Note on adaptive prescribed-time stabilization of nonlinear systems with uncertainty

Mengyuan Dai; Chunyan Zhang; Yingli Zhang; Lichao Feng; Mengyuan Dai; Chunyan Zhang; Yingli Zhang; Lichao Feng

doi:10.3934/nhm.2025034

Networks and Heterogeneous Media

2025, Volume 20, Issue 3: 798-817. doi: 10.3934/nhm.2025034

Previous Article Next Article

Research article Special Issues

Note on adaptive prescribed-time stabilization of nonlinear systems with uncertainty

1.
College of Science, North China University of Science and Technology, Tangshan 063210, China
2.
Hebei Key Laboratory of Data Science and Application, North China University of Science and Technology, Tangshan 063210, China

Received: 23 March 2025 Revised: 17 June 2025 Accepted: 01 July 2025 Published: 09 July 2025

Krishnamurthy et al. investigated the adaptive output feedback control for prescribed-time stability (PTS) of nonlinear uncertain systems on $ [0, T) $, where $ T > 0 $. However, there are special constraints on system structure, and the PTS issue is considered on a finite interval $ [0, T) $. For systems without required constraints, the existing adaptive output feedback control seems not to be applicable; the PTS issue on $ [0, \infty) $ is more practical than that on $ [0, T) $. Motivated by the above, an improved observer and controller were proposed to achieve PTS for nonlinearly uncertain systems with lower-triangular linear growth condition on uncertainties. Compared with the existing work, we emphasized the subsequent contributions: 1) Relax the original structure constraint of objective system; 2) achieve PTS on $ [T, \infty) $; and 3) keep the proposed controller bounded. The effectiveness of the controller was verified by numerical simulations across varying initial conditions and prescribed times.
- prescribed-time stabilization,
- uncertain systems,
- output feedback control
Citation: Mengyuan Dai, Chunyan Zhang, Yingli Zhang, Lichao Feng. Note on adaptive prescribed-time stabilization of nonlinear systems with uncertainty[J]. Networks and Heterogeneous Media, 2025, 20(3): 798-817. doi: 10.3934/nhm.2025034

Related Papers:

Abstract

Krishnamurthy et al. investigated the adaptive output feedback control for prescribed-time stability (PTS) of nonlinear uncertain systems on $ [0, T) $, where $ T > 0 $. However, there are special constraints on system structure, and the PTS issue is considered on a finite interval $ [0, T) $. For systems without required constraints, the existing adaptive output feedback control seems not to be applicable; the PTS issue on $ [0, \infty) $ is more practical than that on $ [0, T) $. Motivated by the above, an improved observer and controller were proposed to achieve PTS for nonlinearly uncertain systems with lower-triangular linear growth condition on uncertainties. Compared with the existing work, we emphasized the subsequent contributions: 1) Relax the original structure constraint of objective system; 2) achieve PTS on $ [T, \infty) $; and 3) keep the proposed controller bounded. The effectiveness of the controller was verified by numerical simulations across varying initial conditions and prescribed times.

References

[1]	M. B. Brilliant, Theory of the Analysis of Nonlinear Systems, Technical report, Massachusetts Institute of Technology, Research Laboratory of Electronics, Cambridge, 1958. https://doi.org/10.21236/ad0216209
[2]	S. Sastry, Nonlinear Systems: Analysis, Stability, and Control, Springer Science & Business Media, New York, 1999. https://doi.org/10.1007/978-1-4757-3108-8
[3]	J. Tsinias, A theorem on global stabilization of nonlinear systems by linear feedback, Syst. Control Lett., 17 (1991), 357–362. https://doi.org/10.1016/0167-6911(91)90135-2 doi: 10.1016/0167-6911(91)90135-2
[4]	W. Yu, S. Liu, F. Zhang, Adaptive output feedback regulation for a class of uncertain nonlinear systems, Int. J. Robust Nonlinear Control, 25 (2015), 2241–2253. https://doi.org/10.1002/rnc.3192 doi: 10.1002/rnc.3192
[5]	L. Xing, C. Wen, Y. Zhu, H. Su, Z. Liu, Output feedback control for uncertain nonlinear systems with input quantization, Automatica, 65 (2016), 191–202. https://doi.org/10.1016/j.automatica.2015.11.028 doi: 10.1016/j.automatica.2015.11.028
[6]	J. Zhai, W. Ai, S. Fei, Global output feedback stabilisation for a class of uncertain nonlinear systems, IET Control Theory Appl., 7 (2013), 305–313. https://doi.org/10.1049/iet-cta.2011.0505 doi: 10.1049/iet-cta.2011.0505
[7]	Y. Liu, Global asymptotic regulation via time-varying output feedback for a class of uncertain nonlinear systems, SIAM J. Control Optim., 51 (2013), 4318–4342. https://doi.org/10.1137/120862570 doi: 10.1137/120862570
[8]	X. Zhang, W. Lin, C. Qian, Universal adaptive control via output feedback for nonlinear systems with parametric and measurement uncertainty, in 2018 IEEE Conference on Decision and Control (CDC), (2018), 5530–5535. https://doi.org/10.1109/CDC.2018.8619162
[9]	H. Li, X. Zhang, L. Chang, Output feedback regulation of a class of triangular structural nonlinear systems with unknown measurement sensitivity, Int. J. Syst. Sci., 50 (2019), 2486–2496. https://doi.org/10.1080/00207721.2019.1671529 doi: 10.1080/00207721.2019.1671529
[10]	C. Qian, W. Lin, Output feedback control of a class of nonlinear systems: A nonseparation principle paradigm, IEEE Trans. Autom. Control, 47 (2002), 1710–1715. https://doi.org/10.1109/TAC.2002.803542 doi: 10.1109/TAC.2002.803542
[11]	C. Qian, H. Du, Global output feedback stabilization of a class of nonlinear systems via linear sampled-data control, IEEE Trans. Autom. Control, 57 (2012), 2934–2939. https://doi.org/10.1109/TAC.2012.2193707 doi: 10.1109/TAC.2012.2193707
[12]	A. A. Prasov, H. K. Khalil, A nonlinear high-gain observer for systems with measurement noise in a feedback control framework, IEEE Trans. Autom. Control, 58 (2012), 569–580. https://doi.org/10.1109/TAC.2012.2218063 doi: 10.1109/TAC.2012.2218063
[13]	C. C. Chen, C. Qian, Z. Y. Sun, Y. W. Liang, Global output feedback stabilization of a class of nonlinear systems with unknown measurement sensitivity, IEEE Trans. Autom. Control, 63 (2017), 2212–2217. https://doi.org/10.1109/TAC.2017.2759274 doi: 10.1109/TAC.2017.2759274
[14]	L. Liang, X. Yan, T. Shen, Global output-feedback adaptive stabilization for uncertain nonlinear systems with polynomial growth nonlinearities, Syst. Control Lett., 165 (2022), 105269. https://doi.org/10.1016/j.sysconle.2022.105269 doi: 10.1016/j.sysconle.2022.105269
[15]	X. Huang, W. Lin, B. Yang, Global finite-time stabilization of a class of uncertain nonlinear systems, Automatica, 41 (2005), 881–888. https://doi.org/10.1016/j.automatica.2004.11.036 doi: 10.1016/j.automatica.2004.11.036
[16]	Y. Shen, Y. Huang, Global finite-time stabilisation for a class of nonlinear systems, Int. J. Syst. Sci., 43 (2012), 73–78. https://doi.org/10.1080/00207721003770569 doi: 10.1080/00207721003770569
[17]	Z. Y. Sun, L. R. Xue, K. Zhang, A new approach to finite-time adaptive stabilization of high-order uncertain nonlinear system, Automatica, 58 (2015), 60–66. https://doi.org/10.1016/j.automatica.2015.05.005 doi: 10.1016/j.automatica.2015.05.005
[18]	S. Ding, A. Levant, S. Li, Simple homogeneous sliding-mode controller, Automatica, 67 (2016), 22–32. https://doi.org/10.1016/j.automatica.2016.01.017 doi: 10.1016/j.automatica.2016.01.017
[19]	Z. Y. Sun, Y. Shao, C. C. Chen, Fast finite-time stability and its application in adaptive control of high-order nonlinear system, Automatica, 106 (2019), 339–348. https://doi.org/10.1016/j.automatica.2019.05.018 doi: 10.1016/j.automatica.2019.05.018
[20]	A. Polyakov, Nonlinear feedback design for fixed-time stabilization of linear control systems, IEEE Trans. Autom. Control, 57 (2011), 2106–2110. https://doi.org/10.1109/TAC.2011.2179869 doi: 10.1109/TAC.2011.2179869
[21]	C. Hu, J. Yu, Z. Chen, H. Jiang, T. Huang, Fixed-time stability of dynamical systems and fixed-time synchronization of coupled discontinuous neural networks, Neural Networks, 89 (2017), 74–83. https://doi.org/10.1016/j.neunet.2017.02.001 doi: 10.1016/j.neunet.2017.02.001
[22]	Z. Zuo, Q. L. Han, B. Ning, X. Ge, X. M. Zhang, An overview of recent advances in fixed-time cooperative control of multiagent systems. IEEE Trans. Ind. Inf., 14 (2018), 2322–2334. https://doi.org/10.1109/TII.2018.2817248 doi: 10.1109/TII.2018.2817248
[23]	B. Ning, Q. L. Han, Z. Zuo, L. Ding, Q. Lu, X. Ge, Fixed-time and prescribed-time consensus control of multiagent systems and its applications: A survey of recent trends and methodologies, IEEE Trans. Ind. Inf., 19 (2022), 1121–1135. https://doi.org/10.1109/TII.2022.3201589 doi: 10.1109/TII.2022.3201589
[24]	Y. Song, Y. Wang, J. Holloway, M. Krstic, Time-varying feedback for regulation of normal-form nonlinear systems in prescribed finite time, Automatica, 83 (2017), 243–251. https://doi.org/10.1016/j.automatica.2017.06.008 doi: 10.1016/j.automatica.2017.06.008
[25]	Y. Song, Y. Wang, M. Krstic, Time-varying feedback for stabilization in prescribed finite time, Int. J. Robust Nonlinear Control, 29 (2019), 618–633. https://doi.org/10.1002/rnc.4084 doi: 10.1002/rnc.4084
[26]	W. Li, M. Krstic, Stochastic nonlinear prescribed-time stabilization and inverse optimality, IEEE Trans. Autom. Control, 67 (2021), 1179–1193. https://doi.org/10.1109/TAC.2021.3061646 doi: 10.1109/TAC.2021.3061646
[27]	F. Gao, Y. Wu, Z. Zhang, Global fixed-time stabilization of switched nonlinear systems: A time-varying scaling transformation approach, IEEE Trans. Circuits Syst. II Express Briefs, 66 (2019), 1890–1894. https://doi.org/10.1109/TCSII.2018.2890556 doi: 10.1109/TCSII.2018.2890556
[28]	H. Ye, Y. Song, Prescribed-time control for linear systems in canonical form via nonlinear feedback, IEEE Trans. Syst. Man Cyber.: Syst., 53 (2022), 1126–1135. https://doi.org/10.1109/TSMC.2022.3194908 doi: 10.1109/TSMC.2022.3194908
[29]	J. Holloway, M. Krstic, Prescribed-time output feedback for linear systems in controllable canonical form, Automatica, 107 (2019), 77–85. https://doi.org/10.1016/j.automatica.2019.05.027 doi: 10.1016/j.automatica.2019.05.027
[30]	P. Krishnamurthy, F. Khorrami, M. Krstic, Prescribed-time stabilization of nonlinear strict-feedback-like systems, in 2019 American Control Conference (ACC), (2019), 3081–3086. https://doi.org/10.23919/ACC.2019.8815272
[31]	P. Krishnamurthy, F. Khorrami, M. Krstic, Robust adaptive prescribed-time stabilization via output feedback for uncertain nonlinear strict-feedback-like systems, Eur. J. Control, 55 (2020), 14–23. https://doi.org/10.1016/j.ejcon.2019.09.005 doi: 10.1016/j.ejcon.2019.09.005
[32]	B. Zhou, Y. Shi, Prescribed-time stabilization of a class of nonlinear systems by linear time-varying feedback, IEEE Trans. Autom. Control, 66 (2021), 6123–6130. https://doi.org/10.1109/TAC.2021.3061645 doi: 10.1109/TAC.2021.3061645
[33]	C. Hua, P. Ning, K. Li, Adaptive prescribed-time control for a class of uncertain nonlinear systems, IEEE Trans. Autom. Control, 67 (2021), 6159–6166. https://doi.org/10.1109/TAC.2021.3130883 doi: 10.1109/TAC.2021.3130883
[34]	P. Krishnamurthy, F. Khorrami, Dynamic high-gain scaling: state and output feedback with application to systems with ISS appended dynamics driven by all states, IEEE Trans. Autom. Control, 49 (2004), 2219–2239. https://doi.org/10.1109/TAC.2004.839235 doi: 10.1109/TAC.2004.839235
[35]	H. Ye, Y. Song, Prescribed-time control of uncertain strict-feedback-like systems, Int. J. Robust Nonlinear Control, 31 (2021), 5281–5297. https://doi.org/10.1002/rnc.5541 doi: 10.1002/rnc.5541
[36]	L. Feng, M. Dai, N. Ji, Y. Zhang, L. Du, Prescribed-time stabilization of nonlinear systems with uncertainties/disturbances by improved time-varying feedback control, AIMS Math., 9 (2024), 23859–23877. https://doi.org/10.3934/math.20241159 doi: 10.3934/math.20241159
[37]	P. Ning, C. Hua, K. Li, R. Meng, Event-triggered control for nonlinear uncertain systems via a prescribed-time approach, IEEE Trans. Autom. Control, 68 (2023), 6975–6981. https://doi.org/10.1109/TAC.2023.3243863 doi: 10.1109/TAC.2023.3243863

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)