Event-triggered impulsive control for input-to-state stability of nonlinear time-delay system with delayed impulse

Yilin Tu; Jin-E Zhang; Yilin Tu; Jin-E Zhang

doi:10.3934/mbe.2025031

Mathematical Biosciences and Engineering

2025, Volume 22, Issue 4: 876-896. doi: 10.3934/mbe.2025031

Previous Article Next Article

Research article

Event-triggered impulsive control for input-to-state stability of nonlinear time-delay system with delayed impulse

Yilin Tu ,
Jin-E Zhang ^,

School of Mathematics and Statistics, Hubei Normal University, Huangshi 435002, China

Academic Editor: Hao Wang

Received: 26 December 2024 Revised: 25 February 2025 Accepted: 27 February 2025 Published: 07 March 2025

The problem of input-to-state stability (ISS) was studied in this paper for a class of nonlinear time-delay systems with delayed impulses under event-triggered impulsive control (ETIC), where delays were flexible and external inputs in continuous and impulse dynamics were different. To avoid Zeno behavior, an event-triggered mechanism (ETM) that used the information of the system state and external disturbances was proposed, and the feasibility was enhanced by introducing a forced impulse sequence. Furthermore, some sufficient conditions were formulated to enable ISS for the considered system by using Lyapunov–Razumikhin-like methods. For a class of nonlinear impulse control systems, an application was introduced that utilizes linear matrix inequalities (LMIs) to design an ETM and control gain. Finally, two numerical examples were given to validate the theoretical results.
- input-to-state stability,
- nonlinear time-delay systems,
- event-triggered impulsive control,
- delayed impulse,
- Zeno behavior
Citation: Yilin Tu, Jin-E Zhang. Event-triggered impulsive control for input-to-state stability of nonlinear time-delay system with delayed impulse[J]. Mathematical Biosciences and Engineering, 2025, 22(4): 876-896. doi: 10.3934/mbe.2025031

Related Papers:

Abstract

The problem of input-to-state stability (ISS) was studied in this paper for a class of nonlinear time-delay systems with delayed impulses under event-triggered impulsive control (ETIC), where delays were flexible and external inputs in continuous and impulse dynamics were different. To avoid Zeno behavior, an event-triggered mechanism (ETM) that used the information of the system state and external disturbances was proposed, and the feasibility was enhanced by introducing a forced impulse sequence. Furthermore, some sufficient conditions were formulated to enable ISS for the considered system by using Lyapunov–Razumikhin-like methods. For a class of nonlinear impulse control systems, an application was introduced that utilizes linear matrix inequalities (LMIs) to design an ETM and control gain. Finally, two numerical examples were given to validate the theoretical results.

References

[1]	E. D. Sontag, Smooth stabilization implies coprime factorization, IEEE Trans. Autom. Control, 34 (1989), 435–443. https://doi.org/10.1109/9.28018 doi: 10.1109/9.28018
[2]	J. P. Hespanha, D. Liberzon, A. R. Teel, Lyapunov conditions for input-to-state stability of impulsive systems, Automatica, 44 (2008), 2735–2744. https://doi.org/10.1016/j.automatica.2008.03.021 doi: 10.1016/j.automatica.2008.03.021
[3]	L. Long, Input/output-to-state stability for switched nonlinear systems with unstable subsystems, Int. J. Robust Nonlinear Control, 29 (2019), 3093–3110. https://doi.org/10.1002/rnc.4539 doi: 10.1002/rnc.4539
[4]	Y. Tang, X. Wu, P. Shi, F. Qian, Input-to-state stability for nonlinear systems with stochastic impulses, Automatica, 113 (2020), 108766. https://doi.org/10.1016/j.automatica.2019.108766 doi: 10.1016/j.automatica.2019.108766
[5]	Z. Wang, J. Sun, J. Chen, Finite-time integral input-to-state stability for switched nonlinear time-delay systems with asynchronous switching, Internat. J. Robust Nonlinear Control, 31 (2021), 3929–3954. https://doi.org/10.1002/rnc.5424 doi: 10.1002/rnc.5424
[6]	P. Wang, W. Guo, H. Su, Improved input-to-state stability analysis of impulsive stochastic systems, IEEE Trans. Automat Control, 67 (2022), 2161–2174. https://doi.org/10.1109/tac.2021.3075763 doi: 10.1109/tac.2021.3075763
[7]	Z. Liang, X. Liu, Input-to-state hybrid impulsive formation stabilization for multi-agent systems with impulse delays, Commun. Nonlinear Sci. Numer. Simul., 139 (2024), 108323. https://doi.org/10.1016/j.cnsns.2024.108323 doi: 10.1016/j.cnsns.2024.108323
[8]	V. Lakshmikantham, D. Bainov, P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, 1989. https://doi.org/10.1142/0906
[9]	F. Cacace, V. Cusimano, P. Palumbo, Optimal impulsive control with application to antiangiogenic tumor therapy, IEEE Trans. Control Syst. Technol., 28 (2020), 106–117. https://doi.org/10.1109/TCST.2018.2861410 doi: 10.1109/TCST.2018.2861410
[10]	X. Tan, J. Cao, L. Rutkowski, G. Lu, Distributed dynamic self-triggered impulsive control for consensus networks: The case of impulse gain with normal distribution, IEEE Trans. Cybern., 51 (2021), 624–634. https://doi.org/10.1109/TCYB.2019.2924258 doi: 10.1109/TCYB.2019.2924258
[11]	F. Wang, J. Cui, J. Zhang, Y. Lu, X. Liu, Stabilization of fractional nonlinear systems with disturbances via sliding mode control, Int. J. Robust Nonlinear Control, 35 (2024), 202–221. https://doi.org/10.1002/rnc.7642 doi: 10.1002/rnc.7642
[12]	M. Pouzesh, S. Mobayen, Event-triggered fractional-order sliding mode control technique for stabilization of disturbed quadrotor unmanned aerial vehicles, Aerosp. Sci. Technol., 121 (2022), 107337. https://doi.org/10.1016/j.ast.2022.107337 doi: 10.1016/j.ast.2022.107337
[13]	X. Lv, Y. Niu, J. H. Park, J. Song, Sliding mode control for 2D FMII systems: A bidirectional dynamic event-triggered strategy, Automatica, 147 (2023), 110727. https://doi.org/10.1016/j.automatica.2022.110727 doi: 10.1016/j.automatica.2022.110727
[14]	T. Yang, Impulsive Control Theory, Springer, 2001. https://doi.org/10.1007/3-540-47710-1
[15]	G. Leen, D. Heffernan, TTCAN: A new time-triggered controller area network, Microprocess. Microsyst., 26 (2002), 77–94. https://doi.org/10.1016/S0141-9331(01)00148-X doi: 10.1016/S0141-9331(01)00148-X
[16]	C. Yi, J. Feng, J. Wang, C. Xu, Y. Zhao, Synchronization of delayed neural networks with hybrid coupling via partial mixed pinning impulsive control, Appl. Math. Comput., 312 (2017), 78–90. https://doi.org/10.1016/j.amc.2017.04.030 doi: 10.1016/j.amc.2017.04.030
[17]	R. Postoyan, P. Tabuada, D. Nešić, A. Anta, A framework for the event-triggered stabilization of nonlinear systems, IEEE Trans. Autom. Control, 60 (2015), 982–996. https://doi.org/10.1109/TAC.2014.2363603 doi: 10.1109/TAC.2014.2363603
[18]	B. Li, Z. Wang, L. Ma, H. Liu, Observer-based event-triggered control for nonlinear systems with mixed delays and disturbances: The input-to-state stability, IEEE Trans. Cybern., 49 (2019), 2806–2819. https://doi.org/10.1109/TCYB.2018.2837626 doi: 10.1109/TCYB.2018.2837626
[19]	J. Huang, W. Wang, C. Wen, G. Li, Adaptive event-triggered control of nonlinear systems with controller and parameter estimator triggering, IEEE Trans. Autom. Control, 65 (2020), 318–324. https://doi.org/10.1109/TAC.2019.2912517 doi: 10.1109/TAC.2019.2912517
[20]	Y. An, Y. Liu, Observer-based dynamic event-triggered adaptive control for uncertain nonlinear strict-feedback systems, Syst. Control Lett., 183 (2024), 105700. https://doi.org/10.1016/j.sysconle.2023.105700 doi: 10.1016/j.sysconle.2023.105700
[21]	X. Tan, J. Cao, X. Li, Consensus of leader-following multiagent systems: A distributed event-triggered impulsive control strategy, IEEE Trans. Cybern., 49 (2019), 792–801. https://doi.org/10.1109/TCYB.2017.2786474 doi: 10.1109/TCYB.2017.2786474
[22]	Y. Zhou, Z. Zeng, Event-triggered impulsive control on quasi-synchronization of memristive neural networks with time-varying delays, Neural Networks, 110 (2019), 55–65. https://doi.org/10.1016/j.neunet.2018.09.014 doi: 10.1016/j.neunet.2018.09.014
[23]	W. Heemels, M. C. F. Donkers, A. R. Teel, Periodic event-triggered control for linear systems, IEEE Trans. Autom. Control, 58 (2013), 847–861. https://doi.org/10.1109/TAC.2012.2220443 doi: 10.1109/TAC.2012.2220443
[24]	R. Goebel, R. G. Sanfelice, A. R. Teel, Hybrid dynamical systems, IEEE Control Syst. Mag., 29 (2009), 28–93. https://doi.org/10.1109/MCS.2008.931718 doi: 10.1109/MCS.2008.931718
[25]	C. Zhang, G. Yang, Event-triggered global finite-time control for a class of uncertain nonlinear systems, IEEE Trans. Autom. Control, 65 (2020), 1340–1347. https://doi.org/10.1109/TAC.2019.2928767 doi: 10.1109/TAC.2019.2928767
[26]	M. Wang, P. Li, X. Li, Event-triggered delayed impulsive control for input-to-state stability of nonlinear impulsive systems, Nonlinear Anal. Hybrid Syst., 47 (2023), 101277. https://doi.org/10.1016/j.nahs.2022.101277 doi: 10.1016/j.nahs.2022.101277
[27]	X. Li, P. Li, Input-to-state stability of nonlinear systems: Event-triggered impulsive control, IEEE Trans. Autom. Control, 67 (2022), 1460–1465. https://doi.org/10.1109/TAC.2021.3063227 doi: 10.1109/TAC.2021.3063227
[28]	X. Li, T. Zhang, J. Wu, Input-to-state stability of impulsive systems via event-triggered impulsive control, IEEE Trans. Cybern., 52 (2021), 7187–7195. https://doi.org/10.1109/TCYB.2020.3044003 doi: 10.1109/TCYB.2020.3044003
[29]	P. Yu, F. Deng, X. Zhao, Y. Huang, Stability analysis of nonlinear systems in the presence of event-triggered impulsive control, Int. J. Robust Nonlinear Control, 34 (2024), 3835–3853. https://doi.org/10.1002/rnc.7165 doi: 10.1002/rnc.7165
[30]	L. Li, J. Zhang, Input-to-state stability of nonlinear systems with delayed impulse based on event-triggered impulse control, AIMS Math., 9 (2024), 26446–26461. https://doi.org/10.3934/math.20241287 doi: 10.3934/math.20241287
[31]	X. Li, W. Liu, S. Gorbachev, J. Cao, Event-triggered impulsive control for input-to-state stabilization of nonlinear time-delay systems, IEEE Trans. Cybern., 54 (2024), 2536–2544. https://doi.org/10.1109/TCYB.2023.3270487 doi: 10.1109/TCYB.2023.3270487
[32]	H. Yu, F. Hao, T. Chen, A uniform analysis on input-to-state stability of decentralized event-triggered control systems, IEEE Trans. Autom. Control, 64 (2019), 3423–3430. https://doi.org/10.1109/TAC.2018.2879764 doi: 10.1109/TAC.2018.2879764
[33]	B. Liu, M. Yang, T. Liu, D. J. Hill, Stabilization to exponential input-to-state stability via aperiodic intermittent control, IEEE Trans. Autom. Control, 66 (2021), 2913–2919. https://doi.org/10.1109/TAC.2020.3014637 doi: 10.1109/TAC.2020.3014637
[34]	J. Liu, X. Liu, W. Xie, Input-to-state stability of impulsive and switching hybrid systems with time-delay, Automatica, 47 (2011), 899–908. https://doi.org/10.1016/j.automatica.2011.01.061 doi: 10.1016/j.automatica.2011.01.061
[35]	G. Ballinger, X. Liu, Existence, uniqueness and boundedness results for impulsive delay differential equations, Appl. Anal., 74 (2000), 71–93. https://doi.org/10.1080/00036810008840804 doi: 10.1080/00036810008840804
[36]	S. Dashkovskiy, M. Kosmykov, A. Mironchenko, L. Naujok, Stability of interconnected impulsive systems with and without time-delays using Lyapunov methods, Nonlinear Anal. Hybrid Syst., 6 (2012), 899–915. https://doi.org/10.1016/j.nahs.2012.02.001 doi: 10.1016/j.nahs.2012.02.001
[37]	P. Feketa, V. Klinshov, L. Lücken, A survey on the modeling of hybrid behaviors: How to account for impulsive jumps properly, Commun. Nonlinear Sci. Numer. Simul., 103 (2021), 105955. https://doi.org/10.1016/j.cnsns.2021.105955 doi: 10.1016/j.cnsns.2021.105955
[38]	Z. Fu, S. Peng, Stability analysis of nonlinear systems with delayed impulses: A generalised average dwell-time scheme, Int. J. Syst. Sci., 55 (2024), 1882–1894. https://doi.org/10.1080/00207721.2024.2322696 doi: 10.1080/00207721.2024.2322696
[39]	S. Dashkovskiy, P. Feketa, Input-to-state stability of impulsive systems and their networks, Nonlinear Anal. Hybrid Syst., 26 (2017), 190–200. https://doi.org/10.1016/j.nahs.2017.06.004 doi: 10.1016/j.nahs.2017.06.004
[40]	S. Dashkovskiy, P. Feketa, Input-to-state stability of impulsive systems and their networks, Nonlinear Anal. Hybrid Syst., 26 (2017), 190–200. https://doi.org/10.1016/j.nahs.2017.06.004 doi: 10.1016/j.nahs.2017.06.004
[41]	S. Dashkovskiy, V. Slynko, Dwell-time stability conditions for inﬁnite dimensional impulsive systems, Automatica, 147 (2023), 110695. https://doi.org/10.1016/j.automatica.2022.110695 doi: 10.1016/j.automatica.2022.110695
[42]	S. Dashkovskiy, V. Slynko, Stability conditions for impulsive dynamical systems, Math. Control Signals Syst., 34 (2022), 95–128. https://doi.org/10.1007/s00498-021-00305-y doi: 10.1007/s00498-021-00305-y
[43]	S. Peng, F. Deng, Y. Zhang, A unified razumikhin-type criterion on input-to-state stability of time-varying impulsive delayed systems, Syst. Control Lett., 116 (2018), 20–26. https://doi.org/10.1016/j.sysconle.2018.04.002 doi: 10.1016/j.sysconle.2018.04.002

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)