Adaptive neural control for non-strict-feedback nonlinear systems with input delay
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The design of nonlinear systems in nonstrict-feedback form incorporating input delay and backlash-like hysteresis remains an unresolved difficulty in prior work.
“The observation that the design of nonlinear systems in nonstrict-feedback form that incorporates input delay and hysteresis similar to backlash is still an unresolved difficulty in the previously discussed outcomes is what inspired this research.”
From Fixed-Time Adaptive Control for Nonstrict-Feedback Nonlinear Systems with Input Delay and Unknown Backlash-Like Hysteresis (2025) · cites this paper as [28]
The presence of input delays can potentially degrade system performance or lead to instability.
“The presence of input delays can potentially degrade system performance or lead to instability [28].”
From Fixed-Time Adaptive Control for Nonstrict-Feedback Nonlinear Systems with Input Delay and Unknown Backlash-Like Hysteresis (2025) · cites this paper as [28]
Radial basis function neural networks with Gaussian function as an activation functions satisfying 0 < P_i^T(•)P_i(•) ≤ N have been implemented to manage such complexities.
“To address this challenge, radial basis function neural networks with Gaussian function as an activation functions satisfying 0 < P T i (•)P i (•) ≤ N have been implemented to manage such complexities [28,47,49].”
From Fixed-Time Adaptive Control for Nonstrict-Feedback Nonlinear Systems with Input Delay and Unknown Backlash-Like Hysteresis (2025) · cites this paper as [28]
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“The following inferences are derived from (14): (1) it is a smooth and strictly increasing function; (2) T(ε) ∈ (−1, 1); (3) limε→+∞T(ε) = 1 and limε→−∞T(ε) = −1.”
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“Substituting (35)-(39) into (34), it is continued as. where Δ = c 𝜆 min ( p ) − 7 4 − 3 2 ‖ p ‖ 83 − 3 n √ n ‖ p ‖ 4 and The RBF neural network is utilized to approximate the function f 1 by Substituting (41) into (40) provides Using Young's inequality and Lemma 2, one concludes (44) (45) Substituting (43)-(47) into (…”
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“According to the inequalities equations (25), (33), (41), and (47) in this paper, we use NNs to approximate the unknown nonlinear function, and this processing approach can also be reflected in Zhao and Chen (2022) and Wang et al. (2020).”
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“Remark 4 Compared with the existing results [50–55, 63,70–72] , which only work on the nonlinear system with small input delay or ignoring input delay for finite time control, the proposed scheme in this paper can tackle the problem of finite time control for non-strict feedback uncertain nonlinear…”
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“An adaptive control approach has been reported using neural network approximation for nonlinear systems with a nonstrict-feedback structure [18].”
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“(2) Different from the works [56], [57] which handled the known delays, our work deals with the unknown delays.”
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“, li,ni] T , Ai,0 is Hurwitz matrix and satisfies: PiAi,0 + Ai,0Pi = −hiIi, (21) with hi>0 being a design constant.”
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“Duetononlinearcharacteristicsanddifficultiesinbuilding an accurate model of the FCHEV system, the control effect of conventional control theory is limited (Wang et al. 2020).”
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Adaptive asymptotical funnel control for nonlinear systems with input quantization
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“1 sin(t)e(−5t), and q(u) denotes the hysteresis input quantizer defined in (2).”
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“j=1 (3 4 b(2)j + 1 2 h(2)j + 1 4 β∗4 j ) + ξ3 i (ξi+1 + ri+1 + αi + ωi+1 − αi + fi − ?̇?i + 1 4 ξi − ṙi ) + 3 2 ξ2 i g T i gi, (35) the following inequalities is verified based on Lemma 2”
Self-citationAdaptive neural control for non-strict feedback stochastic nonlinear systems with input delay
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“…ability, an adaptive fuzzy/neural controller was presented to secure the stability of the closed-loop system (Fu et al., 2022; Jiang et al., 2022; Li et al., 2017, 2019, 2022; Liu et al., 2021; Ma et al., 2018; Qiu et al., 2022; Tehrani et al., 2019; Wang et al., 2020, 2022; Yang et al., 2020).”
BackgroundMethodInfluentialDirect Adaptive Fuzzy Tracking Control of Non-affine Stochastic Nonlinear Time-Delay Systems
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“The recurrent neural network is very helpful to provide system control with high performances (Fei and Wang 2020; Hassanpour et al. 2020; Pan et al. 2020; Wu et al. 2020).”
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Fixed-time neural network composite learning control for uncertain nonlinear systems
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Self-citationXia, X., Pan, J., Zhang, T., & Fang, Y. (2022) · Journal of the Franklin Institute · doi:10.1016/j.jfranklin.2022.10.004 · Cited by 12 · see citing works
Guo, J. Y., Bo, Y., Park, J. H., & Ma, J. (2021) · International Journal of Robust and Nonlinear Control · doi:10.1002/rnc.5929 · Cited by 12 · see citing works
Kharrat, M., & Alhazmi, H. (2024) · AIMS Mathematics · doi:10.3934/math.2024668 · Cited by 11 · see citing works
“The results, as depicted in Table 2, demonstrate a marginal improvement in the proposed control scheme relative to the existing control method [29].”
MethodResultInfluentialXia, X., Pan, J., Zhang, T., & Fang, Y. (2023) · International Journal of Robust and Nonlinear Control · doi:10.1002/rnc.6603 · Cited by 11 · see citing works
“Differentiating (26) to time t, one gets”
An adaptive finite‐time stable control law for manipulator robots with unknown parameters
Boutalbi, O., Benmahammed, K., & Boukezata, B. (2021) · International Journal of Robust and Nonlinear Control · doi:10.1002/rnc.5538 · Cited by 11 · see citing works
“The proposed control law (Equations 20)–(22) make the system (Equation 19) globally uniformly asymptotically stable if the following conditions are respected: r2 ≥ c, r1 ≥ r2 + c, kp ≥ 1 ε2 , kd ≥ 1 ε (43)”
BackgroundZhai, J., Wang, H., Tao, J., & He, Z. (2022) · Transactions of the Institute of Measurement and Control · doi:10.1177/01423312221110437 · Cited by 11 · see citing works
“Compared with the results in Cui et al. (2020), Liu et al. (2020), Sun et al. (2018, 2021b), Wang et al. (2017) and Xu et al. (2019), the method in this paper theoretically guarantees the correctness of the finite time control strategy based on the approximate method.”
MethodResultInfluentialSelf-citationKharrat, M. (2024) · International Journal of Adaptive Control and Signal Processing · doi:10.1002/acs.3936 · Cited by 10 · see citing works
“The results clearly indicate that our proposed method outperforms the strategy described in [5]. in terms of tracking performance.”
InfluentialAntiwindup Compensator-Based Control for Periodic Piecewise Delayed Systems With Input Saturations
Satheesh, T., Sakthivel, R., Harshavarthini, S., Manikandan, R., & Almakhles, D. J. (2022) · Journal of Dynamic Systems Measurement and Control · doi:10.1115/1.4056106 · Cited by 10 · see citing works
Long, L., & Bian, Y. (2020) · International Journal of Robust and Nonlinear Control · doi:10.1002/rnc.5113 · Cited by 10 · see citing works
“0 χ ∂G−1 nk (sn−1, znχ + αn−1) ∂sj dχ ]2 + 1 4ηn , (38)”
Yao, Q. (2021) · Neural Processing Letters · doi:10.1007/s11063-021-10675-8 · Cited by 10 · see citing works
Song, S., Zhang, B., Park, J. H., & Song, X. (2021) · International Journal of Robust and Nonlinear Control · doi:10.1002/rnc.5938 · Cited by 10 · see citing works
“In Reference 36, an NN-based adaptive backstepping control scheme was developed for nonlinear systems with input delay, where an auxiliary system is constructed to eliminate the influence of input delay.”
BackgroundMethodReachable 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
“, 11, δ12 = [0q×11p Iq], τ12 = τ2 − τ1, d12 = d2 − d1, the following theorem is presented to give reachable set estimation for NMJSs (3) with different delay mode and system mode.”
BackgroundMethodInfluentialAbdulkareem, M. M., Attrah, N. H. Y., Alrubaee, S. H., Al-Sabti, S. M. B., Dheyab, A., & Mohammed, A. H. (2022) · International Symposium on Multidisciplinary Studies and Innovative Technologies (ISMSIT) · doi:10.1109/ismsit56059.2022.9932689 · Cited by 8 · see citing works
Ghoreishee, S. A., Shahrokhi, M., Vafa, E., & Moradvandi, A. (2022) · International Journal of Adaptive Control and Signal Processing · doi:10.1002/acs.3541 · Cited by 8 · see citing works
“Based on Lemma 3, it can be concluded that if a RBFNN is used for the approximation of a function, there is a ε∗ > 0 such that F(X) = W TS(X) + ε(X), |ε(X)| ≤ ε∗ (20) and based on the definition of Gaussian functions, (19), there is a positive constant, Λ, such that ||S(X)||2 ≤ Λ (21) Lemma 4.”
BackgroundCongestion Tracking Control for Wireless TCP/AQM Network Based on Adaptive Integral Backstepping
Ma, L., Liu, X., Wang, H., & Zhou, Y. (2020) · International Journal of Control Automation and Systems · doi:10.1007/s12555-019-0724-y · Cited by 8 · see citing works
“Many researches combine backstepping with other control methods, such as minimax [38], H ∞ [39], Prescribed Performance Control (PPC) [40], funnel control, fuzzy logic [41–46], neural network [47–51] and so on, to solve network congestion control problems.”
MethodSelf-citationYang, Y., Chen, D., Yue, W., & Liu, Q. (2022) · ISA Transactions · doi:10.1016/j.isatra.2022.02.030 · Cited by 8 · see citing works
Yao, Q. (2022) · Asian Journal of Control · doi:10.1002/asjc.2771 · Cited by 8 · see citing works
“Over the recent years, some adaptive neural/fuzzy backstepping controllers have been presented for the control of nonstrict-feedback nonlinear systems [16 – 24].”
BackgroundKharrat, M. (2025) · International Journal of Systems Science · doi:10.1080/00207721.2025.2563692 · Cited by 7 · see citing works
“In recent years, the control of nonlinear systems has attracted significant attention due to its widespread applications in robotics, aerospace, process engineering, and other complex systems (Lu et al., 2025; H. Wang et al., 2020; W. J. Zhao et al.,2021 ).”
Wu, P., Liu, W., & Lu, Y. (2023) · Transactions of the Institute of Measurement and Control · doi:10.1177/01423312231157794 · Cited by 7 · see citing works
“In Wang et al. (2020), a novel auxiliary sys-tem was constructed to handle the adverse influence caused by input delay.”
MethodYue, H., Xue, A., & Li, J. (2024) · Journal of low frequency noise, vibration and active control · doi:10.1177/14613484241238595 · Cited by 7 · see citing works
“Subsequently, in literatures, 12,13 the authors cleverly removed the restriction of monotonically increasing nonlinear functions by using the properties of the FLS/NN, and designed a new fuzzy adaptive fi xed-time tracking control strategy and a novel adaptive NN tracking control strategy for…”
Fu, M., Shen, D., Zhao, G., Shang, G., Bai, S., Li, X., & Hao, Z. (2023) · Textile Research Journal · doi:10.1177/00405175221148258 · Cited by 7 · see citing works
Ma, T. (2020) · International Journal of Robust and Nonlinear Control · doi:10.1002/rnc.5001 · Cited by 7 · see citing works
Wang, Z., Zhao, M., Sun, D., Wang, L., Lin, J., & Cheng, S. (2025) · IEEE Transactions on Vehicular Technology · doi:10.1109/tvt.2025.3533114 · Cited by 6 · see citing works
“Other parameter selections were based on the information provided in [1] and [34].”
Event-triggered adaptive neural network control for stochastic nonholonomic systems with input delay
Du, Q., Wang, S., & Zhu, Q. (2025) · Journal of the Franklin Institute · doi:10.1016/j.jfranklin.2025.107955 · Cited by 6 · see citing works
Jia, J., Lan, J., Liu, Y., & Liu, L. (2022) · International Journal of Systems Science · doi:10.1080/00207721.2022.2141596 · Cited by 6 · see citing works
“Several kinds of adaptive neural network control are studied in Liu and Tong (2016), Wang and Huang (2002) and Wang et al. (2020), which involve two kinds of systems: nonlinear pure feedback and nonlinear nonstrict-feedback.”
BackgroundA Differential Hall Effect Based Pressure Sensor
Paliwal, S., & Yenuganti, S. (2021) · Journal of Electrical Engineering and Technology · doi:10.1007/s42835-020-00647-8 · Cited by 6 · see citing works
Kharrat, M., & Alhazmi, H. (2025) · Neural Processing Letters · doi:10.1007/s11063-025-11749-7 · Cited by 5 · see citing works
“The main achievements of this article are summarized as follows: (i) In contrast to established findings [13, 18, 19, 28], this paper explores an adaptive fixed-time tracking control issue within nonstrict-feedback nonlinear systems, contending with both input delay and unknown backlash-like…”
Manual steering and robust adaptive constrained control of ship autopilot
Huang, C., Zhao, Y., Li, B., & Zhang, G. (2024) · Ocean Engineering · doi:10.1016/j.oceaneng.2024.119248 · Cited by 5 · see citing works
Bi, W., Sui, S., Tong, S., & Chen, C. L. P. (2024) · International Journal of Systems Science · doi:10.1080/00207721.2024.2343740 · Cited by 5 · see citing works
“In the case of input delay, a solution to the problem of control design for nonlinear systems is given (Wang et al., 2020).”
BackgroundYang, X., Li, J., Ge, S. S., Liang, X., & Han, T. (2024) · Fractal and Fractional · doi:10.3390/fractalfract8060339 · Cited by 5 · see citing works
Self-citationZeng, C., Su, A., Chen, T., & Chen, S. (2024) · Nonlinear Dynamics · doi:10.1007/s11071-024-10521-1 · Cited by 5 · see citing works
“In [24], the controller design problem was investigated for non-strict-feedback systems with input delay.”
Kharrat, M., Krichen, M., Alhazmi, H., & Mercorelli, P. (2025) · Arabian Journal for Science and Engineering · doi:10.1007/s13369-024-09934-2 · Cited by 5 · see citing works
Fixed-time adaptive fuzzy control for a class of non-strict-feedback uncertain non-linear systems
Yao, Q. (2021) · Transactions of the Institute of Measurement and Control · doi:10.1177/01423312211015124 · Cited by 5 · see citing works
“In this respect, some adaptive neural/fuzzy backstepping controllers have been presented for the control of NF non-linear systems (Chen et al., 2012, 2016, 2018; Tong et al., 2016; Wang et al., 2017, 2018, 2020; Wu et al., 2017; Yang et al., 2016; Zhou et al., 2017).”
BackgroundMethodZhang, X., Zhuang, X., Liu, E., Zhang, A., & Qiu, J. (2023) · Asian Journal of Control · doi:10.1002/asjc.3105 · Cited by 5 · see citing works
Dong, Y., Song, S., Song, X., & Tejado, I. (2024) · Mathematics · doi:10.3390/math12020314 · Cited by 5 · see citing works
Zhou, Y., Zhao, W., Zhu, S., Han, Y., Xing, J.-M., & Qing-hua, Z. (2024) · International Journal of Control · doi:10.1080/00207179.2024.2319752 · Cited by 5 · see citing works
“Meanwhile, according to Wang et al. (2020b), the state variables (cid:20) i of the auxiliary system (13) remain bounded.”
Ju, X., & Zhou, S. (2024) · International Journal of Systems Science · doi:10.1080/00207721.2024.2435575 · Cited by 5 · see citing works
“Some existing methods may not be able to fully estimate these parameters, for example, see (Sun et al., 2021; Tong, Li and Sui, 2016; Tong, Min et al., 2020; H. Wang et al.,2019 ; Xie et al., 2022) and so on.”
InfluentialLiu, X., Gao, C., Chen, M., & Yang, Y. (2020) · IEEE Access · doi:10.1109/access.2020.3027686 · Cited by 5 · see citing works
Predictor-based sampled-data output-feedback control for feedforward nonlinear time-delay systems
Zhang, W., Pan, W., Zhang, X., & Liu, Q. (2024) · Control Theory and Technology · doi:10.1007/s11768-024-00237-3 · Cited by 4 · see citing works
“Specifi-cally, existing results in the open literature involved input delays [14–16], state delays [17–19], output delays [20–23] or any combination of them [24, 25].”
Adaptive Dynamic Surface Control of Epileptor Model Based on Nonlinear Luenberger State Observer
Dolatabadi, M. K., Kamali, M., & Shayegh, F. (2025) · International Journal of Neural Systems · doi:10.1142/s0129065725500224 · Cited by 4 · see citing works
Kharrat, M. (2026) · Communications in Nonlinear Science and Numerical Simulation · doi:10.1016/j.cnsns.2026.109690 · Cited by 4 · see citing works
- Authors
- Wang, H., Liu, S., & Yang, X.
- Affiliations
- Bohai University (CN)
- Harbin Institute of Technology (CN)
- Corresponding author
- Wang, Huanqing
- Venue
- Information Sciences
- Publisher
- Elsevier BV
- Published
- 2020-04
- Volume
- 514
- Pages
- 605-616
- ISSN
- 0020-0255 (ISSN-L)
- OpenAlex
- W2988820939
- Cited by
- 152
- References
- 47
- Citations by year
- 2026: 9 · 2025: 23 · 2024: 24 · 2023: 19 · 2022: 15 · 2021: 23 · 2020: 39
- FWCI
- 10.66
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- Information Sciences (publishedVersion)
- Funders
- Program for Liaoning Innovative Talents in University (LR2016040)
- Taishan Scholar Project of Shandong Province (2015162, tsqn201812093)
- Natural Science Foundation of Liaoning Province (20180550590, 20180550691)
- National Natural Science Foundation of China (61773072, 61773073)
- 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, input delay, auxiliary systems, backstepping control, state feedback stabilization, semi-global uniform ultimate boundedness, intelligent control
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