{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,30]],"date-time":"2025-10-30T22:30:37Z","timestamp":1761863437272,"version":"build-2065373602"},"reference-count":36,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2015,6,2]],"date-time":"2015-06-02T00:00:00Z","timestamp":1433203200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>This paper studies consensus and \\(H_\\infty\\) consensus problems for heterogeneous multi-agent systems composed of first-order and second-order integrator agents. We first rewrite the multi-agent systems into the corresponding reduced-order systems based on the graph theory and the reduced-order transformation. Then, the linear matrix inequality approach is used to consider the consensus of heterogeneous multi-agent systems with time-varying delays in directed networks. As a result, sufficient conditions for consensus and \\(H_\\infty\\) consensus of heterogeneous multi-agent systems in terms of linear matrix inequalities are established in the cases of fixed and switching topologies. Finally, numerical simulations are given to illustrate the effectiveness of the theoretical results.<\/jats:p>","DOI":"10.3390\/e17063631","type":"journal-article","created":{"date-parts":[[2015,6,2]],"date-time":"2015-06-02T11:38:36Z","timestamp":1433245116000},"page":"3631-3644","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["Consensus Analysis of Heterogeneous Multi-Agent Systems with Time-Varying Delay"],"prefix":"10.3390","volume":"17","author":[{"given":"Beibei","family":"Wang","sequence":"first","affiliation":[{"name":"School of Mathematical Sciences, University of Jinan, Jinan 250022, China"}]},{"given":"Yuangong","family":"Sun","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, University of Jinan, Jinan 250022, China"}]}],"member":"1968","published-online":{"date-parts":[[2015,6,2]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"169","DOI":"10.1109\/TAC.2004.841888","article-title":"Stability of multi-agent systems with time-dependent communication links","volume":"50","author":"Moreau","year":"2005","journal-title":"IEEE Trans. Autom. Control."},{"key":"ref_2","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."},{"doi-asserted-by":"crossref","unstructured":"Ren, W., and Beard, R.W. (2008). Distributed Consensus in Multi-Vehicle Cooperative Control, Springer-Verlag.","key":"ref_3","DOI":"10.1007\/978-1-84800-015-5"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"2343","DOI":"10.1016\/j.jfranklin.2012.05.003","article-title":"Consensus of networked multi-agent systems via the networked predictive control and relative outputs","volume":"349","author":"Tan","year":"2012","journal-title":"J. Frankl. Inst."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1986","DOI":"10.1016\/j.automatica.2013.03.015","article-title":"Distributed consensus of linear multi-agent systems with adaptive dynamic protocols","volume":"49","author":"Li","year":"2013","journal-title":"Automatica"},{"key":"ref_6","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":"Saber","year":"2004","journal-title":"IEEE Trans. Autom. Control."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"655","DOI":"10.1109\/TAC.2005.846556","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_8","doi-asserted-by":"crossref","first-page":"918","DOI":"10.1016\/j.sysconle.2006.06.005","article-title":"Non-linear protocols for optimal distributed consensus in networks of dynamic agents","volume":"55","author":"Bauso","year":"2006","journal-title":"Syst. Control Lett."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"2375","DOI":"10.1016\/j.automatica.2008.01.011","article-title":"Distributed nonlinear control algorithms for network consensus","volume":"44","author":"Hui","year":"2008","journal-title":"Automatica."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"152","DOI":"10.1016\/j.sysconle.2012.11.003","article-title":"Consensus for multi-agent systems with inherent nonlinear dynamics under directed topologies","volume":"62","author":"Liu","year":"2013","journal-title":"Syst. Control Lett."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"50","DOI":"10.1088\/1674-1056\/22\/4\/040203","article-title":"Consensus problems of first-order dynamic multi-agent systems with multiple time delays","volume":"22","author":"Ji","year":"2013","journal-title":"Chin. Phys. B."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1937","DOI":"10.1080\/00207170902838269","article-title":"Consensus problems in networks of agents with double-integrator dynamics and time-varying delays","volume":"82","author":"Sun","year":"2009","journal-title":"Int. J. Control."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1706","DOI":"10.1016\/j.automatica.2011.02.045","article-title":"Finite-time consensus algorithm for multi-agent systems with double-integrator dynamics","volume":"47","author":"Li","year":"2011","journal-title":"Automatica"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1419","DOI":"10.1016\/j.automatica.2013.01.058","article-title":"Distributed coordination for second-order multi-agent systems with nonlinear dynamics using only relative position measurements","volume":"49","author":"Mei","year":"2013","journal-title":"Automatica"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1881","DOI":"10.1049\/iet-cta.2011.0033","article-title":"Consensus of heterogeneous multi-agent systems","volume":"5","author":"Zheng","year":"2011","journal-title":"IET Control Theory Appl"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"906","DOI":"10.1080\/00207179.2012.669048","article-title":"Consensus of heterogeneous multi-agent systems without velocity measurements","volume":"85","author":"Zheng","year":"2012","journal-title":"Int. J. Control."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"871","DOI":"10.1016\/j.sysconle.2012.05.009","article-title":"Finite-Time consensus of heterogeneous multi-agent systems with and without velocity measurements","volume":"61","author":"Zheng","year":"2012","journal-title":"Syst. Control Lett."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1277","DOI":"10.1080\/00207170600825097","article-title":"State consensus for multi-agent systems with switching topologies and time-varying delays","volume":"79","author":"Xiao","year":"2006","journal-title":"Int. J. Control."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1607","DOI":"10.1109\/TAC.2009.2017963","article-title":"Consensus of multi-agent systems in directed networks with nonuniform time-varying delays","volume":"54","author":"Sun","year":"2009","journal-title":"IEEE Trans. Autom. Control."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"175","DOI":"10.1016\/j.sysconle.2007.08.009","article-title":"Average consensus in networks of dynamic agents with switching topologies and multiple time-varying delays","volume":"57","author":"Sun","year":"2008","journal-title":"Syst. Control Lett."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1061","DOI":"10.1016\/j.jfranklin.2011.12.007","article-title":"Average consensus in networks of dynamic agents with uncertain topologies and time-varying delays","volume":"349","author":"Sun","year":"2012","journal-title":"J. Frankl. Inst."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"848","DOI":"10.1016\/j.automatica.2011.01.053","article-title":"Multi-Agent consensus with diverse time-delays and jointly-connected topologies","volume":"47","author":"Lin","year":"2011","journal-title":"Automatica"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"293","DOI":"10.1016\/j.jfranklin.2011.11.002","article-title":"A new approach to average consensus problems with multiple time-delays and jointly-connected topologies","volume":"349","author":"Lin","year":"2012","journal-title":"J. Frankl. Inst."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"853","DOI":"10.1016\/j.physa.2006.08.015","article-title":"Leader-Following coordination of multi-agent systems with coupling time delays","volume":"374","author":"Hu","year":"2007","journal-title":"Physica A."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"5020","DOI":"10.3390\/e16095020","article-title":"Adaptive Leader-Following Consensus of Multi-Agent Systems with Unknown Nonlinear Dynamics","volume":"16","author":"Wang","year":"2014","journal-title":"Entropy"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1016\/j.sysconle.2014.05.010","article-title":"Leader-Following consensus of multi-agent systems under directed communication topology via distributed adaptive nonlinear protocol","volume":"70","author":"Wang","year":"2014","journal-title":"Syst. Control Lett."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"2577","DOI":"10.1016\/j.automatica.2008.02.017","article-title":"Consensus protocols for discrete-time multi-agent systems with time-varying delays","volume":"44","author":"Xiao","year":"2008","journal-title":"Automatica"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"883","DOI":"10.1016\/j.automatica.2013.12.008","article-title":"Distributed robust consensus control of multi-agent systems with heterogeneous matching uncertainties","volume":"50","author":"Li","year":"2014","journal-title":"Automatica"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"6524","DOI":"10.3390\/e16126524","article-title":"Consensus of Discrete Multiagent System with Various Time Delays and Environmental Disturbances","volume":"16","author":"Yan","year":"2014","journal-title":"Entropy"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.automatica.2010.10.002","article-title":"Distributed robust filtering with H\u221e consensus of estimates","volume":"47","author":"Ugrinovskii","year":"2011","journal-title":"Automatica"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"857","DOI":"10.1016\/j.sysconle.2012.05.007","article-title":"H\u221e consensus of second-order multi-agent systems with asymmetric delays","volume":"61","author":"Sun","year":"2012","journal-title":"Syst. Control Lett."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"487","DOI":"10.1049\/iet-cta.2008.0492","article-title":"Robust H\u221e consensus analysis of a class of second-order multi-agent systems with uncertainty","volume":"4","author":"Lin","year":"2010","journal-title":"IET Control Theory Appl."},{"doi-asserted-by":"crossref","unstructured":"Boyd, B., Ghaoui, L.E., Feron, E., and Balakrishnan, V. (1994). Linear Matrix Inequalities in System and Control Theory, SIAM.","key":"ref_33","DOI":"10.1137\/1.9781611970777"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1016\/j.automatica.2014.10.092","article-title":"Event-Based H\u221e filtering for sampled-data systems","volume":"51","author":"Zhang","year":"2014","journal-title":"Automatica"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"157","DOI":"10.1016\/j.automatica.2007.04.024","article-title":"Robust image filtering for a class of uncertain linear systems with time-varying delay","volume":"44","author":"Zhang","year":"2008","journal-title":"Automatica"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"1054","DOI":"10.1049\/iet-cta.2013.0840","article-title":"New stability criterion using a matrix-based quadratic convex approach and some novel integral inequalities","volume":"8","author":"Zhang","year":"2014","journal-title":"IET Control Theory Appl"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/17\/6\/3631\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T20:47:19Z","timestamp":1760215639000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/17\/6\/3631"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,6,2]]},"references-count":36,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2015,6]]}},"alternative-id":["e17063631"],"URL":"https:\/\/doi.org\/10.3390\/e17063631","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2015,6,2]]}}}