Auomaca 49 (203) 225 232 Conens lss avalable a ScVerse ScenceDrec Auomaca journal homepage: www.elsever.com/locae/auomaca Bref paper Even based agreemen proocols for mul-agen neworks Xangyu Meng, Tongwen Chen Deparmen of Elecrcal and Compuer Engneerng, Unversy of Albera, Edmonon, AB, T6G 2V4, Canada a r c l e n f o a b s r a c Arcle hsory: Receved 9 Aprl 202 Receved n revsed form 4 January 203 Acceped 27 February 203 Avalable onlne 9 Aprl 203 Keywords: Even-rggered conrol Cooperave conrol Mul-agen sysems Sampled-daa conrol Neworked conrol sysems Ths paper consders an average consensus problem for mulple negraors over fxed, or swchng, undreced and conneced nework opologes. Even based conrol s used on each agen o drve he sae o her nal average evenually. An even rggerng scheme s desgned based on a quadrac Lyapunov funcon. The dervave of he Lyapunov funcon s made negave by an approprae choce of he even condon for each agen. The even condon s sampled-daa and dsrbued n he sense ha he even deecor uses only neghbor nformaon and local compuaon a dscree samplng nsans. The even based proocol desgn s llusraed wh smulaons. 203 Elsever Ld. All rghs reserved.. Inroducon Numerous conrbuons have been gven n he leraure for mul-agen sysems by research papers (Arcak, 2007; Corés, 2008; Jadbabae, Ln, & Morse, 2003; Ln, Broucke, & Francs, 2004; Moreau, 2005; Olfa-Saber & Murray, 2004; Tanner, Jadbabae, & Pappas, 2007; Xao & Wang, 2008) and monographs (Mesbah & Egersed, 200; Ren & Beard, 2008). Connuous communcaon beween neghborng agens s ofen used for dsrbued consensus proocol desgn. Whle connuous communcaon s an deal assumpon, s more realsc o nerac nermenly a dscree samplng nsans (Chen & Francs, 995). One choce s o use perodc synchronous samplng (Xe, Lu, Wang, & Ja, 2009a,b); however, s undesrable and unnecessary o updae he conrol acons for all agens a he same me. Even based conrol s an alernave o me rggered conrol (Hennngsson, Johannesson, & Cervn, 2008; Lunze & Lehmann, 200). The dsnc feaure of even based conrol s ha conrol acon s updaed only when some specfc even occurs. For example, a logc condon s volaed or he nework opology s changed. By comparson wh me rggered conrol, even based conrol has he ofen ced advanage on communcaon reducon. Snce he poneerng paper (Åsröm & Bernhardsson, 2002), even based Ths work was suppored by NSERC and an CORE Ph.D. Recrumen Scholarshp from he Provnce of Albera. The maeral n hs paper was no presened a any conference. Ths paper was recommended for publcaon n revsed form by Assocae Edor Hdeak Ish, under he drecon of Edor Ian R. Peersen. E-mal addresses: xmeng2@ualbera.ca (X. Meng), chen@ualbera.ca (T. Chen). Tel.: + 780 492 3940; fax: + 780 492 8. conrol has been suded exensvely n neworked conrol sysems (Wang & Hovakmyan, 202), decenralzed sysems (Mazo & Tabuada, 20; Wang & Lemmon, 20), and n many cases ouperforms he radonal me rggered conrol (Meng & Chen, 202). I has also been proved especally useful n mul-agen sysems, such as consensus algorhm (Dmarogonas, 20; Dmarogonas & Frazzol, 2009; Dmarogonas, Frazzol, & Johansson, 202; Dmarogonas & Johansson, 2009; Lu & Chen, 200, 20; Seyboh, Dmarogonas, & Johansson, 203; Sh & Johansson, 20), formaon conrol (Tang, Lu, & Chen, 20), rackng conrol (Hu, Chen, & L, 20a,b), and pah plannng (Texera, Dmarogonas, Johansson, & Sousa, 200a,b). The focus here s he even based consensus problem, whch arses n a varey of domans ncludng cooperave conrol of mulple auonomous vehcles, cooperave robocs, and wreless sensor neworks. Ineresed readers are referred o he above ced references on heorec research on even based consensus proocols. A common feaure of hese references s connuous communcaon and even based conrol updang. Such connuous deecon and updang do no mee he orgnal purpose of nroducng even based conrol as a means for reducng communcaon requremens beween nerconneced subsysems, snce o mplemen he connuous even deecor requres delcae hardware o monor and check he even condon consanly, whch may also become a major source of energy consumpon. Based on he above observaon, he concep of sampled-daa even deecon s defned as perodc evaluaon of he even condon. Ths paper s devoed o he developmen and analyss of dsrbued even based algorhms wh sampled-daa even deecon for solvng average consensus problems ha are defned 0005-098/$ see fron maer 203 Elsever Ld. All rghs reserved. hp://dx.do.org/0.06/j.auomaca.203.03.002
226 X. Meng, T. Chen / Auomaca 49 (203) 225 232 over undreced, conneced nework opologes. The analyss s begun wh consensus problems over a fxed opology. A relavely sraghforward exenson o he analyss of swchng opologes s also presened. To he bes knowledge of he auhors, hs paper s he frs o address consensus problems of mul-agen sysems usng a sampled-daa even deecor, whch s an mprovemen over connuous even deecors. Besdes he sampled-daa even deecor here adms a mnmum ner-even me whch s lower bounded by he synchronous samplng perod. Ths s benefcal for he even deecor desgn of each agen o reduce communcaon beween neghborng agens and save sensor energy for even deecon. A Lyapunov-based approach s used whch s nsrumenal n recen sudes on he consensus of mul-agen sysems usng even drven communcaon. In conras o commonly used Lyapunov funcons n exsng work, a new Lyapunov funcon s nroduced as absracon of he dealed dynamcal models. I s shown ha he parameers of he even deecor can be seleced so ha he me dervave of he Lyapunov funcon calculaed along he rajecores of he closed-loop sysem s negave sem-defne. Wh he ad of LaSalle s nvarance prncple, each agen can be shown o converge o he nal average of all agens. There are wo man conrbuons n hs paper. The frs one s o provde a new even based consensus algorhm wh sampleddaa even deecon for mul-agen sysems. Ths approach s fundamenally dfferen from prevously developed mehods, and he dfferences faclae our mplemenaon of even deecors n a sampled-daa fashon. The second man conrbuon s he proposal of new even based consensus algorhms for swchng nework opologes wh dsrbued and sampled-daa even deecon ha demonsraes a close lnk beween he fxed opology and swchng opology. The remander of hs paper s organzed as follows. Secon 2 s devoed o an nroducon of some conceps n algebrac graph heory and a formal saemen of he problem; whereas Secon 3 saes he man resuls, whch wll be exended o swchng opologes n Secon 4. In Secon 5, he smulaon resuls are presened o valdae our analyss resuls. Fnally, Secon 6 dscusses conclusons and possble exensons. 2. Prelmnares and problem formulaon 2.. Algebrac graph heory Some conceps and facs abou algebrac graph heory wll be examned snce he neracon opology of mul-agen neworks can be modeled by an undreced graph G = {V, E}, whch consss of a fne verex se V = {v, v 2,..., v n }, represenng n agens, and an edge se E V V, correspondng o he communcaon lnks beween agens (Godsl & Royle, 200). If v v j E s an edge, hen v and v j are adjacen or v j s a neghbor of v, and for an undreced graph, v v j E ff v j v E. Analogously, he neghborhood N (G) of agen v can be mahemacally defned as N (G) = j v v j E, j, whch conans all ndexes of agens ha agen v can communcae wh. A pah of lengh r from v 0 o v r n a graph s a sequence of r + dsnc verces sarng wh v 0 and endng wh v r v 0, v,..., v r, such ha for k = 0,,..., r, he consecuve verces v k and v k+ are adjacen. Graph G s conneced f here s a pah beween any wo verces of a graph G. A graph also adms marx represenaons. Some of hese marces, such as he adjacency marx, he degree marx, and he Laplacan marx, wll be revewed subsequenly. The adjacency marx A(G) encodng of he adjacency relaonshp n he graph G s defned such ha f v v a j = j E, 0 oherwse, where a j s he (, j) enry of he adjacency marx A(G) R n n. The adjacency marx of an undreced graph s symmerc because a j = a j for all j. The degree marx D(G) for an undreced graph G s a dagonal marx dag {d, d 2,..., d n } wh d beng he cardnaly of agen v s neghbor se N (G). The Laplacan marx L(G) assocaed wh an undreced graph G s defned as L(G) = D(G) A(G), where D(G) s he degree marx of G and A(G) s s adjacency marx. For undreced graphs, he Laplacan marx L(G) s symmerc and posve sem-defne, ha s, L(G) = L(G) T 0; hence s egenvalues are real and can be ordered as λ λ 2 λ n wh λ = 0 and λ 2 s he smalles nonzero egenvalue for conneced graphs. The vecor, wh all enres equal o, s an egenvecor of L(G) assocaed wh egenvalue 0. 2.2. Consensus problem The dynamcs assocaed wh each agen v V s descrbed by he followng equaon: ẋ () = u (), () where x R s he sae, u R s he conrol npu of he h agen. The followng remarks are n order. Remark. In order no o overshadow he man dea and complcae he noaon, he case ha scalar agens over unweghed graphs s consdered. However, he framework proposed n hs paper can be exended o desgn even based consensus proocols for mul-agen sysems over weghed opology and wh hgher dmensonal agens, ha s, x R p. The overall goal s o propose an even based conrol mechansm o reduce communcaon beween neghborng agens along wh he energy consumpon of even deecon for each agen whle preservng asympoc propery of consensus. Therefore, an even deecor s confgured a each agen whch s used o deermne when he sampled local nformaon should be used o updae he conrol acons of self and s neghbors. The even condon for agen v has he followng form e k + lh 2 2 σ z + lh 2 k 2, l =, 2,... (2) where σ s a posve scalar o be deermned laer, k s he kh even nsan for agen v and s an neger mulple of h, h s he samplng perod for all agens synchronzed physcally by a clock, e + lh k s defned as he dfference beween he sae a he las even me and he currenly sampled sae e + lh k = x k x + lh k, and z k + lh = j N (G) x ( + lh) k x j + lh k. Remark 2. A each samplng nsan, each agen broadcass s sae nformaon o he neghbors and also receves sae nformaon from s neghbors for even deecon. If he condon n (2) s sasfed, no furher acon s requred for agen v ; oherwse, agen v wll updae s own conrol acon and nofy s neghbors o updae her conrol acons by usng s curren sae nformaon. The volaon of he nequaly n (2) has he effec of reseng
X. Meng, T. Chen / Auomaca 49 (203) 225 232 227 he error e k + lh o zero; a he same me, he even condon s sasfed agan. The even nsans for agen v are hus defned eravely by k+ = k + h nf l : e ( k + lh) 2 2 > σ z ( k + lh) 2 2, where 0 = 0 s he nal me. Obvously, all he measuremens x k are subsequence of he sampled sae x (kh), ha s o say, he even nsans, 0,... {0, h, 2h,...}. Ths means ha he ner-even mes k+ k, k = 0,,... are a leas lower bounded by he samplng perod h for all agens. Remark 3. Whle he proposed even based consensus scheme and he sampled-daa consensus n Xe e al. (2009a,b) share a common samplng nerval n nformaon exchange, hey are fundamenally dfferen. For he sampled-daa consensus, all he daa sampled are used for acuaon; for he even based consensus, all he daa sampled are used for even deecon; f he even condon of agen v s sasfed a he samplng nsan kh, hen he sae nformaon x (kh) wll no be used for updang s own and neghbors conrol laws. However, agen v j wh j N (G) may updae s acuaon a he samplng nsan kh. Therefore, he average acuaor updang perod s larger han he samplng perod h snce only a par of he daa sampled are used for acuaon. Moreover, he proposed even based acuaor updaes are asynchronous n general. Ths s n conras o he sampleddaa consensus n whch he acuaor updaes are synchronous. Specally, when σ < 0, he even condon n (2) s no sasfed a each samplng nsan, and he even based consensus hus reduces o he sampled-daa consensus. Remark 4. The advanages of he even condon n (2) over exsng ones are obvous. Frsly, dfferen from cenralzed even deecors n Dmarogonas and Johansson (2009), Dmarogonas e al. (202), and Lu and Chen (200), ha s, every agen has o be aware of he global nformaon, he even deecor n (2) s dsrbued n he sense ha each agen needs only he nformaon from s neghbors o decde he updang nsans. Secondly, dfferen from he dsrbued even deecor n Dmarogonas and Johansson (2009), he even deecor n (2) does no need o know he rendezvous locaon n advance and access o s global poson. Each agen needs only he relave dsplacemens wh respec o s neghbors and he relave dsplacemen self a dfferen mes. Thrdly, dfferen from he connuous even deecor n Seyboh e al. (203), whch requres connuous local even deecon and he connuous even deecors n Dmarogonas and Frazzol (2009); Dmarogonas e al. (202); Dmarogonas and Johansson (2009); Lu and Chen (200), whch requre boh connuous local even deecon and connuous communcaon beween neghborng agens, he even deecor n (2) can grealy reduce he sensor energy consumpon and nework bandwdh usage by checkng he even condon a dscree samplng nsans only. Fnally, s worh nong ha exsng resuls on dsrbued mehods can only guaranee he nonexsence of accumulaon pons, bu fal o provde he mnmum ner-even me. However, he even deecor n (2) nherenly adms a mnmum ner-even me h as menoned prevously. To reduce cluer n he noaon, defne ˆx () x k, for k <, k+ whch convers he dscree-me sgnal x k no he connuousme sgnal ˆx () smply by holdng consan unl he nex even occurs. Wh he noaon defned above, an even based consensus algorhm s gven by u () = ˆx () ˆx j (). (3) j N (G) Remark 5. Noe ha he conrol law s no pecewse consan beween he even mes, 0,... bu pecewse consan beween he samplng nsans {0, h, 2h,...} snce he conrol law wll be updaed boh a s own even mes, 0,... as well as j he even mes of s neghbors j N (G), 0 j,..., bu a dscree samplng nsans only. The asympoc consensus problem s sad o be solved f one can fnd an even based proocol such ha for all x (0), and all, j =,..., n, x () x j () 0 as. 2 3. Mul-agen neworks wh fxed opology Tenavely, he opology s assumed o be fxed, hen he dependence on he graph G can be dropped n he correspondng noaon. Under he conrol law gven n he prevous secon, he closedloop sysem for agen v can be obaned ha ẋ () = j N ˆx () ˆx j (). Combnng he defnon of e k + lh, he dynamcs of agen v for + lh, k + k lh + h s hen gven by ẋ () = x xj j k j N k = j N x k + lh x j k + lh x k x + lh k j N + j N x j j k x j k + lh = j N x k + lh x j k + lh j N e k + lh e j k + lh, where j k s defned as j k = max { j, k k = 0,,...}, + k lh. The equaons above for [kh, (k + )h) can also be wren n compac form as ẋ () = Lx (kh) Le (kh), (4) where x = [x,..., x n ] T, e = [e,..., e n ] T, and L s he Laplacan marx. Denoe he sae average of agens as x () = n x (), n = hen under he even based proocol n (3) x () = n ẋ () = n n T ẋ () = n T Lˆx () 0 = snce T L = 0 T. Therefore, s me-nvaran, and defne he dsagreemen vecor as δ () = x () x () = x () x. Gven a conneced graph G, consder he followng Lyapunov funconal canddae: V (x ()) = 2 xt () x (), (5) ha s, half of he sum of squares of he saes.
228 X. Meng, T. Chen / Auomaca 49 (203) 225 232 Remark 6. I s worh menonng ha exsng resuls on even based consensus algorhm resor mosly o a Lyapunov-ype argumen, ha s, defne he followng Lyapunov funcon V() = 2 δt () δ () or V() = 2 x ()T Lx () and assess he convergence o he orgn. Dfferen from exsng resuls, LaSalle s nvarance prncple wll be nroduced o analyze he convergence of an even based agreemen proocol o he agreemen subspace nsead of he orgn, where he meeng locaon for mul-agen sysems over an undreced, conneced graph s exacly x = x 2 = = x n = x. Remark 7. A clam s made ha he funcon n (5) mus decrease o reach he agreemen subspace, and one can never ncrease he funcon o acheve consensus a her nal average. To see hs, apply he Jensen s nequaly o he convex funcon f (y) = y 2, 2 V (x) = n n 2 n x2 n n 2 n x = n 2 x2 = V ( x). = = Therefore, a vald even based proocol canddae would be he one whch can make he funcon n (5) decrease wh respec o. Now consder he me evoluon of he funcon V (x ()) n (5) along he rajecory generaed by (4) for any [kh, (k + )h), whch s gven by V () = x T () L (x (kh) + e (kh)) = ( kh) (x (kh) + e (kh)) T L 2 (x (kh) + e (kh)) x T (kh) L (x (kh) + e (kh)) x T (kh) L (x (kh) + e (kh)) + hλ n (x (kh) + e (kh)) T L (x (kh) + e (kh)) = ( hλ n )x T (kh) Lx (kh) x T (kh) Le (kh) + hλ n e T (kh) Le (kh) + 2hλ n x T (kh) Le (kh). Usng he nequaly x T (kh) Le (kh) 2 xt (kh) Lx (kh) + et (kh) Le (kh) V () can be bounded as V () 2 xt (kh) Lx (kh) + 2 et (kh) Le (kh) wh 2hλ n. Combnng he even condon n (2), we ge V () 2 ( λ2 n σ max)x T (kh) Lx (kh) where σ max = max {σ, =,..., n}. Thereby V () 0 for any k {0,, 2,...} and [kh, (k + ) h) f 0 < h and 0 < σ max <. 2λ n λ 2 n Moreover, based on he fac ha he underlyng communcaon opology G s conneced, he larges nvaran se conaned n he se s x R n V () = 0 = span {}. Thus, from LaSalle s nvarance prncple, V () 0 for 0 mples consensus for all agens. Hence, he followng heorem can be concluded. 2 Theorem 8. Consder he sysem n () over a conneced communcaon graph wh he proocol n (3) drven by even condon n (2). Then all agens are asympocally convergng o her nal average f 0 < h and 0 < σ max <. 2λ n λ 2 n Remark 9. The choces of he samplng perod h and he parameers σ, =, 2,..., n requre some global nformaon abou he opology. An upper bound on he larges egenvalue λ n can be found by λ n 2d max 2(n ), based on he resul n Grone and Merrs (994) and he fac ha d max n. Therefore, he samplng perod h and he parameers σ can be chosen wh he consrans 0 < σ max < and 0 < h 4 (n ) 2, 4 (n ). There s a way o choose he samplng perod h locally and realze samplng synchronzaon for all agens f each agen knows n, he oal number of agens. Ths can be done by scalng he maxmum samplng perod by a common scalar α wh 0 < α < known by all agens, ha s, each agen chooses α h = 4(n ) as s local samplng perod. Also noce ha h and σ, =, 2,..., n, have only upper bound consrans; herefore, small enough α and σ are always approprae. I s more realsc o approxmae he connuous even deecon by a hgh fas rae sampled-daa even deecon. Inuvely speakng, he smaller σ wll lead o hgher frequency of conrol updae and faser convergence rae for he sysem, so here s a rade-off beween he performance and conrol updang cos n hs sense. Remark 0. Accordng o Xe e al. (2009a), he maxmum sablzng samplng perod o solve he average consensus problem for undreced and conneced graphs s 2/λ n. Alhough hs maxmum samplng perod s four mes hgher han he one presened n Theorem 8, he average acuaor updang perod n hs paper s deermned by boh he samplng perod and even deecors, and s larger han he samplng perod n general. In addon, our desgn s performed n connuous me, whereas he sampleddaa consensus approach s a purely dscree-me desgn, whch compleely gnores wha s happenng beween samplng nsans. Therefore, here mgh be large ner-sample ampludes. 4. Mul-agen neworks wh swchng opology In hs secon, he even based proocol wll be exended o he case when he underlyng undreced communcaon opology G swches among possble conneced graphs wh he same fne verex se: {G, G 2,..., G m } wh he ndex se J = {,..., m}. The swchng neworks can be modeled usng a pecewse consan swchng sgnal s () : [0, + ) J.
The swchng mes are defned by 0 = T 0 < T < T 2 <. Denoe he acve opology a he samplng nsan kh as G s(kh) and he correspondng Laplacan marx by L(G s(kh) ). X. Meng, T. Chen / Auomaca 49 (203) 225 232 229 Remark. The opology can swch no only a samplng nsans bu also beween samplng nsans. There may be several swches akng place beween wo consecuve samplng nsans, bu only he recen one o he curren samplng nsan has nfluence on conrollers and even deecors. Agens whose neghborhood relaon reman he same a wo consecuve samplng nsans wll no be affeced by swchng; agens wh communcaon lnk changes from wo consecuve samplng nsans have o evaluae her even condons and conrol laws usng he curren se of neghbors. Fg.. Communcaon opology. In he case of swchng opology, he even condon and even based consensus proocol can be defned smlarly as he one n (2) and (3), respecvely. The common Lyapunov funcon V () = 2 xt () x () can be used o nvesgae he convergence of he even based consensus proocol for swchng opologes over undreced and conneced graphs. Then, wh respec o (), he dervave of V () n he me nerval [kh, (k + )h) s gven by V () = x T () L(G s(kh) ) (x (kh) + e (kh)). If 0 < σ max < and 0 < h λ 2 n, Gs(kh) 2λ n Gs(kh), hen smlar o he fxed opology case, can be proved V () λ 2 n Gs(kh) σmax x T (kh) L(G s(kh) )x (kh), 2 wh λ n Gs(kh) beng he larges egenvalue of he Laplacan marx L G s(kh). Snce he se x R n V () = 0 = span {} s ndependen of any ndvdual opology as swches among a number of conneced graphs, he followng heorem can hus be obaned. Theorem 2. Consder he sysem n () swches over a number of conneced graphs wh he proocol n (3) drven by he even condon n (2). Then all agens are asympocally convergng o her nal average f 0 < σ max <, and 0 < h, λ 2 max 2λ max where λ max = max {λ n (G), G {G, G 2,..., G m }} wh λ n (G) beng he larges egenvalue of he Laplacan marx L (G). 5. Numercal smulaons The even based consensus proocols proposed are now llusraed by compuer smulaons. Example 3. Consder a scenaro where four agens are o mee a a sngle locaon. Fg. shows he correspondng communcaon Fg. 2. Evoluon of each agen. opology among hese agens, whch s used n Dmarogonas e al. (202) as well. Noe ha he graph s conneced. Based on he communcaon opology, he adjacency marx A and he degree marx D are 0 0 2 0 0 0 0 0 0 2 0 0 A = 0 0 0 0, D = 0 0 3 0 0 0 0 and he Laplacan marx s hus gven by 2 0 2 0 L = 3. 0 0 The larges egenvalue of he Laplacan marx s λ n = 4. The parameers of he even deecor for each agen and he samplng perod for all agens are chosen as σ = σ 2 = 0.033, σ 3 = 0.02, σ 4 = 0.06, h = 0.002, whch sasfy he condons ha σ max < 0.0625, h 0.25. The nal values of agens are chosen as x (0) = [0.4773 0.3392 0.5 0.638] T. Usng he even condon n (2), a smulaon s conduced for [0, 0). The evoluon of he sae and he norm of he dsagreemen vecor x () x usng even based consensus proocol are shown n Fgs. 2 and 3, respecvely. I can be seen n boh fgures ha he agens reach consensus a her nal average. The conrol sgnal and he me nsans when he evens occur for each agen are shown n Fgs. 4 and 5, respecvely. I can be seen ha he number of acuaor conrol updaes s grealy reduced o reach average consensus compared wh connuous communcaon scheme. Fg. 6 shows he evoluon of e (kh) for =, 2, 3, 4. In hese fgures, an even s generaed when he error sgnal norm reaches he hreshold σ z (kh), and herefore he error sgnal e (kh) s rese o zero mmedaely. In addon, he,
230 X. Meng, T. Chen / Auomaca 49 (203) 225 232 Fg. 3. Evoluon of x () x. Fg. 4. Conrol npus for he agens. Fg. 5. Even mes for each agen. smulaon resul for each agen s also repored n Table. I can be seen from he able ha he acual mnmum ner-even mes for agens v and v 4 are greaer han he samplng perod 0.002 excep agens v 2 and v 3 whose mnmum ner-even me s equal o 0.002. Noe ha he acuaon updaes are no nvoked when he sysem s n seady sae. Example 4. Fve agens swchng over hree possble neracon opologes are llusraed n Fg. 7. Noe ha all he graphs are Fg. 6. Evoluon of error sgnals for each agen. conneced. The nal value of each agen s generaed randomly from he unform dsrbuon on he nerval [ 0, 0], and he
X. Meng, T. Chen / Auomaca 49 (203) 225 232 23 Table Even nervals for he agens. Agen v v 2 v 3 v 4 Even mes 66 69 75 52 Mn nerval 0.320 0.002 0.002 0.600 Mean nerval 0.56 0.453 0.309 0.955 Max nerval 0.560 0.3460 0.640 0.980 (a) G. (b) G 2. (c) G 3. Fg. 7. Swchng communcaon opology. Fg. 0. Even mes for each agen. even condons work well n he case of swchng opology. The smulaon resul of even mes for each agen s shown n Fg. 0, where he sold vercal lne denoes swchng o opology G, he dashed vercal lne denoes swchng o opology G 2, and he dash-doed vercal lne denoes swchng o opology G 3. 6. Conclusons Fg. 8. Evoluon of each agen. In hs paper, even based conrol algorhms have been proposed o make mul-agen sysems wh fxed opology conracve n he sense of consensus. A new Lyapunov funcon was nroduced, and he me dervave of he Lyapunov funcon was made negave sem-defne by an approprae choce of even condons. Based on hs Lyapunov funcon, sampled-daa even deecors were desgned o drve he saes o her nal average. Based on he resuls for fxed opologes, an even based consensus algorhm for swchng opology was also gven. These desgns were llusraed wh smulaons. Fuure work wll address he generalzaon o dreced opology neworks wh communcaon delays as well as dsurbances. Moreover, he ulzaon of a common samplng perod for all agens mgh be resrcve n dsrbued neworks. Employng dfferen samplng perods for dfferen agens would be an neresng exenson bu may requre new ools for he analyss. References Fg. 9. Evoluon of x () x. nal nework opology s G. Afer he dwell me whch s randomly chosen from he unform dsrbuon on he nerval [0., 0.5], he nework opology swches o anoher graph whch s chosen randomly from he unform dsrbuon on he ndex se J = {, 2, 3}. Such randomly swchng process connuous unl he end of smulaon. The parameers used of he even deecor for each agen and he samplng perod for all agens are σ = 0.02, =, 2, 3, 4, 5, and h = 0.05, respecvely. The evoluon of he sae and he norm of he dsagreemen vecor x () x are shown n Fgs. 8 and 9, respecvely. I can be seen ha he Arcak, M. (2007). Passvy as a desgn ool for group coordnaon. IEEE Transacons on Auomac Conrol, 52(8), 380 390. Åsröm, K. J., & Bernhardsson, B. M. (2002). Comparson of Remann and Lebesgue samplng for frs order sochasc sysems. In: Proc. of he 4s IEEE conf. on decson and conrol (pp. 20 206). Las Vegas, Nevada USA. December. Chen, T., & Francs, B. (995). Opmal sampled-daa conrol sysems. Sprnger. Corés, J. (2008). Dsrbued algorhms for reachng consensus on general funcons. Auomaca, 44(3), 726 737. Dmarogonas, D. (20). L 2 gan sably analyss of even-rggered agreemen proocols. In: Proc. of he 50h IEEE conf. on decson and conrol (pp. 230 235). Orlando, FL, USA. December. Dmarogonas, D., & Frazzol, E. (2009). Dsrbued even-rggered conrol sraeges for mul-agen sysems. In Proc. of he 47h annual Alleron conf. on communcaon, conrol, and compung (pp. 906 90). Illnos, USA: Alleron House, UIUC. Dmarogonas, D., Frazzol, E., & Johansson, K. (202). Dsrbued even-rggered conrol for mul-agen sysems. IEEE Transacons on Auomac Conrol, 57(5), 29 297. Dmarogonas, D., & Johansson, K. (2009). Even-rggered cooperave conrol. In: Proc. of he European conrol conf. 2009 (pp. 305 3020). Budapes, Hungary. Augus. Godsl, C., & Royle, G. (200). Algebrac graph heory. Sprnger. Grone, R., & Merrs, R. (994). The Laplacan specrum of a graph II. SIAM Journal on Dscree Mahemacs, 7, 22. Hennngsson, T., Johannesson, E., & Cervn, A. (2008). Sporadc even-based conrol of frs-order lnear sochasc sysems. Auomaca, 44(), 2890 2895.
232 X. Meng, T. Chen / Auomaca 49 (203) 225 232 Hu, J., Chen, G., & L, H. (20a). Dsrbued even-rggered rackng conrol of leader follower mul-agen sysems wh communcaon delays. Kyberneka, 47(4), 630 643. Hu, J., Chen, G., & L, H. (20b). Dsrbued even-rggered rackng conrol of second-order leader follower mul-agen sysems. In: Proc. of he 30h Chnese conrol conf. (pp. 489 4824). Yana, Chna. July. Jadbabae, A., Ln, J., & Morse, A. S. (2003). Coordnaon of groups of moble auonomous agens usng neares neghbor rules. IEEE Transacons on Auomac Conrol, 48(6), 988 00. Ln, Z., Broucke, M., & Francs, B. (2004). Local conrol sraeges for groups of moble auonomous agens. IEEE Transacons on Auomac Conrol, 49(4), 622 629. Lu, Z., & Chen, Z. (200). Even-rggered average-consensus for mul-agen sysems. In: Proc. of he 29h Chnese conrol conf. (pp. 4506 45). Bejng, Chna. July. Lu, Z., & Chen, Z. (20). Reachng consensus n neworks of agens va evenrggered conrol. Journal of Informaon & Compuaonal Scence, 8(3), 393 402. Lunze, J., & Lehmann, D. (200). A sae-feedback approach o even-based conrol. Auomaca, 46(), 2 25. Mazo, M., & Tabuada, P. (20). Decenralzed even-rggered conrol over wreless sensor/acuaor neworks. IEEE Transacons on Auomac Conrol, 56(0), 2456 246. Meng, X., & Chen, T. (202). Opmal samplng and performance comparson of perodc and even based mpulse conrol. IEEE Transacons on Auomac Conrol, 57(2), 3252 3259. Mesbah, M., & Egersed, M. (200). Graph heorec mehods n mulagen neworks. Prnceon Unversy Press. Moreau, L. (2005). Sably of mulagen sysems wh me-dependen communcaon lnks. IEEE Transacons on Auomac Conrol, 50(2), 69 82. Olfa-Saber, R., & Murray, R. (2004). Consensus problems n neworks of agens wh swchng opology and me-delays. IEEE Transacons on Auomac Conrol, 49(9), 520 533. Ren, W., & Beard, R. (2008). Dsrbued consensus n mul-vehcle cooperave conrol: heory and applcaons. Sprnger. Seyboh, G., Dmarogonas, D., & Johansson, K. (203). Even-based broadcasng for mul-agen average consensus. Auomaca, 49(), 245 252. Sh, G., & Johansson, K.H. (20). Mul-agen robus consensus-par II: applcaon o dsrbued even-rggered coordnaon. In: Proc. of he 50h IEEE conf. on decson and conrol and European conrol conf. (pp. 5738 5743). Orlando, FL, USA. December. Tang, T., Lu, Z., & Chen, Z. (20). Even-rggered formaon conrol of mul-agen sysems. In: Proc. of he 30h Chnese conrol conf. (pp. 4783 4786). Yana, Chna. July. Tanner, H. G., Jadbabae, A., & Pappas, G. J. (2007). Flockng n fxed and swchng neworks. IEEE Transacons on Auomac Conrol, 52(5), 863 868. Texera, P., Dmarogonas, D., Johansson, K., & Sousa, J. (200a). Even-based moon coordnaon of mulple underwaer vehcles under dsurbances. In Proc. of he OCEANS 200 (pp. 6). Sydney, Ausrala: IEEE. Texera, P., Dmarogonas, D., Johansson, K., & Sousa, J. (200b). Mul-agen coordnaon wh even-based communcaon. In Proc. of he 200 Amercan conrol conf. (pp. 824 829). MD, USA: Marro Waerfron, Balmore. Wang, X., & Hovakmyan, N. (202). Performance predcon n unceran mulagen sysems usng L adapaon-based dsrbued even-rggerng. In Rolf Johansson, & Anders Ranzer (Eds.), Lecure noes n conrol and nformaon scences: vol. 47. Dsrbued decson makng and conrol (pp. 7 93). Berln, Hedelberg: Sprnger. Wang, X., & Lemmon, M. (20). Even-rggerng n dsrbued neworked conrol sysems. IEEE Transacons on Auomac Conrol, 56(3), 586 60. Xao, F., & Wang, L. (2008). Asynchronous consensus n connuous-me mulagen sysems wh swchng opology and me-varyng delays. IEEE Transacons on Auomac Conrol, 53(8), 804 86. Xe, G., Lu, H., Wang, L., & Ja, Y. (2009a). Consensus n neworked mul-agen sysems va sampled conrol: fxed opology case. In: Proc. of he 2009 Amercan conrol conf. (pp. 3902 3907). S. Lous, MO, USA. June. Xe, G., Lu, H., Wang, L., & Ja, Y. (2009b). Consensus n neworked mul-agen sysems va sampled conrol: swchng opology case. In: Proc. of he 2009 Amercan conrol conf. (pp. 4525 4530). S. Lous, MO, USA. June. Xangyu Meng was born n Changchun, Chna. He receved hs B.S. degree n Informaon and Compuaonal Scence from Harbn Engneerng Unversy n 2006, and M.Sc. degree n Conrol Scence and Engneerng from Harbn Insue of Technology n 2008. He was a Research Assocae n he Deparmen of Mechancal Engneerng a he Unversy of Hong Kong beween June 2007 and July 2007, and beween November 2007 and January 2008. He was a Research Award Recpen n he Deparmen of Elecrcal and Compuer Engneerng a he Unversy of Albera beween February 2009 and Augus 200. Currenly, he s a Ph.D. suden n he Deparmen of Elecrcal Engneerng a he Unversy of Albera. Hs research neress nclude even based conrol, esmaon, and opmzaon. Tongwen Chen s presenly a Professor of Elecrcal and Compuer Engneerng a he Unversy of Albera, Edmonon, Canada. He receved he B.Eng. degree n Auomaon and Insrumenaon from Tsnghua Unversy (Bejng) n 984, and he M.A.Sc. and Ph.D. degrees n Elecrcal Engneerng from he Unversy of Torono n 988 and 99, respecvely. Hs research neress nclude compuer and nework based conrol sysems, process safey and alarm sysems, and her applcaons o he process and power ndusres. He has served as an Assocae Edor for several nernaonal journals, ncludng IEEE Transacons on Auomac Conrol, Auomaca, and Sysems and Conrol Leers. He s a Fellow of IEEE.