{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,22]],"date-time":"2025-11-22T11:35:46Z","timestamp":1763811346566,"version":"build-2065373602"},"reference-count":42,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2024,1,26]],"date-time":"2024-01-26T00:00:00Z","timestamp":1706227200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Guangdong Basic and Applied Basic Research Foundation","award":["2019A1515010830","2023A1515010168","2022ZDZX1018","2022S043","2019BS001"],"award-info":[{"award-number":["2019A1515010830","2023A1515010168","2022ZDZX1018","2022S043","2019BS001"]}]},{"name":"Key Special Foundation for General Universities in Guangdong Province","award":["2019A1515010830","2023A1515010168","2022ZDZX1018","2022S043","2019BS001"],"award-info":[{"award-number":["2019A1515010830","2023A1515010168","2022ZDZX1018","2022S043","2019BS001"]}]},{"name":"Maoming Science and Technology Plan Foundation","award":["2019A1515010830","2023A1515010168","2022ZDZX1018","2022S043","2019BS001"],"award-info":[{"award-number":["2019A1515010830","2023A1515010168","2022ZDZX1018","2022S043","2019BS001"]}]},{"name":"Guangdong University of Petrochemical Technology Doctoral Initiation Foundation","award":["2019A1515010830","2023A1515010168","2022ZDZX1018","2022S043","2019BS001"],"award-info":[{"award-number":["2019A1515010830","2023A1515010168","2022ZDZX1018","2022S043","2019BS001"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>This paper investigates achieving leader-following consensus in a class of multi-agent systems with nonlinear dynamics. Initially, it introduces a dynamic event-triggered strategy designed to effectively alleviate the strain on the system\u2019s communication resources. Subsequently, a distributed control strategy is proposed and implemented in the nonlinear leader-follower system using the dynamic event-triggered mechanism, aiming to ensure synchronization across all nodes at an exponential convergence speed. Thirdly, the research shows that under the dynamic event-triggered strategy the minimum event interval of any two consecutive triggers guarantees the elimination of Zeno behavior. Lastly, the validity of the calculation results is verified by a simulation example.<\/jats:p>","DOI":"10.3390\/e26020113","type":"journal-article","created":{"date-parts":[[2024,1,26]],"date-time":"2024-01-26T04:05:46Z","timestamp":1706241946000},"page":"113","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Distributed Dynamic Event-Triggered Control to Leader-Following Consensus of Nonlinear Multi-Agent Systems with Directed Graphs"],"prefix":"10.3390","volume":"26","author":[{"given":"Jia-Cheng","family":"Guan","sequence":"first","affiliation":[{"name":"College of Automation, Guangdong University of Petrochemical Technology, Maoming 525000, China"},{"name":"College of Information and Control Engineering, Jilin Institute of Chemical Technology, Jilin 132022, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4783-1261","authenticated-orcid":false,"given":"Hong-Wei","family":"Ren","sequence":"additional","affiliation":[{"name":"College of Automation, Guangdong University of Petrochemical Technology, Maoming 525000, China"}]},{"given":"Guo-Liang","family":"Tan","sequence":"additional","affiliation":[{"name":"College of Automation, Guangdong University of Petrochemical Technology, Maoming 525000, China"},{"name":"School of Automation, Guangdong University of Technology, Guangzhou 510006, China"}]}],"member":"1968","published-online":{"date-parts":[[2024,1,26]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"4043","DOI":"10.1109\/TCYB.2022.3224386","article-title":"Sampled Memory-Event-Triggered Fuzzy Load Frequency Control for Wind Power Systems Subject to Outliers and Transmission Delays","volume":"53","author":"Yan","year":"2023","journal-title":"IEEE Trans. Cybern."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"988","DOI":"10.1109\/TAC.2003.812781","article-title":"Coordination of groups of mobile autonomous agents using nearest neighbor rules","volume":"48","author":"Jadbabaie","year":"2003","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Oyinloye, D.P., Teh, J.S., Jamil, N., and Alawida, M. (2021). Blockchain Consensus: An Overview of Alternative Protocols. Symmetry, 13.","DOI":"10.3390\/sym13081363"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1109\/JPROC.2006.887293","article-title":"Consensus and cooperation in networked multi-agent systems","volume":"95","author":"Fax","year":"2007","journal-title":"Proc. IEEE"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1428","DOI":"10.1016\/j.automatica.2013.01.060","article-title":"Network-based H\u221e fifiltering using a logic jumping-like trigger","volume":"49","author":"Zhang","year":"2013","journal-title":"Automatica"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"755","DOI":"10.1007\/s11071-017-3503-4","article-title":"Recent advances in vibration control of offshore platforms","volume":"89","author":"Zhang","year":"2017","journal-title":"Nonlin. Dyn."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1357","DOI":"10.1109\/TAC.2009.2015555","article-title":"Digital Variable Sampling Integral Control of Infinite Dimensional Systems Subject to Input Nonlinearity","volume":"54","author":"Ozdemir","year":"2009","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1044","DOI":"10.1109\/TAC.2005.851456","article-title":"Optimality analysis of adaptive sampled control of hybrid systems with quadratic index","volume":"50","author":"Tan","year":"2005","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_9","first-page":"3440","article-title":"Dynamic Event-Triggered Consensus of General Linear Multi-Agent Systems with Adaptive Strategy","volume":"69","author":"Liu","year":"2022","journal-title":"IEEE Trans. Circuits Syst. II Express Briefs"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Qie, S., Dou, J., Liu, X., Tang, Y., Zhang, Y., and Zheng, Y. (2022, January 27\u201330). Random Number Generation Based DoS Attack-resilient Distributed Secondary Control Strategy. Proceedings of the 2022 4th International Conference on Smart Power and Internet Energy Systems (SPIES), Beijing, China.","DOI":"10.1109\/SPIES55999.2022.10082111"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Miao, P., and Dong, L. (2021, January 26\u201328). Attack Effect Observer-Based Security Control for Cyber-Physical Systems Subjected to False Data Injection Attack. Proceedings of the 2021 40th Chinese Control Conference (CCC), Shanghai, China.","DOI":"10.23919\/CCC52363.2021.9550577"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1048","DOI":"10.1109\/JSYST.2020.2992702","article-title":"Finite-Time Stability Under Denial of Service","volume":"15","author":"Doostmohammadian","year":"2021","journal-title":"IEEE Syst. J."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"837","DOI":"10.1109\/TSG.2021.3115180","article-title":"Distributed Event-Triggered Secondary Control for Islanded Microgrids with Proper Trigger Condition Checking Period","volume":"13","author":"Chen","year":"2022","journal-title":"IEEE Trans. Smart Grid"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Ke, X., and Zong, X. (2021, January 15\u201317). Formation Control of Second-order Multi-agent Systems with Multiplicative Noises. Proceedings of the 2021 IEEE International Conference on Unmanned Systems (ICUS), Beijing, China.","DOI":"10.1109\/ICUS52573.2021.9641317"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1954","DOI":"10.1109\/TCNS.2021.3092832","article-title":"Distributed Optimization of Second-Order Nonlinear Multiagent Systems with Event-Triggered Communication","volume":"8","author":"Li","year":"2021","journal-title":"IEEE Trans. Control. Netw. Syst."},{"key":"ref_16","first-page":"2112","article-title":"Group Consensus of Second-Order Sample MASs via Distributed Event-Triggered Mechanism","volume":"70","author":"Sun","year":"2023","journal-title":"IEEE Trans. Circuits Syst. II Express Briefs"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"3006","DOI":"10.1109\/TCSI.2022.3164552","article-title":"Fixed-Time Stabilization for Nonlinear Systems with Low-Order and High-Order Nonlinearities via Event-Triggered Control","volume":"69","author":"Meng","year":"2022","journal-title":"IEEE Trans. Circuits Syst. I Regul. Pap."},{"key":"ref_18","first-page":"3436","article-title":"Dynamic Event-Triggered Output Feedback Control for a Class of High-Order Feedforward Nonlinear Time-Delay Systems","volume":"68","author":"Yuan","year":"2021","journal-title":"IEEE Trans. Circuits Syst. II Express Briefs"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"5468","DOI":"10.1109\/TSMC.2021.3127825","article-title":"Fuzzy Adaptive Constrained Consensus Tracking of High-Order Multi-agent Networks: A New Event-Triggered Mechanism","volume":"52","author":"Wang","year":"2022","journal-title":"IEEE Trans. Syst. Man Cybern. Syst."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1724","DOI":"10.1109\/LCSYS.2021.3132654","article-title":"Robust Resource-Aware Self-Triggered Model Predictive Control","volume":"6","author":"Lian","year":"2022","journal-title":"IEEE Control Syst. Lett."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"8036","DOI":"10.1109\/TAC.2023.3282066","article-title":"Periodic Event-Triggered MPC for Continuous-Time Nonlinear Systems with Bounded Disturbances","volume":"68","author":"Wang","year":"2023","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_22","first-page":"2447","article-title":"Wireless Alarm Microsystem Self-Powered by Vibration-Threshold-Triggered Energy Harvester","volume":"63","author":"Tang","year":"2016","journal-title":"IEEE Trans. Ind. Electron."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"3902","DOI":"10.1109\/TCYB.2020.3020253","article-title":"Model-Based Event-Triggered Sliding-Mode Control for Multi-Input Systems: Performance Analysis and Optimization","volume":"52","author":"Song","year":"2022","journal-title":"IEEE Trans. Cybern."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Zhang, Z., Liu, J., Zhang, S., Zhu, H., and Zhang, B. (2021, January 14\u201316). Dynamic Event-triggered Scheme and Output Feedback Control for CPS under Multiple Cyber Attacks. Proceedings of the 2021 IEEE 10th Data Driven Control and Learning Systems Conference (DDCLS), Suzhou, China.","DOI":"10.1109\/DDCLS52934.2021.9455613"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"5843","DOI":"10.1109\/TSMC.2021.3131709","article-title":"Event-Triggered Adaptive Bipartite Containment Control for Stochastic Multiagent Systems","volume":"52","author":"Zhang","year":"2022","journal-title":"IEEE Trans. Syst. Man Cybern. Syst."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"366","DOI":"10.1109\/TNNLS.2021.3094901","article-title":"Event-Triggered Multigradient Recursive Reinforcement Learning Tracking Control for Multiagent Systems","volume":"34","author":"Bai","year":"2023","journal-title":"IEEE Trans. Neural Netw. Learn. Syst."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Kenyeres, M., and Kenyeres, J. (2021). Comparative Study of Distributed Consensus Gossip Algorithms for Network Size Estimation in Multi-Agent Systems. Future Internet, 13.","DOI":"10.3390\/fi13050134"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"3894","DOI":"10.1109\/TFUZZ.2021.3133892","article-title":"Adaptive Memory-Event-Triggered Static Output Control of T\u2013S Fuzzy Wind Turbine Systems","volume":"30","author":"Yan","year":"2022","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1992","DOI":"10.1109\/TAC.2014.2366855","article-title":"Dynamic triggering mechanisms for event-triggered control","volume":"60","author":"Girard","year":"2015","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"869","DOI":"10.1109\/TCYB.2018.2868778","article-title":"A Distributed Dynamic Event-Triggered Control Approach to Consensus of Linear Multiagent Systems with Directed Networks","volume":"50","author":"Hu","year":"2020","journal-title":"IEEE Trans. Cybern."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1979","DOI":"10.1109\/TCYB.2018.2818941","article-title":"Synchronization control for a class of discrete time-delay complex dynamical networks: A dynamic event-triggered approach","volume":"49","author":"Li","year":"2018","journal-title":"IEEE Trans. Cybern."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Chen, J., Chen, B., and Zeng, Z. (2022, January 15\u201317). Leader-Following Consensus of Linear Multi-Agent Systems via Dynamic Event-Triggered Adjustable Control Protocol. Proceedings of the 2022 14th International Conference on Advanced Computational Intelligence (ICACI), Wuhan, China.","DOI":"10.1109\/ICACI55529.2022.9837747"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"3243","DOI":"10.1109\/TSMC.2018.2866853","article-title":"Dynamic event-triggered control for leader-following consensus of multiagent systems","volume":"50","author":"Du","year":"2020","journal-title":"IEEE Trans. Syst. Man Cybern. Syst."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.automatica.2018.10.032","article-title":"Distributed event-triggered control for global consensus of multi-agent systems with input Saturation","volume":"100","author":"Yi","year":"2019","journal-title":"Automatica"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"1204","DOI":"10.1109\/JSYST.2022.3173939","article-title":"Dynamic Periodic Event-Triggered Consensus Protocols for Linear Multiagent Systems with Network Delay","volume":"17","author":"Lin","year":"2023","journal-title":"IEEE Syst. J."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"3300","DOI":"10.1109\/TAC.2018.2874703","article-title":"Dynamic event-triggered and self-triggered control for multi-agent systems","volume":"64","author":"Yi","year":"2019","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"3702","DOI":"10.1016\/j.jfranklin.2020.02.036","article-title":"Output-based event-triggered consensus of general linear multi-agent systems with communication delay under directed graphs","volume":"357","author":"Wu","year":"2020","journal-title":"J. Frankl. Inst."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"148","DOI":"10.1109\/TCYB.2015.2398892","article-title":"Consensus of linear multi-agent systems by distributed event-triggered strategy","volume":"46","author":"Hu","year":"2016","journal-title":"IEEE Trans. Cybern."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"4391","DOI":"10.1109\/TAC.2018.2817160","article-title":"Event-triggered leader-following consensus for nonlinear multiagent systems subject to actuator saturation using dynamic output feedback method","volume":"63","author":"You","year":"2018","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"110285","DOI":"10.1109\/ACCESS.2022.3214849","article-title":"Dynamic Event-Triggered Consensus Control for Multi-Agent Systems Using Adaptive Dynamic Programming","volume":"10","author":"Zhang","year":"2022","journal-title":"IEEE Access"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"51261","DOI":"10.1109\/ACCESS.2022.3173648","article-title":"Robust Dynamic Output Feedback Event-Triggering Synchronization for Complex Dynamical Networks","volume":"10","author":"Jeong","year":"2022","journal-title":"IEEE Access"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"6986","DOI":"10.1109\/TSMC.2019.2962827","article-title":"Event-triggered control for a class of nonlinear multiagent systems with directed graph","volume":"51","author":"Li","year":"2021","journal-title":"IEEE Trans. Syst. Man Cybern. Syst."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/26\/2\/113\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T13:49:33Z","timestamp":1760104173000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/26\/2\/113"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,1,26]]},"references-count":42,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2024,2]]}},"alternative-id":["e26020113"],"URL":"https:\/\/doi.org\/10.3390\/e26020113","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2024,1,26]]}}}