Adaptive Neural Control of MIMO Nonstrict-Feedback Nonlinear Systems With Time Delay
Abstract
In this paper, an adaptive neural output-feedback tracking controller is designed for a class of multiple-input and multiple-output nonstrict-feedback nonlinear systems with time delay. The system coefficient and uncertain functions of our considered systems are both unknown. By employing neural networks to approximate the unknown function entries, and constructing a new input-driven filter, a backstepping design method of tracking controller is developed for the systems under consideration. The proposed controller can guarantee that all the signals in the closed-loop systems are ultimately bounded, and the time-varying target signal can be tracked within a small error as well. The main contributions of this paper lie in that the systems under consideration are more general, and an effective design procedure of output-feedback controller is developed for the considered systems, which is more applicable in practice. Simulation results demonstrate the efficiency of the proposed algorithm.
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Traditional methods based on Lyapunov-Krasovskii functionals are generally less effective in managing input delays compared to state delays.
“Traditional methods based on Lyapunov-Krasovskii functionals are generally less effective in managing input delays compared to state delays [25][26][27].”
From Fixed-Time Adaptive Control for Nonstrict-Feedback Nonlinear Systems with Input Delay and Unknown Backlash-Like Hysteresis (2025) · cites this paper as [25]
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“Substituting (36)-(39) into (35) results in V̇e ≤ eT ( A T P + PA + P + (b1 + b2 + b4)I ) e − n‖P‖ n ∑”
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“1 (Zhao et al., 2016): For unknown function Fi, j(z̄i, j + ᾱi, j−1 + ēi, j), there exist positive unknown smooth functions βi, jl (z̄i, j + ᾱi, j−1 + ēi, j) such that”
BackgroundMethodInfluentialXi, C., & Dong, J. (2018) · Fuzzy Sets and Systems · doi:10.1016/j.fss.2018.12.003 · Cited by 22 · see citing works
Li, S., An, X., Ding, L., Wang, Q., Gao, H., Hou, Y., & Deng, Z. (2019) · IEEE Transactions on Systems Man and Cybernetics Systems · doi:10.1109/tsmc.2019.2944275 · Cited by 22 · see citing works
“Inthe future research work, the author will fully consider the influence of the constraint [54] or time delay [55] on the target system based on the research of this article, and overcome or effectively suppress the impact caused by the corresponding technical means, and finally realize finite-time…”
BackgroundWang, P., Zhang, D., & Lu, B. (2019) · Industrial Robot the international journal of robotics research and application · doi:10.1108/ir-04-2019-0074 · Cited by 22 · see citing works
Adaptive finite-time control of a class of non-triangular nonlinear systems with input saturation
Cai, M., & Xiang, Z. (2016) · Neural Computing and Applications · doi:10.1007/s00521-016-2540-0 · Cited by 21 · see citing works
“In [32], NNs were used to approximate the unknown functions and an adaptive tracking controller was developed for a class of MIMO non-strict feedback systems by constructinga input-driven filter.”
BackgroundMethodDai, A., Zhou, X., Dang, H., Sun, M., & Wu, Z. (2018) · IEEE Access · doi:10.1109/access.2018.2806370 · Cited by 21 · see citing works
“For instance the ANN application in the wind turbine control shown in [18], and in [19] an adaptive fuzzy control method is presented for a class of nonaffine stochastic nonlinear systems via backstepping; in [20] an observer-based fuzzy output feedback control algorithm is proposed by using backstepping and fuzzy sys…”
MethodXu-dong, L., Chen, X., & Zhou, C. (2018) · Journal of the Franklin Institute · doi:10.1016/j.jfranklin.2018.05.050 · Cited by 21 · see citing works
Xin-jun, W., Yin, X., Wu, Q., & Meng, F. (2018) · International Journal of Robust and Nonlinear Control · doi:10.1002/rnc.4303 · Cited by 20 · see citing works
“Then, other works 16 - 19 extended the research to multiple-input–multiple-output nonlinear systems.”
BackgroundZhang, A., Lv, B. C., Zhang, C. Z., & She, D. Z. (2017) · IEEE Access · doi:10.1109/access.2017.2745701 · Cited by 19 · see citing works
“In the recent references [12], [13], the authors not only focus on the nonlinear Hamiltonian multi-input multi-output systems but also focus on a class of uncertain nonlinear systems, the uncertainty of the system are all dealt with adaptive tracking control.”
BackgroundWang, X., Yin, X., & Shen, F. (2018) · Neurocomputing · doi:10.1016/j.neucom.2018.02.082 · Cited by 19 · see citing works
A Runge–Kutta neural network-based control method for nonlinear MIMO systems
Uçak, K. (2018) · Soft Computing · doi:10.1007/s00500-018-3405-5 · Cited by 19 · see citing works
Zhang, X., Lan, J., Liu, Y., & Liu, L. (2021) · IET Control Theory and Applications · doi:10.1049/cth2.12183 · Cited by 17 · see citing works
Ye, D., Wang, K., Yang, H., & Zhao, X. (2020) · International Journal of Adaptive Control and Signal Processing · doi:10.1002/acs.3172 · Cited by 17 · see citing works
Self-citationVan, M., & Phu, D. X. (2020) · European Journal of Control · doi:10.1016/j.ejcon.2019.12.005 · Cited by 16 · see citing works
Ji, N., & Liu, J. (2019) · International Journal of Robust and Nonlinear Control · doi:10.1002/rnc.4658 · Cited by 16 · see citing works
“In the work of Zhao et al, 30 an adaptive neural output-feedback tracking controller is developed for a class of MIMO nonstrict-feedback nonlinear systems with time delay and unknown function entries are approximated by employing neural networks.”
MethodSu, H., & Zhang, W. (2018) · Nonlinear Dynamics · doi:10.1007/s11071-018-4645-8 · Cited by 15 · see citing works
Yang, Y., Yu, Z., Li, S., & Sun, J. (2017) · Neurocomputing · doi:10.1016/j.neucom.2017.12.023 · Cited by 15 · see citing works
Cao, X., Gu, L., Qiu, H., Lai, C., & Qin, Y. (2017) · Proceedings of the Institution of Mechanical Engineers Part I Journal of Systems and Control Engineering · doi:10.1177/0959651817731250 · Cited by 15 · see citing works
“Then, the error term in equation (7) can be replaced by contour error term”
BackgroundConsensus of Multi-agent Systems with Feedforward Nonlinear Dynamics and Digraph
Yu, S., Jiang, H., Yu, Z., & Hu, C. (2018) · International Journal of Control Automation and Systems · doi:10.1007/s12555-017-0702-1 · Cited by 14 · see citing works
Predictor-Based Neural Dynamic Surface Control of a Nontriangular System With Unknown Disturbances
Yang, Y., Chen, D., Liu, Q., Zhang, T., Liu, A., & Yue, W. (2022) · IEEE Transactions on Circuits and Systems I Regular Papers · doi:10.1109/tcsi.2022.3166988 · Cited by 14 · see citing works
Yoo, S. J. (2016) · International Journal of Systems Science · doi:10.1080/00207721.2016.1256450 · Cited by 13 · see citing works
“1 for nonstrict-feedback nonlinear systems are not provided in existing literatures (Bechlioulis & Rovithakis, 2009, 2014; Chen et al., 2012, 2014, 2016;Hua et al., 2016; Li & Tong, 2015, 2016a, 2016b; Liu et al., 2014; Theodorakopoulos & Rovithakis, 2015; Tong et al., 2015; Wang et al., 2014, 2016a, 2016b, 2015; Yu e…”
BackgroundMethodInfluentialChen, C., Tang, Y., Wu, L., Lu, M. L., Zhan, X., Li, X., Huang, C., & Gui, W. (2018) · IEEE Access · doi:10.1109/access.2018.2889969 · Cited by 13 · see citing works
“In [21], [29], and [33], the virtual control coefficients φi is an unknown constant.”
BackgroundYang, Y., Xu, C., Fan, X., Yue, D., Zhang, T., & Hou, X. (2020) · Journal of the Franklin Institute · doi:10.1016/j.jfranklin.2019.12.025 · Cited by 13 · see citing works
Askari, M. R., Shahrokhi, M., Talkhoncheh, M. K., & Moradvandi, A. (2020) · Journal of the Franklin Institute · doi:10.1016/j.jfranklin.2020.05.006 · Cited by 13 · see citing works
Shi, X., Lim, C., Xu, S., & Shi, P. J. (2018) · International Journal of Robust and Nonlinear Control · doi:10.1002/rnc.4081 · Cited by 13 · see citing works
“Putting together (28), (29), (30), and (31) yields V̇ei ≤ ei [ 3n 2 ei + i−1 ∑”
Funnel tracking control for nonlinear servo drive systems with unknown disturbances
Wang, S., Li, S., Chen, Q., Ren, X., & Yu, H. (2021) · ISA Transactions · doi:10.1016/j.isatra.2021.08.047 · Cited by 12 · see citing works
Finite-Time Stability of Large-Scale Systems with Interval Time-Varying Delay in Interconnection
La-inchua, T., Niamsup, P., & Liu, X. (2017) · Complexity · doi:10.1155/2017/1972748 · Cited by 11 · see citing works
Shi, X., Xu, S., Chen, W., Li, Y., & Zhang, Z. (2017) · Journal of the Franklin Institute · doi:10.1016/j.jfranklin.2017.10.011 · Cited by 11 · see citing works
Bai, L., & Gao, Z. (2019) · Journal of Control Automation and Electrical Systems · doi:10.1007/s40313-019-00498-3 · Cited by 11 · see citing works
Lee, S.-D., & Jung, S. (2018) · International Journal of Control Automation and Systems · doi:10.1007/s12555-017-0187-y · Cited by 11 · see citing works
Control of twin-double pendulum lower extremity exoskeleton system with fuzzy logic control method
Tanyıldızı, A. K., Yakut, O., Taşar, B., & Tatar, A. B. (2021) · Neural Computing and Applications · doi:10.1007/s00521-020-05554-7 · Cited by 10 · see citing works
Lin, Q., Zhou, P., & Duan, D. (2022) · Transactions of the Institute of Measurement and Control · doi:10.1177/01423312221124648 · Cited by 10 · see citing works
“The control problem of time delay has attracted the attention of many researchers (Li et al., 2019b; Liu et al., 2014; Wang et al., 2015; Xia et al., 2019, 2020, 2021a; Zhao et al., 2016).”
BackgroundLv, G., Shan, Y., Li, F., & Zhang, Y. (2025) · Journal of the Franklin Institute · doi:10.1016/j.jfranklin.2025.107751 · Cited by 9 · see citing works
Liu, W., Lu, J., Zhang, Z., & Xu, S. (2016) · IET Control Theory and Applications · doi:10.1049/iet-cta.2016.0789 · Cited by 9 · see citing works
Du, P., Liang, H., Huang, T., & Li, T. (2019) · Neurocomputing · doi:10.1016/j.neucom.2019.07.067 · Cited by 9 · see citing works
Command filtered event‐triggered adaptive control for MIMO stochastic multiple time‐delay systems
Zhu, Z., Liang, H., Liu, Y., & Xue, H. (2021) · International Journal of Robust and Nonlinear Control · doi:10.1002/rnc.5868 · Cited by 9 · see citing works
Finite‐time exact tracking control for a class of non‐linear dynamical systems
Hussian, A., Zhao, X., & Zong, G. (2017) · IET Control Theory and Applications · doi:10.1049/iet-cta.2017.0093 · Cited by 9 · see citing works
Self-citationAdaptive fuzzy constraint control for switched nonlinear systems in nonstrict feedback form
Tang, L. D., Yang, M., & Sun, J.-G. (2021) · International Journal of Adaptive Control and Signal Processing · doi:10.1002/acs.3277 · Cited by 8 · see citing works
Liu, Z., Sun, W., & Zhang, W. (2021) · ISA Transactions · doi:10.1016/j.isatra.2021.11.007 · Cited by 8 · see citing works
Yang, Y., Chen, D., Yue, W., & Liu, Q. (2022) · ISA Transactions · doi:10.1016/j.isatra.2022.02.030 · Cited by 8 · see citing works
Duan, C., & Tian, M. (2024) · IEEE Access · doi:10.1109/access.2024.3360107 · Cited by 7 · see citing works
“Aswe all know, the network bandwidth is limited [25], [26], therefore, in order to avoid unnecessary waste of resources and reduce network load capacity, the event-triggered mechanism [27] was proposed as a powerful tool and has become a subject of intense research activities [28], [29], [30],…”
Background
- Authors
- Zhao, X., Yang, H., Karimi, H. R., & Zhu, Y.
- Affiliations
- Bohai University (CN)
- University of Agder (NO)
- Harbin Institute of Technology (CN)
- Venue
- IEEE Transactions on Cybernetics
- Published
- 2016-06
- Volume
- 46
- Issue
- 6
- Pages
- 1337-1349
- ISSN
- 2168-2267 (ISSN-L)
- 2168-2275
- OpenAlex
- W2353924536
- Cited by
- 130
- References
- 44
- Citations by year
- 2025: 2 · 2024: 2 · 2023: 2 · 2022: 8 · 2021: 15 · 2020: 11 · 2019: 23 · 2018: 30 · 2017: 30 · 2016: 7
- FWCI
- 20.68
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- IEEE Transactions on Cybernetics (publishedVersion)
- PubMed (publishedVersion)
- Virtual Community of Pathological Anatomy (University of Castilla La Mancha) (submittedVersion)
- Funders
- National Natural Science Foundation of China (61203123, 61333012)
- Liaoning Excellent Talents in University (LR2014035)
- Shandong Provincial Natural Science Foundation, China (ZR2012FQ019)
- Topics
- Physical Sciences › Engineering › Control and Systems Engineering › Adaptive Control of Nonlinear Systems
- Physical Sciences › Engineering › Control and Systems Engineering › Stability and Control of Uncertain Systems
- Physical Sciences › Computer Science › Computer Networks and Communications › Neural Networks Stability and Synchronization
- Keywords
- adaptive neural network control, MIMO nonlinear systems, nonstrict-feedback systems, time delay, output feedback tracking control, backstepping design, neural network approximation, ultimate boundedness
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