), 2071-2076.  doi: 10.1109/TAC.2016.2594204. [27] W. Q. Xu, X. P. Liu, H. Q. Wang and Y. C. Zhou, Event-triggered adaptive NN tracking control for MIMO nonlinear discrete-time systems, IEEE Transactions on Neural Networks and Learning Systems, (2021), 1–11. doi: 10.1109/TNNLS.2021.3084965. [28] J. P. Yu, P. Shi and L. Zhao, Finite-time command filtered backstepping control for a class of nonlinear systems, Automatica, 92 (2018), 173-180.  doi: 10.1016/j.automatica.2018.03.033. [29] S. Zeghlache, L. Benyettou, A. Djerioui and M. Z. Ghellab, Twin rotor MIMO system experimental validation of robust adaptive fuzzy control against wind effects, IEEE Systems Journal, (2020), 1–11. doi: 10.1109/JSYST.2020.3034993. [30] Y. Zhang, X. H. Su, Z. Liu and C. L. P. Chen, Event-triggered adaptive fuzzy tracking control with guaranteed transient performance for MIMO nonlinear uncertain systems, IEEE Transactions on Cybernetics, 51 (2021), 736-749.  doi: 10.1109/TCYB.2019.2894343.
System output ${y_1}$ and the reference signal ${y_{1,r}}$
System tracking error ${y_1} - {y_{1,r}}$
Control signal
The time intervals ${t_{kk + 1}} - {t_k}$ of triggering events
System output ${y_2}$ and the reference signal ${y_{2,r}}$
System tracking error ${y_2} - {y_{2,r}}$
Control signal
The time intervals ${t_{kk + 1}} - {t_k}$ of triggering events
Tracking performance
 [1] 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 [2] Hongru Ren, Shubo Li, Changxin Lu. Event-triggered adaptive fault-tolerant control for multi-agent systems with unknown disturbances. Discrete and Continuous Dynamical Systems - S, 2021, 14 (4) : 1395-1414. doi: 10.3934/dcdss.2020379 [3] Ramalingam Sakthivel, Palanisamy Selvaraj, Yeong-Jae Kim, Dong-Hoon Lee, Oh-Min Kwon, Rathinasamy Sakthivel. Robust $H_\infty$ resilient event-triggered control design for T-S fuzzy systems. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022028 [4] Masashi Wakaiki, Hideki Sano. Stability analysis of infinite-dimensional event-triggered and self-triggered control systems with Lipschitz perturbations. Mathematical Control and Related Fields, 2022, 12 (1) : 245-273. doi: 10.3934/mcrf.2021021 [5] Yuan Xu, Xin Jin, Saiwei Wang, Yang Tang. Optimal synchronization control of multiple euler-lagrange systems via event-triggered reinforcement learning. Discrete and Continuous Dynamical Systems - S, 2021, 14 (4) : 1495-1518. doi: 10.3934/dcdss.2020377 [6] Liqiang Jin, Yanyan Yin, Kok Lay Teo, Fei Liu. Event-triggered mixed $H_\infty$ and passive control for Markov jump systems with bounded inputs. Journal of Industrial and Management Optimization, 2021, 17 (3) : 1343-1355. doi: 10.3934/jimo.2020024 [7] Tayel Dabbous. Adaptive control of nonlinear systems using fuzzy systems. Journal of Industrial and Management Optimization, 2010, 6 (4) : 861-880. doi: 10.3934/jimo.2010.6.861 [8] Kun Liang, Wangli He, Yang Yuan, Liyu Shi. Synchronization for singularity-perturbed complex networks via event-triggered impulsive control. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022068 [9] Qian Zhang, Huaicheng Yan, Jun Cheng, Xisheng Zhan, Kaibo Shi. Fault detection filtering for continuous-time singular systems under a dynamic event-triggered mechanism. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022023 [10] Zbigniew Bartosiewicz, Ülle Kotta, Maris Tőnso, Małgorzata Wyrwas. Accessibility conditions of MIMO nonlinear control systems on homogeneous time scales. Mathematical Control and Related Fields, 2016, 6 (2) : 217-250. doi: 10.3934/mcrf.2016002 [11] Jin-Zi Yang, Yuan-Xin Li, Ming Wei. Fuzzy adaptive asymptotic tracking of fractional order nonlinear systems with uncertain disturbances. Discrete and Continuous Dynamical Systems - S, 2022, 15 (7) : 1615-1631. doi: 10.3934/dcdss.2021144 [12] Chao Chen, Shanlin Yi, Feng Wang, Chengxi Zhang, Qingmin Yu. Prescribed performance tracking control of multi-link robotic manipulator with uncertainties. Mathematical Foundations of Computing, 2022  doi: 10.3934/mfc.2022012 [13] Qiying Hu, Wuyi Yue. Optimal control for resource allocation in discrete event systems. Journal of Industrial and Management Optimization, 2006, 2 (1) : 63-80. doi: 10.3934/jimo.2006.2.63 [14] Aliki D. Muradova, Georgios K. Tairidis, Georgios E. Stavroulakis. Adaptive Neuro-Fuzzy vibration control of a smart plate. Numerical Algebra, Control and Optimization, 2017, 7 (3) : 251-271. doi: 10.3934/naco.2017017 [15] Qiying Hu, Wuyi Yue. Optimal control for discrete event systems with arbitrary control pattern. Discrete and Continuous Dynamical Systems - B, 2006, 6 (3) : 535-558. doi: 10.3934/dcdsb.2006.6.535 [16] Qi Li, Hong Xue, Changxin Lu. Event-based fault detection for interval type-2 fuzzy systems with measurement outliers. Discrete and Continuous Dynamical Systems - S, 2021, 14 (4) : 1301-1328. doi: 10.3934/dcdss.2020412 [17] Stefan Jerg, Oliver Junge, Marcus Post. Global optimal feedbacks for stochastic quantized nonlinear event systems. Journal of Computational Dynamics, 2014, 1 (1) : 163-176. doi: 10.3934/jcd.2014.1.163 [18] Xiao-Li Hu, Han-Fu Chen. Optimal Adaptive Regulation for Nonlinear Systems with Observation Noise. Journal of Industrial and Management Optimization, 2007, 3 (1) : 155-164. doi: 10.3934/jimo.2007.3.155 [19] Qiying Hu, Chen Xu, Wuyi Yue. A unified model for state feedback of discrete event systems II: Control synthesis problems. Journal of Industrial and Management Optimization, 2008, 4 (4) : 713-726. doi: 10.3934/jimo.2008.4.713 [20] James P. Nelson, Mark J. Balas. Direct model reference adaptive control of linear systems with input/output delays. Numerical Algebra, Control and Optimization, 2013, 3 (3) : 445-462. doi: 10.3934/naco.2013.3.445

2021 Impact Factor: 1.865