Prescribed-time stabilization of uncertain nonlinear impulsive systems with multiple delays

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Abstract

Researchers have reported on prescribed-time stability (PTS) for delay systems, but they do not take into account the significant impulses phenomenon and focus on only single delay. To tackle these two aspects, we aimed to investigate PTS issues for nonlinear impulsive systems with multiple time-varying delays and uncertainties. By the Lyapunov-Krasovskii functional method, PTS criteria were established for multi-delay systems. Specifically speaking, an adaptive control strategy was proposed for multi-delay systems to achieve PTS through the integrated design of time-varying delay compensation and uncertainty handling, which could ensure system states and control inputs to precisely converge to the origin within the prescribed time frame. Furthermore, the proposed method significantly enhanced the system’s robustness against time-varying delays and uncertainties, thus overcoming the limitations of traditional methods in terms of convergence time and disturbance rejection capability. Finally, a simulation result was given to verify the feasibility and effectiveness of the proposed method.

Keywords

uncertain impulsive systems / multiple delays / Lyapunov-Krasovskii functional / prescribed time stability / settling time

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References

[1] S. Kanakalakshmi, R. Sakthivel, S. A. Karthick, A. Leelamani, A. Parivallal, Finite-time decentralized event-triggering non-fragile control for fuzzy neural networks with cyber-attack and energy constraints, Eur. J. Control, 57 (2021), 135-146. https://doi.org/10.1016/j.ejcon.2020.05.001

[2] 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

[3] D. Cui, Z. R. Xiang, Nonsingular fixed-time fault-tolerant fuzzy control for switched uncertain nonlinear systems, IEEE. Trans. Fuzzy. Syst., 31 (2022), 174-183. https://doi.org/10.1109/TFUZZ.2022.3184048

[4] C. Q. Guo, J. P. Hu, Time base generator-based practical predefined-time stabilization of high-order systems with unknown disturbance, IEEE Trans. Circuits Syst. II Express Briefs, 70 (2023), 2670-2674. https://doi.org/10.1109/TCSII.2023.3242856

[5] Y. D. Song, Y. J. 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

[6] Y. D. Song, Y. J. 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

[7] Y. D. Song, H. F. Ye, F. L. Lewis, Prescribed-time control and its latest developments, IEEE Trans. Syst. Man Cybern.: Syst., 53 (2023), 4102-4116. https://doi.org/10.1109/TSMC.2023.3240751

[8] W. Q. Li, M. Krstic, Prescribed-time output-feedback control of stochastic nonlinear systems, IEEE Trans. Autom. Control, 68 (2022), 1431-1446. https://doi.org/10.1109/TAC.2022.3151587

[9] W. Q. 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.2021.3061649

[10] Y. Zhang, Z. R. Xiang, Prescribed-time optimal control for a class of switched nonlinear systems, IEEE Trans. Autom. Sci. Eng., 22 (2024), 3033-3043. https://doi.org/10.1109/TASE.2024.3388456

[11] L. C. Feng, M. Y. Dai, N. Ji, Y. L. Zhang, L. P. 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

[12] C. C. Hua, P. J. 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

[13] Y. Ding, B. Zhou, K. K. Zhang, W. Michiels, Strong prescribed-time stabilization of uncertain nonlinear systems by periodic delayed feedback, IEEE Trans. Autom. Control, 69 (2023), 4072-4079. https://doi.org/10.1109/TAC.2023.3347703

[14] Z. H. Guan, Z. W. Liu, G. Feng, M. Jian, Impulsive consensus algorithms for second-order multi-agent networks with sampled information, Automatica, 48 (2012), 1397-1404. https://doi.org/10.1016/j.automatica.2012.05.005

[15] X. R. Ji, J. Q. Lu, B. X. Jiang, J. Zhong, Network synchronization under distributed delayed impulsive control: Average delayed impulsive weight approach, Nonlinear Anal. Hybrid Syst., 44 (2022), 101148. https://doi.org/10.1016/j.nahs.2021.101148

[16] L. Z. Zhang, J. Q. Lu, F. Y. Liu, J. G. Lou, Synchronization of time-delay coupled neural networks with stabilizing delayed impulsive control, IEEE Trans. Neural Networks, 35 (2023), 18899-18906. https://doi.org/10.1109/TNNLS.2023.3320651

[17] C. C. Li, C. Y. Zhang, L. C. Feng, Z. H. Wu, Note on prescribed-time stability of impulsive piecewise-smooth differential systems and application in networks, Networks Heterogen. Media, 19 (2024), 970-991. https://doi.org/10.3934/nhm.2024043

[18] L. C. Feng, C. C. Li, M. Abdel-Aty, J. D. Cao, Prescribed-time stability of nonlinear impulsive piecewise systems and synchronization for dynamical networks, Qual. Theory Dyn. Syst., 24 (2025), 165. https://doi.org/10.1007/s12346-025-01302-1

[19] A. Mapui, S. Mukhopadhyay, Prescribed-time stabilisation of nonlinear time-varying systems exhibiting impulsive behaviour, Int. J. Control, (2025), 1-14. https://doi.org/10.1080/00207179.2025.2481592

[20] T. X. Zhang, J. D. Cao, X. D. Li, Prescribed-time control of switched systems: an event-triggered impulsive control approach, IEEE Trans. Syst. Man Cybern.: Syst., 54 (2024), 4284-4293. https://doi.org/10.1109/TSMC.2024.3380063

[21] Z. D. Tian, Y. H. Wang, Predictive control compensation for networked control system with time-delay, Proc. Inst. Mech. Eng., Part I: J. Syst. Control Eng., 236 (2022), 107-124. https://doi.org/10.1177/09596518211018298

[22] X. M. Zhang, Q. L. Han, X. H. Ge, A novel approach to H_∞ performance analysis of discrete-time networked systems subject to network-induced delays and malicious packet dropouts , Automatica, 136 (2022), 110010. https://doi.org/10.1016/j.automatica.2021.110010

[23] M. P. Sun, S. Y. Zou, X. H. Nian, Y. Chen, Distributed power economic dispatch algorithm based on hybrid time-delay system framework under DoS attacks, J. Franklin Inst., 362 (2025), 107454. https://doi.org/10.1016/j.jfranklin.2024.107454

[24] S. N. Campos-Martínez, O. Hernández-González, M. E. Guerrero-Sánchez, G. Valencia-Palomo, B. Targui, F. R. López-Estrada, Consensus tracking control of multiple unmanned aerial vehicles subject to distinct unknown delays, Machines, 12 (2024), 337. https://doi.org/10.3390/machines12050337

[25] L. C. Feng, L. Liu, J. D. Cao, L. Rutkowski, G. P. Lu, General decay stability for nonautonomous neutral stochastic systems with time-varying delays and Markovian switching, IEEE Trans. Cybern., 52 (2022), 5441-5453. https://doi.org/10.1109/TCYB.2020.3031992

[26] H. B. Zeng, Y. J. Chen, Y. He, X. M. Zhang, A delay-derivative-dependent switched system model method for stability analysis of linear systems with time-varying delay, Automatica, 175 (2025), 112183. https://doi.org/10.1016/j.automatica.2025.112183

[27] R. M. Yang, L. Y. Sun, Finite-time robust control of a class of nonlinear time-delay systems via Lyapunov functional method, J. Franklin Inst., 356 (2019), 1155-1176. https://doi.org/10.1016/j.jfranklin.2018.08.029

[28] K. Xu, H. Wang, P. X. Liu, Adaptive fixed-time control for high-order stochastic nonlinear time-delay systems: An improved Lyapunov-Krasovskii function, IEEE Trans. Cybern., 54 (2023), 776-786. https://doi.org/10.1109/TCYB.2023.3337792

[29] P. J. Ning, C. C. Hua, K. Li, H. Li, A novel theorem for prescribed-time control of nonlinear uncertain time-delay systems, Automatica. 152 (2023), 111009. https://doi.org/10.1016/j.automatica.2023.111009

[30] P. J. Ning, S. N. Dashkovskiy, C. C. Hua, K. Li, Dual-gain function based prescribed-time output feedback control nonlinear time-delay systems, SIAM J. Control Optim., 62 (2024), 2254-2272. https://doi.org/10.1137/23M1556496

[31] C. C. Hua, H. Li, K. Li, P. J. Ning, Adaptive prescribed-time control of time-delay nonlinear systems via a double time-varying gain approach, IEEE Trans. Cybern., 53 (2022), 5290-5298. https://doi.org/10.1109/TCYB.2022.3192250

[32] B. Targui, O Hernández-González, C. M. Astorga-Zaragoza, M. E. Guerrero-Sánchez, G. Valencia-Palomo, Observer for a class of Lipschitz nonlinear systems with multiple time-varying delays in the nonlinear measured outputs, Asian J. Control, 24 (2022), 1122-1132. https://doi.org/10.1002/asjc.2537

[33] O. Hernández-González, B. Targui, G. Valencia-Palomo, M. E. Guerrero-Sánchez, Robust cascade observer for a disturbance unmanned aerial vehicle carrying a load under multiple time-varying delays and uncertainties, Int. J. Syst. Sci., 55 (2024), 1056-1072. https://doi.org/10.1080/00207721.2023.2301493

[34] S. S. Wang, B. S. Chen, T. P. Lin, Robust stability of uncertain time-delay systems, Int. J. Control, 46 (1987), 963-976. https://doi.org/10.1080/00207178708547406

[35] A. G. Wu, S. L. Shen, J. Zhang, J. Mei, Predictor feedback control for discrete-time systems with input delays and multiple state delays via bivariant fundamental matrices, IEEE Trans. Circuits Syst. I Regul. Pap., 17 (2024), 1800-1812. https://doi.org/10.1109/TCSI.2024.3469931

[36] S. A. Samy, Y. Cao, R. Ramachandran, J. Alzabut, M. Niezabitowski, C. P. Lim, Globally asymptotic stability and synchronization analysis of uncertain multi-agent systems with multiple time-varying delays and impulses, Int. J. Robust Nonlinear Control, 32 (2022), 737-773. https://doi.org/10.1002/rnc.5851

[37] Z. C. Wang, G. L. Chen, J. W. Xia, Finite-time input/output-to-state stability of impulsive switched nonlinear time-delay systems, IEEE Trans. Circuits Syst. II Express Briefs, 69 (2022), 3585-3589. https://doi.org/10.1109/TCSII.2022.3167056

[38] S. C. Wu, X. D. Li, Finite-time stability of nonlinear systems with delayed impulses, IEEE Trans. Syst., Man, Cybern.: Syst., 53 (2023), 7453-7460. https://doi.org/10.1109/TSMC.2023.3298071

[39] G. Palanisamy, A. Kashkynbayev, R. Rajan, Finite-time stability of fractional-order discontinuous nonlinear systems with state-dependent delayed impulses, IEEE Trans. Syst. Man Cybern.: Syst., 54 (2023), 1312-1324. https://doi.org/10.1109/TSMC.2023.3326612

[40] Y. Wang, J. Lu, Y. Lou, W. X. Zheng, Fixed-time stability of nonlinear systems with destabilizing delayed impulses: necessary and sufficient conditions, IEEE Trans. Circuits Syst. I Regul. Pap., (2025), 1-14. https://doi.org/10.1109/TCSI.2025.3550551

[41] V. S. Muni, R. K. George, Controllability of semilinear impulsive control systems with multiple time delays in control, IMA J. Math. Control Inf., 36 (2019), 869-899. https://doi.org/10.1093/imamci/dny011

[42] Y. Liu, N. Li, Y. L. Huang, Stability of highly nonlinear impulsive coupled networks with multiple time delays, Nonlinear Dyn., 113 (2025), 8741-8756. https://doi.org/10.1007/s11071-024-10745-1

[43] P. Krishnamurthy, F. Khorrami, M. Krstic, A dynamic high-gain design for prescribed-time regulation of nonlinear systems, Automatica, 115 (2020), 108860. https://doi.org/10.1016/j.automatica.2020.108860

[44] B. Zhou, Finite-time stability analysis and stabilization by bounded linear time-varying feedback, Automatica, 121 (2020), 109191. https://doi.org/10.1016/j.automatica.2020.109191

[45] X. Y. He, X. D. Li, S. J. Song, Prescribed-time stabilization of nonlinear systems via impulsive regulation, IEEE Trans. Syst. Man Cybern.: Syst., 53 (2022), 981-985. https://doi.org/10.1109/TSMC.2022.3188874

[46] T. Wei, X. Li, Fixed-time and predefined-time stability of impulsive systems, IEEE/CAA J. Autom. Sin., 10 (2023), 1086-1089. https://doi.org/10.1109/JAS.2023.123147

[47] P. J. Ning, C. 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

[48] G. Ballinger, X. Z. Liu, Existence, uniqueness and boundedness results for impulsive delay differential equations, Appl. Anal., 74 (2000), 71-93. https://doi.org/10.1080/00036810008840804

[49] X. N. Li, H. Q. Wu, J. D. Cao, A new prescribed-time stability theorem for impulsive piecewise-smooth systems and its application to synchronization in networks. Appl. Math. Modell., 115 (2023), 385-397. https://doi.org/10.1016/j.apm.2022.10.051

[50] Z. C. Wang, G. L. Chen, J. W. Xia, Finite-time input/output-to-state stability of impulsive switched nonlinear time-delay systems. IEEE Trans. Circuits Syst. II Express Briefs, 69 (2022), 3585-3589. https://doi.org/10.1109/TCSII.2022.3167056