{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,27]],"date-time":"2026-03-27T09:57:13Z","timestamp":1774605433946,"version":"3.50.1"},"reference-count":38,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2025,9,11]],"date-time":"2025-09-11T00:00:00Z","timestamp":1757548800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,9,11]],"date-time":"2025-09-11T00:00:00Z","timestamp":1757548800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["Grant 62466011"],"award-info":[{"award-number":["Grant 62466011"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Natural Science Foundation of Guangxi","award":["Grant 2023GXNSFBA026141"],"award-info":[{"award-number":["Grant 2023GXNSFBA026141"]}]},{"name":"Science and Technology Project of Guangxi","award":["Grant Guike AD23023002"],"award-info":[{"award-number":["Grant Guike AD23023002"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Circuits Syst Signal Process"],"published-print":{"date-parts":[[2026,3]]},"DOI":"10.1007\/s00034-025-03322-3","type":"journal-article","created":{"date-parts":[[2025,9,11]],"date-time":"2025-09-11T14:53:12Z","timestamp":1757602392000},"page":"1848-1864","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Adaptive Hierarchical Pinning Control of Lag Consensus for Multiagent Systems on Two-Layer Dynamical Networks"],"prefix":"10.1007","volume":"45","author":[{"given":"Junsheng","family":"Yu","sequence":"first","affiliation":[]},{"given":"Zhongjun","family":"Ma","sequence":"additional","affiliation":[]},{"given":"Huizhi","family":"Xu","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,9,11]]},"reference":[{"issue":"4","key":"3322_CR1","doi-asserted-by":"publisher","first-page":"933","DOI":"10.1109\/TFUZZ.2021.3050847","volume":"30","author":"Y Arefi","year":"2022","unstructured":"Y. Arefi, A. Khayatian, O. Kaynak, Event-triggered fuzzy adaptive leader-following tracking control of nonaffine multiagent systems with finite-time output constraint and input saturation. IEEE Trans. Fuzzy Syst. 30(4), 933\u2013944 (2022)","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"3322_CR2","doi-asserted-by":"publisher","first-page":"115485","DOI":"10.1016\/j.chaos.2024.115485","volume":"188","author":"Y Cen","year":"2024","unstructured":"Y. Cen, L. Cao, L. Zhang, Y. Pan, H. Liang, GPIO-based optimal containment control for autonomous underwater vehicles with external disturbances. Chaos Soli. Fract. 188, 115485 (2024)","journal-title":"Chaos Soli. Fract."},{"issue":"2","key":"3322_CR3","doi-asserted-by":"publisher","first-page":"611","DOI":"10.1007\/s00034-018-0872-y","volume":"38","author":"S Chen","year":"2019","unstructured":"S. Chen, B. Chen, F. Shi, Distributed fault-tolerant consensus protocol for fuzzy multi-agent systems. Circuits Syst. Signal Process. 38(2), 611\u2013624 (2019)","journal-title":"Circuits Syst. Signal Process."},{"issue":"4","key":"3322_CR4","first-page":"373","volume":"14","author":"Y Cao","year":"2024","unstructured":"Y. Cao, A.R. Subhashri, A. Chandrasekar, T. Radhika, V. Przybyszewski, Exponential state estimation for delayed competitive neural network via stochastic sampled-data control with Markov jump parameters under actuator failure. J. Artif. Intell. Soft. 14(4), 373\u2013385 (2024)","journal-title":"J. Artif. Intell. Soft."},{"key":"3322_CR5","doi-asserted-by":"publisher","first-page":"174","DOI":"10.1016\/j.matcom.2023.08.007","volume":"222","author":"Y Cao","year":"2024","unstructured":"Y. Cao, A. Chandrasekar, T. Radhika, V. Vijayakumar, Input-to-state stability of stochastic Markovian jump genetic regulatory networks. Math. Comput. Simul. 222, 174\u2013187 (2024)","journal-title":"Math. Comput. Simul."},{"issue":"3","key":"3322_CR6","doi-asserted-by":"publisher","first-page":"576","DOI":"10.1109\/TCSI.2010.2072270","volume":"58","author":"P DeLellis","year":"2011","unstructured":"P. DeLellis, M. Bernardo, G. Russo, On QUAD, Lipschitz, and Contracting vector fields for consensus and synchronization of networks. IEEE Trans. Circuits Syst. I Regul. Pap. 58(3), 576\u2013583 (2011)","journal-title":"IEEE Trans. Circuits Syst. I Regul. Pap."},{"key":"3322_CR7","doi-asserted-by":"publisher","first-page":"78","DOI":"10.1016\/j.neucom.2019.06.057","volume":"363","author":"X Du","year":"2019","unstructured":"X. Du, W. Yu, Consensus of multi-agent systems with delayed sampled-data and directed topologies. Neurocomputing 363, 78\u201387 (2019)","journal-title":"Neurocomputing"},{"issue":"9","key":"3322_CR8","doi-asserted-by":"publisher","first-page":"1465","DOI":"10.1109\/TAC.2004.834433","volume":"49","author":"J Fax","year":"2004","unstructured":"J. Fax, R. Murray, Information flow and cooperative control of vehicle formations. IEEE Trans. Autom. Control 49(9), 1465\u20131476 (2004)","journal-title":"IEEE Trans. Autom. Control"},{"issue":"6","key":"3322_CR9","doi-asserted-by":"publisher","first-page":"5470","DOI":"10.1109\/TITS.2021.3054177","volume":"23","author":"X Hu","year":"2021","unstructured":"X. Hu, X. Wei, Y. Kao, J. Han, Robust synchronization for under-actuated vessels based on disturbance observer. IEEE Trans. Intell. Transp. Syst. 23(6), 5470\u20135479 (2021)","journal-title":"IEEE Trans. Intell. Transp. Syst."},{"issue":"10","key":"3322_CR10","doi-asserted-by":"publisher","first-page":"4759","DOI":"10.1007\/s00034-021-01701-0","volume":"40","author":"D Huong","year":"2020","unstructured":"D. Huong, Event-triggered guaranteed cost control for uncertain neural networks systems with time delays. Circuits Syst. Signal Process. 40(10), 4759\u20134778 (2020)","journal-title":"Circuits Syst. Signal Process."},{"issue":"13","key":"3322_CR11","doi-asserted-by":"publisher","first-page":"5004","DOI":"10.1002\/rnc.5032","volume":"30","author":"X Hu","year":"2020","unstructured":"X. Hu, X. Wei, H. Zhang, J. Han, X. Zhu, Adaptive disturbance estimation and cancelation for ships under thruster saturation. Int. J. Robust Non. Cont. 30(13), 5004\u20135020 (2020)","journal-title":"Int. J. Robust Non. Cont."},{"issue":"3","key":"3322_CR12","doi-asserted-by":"publisher","first-page":"2293","DOI":"10.1007\/s11071-018-4692-1","volume":"95","author":"X Hu","year":"2019","unstructured":"X. Hu, X. Wei, H. Zhang, J. Han, X. Liu, Global asymptotic regulation control for MIMO mechanical systems with unknown model parameters and disturbances. Nonlinear Dyn. 95(3), 2293\u20132305 (2019)","journal-title":"Nonlinear Dyn."},{"key":"3322_CR13","doi-asserted-by":"publisher","first-page":"107920","DOI":"10.1016\/j.oceaneng.2020.107920","volume":"216","author":"X Hu","year":"2020","unstructured":"X. Hu, X. Wei, G. Zhu, D. Wu, Adaptive synchronization for surface vessels with disturbances and saturated thruster dynamics. Ocean Eng. 216, 107920 (2020)","journal-title":"Ocean Eng."},{"issue":"3","key":"3322_CR14","doi-asserted-by":"publisher","first-page":"1134","DOI":"10.1109\/TCST.2015.2478078","volume":"24","author":"R Haghighi","year":"2016","unstructured":"R. Haghighi, C. Cheah, Two-layered framework for distributed multiagent formation following. IEEE Trans. Cont. Syst. Tech. 24(3), 1134\u20131139 (2016)","journal-title":"IEEE Trans. Cont. Syst. Tech."},{"key":"3322_CR15","doi-asserted-by":"publisher","first-page":"105930","DOI":"10.1016\/j.sysconle.2024.105930","volume":"193","author":"Y Lin","year":"2024","unstructured":"Y. Lin, Y. Ai, L. Mo, P. Lin, Y. Jia, Containment control for high-order multi-agent systems with nonconvex control input constraints and non-particle leaders. Syst. Cont. Lett. 193, 105930 (2024)","journal-title":"Syst. Cont. Lett."},{"issue":"9","key":"3322_CR16","doi-asserted-by":"publisher","first-page":"9490","DOI":"10.1109\/TCYB.2021.3061700","volume":"52","author":"K Li","year":"2022","unstructured":"K. Li, Y. Bai, Z. Ma, J. Cao, Feedback pinning control of successive lag synchronization on a dynamical network. IEEE Trans. Cybern. 52(9), 9490\u20139503 (2022)","journal-title":"IEEE Trans. Cybern."},{"issue":"12","key":"3322_CR17","doi-asserted-by":"publisher","first-page":"123120","DOI":"10.1063\/1.5122699","volume":"29","author":"C Ma","year":"2019","unstructured":"C. Ma, Q. Yang, X. Wu, J. Lu, Cluster synchronization: from single-layer to multi-layer networks. Chaos 29(12), 123120 (2019)","journal-title":"Chaos"},{"key":"3322_CR18","doi-asserted-by":"publisher","first-page":"126888","DOI":"10.1016\/j.neucom.2023.126888","volume":"561","author":"F Mu\u00f1oz","year":"2023","unstructured":"F. Mu\u00f1oz, J. Valdovinos, J. Cervantes-Rojas, S. Cruz, A. Santana, Leader-follower consensus control for a class of nonlinear multi-agent systems using dynamical neural networks. Neurocomputing 561, 126888 (2023)","journal-title":"Neurocomputing"},{"issue":"3","key":"3322_CR19","doi-asserted-by":"publisher","first-page":"1303","DOI":"10.1007\/s11071-015-2403-8","volume":"83","author":"Z Ma","year":"2016","unstructured":"Z. Ma, Y. Wang, X. Li, Cluster-delay consensus in first-order multi-agent systems with nonlinear dynamics. Nonlinear Dyn. 83(3), 1303\u20131310 (2016)","journal-title":"Nonlinear Dyn."},{"key":"3322_CR20","first-page":"128522","volume":"468","author":"R Peng","year":"2024","unstructured":"R. Peng, R. Guo, B. Zheng, Z. Miao, J. Zhou, Neural network-based robust consensus tracking for uncertain networked Euler\u2013Lagrange systems with time-varying delays and output constraints. Appl. Math. Comput. 468, 128522 (2024)","journal-title":"Appl. Math. Comput."},{"key":"3322_CR21","doi-asserted-by":"publisher","first-page":"49","DOI":"10.1016\/j.isatra.2020.09.012","volume":"109","author":"Y Ren","year":"2021","unstructured":"Y. Ren, W. Zhou, Z. Li, L. Liu, Y. Sun, Prescribed-time cluster lag consensus control for second-order non-linear leader-following multiagent systems. ISA Trans. 109, 49\u201360 (2021)","journal-title":"ISA Trans."},{"key":"3322_CR22","doi-asserted-by":"publisher","first-page":"1303","DOI":"10.1016\/j.jfranklin.2023.12.054","volume":"361","author":"Y Sun","year":"2024","unstructured":"Y. Sun, C. Hu, J. Yu, Bipartite leaderless synchronization of fractional-order coupled neural networks via edge-based adaptive pinning control. J. Frankl. Inst. 361, 1303\u20131317 (2024)","journal-title":"J. Frankl. Inst."},{"key":"3322_CR23","doi-asserted-by":"publisher","first-page":"450","DOI":"10.1016\/j.isatra.2021.10.016","volume":"128","author":"S Sun","year":"2022","unstructured":"S. Sun, T. Ren, Y. Xu, Pinning synchronization control for stochastic multi-layer networks with coupling disturbance. ISA Trans. 128, 450\u2013459 (2022)","journal-title":"ISA Trans."},{"issue":"8","key":"3322_CR24","doi-asserted-by":"publisher","first-page":"3490","DOI":"10.1109\/TNNLS.2017.2726354","volume":"29","author":"Y Wang","year":"2018","unstructured":"Y. Wang, Z.M.G. Chen, Avoiding congestion in cluster consensus of the second-order nonlinear multiagent systems. IEEE Trans. Neu. Netw. Learn Syst. 29(8), 3490\u20133498 (2018)","journal-title":"IEEE Trans. Neu. Netw. Learn Syst."},{"issue":"6","key":"3322_CR25","doi-asserted-by":"publisher","first-page":"2093","DOI":"10.1109\/TSMC.2018.2797542","volume":"50","author":"Y Wang","year":"2020","unstructured":"Y. Wang, Y. Li, Z. Ma, G. Cai, G. Chen, Cluster lag consensus for second-order multiagent systems with nonlinear dynamics and switching topologies. IEEE Trans. Syst. Man Cybern. 50(6), 2093\u20132100 (2020)","journal-title":"IEEE Trans. Syst. Man Cybern."},{"issue":"8","key":"3322_CR26","doi-asserted-by":"publisher","first-page":"2203","DOI":"10.1109\/TCYB.2016.2591518","volume":"47","author":"Y Wang","year":"2017","unstructured":"Y. Wang, Z. Ma, S. Zheng, G. Chen, Pinning control of lag consensus for second-order nonlinear multiagent systems. IEEE Trans. Cybern. 47(8), 2203\u20132211 (2017)","journal-title":"IEEE Trans. Cybern."},{"issue":"5","key":"3322_CR27","doi-asserted-by":"publisher","first-page":"2852","DOI":"10.1109\/TCYB.2021.3120847","volume":"53","author":"Y Wang","year":"2021","unstructured":"Y. Wang, H. Song, G. Chen, Z. Ma, J. Cao, $$p$$ Components of cluster-lag consensus for second-order multiagent systems with adaptive controller on cooperative-competitive networks. IEEE Trans. Cybern. 53(5), 2852\u20132863 (2021)","journal-title":"IEEE Trans. Cybern."},{"key":"3322_CR28","doi-asserted-by":"publisher","first-page":"126899","DOI":"10.1016\/j.physa.2022.126899","volume":"593","author":"M Wang","year":"2022","unstructured":"M. Wang, X. Lu, Q. Yang, Z. Ma, J. Cheng, K. Li, Pinning control of successive lag synchronization on a dynamical network with noise perturbation. Phys. A 593, 126899 (2022)","journal-title":"Phys. A"},{"key":"3322_CR29","doi-asserted-by":"publisher","first-page":"231","DOI":"10.1016\/j.neucom.2016.03.073","volume":"207","author":"B Wang","year":"2016","unstructured":"B. Wang, J. Wang, B. Zhang, H. Lin, X. Li, H. Wang, Leader-follower consensus for multi-agent systems with three-layer network framework and dynamic interaction jointly connected topology. Neurocomputing 207, 231\u2013239 (2016)","journal-title":"Neurocomputing"},{"issue":"9","key":"3322_CR30","doi-asserted-by":"publisher","first-page":"3152","DOI":"10.1109\/TCSI.2020.2986953","volume":"67","author":"G Wen","year":"2020","unstructured":"G. Wen, P. Wang, T. Huang, J. Lu, Distributed consensus of layered multi-agent systems subject to attacks on edges. IEEE Trans. Circuits Syst. I Regul. Pap. 67(9), 3152\u20133162 (2020)","journal-title":"IEEE Trans. Circuits Syst. I Regul. Pap."},{"key":"3322_CR31","doi-asserted-by":"publisher","first-page":"130","DOI":"10.1016\/j.neucom.2022.02.081","volume":"488","author":"Q Wang","year":"2022","unstructured":"Q. Wang, W. He, L. Zino, D. Tan, W. Zhong, Bipartite consensus for a class of nonlinear multi-agent systems under switching topologies: a disturbance observer-based approach. Neurocomputing 488, 130\u2013143 (2022)","journal-title":"Neurocomputing"},{"key":"3322_CR32","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.physrep.2024.01.005","volume":"1060","author":"X Wu","year":"2024","unstructured":"X. Wu, X. Wu, C. Wang, B. Mao, J. Lu, J. L\u00fc, Y. Zhang, L. L\u00fc, Synchronization in multiplex networks. Phy. Rep. 1060, 1\u201354 (2024)","journal-title":"Phy. Rep."},{"issue":"4","key":"3322_CR33","first-page":"17","volume":"63","author":"Y Xie","year":"2014","unstructured":"Y. Xie, Y. Wang, Z. Ma, Delay consensus of leader-following multi-agent system. Acta Phys. Sin. 63(4), 17\u201321 (2014)","journal-title":"Acta Phys. Sin."},{"issue":"6","key":"3322_CR34","doi-asserted-by":"publisher","first-page":"3500","DOI":"10.1007\/s00034-024-02646-w","volume":"43","author":"J Yu","year":"2024","unstructured":"J. Yu, Z. Ma, K. Li, M. Chang, Event-triggered control of lag consensus for leader-following systems with nonlinear dynamics. Circuits Syst. Signal Process. 43(6), 3500\u20133515 (2024)","journal-title":"Circuits Syst. Signal Process."},{"issue":"7","key":"3322_CR35","doi-asserted-by":"publisher","first-page":"2107","DOI":"10.1016\/j.automatica.2013.03.005","volume":"49","author":"W Yu","year":"2013","unstructured":"W. Yu, W. Ren, W. Zheng, G. Chen, Distributed control gains design for consensus in multi-agent systems with second-order nonlinear dynamics. Automatica 49(7), 2107\u20132115 (2013)","journal-title":"Automatica"},{"issue":"12","key":"3322_CR36","doi-asserted-by":"publisher","first-page":"10009","DOI":"10.1007\/s11071-024-09574-z","volume":"112","author":"Y Yan","year":"2024","unstructured":"Y. Yan, X. He, L. Wu, Q. Yu, Adaptive event-triggered output-feedback consensus tracking control for switched multi-agent systems. Nonlinear Dyn. 112(12), 10009\u201310024 (2024)","journal-title":"Nonlinear Dyn."},{"issue":"8","key":"3322_CR37","first-page":"1377","volume":"40","author":"J Yang","year":"2023","unstructured":"J. Yang, Z. Ma, K. Li, Partial component consensus for two-layer multi-agent systems. Contl. Theor. Appl. 40(8), 1377\u20131383 (2023)","journal-title":"Contl. Theor. Appl."},{"issue":"4","key":"3322_CR38","doi-asserted-by":"publisher","first-page":"2068","DOI":"10.1109\/TCYB.2019.2931706","volume":"51","author":"H Zhang","year":"2021","unstructured":"H. Zhang, D. Yue, C. Dou, K. Li, X. Xie, Event-triggered multiagent optimization for two-layered model of hybrid energy system with price bidding-based demand response. IEEE Trans. Cybern. 51(4), 2068\u20132079 (2021)","journal-title":"IEEE Trans. Cybern."}],"container-title":["Circuits, Systems, and Signal Processing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00034-025-03322-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00034-025-03322-3","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00034-025-03322-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,3,27]],"date-time":"2026-03-27T09:14:11Z","timestamp":1774602851000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00034-025-03322-3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,9,11]]},"references-count":38,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2026,3]]}},"alternative-id":["3322"],"URL":"https:\/\/doi.org\/10.1007\/s00034-025-03322-3","relation":{},"ISSN":["0278-081X","1531-5878"],"issn-type":[{"value":"0278-081X","type":"print"},{"value":"1531-5878","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,9,11]]},"assertion":[{"value":"19 January 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"16 August 2025","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"18 August 2025","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"11 September 2025","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}