This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and

Transcription

1 Ths artcle appeared n a ournal publshed by Elsever The attached copy s furnshed to the author for nternal non-commercal research and educaton use, ncludng for nstructon at the authors nsttuton and sharng wth colleagues Other uses, ncludng reproducton and dstrbuton, or sellng or lcensng copes, or postng to personal, nsttutonal or thrd party webstes are prohbted n most cases authors are permtted to post ther verson of the artcle (eg n Word or Tex form to ther personal webste or nsttutonal repostory Authors requrng further nformaton regardng Elsever s archvng and manuscrpt polces are encouraged to vst:

2 Systems & Control Letters 59 ( Contents lsts avalable at ScenceDrect Systems & Control Letters ournal homepage: wwwelsevercom/locate/sysconle Partal state consensus for networs of second-order dynamc agents Feng Xao a,,, Long Wang b, Je Chen a a School of Automaton, Beng nsttute of Technology, Beng 0008, Chna b ntellgent Control Laboratory, Center for Systems and Control, College of Engneerng, and Key Laboratory of Machne Percepton (Mnstry of Educaton, Peng Unversty, Beng 0087, Chna a r t c l e n f o a b s t r a c t Artcle hstory: Receved 4 March 00 Receved n revsed form 3 August 00 Accepted 4 September 00 Avalable onlne 3 October 00 Keywords: Mult-agent systems Partal state consensus Tme-delay Swtchng topology Ths paper addresses the partal state consensus problem of mult-agent systems wth second-order agent dynamcs and proposes an asynchronous dstrbuted consensus protocol for the case wth swtchng nteracton topology, tme-varyng delays and ntermttent nformaton transmsson Partal state consensus means reachng an agreement asymptotcally wth each other on part, but not all, of each ndvdual s states, where the concerned states usually cannot be decoupled from the other ones Partal state consensus has ts broad applcatons n the coordnaton of mult-robot systems, dstrbuted tas management, and dstrbuted estmaton for sensor networs, etc Ths paper assumes that poston-le states are the only detectable nformaton transmtted over the networ and velocty-le states are the ey quanttes of nterest, whch are requred to be equalzed We frst gve the asynchronous dstrbuted protocol based on the delayed poston-le state nformaton and then provde ts convergence result wth respect to velocty-le states t s shown that f the unon of the nteracton topology across the tme nterval wth a gven length always contans a spannng tree, then the proposed protocol wll solve the partal state (velocty-le state consensus problem asymptotcally 00 Elsever BV All rghts reserved ntroducton Consensus theory, as a basc and fundamental research topc n dstrbuted coordnaton of mult-agent systems, has receved consderable attenton from researchers n recent years because of ts potental applcatons n cooperatve control of unmanned ar vehcles, formaton control of moble robots, desgn of sensor networs, swarm-based computng, etc [ 5] t requres that all agents reach an agreement on certan quanttes of nterest The shared common value may be the antcpated atttude n multple spacecraft algnment, poston and velocty n flocng control, or processng rate n dstrbuted tas management Due to the complexty of networ dynamcs, a large amount of wors assume that each ndvdual follows a frst-order dfferental equaton [6,7] And there s also a fracton of the lterature concentratng on the networs wth second-order agent dynamcs For nstance, n [8], Xe and Wang consdered the state consensus problem n networs of multple double-ntegrators Ths wor was supported by NSFC ( , and and the Beng Educaton Commttee Cooperaton Buldng Foundaton Proect Correspondng author E-mal addresses: fxao@eceualbertaca, fengxao@pueducn (F Xao He s now vstng the Department of Electrcal and Computer Engneerng, Unversty of Alberta, Edmonton, Alberta T6G V4, Canada under fxed and swtchng topologes and desgned the protocols ensurng that poston-le states converge to a common statc value asymptotcally n [9], Hong et al used a set of frstorder agents to trac an actve second-order leader, where the consensus state may be tme-varyng Ren proposed and analyzed consensus algorthms for networs of double-ntegrators n [0,] and second-order lnear harmonc oscllators n [] under the assumpton that each agent can fully or partally access ts neghbors relatve states Along ths lne, fnte-tme agreement for multple double-ntegrators was dscussed n [3], where the proposed protocols are non-smooth and ther consensus property was proved by the theory of fnte-tme homogeneous systems Under these algorthms, all agents states, ncludng postonle and velocty-le states, wll reach consensus asymptotcally However, n many practcal stuatons, t often occurs that only a small part of state varables of each agent are the ey quanttes of nterest that are requred to be coordnated and usually cannot be decoupled from the other ones n these cases, the concept of consensus n the tradtonal sense wll not be sutable Partal state consensus means reachng an agreement asymptotcally wth each other on part, but not all, of each ndvdual s states, and n ts studes, we may face many new challenges caused by the constrant of coupled agreement and nonagreement varables, such as characterzng the partal state consensus problem n the framewor of coordnaton of full state varables, dstrbuted estmatons for nterestng states of neghborng agents va the /$ see front matter 00 Elsever BV All rghts reserved do:006/sysconle

3 776 F Xao et al / Systems & Control Letters 59 ( nonagreement coupled nformaton, and so on Those challenges greatly ncrease the dffculty of desgn and analyss of partal state consensus protocols As the frst step towards the general study on partal state consensus, ths paper consders the smplest case and nvestgates the networs of second-order dynamc agents t s assumed that poston-le states are the only detectable nformaton transmtted over the networ and velocty-le states are the quanttes of nterest, whch are requred to be equalzed Next, we show several scenaros where the velocty-le state s the only quantty of nterest The frst example ncludes the varous versons of the Vcse model [7,4,5], where the velocty consensus s a prerequste for further study Also n the theoretcal study, n [6], Barbarossa and Scutar proposed a decentralzed sensor networ scheme capable of reachng a globally optmum maxmum-lelhood estmate through self-synchronzaton of nonlnearly coupled dynamcal systems Although each node n the networ they studed s a frst-order dynamcal system, the fnal agreement estmate s related to the state dervatve of each sensor, namely, the authors studed the velocty-le state consensus problem n essence n applcatons, one example s the congeston control of the nternet t s desrable that each router coordnates ts data-processng rate to be consstent wth ts neghbors accordng to the amount of date processed n the latest tme, snce velocty coordnaton can greatly reduce the queung delay and pacage loss rate, and enhance the processng effcency Another example s the decomposton of complex tass Each agent also should coordnate ts processng rate to mprove wor effcency The man contrbuton of ths paper s to provde an effectve partal state consensus control strategy, vald for the case wth swtchng nteracton topology, tme-varyng delays and ntermttent nformaton transmsson We frst gve the desgn result of the dstrbuted coordnaton protocol based on delayed poston-le state nformaton Then by usng the tools from graph theory and nonnegatve matrx theory, we show that f the unon of the nteracton topology across the tme nterval wth some gven length always contans a spannng tree, the partal state consensus problem wll be solvable Moreover, the studed system s an asynchronous one, whch means that each agent does not necessarly approxmate ts neghbors velocty-le states for ts local feedbac at the same tme-steps by a global cloc Ths paper s organzed as follows Prelmnary notons are assembled n Secton The consdered problem s formulated n Secton 3 The man result s presented n Secton 4 and ts techncal proof s postponed to Secton 5 Fnally, concludng remars are summarzed n Secton 6 Notatons: Throughout ths paper, let = A = A A A, denotng the left product of matrces, let be the dentty matrx and let = [,,, ] T wth compatble dmensons We wrte A B f A B s nonnegatve Prelmnary notons n graph theory A drected graph G conssts of vertex set V(G = {v, v,, v n } and edge set E(G V(G V(G The edges such as (v, v are called self-loops A path n drected graph G from v to v s a sequence v, v,, v of fnte vertces such that (v l, v l+ E(G for l =,,, Drected graph G s sad to have a spannng tree f there exsts a vertex, called the root, such that t can be connected to any other vertces through paths The unon of a group of drected graphs G,, wth a common vertex set V, s also a drected graph wth the vertex set V and wth the edge set gven by E(G, where s the ndex set of the group A weghted drected graph G(C s a drected graph G together wth a nonnegatve weght matrx C = [c ] R n n such that (v, v E(G c > 0 And n ths case, c s called the weght of edge (v, v 3 Problem formulaton The system studed n ths paper conssts of n autonomous agents, labeled through n All the agents share a common state space R Let x and v denote the poston-le and veloctyle states of agent respectvely and suppose that agent, =,,, n, s wth the followng second-order dynamcs ẋ (t = v (t v (t = u (t where u (t s a local state feedbac, called protocol, to be desgned based on the nformaton receved by agent from ts neghbors Poston-le varables x may represent postons, worloads, etc, and they are the only nformaton transmtted over the networ Suppose that each agent can communcate wth some other agents whch are defned as ts neghbors We use a drected graph G wth vertex set V(G = {v, v,, v n } to represent the communcaton topology Vertex v represents agent and edge (v, v E(G f and only f there exsts an avalable nformaton channel from agent to agent Because of lmted detecton range of agents, exstence of obstacles, or external nterference n sgnals, the communcaton topology s usually dynamcally changng We denote the changng topology by G(t Defnton (Partal State Consensus n the Second-Order Case Gven protocol u, =,,, n, u or ths mult-agent system s sad to solve the velocty-le state consensus problem, that s, a partal state consensus problem, f for any ntal states, there exsts a common asymptotcally stable equlbrum pont v R for all agents wth respect to velocty-le states, such that lm t v (t = v for all Remar n the case of the networs of second-order dynamc agents, governed by Eq (, the poston-le state consensus mples the velocty-le state consensus, and thus the veloctyle state consensus s the only partal state consensus n the strct sense So ths paper focuses on the latter case only 3 Velocty-le state estmaton The obectve of ths paper s to propose an effectve protocol ensurng the solvablty of the partal state consensus problem under relaxable condtons To acheve ths end, we next gve a smple velocty-le state estmaton strategy based on the receved poston-le state nformaton Assume that agent detects ts neghbors poston-le states ntermttently at ts update tmes t, 0 t,, t,, and assume that the update tme sequence satsfes the followng assumpton: (A there exst common lower and upper bounds Ť u, ˆTu for the length of tme nterval between any two consecutve update tmes, such that 0 < Ť u t + t ˆTu for any N and any {,,, n} The above assumpton mples that the update tme sequence s strctly ncreasng, unbounded and wth no fnte accumulaton ponts, namely, lm t = By the propertes of the update tme sequence, the studed system can be classfed nto two categores: the one wth the property that t = t for all,, s called the synchronous system, and the other one wthout the precedng property s called the asynchronous system Ths paper wll focus on the asynchronous case At update tme t, agent may get only some of ts neghbors states because of the exstence of communcaton tme-delays Assume that each agent s equpped wth on-board memory to store nformaton receved from neghbors The nformaton may (

4 F Xao et al / Systems & Control Letters 59 ( be wth tme-delays To estmate the velocty-le states, t s further assumed that the tme-delays are detectable, e, the nformaton s tme-stamped Denote the avalable data of agent at tme t by D (t, whch s composed of the nformaton receved at prevous fnte update tmes To be consstent wth the protocol proposed n the next subsecton, we gve another defnton of neghbors: Defnton (Neghbor Set N E (t and Estmaton of Velocty-Le States Let T E > 0 be gven For any and any update tme t, f there exsts at least one tme-par (t, t such that (A x (t, x (t D (t ; (A3 t T E t < t t, then agent s called a neghbor of agent on the tme nterval [t, t + Denote the neghbor set of agent n ths sense at tme t by N E (t At update tme t, f N E (t, then agent selects one such tme-par (t, t, satsfyng Assumptons (A and (A3, and updates the estmaton about the velocty-le state of agent by v E (t = x (t x (t t t, t [t, t + ( n the above defnton, parameter T E s a pre-gven parameter, nown by all agents and ndependent of parameters Ť u and ˆTu n Assumpton (A, and t represents the maxmum allowable tmedelay, guaranteeng that the data used n the desgned protocol are suffcently new Moreover, the selecton of T E affects the number of elements n the neghbor set N E (t and the structure of nteracton topology G E (t, defned n the next subsecton And t also affects the fnal value of the consensus state n the process of data selecton n approxmatng velocty-le states, t may happen that there exst more than one tme-pars, le (t, t, satsfyng Assumptons (A and (A3 n ths case, agent can choose one tme-par randomly or by the most-recentdata strategy n the latter case, x (t s the most recent postonle state nformaton about agent n D (t and x (t s the most recent poston-le state nformaton satsfyng Assumptons (A and (A3 Obvously, n ths case, f N E (t N E (t +, then t,+ t and t,+ t t can be shown by smulatons that the most-recent-data strategy s more lely to result n a hgher convergence rate Notce that the choce of tme-par (t, t n fact affects the fnal value of consensus state and dfferent agents may choose dfferent polces n a dstrbuted manner for the choce among multple tme-pars, satsfyng Assumptons (A and (A3 Moreover, a lower bound requrement of t t can be added to reduce the effect of nose Remar Note that estmaton v E (t and thus protocol (3 are only dependent on the tme dfference t t and state dsplacement across the tme nterval [t ] Therefore, synchronzaton of all, t agents clocs s not a necessary condton, and t and t can be replaced by the ones decded by the local cloc of agent or accordng to practcal stuatons However, t s requred that all agents should evolve n the same tme scale From ths vewpont, the update tme sequence t, 0 t,, =,,, n, whch s decded by the global cloc, can be seen as the one decded by local clocs of agents, wthout losng the consensus property of protocol (3 Remar f all agents can get ther neghborng agents postonle states wthout tme-delays, then the assumpton of tmestamped nformaton can be removed One example s the velocty consensus control of multple robots, where x (t represents the poston of agent n ths case, we can suppose that at detectng tmes t and t, agent can detect the relatve postons of ts neghborng agent, namely, x (t x (t and x (t x (t, respectvely f the dsplacement of agent over the tme nterval [t, t] s obtanable by agent, then x (t x (t can be gotten by (x (t x (t (x (t x (t + (x (t x (t 3 Partal state consensus protocol Wth the above preparatons, we now propose the followng dstrbuted partal state consensus protocol : u (t = N E (t W (t N E (t W (t (ve (t v (t, t [t, t +, (3 where W (t are called weghtng factors [7], taen from a gven compact set W consstng of postve real numbers Obvously, u (t and v (t are not smooth but they are pecewse dfferentable wth respect to tme t By ths fact and by the asynchrony of update tmes, the dfferental Md-Value Theorem does not hold, n other words, for the estmaton v E (t gven by Eq (, there may not exst t [t, t ] such that v (t = v E (t Ths shows that system ( s not equvalent to the frst-order case wth tme-delays studed n [4] We end ths secton wth a further dscusson on the communcaton topology We now that the communcaton topology G(t only represents the avalable nformaton flow among agents, whereas t does not ndcate whether the neghbors nformaton s used n the feedbac The followng defnton of nteracton topology reflects the relatonshp determned by the nter-usage of nformaton Defnton 3 (nteracton Topology G E (t The vertex set of nteracton topology G E (t s {v, v,, v n }, representng the n agents respectvely, and (v, v E(G E (t f and only f N E (t 4 Convergence result Ths secton presents the convergence result about the system under protocol (3 and ts techncal proof s postponed to the next secton Theorem 4 f there exsts some T > 0, such that for any t 0, the unon of nteracton topology G E (t over tme nterval [t, t+t] always contans a spannng tree, and asynchronous system ( satsfes Assumptons (A (A3, then protocol (3 solves the partal state (velocty-le state consensus problem asymptotcally Remar The suffcent condton provded n the above theorem s a mld and less conservatve one Here, contanng a spannng tree s n fact to guarantee that the nformaton of at least one agent can flow to the entre networs drectly or ndrectly n the lterature, the typcal nteracton graph condtons ensurng solvablty of consensus problems under the assocated proposed protocols can be generally classfed nto three categores The frst s that the nteracton topology s always connected (n the bdrectonal case or always has a spannng tree (n the undrectonal case Ths s the most conservatve one The second s that the perodcal unon of nteracton graph s always Ths paper assumes that f N E (t =, then u (t = 0

5 778 F Xao et al / Systems & Control Letters 59 ( connected or always contans a spannng tree The suffcent condton stated n Theorem 4 belongs to ths category The last one s the unon of all forthcomng nteracton graphs contans a spannng tree and ths s the mldest one for the case under tme-dependent nteracton topology On the other hand, there s a tradeoff between the nteracton graph condton and the basc setup of the studed system Generally speang, the stronger basc assumpton s expected to have a relatvely mlder suffcent nteracton graph condton Tll now, to the best of our nowledge, the only few results that can get the last mldest condton usually concern the specal system n the bdrectonal case, see the wor of Moreau [7] To determne whether the partal state consensus problem s solvable by the topology of nformaton channel G(t, we have the followng corollary: Corollary 5 Assume that the communcaton topology G(t s tmenvarant and contans a spannng tree, and assume that each agent can obtan ts neghbors (determned by G(t poston-le states wth bounded tme-delays for any tme, that s, there exsts a maxmum transmsson tme-delay T max and f there exsts an nformaton channel from agent to agent, then, at any tme t, agent can obtan agent s poston-le state, denoted by x (t, wth the property that 0 t t T max Then there exsts an avalable dstrbuted control rule n the form of protocol (3, satsfyng Assumptons (A (A3, and t solves the partal state (velocty-le state consensus problem asymptotcally Proof To prove the result, t suffces to fnd a possble dstrbuted control rule, satsfyng the assumptons assumed by Theorem 4 Suppose that the maxmum transmsson tme-delay s T max and let the update tme sequence of agent, =,,, n, be t 0, t = t 0 + T max,, t + = t + T max, Then t satsfes Assumpton (A wth Ť u = ˆTu = T max We further assume that agent measures all ts neghbors poston-le state nformaton n the tme ntervals (t, t, =,, For wth (v, v E(G, denote the obtaned nformaton related to agent n tme nterval (t, t by x (t Then t < t < t and thus t > t For t [t, t +, 4, let v E (t = x (t x (t t t < t < t t, we have, satsfy Assumptons (A Then t s well defned Snce t 4 that t and t, n the place of t and t and (A3 wth T E = 4T max Furthermore, the above assumpton also mples that for t max t 4, E(GE (t = E(G(t Therefore, under the above proposed control rule, the system solves the partal state consensus problem asymptotcally 5 Techncal proof Ths secton performs the convergence analyss on asynchronous system ( based on the nonnegatve matrx approach, whch s an effectve way to show the consensus property of multagent systems wth swtchng topology and tme-varyng delays The frst subsecton summarzes some ey lemmas, establshed n [4,5,8] They descrbe the convergence property of the product of a compact set of SA (Stochastc, ndecomposable and Aperodc matrces and gve relaxable condtons ensurng a stochastc matrx to be an SA matrx, respectvely The second subsecton collects all event tmes and merges them nto a sngle ordered tme sequence, denoted by t 0, t,, t, The evoluton of velocty-le states and ther estmaton values s studed wth respect to the above newly defned tme sequence Ths approach s called the Analytc Synchronzaton method n [9] Then we defne a set of new varables v A (t + by (x (t + x (t /(t + t, =,,, n t s shown that the estmated velocty-le states v E ( can be represented by a convex combnaton of varables v A ( By nvestgatng the relatonshp among these varables, we ntroduce an augmented (m + n-dmensonal state varable y(, and then transform the contnuous-tme system ( nto ts equvalent dscrete-tme system (7, where m s a nonnegatve nteger, determned by Assumptons (A and (A3 Thus the partal state consensus problem can be treated as a full state consensus problem equvalently However, the state matrx Ξ( of the dscretetme system has ts specal structure, whch s dfferent from that of the exstng ones nvestgated n the lterature [4,5,7,5,8] Fortunately, by constructng and characterzng two compact sets M and H n the thrd subsecton, whch nclude all possble state matrces of system (7 and all possble products of a fnte number of state matrces at consecutve tme-steps of system (7, respectvely, we can apply the ey lemmas, presented n the frst subsecton, and get the convergence result Here, we emphasze that although the proof steps are smlar to that taen n [4], n other words, they are all based on the presented ey lemmas, the dfferences between the contrbutons of the two papers are also obvous Frst, they study two dstnct nds of problems The feasblty of the proof wth the help of Lemmas 6 8 owes much to the sllful choce of state varable y( Second, the proof detals are also dfferent Ths paper gves the only arguments that are needed to be clarfed and omts the obvous ones that can be learnt from other lterature 5 Key lemmas n ths subsecton, we frst gve the defnton of SA matrx and then lst three mportant lemmas, whch are useful n provng the man result A stochastc matrx A s called ndecomposable and aperodc (SA f there exsts a column vector ν such that lm A = ν T Lemma 6 ([5, Lemma 5] Let A be a compact set, consstng of n n SA matrces f for any and any A, A,, A A (repettons permtted, = A s SA, then for any gven nfnte matrx sequence A, A,, A, (repettons permtted, there exsts a column vector ν such that lm A = ν T = Lemma 7 ([8, Lemma ] Let A be a stochastc matrx f G(A contans a spannng tree wth the property that the assocated root vertex has a self-loop n G(A, then A s SA Lemma 8 ([4, Lemma 8] Let A 0, A,, A m be n n nonnegatve matrces, let A 0 A A m D = (m+n (m+n let 0 Q 0 = 0 0 (m+n (m+n

6 and let Q = D + Q 0 for any {,,, m} f G( m = A contans a spannng tree, then G(Q also contans a spannng tree wth the property that the assocated root vertex has a self-loop n G(Q 5 Equvalent representaton n ths subsecton, we employ the Analytc Synchronzaton method and get the dscrete-tme Eq (7, that s, an equvalent representaton of the orgnal contnuous-tme system The basc dea of Analytc Synchronzaton s to study the asynchronous system by usng a sutably defned dscrete-tme synchronous system, evolvng on the collecton of event tmes of all orgnal subsystems [9] Frst, for symbolc smplcty, we generalze the defnton of v E (t and ntroduce a weght matrx A(t = [a (t] R n n, where v E (t s generalzed by f N E (t, then v E (t s defned by Eq (; f =, then v E (t = v (t; 3 n other cases, v E (t = 0, and A(t s defned by f N E (t, W (t a (t = W s (t, N E (t s N E (t 0, otherwse, f N E (t =,, = a (t = 0, otherwse, t [t, t + F Xao et al / Systems & Control Letters 59 ( t [t, t + By the above defnton, v E (t,, s a pece-wse constant functon of tme t, and f gnore the weght of each edge and selfloops n G(A(t, then G(A(t and G E (t represent the same nteracton topology Notcng that W (t W, we get that all possble A(t consttute a compact set A and ther nonnegatve entres are not less than mn{w : w W}/((n max{w : w W} Now collect all tme {t, t, t : =,,, n, N E (t, N} and relabel the nonnegatve elements of them by t 0, t,, t,, n ncreasng order such that t 0 = 0, t < t + Here, we assume that t 0 = 0 for all {,,, n} ndeed, wthout ths assumpton, we can consder the dynamcs of agents after tme max t 0 and get the same convergence result For smplcty, denote t + t by τ and denote (x (t + x (t /(t + t by v A (t + (superscrpt A means average velocty, for N, =,,, n Next we study the evoluton of varables v (t and v A (t wth respect to We wll prove that ther reachng an agreement mples the solvablty of the partal state consensus problem Solvng Eq (3 gves that n v (t = e (t t v (t + ( e (t t a (t ve (t x (t x (t t t = e (t t t t + v (t e (t t t t = n = a (t ve (t where t < t t + t can be observed that 0 < ( e (t t /(t t < for any t < t t + The above equaton mples that (4 n v (t + = e τ v (t + ( e τ = v A (t + = e τ v (t τ + e τ n τ = a (t s v E (t s a (t s v E (t s where s N such that t s t < t + t s + Ths part gves an expresson v E (t s n Eq (5 n terms of v A ( Suppose that N E (t s, t s = t l and t s = t p By Assumptons (A and (A3, there exsts an m N, ndependent of,, (cf Lemma n [4], such that t m t l < t p t s for m Then v E (t s = x (t p x (t l t p t l = τ p v A (t p + τ p v A (t p + + τ l v A (t l + τ p + τ p + + τ l (6 whch means that v E (t s s a convex combnaton of v A (t l +, v A (t l +,, v A (t p From the fact that p l m, t follows that some of ts coeffcents are not less than /m To represent the evoluton Eq (5 n matrx form, we ntroduce the augmented state varable y( = [v(t T, v A (t T, v A (t T,, v A (t m+ T ] T for m, where v(t = [v (t, v (t,, v n (t ] T and v A (t = [v A (t, v A (t,, v A n (t ] T Combnng Eqs (5 and (6 yelds that y( + = Ξ(y(, (7 where Ξ( R (m+n (m+n s defned n Box and A γ ( = ], γ {0,,, m}, are defned by [a γ f N E (t s, τ γ a (t s, a γ τ = p + τ p + + τ l N E (t s, p + γ l 0, otherwse f N E (t s =, a γ, =, γ = 0 = 0, otherwse n the followng, the matrx gven n Box, wth the above structure, s denoted by M(τ, A 0 (, A (,, A m ( to ndcate ts dependence on parameters τ, A 0 (, A (,, A m ( By the above defnton, t can be easly obtaned that m Lemma 9 γ =0 A γ ( = A(t and thus Ξ( s stochastc; f N E (t, there exsts γ m, such that aγ mn{w : w W}/(m(n max{w : w W} Remar Eq (7 can be seen as the dynamcal equaton of a dscrete-tme mult-agent system consstng of (m + n agents under the tme-varyng nteracton topology G(Ξ(, and varable y( s the column vector, staced wth the states of the (m + n agents These states are the veloctyle states v (t, =,,, n, and the average velocty v A (t, v A (t,, v A (t m+, =,,, n n Lemma, we wll show that ther convergence to a consensus state as wll lead to that v (t, =,,, n, reach consensus asymptotcally as t Consensus problems of dscretetme mult-agent systems were wdely studed by researchers (5

7 780 F Xao et al / Systems & Control Letters 59 ( e τ + ( e τ A0 ( ( e τ A ( ( e τ Am ( ( e τ Am ( e τ τ + ( e τ A τ 0 ( ( e τ A τ ( ( e τ A τ m ( ( e τ A τ m ( Ξ( = 0 0 Box However, ths dscrete-tme model cannot be covered by the exstng ones, such as those studed n [7,7,0], because the dagonal entres of state matrx Ξ( are not all larger than zero and parameters τ may tae any value n (0, ˆTu ], see [4] for detaled dscussons Defnton 0 (Full State Consensus The dscrete-tme system (7 s sad to solve a (full state consensus problem f for any ntal state, there exsts an asymptotcally stable equlbrum pont y, y R, such that lm y( = y, n other words, lm v (t = lm v A (t = y, =,,, n The followng lemma states the relatonshp between contnuous-tme system ( and dscrete-tme system (7 Lemma System ( solves a partal state (velocty-le state consensus problem f and only f system (7 solves a full state consensus problem Proof The necessty s obvous and we only prove the suffcency Suppose that lm v (t = lm v A (t = v for all =,,, n Thus by Eqs (4 and (6 and Assumpton (A, lm t v (t = v The next subsecton wll prove Theorem 4 by showng system (7 solves a full state consensus problem 53 Convergence analyss of system (7 Ths subsecton conssts of three parts The frst part gves an equvalent representaton of the condton assumed n Theorem 4 The second part characterzes the propertes of state matrx Ξ( of system (7 by two compact matrx sets M and H The last part gves the proof of Theorem 4 Frst, the followng lemma restates the condton provded n Theorem 4 n an equvalent way Lemma f there exsts T > 0, such that for any t 0, the unon of nteracton topology G E (t across the tme nterval [t, t + T] always contans a spannng tree, then there exst a postve nteger h and a postve real number T h wth the followng property: for any N, there exsts a subset of {t, t +,, t +h }, denoted by T, such that the unon of G E (t on T contans a spannng tree, and for any t l T, T h τ l ˆTu Proof t s a straghtforward consequence of Assumptons (A and (A3 and the detals are omtted, see Lemma 9 n [4] for smlar dscussons n order to mae use of Lemma 6 to perform the convergence analyss, we next construct two compact sets M and H The frst compact set M ncludes all possble state matrces of system (7, whch s defned by M = M(ς, N 0, N,, N m : 0 ς ˆTu, N 0, N,, N m are nonnegatve, and there exsts some A A, such that N 0 + N + + N m = A where we use the conventon ( e ς /ς ς=0 = lm ς 0 ( e ς /ς = The second compact set H ncludes all possble products of h state matrces at consecutve tme-steps of system (7, whch s defned by h H = M(ς, N, 0 N,, N : m = M(ς, M, and there exsts a subset of {,,, h}, denoted by T, such that for any m T, T h ς ˆTu and G contans a spannng tree T where h and T h are gven n Lemma Lemma 3 M and H are compact sets, and for any m, Ξ( M +h and l= Ξ(l H; for any H H, H s SA, and for any N, f H, H,, H H (repettons permtted, then = H s SA Proof (a The compactness of M follows from the fact that set A and nterval [0, Ť u ] are compact; the compactness of H follows from that M s compact; all possble choces of T and the spannng tree are fnte; 3 the product of fnte matrces s a contnuous functon; 4 all the nonnegatve entres of matrces n A are lower bounded by mn{w : w W}/((n max{w : w W} (b The concluson that Ξ( M follows drectly from the defntons of Ξ( and M, and Lemma 9 By Lemmas 9 and, G( m l T γ =0 (A γ (l contans a spannng tree, and thus +h l= Ξ(l H (c Let H = h M(ς =, N, 0 N,, N m and T be the assocated subset of {,,, h} such that for any T, T h ς ˆTu and G m T =0 N contans a spannng tree Let Q0 be the same as the Q 0 n Lemma 8, let N 0 N N m D = and let ε = nf{e ς, ( e ς /ς : ς (0, ˆTu ]} Then ε > 0 and h M(ς, = =0 h εq0 + ( e ς D = ε h Q 0 h + ε h N h ( e ς Q h 0 D Q 0 = mn{ε h, ε h ( e T h } Q 0 h + T D Q 0 h

8 F Xao et al / Systems & Control Letters 59 ( where the last nequalty follows from Q 0 D D and from the fact that T h ς ˆTu for T Let the frst n rows of D Q h 0 be B, 0 B,, m B, where B R n n, = 0,,, m m Then = B = m =0 N and thus m T =0 B = m T =0 N Snce G m T =0 N contans a spannng tree, G m T =0 B also contans a spannng tree Let D R (m+n (m+n, wth the same frst n rows as T D Q 0 h and wth the other rows beng zeros Then h M(ς, mn{ε h, ε h ( e T h }(Q h 0 + D = By Lemma 8 and the fact that Q 0 = Q 0 m f m, G(Q 0 h + D contans a spannng tree wth the property that the assocated root vertex has a self-loop, and so s G(H Snce H s stochastc, by Lemma 7, H s SA By the same arguments, we have that for any H, H,, H H, = H s SA Proof of Theorem 4 Ths part only proves that system (7 solves a full state consensus problem t follows from Lemmas 6 and 3 that there exsts ν R (m+n such that ph lm p Ξ(m + l = ν T 3 (8 For any N, there exsts p such that p h < (p + h And snce matrx Ξ(m + l s stochastc, lm Ξ(m + l ν T = lm Ξ(m + l l=p h p h Ξ(m + l ν T Snce M s compact, we get that matrx l=p h Ξ(m + l n the above equaton belongs to a bounded set Furthermore, t follows from Eq (8 that lm p h p h Ξ(m + l ν T = lm p Thus, lm Ξ(m + l ν T = 0, whch yelds that lm y( = lm Ξ(m + ly(m = ν T y(m, Ξ(m + l ν T = 0 that s, system (7 solves a full state consensus problem 6 Concluson Ths paper consdered the partal state consensus problem n networs of second-order agents wth dynamcally changng nteracton topologes and tme-varyng communcaton tmedelays and proposed an asynchronous partal state consensus protocol, whose valdty can be guaranteed under the relaxable condton that poston-le states as the only nformaton are transmtted ntermttently over the networ Nevertheless, there stll exst some other nterestng problems that need to be addressed, such as the desgn and analyss of a full state consensus protocol n the framewor of ths paper References [] R Olfat-Saber, JA Fax, RM Murray, Consensus and cooperaton n networed mult-agent systems, Proceedngs of the EEE 95 ( [] W Ren, Mult-vehcle consensus wth a tme-varyng reference state, Systems & Control Letters 56 ( [3] Z J, Z Wang, H Ln, Z Wan, nterconnecton topologes for multagent coordnaton under leader follower framewor, Automatca 45 ( [4] F Xao, L Wang, Asynchronous consensus n contnuous-tme mult-agent systems wth swtchng topology and tme-varyng delays, EEE Transactons on Automatc Control 53 ( [5] F Xao, L Wang, Consensus protocols for dscrete-tme mult-agent systems wth tme-varyng delays, Automatca 44 ( [6] R Olfat-Saber, RM Murray, Consensus problems n networs of agents wth swtchng topology and tme-delays, EEE Transactons on Automatc Control 49 ( [7] W Ren, RW Beard, Consensus seeng n multagent systems under dynamcally changng nteracton topologes, EEE Transactons on Automatc Control 50 ( [8] G Xe, L Wang, Consensus control for a class of networs of dynamc agents, nternatonal Journal of Robust and Nonlnear Control 7 ( [9] Y Hong, J Hu, L Gao, Tracng control for mult-agent consensus wth an actve leader and varable topology, Automatca 4 ( [0] W Ren, On consensus algorthms for double-ntegrator dynamcs, EEE Transactons on Automatc Control 53 ( [] W Ren, Collectve moton from consensus wth Cartesan coordnate couplng, EEE Transactons on Automatc Control 54 ( [] W Ren, Synchronzaton of coupled harmonc oscllators wth local nteracton, Automatca 44 ( [3] X Wang, Y Hong, Fnte-tme consensus for mult-agent networs wth second-order agent dynamcs, n: Proceedngs of the 7th World Congress, The nternatonal Federaton of Automatc Control, 008, pp [4] T Vcse, A Czro, E Ben-Jacob, Cohen, O Schochet, Novel type of phasetranston n a system of self-drven partcles, Physcal Revew Letters 75 ( [5] A Jadbabae, J Ln, AS Morse, Coordnaton of groups of moble autonomous agents usng nearest neghbor rules, EEE Transactons on Automatc Control 48 ( [6] S Barbarossa, G Scutar, Decentralzed maxmum-lelhood estmaton for sensor networs composed of nonlnearly coupled dynamcal systems, EEE Transactons on Sgnal Processng 55 ( [7] L Moreau, Stablty of multagent systems wth tme-dependent communcaton lns, EEE Transactons on Automatc Control 50 ( [8] F Xao, L Wang, State consensus for mult-agent systems wth swtchng topologes and tme-varyng delays, nternatonal Journal of Control 79 ( [9] J Ln, AS Morse, BDO Anderson, The mult-agent rendezvous problem Part : the asynchronous case, SAM Journal on Control and Optmzaton 46 ( [0] D Angel, PA Blman, Stablty of leaderless dscrete-tme mult-agent systems, Mathematcs of Control, Sgnals, and Systems 8 ( f <, then l= Ξ(l =