Adaptive Fuzzy Practical Fixed-Time Tracking Control of Nonlinear Systems
Abstract
This article investigates an adaptive practical fixed-time control strategy for the output tracking control of a class of strict feedback nonlinear systems. By utilizing a backstepping algorithm, finite-time Lyapunov stable theory, and fuzzy logic control, a novel adaptive practical fixed-time controller is constructed. Fuzzy logic systems are introduced to approximate the unknown items of the system. Theoretical analysis proves that under the presented control strategy, the closed-loop system is practically fixed-time stable, and the tracking error converges to a small neighborhood of the origin within a fixed-time interval, in which the convergence time has no connection with the initial states of the system. In the meantime, all the signals of the closed-loop system are bounded. Finally, a numerical example is presented to indicate the feasibility and effectiveness of the proposed method.
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Fixed-time control methods offer a viable approach to address the issue of swift convergence without relying on knowledge of the initial states.
“The advent of fixed-time control methods offers a viable approach to address the issue of swift convergence without relying on knowledge of the initial states [41][42][43].”
From Fixed-Time Adaptive Control for Nonstrict-Feedback Nonlinear Systems with Input Delay and Unknown Backlash-Like Hysteresis (2025) · cites this paper as [41]
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Zhang, C., Zhou, M., Nie, L., Zhang, X., & Su, C. (2023) · Mechanical Systems and Signal Processing · doi:10.1016/j.ymssp.2023.110562 · Cited by 41 · see citing works
Jiang, H., Ma, Q., & Guo, J. C. (2023) · IEEE Transactions on Fuzzy Systems · doi:10.1109/tfuzz.2023.3318572 · Cited by 40 · see citing works
“Although the fixed-time stability results [20], [21], [22] have decreased convergence time, the singularity problem is typically unavoidable in backstepping-based fixed-time controller design.”
BackgroundYu, Y., Zhang, Z., Zhong, M., & Wang, Z. (2021) · Journal of the Franklin Institute · doi:10.1016/j.jfranklin.2021.11.036 · Cited by 39 · see citing works
Liu, W., Ma, Q., & Xu, S. (2023) · IEEE Transactions on Fuzzy Systems · doi:10.1109/tfuzz.2023.3325529 · Cited by 38 · see citing works
“In [12], a novel adaptive practical fixed-time control scheme for a kind of nonlinear systems in strict-feedback form via output-feedback tracking regulation was constructed.”
BackgroundWu, Y., Zhang, G., & Wu, L. (2021) · IEEE Transactions on Fuzzy Systems · doi:10.1109/tfuzz.2021.3065538 · Cited by 38 · see citing works
Brain-Computer Interface Using Brain Power Map and Cognition Detection Network During Flight
Wu, E. Q., Cao, Z., Xiong, P., Song, A., Zhu, L., & Yu, M. (2022) · IEEE/ASME Transactions on Mechatronics · doi:10.1109/tmech.2022.3148141 · Cited by 37 · see citing works
“More complex approximation methods can be used to further reduce the error [33], [34], but it will increase the running cost of the model.”
MethodWang, C., Wang, J., Guo, Q., Liu, Z., & Chen, C. L. P. (2025) · IEEE Transactions on Industrial Electronics · doi:10.1109/tie.2024.3429638 · Cited by 36 · see citing works
Kang, S., Liu, P. X., & Wang, H. (2022) · Nonlinear Dynamics · doi:10.1007/s11071-022-07731-w · Cited by 36 · see citing works
“According to Lemmas 3 and 4 in [32], we have”
BackgroundSelf-citationWang, K., Liu, X., & Jing, Y. (2021) · IEEE Transactions on Neural Networks and Learning Systems · doi:10.1109/tnnls.2021.3083800 · Cited by 36 · see citing works
“In comparison with the existing results [43]–[45], the adopted control techniques are different.”
BackgroundMethodResultSelf-citationCommand filter-based adaptive fuzzy decentralized control for large-scale nonlinear systems
Kang, S., Liu, P. X., & Wang, H. (2021) · Nonlinear Dynamics · doi:10.1007/s11071-021-06779-4 · Cited by 36 · see citing works
Self-citationLv, J., Ju, X., & Wang, C. (2023) · IEEE Transactions on Neural Networks and Learning Systems · doi:10.1109/tnnls.2023.3296194 · Cited by 35 · see citing works
“Unfortunately, finite-time and fixed-time controllers [1], [2], [3], [4], [5], [6], [8], [9], [10], [11], [12], [13], [14] fail to do so.”
MethodNaderolasli, A., Shojaei, K., & Chatraei, A. (2023) · European Journal of Control · doi:10.1016/j.ejcon.2023.100903 · Cited by 34 · see citing works
Wang, C., Guo, Q., Wang, J., Liu, Z., & Chen, C. L. P. (2024) · IEEE Transactions on Circuits and Systems I Regular Papers · doi:10.1109/tcsi.2024.3371050 · Cited by 34 · see citing works
“For example, Chen et al. [16] investigated the fixed-time tracking control for strict-feedback nonlinear systems.”
BackgroundLai, G., Zhang, Y., Liu, Z., Wang, J., Chen, K., & Chen, C. L. P. (2021) · IEEE Transactions on Fuzzy Systems · doi:10.1109/tfuzz.2021.3049902 · Cited by 34 · see citing works
Zhou, Y., Dong, W., Liu, Z., Lv, M., Zhang, W., & Chen, Y. (2022) · IEEE Transactions on Vehicular Technology · doi:10.1109/tvt.2022.3223121 · Cited by 33 · see citing works
“• To address the singularity issue that appears in the derivatives of fractional power terms by incorporating fixed-time control method under backstepping framework [25], [34], this article proposes a non-singular fixed-time tracking control method for UAV, which can keep the control variable to be finite when the tra…”
MethodZhang, J., Fu, Y., & Fu, J. (2023) · IEEE Transactions on Fuzzy Systems · doi:10.1109/tfuzz.2023.3308573 · Cited by 33 · see citing works
Adaptive neural tracking control of high-order nonlinear systems with quantized input
Wang, H., Liu, S., Wang, D., Niu, B., & Chen, M. (2021) · Neurocomputing · doi:10.1016/j.neucom.2021.05.054 · Cited by 33 · see citing works
Self-citationLv, M., Li, Y., Wan, L., Dai, J., & Chang, J. (2022) · IEEE Transactions on Fuzzy Systems · doi:10.1109/tfuzz.2022.3157393 · Cited by 33 · see citing works
“The peculiar feature of fixed-time stability consists in that convergence rate can be prescribed a priori via the so-called fixed-time tracking concept: appropriate modifications to standard design methodologies need to be made to achieve fixed-time tracking [21]–[24], [26].”
BackgroundNovel fast fixed-time sliding mode trajectory tracking control for manipulator
Zhang, X., & Shi, R. (2022) · Chaos Solitons & Fractals · doi:10.1016/j.chaos.2022.112469 · Cited by 31 · see citing works
Xie, H., Jing, Y., Zhang, J., & Dimirovski, G. M. (2024) · IEEE Transactions on Cybernetics · doi:10.1109/tcyb.2024.3390440 · Cited by 30 · see citing works
“…1 − δ 2 ] / 2 ) [21], [27], [34]; 3) different from the practical finite-and fixed-time control schemes with unknown tracking accuracy [3], [4], [5], [6], [7], our approach fulfills trajectory tracking with the predefined settling time and accuracy; and 4) in comparison with the tuning function-based…”
BackgroundResultInfluentialZhu, Z., & Zhu, Q. (2022) · IEEE Transactions on Fuzzy Systems · doi:10.1109/tfuzz.2022.3229073 · Cited by 30 · see citing works
“Lemma 4 (See [25]): For any k1 ∈ and k2 ∈ , the following inequality holds: 0 ≤ |k1| < k2 + k(2) 1(k(2) 1 + k(2) 2) 1 2 .”
Solving Time-Varying Nonsymmetric Algebraic Riccati Equations With Zeroing Neural Dynamics
Simos, T. E., Katsikis, V. N., Mourtas, S. D., & Stanimirović, P. S. (2023) · IEEE Transactions on Systems Man and Cybernetics Systems · doi:10.1109/tsmc.2023.3284533 · Cited by 29 · see citing works
“The tracking control has become one of the most important schemes in past studies [24], [25], [26], [27], [28].”
BackgroundMethodFixed-Time Neural Control of a Quadrotor UAV With Input and Attitude Constraints
Gao, B., Liu, Y., & Liu, L. (2023) · IEEE/CAA Journal of Automatica Sinica · doi:10.1109/jas.2023.123045 · Cited by 28 · see citing works
Chen, J., & He, C. (2021) · IEEE Transactions on Fuzzy Systems · doi:10.1109/tfuzz.2021.3099470 · Cited by 27 · see citing works
Yang, J. L., Liu, B., Zhang, T., Hong, J., & Zhang, H. (2022) · Energy · doi:10.1016/j.energy.2022.125877 · Cited by 27 · see citing works
Fang, X., Fan, H., Liu, L., & Wang, B. Y. (2022) · Nonlinear Dynamics · doi:10.1007/s11071-022-07784-x · Cited by 26 · see citing works
“An adaptive fuzzy fixed-time control strategy was proposed in [33] for a class of SISO nonlinear systems.”
MethodFan, Y., Shi, Y., Mu, D., Liu, J., & Li, Z. (2023) · Ocean Engineering · doi:10.1016/j.oceaneng.2023.115533 · Cited by 26 · see citing works
Fixed-time tracking control for high-order nonlinear systems with unknown time-varying input delay
Zhai, J., Wang, H., & He, Z. (2023) · Applied Mathematics and Computation · doi:10.1016/j.amc.2023.128027 · Cited by 25 · see citing works
Self-citationZeng, D., Cai, C., Zhao, J., & Liu, Y. (2023) · Control Engineering Practice · doi:10.1016/j.conengprac.2023.105789 · Cited by 25 · see citing works
Fuzzy synchronization of fractional-order chaotic systems using finite-time command filter
Alassafi, M. O., Ha, S., Alsaadi, F. E., Ahmad, A. M., & Cao, J. (2021) · Information Sciences · doi:10.1016/j.ins.2021.08.005 · Cited by 25 · see citing works
Command filtered fixed‐time control for a class of multi‐agent systems with sensor faults
Dong, G., Li, X., Yao, D., Li, H., & Lu, R. (2021) · International Journal of Robust and Nonlinear Control · doi:10.1002/rnc.5793 · Cited by 25 · see citing works
“Lemma 1 (See the work of Chen et al. 47 ).”
Yuan, F., Ma, Y., Liu, Y., Lan, J., & Xu, T. (2021) · Journal of the Franklin Institute · doi:10.1016/j.jfranklin.2021.09.027 · Cited by 25 · see citing works
Position Control of a Mobile Robot through Deep Reinforcement Learning
Quiroga, F., Hermosilla, G., Farías, G., Fábregas, E., & Montenegro, G. (2022) · Applied Sciences · doi:10.3390/app12147194 · Cited by 25 · see citing works
“Additionally, different approaches are available, such as finite-time adaptive fault-tolerant control [8] and adaptive fuzzy control algorithms [9], which have been used for nonlinear systems in mobile robotics, as in [10].”
MethodWang, C., Wang, J., Du, Y., Zhang, C., Liu, Z., & Chen, C. L. P. (2023) · Information Sciences · doi:10.1016/j.ins.2023.03.010 · Cited by 25 · see citing works
Yao, Y., Tan, J., Yao, Y., Zhang, X., & Chen, P. (2023) · Neurocomputing · doi:10.1016/j.neucom.2023.127200 · Cited by 24 · see citing works
Non-singular terminal sliding mode cooperative guidance law under impact angle constraint
Dong, F., Zhang, X., & Tan, P. (2024) · Journal of the Franklin Institute · doi:10.1016/j.jfranklin.2024.107079 · Cited by 24 · see citing works
Cuan, Z., & Ding, D. (2023) · IEEE Transactions on Network Science and Engineering · doi:10.1109/tnse.2023.3292843 · Cited by 23 · see citing works
“6 reveals simulation results using the method in [26], which inferred that the tracking performance can not be assured and the control objective fails to be accomplished.”
MethodResultInfluentialLi, J., Wang, Q., & Fang, Y. (2024) · Communications in Nonlinear Science and Numerical Simulation · doi:10.1016/j.cnsns.2024.108085 · Cited by 23 · see citing works
Adaptive fixed-time control for Lorenz systems
Wang, H., Yue, H., Liu, S., & Li, T. (2020) · Nonlinear Dynamics · doi:10.1007/s11071-020-06061-z · Cited by 23 · see citing works
Self-citationWu, Z., Guo, J., Liu, B., Ni, J., & Bu, X. (2021) · International Journal of Robust and Nonlinear Control · doi:10.1002/rnc.5582 · Cited by 23 · see citing works
Adaptive fixed‐time tracking control for stochastic pure‐feedback nonlinear systems
Liang, Y., Li, Y., & Hou, Z. (2021) · International Journal of Adaptive Control and Signal Processing · doi:10.1002/acs.3285 · Cited by 23 · see citing works
“s [?̇?(x(v)) + k1ξ(x(v)) + k2ξq(x(v)) − ρ]dv ≤ 0 (13) According to Lemma 1, for ∀t ≥ 0, we have ?̇?(x(t)) ≤ −k1ξ(x(t)) − k2ξ(x(t)) + ρ (14) Therefore, for ∀φ ∈ (0, 1), one has ?̇?(x(t)) ≤ − k1ξ(x(t)) − φk2ξ(x(t)) − (1 − φ)k2ξ(x(t)) + ρ (15) or ?̇?(x(t)) ≤ − φk1ξ(x(t)) − (1 − φ)k1ξ(x(t)) − k2ξ(x(t)) + ρ (16)”
Song, Z., & Sun, K. (2021) · Journal of the Franklin Institute · doi:10.1016/j.jfranklin.2021.08.036 · Cited by 23 · see citing works
Wang, J., Wang, C., Zhang, C., Liu, Z., & Chen, C. L. P. (2023) · Fuzzy Sets and Systems · doi:10.1016/j.fss.2023.108563 · Cited by 23 · see citing works
An, S., Wang, X., Wang, L., & He, Y. (2023) · Ocean Engineering · doi:10.1016/j.oceaneng.2023.115158 · Cited by 23 · see citing works
Adaptive Approximation Tracking Control of a Continuum Robot With Uncertainty Disturbances
Xu, S., & He, B. (2022) · IEEE Transactions on Cybernetics · doi:10.1109/tcyb.2022.3202540 · Cited by 22 · see citing works
“Considering the feed-back nonlinearity of systems, an adaptive fixed-time controller was proposed to achieve effective tracking [30].”
MethodYang, Y., Niu, Y., & Zhang, Z. (2020) · IET Control Theory and Applications · doi:10.1049/iet-cta.2020.0689 · Cited by 22 · see citing works
He, Z., Wang, G., & Fan, Y. (2023) · Journal of the Franklin Institute · doi:10.1016/j.jfranklin.2023.07.043 · Cited by 21 · see citing works
Zhu, F., Zhao, Y., Fu, Y., & Dinh, T. N. (2023) · IEEE Transactions on Network Science and Engineering · doi:10.1109/tnse.2023.3346442 · Cited by 21 · see citing works
“For example, the fuzzy logic system is taken to deal with nonlinear terms existing in the strict-feedback systems in [6], [7], [8], [9], where the controllers with adaptive parameters is constructed by backstepping scheme.”
Method
- Authors
- Chen, M., Wang, H., & Liu, X.
- Affiliations
- University of Science and Technology Liaoning (CN)
- Bohai University (CN)
- Lakehead University (CA)
- Venue
- IEEE Transactions on Fuzzy Systems
- Published
- 2021-03
- Volume
- 29
- Issue
- 3
- Pages
- 664-673
- ISSN
- 1063-6706 (ISSN-L)
- 1941-0034
- OpenAlex
- W2995874545
- Cited by
- 405
- References
- 38
- Citations by year
- 2026: 35 · 2025: 77 · 2024: 65 · 2023: 86 · 2022: 65 · 2021: 66 · 2020: 11
- FWCI
- 19.60
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- IEEE Transactions on Fuzzy Systems (publishedVersion)
- Funders
- National Natural Science Foundation of China (61403177, 61773072)
- Natural Science Foundation of Liaoning Province (20180550319)
- Education Department Project of Liaoning Province of China (2019LNJC09)
- Taishan Scholar Project of Shandong Province of China (tsqn201812093)
- 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 › Physics and Astronomy › Statistical and Nonlinear Physics › Chaos control and synchronization
- Keywords
- output tracking control, strict-feedback nonlinear systems, adaptive control, fuzzy systems, backstepping control, finite-time Lyapunov stability, signal boundedness
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