A Novel Composite-rotating Consensus for Multi-agent System

Transcription

1 d Itertol Symposum o Computer, Commuto, Cotrol d Automto (CA ) A Novel Composte-rottg Cosesus for Mult-get System Gu L Shool of Aerouts d Astrouts Uversty of Eletro See d ehology of Ch ChegDu, Ch lgu@uestedu Abstrt We develop dstrbuted otrol poly to heve omposte-rottg osesus roud the movg eters whose trjetores form rle ths pper It s ssumed tht the ommuto topology mog gets s udreted etwor d ll gets move o ple perpedulr to spefed vetor outerlowse dreto he dstrbuted otrol protool s dvded to two prts: oe prt s the lol veloty feedb tht mes eh get surroud the movg eters d the other s the dstrbuted feedb tht elmtes the dsgreemet mog the gets he otrol protool s exteded to -D spe by trodued rottg mtrx Bsed o the stble theory of Hurwtz, we verfy the stblty of the mult-get system Flly, Smulto exmples re provded to demostrte the effetveess of our theoretl results omposte-rottg osesus rulr orbt d exteded t to -D spe II PRELIMINARIES A Grph theory For oveee, we suppose the ommuto mult-get system s modeled through udreted weghted grph G = ( S, e) of order wth set of odes S = (s, s,l, s ), d set e S S of edges, d the edges represet the ommuto ls f ode get formto from ode he set of eghbors of ode s s deoted by N = { (, ) e, } he relto of the ommuto ls be desrbed by weghted wth o-egtve djey mtrx Α = [ ] R elemets for ll, I = {,,L, } he djey elemets ssoted wth the edges re postve, tht s, e e > Moreover, we ssume = d for ll, I A dgol mtrx = Δ = dg[δ, Δ,K, Δ ] s degree mtrx of G, whose Keywords-mult-get system, omposte-rottg osesus, -D spe, movg eters I INRODUCION Cosesus problem of mult-get system hs reeved osderble tteto by my reserhers Pvoe et l [] proposed deetrlzed otrol poly for symmetr formtos mult-get systems tht ll gets evetully overged to sgle pot, rle or logrthm sprl ptter Motvted by the pplto of utoomous uderwter vehles (AUVs) s moble sesors to ollet oeogrph mesuremets formtos or ptters tht yeld mxmlly formto-rh dt sets, Sepulhre[] et l studed otrol strtegy to me ll mult-gets move ple t ut speed However, these otet of reserhes re ofed to -D dmeso Motvted by the wor of [], Re[] et l provded the olletve motos of tem of vehles D by trodug rotto mtrx I [4], So Herdez et l studed deetrlzed otrols to stblze three-dmesol olletve moto of utoomous vehles d heved prllel, hell, d rulr formtos YANG et l studed dstrbuted rottg osesus etwors of seod-order mult-get systems usg oly lol posto formto three-dmesol (-D) spe [5] L[6] et l vestgted olletve rottg motos of seod-order mult-get system However, ths pper s just -D spe L et l ddressed olletve rottg motos of seod-order mult-get systems -D spe [7] I ths pper, our m objetve s to desg dstrbuted otrol protool tht hs gurteed ll gets heve he uthors - Publshed by Atlts Press dgol elemets Δ = for ll =,,L, he, the = Lpl of the udreted grph G s defed s L = Δ A Lemm : If udreted grph G s oeted, the the mtrx L s postve Lemm : Let єr be ut vetor whose oordte represetto Γ s = [ x, y, z ], d Γ be ew oordte system obted by rottg Γ bout the xs wth gle θ he the rotto mtrx of Γ wth respet to Γ s R (θ ) = (os θ ) I + ( os θ ) + (s θ ) S ( ) S ( ) - z y z - x - y x B rsformto of Coordtes I ths subseto, we trodue some oepts d results of trsformto of oordtes Suppose two oordtes three spe d shre ommo org, 4

2 deoted by Γ ( x m, yz, ) d Γ (,, ) α βγ Cosder rbtrry vetor expressed terms of the ompoets log the x, yzxes, d the αβγ,, xes hus we hve the equvlet represettos: m = xx + yy + zz = αα + ββ + γγ () d,, x y z α, β, γ re two sets of orthogol ut vetors prllel to ther respetve oordte xes he subsrpt of s orrespodg to the relevt oordte system For oveee, the vetor be wrtte the mer of mtrx: = [ x, y, z ] = [ α, β, γ ] () m m m m he two vetors bove trsform mutully by rotto mtrx R he, we obt Γ = R Γ or Γ = R Γ () m m m m R d t hs the property tht RR = R R = I III PROBLEM DESCRIPION he dstrbuted system vestgted ths pper ossts of utoomous gets Aget s be regrd s ode udreted grph G d edge e S S of grph G orrespods to formto l betwee get s d s Suppose the dyms of the th get re gve by follows: r& = v, v& = u, =,, L, (4) r [,, ] = r r r d v [,, ] = v v v re respetvely the posto d speed of get s three dmeso d u s the otrol protool Note tht ll the vetors re represeted ertl retgulr oordte system, deoted by Γ o ths pper Due to the pplto of dstrbuted system stellte d umed erl vehle otrol, et, omposte-rottg osesus of seod-order mult-get system s studed ths pper d exteded to three dmeso ht s to sy,eh get ot oly moves roud ommo xs but lso o ts ow xs Defto he mult-get system (4) heves omposte-rottg osesus f every get stsfes lm ρυ ( t) = (5) lm[ rt ( ) rt ( )] = (6) lm[ υ( t) υ ( t)] = ( 7) lm[ r& () t + ω RoR R & oυ() t ωror Ro( r() t (8) + ω RoR Roυ ())] t = for y, I, ρ s spefed vetor, ω ω re postve ostts d ω ω, - R = For oveee dsusso, we ssume tht ll gets move tlowse dreto Besdes, we reommed ew retgulr oordte system Γ whose thrd oordte xs s prllel to the spefed vetor ρ Mewhle, Γo d Γ shre the ommo org hus the rotto mtrx tht trsforms Γ o to Γ s deoted s R o Here, we do ot gve y defto of the other two xes for Γ so tht the rotto mtrx Ro te dfferet vlues whh hve o fluee o results IV MAIN RESULS I ths seto, we vestgte the omposte- rottg osesus otrol of the mult-get system (4) d obt the followg osesus protool: u = u + u (9) u = ωror Roυ ωω ( r + ω RoR Roυ) d u = [ υ υ ] [( r + ω R R R υ ) o o s N s N ( r ω RoR Roυ )] +, for ll, I = {,, L, } - d R = I the protool (9), u s lol - veloty feedb tht mes ll gets surroud movg pot o ple perpedulr to the vetor ρ d the trjetory of the pot s rle, whle u s dstrbuted stte formto feedb tht elmtes the dsgreemet dyms of the mult-get system Let = [ r, υ, L, r, υ ], I A = ωω I ( ω ω) RoR Ro Ad B = I I ω RoR + Ro he usg the protool (9) for system (4), the losed-loop system be wrtte vetor form s 4

3 & = ( I A L B) () L s the Lpl of the grph G Lemm: Deote = ( I Ro) = [ r, r, r, υ, υ, υ, L, r, r, r,, 6 υ, υ, υ ] rj, υj he the system () be repled by & = ( I A L B) Where I A = ωω I ( ω ω) R d B = I I ω R + Further, 4 ρ = [ r, r, υ, υ, L, r, r, υ, υ ] = [ r,, L, r, ] () let d ρ υ υ he the system () be deomposed to the followg two systems, & ρ = ( I C L D) ρ () & ρ = ( I E L F) ρ () C =, ωω ω- ω ωω ω- ω D = -, ω ω E = ωω ω- ω d F = + ω I ft, the system () d () re the substtutes for system () the retgulr oordte System Γ Lemm 4: All roots of the equto + ( + + ) + + = (4) s b s b hve egtve rel prts, > b >, > Proof : e s = jω to the equto (4), ω s postve umber he Eq (4) s trsformed to the followg two equtos, ω + b + = (5) ( b+ + ) ω = (6) Form (6), we obt ω = d substtute t to (5), we get b + = (7) It s esy to see tht the equto (5) ot hold whe > b>, > ht s, the Eq (4) hs o roots o the mgry xs Wht s more, se > b>, >, the oeffets of Eq (4) re postve Aordg to the theory of Hurwtz, ll roots of Eq (4) hve egtve rel prts Lemm5: If the grph G s oeted, the lm[ r() t r()] t = d lm υ( t) = for y, Iht s, ll gets flly overge to ple perpedulr to the vetor ρ Proof: Cosder the system () By Lemm, there exst orthogol mtrx W R suh tht W LW = dg{ λ, λ, K, λ } wth < λ λ L λ It follows tht ( W I)( I E L F)( W I) = dg{ E λfe, λ F, L, E λ F} (8) he hrterst polyoml of eh E λ F s λ det( s I E+ λf) = s + ( ω ω + + λ) s+ ωω + λ ω =, =,, K, he, ordg to lemm 4, system (8) s Hurwtz stble I ddto, by smple lulto we obt tht ( I E L F ) ( [,]) = he by ler system theory, we hve lm υ( t) = d lm[ r ( t) r ()] t = for y, I hus, we ow ll gets flly move o ple perpedulr to the vetor ρ Lemm 6: All roots of the equto s + s + [ + ( b + ) + ( b ) ] s + ( b+ s ) + ( b+ ) = 4 (9) hve egtve rel prts, > b >, > Proof : e s = jω to the equto (9), ω 4

4 s postve umber he Eq (9) s trsformed to the followg two equtos, 4 ω [ + ( b+ ) + ( b ) ] ω + ( b+ ) = () ω + ( b+ ) ω = () From (), we hve ω = or ω = b + Substtutg ω = or ω = b + to (), we get ( b + ) = or + ( b ) = () It s esy to see tht the equto () ot hold whe > b >, > ht s, the equto (9) hs o roots o the mgry xs Addtolly, wth regrd to (9), we get the ll oeffets of equto (9) s follows =, =, = + ( b+ ) + ( b ), = ( b+ ), 4 = ( b+ ) the we obt () d (4) d Δ = = + b+ + ( b ) ( b+ ) > *( ) 4 4 ( b+ ) Δ > = = ( b+ ) ( b+ ) () (4) herefore, by the theory of Hurwz, the system (9) s stble heorem : Cosder mult-get system wth fxed topology tht s oeted he mult-get system (4) wth protool (9) reh omposte- rottg osesus Proof : By Lemm, we express L to < λ λ L λ he, there exst orthogol mtrx W R suh tht W LW = dg{ λ, λ, K, λ } he, we hve for y I hus, the mult-get system (4) wth the protool (9) heve omposte rottg osesus V SIMULAION I ths seto, we study the omposte-rottg osesus usg protool (9) Suppose tht the ommuto topology of udreted etwor wth four gets s gve by Fg he problem of omposterottg osesus s exteded to -D spe by trodued rottg mtrx he tl odto of system (4) s gve s [ r, r, r, υ, υ, υ, r, r, r, υ, υ, υ, r, r, r, υ, υ, υ, r4, r4, r4, υ4, υ4, υ4] = [ ] d ρ s te s [/,/,/ ] Let ω = d ω =, the we obt tht, 6 5 C = , D = E 5 = F = - 6 6,, Clerly, t hs bee show Fg tht the movg eters of the four gets reh rottg osesus rulr orbt Besdes, the mult-get system flly heve omposte- rottg osesus rulr orbt roud the movg eters d ll the gets move the outerlowse dreto o the ple perpedulr to the vetor ρ herefore, smulto results hve show the effetveess of theoretl results ( W I )( I C L D)( W I ) = dg{ C λd, C λ D, L, C λ D} he hrterst polyoml of eh C λ D s (5) 4 det( si C+ λd) = s + λs + [ λ + ( ωω + λ) + ( ω ω λ )] s + λ( ωω + λ) s+ ( ωω + λ) =, =,, K, ω he, ordg to Lemm 6, the system (5) s Hurwtz stble Besdes, by smple lultos, we hve lm[ r& () t + ω R R R & υ () t ω R R R ( r() t + ω R R R t o o o o o o υ ())] t = Fgure he ommuto topology of udreted etwor VI CONCLUSIONS he m otrbuto of ths pper s to provde dstrbuted otrol poly tht llows the gets to heve omposte-rottg osesus rulr orbt roud the movg eters whose trjetores lso form ommo rle he otrol poly s dvded to two prts: oe prt s feedb from tself so tht the trjetores of ll gets 4

5 flly form rle d the other s dstrbuted feedb to llevte dvergee mog dfferet gets It s studed -D spe by trodued rottg mtrx d ll the gets move o ple perpedulr to the vetor ρ We hve proved the effetveess of the poly proposed ths pper by Hurwtz theory REFERENCES [] M Pvoe d E Frzzol, Deetrlzed poles for geometr ptter formto d pth overge[j], ASME Jourl of Dym Systems, Mesuremet, d Cotrol, 7,9(5): 6-64 [] Sepulhre R, Pley D d Leord N, Stblzto of Plr Colletve Moto Wth Lmted Commuto[J], IEEE RANSACIONS ON AUOMAIC CONROL, 8,5(): [] We Re, Colletve Moto From Cosesus Wth Crtes Coordte Couplg[J], IEEE RANSACIONS ON AUOMAIC CONROL,9,54(6):-5 [4] So Herdez d Dere A Pley, hree-dmesol Moto Coordto me-ivrt Flowfeld[C],Jot 48th IEEE Coferee o Deso d Cotrol d 8th Chese Cotrol Coferee,9: [5] YANG, WANG Weyog, HUANG Lshe, Mult-get Rottg Cosesus Usg Oly Lol Posto Iformto hree-dmesol Spe[C] Proeedgs of the th Chese Cotrol Coferee,: [6] P L, Y J, Dstrbuted rottg formto otrol of mult-get systems, Systems d Cotrol Letters[J], Systems & Cotrol Letters,,59(): [7] Peg L,Kyu Q, Zhogu L,We Re, Colletve rottg motos of seod-order mult-get systems three-dmesol spe[j], Systems & Cotrol Letters,,6:65-7 he trjetores of the postos of the four gets he trjetores of the movg eters Fgure he posto trjetores of the mult-gets system 44