Published in IET Control Theory and Applications Received on 15th November 2009Revised on 7th May 2010doi:10.1049/iet-cta.2009.0589ISSN 1751-8644Brief PaperSwarm stability of high-order linear time-invariant swarm systemsN.Cai 1,2J.-X.Xi 1,3Y.-S.Zhong 11Department of Automation,Tsinghua University,Beijing 100084,People’s Republic of China 2School of Electrical Engineering and Automation,Tianjin Polytechnic University,Beijing 100084,People’s Republic of China 3High-Tech Institute of Xi’an,Xi’an 710025,People’s Republic of China E-mail:cain04@mails.tsinghua.edu.cnAbstract:In this study the swarm stability problem of high-order linear time-invariant (LTI)swarm systems with directed graph topology is dealt with.Consensus can be regarded as a specific type of swarm stability problem.Necessary and sufficient conditions for both swarm stability and consensus are presented.These conditions depend on the graph topology,the dynamics of agents and the interactions among the neighbours.Simulation instances are shown to illustrate the theoretical results.1IntroductionSwarm systems are multi-agent dynamic systems composed of autonomous agents interacting with each other.Models of these systems originated from bioscience.Studies on them have aroused great interest in the field of automatic control theory since about a decade ago.Stability is usually a basic requirement for control systems.Stability of swarm systems is special because of the configuration difference between these systems and isolated dynamic systems.The research on stability problems of swarm systems has been performed for years.Researchers have endeavoured to give criteria for checking whether a specific swarm system is stable.For instance,Jin et al.[1]paid attention to the stability of a simple swarm system rather early.The background of their research came from engineering systems such as multi-agent supporting system [2].Based on the work of Jin,Liu et al.[3,4]tried to extend the research to more general and critical situations,and they studied the so-called ‘asynchronous’swarm systems.However,the technical setting of their model is complicated and the results are restrictive.Parallel to the research of Liu et al.,Gazi and Passino [5,6]and Gazi [7]studied the stability problem of some first-order models abstracted from biological systems.Chu et al.[8]extended the discussions in [5]to certain ‘anisotropic’model.Recently,Li [9]extended the results in [5]to the situation with general graph topology instead of a complete graph.Besides,there were also other studies related to the stability of swarm systems,for example the study about ‘string stability’[10,11].The consensus problems originated in computer science long ago and in some sense formed the foundation of distributed computing [12].Since about five years ago,consensus has become a major topic in swarm system study.Olfati-Saber et al.first introduced the term ‘consensus’into control theory,and performed inaugurating research on it.For instance [13],they studied some first-order swarm systems and discovered that strong connection of the digraph is a sufficient condition for consensus achievement.Ren and Beard [14]relaxed the condition in [13]and proved that the requirement for the digraph to include a spanning tree is necessary and sufficient.An approach based on local convex analysis was developed by Moreau [15].He presented a condition for the consensus of specific swarm systems.Until about 2007,the majority of the researchers studying consensus problems had dealt with first-order models without agent dynamics whereas the analysis of high-order swarm systems is much more involved and challenging.Moreau’s model [15]is of high order and non-linear,whereas the technical assumptions are strict.Recently,there have appeared a few papers on the consensus problems for high-order swarm systems.For instance,Xiao and Wang [16]proposed a criterion based on the structure of certain high-dimensional matrices.Ren et al.[17]and Wieland et al.[18],respectively,studied linear time-invariant (LTI)swarm systems with specific structure.Ren et al.[17]gave a necessary and sufficient condition relying on the spectrum of a high-dimensional matrix whereas Wieland et al.[18]gave a criterion that checks whether a given polynomial is Hurwitz.Wang et al.[19]endeavoured to determine whether an appropriate linear high-order consensus algorithm exists under a given undirected graph topology.Cai and Zhong [20]defined the concept ‘formation controllability’on the basis of consensus and presented a necessary and sufficient condition for controllability of high-order swarm systems.They [21,22]also analysed the consensus conditions for402IET Control Theory Appl.,2011,Vol.5,Iss.2,pp.402–408&The Institution of Engineering and Technology 2011doi:10.1049/iet-cta.2009.0589www.ietdl.org
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