Average Consensus in Networks of Multi-Agent with Multiple Time-Varying Delays
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1 I. J. Commucaos ewor ad Sysem Sceces do:.436/jcs..38 Publshed Ole February (hp:// Average Cosesus ewors of Mul-Age wh Mulple me-varyg Delays echeg ZHAG Hu YU Isue of olear ad Complex Sysems Cha hree Gorges Uversy Ychag Cha Emal: zchogcgu@yahoo.c yuhu@cgu.edu.c Receved ovember 9 9; revsed December 5 9; acceped Jauary Absrac he average cosesus udreced ewors of mul-age wh boh fxed ad swchg opology couplg mulple me-varyg delays s suded. By usg orhogoal rasformao echques he orgal sysem ca be ured o a reduced dmesoal sysem ad he LMI-based mehod ca be appled coveely. Covergece aalyss s coduced by cosrucg Lyapuov-Krasovs fuco. Suffce codos o average cosesus problem wh mulple me-varyg delays udreced ewors are obaed va lear marx equaly (LMI) echques. I parcular he maxmal admssble upper boud of me-varyg delays ca be easly obaed by solvg several smple ad feasble LMIs. Fally smulao examples are gve o demosrae he effecveess of he heorecal resuls. Keywords: Average Cosesus Mul-Age Sysem Mulple me-varyg Delays Lear Marx Iequaly. Iroduco Recely more ad more researchers have pad a grea deal of aeo o dsrbued coordaed corol of ewors of dyamc ages wh he corol commuy. Especally he cosesus problem was dscussed wdely whch ca be arbued o he broad applcaos of mul-age sysems may areas cludg cooperave corol of umaed ar vehcles formao corol of mul-robo flocg swarmg dsrbuo sesor fuso aude algme ad cogeso corol commucao ewors ec. I cooperave corol of mul-age sysem a crcal ssue s o desg approprae proocol ad algorhms such ha all ages ca reach a commo cosesus value. hs problem s called cosesus problem. I he pas decades some heorecal resuls have bee esablshed [ ] o ame a few. I [] Vcse e al. proposed a smple model bu eresg dscree-me model of auoomous ages all movg he plae wh he same speed bu wh dffere headgs. Smulao resuls provded [] show ha all ages ca eveually move he same dreco whou ceralzed coordao. he frs paper provdg a heorecal explaao for hese observed behavors Vcse model s []. he heorecal resuls Idefy applcable sposor/s here. (sposors) [] are exeded o he case of dreced graph by Re e al. [3] whch marx aalyss ad algebrac graph heory were used. Moreau [4] used a se-valued Lyapuov approach o sudy cosesus problem wh udrecoal me-depede commucao ls. Saber e al. [5] dscussed average cosesus problem. Whe ewor commucao s affeced by me delay he cosesus problem s vesgaed [5 7]. I [8] Saber provded a heorecal framewor for aalyss of cosesus algorhms for mul-age ewored sysems wh a emphass o he role of dreced formao flow robusess o chages ewor opology due o l/ode falures me-delays ad performace guaraees. Re e al. provded a uoral overvew of formao cosesus mulvehcle cooperave corol [9]. Yu e al. [] proposed weghed average cosesus dreco ewors ad udrecoal ewors wh me-delay. Mul-vehcle cosesus wh me-varyg referece sae was dscussed []. Some oher ssues o cosesus problem ca be foud [ 6]. Currely cosesus problem for mul-age ewors wh me delay was suded usg lear marx equaly mehod for example [7 9] ad []. I me delay sysems of mul-age he ewor opology of mul-age s a ey facor he aalyss of sably of mul-age sysem. Average cosesus problem for mul-age ewors wh boh cosa
2 . C. ZHAG E AL. 97 ad me-varyg delay has bee exesvely cosdered for he cases of mul-age udreced ad/or dreco ewors wh fxed ad/or swchg opology [78] ad [9]. However he sudes o cosesus problem of mul-age ewors wh mulple mevaryg delays are sll sparse. I hs case heorecal aalyss s more challegg. I [] he auhors proposed a cosesus proocol for mul-age udreced ewor wh mulple me-varyg delays based o LMI mehod. I hs paper we sudy he average cosesus problem for couous me udreced ewors of mulage couplg mulple me-varyg delays. Because he closed-loop sysem marx s sgular ad radoal LMI-based corol heory s vald. herefore heorecal aalyss for hs case s a challegg as. By usg orhogoal rasformao echques suffce codos based o LMI for mul-age achevg average cosesus s obaed. Compared o [] a dffere approach s used hs paper. Frs of all he orgal sysem s ured o a reduced dmesoal sysem by orhogoal rasformao; Secodly va cosrucg a dffere Lyapuov-Krasovs fuco a suffce codo expressed as LMI s proposed o guaraee all ages reach average cosesus fxed ad swchg ewor. Fally he maxmal admssble upper boud of mulple me-varyg delays ca be easly obaed by solvg several smple ad feasble LMIs. hs paper s orgazed as follows. Seco s he oao ad formally saes he problem. Seco 3 coas our ma resuls. Smulao resuls are preseed Seco 4. he cocludg remars are made Seco 5.. Problem Saeme ad Prelmares I hs seco we provde a bref roduco abou algebrac graph heory [4] ad sae he problem... Algebrac Graph heory Le GVEA be a weghed udreced graph of order where V v v v 3 v s he se of odes E VV s he se of edges ad A a j s a weghed adjacecy marx. he ode dexes belog o a fe dex se I. I udreced graph ej ej. ej E f ad oly f aj. Moreover we assume aj for all I. A udreced graph s always coeced. he se of eghbors of odes s deoed by v v V : v v E. j j he ou-degree of ode v s defed as follows: deg ou ( v ) a. he degree marx of graph G s a j dagoal marx D d j where dj for all j ad d deg ou ( v ). he Laplaca marx assocaed wh he graph s defed as aj j L lj D A j j aj j. A mpora fac of L s ha all he row sums of L are zero ad hus s a egevecor of L assocaed wh he egevalue. Lemma [5]. If he udreced graph G s coeced he s Laplaca marx L sasfes: ) L s symmerc ad ra ( L) ; ) zero s oe egevalue of L ad ad are he correspodg lef ad rgh egevecor respecvely he L ad L ; 3) he res egevalues are all posve ad real... Cosesus Problem o ewor Cosder a group of ages wh dyamcs gve by x u() I () where x s he sae of he h age a me whch mgh represe physcal quaes such as aude poso emperaure volage ad so o ad u () s he corol pu (or proocol) a me. We say proocol u asympocally solves he cosesus problem f ad oly f he saes of ages sasfy lm x ( ) x ( ) for all j I. Furhermore f j lm x( ) x Ave( x()) We say proocol u asympocally solves he average cosesus problem..3. Corol Proocol for me Delay Le j oes he me delay for formao commus proocols have bee suded. Oe s caed from age j o age. Because of me delay wo dffcul cosesu u aj[ xj( j()) x( j())] I () vj ad he oher s
3 98. C. ZHAG E AL. u aj[ xj( j x( )] I. (3) vj Leraure [5] have ae he proocol () wh j o accou ad obaed ha ca asympocally solves he average cosesus problem he fxed ad udreced ewor opology f ad oly f max L. Some coclusos also were vesgaed [6-8]. For cosesus proocol (3) commucao delay oly affecs he ages who are rasmed. Leraure [4] has esablshed he cosesus resuls he dreced ad dyamcally swchg graph. Some resuls also appeared [ 3]. I hs paper we are eresed dscussg he aver- age cosesus problem ewor of dyamc ages wh fxed ad swchg opology couplg mulple me-varyg delays where he formao passes hrough dffere edge wh dffere me-varyg delays. o solve such a problem we assume he me delay sasfes j j udreced graph.e. he delays rasmsso from x o x j ad x j o x cocde. So we use he followg proocol: u a [ x ( ()) x ( ())] I (4) j j vj Wh (4) () ca be wre marx form: x L x( (5) where x x x x () ad L l j s he marx defed by aj j ( ) j ( ) lj j ( ) j ( ) aj j. j Because of A s symmerc ad j j he udreced ewor we ca oba s a symmerc ad L L. he me-varyg delay () s assumed o sasfy he follow equao s: L ( ) d () h h (6) d h where ad are cosas. d ad h are he upper boud of d ad h am ly d m ax d h max h. e I he udreced ewor wh swchg opology we address he follow hybrd sysem: x L x( s s () I. (7) where he map ():[ ) I M s a swchg sgal ha deermes he ewor opology ad M deoes he oal umber of all possble swchg udreced graphs. L L ( G ) s he Laplaca marx of he graph G o a se : s s s V E A ha belogs s s s s G I whch s obvously fe. Lemma (Schur compleme [5]). Le S S S be gve symmerc marces such ha S S S he S S S SS S. Lemma 3. For ay Laplaca marx L of udreced coeced ewor here exss a orhogoal marx W such ha L W LW where he las colum of marx ( ) ( ). W s L Proof: Because ad are he correspodg lef ad rgh egevecor respecvely. he proof of hs lemma s sraghforward. 3. Ma Resuls I hs seco we provde he covergece aalyss of he average cosesus problem udreced ewor wh fxed ad swchg opology couplg mulple me-varyg delays. Suffce codos expressed as LMI are preseed for udreced ewors of mulage wh fxed ad swchg opology respecvely. 3.. ewors wh Fxed opology If he fxed commucao opology GVEA ep coeced we have L ad L whch mply x u. he Ave( x()) a vara quay. hus we have he decomposed equao x() () where 3 sasfes.e.. he (5) s equvale o s s
4 . C. ZHAG E AL. 99 he L ( ( ). (8) By Lemma 3 we have ) WW LWW ( W W LWW (. Le W wher e (8) ca be rasformed o he followg equao: where L sasfes Lemma 4. If. he L ( ( 9) L W LW. m he lm. l Proof: From W W we have. herefore whe l we have lm. hs complees he proof. m I he followg seco we wll dscuss he coverdyamcal sy sem (9) ha s lm gece of. heorem. Cosder a udreced ewor of mul-age wh fxed opology couplg mulple me-varyg delays sasfes (6). Assume he commucao opology G s ep coeced. he sysem (5) asympocally solves he average cosesus problem f here exs posve defe marces PQ R where sasfyg L P PL ( ) ( ) R ( ) ( ) d ( ) ( ) ( ) ( ) () PL L P Q dag[ ( ) L R L h Q () L RL h Q]. Proof: Defe a Lyapuov-Krasovs fuco for sysem (9) as follows: V () V() V() V() 3 V () () P () () V () () s Q () s ds V () () s R () s dsd 3 d where P Q ad R are posve defe marx. Alog he rajecory of sysem (9) we have 3 V ( ) () PL( ()) V () { () Q () ( ()) ( ()) Q ( ())} { ( Q ) ( ) ( h ) ( ()) Q ( ())} V () { d ( ()) LR L( ()) () s R () s ds}. By ewo-lebz formula ad oe ha ( ( ) ( s) ds x y x F x () y Fy hold for ay approprae posve defe marx F we have: ( PL ) ( () { PL() [ L P()] ( sds ) } { () PL () d () PL R L P () () s R () s ds}. ( ) Cosequely () { () () () () V PL d PLR L P d ( L R L ( ( ) Q ( ) ( h ) ( ()) Q ( ())} { ( ) PL L P Q dplr L P ( ) ( ( d L R L ( h ) Q ) ( }. he a suffce codo for V () s
5 . C. ZHAG E AL. { ( ) PL L PQ dplr L P ( )} ad () { ( ( dl RL ( h) Q) ( }. () As a resul he marx equales () hold f ad oly f PL L P Q d PL R L P he by Schur compleme formula he marx equaly (3) s equvale o L P d PL R (3) (4) where s defed (). herefore from Lemma 4 average cosesus ca be acheved f he marx equaly (6) holds. hs com- plees he proof. 3.. ewors wh Swchg opology Cosderg he swchg commucao opology Gs ( s I ) wh sysem (7) we ca oba he followg dsagreeme swchg sysem: where L ( s ( ) I (5) s L s sasfes L s W LsW. heorem. Cosder a udreced ewor of mul-age wh swchg opology couplg mulple me-varyg delays sasfes (6). Assume he commucao opology Gs ( s I ) s ep coeced. he sysem (7) asympocally solves he average cosesus problem f here exs posve defe marces PQ R sasfyg PL s ( ) ( ) R Ls P ( ) d (6) ( ) ( ) ( ) ( ) where PLs Ls P Q dag[ ( ) L R L h Q s s () L RL h Q]. s s Proof: he proof of heorem s smlar o ha of heorem so s omed here. Remar: Whe h of he codo (6) s replaced wh h we ca oba correspodg resuls f choosg V () V() V(). 4. Smulao 3 I hs seco smulao examples wll be gve o valdae he heorecal resuls obaed he prevous seco. Cosder a group of ages labeled hrough. Fgure shows four examples of udreced graph whch are all coeced ad he correspodg adja- lmed o - marces. Le he or- cecy marces are hogoal marx W s where he las colum of marx W s. For smplcy we assume udreced ewors of mul-age wh boh fxed ad swchg opology..
6 . C. ZHAG E AL. 4.. Examples of ewors wh Fxed opology me Delay=.7 cos.534 Cosder a udreced ewor wh fxed opology G Fgure. Employg heorem we have: ) for h.e. d d.5. Fgure shows he correspodg error sysem coverges zero asympocally. ) for h.5 d.7. Fgure 3 shows he correspodg error sysem coverges zero asympocally error of sysem G me(s) G G G4 Fgure. Examples of coeced udreced graph. 5 me Delay= Fgure 3. Error sysem wh fxed opology ad mevaryg delay ( ).7 cos.534 s wh h.5 coverges o zero asympocally. 4.. Examples of ewors wh Swchg opology A fe auomao wh se of saes G G G s 3 4 show Fgure 4 whch represes he dscree saes of a ewor wh swchg opology a d me delay as a hybrd sysem. I sars a he dsc ree sae G ad swches every smulao me sep o he ex sae accordg o he sae mache Fgure 4. For mul-age sysem wh me-varyg commucao me delay we ae he dervave of delay h.5. Fr om heorem he feasble maxmum delay boud of he sysem s d.s. Ad he correspodg feasble soluo P Q ad R ca b e obaed by employg he LM I ool box Malab. Assume he me-varyg delay of h e error sysem s ( ). cos.534 s. Fgure 5 shows he correspodg error sysem coverges zero asympocally. error of sysem 5 = G me(s) Fgure. Error sysem wh fxed opology ad cosa me-delay d.5s wh h coverges o zero asympocally. G4 G3 Fgure 4. A fe auomao wh hree saes.
7 . C. ZHAG E AL. error of sysem me Delay=. cos me(s) Fgure 5. Error sysem wh swchg opology ad mevaryg delay ( ). cos.534 s coverges o zero asympocally. 5. Coclusos 6. Acowledgme hs wor was suppored by aoal aural Scece Foudao of Cha uder he gra 664 ad Hube Provcal aural Scece Foudao uder he gra 8CDB36 ad 8CDZ46 ad Scefc Iovao eam Projec of Hube Provcal College uder he gra Refereces []. Vcse A. Czro E. Bejacob I. Cohe e al. ovel ype of phaseraso a sysem of self-drve parcles Physcal Revew Leers Vol. 75 o. 6 pp [] A. Jadbabae J. L ad A. S. Morse Coordao of groups of moble auoomous ages usg eares eghbor rules IEEE rasacos o Auomac Corol Vol. 48 o. 6 pp [3] W. Re ad R. W. Beard Cosesus seeg mul- age sysems uder dyamcally chagg eraco opologes IEEE rasacos o Auomac Corol Vol. 5 o. 5 pp hs paper addresses a average cosesus problem of mulage sysems. Udreced ewors wh fxed ad swchg ewor opology couplg mulple me- varyg commucao delays are cosdered hs paper. A orhogoal marx s roduced ad he orgal sysem s ured o a reduced dmesoal sysem. A he same me a Lyapuov-Krasovs fuco s cosruced he sable aalyss. Suffce codos erms of LMI are gve o guaraee he sysem reach average cosesus. Moreover umercal smulao ex- amples are show o verfy he heorecal aalyss. [4] L. Moreau Sably of mulage sysems wh medepede commucao ls IEEE rasacos o Auomac Corol pp [5] R. O. Saber ad R. M. Murray Cosesus problems ewors of ages wh swchg opology ad me-delays IEEEE rasacos o Auomac Corol Vol. 49 o. 9 pp [6] R. O. Saber ad R. M. Murray Cosesus proocols for ewors of dyamc ages Proceedgs of Amerca Corol Coferece pp Jue 3. [7] Y. a ad C. Lu Cosesus of mul-age sysems wh dverse pu ad commucao delays IEEE rasacos o Auomac Corol Vol. 53 o. 9 pp. 8 Ocober 8. [8] R. O. Saber J. A. Fax ad R. M. Murray Cosesus ad cooperao ewored mul-age sysems Proceedgs of IEEE Vol. 95 o. pp Jauary 7. [9] W. Re R. W. Beard ad E. M. As Iformao cosesus mulvehcle cooperave corol IEEE Corol Sys. Mag. Vol. 7 o. pp. 7 8 Aprl 7. [] H. Yu J. G. Ja ad Y. J. Wag Weghed average cosesus for dreced ewors of mul-age Mcrocompuer Iformao Vol. 3 o. 7 pp Auges 7. [] W. Re Mul-vehcle cosesus wh a me-varyg referece sae Sysems & Corol Leers Vol. 56 o. 7-8 pp July 7. [] D. P. Spaos R. O. Saber ad R. M. Murray Dyamc cosesus o moble ewors I IFAC World Cogr. Prague Czech Republc 5. [3] Y. Haao ad M. Mesbah Agreeme over radom ewors IEEE rasacos o Auomac Corol Vol. 5 o. pp ovember 5. [4] R. O. Saber Flocg for mul-age dyamc sysems: algorhms ad heory IEEE rasacos o Auomac Corol Vol. 5 o. 3 pp. 4 4 March 6. [5] A.. Saleh ad A. Jadbabae A ecessary ad suffce codo for cosesus over radom ewors IEEE rasacos o Auomac Corol vol. 53 o. 4 pp Aprl 8. [6] G. M. Xe ad L. Wag Cosesus corol for a class of ewors of dyamc ages Robus oler Corol Vol. 7 pp [7] Y. G. Su L. Wag ad G. M. Xe Average cosesus dreced ewors of dyamc ages wh me-varyg delays I Proceedg of IEEE Coferece Decso & Corol Vol. 57 o. pp December 6. [8] P. L ad Y. Ja Average cosesus ewors of mul-ages wh boh swchg opology ad couplg me-delay Physca A Vol. 387 o. pp Jauary 8. [9] P. L Y. Ja ad L. L Dsrbued robus H cosesus corol dreced ewors of ages wh me-delay Sysems & Corol Leers Vol. 57 o. 8 pp Auges 8. [] Y. G. Su L. Wag ad G. M. Xe Average cosesus
8 . C. ZHAG E AL. 3 ewors of dyamc ages wh swchg opologes ad mulple me-varyg delays Sysems & Corol Leers Vol. 57 o. pp February 8. [] D. Agel ad P. A. Blma Sably of leaderless ds- cree-me mulage sysems Mah. Corol Sgs Sysems Vol. 8 pp [] A. Papachrsodoulou ad A. Jadbabae Sychrozao oscllaor ewors: Swchg opologes ad o-homogeeous delays I Proceedgs of he 44h IEEE Coferece Decso ad Corol ad he Europea Corol Coferece pp December 5. [3] A. Papachrsodoulou ad A. Jadbabae Sychrozao oscllaor ewors wh heerogeeous delays swchg opologes ad olear dyamcs I Pro- ceedgs of he 45h IEEE Coferece Decso ad Corol pp December 6. [4]. Bggs Algebrac Graph heory Cambrdge Uversy Press Cambrdge U. K [5] R. A. Hor ad C. R. Johso Marx Aalyss Cambrdge Uversy Press Cambrdge ew Yor 985.
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