Group Consensus of Second-Order Multi-agent Networks with Multiple Time Delays

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1 Itratoal Cofrc o Appld Mathmatcs, Smulato ad Modllg (AMSM 6) Group Cossus of Scod-Ordr Mult-agt Ntworks wth Multpl Tm Dlays Laghao J* ad Xyu Zhao Chogqg Ky Laboratory of Computatoal Itllgc, Chogqg Uvrsty of Posts ad Tlcommucatos, Chogqg 465, Cha * Corrspodg Author cossus of frst-ordr tworks wth coctd bpartt topology wr dscussd, rspctvly. Ad th boud of tm dlay was obtad aalytcally. Mawhl, wghtd group cossus of agt wth tm dlays was vstgatd ad a uppr boud of maxmum tm dlay was obtad [6]. Cosdrg th coxstc of put ad commucato tm dlays, group cossus for mult-agt tworks wth udrctd ad coctd bpartt topology wr studd, rspctvly. Ad som suffct codtos wr proposd [7]. Abstract I ths papr, th problm of group cossus of scod-ordr mult-agt tworks wth multpl tm dlays s vstgatd. Basd o th thory of frqucy-doma, som algbrac crtra ar proposd aalytcally whch ca guarat th mult-agt tworks to achv group cossus. Rsults show that put tm dlays of th agts, couplg wghts ad couplg strgths btw th agts play a ky rol rachg group cossus. Mawhl, commucato tm dlays ca oly affct th covrgc rat of th systms. Fally, th valdty ad corrctss of our thortcal rsults ar vrfd by svral umrcal smulatd xampls. It s worth otg that thr ar two dlays xstd tworks, put ad commucato tm dlays. From th rlvat works mtod abov, t s ot dffcult to fd out thr ar som mor cosrvatv assumptos thm. Frst, som rlatd rsarch works oly cosdr commucato dlays, or oly aalyz th sam commucato ad put dlays, such [-3, 5-7]. As mtod abov, th two dlays btw agts should b dffrt ad coxstg. Scod, most of th works oly focus o th tworks wth symmtrcal topology, such as [8,,, 4-7]. Kywords-tm dlays; group cossus; mult-agt twork; complx tworks I. INTRODUCTION Th cossus problm ams to dsg cotrol stratgs ad protocols such that th whol twork ca achv stablty. Up to ow, t has attractd crasg attto of rsarchrs dffrt flds. So far, may rlatd rsarch works hav b succssvly rportd, whch ca b s [-6] ad th rfrcs thr. Motvatd by th rlatd work, w wll dscuss th group cossus of scod-ordr mult-agt tworks wth multpl tm dlays. Ad th topology of th systm s mor gral whch thr ds to b udrctd or strogly coctd dgraph. I coopratv cotrol, udr th fluc of vromts, stuatos, coopratv tasks or v tm, th cossus stats of th systm wll b lkly to chag. Ths phomo ca b dscrbd as group cossus of multagt tworks, whch mas that all agts wth th sam clustr ca rach a cosstt stat, whl dffrt clustrs achv dstct stats. Up to dat, lots of rsarch work about group cossus has b rportd. I [7], th group cossus problms of scod-ordr mult-agt systms wth tm dlays wr vstgatd. I[8], a ovl hybrd protocol was dsgd, ad th avrag group cossus problm wth udrctd topologs s studd. I [9], couplgroup cossus problms of scod-ordr tworks wth fxd ad stochastc swtchg topologs wr dscussd, ad th authors provdd th suffct ad cssary codto for ma-squar coupl-group cossus. I [, ], Yu studd group cossus of tworks wth udrctd graphs ad strogly coctd ad balacd topology. Furthrmor, by applyg doubl-tr-form trasformato, thy xtdd group cossus wth commucato dlays ad swtchg topologs[, 3]. I [4], group cossus of tworks wth commucato dlays was vstgatd ad th topology of th systm was strogly coctd ad balacd. I [5], wth ad wthout tm dlays, group 6. Th authors - Publshd by Atlats Prss Th rst of ths papr s orgazd as follows as I Scto, som prlmars ar brfly outld. Th ma rsults ar addrssd Scto 3. I Scto 4, som xampls ar llustratd to vrfy our thortcal rsults ad coclusos ar draw Scto 5. II. PRELIMINARIES I a mult-agt twork, w ca rprst ach agt ad th formato xchag amog thm as a od ad a dg of a wghtd drctd graph, rspctvly. For covc, lt G = (V, E, A) dots a wghtd drctd graph, whr V = {v, v,..., vn } s th od st, th st of { } ghbors of v s dscrbd by N = v V : ( v, v ) E, = {,,..., N } dots th od dx st, E V V s th dg st ad A = ( a ) N N s th wghtd adaccy matrx. If th dg E, a >, whch mas od v 437

2 v. Othrws, ca rcv formato from od a =. I ths papr, w assumd that a = for all. For scod-ordr mult-agt tworks, th dyamcs ar lstd as (), () = ( ) () = (), x& t v t v& t u t whr x (), t v (), t u () t dots th posto, vlocty stat, cotrol put of th agt. For covc, w assum =.W oly cosdr x (), t v (), t u () t. Th rsults w proposd ca b asly xpadd for x (), t u () t by usg th Krockr product. Nxt, w wll lst som rlatd dftos ad lmma. A. Dfto Cossus scod-ordr systm () s sad to b rachd asymptotcally f ad oly f () () =, v () t v () t lm x t x t t B. Dfto lm = t If thr xsts a path G from od v to od v, th v s sad to b rachabl from v. If a od s rachabl from vry othr od G, th t s tratd as a globally rachabl od. Assums that th twork cotas + m(, m> ) agts, ts topology G cossts of two sub-graphs, G = ( V, E, A ) ad G = ( V, E, A ), whr V = { v, v,... v }, V = { v, v,..., v }. Th st of ft dx m s L = { }, L = { m}.,,...,,,..., = { ( ) }, N { v V ( v, v ) E} : N v V : v, v E () = rprst th ghbor st of v th two sub-graphs, rspctvly. If th frst agts covrg to a cosstt stat, ad th othr m agts covrg to a dffrtly cosstt stat, th w ca say that th twork ralzs group cossus. C. Lmma [8] If th graph G xsts a globally rachabl od, ts Laplaca matrx wll obta a smpl gvalu. D. Lmma [9] For γ [,), wh ω, covx hull ( U{ ( ), }) dos ot cota (, ), γco E T whr E ( ) = ( π / T) ( / ), ad T dots th systm dlay. E. Lmma 3 [] Th st U G s cludd th covx hull ( U { ( ), }) for ω. γco E III. GROUP CONSENSUS OF MULTI-AGENT NETWORKS WITH MULTIPLE DELAYS I ths scto w wll dscuss group cossus of multagt tworks wth multpl tm dlays. I [-3], basd o th -dgr balac codtos(a) ad (A), avrag group cossus problms of mult-agt tworks wth udrctd, strogly coctd ad balacd topologs wr vstgatd, rspctvly. Th put protocol s as follows: u a ( x ( t) x ( t)) + a x ( t), L V N V N () t = V N V N a ( x ( t) x ( t)) + a x ( t), L I (), for, L, a ;, L, a ;, φ = {(, ) : L, L } U {(, ) : L, L } ad th -dgr balac assumptos ar + m (A): a =, L ;(A): a =, L. = + = (), a, Basd o (3), R t al.[8] aalyzd th group cossus of scod-ordr mult-agt systms wth tm dlays. ( ) = α ( ( ) ( )) V N + β a ( v ( t T ) v ( t T)) u t a x t T x t T V N. (3) whr T, T dot commucato dlays ad put dlays,, ad α >, β > ar th couplg strgths of systms. Motvatd by th rlatd work, w wll vstgat th group cossus problms of scod-ordr mult-agt tworks wth multpl dlays. Cosdr th followg cotrol protocol (4), α a ( x ( t T ) x ( t T )) + β a ( v ( t T ) v ( t T )) V N V N + α ax( t T) + β ( ), av t T L V N V N u () t = α a ( x ( t T ) x ( t T )) + β a ( v ( t T ) v ( t T )) V N V N + α ax( t T) + β av( ), t T L V N V N whr T, T dots th put dlay ad th commucato dlay of a od v, rspctvly. (4) 438

3 A. Thorm Suppos th mult-agt tworks () wth cotrol protocol (4) cota + m(, m > ) agts, ad ts topology cluds a globally rachabl od, f d % ( α cos ω T + βw s ω T) < ω holds, th th systm ca o achv group cossus asymptotcally. Whr d% = a, k ω s th trscto btw th Nyqust curv of ( ) G ad th gatv ral axs of th complx pla, T ( ω) = % ad ( ω T ) G d α + β β = ω. α ta o o B. Proof: Applyg th Laplac trasform to () wth (4), t s asy to gt ts charactrstc quato, whch s dt s I + αl( s) + βl( s) s =, whr I s dtty matrx ( ) k=, k k =, k st a, L( s) = ( l ( s) ) = st a, = k. (5) ( ) For covc, lt F ( s) = dt s I + αl( s) + βl( s) s. Basd o th thory of frqucy-doma, w wll prov that all zros of F ( s ) hav gatv ral parts or F() s has a smpl zro at s =. ) Lt s =, F ( ) = dt( α L). Basd o Lmma, w kow that F() s dd xsts a smpl zro at s =. ) Lt s ad P( s) = dt ( I + G( s) ), whr () = ( α ( ) + β ( ) )/. G s L s L s s s Basd o th gral Nyqust stablty crtro, f th track of th λ ( G( )) dos ot clos th pot (, ), all zros of P ( s ) hav gatv ral parts. Lt s =. Accordg to th Grschgor dsk crtro, th gvalu of G( ) satsfs λ( G( ) ) U G, whr N T T α + β α + β x d% > d% (7) By calculatos, w ca obta th qualty asly, % ω ( α ω βω ω ) (8) x ( d / ) cos T + s T > Wh x, th followg qualty holds ( d% / ω ) ( α cosωt + βωsωt) <. At th momt, w obta that th track of λ ( G( )) dos ot clos th pot (, ), so all zros of P ( s ) hav gatv ral parts. From (6), w ca gt th ctr of th dsk s T α + β ( ω) = %.Suppos ω s th trscto G d pot btw th Nyqust curv of G ( ) ad th gatv ral axs of th complx pla, thus w gt ( ω T ) Thorm s provd. β = ω. α ta o o C. Rmark From Thorm, w ca kow that group cossus of scod-ordr systms s rlatd to put dlays ad couplg strgths of agts or systms, ad s dpdt of commucato dlays btw th agts. IV. SIMULATION EXAMPLES Accordg to Thorm, som smulato xampls ar gv to vrfy th ffctvss ad th corrctss of th crtra stablshd scto 3. A. Exprmt Cosdr a dyamcal twork () wth 5 ods, th topology ad cocto wghts btw ods llustratd Fg.. St agts,,3 a group ad agts 4,5 aothr group. Mawhl, th topology ows a globally rachabl od ad (A) ad (A) ar also satsfd. T α + β G = : ζ ζ C ζ ak k=, k, (6) T α + β ak k=, k ad C dots complx umbr st, d% = a. k k=, k From (6), by Lmma ad 3, t s clar that f ( x, ) wth x s ot G, th followg qualty ca hold FIGURE I. INTERCONNECTION GRAPH OF MULTI-AGENT SYSTEM () St α =.6, β =.3, rspctvly ad radomly grat th tal stats of th agts. Mawhl, th put dlay ar.4s,.s,.s,.s,.s, rspctvly. It s ot 439

4 dffcult to vrfy that th codto of group cossus Thorm ca b satsfd. For smplcty, assum all th commucato dlays btw th ods ar qual to.s. Th tractors of th postos ad th vlocts systm () ar show Fgur. From th rsults, w ca asly fd that th group cossus of th systm s achvd. Postos of agts Vlocts of agts Agt Agt Agt Agt4 Agt (a) Posto stat Agt Agt Agt Agt4 Agt (b) Vlocty stat FIGURE II. TRAJECTORIES OF ALL NODES IN THE SYSTEM (), WHERE T =.4S Nxt, rst T =.s ad kp all th othr paramtrs ot chagd, t s asy to foud that th codto caot b satsfd Thorm. I ths cas, th tractors of th agts systm () ar llustratd Fgur 3. Postos of agts Agt Agt Agt Agt4 Agt (a) Posto stat Vlocts of agts Agt Agt Agt Agt4 Agt (b) Vlocty stat FIGURE III. TRAJECTORIES OF ALL NODES IN THE SYSTEM (), WHERE T =.S V. CONCLUSION For scod-ordr systms, ths tchcal ot s amd at xplorg th ssu of group cossus of mult-agt tworks wth dvrs commucato dlays ad put dlays. By applyg th thory of frqucy-doma, som algbrac crtra of th group cossus ar drvd. It ca b show that th group cossus of systms s dtrmd by put dlays, couplg strgths ad cocto strgths btw agts, dpdt of commucato dlays. ACKNOWLEDGMENT Ths work was supportd by th Natoal Natural Scc Foudato of Cha (Grat Nos ), th Natural Scc Foudato Proct of Chogqg Scc ad Tchology Commsso (Grat Nos. cstc4cya447) ad th Sctfc ad Tchologcal Rsarch Program of Chogqg Mucpal Educato Commsso (Grat No. KJ443), part awardd by Stat Scholarshp Fud of Cha Scholarshp Coucl. Th authors also would lk to thak aoymous rvwrs who hlpd us gvg commts to ths paprad. REFERENCES [] Yu W.W., G.R. Ch, Z.D. Wag ad W. Yag, "Dstrbutd cossus fltrs ssor tworks", IEEE Trasacto o Systm, Ma, ad Cybrtcs, Vol. 39, No. 6, pp ,9. [] R. Olfat-Sabr ad R. M. Murray, Cossus problms tworks of agts wth swtchg topology ad tm-dlays, IEEE Tras. Automat. Cotr., vol. 49, o. 9, pp , 4. [3] W. Yu, G. Ch ad M. Cao, Som cssary ad suffct codtos for scod-ordr cossus mult-agt dyamcal systms, Automatca, vol. 46, o. 6, pp ,. [4] W. Yu, W. Zhg, G. Ch, W. R ad J. Cao, Scod-ordr cossus mult-agt dyamcal systms wth sampld posto data, Automatca, vol. 47, o. 7, pp ,. [5] Lu B. ad X.L. Wag, "Adaptv scod-ordr cossus of multagt systms wth htrogous olar dyamcs ad tmvaryg dlays", Nurocomputg, Vol. 8, pp. 89-3(3). [6] Ya H.C., Y.C. Sh, ad H. Zhag, "Dctralzd vt-trggrd cossus cotrol for scod-ordr mult-agt systms", Nurocomputg, Vol. 33, pp. 8-4(4). 44

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