{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:10:59Z","timestamp":1760235059082,"version":"build-2065373602"},"reference-count":44,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2021,7,14]],"date-time":"2021-07-14T00:00:00Z","timestamp":1626220800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["61973139, 61473138"],"award-info":[{"award-number":["61973139, 61473138"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012226","name":"Fundamental Research Funds for the Central Universities","doi-asserted-by":"publisher","award":["JUSRP22014"],"award-info":[{"award-number":["JUSRP22014"]}],"id":[{"id":"10.13039\/501100012226","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>This paper investigates the containment control problem of discrete-time first-order multi-agent system composed of multiple leaders and followers, and we propose a proportional-integral (PI) coordination control protocol. Assume that each follower has a directed path to one leader, and we consider several cases according to different topologies composed of the followers. Under the general directed topology that has a spanning tree, the frequency-domain analysis method is used to obtain the sufficient convergence condition for the followers achieving the containment-rendezvous that all the followers reach an agreement value in the convex hull formed by the leaders. Specially, a less conservative sufficient condition is obtained for the followers under symmetric and connected topology. Furthermore, it is proved that our proposed protocol drives the followers with unconnected topology to converge to the convex hull of the leaders. Numerical examples show the correctness of the theoretical results.<\/jats:p>","DOI":"10.3390\/a14070209","type":"journal-article","created":{"date-parts":[[2021,7,14]],"date-time":"2021-07-14T02:28:01Z","timestamp":1626229681000},"page":"209","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Containment Control of First-Order Multi-Agent Systems under PI Coordination Protocol"],"prefix":"10.3390","volume":"14","author":[{"given":"Mingyang","family":"Huang","sequence":"first","affiliation":[{"name":"Key Laboratory of Advanced Process Control for Light Industry (Ministry of Education), Institute of Automation, Jiangnan University, Wuxi 214122, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9032-2991","authenticated-orcid":false,"given":"Chenglin","family":"Liu","sequence":"additional","affiliation":[{"name":"Key Laboratory of Advanced Process Control for Light Industry (Ministry of Education), Institute of Automation, Jiangnan University, Wuxi 214122, China"}]},{"given":"Liang","family":"Shan","sequence":"additional","affiliation":[{"name":"School of Automation, Nanjing University of Science and Technology, Nanjing 210094, China"}]}],"member":"1968","published-online":{"date-parts":[[2021,7,14]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1520","DOI":"10.1109\/TAC.2004.834113","article-title":"Consensus problems in networks of agents with switching topology and time-delays","volume":"49","author":"Murray","year":"2004","journal-title":"IEEE Trans. Automat. Control"},{"key":"ref_2","unstructured":"Olfati-Saber, R. (2005, January 15). Distributed Kalman filter with embedded consensus filters. Proceedings of the 44th IEEE Conference on Decision and Control, Seville, Spain."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Freeman, R.A., Yang, P., and Lynch, K.M. (2006, January 13\u201315). Stability and convergence properties of dynamic average consensus estimators. Proceedings of the 45th IEEE Conference on Decision and Control, San Diego, CA, USA.","DOI":"10.1109\/CDC.2006.377078"},{"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","unstructured":"Ren, W., and Beard, R.W. (2008). Distributed Consensus in Multi-Vehicle Cooperative Control, Springer.","DOI":"10.1007\/978-1-84800-015-5"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Liu, D.J., and Liu, C.L. (2009, January 21\u201322). Consensus problem of discrete-time second-order multi-agent network with communication delays. Proceedings of the 2009 Third International Symposium on Intelligent Information Technology Application, Nanchang, China.","DOI":"10.1109\/IITA.2009.300"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"2019","DOI":"10.1002\/rnc.3187","article-title":"Distributed consensus filter on directed switching graphs","volume":"25","author":"Li","year":"2015","journal-title":"Int. J. Robust Nonlinear Control"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"224","DOI":"10.1016\/j.ins.2018.04.089","article-title":"Adaptive consensus for high-order unknown nonlinear multi-agent systems with unknown control directions and switching topologies","volume":"459","author":"Rezaei","year":"2018","journal-title":"Inf. Sci."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"847","DOI":"10.1049\/iet-cta.2017.0877","article-title":"Adaptive output stationary average consensus for heterogeneous unknown linear multi-agent systems","volume":"12","author":"Rezaei","year":"2018","journal-title":"IET Control Theory Appl."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Liu, C.L., Gu, X.Y., and Shan, L. (2019, January 3\u20135). Average-consensus tracking of ramp inputs via second-order multi-agent systems. Proceedings of the 2019 Chinese Control And Decision Conference (CCDC), Nanchang, China.","DOI":"10.1109\/CCDC.2019.8832549"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"132","DOI":"10.1109\/LCSYS.2018.2856105","article-title":"Average-consensus tracking of sensor network via distributed coordination control of heterogeneous multi-agent systems","volume":"3","author":"Zheng","year":"2019","journal-title":"IEEE Control Syst. Lett."},{"key":"ref_12","unstructured":"Ren, W., Beard, R.W., and Atkins, E.M. (2005, January 8\u201310). A survey of consensus problems in multi-agent coordination. Proceedings of the 2005 American Control Conference, Portland, OR, USA."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"284","DOI":"10.1080\/00207721.2017.1410250","article-title":"Stationary average consensus protocol for a class of heterogeneous high-order multi-agent systems with application for aircraft","volume":"49","author":"Rezaei","year":"2018","journal-title":"Int. J. Syst. Sci."},{"key":"ref_14","first-page":"1859","article-title":"Consensus seeking in multiagent systems under dynamically changing interaction topologies","volume":"50","author":"Ren","year":"2005","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Jiang, W., Wen, G., Meng, Y., and Rahmani, A. (2017, January 26\u201328). Distributed adaptive time-varying formation tracking for linear multi-agent systems: A dynamic output approach. Proceedings of the 2017 36th Chinese Control Conference (CCC), Dalian, China.","DOI":"10.23919\/ChiCC.2017.8028717"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"465","DOI":"10.1016\/j.neucom.2018.07.040","article-title":"Distributed consensus of linear MASs with an unknown leader via a predictive extended state observer considering input delay and disturbances","volume":"315","author":"Jiang","year":"2018","journal-title":"Neurocomputing"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1586","DOI":"10.1016\/j.automatica.2012.05.071","article-title":"Distributed containment control with multiple stationary or dynamic leaders in fixed and switching directed networks","volume":"48","author":"Cao","year":"2012","journal-title":"Automatica"},{"key":"ref_18","unstructured":"Li, B., Cheng, Z.Q., Liu, Z.X., and Zhang, Q. (2015, January 28\u201330). Containment control of discrete-time multi-agent systems with multiple stationary leaders and time-delays. Proceedings of the 2015 34th Chinese Control Conference(CCC), Hangzhou, China."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1016\/j.sysconle.2018.10.006","article-title":"Formation-containment tracking for general linear multi-agent systems with a tracking-leader of unknown control input","volume":"122","author":"Hua","year":"2018","journal-title":"Syst. Control Lett."},{"key":"ref_20","unstructured":"Jiang, W. (2018). Fully Distributed Time-Varying Formation and Containment Control for Multi-Agent\/Multi-Robot Systems. [Ph.D. Thesis, Ecole Centrale de Lille]."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"8706","DOI":"10.1016\/j.jfranklin.2018.11.057","article-title":"Containment control for second-order nonlinear multi-agent systems with aperiodically intermittent position measurements","volume":"356","author":"Wang","year":"2019","journal-title":"J. Frankl. Inst."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"2546","DOI":"10.1109\/TCYB.2019.2915941","article-title":"Containment control of asynchronous discrete-time general linear multiagent systems with arbitrary network topology","volume":"50","author":"Shi","year":"2020","journal-title":"IEEE Trans. Cybern."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"104673","DOI":"10.1016\/j.sysconle.2020.104673","article-title":"Sampled-data containment control for double-integrator agents with dynamic leaders with nonzero inputs","volume":"139","author":"Ding","year":"2020","journal-title":"Syst. Control Lett."},{"key":"ref_24","first-page":"550","article-title":"Containment control for general second-order multiagent systems with switched dynamics","volume":"50","author":"Wang","year":"2020","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1415","DOI":"10.1016\/j.automatica.2012.05.010","article-title":"Necessary and sufficient conditions for containment control of networked multi-agent systems","volume":"48","author":"Liu","year":"2012","journal-title":"Automatica"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1922","DOI":"10.1016\/j.automatica.2014.05.018","article-title":"Containment control of multi-agent systems in a noisy communication environment","volume":"50","author":"Wang","year":"2014","journal-title":"Automatica"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"3600","DOI":"10.1016\/j.jfranklin.2014.10.006","article-title":"Distributed PD-type protocol based containment control of multi-agent systems with input delays","volume":"352","author":"Rong","year":"2015","journal-title":"J. Frankl. Inst."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"2232","DOI":"10.1109\/TAC.2015.2482959","article-title":"Containment control of single-integrator network with limited communication data rate","volume":"61","author":"Mu","year":"2016","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"6956","DOI":"10.1016\/j.jfranklin.2017.08.010","article-title":"Event-triggered containment control for multi-agent systems with constant time delays","volume":"354","author":"Miao","year":"2017","journal-title":"J. Frankl. Inst."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Shao, J.L., Shi, L., and Gong, L.S. (2017, January 25\u201326). Analysis of asynchronous containment control problem for discrete-time multi-agent systems. Proceedings of the 2017 IEEE 2nd Advanced Information Technology, Electronic and Automation Control Conference (IAEAC), Chongqing, China.","DOI":"10.1109\/IAEAC.2017.8054015"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"2919","DOI":"10.1016\/j.jfranklin.2019.01.034","article-title":"Containment control of multi-agent systems via a disturbance observer-based approach","volume":"35","author":"Han","year":"2019","journal-title":"J. Frankl. Inst."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"78","DOI":"10.1016\/j.automatica.2015.04.005","article-title":"Containment control of continuous-time linear multi-agent systems with aperiodic sampling","volume":"57","author":"Liu","year":"2015","journal-title":"Automatica"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"108992","DOI":"10.1016\/j.automatica.2020.108992","article-title":"Distributed containment control of multi-agent systems with velocity and acceleration saturations","volume":"117","author":"Liu","year":"2020","journal-title":"Automatic"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"418","DOI":"10.1016\/j.automatica.2018.12.029","article-title":"Necessary and sufficient conditions for containment control of multi-agent systems with time delay","volume":"103","author":"Wang","year":"2019","journal-title":"Automatica"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"8957","DOI":"10.1016\/j.jfranklin.2018.10.007","article-title":"Average-consensus tracking of multi-agent systems with additional interconnecting agents","volume":"355","author":"Shan","year":"2018","journal-title":"J. Frankl. Inst."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1016\/j.ifacol.2019.12.137","article-title":"Distributed dynamic containment control over a strongly connected and weight-balanced digraph","volume":"52","author":"Chung","year":"2019","journal-title":"IFAC-PapersOnLine"},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Huang, M.Y., and Liu, C.L. (2020, January 6\u20138). Containment-rendezvous control for first-order multi-agent systems. Proceedings of the 2020 Chinese Automation Congress (CAC), Shanghai, China.","DOI":"10.1109\/CAC51589.2020.9327418"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1109\/TAC.2004.841121","article-title":"Necessary and sufficient graphical conditions for formation control of unicycles","volume":"50","author":"Lin","year":"2005","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_39","unstructured":"Parks, P.C., and Hahn, V. (1993). Stability Theory, Prentice Hall."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"185","DOI":"10.3182\/20050703-6-CZ-1902.00975","article-title":"Scalable robustness for consensus protocols with heterogeneous dynamics","volume":"38","author":"Lestas","year":"2005","journal-title":"IFAC Proc. Vol."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1109\/TAC.1980.1102280","article-title":"On the generalized Nyquist stability criterion","volume":"50","author":"Desoer","year":"1980","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"401","DOI":"10.1109\/TAC.2005.864190","article-title":"Flocking for multi-agent dynamic systems: Algorithms and theory","volume":"51","year":"2006","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"680","DOI":"10.1109\/TCNS.2016.2542978","article-title":"Formation control and obstacle avoidance of multiple rectangular agents with limited communication ranges","volume":"4","author":"Nguyen","year":"2016","journal-title":"IEEE Trans. Control Netw. Syst."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"104245","DOI":"10.1016\/j.conengprac.2019.104245","article-title":"Switching time-invariant formation control of a collaborative multi-agent system using negative imaginary systems theory","volume":"95","author":"Tran","year":"2020","journal-title":"Control Eng. Pract."}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/14\/7\/209\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:30:08Z","timestamp":1760164208000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/14\/7\/209"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,7,14]]},"references-count":44,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2021,7]]}},"alternative-id":["a14070209"],"URL":"https:\/\/doi.org\/10.3390\/a14070209","relation":{},"ISSN":["1999-4893"],"issn-type":[{"type":"electronic","value":"1999-4893"}],"subject":[],"published":{"date-parts":[[2021,7,14]]}}}