Event-triggered proportional consensus with input constraints in adversarial networks

Min Wang; Zhanheng Chen; Zhiyong Yu; Haijun Jiang; Min Wang; Zhanheng Chen; Zhiyong Yu; Haijun Jiang

doi:10.3934/math.20251047

AIMS Mathematics

2025, Volume 10, Issue 10: 23564-23589. doi: 10.3934/math.20251047

Previous Article Next Article

Research article

Event-triggered proportional consensus with input constraints in adversarial networks

1.
College of Mathematics and Statistics, Yili Normal University, Yining 835000, China
2.
Institute of Applied Mathematics, Yili Normal University, Yining 835000, China
3.
College of Mathematics and System Sciences, Xinjiang University, Urumqi 830017, China

Received: 16 August 2025 Revised: 22 September 2025 Accepted: 26 September 2025 Published: 16 October 2025
MSC : 93A16, 93C10, 93D50

This paper addresses the problem of proportional consensus for multi-agent systems (MASs) under input constraints in adversarial network environments. A dynamic event-triggered proportional consensus control algorithm based on distributed observers was proposed. An input constraint mechanism was introduced to model actuator limitations, reflecting practical applications and ensuring reliable agent operation. On this basis, a dynamic event-triggered scheme was designed to significantly reduce communication load, improve resource utilization, and enhance resilience against denial-of-service (DoS) attacks. Leveraging fully distributed observers, each agent achieved proportional consensus using only local information. Theoretical analysis based on Lyapunov stability theory and graph theory provided sufficient conditions for achieving proportional consensus under DoS attacks, offering rigorous support for the method's feasibility. Simulation results demonstrated that the proposed approach effectively achieves proportional consensus for MASs under dual constraints of input saturation and adversarial attacks, while avoiding Zeno behavior.
- proportional consensus,
- denial-of-service attack,
- input saturation,
- dynamic event-triggered,
- fully distributed observer
Citation: Min Wang, Zhanheng Chen, Zhiyong Yu, Haijun Jiang. Event-triggered proportional consensus with input constraints in adversarial networks[J]. AIMS Mathematics, 2025, 10(10): 23564-23589. doi: 10.3934/math.20251047

Related Papers:

Abstract

This paper addresses the problem of proportional consensus for multi-agent systems (MASs) under input constraints in adversarial network environments. A dynamic event-triggered proportional consensus control algorithm based on distributed observers was proposed. An input constraint mechanism was introduced to model actuator limitations, reflecting practical applications and ensuring reliable agent operation. On this basis, a dynamic event-triggered scheme was designed to significantly reduce communication load, improve resource utilization, and enhance resilience against denial-of-service (DoS) attacks. Leveraging fully distributed observers, each agent achieved proportional consensus using only local information. Theoretical analysis based on Lyapunov stability theory and graph theory provided sufficient conditions for achieving proportional consensus under DoS attacks, offering rigorous support for the method's feasibility. Simulation results demonstrated that the proposed approach effectively achieves proportional consensus for MASs under dual constraints of input saturation and adversarial attacks, while avoiding Zeno behavior.

References

[1]	T. Schetter, M. Campbell, D. Surka, Multiple agent-based autonomy for satellite constellations, Artif. Intell., 145 (2003), 147–180. https://doi.org/10.1016/S0004-3702(02)00382-X doi: 10.1016/S0004-3702(02)00382-X
[2]	E. Nuño, A. Loría, E. Panteley, Leaderless consensus formation control of cooperative multi-agent vehicles without velocity measurements, IEEE Contr. Syst. Lett., 6 (2021), 902–907. https://doi.org/10.1109/LCSYS.2021.3087095 doi: 10.1109/LCSYS.2021.3087095
[3]	A. Jadbabaie, J. Lin, A. S. Morse, Coordination of groups of mobile autonomous agents using nearest neighbor rules, IEEE T. Automat. Contr., 48 (2003), 988–1001. https://doi.org/10.1109/TAC.2003.812781 doi: 10.1109/TAC.2003.812781
[4]	A. Das, F. L. Lewis, Distributed adaptive control for synchronization of unknown nonlinear networked systems, Automatica, 46 (2010), 2014–2021. https://doi.org/10.1016/j.automatica.2010.08.008 doi: 10.1016/j.automatica.2010.08.008
[5]	H. Su, M. Z. Q. Chen, X. Wang, J. Lam, Semiglobal observer-based leader-following consensus with input saturation, IEEE T. Ind. Electron., 61 (2014), 2842–2850. https://doi.org/10.1109/TIE.2013.2275976 doi: 10.1109/TIE.2013.2275976
[6]	Z. Li, Z. Duan, G. Chen, L. Huang, Consensus of multiagent systems and synchronization of complex networks: A unified viewpoint, IEEE T. Circuits-I, 57 (2010), 213–224. https://doi.org/10.1109/TCSI.2009.2023937 doi: 10.1109/TCSI.2009.2023937
[7]	Z. Zhang, S. Chen, H. Su, Scaled consensus of second-order nonlinear multiagent systems with time-varying delays via aperiodically intermittent control, IEEE T. Cybernetics, 50 (2020), 3503–3516. https://doi.org/10.1109/TCYB.2018.2883793 doi: 10.1109/TCYB.2018.2883793
[8]	X. Li, H. Jiang, C. Hu, Y. Ren, Scaled consensus of second-order nonlinear multi-agent systems with fully distributed adaptive aperiodically intermittent communication: A non-reduced order approach, Nonlinear Dyn., 112 (2024), 15377–15397. https://doi.org/10.1007/s11071-024-09655-z doi: 10.1007/s11071-024-09655-z
[9]	S. Wang, L. Li, H. Peng, Y. Yang, M. Zheng, Global asymptotic synchronization of fractional order multi‐linked memristive neural networks with time‐varying delays via discontinuous control, Math. Method. Appl. Sci., 68 (2021), 4642–4652. https://doi.org/10.1002/mma.7523 doi: 10.1002/mma.7523
[10]	A. Girard, Dynamic triggering mechanisms for event-triggered control, IEEE T. Automat. Contr., 60 (2015), 1992–1997. https://doi.org/10.1109/TAC.2014.2366855 doi: 10.1109/TAC.2014.2366855
[11]	D. Liu, B. Wang, K. Zhou, X. Cui, K. Shi, Distributed event-triggered collaborative control for multiagent systems against DoS attacks, J. Franklin I., 361 (2024), 106959. https://doi.org/10.1016/j.jfranklin.2024.106959 doi: 10.1016/j.jfranklin.2024.106959
[12]	X. Du, S. Qu, H. Zhang, W. Xu, Q. Tang, Distributed bipartite consensus for multi-agent systems with dynamic event-triggered mechanism, J. Franklin I., 360 (2023), 8877–8897. https://doi.org/10.1016/j.jfranklin.2022.05.022 doi: 10.1016/j.jfranklin.2022.05.022
[13]	W. Xu, W. He, D. W. Ho, J. Kurths, Fully distributed observer-based consensus protocol: Adaptive dynamic event-triggered schemes, Automatica, 139 (2022), 110188. https://doi.org/10.1016/j.automatica.2022.110188 doi: 10.1016/j.automatica.2022.110188
[14]	K. Liu, Z. Ji, Dynamic event-triggered consensus of general linear multi-agent systems with adaptive strategy, IEEE T. Circuits-II, 69 (2022), 3440–3444. https://doi.org/10.1109/TCSII.2022.3144280 doi: 10.1109/TCSII.2022.3144280
[15]	X. Ruan, J. Cai, Z. Wang, C. Wang, H. Yang, Observer-based dynamic event-triggered tracking consensus for switched multi-agent systems, Mathematics, 11 (2023), 2861. https://doi.org/10.3390/math11132861 doi: 10.3390/math11132861
[16]	Y. Li, W. Meng, B. Fan, S. Zhao, Q. Yang, Distributed aperiodic control of multibus DC microgrids with DoS-attack resilience, IEEE T. Smart Grid, 13 (2022), 4815–4827. https://doi.org/10.1109/TSG.2022.3180502 doi: 10.1109/TSG.2022.3180502
[17]	Q. Hu, N. Zheng, M. Xu, Y. Wu, X. He, Mean consensus control of multi-agent systems with privacy protection under DoS attacks, Automatica, 48 (2022), 1961–1971. https://doi.org/10.16383/j.aas.c201019 doi: 10.16383/j.aas.c201019
[18]	S. Hu, P. Yuan, D. Yue, C. Dou, Z. Cheng, Y. Zhang, Attack-resilient event-triggered controller design of DC microgrids under DoS attacks, IEEE T. Circuits-I, 67 (2020), 699–710. https://doi.org/10.1109/TCSI.2019.2948015 doi: 10.1109/TCSI.2019.2948015
[19]	W. Xu, G. Hu, D. W. C. Ho, Z. Feng, Distributed secure cooperative control under denial-of-service attacks from multiple adversaries, IEEE T. Cybernetics, 50 (2020), 3458–3467. https://doi.org/10.1109/TCYB.2019.2896160 doi: 10.1109/TCYB.2019.2896160
[20]	Y. S. Ma, W. W. Che, C. Deng, Z. G. Wu, Observer-based event-triggered containment control for MASs under DoS attacks, IEEE T. Cybernetics, 52 (2022), 13156–13167. https://doi.org/10.1109/TCYB.2021.3104178 doi: 10.1109/TCYB.2021.3104178
[21]	J. Liang, Z. Chen, Z. Yu, H. Jiang, Fixed-time consensus of second-order multi-agent systems based on event-triggered mechanism under DoS attacks, AIMS Mathematics, 10 (2025), 1501–1528. https://doi.org/10.3934/math.2025070 doi: 10.3934/math.2025070
[22]	P. Yang, A. Zhang, W. Bi, Finite-time group consensus for second-order multi-agent systems with input saturation, Neural Process. Lett., 54 (2022), 4211–4228. https://doi.org/10.1007/s11063-022-10805-w doi: 10.1007/s11063-022-10805-w
[23]	W. Zhang, B. Gong, X. Wang, H. Li, Leader-following consensus of stochastic delayed multiagent systems with input saturation under self-triggered impulsive control, Sci. China Technol. Sci., 68 (2025), 1420404. https://doi.org/10.1007/s11431-024-2715-5 doi: 10.1007/s11431-024-2715-5
[24]	H. Zhou, S. Tong, Fuzzy adaptive event-triggered resilient formation control for nonlinear multiagent systems under DoS attacks and input saturation, IEEE T. Syst. Man Cy., 54 (2024), 3665–3674. https://doi.org/10.1109/TSMC.2024.3369093 doi: 10.1109/TSMC.2024.3369093
[25]	S. Sui, C. L. P. Chen, S. Tong, Command filter-based predefined time adaptive control for nonlinear systems, IEEE T. Automat. Contr., 69 (2024), 7863–7870. https://doi.org/10.1109/TAC.2024.3399998 doi: 10.1109/TAC.2024.3399998
[26]	H. Xu, D. Yu, Z. Wang, K. H. Cheong, C. P. Chen, Nonsingular predefined time adaptive dynamic surface control for quantized nonlinear systems, IEEE T. Syst. Cy., 54 (2024), 5567–5579. https://doi.org/10.1109/TSMC.2024.3407150 doi: 10.1109/TSMC.2024.3407150
[27]	S. Tong, S. Sui, Y. Li, Fuzzy adaptive output feedback control of MIMO nonlinear systems with partial tracking errors constrained, IEEE T. Fuzzy Syst., 23 (2015), 729–742. https://doi.org/10.1109/TFUZZ.2014.2327987 doi: 10.1109/TFUZZ.2014.2327987
[28]	Z. Lu, L. Zhang, Z. Ji, L. Wang, Controllability of discrete-time multi-agent systems with directed topology and input delay, Int. J. Control, 89 (2016), 179–192. https://doi.org/10.1080/00207179.2015.1063165 doi: 10.1080/00207179.2015.1063165
[29]	Y. Si, H. Long, Controllability of second-order multi-agent systems with impulse-effected switching topologies, Int. J. Control, 2025, 1–12. https://doi.org/10.1080/00207179.2025.2485159
[30]	C. De Persis, P. Tesi, Input-to-state stabilizing control under denial-of-service, IEEE T. Automat. Contr., 60 (2015), 2930–2944. https://doi.org/10.1109/TAC.2015.2416924 doi: 10.1109/TAC.2015.2416924
[31]	Z. Zuo, R. Ke, Q. L. Han, Fully distributed adaptive practical fixed-time consensus protocols for multi-agent systems, Automatica, 157 (2023), 111248. https://doi.org/10.1016/j.automatica.2023.111248 doi: 10.1016/j.automatica.2023.111248
[32]	Y. Sun, X. Yang, H. Su, Fully distributed observer-based scaled consensus of multi-agent systems with actuator saturation and edge-based event-triggered communication, Neurocomputing, 600 (2024), 128134. https://doi.org/10.1016/j.neucom.2024.128134 doi: 10.1016/j.neucom.2024.128134
[33]	M. Franceschelli, A. Gasparri, A. Giua, C. Seatzu, Decentralized estimation of Laplacian eigenvalues in multi-agent systems, Automatica, 49 (2013), 1031–1036. https://doi.org/10.1016/j.automatica.2013.01.029 doi: 10.1016/j.automatica.2013.01.029
[34]	Z. Lin, Low gain feedback, London: Springer, 1999. https://doi.org/10.1007/BFb0119075
[35]	H. Zhang, F. L. Lewis, A. Das, Optimal design for synchronization of cooperative systems: State feedback, observer and output feedback, IEEE T. Automat. Contr., 56 (2011), 1948–1952. https://doi.org/10.1109/TAC.2011.2139510 doi: 10.1109/TAC.2011.2139510

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)