Group Consensus of Multi-agent Networks With Multiple Time Delays

Transcription

1 Intrnatonal Confrnc on Artfcal Intllgnc: Tchnologs and Applcatons (ICAITA 06 Group Consnsus of Mult-agnt Ntworks Wth Multpl Tm Dlays Langhao J* Xnyu Zhao Qun Lu and Yong Wang Chongqng Ky Laboratory of Computatonal Intllgnc Chongqng Unvrsty of Posts and Tlcommuncatons Chongqng P R Chna School of Computr Scnc and Tchnology Chongqng Unvrsty of Posts and Tlcommuncatons Chongqng P R Chna wth connctd undrctd and connctd bpartt graphs rspctvly n [3] In th connctd bpartt graph th rfrnc [4] addrssd group consnsus of frst-ordr ntworks wth and wthout dlays rspctvly by a control protocol Morovr Du t al [5] xtndd th conclusons proposd n [4] Abstract Ths papr nvstgats th group consnsus problm of frst-ordr mult-agnt ntworks wth multpl tm dlays By applyng th thory of frquncy-doman w am to propos som algbrac crtra such that th mult-agnt ntworks can rach group consnsus From th rsults t can b shown that group consnsus of ntworks s dtrmnd by ownng nput tm dlays and conncton strngths btwn agnts ndpndnt of communcaton dlays Howvr th xstnc of communcaton dlays wll affct th convrgnc rat of mult-agnt ntworks Fnally svral numrcal smulatd xampls ar gvn to show th valdty and corrctnss of our thortcal rsults As w know du to th communcaton lnk qupmnt tc thr xst two dffrnt knds of dlays n ntworks that s communcaton dlays and nput dlays In fact ths two dlays obctvly xst and ar dffrnt from ach othr Thus th rsarch towards group consnsus of ntworks wth multpl dlays bcom mor ralstc Comparng wth xstng nvstgatons thr xst two man shortcomngs: Frst th ffct of mult-agnt ntworks wth th two knds of dlays ar consdrd nsuffcntly Som rlatd rsarchs only nvolv communcaton dlays or only analyz th sam communcaton and nput dlays [5-73-5] Scond most of survys only focus on ntworks wth spcal topologs such as undrctd strongly connctd as wll as strongly connctd and balancd graphs tc [5-6-5] Insprd by rlatd rsarchs mor gnral topologs ar nvstgatd W addrss th group consnsus ssu wth multpl dlays on frstordr ntworks Thraftr algbrac crtra ar drvd that nsur group consnsus to b achvd Evntually rlatd rsults for smlar ssus ar vwd as spcal cass of ths papr or analyss crtra ar rlatvly lss consrvatv Kywords-group consnsus; mult-agnt ntworks; tm dlays; complx ntwork I INTRODUCTION Partly du to broad applcatons of mult-agnt ntworks n many aras such as congston control[] and flockng [] th coopratv control has attractd many rsarchrs ntrst As a fundamntal branch of coopratv control th consnsus ssus hav bcom hot spots Rcntly many rsults about consnsus problms hav bn stablshd Spcfc contnts can b found n survy paprs [3-4] and rfrncs thrn tc In coopratv control n ordr to nsur targt tasks to b compltd harmonously t rqurs that stats of all agnts kp consstnt wth tm Howvr wth th changs of nvronmnts stuatons or vn tm th consnsus stats ar dffrnt Gnrally t can b dscrbd by group consnsus problm Up to dat group consnsus of mult-agnt ntworks has achvd som progrss Yu t al [56] addrssd group consnsus wth undrctd or strongly connctd and balancd graphs on th bass of th matrx thory Furthrmor Yu t al by th da of doubl-tr-form transformaton xtndd group consnsus problms wth communcaton dlays and swtchng topologs [7] For spcal topologs wth th sam and dffrnt slf-dynamc ntworks th group consnsus problms wr nvstgatd n [8] Morovr n ordr to rduc th cost of ntwork control pnnng control stratgs hav bn ntroducd nto mult-agnt ntworks Partcularly th analyss of pnnng group consnsus startd n [9-0] and rfrncs thrn tc Addtonally undr th strongly connctd and balancd graph Wang t al [] took group consnsus wth communcaton dlays nto consdraton Hu t al[] gav an ovrvw of avrag-group consnsus problms on ntworks wth undrctd topologs J t al provdd group consnsus 06 Th authors - Publshd by Atlants Prss Th rst of th papr s organzd as follows In Scton rlvant prlmnars on graph thory and modl formulaton ar summarzd Th problm of group consnsus of ntworks wth multpl tm dlays s dscussd n Scton 3 By a pluralty of numrcal xprmnts th valdty and accuracy of conclusons s vrfd n Scton 4 Fnally concludng rmarks and futur trnds ar statd II PRELIMINARIES In mult-agnt ntworks th topology can b dscrbd by a drctd graph Th nod st s dfnd by V v v vn Dnot th nod ndxs blong to a fnt ndx st by N and th dgs st s E V V Th nghbor st of v s A {a } 54 N N N v V : v v E Morovr s th wghtd adacncy matrx Whn

2 v N a 0 For smplcty w assum a 0 Lt D dag d b th dgr matrx of G L D A s th Laplacan matrx of G For frst-ordr ntworks th dynamc s lstd as (: x t u t n whr x( t u( t dnots th poston control nput of th agnt Wthout loss of gnralty w assum n w ust only consdr x( t u( t Whn n by th Kronckr algorthm t can b asly gnralzd Usually for convnnc w rstat som rlatd lmma and dfntons as follows Dfnton Th frst-ordr mult-agnt ntworks ( s sad to ralz consnsus asymptotcally f for any t follows that x t x t lm 0 t Dfnton [6] For G V E A f V s dvdd nto two dsont subgroups { V V } and th two vrtcs v assocatd wth ach dg ( v v rspctvly blongs to two dffrnt sts of vrtcs w call G th bpartt graph Dfnton 3 If thr xsts a path n G from v to v thn v s sad to b rachabl from v If a nod s rachabl from vry othr nod n G thn t s tratd as a globally rachabl nod Lmma [4] If th topology of G s a connctd bpartt graph th rank of D A s n Lmma [7] If th graph G xsts a globally rachabl nod ts Laplacan matrx wll hav a smpl gnvalu 0 Lmma 3 [8] For 0 0 U dos not contan 0 Co E E whn convx hull T and T dnots th systm dlay T Lmma 4 [9] For th st convx hull III Co 0 U E whr U G s ncludd n th GROUP CONSENSUS OF MULTI-AGENT NETWORKS WITH MULTIPLE DELAYS A Group Consnsus of Dlayd Mult-agnt Ntworks wth th Connctd Bpartt Topologs For th ntwork wth th connctd bpartt topology rf [5] by dsgnng protocol ( drvd consstnt stats of convrgnc vntually At th sam tm t also dscussd th maxmum allowd dlay such that th systm ( undr protocol (3 can achv group consnsus ( ( ( u t a x t x t vn ( ( ( u t a x t x t vn whr dnots th systm dlay By protocol (3 rfrncs [4-6] only dscussd a spcal cas of th sam communcaton and nput dlays Rfrnc [5] dd not gv th condtons whr ntworks rachd group consnsus n th xstnc of tm dlays In gnral th two dffrnt knds of dlays xst obctvly so w drv som algbrac crtra Ths algbrac crtra can guarant group convrgnc of systm ( wth protocol (4 ( ( ( u t a x tt x tt vn whr T dnots communcaton dlay and T ndcats nput dlay Wth algorthm (4 th closd-loop form of ( s ( ( ( x t a x tt x tt vn Thorm Assum th systm (5 of N agnts wth undrctd bpartt topology For f max dt / 4 systm (5 can rach group consnsus asymptotcally Proof: By takng th Laplac transformaton of (5 w st st dt si D A 0 For smplcty w obtan dfn Fs dt si st D st A By th gnral Nyqust stablty crtron th followng two cass ar dscussd rspctvly: Whn 0 basd on Thorm w can drv that F s has a smpl zro at s 0 st st D A Whn 0 s s Th dscusson about th zros of Ps So f all zros of s Fs dt D A Fs s lt Ps Gs F s quals to th zros of Ps bng on th opn lft complx plan systm (5 wll hav group consnsus Lt s From th Grshgorn dsk thorm th quvalnt of G satsfs 55

3 G G T T G : C a a vn vn whr C dnots complx fld On th bass of (7 th cntr of th dsk G s T G0 a Thn ntrscton pont s dfnd vn by W whch s mad by th boundary of th dsk and th orgn pont of th complx plan O W can s that th track T W a From Lmma 3 notng of pont s vn w know that for any gvn t s asy to know that at /4 that W E vn Now lttng max obvously whn for any t s asy to obtan that th nxt quaton holds Co 0 E 0 E Co 0 W In th vw of Lmma 3 snc ( 0 Co 0 E ( U w also can conclud that ( 0 U G Notng that Co(0 U W ( U G accordng to Lmma 4 w hav ( 0 U G That s to say ( G( w dos not contan ( 0 Thrfor w conclud that f th gnral Nyqust stablty crtron s appld all zros of ngatv ral parts Th proof of Thorm s compltd Ps hav Rmark Th allowabl uppr bounds of dlays ar proposd analytcally whch can guarant group consnsus n [4] In th sharp contrast to Lmma th concluson about th bound of dlays s too broad Th followng compard rsults of xprmnts vrfy th concluson Manwhl from th concluson of Thorm t shows that group convrgnc s subct to th nput dlays and adacnt wghts and s ndpndnt of communcaton dlays Corollary Supposd th systm (5 of N agnts wth drctd bpartt topology for f max dt / 4 s satsfd systm (5 can rach group consnsus asymptotcally Th procss of th proof s smlar wth Thorm w omt t du to th lmtaton of spac followng two assumptons of n-dgr balanc consdrng th followng protocol (8 wth multpl tm dlays u nm n a a 0 L ;b a 0 L t n a ( x ( tt x ( tt a x ( tt VN VN a ( x ( tt x ( tt a x ( tt VN VN In (8 for L a 0 ; L a 0 ; U L L L L a : : Wth (8 th closd-loop form of ( s xt a ( x ( tt x ( tt a x ( tt VN VN a ( x ( tt x ( tt a x ( tt VN VN Thorm Undr assumptons of n-dgr balanc w consdr systm (9 of n m( n m agnts s a dgraph whch owns a globally rachabl nod Thn th ntwork wll achv group consnsus asymptotcally f and only f mn max dt / 4 n m whr d ak k k Th proof progrss of Thorm s vry smlar to Thorm and s omttd from ths not du to th lmtaton of spac IV SIMULATION EXAMPLES Accordng to Thorm and Thorm rspctvly som smulaton xampls ar gvn to vrfy th ffctvnss and th corrctnss of th crtra stablshd abov A Exprmnt I W consdr systm (5 wth th topology and couplng wghts btwn agnts dscrbd n Fgur I Lt v v n a group 3 v 4 v5 n anothr group Th ntal stats of agnts ar x(0 [ ] T Input dlays ar T T 0 s T3 T4 0 s T5 0s In Fgur II t shows that th ntwork can achv group consnsus Comparng th ntwork curvs wth dffrnt dlays t s obvous that th xstnc of communcaton dlays can mpact th convrgnc spds of ntworks B Group Consnsus of Dlay Systms wth th Topology Ownng a Globally Rachabl Nod Suppos th ntwork conssts of n m agnts and L { n} L { n n n m} Basd on th 56

4 group consnsus s fal to b ralzd Th corrct and ffctv of Thorm s vrfd G 4 G FIGURE I TOPOLOGY OF SYSTEM (5 3 5 FIGURE IV INTERCONNECTION GRAPH OF SYSTEM ( a T 0s b T 04s FIGURE II TRAJECTORIES OF SYSTEM (5 From Fgur I th dgrs of v ar 3 By Thorm n ordr to rach th group consnsus of ntworks th allowd nput dlays of v should hold T / 06s Basd on th abov xprmnts w conduct xprmnts on th followng stuatons Th stat tractors of systm (5 ar plottd n FgurIII a T 06s b T 06s FIGURE III STATE TRAJECTORIES OF SYSTEM (5 From FgurIII t llustrats that systm (5 wll not rach group consnsus Compard wth th rsult n [5] t s clar that th uppr bound of th tm dlay w drvd s mor accurat B Exprmnt II W consdr th topology of th ntwork ( wth 5 nods plottd n Fgur IV Th ntal stats of agnts ar x(0 [ ] T Lt T 04s and nput dlays of ach nod ar 06s 07s 03s 04s and 0s rspctvly Th stat tractors of ( ar shown n Fgur V (a Th group consnsus s achvd By th condton of Thorm w can larn that d% and th nput dlay satsfs T / s If st T 079s and th dlays of othr nods kpng unchangd from Fgur V (b w larn that th a T 06s b T 079s FIGURE V DYNAMIC BEHAVIORS OF SYSTEM ( V CONCLUSION For frst-ordr ntworks ths tchncal survy s amd at xplorng th ssu of group consnsus of mult-agnt ntworks wth dvrs communcaton and nput dlays By th thory of frquncy-doman som algbrac crtra of th group consnsus ar drvd It can b shown that th group consnsus of ntworks s dtrmnd by ownng nput dlays and conncton strngths ndpndnt of communcaton dlays Howvr th xstnc of communcaton dlays wll affct th convrgnc spd of ntworks Du to varous rasons th topology of complx ntworks typcally changs thus our futur work wll nvstgat th group consnsus ssu for dvrs dlays undr swtchng topologs ACKNOWLEDGMENT Ths work was supportd by th Natonal Natural Scnc Foundaton of Chna (Grant No th Natural Scnc Foundaton Proct of Chongqng Scnc and Tchnology Commsson (Grant Nos cstc04cya40047 cstc03cya4007 and th Scntfc and Tchnologcal Rsarch Program of Chongqng Muncpal Educaton Commsson (Grant No KJ Th authors also would lk to thank anonymous rvwrs REFERENCES [] P Yang R Frman G Gordon K Lynch S Srnvasa R Sukthankar Dcntralzd stmaton and control of graph connctvty for mobl snsor ntworks Automatca 00 vol 46 pp [] S Charalambos V Vasos P Arstodmos Congston control n wrlss snsor ntworks through dynamc altrnatv path slcton Computr Ntworks 04 vol 75 pp

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