Adaptive Intelligent Control of Nonaffine Nonlinear Time-Delay Systems With Dynamic Uncertainties
Abstract
Adaptive neural intelligent control is investigated for a class of pure-feedback nonlinear time-delay systems with unmodeled dynamics in nonlower-triangular form, which views the lower-triangular structure as a special structure. A variable partition technique is applied to surmount the difficulty in the nonlinear functions of whole state variables. By utilizing the backstepping recursive design approach and the universal approximation capability of neural networks, an adaptive neural controller is systemically designed. Then, based on the utilization of Lyapunov-Krasovskii functionals, the semiglobally uniform boundedness of all closed-loop signals is guaranteed. Finally, the suggested control method is verified through a numerical example. The main advantage of this paper is that an intelligent control method is developed for pure-feedback nonlinear systems with state time delay, unmodeled dynamics and nonlower triangular form. Further developments will focus on how to deal with the problem of output feedback control of pure-feedback nonlinear time-delay systems with unmodeled dynamics and nonlower-triangular structure.
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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 [26]
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“Substituting (53)-(56) into (52) and using the definitions of θbi, ρbi, ξbi, θbi, ?̃?bi, and ξbi, it gives V̇si = T2 i+1 ( si+1(t),Ki+1(t),Ki+1(t) ) + y(2)i + siri(t)Δαi − kis 2 i + 1 2 e(2)i + siri(t)θbifi(xi) + ?̃?biri(t)siφi(xi) tanh ( ri(t)siφi(xi) ε ) − ξbisiri(t) ( ?̇?i,d − 1 ri(t) ηi(t) ) + ρbiζε.”
Wang, W., Pan, Y., & Liang, H. (2020) · Neurocomputing · doi:10.1016/j.neucom.2020.01.077 · Cited by 26 · see citing works
Motion Planning of Robot Manipulator Based on Improved NSGA-II
Huang, Y., & Fei, M. (2018) · International Journal of Control Automation and Systems · doi:10.1007/s12555-016-0693-3 · Cited by 25 · see citing works
Detection and estimation of mental fatigue in manual assembly process of complex products
Xiao, H. L., Duan, Y., Zhang, Z., & Li, M. (2017) · Assembly Automation · doi:10.1108/aa-03-2017-040 · Cited by 24 · see citing works
Ma, H., Liu, Y., & Ye, D. (2017) · IET Control Theory and Applications · doi:10.1049/iet-cta.2017.0772 · Cited by 24 · see citing works
Jin, D., Niu, B., Wang, H., & Yang, D. (2020) · IEEE Transactions on Systems Man and Cybernetics Systems · doi:10.1109/tsmc.2020.3009405 · Cited by 23 · see citing works
Self-citationChien, Y., Chen, J.-W., & Xu, S. S. (2018) · IEEE Access · doi:10.1109/access.2018.2825239 · Cited by 23 · see citing works
Finite-Time Stability and Boundedness of Switched Systems with Finite-Time Unstable Subsystems
Tan, J., Wang, W., & Yao, J. (2018) · Circuits Systems and Signal Processing · doi:10.1007/s00034-018-1001-7 · Cited by 23 · see citing works
Li, W., Zhang, Z., & Ge, S. S. (2022) · IEEE Transactions on Cybernetics · doi:10.1109/tcyb.2022.3178385 · Cited by 22 · see citing works
“After long-term exploration by scholars, an adaptive backstepping neural-network (NN) control method has been widely used in various time-delay systems, for example, strict-feedback systems [12]–[14] and pure-feedback systems [15]–[18].”
MethodZhang, S., Kong, L., Qi, S., Jing, P., He, W., & Xu, B. (2019) · Nonlinear Dynamics · doi:10.1007/s11071-018-4629-8 · Cited by 22 · see citing works
Adaptive neural control of state-constrained MIMO nonlinear systems with unmodeled dynamics
Shi, X., Xu, S., Jia, X., Chu, Y., & Zhang, Z. (2022) · Nonlinear Dynamics · doi:10.1007/s11071-021-07124-5 · Cited by 21 · see citing works
“In [37,38], two robust adaptive control schemes formulated to achieve semiglobal tracking stability were proposed for nonaffine pure-feedback nonlinear systems in presence of unmodeled dynamics.”
BackgroundDai, 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…”
MethodDing, R., Zhang, J., & Bing Hong, X. U. (2018) · IEEE Access · doi:10.1109/access.2018.2863554 · Cited by 21 · see citing works
“Various adaptive control algorithms are recommended to solve the nonlinearity and uncertainty [21]–[24]; however, it is complex to design these adaptive control algorithms for excavator applications.”
BackgroundA New Approach to Characterize Successive Packet Losses in Stochastic Networked Systems
Hu, Z., Deng, F., Shi, P. J., & Lim, C. (2018) · IEEE Transactions on Systems Man and Cybernetics Systems · doi:10.1109/tsmc.2018.2876675 · Cited by 21 · see citing works
Li, Y., Lei, M.-Y., Guo, Y., Hu, Z., & Wei, H. (2018) · IEEE Access · doi:10.1109/access.2018.2818789 · Cited by 20 · see citing works
“Additionally, the fundamental parametric models and causal relationships of signals may be studied by stochastic nonlinear multiple timedelay systems [14]–[17].”
BackgroundAsynchronous $L_{1}$ Tracking Control of Switched Positive Systems With Actuator Saturation
Dong, X., & Chang, L. (2017) · IEEE Access · doi:10.1109/access.2017.2748364 · Cited by 19 · see citing works
“Intelligent control problems are interesting and important, and these problems have attracted much attention [1]–[3].”
BackgroundCai, Z., & Huang, L. (2017) · IEEE Transactions on Systems Man and Cybernetics Systems · doi:10.1109/tsmc.2017.2657784 · Cited by 18 · see citing works
New Results on Finite-time Stabilization for Stochastic Systems with Time-varying Delay
Zhang, L., Qi, W., Kao, Y., Gao, X., & Zhao, L. (2018) · International Journal of Control Automation and Systems · doi:10.1007/s12555-017-0020-7 · Cited by 18 · see citing works
Fault Detection for a Class of Nonlinear Networked Control Systems with Communication Constraints
Mao, Z., Pan, Y., Jiang, B., & Chen, W. (2018) · International Journal of Control Automation and Systems · doi:10.1007/s12555-017-0258-0 · Cited by 18 · see citing works
Dai, A., Zhou, X., & Wu, Z. (2020) · Food Science & Nutrition · doi:10.1002/fsn3.1340 · Cited by 18 · see citing works
Jiang, K., Niu, B., Wang, X., Xiang, Z., Li, J., Duan, P., & Yang, D. (2019) · ISA Transactions · doi:10.1016/j.isatra.2019.11.028 · Cited by 17 · see citing works
Yu, L.-Y., Sun, Z., Meng, Q., & Chen, C. (2022) · IEEE Transactions on Systems Man and Cybernetics Systems · doi:10.1109/tsmc.2022.3182395 · Cited by 17 · see citing works
Cui, D., Zou, W., Guo, J. C., & Xiang, Z. (2022) · International Journal of Adaptive Control and Signal Processing · doi:10.1002/acs.3491 · Cited by 17 · see citing works
“5,41,42,44 For the available and bounded y ( i ) j , d ( t ) , the main objective is to construct an adaptive neural fault-tolerant decentralized controller for the switched stochastic nonlinear large-scale system (3) such that: (1) for the desired trajectory y j , d ( t ) , the tracking error z j…”
BackgroundMethodZhou, S., Song, Y., & Zhao, K. (2017) · International Journal of Adaptive Control and Signal Processing · doi:10.1002/acs.2823 · Cited by 17 · see citing works
“(34) To achieve accelerated tracking with adjustable convergence speed and on the basis of the commonly used tracking errors (33), we introduce the following speed transformation: ζi = βzi, i = 1, · · · ,n (35) with β = 1 (1 − bf )e−λt + bf , (36) where 0 < bf ≪ 1 and λ > 0 are design parameters chosen by the designer.”
MethodBessa, W. M., Brinkmann, G., Duecker, D. A., Kreuzer, E. J., & Solowjow, E. (2018) · Mathematical Problems in Engineering · doi:10.1155/2018/9648126 · Cited by 17 · see citing works
Liu, Y., Liu, Y., Su, C., Liu, Y., Zhou, Q. F., & Lu, R. (2021) · IEEE Transactions on Neural Networks and Learning Systems · doi:10.1109/tnnls.2021.3131364 · Cited by 16 · see citing works
Zhou, L., & She, J. (2018) · IEEE Access · doi:10.1109/access.2018.2876398 · Cited by 16 · see citing works
“In general, robustness is achieved at the price of sacrificing nominal tracking performance [15]–[20].”
BackgroundLi, R.-B., Niu, B., Feng, Z., Li, J., Duan, P., & Yang, D. (2019) · Journal of the Franklin Institute · doi:10.1016/j.jfranklin.2019.09.019 · Cited by 16 · 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
“The contouring controller parameters are chosen as follows: Mxt = diag(0:001, 0:001), a= diag(1:8, 1:8), b= diag(0:8, 0:8), K= diag(1, 2), k1 = diag(1, 2), k2 = diag(1, 2), while L=0:007 is the time delay.”
Xue, L., Liu, Z., & Zhang, W. (2023) · International Journal of Robust and Nonlinear Control · doi:10.1002/rnc.6794 · Cited by 14 · see citing works
“, gn−1 and transformations S1 = x1 − x∗ 1 , Si = xi − x ∗ i , 2 ≤ i ≤ k, x∗ 1 = 0, x ∗ i = −gi−1(⋅)Si−1, (15) which guarantee that”
Xi, C., Zhai, D., Dong, J., & Zhang, Q. (2017) · International Journal of Systems Science · doi:10.1080/00207721.2017.1322641 · Cited by 14 · see citing works
“…of the study for pure-feedback systems have been given by assuming that nonaffine terms can be linearly parameterised (Ferrara & Giacomini, 1998; Seto, Annaswamy, & Baillieul, 1994), and by developing approximation-based adaptive control approaches (Wang, Liu, & Liu, 2016; Wang, Sun, & Liu, 2016).”
BackgroundAdaptive fuzzy tracking control for switched uncertain strict-feedback nonlinear systems
Yang, Z., Zhang, H., & Cui, Y. (2017) · Journal of the Franklin Institute · doi:10.1016/j.jfranklin.2017.11.038 · Cited by 14 · see citing works
Robust Adaptive Synchronization of a Class of Uncertain Chaotic Systems with Unknown Time-Delay
Zare, A., Mirrezapour, S. Z., Hallaji, M., Shoeibi, A., Jafari, M., Ghassemi, N., Alizadehsani, R., & Mosavi, A. H. (2020) · Applied Sciences · doi:10.3390/app10248875 · Cited by 14 · see citing works
“In [22], an adaptive intelligent controller is introduced to control uncertain systems with time delay.”
BackgroundSimple adaptive robust control schemes of uncertain strict‐feedback non‐linear time‐delay systems
Wu, H. (2017) · IET Control Theory and Applications · doi:10.1049/iet-cta.2017.0220 · Cited by 14 · see citing works
Sui, S., Tong, S., & Sun, K. (2017) · Optimal Control Applications and Methods · doi:10.1002/oca.2379 · Cited by 14 · see citing works
Pang, J., Zhang, J., Li, M., & Li, S. (2025) · Journal of the Franklin Institute · doi:10.1016/j.jfranklin.2025.107527 · Cited by 13 · see citing works
Chen, 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
“INTRODUCTION In recent years, a hot topic is adaptive control andmany effective control strategies have been proposed, including adaptive backstepping design [2], [3], intelligent control [5], [6], sliding mode control [7], distributed control [8], [9] and more.”
BackgroundDalir, M., & Bigdeli, N. (2019) · Applied Soft Computing · doi:10.1016/j.asoc.2019.106000 · Cited by 13 · see citing works
Improved Minmax Control for Industrial Networked Systems Over Imperfect Communication
Jin, Q., Wu, S., Zhang, R., & Lu, R. (2017) · IEEE Transactions on Systems Man and Cybernetics Systems · doi:10.1109/tsmc.2017.2721111 · Cited by 13 · see citing works
Zhang, T., Feng, C., & Xia, X. (2024) · Journal of the Franklin Institute · doi:10.1016/j.jfranklin.2024.107453 · Cited by 12 · see citing works
Su, W., Zhao, X., Niu, B., Zhang, G., & Wang, H. (2022) · Neurocomputing · doi:10.1016/j.neucom.2021.12.103 · Cited by 12 · see citing works
Self-citationLiu, H., & Xie, G. (2018) · Neurocomputing · doi:10.1016/j.neucom.2018.01.045 · Cited by 12 · see citing works
Pattern-based NN control for uncertain pure-feedback nonlinear systems
Zhang, F., Wang, C., & Yang, F. (2019) · Journal of the Franklin Institute · doi:10.1016/j.jfranklin.2019.01.014 · Cited by 12 · see citing works
Adaptive dynamic surface control of nonaffine nonlinear system with time‐varying state constraints
Zhao, S., Liang, H., Du, P., & Ma, H. (2019) · International Journal of Robust and Nonlinear Control · doi:10.1002/rnc.4793 · Cited by 12 · see citing works
Liu, Z., Gong, Q., & Hu, X. (2025) · Ocean Engineering · doi:10.1016/j.oceaneng.2025.120503 · Cited by 11 · see citing works
Dinh, T. X., & Ahn, K. K. (2018) · Proceedings of the Institution of Mechanical Engineers Part I Journal of Systems and Control Engineering · doi:10.1177/0959651818771810 · Cited by 11 · see citing works
“where the equivalent control signal and FCMAC estimator are the same as that in equations (35) and (31), respectively.”
BackgroundResultStatic Output Feedback Stabilization of a Class of Switched Linear Systems with State Constraints
Su, Q., Zhu, H., & Li, J. (2018) · International Journal of Control Automation and Systems · doi:10.1007/s12555-016-0800-5 · Cited by 11 · see citing works
Li, M., Li, S., Zhang, J., Wu, F., & Zhang, T. (2022) · Entropy · doi:10.3390/e24081028 · Cited by 11 · see citing works
Yu, J., Sun, Y., & Li, Z. (2018) · Signal Processing · doi:10.1016/j.sigpro.2018.01.023 · 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
Mean First Passage Time of Preferential Random Walks on Complex Networks with Applications
Zheng, Z., Xiao, G., Wang, G., Zhang, G., & Jiang, K. (2017) · Mathematical Problems in Engineering · doi:10.1155/2017/8217361 · Cited by 10 · see citing works
“Such processes have potential applications in the control of stochastic systems [7–10].”
BackgroundJiang, K., Niu, B., Li, J., Duan, P., Wang, J., & Yang, D. (2018) · IEEE Access · doi:10.1109/access.2018.2877798 · Cited by 10 · see citing works
“For example, the paper [55] applied the variable separation technology to overcome the dsign difficulties of non-strict-feedback nonlinear functions and successfully constructed an adaptive intelligent controller for a class of nonaffine nonlinear time-delay systems.”
MethodLiu, J., Zhao, M., Li, G., & Chen, J. (2020) · IEEE Access · doi:10.1109/access.2020.3020980 · Cited by 10 · see citing works
“Adaptive control method is widely applied to uncertain systems, especially for controlling strict-feedback nonlinear systems [12].”
MethodLi, Z., Cao, L., Pan, Y., & Zhang, P. (2022) · Journal of the Franklin Institute · doi:10.1016/j.jfranklin.2022.09.033 · Cited by 10 · see citing works
Random Search Enhancement of Incremental Regularized Multiple Hidden Layers ELM
Liu, J., Liu, X., Liu, C., Le, B. T., & Xiao, D. (2019) · IEEE Access · doi:10.1109/access.2019.2905077 · Cited by 10 · see citing works
Optimal Control Method for Robot-Tracking Based on Control-Lyapunov-Function
Wu, Y., Liu, L., Yang, Y., & Dai, S. (2019) · IEEE Access · doi:10.1109/access.2019.2923658 · Cited by 10 · see citing works
“Want H et al employ Adaptive Neural Output-Feedback Control to deal these problems with unknown external disturbance [8].”
MethodReachable set estimation for neutral Markovian jump systems with mode‐dependent time‐varying delays
Jiang, X., Xia, G., Feng, Z., & Jiang, Z. (2020) · Optimal Control Applications and Methods · doi:10.1002/oca.2670 · Cited by 9 · see citing works
“P{vt+Δt = n|vt = s} = {σsnΔt + o(Δt), s ≠ n, 1 + σssΔt + o(Δt), s = n, (2)”
- Authors
- Wang, H., Sun, W., & Liu, P. X.
- Affiliations
- Carleton University (CA)
- Bohai University (CN)
- Venue
- IEEE Transactions on Systems, Man, and Cybernetics: Systems
- Published
- 2017-07
- Volume
- 47
- Issue
- 7
- Pages
- 1474-1485
- ISSN
- 2168-2216 (ISSN-L)
- 2168-2232
- OpenAlex
- W2558651951
- Cited by
- 151
- References
- 63
- Citations by year
- 2026: 4 · 2025: 10 · 2024: 4 · 2023: 7 · 2022: 14 · 2021: 7 · 2020: 13 · 2019: 20 · 2018: 33 · 2017: 38 · 2016: 1
- FWCI
- 25.80
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- IEEE Transactions on Systems Man and Cybernetics Systems (publishedVersion)
- Funders
- National Natural Science Foundation of China (61304002, 61175072, 61473160)
- Natural Science Foundation of Liaoning Province of China (2015020058)
- 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, pure-feedback nonlinear systems, time-delay systems, unmodeled dynamics, backstepping control, Lyapunov-Krasovskii functional, state delay
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