Adaptive Finite-Time Tracking Control of Nonlinear Systems With Dynamics Uncertainties
Abstract
In this article, the problem of adaptive backstepping finite-time tracking control is investigated for a class of strict-feedback nonlinear systems with unmodeled dynamics and dynamic disturbances. A modified finite-time dynamic signal is first introduced to dominate the dynamic disturbance. By using the adaptive control, backstepping technique, and finite-time stability theory, an adaptive finite-time tracking controller is developed. Under the proposed control scheme, the finite-time tracking performance and the boundedness property of all signals in the closed-loop system are ensured. Finally, simulation results check the effectiveness of the suggested approach.
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Traditional adaptive finite-time control methods depend on initial conditions, which makes the settling time difficult to predict in advance.
“Additionally, a significant limitation of traditional adaptive finite-time control methods is their dependency on initial conditions, which makes it difficult to predict the settling time in advance.”
From Fixed-Time Adaptive Control for Nonstrict-Feedback Nonlinear Systems with Input Delay and Unknown Backlash-Like Hysteresis (2025) · cites this paper as [38]
The proposed fixed-time controller overcomes the prevailing limitations associated with convergence time dependence on initial states in prior work.
“This work overcomes the prevailing limitations associated with convergence time dependence on initial states in [38,39] by devising a fixed-time controller.”
From Fixed-Time Adaptive Control for Nonstrict-Feedback Nonlinear Systems with Input Delay and Unknown Backlash-Like Hysteresis (2025) · cites this paper as [38]
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“There are methods such as neural network identifier [12], fuzzy logic system [13], Adaptive Fixed-time Tracking [14], [15], etc. to deal with the nonlinear system with dynamic uncertainty of the system.”
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“…a systematic approach to addressing the control issue for strict feedback nonlinear systems of integer order [17], and adaptive control has been demonstrated to be a reliable and efficient method of controlling systems with uncertain parameters [18], [19], [20], [21], [22], [23], [24], [25].”
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“For deterministic systems, FTS has been extensively studied in the pursuit of faster convergence, higher accuracy, and better anti-disturbances [31], [32], [33], [34], [35].”
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“For strict-feedback nonlinear systems with dynamics uncertainties, a nonsingularity finite-time adaptive tracking control scheme was proposed in [20].”
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“In the realm of adaptive PFT control, significant exploration has been conducted on sliding mode control (SMC) [9] and backstepping control [10] due to their well-established robustness against perturbations and uncertainties.”
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“Remark 4: Unlike [42] where a class of disturbances satisfying Lipschitz restrictive conditions are considered, the disturbances considered in this article is more general.”
BackgroundInfluentialAdaptive Finite Time Output Feedback Bipartite Tracking Control for Nonlinear Multiagent Systems
Wang, X., Niu, B., Gao, Y., & Shang, Z. (2023) · IEEE Transactions on Automation Science and Engineering · doi:10.1109/tase.2023.3325064 · Cited by 23 · see citing works
“Instead, the finite-time control method [23], [24], [25] can rapidly make system transient performances achieved and has the shorter convergent time.”
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“In addition, numerous con-trol schemes have been successfully implemented in nonlinear systems with uncertainties to improve the performance of the system [26], [27], [28], [29], [30], [31], [32].”
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“…of treatment processes and the need to satisfy multi-objective demands, compared to the traditional PID controllers, intelligent control methods hold greater promise in industrial applications, which has attracted substantial attention [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13].”
BackgroundLi, N., Liu, X., Liu, C., He, W., & Wang, H. (2024) · IEEE Transactions on Automation Science and Engineering · doi:10.1109/tase.2024.3392877 · Cited by 21 · see citing works
“Recently, with the development of finite-time control, a large number of adaptive methods have been proposed for uncertain nonlinear systems, which are further extensions of backstepping method (see [39], [40], [46]).”
Self-citationHong, M., Gu, X., Xia, L., & Guo, Y. H. (2024) · Journal of the Franklin Institute · doi:10.1016/j.jfranklin.2024.107183 · Cited by 21 · see citing works
Li, R.-B., Feng, Z., & Shi, Y. (2025) · IEEE Transactions on Automatic Control · doi:10.1109/tac.2025.3595002 · Cited by 21 · see citing works
Zhu, Y., Liu, J., Yu, J., & Wang, Q. (2023) · IEEE Transactions on Circuits & Systems II Express Briefs · doi:10.1109/tcsii.2023.3295840 · Cited by 21 · see citing works
“Many scholars have conducted extensive research in recent years, proposing adaptive control [3], [4], finite-time control [5], [6], backstepping control [7], [8], and other control methods.”
BackgroundHan, Y., & Sun, J. (2024) · International Journal of Robust and Nonlinear Control · doi:10.1002/rnc.7266 · Cited by 21 · see citing works
Li, T., Chen, X., Liu, J., & Yu, J. (2024) · IEEE Transactions on Industrial Electronics · doi:10.1109/tie.2024.3379644 · Cited by 21 · see citing works
“For example, adaptive control [6], [7], [8] and fuzzy logic system (FLS) control [9], [10], [11] can effectively deal with the unknown nonlinear functions in the systems.”
BackgroundQian, W., Wu, Y., & Shen, B. (2024) · IEEE/CAA Journal of Automatica Sinica · doi:10.1109/jas.2024.124419 · Cited by 21 · see citing works
Adaptive fixed-time tracking control of nonlinear systems with unmodeled dynamics
Wang, H., & Ai, Z. (2024) · Nonlinear Dynamics · doi:10.1007/s11071-024-10048-5 · Cited by 20 · see citing works
“Assumption 1 [27] The given anticipant signal y d and its derivative with respect to time up to the n th order are bounded and smooth.”
InfluentialSelf-citationAdaptive Prescribed-Time Filtered Control Design for a Full-State Constrained Nonlinear System
Wang, F., Gao, Z., Xie, X., Zhou, C., & Hua, C. (2024) · IEEE Transactions on Cybernetics · doi:10.1109/tcyb.2024.3486721 · Cited by 20 · see citing works
Predefined-Time Event-Triggered Adaptive Tracking Control for Uncertain Time Delay Nonlinear Systems
Gao, L., Wang, Z., Huang, X., & Shen, H. (2025) · IEEE Transactions on Automation Science and Engineering · doi:10.1109/tase.2025.3566407 · Cited by 20 · see citing works
“For two types of uncertain nonlinear systems, [17] explored adaptive backstepping FT tracking control, while [18] designed a continuous adaptive global FT stabilization controller.”
Liu, Y., Liu, J., & Yu, J. (2023) · IEEE Transactions on Circuits & Systems II Express Briefs · doi:10.1109/tcsii.2023.3314730 · Cited by 20 · see citing works
“In [10], a new singularity avoidance control method is proposed by combining a novel piecewise function and adaptive control.”
MethodSu, Z., Huang, B., Miao, J., & Lin, X. (2024) · International Journal of Robust and Nonlinear Control · doi:10.1002/rnc.7363 · Cited by 20 · see citing works
Hou, Q., & Dong, J. (2024) · IEEE Transactions on Automation Science and Engineering · doi:10.1109/tase.2024.3408161 · Cited by 20 · see citing works
“On the other hand, finite time control has achieved impressive results [11], [12], [13], [14].”
Aghayan, Z. S., Alfi, A., Mousavi, Y., & Fekih, A. (2025) · IEEE Transactions on Systems Man and Cybernetics Systems · doi:10.1109/tsmc.2024.3508573 · Cited by 19 · see citing works
“Additionally, failure to take into account system uncertainties, such as parameter perturbations, can seriously hamper the controller ability to mitigate their effects on system stability and dynamic behavior [35], [36], [37], [38], [39].”
Kang, S., Liu, P. X., & Wang, H. (2023) · IEEE Transactions on Circuits & Systems II Express Briefs · doi:10.1109/tcsii.2023.3291360 · Cited by 19 · see citing works
“Therefore, a great number of finite-time control schemes have been established for kinds of nonlinear systems in recent years [13], [14], [15], [16], [17].”
BackgroundSelf-citationWang, C., Zhan, H., Guo, Q., & Li, T. (2024) · IEEE/ASME Transactions on Mechatronics · doi:10.1109/tmech.2024.3410299 · Cited by 19 · see citing works
Zhang, C., Wang, J., Fan, S., & Chen, Y. (2025) · Nonlinear Dynamics · doi:10.1007/s11071-024-10845-y · Cited by 19 · see citing works
“Then the function V will converge to the origin in fast finite time under any initial conditions, and the convergence time satisfies Lemma 2 [43] For the system ˙ x = f ( x ) , if there exist and a positive definite function V ( x ) such that for all x satisfies the inequality then the system ˙ x = f (…”
Parameter tuning of modified adaptive backstepping controller for strict-feedback nonlinear systems
Liu, X., Li, N., Liu, C., Fu, J., & Wang, H. (2024) · Automatica · doi:10.1016/j.automatica.2024.111726 · Cited by 18 · see citing works
Self-citationWang, F., & Long, L. (2025) · IEEE Transactions on Systems Man and Cybernetics Systems · doi:10.1109/tsmc.2025.3549528 · Cited by 19 · see citing works
“For example, adaptive finite-time and fixed-time control problems have been studied for non-switched nonlinear systems in [40] and [41] ], p , ı ∈ N , ˙ α i = 0 , i = 1 , . . . , n − 1 hold.”
Practical Prescribed-Time Trajectory Tracking Control for Marine Surface Vehicles
Zeng, D., Zeng, B., Liu, Y., Zhao, J., & Cai, C. (2024) · IEEE Transactions on Circuits & Systems II Express Briefs · doi:10.1109/tcsii.2024.3410466 · Cited by 18 · see citing works
Zhang, Z., Wen, C., Chen, L., Song, Y., Peng, B., & Feng, G. (2024) · IEEE Transactions on Cybernetics · doi:10.1109/tcyb.2024.3498071 · Cited by 18 · see citing works
“Lemma 1 [40]: For any x , y ∈ R , one has xy ≤ ( r p / p ) | x | p + ( 1 / qr q ) | y | q , with r > 0, p , q > 1, and The primary objective of this scheme is to effectively schedule the sampling and communication of nonlinear interconnected systems, especially under sensor and actuator faults, and…”
Zhao, J., Zhang, B., Hu, Y., Yu, D., Sun, Z., & Chen, C. L. P. (2025) · IEEE Transactions on Systems Man and Cybernetics Systems · doi:10.1109/tsmc.2024.3515040 · Cited by 18 · see citing works
“2785 γ. Lemma 7 [22]: Take into account the set (cid”
InfluentialLuo, S., Song, Y., Lewis, F. L., & Deng, G. (2023) · IEEE Transactions on Fuzzy Systems · doi:10.1109/tfuzz.2023.3331386 · Cited by 17 · see citing works
“Besides, the prescribed performance function (PPF) [29], [30], [31], [32] which guarantees the constrained boundary of error are proposed to put into the backstepping framework.”
BackgroundPreset-Trajectory-Based Tracking Control of a Class of Mismatched Uncertain Systems
Shi, W., Keliris, C., Hou, M., Duan, G., & Polycarpou, M. M. (2024) · IEEE Transactions on Automatic Control · doi:10.1109/tac.2024.3433070 · Cited by 17 · see citing works
“Assumption 2 is commonly used in the backstepping method [4], [13], [15].”
Predefined-Time Adaptive Neural Control for Nonlinear Systems With Unknown Interconnections
Han, H., Ji, W., Liu, Z., Sun, H., & Qiao, J. (2025) · IEEE Transactions on Cybernetics · doi:10.1109/tcyb.2025.3552926 · Cited by 16 · see citing works
“Table I shows the detailed comparison results of the control performance metrics for PID control [41], FITC [15], and FXTC [20], and PTANC.”
InfluentialYu, Z., Liu, P. X., Ling, S., & Wang, H. (2023) · International Journal of Systems Science · doi:10.1080/00207721.2023.2271621 · Cited by 16 · see citing works
“Typically, the designer anticipates that the control system will achieve the desired performance within a finite time (Gao et al., 2023; Kang et al.,2023 ; Li et al., 2023; Wang et al., 2022).”
BackgroundSelf-citationWan, L., Zhang, H., Sun, J., Liu, Z., & Xie, X. (2024) · IEEE Transactions on Cybernetics · doi:10.1109/tcyb.2024.3414650 · Cited by 16 · see citing works
“As an important research problem, the tracking con-trol schemes were studied for many nonlinear uncertainty systems [18], [19], [20], [21], [22], [23].”
BackgroundSelf-citationActive Disturbance Rejection Based Adaptive Dynamic Surface Control for Nonlinear Systems
Zhou, L., Sun, Y., He, Y. M., Li, H., & Zhang, X. (2024) · IEEE Transactions on Automation Science and Engineering · doi:10.1109/tase.2024.3487131 · Cited by 15 · see citing works
“Remark 5: One critical problem with conventional back-stepping control method [37], [38], [39], [40], [41] is that the complexity of controller implementation increases explosively as the system order grows.”
Xu, K. W., Wang, H., & Liu, P. X. (2023) · International Journal of Robust and Nonlinear Control · doi:10.1002/rnc.6636 · Cited by 15 · see citing works
“Compared with (30), when |z1| < ε10, there appears an additional term Λ = K11(z(4) 1∕4) 3∕4(1 − ∑n+1 j=1 bj(z(4) 1∕ε 4 10) j−3∕4) in (31).”
Self-citationAdaptive fault-tolerant consensus control of multi-agent systems with event-triggered inputs
Wang, Y., Yang, Y., & Wu, L. (2023) · Information Sciences · doi:10.1016/j.ins.2023.119594 · Cited by 15 · see citing works
Zhao, B., & Yao, X. (2024) · IEEE Transactions on Automation Science and Engineering · doi:10.1109/tase.2024.3506931 · Cited by 14 · see citing works
“In [17], finite-time dynamic signals are introduced to control external disturbances.”
Adaptive Control of Strict-Feedback Nonlinear Systems Under Denial-of-Service: A Synthetic Analysis
Gao, R., Huang, J., Su, X., & Zhao, L. (2023) · IEEE Transactions on Information Forensics and Security · doi:10.1109/tifs.2023.3347969 · Cited by 14 · see citing works
“can be divided into denial-of-service (DoS) attacks ([5], [6], [7], [8], [9], [10]), replay attacks ([11], [12], [13], [14]) and false data injection (FDI) attacks ([15], [16], [17], [18], [19]).”
BackgroundZhang, H., Yan, Y., & Sun, J. (2024) · IEEE Transactions on Systems Man and Cybernetics Systems · doi:10.1109/tsmc.2024.3355909 · Cited by 14 · see citing works
“Al-Dabooni and Wunsch [12], [13] presented a novel powerful model-free online [14] action dependence heuristic dynamic programming design that implements powerful temporal difference (TD)-learning over regular TD learning.”
BackgroundSelf-citationXie, Y., Ma, Q., & Ahn, C. K. (2024) · IEEE Transactions on Systems Man and Cybernetics Systems · doi:10.1109/tsmc.2024.3417977 · Cited by 14 · see citing works
Kong, L., He, W., & Silvestre, C. (2025) · Automatica · doi:10.1016/j.automatica.2025.112123 · Cited by 14 · see citing works
Adaptive control and almost disturbance decoupling for uncertain HOFA nonlinear systems
Wang, N., Liu, X., Liu, C., Wang, H., & Zhou, Y. (2023) · International Journal of Adaptive Control and Signal Processing · doi:10.1002/acs.3630 · Cited by 14 · see citing works
Self-citationQin, C., Hou, S., Pang, M., Wang, Z., & Qiu, Q. (2026) · IEEE Internet of Things Journal · doi:10.1109/jiot.2026.3705869 · Cited by 13 · see citing works
“Tracking control [5], [6], [7] plays an important role in ensuring stability and performance by enabling system outputs to follow desired trajectories despite uncertainties and disturbances.”
Distributed Finite-Time Observer for Multiple Line Outages Detection in Power Systems
Chen, Y., Liu, Z., Wen, G., & Wang, Y. (2023) · IEEE Transactions on Systems Man and Cybernetics Systems · doi:10.1109/tsmc.2023.3329582 · Cited by 13 · see citing works
“Recently, some efforts have been made in the field of finite-time control for nonlinear complex systems with uncertainties, including feedback control [23], sliding mode control [24], tracking control [25], and more.”
BackgroundShi, M., Yu, J., & Zhang, T. (2023) · Asian Journal of Control · doi:10.1002/asjc.3177 · Cited by 13 · see citing works
“In addition, adaptive control of MIMO nonlinear systems with block-structure including robotic systems had also been widely studied in other previous research [36–40].”
ResultYang, X., Ju, X., Shi, P. J., & Wen, G. (2023) · IEEE Transactions on Neural Networks and Learning Systems · doi:10.1109/tnnls.2023.3321761 · Cited by 13 · see citing works
Prescribed-Time Stabilization of High-Order Polynomial Time-Varying Nonlinear Systems
Li, J.-J., Sun, Z., Wen, C., & Chen, C. (2025) · IEEE Transactions on Cybernetics · doi:10.1109/tcyb.2024.3524366 · Cited by 12 · see citing works
“…which is able to make states approach zero within a finite-time horizon and stay there thereafter, has been typically researched for various nonlinear systems due to its advantages of rapidness, robustness and precision (see [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16]).”
Yao, Z., Li, D., Kan, Z., & Xiang, J. (2025) · IEEE Transactions on Aerospace and Electronic Systems · doi:10.1109/taes.2025.3540785 · Cited by 12 · see citing works
Gao, H., Wang, J., Xia, Y., Zhang, J., & Cui, B. (2024) · IEEE Transactions on Automation Science and Engineering · doi:10.1109/tase.2024.3390007 · Cited by 12 · see citing works
“In order to achieve this aim, the finite-time control strategy with strong robustness and fast convergence rate was proposed for nonlinear uncertain systems [20], [21], [22], [23], [24], [25].”
Fixed-Time Angle Tracking Control for Multi-DOF Manipulator Driven by Pneumatic Artificial Muscles
Liu, X., Zhang, X., Xu, F. Y., Gu, S., & Zhang, J. (2024) · IEEE Transactions on Industrial Electronics · doi:10.1109/tie.2024.3451156 · Cited by 12 · see citing works
“To deal with the unmodeled dynamics and unknown disturbances, an adaptive finite-time tracking control method is designed for a class of strict-feedback nonlinear systems [19].”
Qin, Y. F., Cao, L., Ren, H., Liang, H., & Pan, Y. (2023) · Journal of the Franklin Institute · doi:10.1016/j.jfranklin.2023.12.024 · Cited by 11 · see citing works
Amirparast, A., & Sani, S. K. H. (2024) · International Journal of Robust and Nonlinear Control · doi:10.1002/rnc.7278 · Cited by 11 · see citing works
“9 One of the most effective control techniques for solving the robust control problem of affine systems is through optimal approaches.”
BackgroundWang, H., Meng, Z., Ma, J., & Zhao, X. (2024) · Neurocomputing · doi:10.1016/j.neucom.2024.127590 · Cited by 11 · see citing works
Self-citationAdaptive Dynamic Programming-Based Fixed-Time Optimal Control for Wheeled Mobile Robot
Wang, C., Zhan, H., Guo, Q., & Li, T. (2024) · IEEE Robotics and Automation Letters · doi:10.1109/lra.2024.3504314 · Cited by 11 · see citing works
“It has been demonstrated in numerous studies that finite-time control exhibits better robustness, faster convergence, and superior anti-interference capability compared to non-finite-time control methods [13], [14], [15].”
Li, S., Zhou, B., Shi, Y., & Duan, G. (2024) · IEEE Transactions on Cybernetics · doi:10.1109/tcyb.2024.3498838 · Cited by 11 · see citing works
Borah, K. J., & Kumar, K. D. (2023) · International Journal of Robust and Nonlinear Control · doi:10.1002/rnc.6913 · Cited by 11 · see citing works
“Researchers, 22 investigate the neural-network-based adaptive predefined-time tracking control problem for switched nonlinear systems.”
BackgroundShao, X., & Ye, D. (2023) · International Journal of Robust and Nonlinear Control · doi:10.1002/rnc.6821 · Cited by 10 · see citing works
“+ i,j2 ( i,j2 + 1 2 i,j2 ) − β i,j ̇̂ β i,j + 1 2 b(2)i,jd ∗2 i,j } (48)”
Background
- Authors
- Wang, H., Xu, K., & Zhang, H.
- Affiliations
- Bohai University (CN)
- Beijing Jiaotong University (CN)
- State Key Laboratory of Synthetical Automation for Process Industries (CN)
- Venue
- IEEE Transactions on Automatic Control
- Published
- 2023-09
- Volume
- 68
- Issue
- 9
- Pages
- 5737-5744
- ISSN
- 0018-9286 (ISSN-L)
- 1558-2523
- 2334-3303
- OpenAlex
- W4312809224
- Cited by
- 342
- References
- 36
- Citations by year
- 2026: 69 · 2025: 108 · 2024: 111 · 2023: 54
- FWCI
- 27.48
- Type
- journal article
- Language
- en
- Open access
- closed
- Licence
- Locations
- IEEE Transactions on Automatic Control (publishedVersion)
- Funders
- National Natural Science Foundation of China (62173046, 51939001, 61976033)
- National Key R&D Program of China (2018YFA0702200)
- Fundamental Research Funds for the Central Universities (2022JBMC029)
- 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
- finite-time tracking control, adaptive control, backstepping control, strict-feedback nonlinear systems, unmodeled dynamics, dynamic disturbances, finite-time stability, signal boundedness
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