Consensus Tracking of Multi-Agent Systems with Constrained Time-delay by Iterative Learning Control

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1 Coeu Trackg of Mult-Aget Sytem wth Cotrae Tme-elay by Iteratve Learg Cotrol Yogl R Yog Fag, Hogwe Yu School of Commucato a Iformato Egeerg, Shagha Uverty, Shagha 00444, Cha E-mal: ryogl@163.com yfag@taff.hu.eu.c hogweyu@hu.eu.c Abtract: Th paper vetgate the coeu trackg problem for the crete mult-aget ytem (DMAS) wth tme-elay. By ug teratve learg cotrol (ILC) metho a covergece crete ytem, the covergece properte for total ytem ca be aalyze bae o the tme-oma aaly, frequecy-oma aaly a matrx theory. The reult how that the propoe metho epe o each aget elf-elay tme, the weght of the ege to each aget eghbor, a the tercoecto topology of the etwork. The teratve umber of learg cotrol rele o the amplg pero a elf-elay tme for the crete mult-aget ytem. Fally, umercal example are preete to emotrate the theoretcal reult. Key Wor: Coeu, Covergece, Mult-aget Sytem, Self- Delay, Dcrete Sytem 1 INTRODUCTION Mult-aget coeu ha gae great atteto recet year, a t ha a qute we applcato, uch a commucato etwork, eor etwork, mult-robot ytem a o o [1-16]. Reearcher have oe a lot of work o mult-aget coeu. I [3], a mple moel for phae trato of a group of elf-rve partcle wa propoe a complex yamc of the moe were umercally emotrate. Moreover, a theoretcal explaato for the coeu behavor of the Vcek moel ug graph theory wa prove [4]. Some reult have bee got recet paper about mult-aget wth fferet moto moel a commucato ytem [1, 6]. Re et al tue the coeu problem of collecto moto, a erve ome coto about pag tree to guaratee aymptotcal coeu of the urecte tegrate ytem [7 9]. Some other reearche about coeu o robute, olearty, a averagg equlbrum mult-aget ytem ca be fou [3]. A mlar moel wa cue curret paper wa ue [3, 4]. The author tue ymmetrcal, fxe a urecte etwork topology wth tme-elay, but the o-ymmetrcal recte etwork wa le coere. I [9, 10], a o-mooth Lyapuov approach wa propoe for the cae of tme-varyg graph wth tme-elay. Sometme-elaye coeu reearch etal were alo preete [11 16]. However, the tme-elay the above-metoe paper were tme-varat, a the tme-varyg elay are ot take to coerato. A kow to all, there are alway ue tuato t exteral evromet for mult-aget, uch a the topology lk falure may occur by etructve attack. Therefore, t worthy to tuy the mult-aget coeu wth wtchg topologe. Some reearch work o th problem ha bee regare recet year. I [3], the average-coeu problem of mult-aget vetgate wth fxe topologe a tme-varyg topologe. Referece [1, 15] tue the coeu problem wth a tme-varyg referece tate where the formato tate of all vehcle approach a tme-varyg referece tate uer the gve coto. However, the cae of tme-varyg (wtchg) topologe wth tme-elay rarely coere. Recetly, may reearcher have vetgate the ILC problem MAS, but the etermg umber of repetto of the terato (put elay) wa ot metoe. I partcular, put elay oe of the mot mportat compoet of the total tme-elay, o we focu o the elf-elay (put elay) of aget, ILC rule a covergece coto by teratve learg cotrol (ILC) the MAS. I th paper, we coer the covergece crtera crete ytem, aalyze t ample pero a the tme-elay. From the mulato reult, t ca be ee that the propoe metho meet the requremet of the theoretcal reult. Th paper orgaze a follow: Secto a Secto 3 trouce the kowlege of algebra graph theory a ILC coeu rule. Covergece aaly of the crete ytem wth tme-elay gve Secto 4. Smulato a cocluo are lte Secto 5 a Secto 6. Th work upporte by Natoal Nature Scece Fouato uer Grat (617113)a the Ph.D. Program Fouato of Mtry of Eucato of Cha( ) /16/$31.00 c 016 IEEE 6816

2 PRELIMINARIES ON GRAPH THEORY Graph theory ha bee aopte mult-aget coorato problem for may ecae, a t wll be ue th work to ecrbe the commucato amog aget. Therefore, the bac termologe graph theory are brefly revewe below. Let G = ( V, E) be a weghte urecte graph wth the et of vertce V = {1,,, N} a the et of ege E V V. Let V alo be the ex et repreetg the aget the etworke ytem. A urecte ege from k to ca be eote by a orere par ( k, ) E, whch mea that aget ca receve formato from aget k. I th cae, the k calle the paret of. The et of eghbor of the kthaget eote by N = { V (, k) E}. N N A= ( ak, ) R the weghe aacecy matrx of G wth oegatve etre. I partcular, a = 0, a = 1 f ( k, ) E, kk, k, otherwe a = 0. The -egree of vertex k efe a, k, k = N = a k, k a the Laplaca of G efe a L = D A, where D = ag(,, ) 1 N. A pag tree a graph, a each vertex ha exactly oe paret except oe vertex calle the root whch ha o paret. A graph a to have or cota a pag tree f the vertce et V a a ubet of the ege et E ca form a pag tree. The, we trouce the λ -orm, whch eetally a expoetally tme weghte orm. N Defto 1: Gve a vector fucto f : [0, T] R, - t λ -orm efe by f max e λt = f ( t ), where λ t [0, T] a potve cotat, a ay geerc vector orm. I the ILC covergece aaly, λ -orm ca be ue to uppre the effect of ytem yamc a reveal the put/output relato rectly, whch wll make the covergece mpler to prove. 3 ITERATIVE LEARNING CONTROL Coer a group of N homogeeou yamc aget, a the th aget govere by the followg lear tme-varat moel, x t u t V () = (),, (1) where eote the terato umber, x R the m tate vector, u R the cotrol put. For mplcty, the tme argumet t roppe whe o cofuo are. The leaer traectory, or the ere coeu traectory x () t efe by a fte-tme terval[0, T ], a geerate by the followg yamc, x = u () where u the cotuou a uque ere cotrol put. Due to commucato a eor lmtato, the leaer traectory oly acceble to a mall porto of the follower. Let the commucato amog follower be ecrbe by the graph G. If the leaer labele by vertex 0, the the complete formato flow amog all the aget ca be characterze by a ew graph G = {0 V, E}, where E the ew ege et. The maor tak to eg a et of trbute ILC rule uch that each vual aget the etwork whch able to track the leaer traectory uer the pare commucato graph G. To mplfy the cotroller eg a covergece aaly, followg aumpto are mpoe. Aumpto 1: The tal tate of all aget are reet to ere tal tate at every terato,.e., x (0) = x (0) for all 1. Aumpto 1 referre a the etcal talzato coto (..c.), whch wely ue the ILC lterature [3]. May effort have bee ecate to the relaxato of th aumpto, but trae-off alway ext. For example, f the tal tate mapulatable a ome of ytem parameter are kow, the tal tate learg rule [15] ca be apple to remove the..c. aumpto. The tal tate learg rule aopte [9] for mult-aget ytem coeu trackg. Whe the tal tate caot be mapulate, whle a fxe poto houl be reet after each terato, rectfyg acto [6] ca be apple to eure atfactory performace. However, the ere traectory lghtly mofe. Let ξ be the trbute meauremet by aget at the th terato over the graph G, a t efe a ξ = ( ) ( ) a x x + x x (3), k k k N where = 1 f (0, ) E, otherwe = 0. Note that the actual trackg error e = x x caot be utlze the cotroller eg whe oly a mall umber of follower have acce to the leaer. Therefore, e ot avalable ome of the follower. It atural to corporate the trbute meauremet ξ the ILC eg. Hece, the followg ILC rule aopte th work, ξ 1, 0, u = u + Q, u = 0, v, + (4) 016 8th Chee Cotrol a Deco Coferece (CCDC) 6817

3 where Q the learg ga. For mplcty, the tal cotrol put u 0, et to zero. However, practcal mplemetato, the tal cotrol put ca be geerate by certa feeback mecham o that the ytem table. Th may mprove the traet performace of the learg cotroller. Note that ξ alreay avalable at the + 1th terato. Therefore, the ervatve of ξ ca be obtae by ay ophtcate umercal fferetato that oe ot geerate a large amout of oe. The tal tate learg rule ca be expree a x (0) x (0) ξ (0). + 1, = + (5) Referece [15] come to a cocluo that mult-aget ytem (1), uer the commucato graph G, a ILC rule (4), f the learg ga Q choe uch that I H Q μ < 1 (6) mn where I () the etty matrx wth the ubcrpt eotg t meo, μ a cotat, H = L+ D, L the Laplaca matrx of G, D = ag(,,, ), a repreet the 1 N Kroecker prouct, whch make the covergece proof mpler. It more coveece whe we cretze the plat by ug zero-orer hol crcut a mpule ampler, we ca obta the correpog crete-tme formula a δ x [ k] = x [ k] + u [ k], x [0] = x (7) 0 where T the amplg pero,t R a T > 0. The ex k Z eote cretze tme,.e., t = kt, a δ eote the elta operator [7] 1 δ x [ k] = ( x [ k+ 1] x [ k]) (8). T Furthermore, x[ k] a uk [ ] are the repectve tate whch put at the cretze tme k.hece, the followg ILC rule ca be expree crete-tme ytem, u+ 1, [ k] = u [ k] + Q( ξ [ k] ξ [ k 1])/ T (9) 4 CONVERGENCE CONDITION IN DISCRETE SYSTEM I practce, we uually coer the aalog gal to gtal gal tramo, a tme-elay ca be coere eparately crete ytem. zero-orer mult-aget ytem crete amplg cotrol moel a follow x ( kt + T ) = x ( kt ) + Tu ( kt ) (10) ( k = 0,1, ; I) where amplg pero T > 0 ; k crete ex. The coeu cotrol protocol wthout tme elay [16] ca be wrtte [ ] [ ] [ ] [ ] u ( kt) = b a ( x kt x kt ) = 1 ba ( x kt x kt ) 0 (11) where cotrol ga b > 0, If leaer the vual aget eghbor, the a 0 > 0, otherwe a = 0. 0 Whle coerg the effect of tme elay τ,( τ (0, T )), th paper exte the coeu trackg protocol (11) to the followg coeu trackg protocol: b a ( x[ kt T] x [ kt T]) ba 0 = 1 ( x[ kt T] x [ kt T]), t [ kt, kt + τ ) (1) u ( kt) = b a ( x[ kt] x [ kt]) ba 0 = 1 ( x [ kt ] x [ kt ]), t [ kt + τ, kt + T ) Defto 1: Protocol (1) calle a boue coeu trackg protocol, f the ytem (10) ug the protocol (1) ha the property: lm x [ kt ] x [ kt ] C <, I, (13) k where C a boue potve cotat epeet o k. Lemma 1: [15] take a fxe, urecte, coecte etwork topologcal graph G whch combe to mult-aget, at leat ext oe path betwee the aget a the vrtual leaer. Whe the mult-aget ytem (10) ue coeu trackg protocol (1), f a oly f the followg two coto are atfe, the ytem ca acheve coeu boue trackg: τ < 1/ λmax ( H ) (14) τ < T < τ + / λ ( H) (15) The followg lemma play a mportat role o the covergece aaly of boue a coeu trackg. Lemma : If a oly f all feature of formula root the left half ope plae, the root of all feature are locate wth the ut crcle the equato, amog them ab, R. max Lemma 3 : The equato + a+ b = 0 ha all root wth the ut crcle, f a oly f all root of (1 + a + bt ) + (1 bt ) + 1 a+ b= 0are ope left half plae[11], where ab, R. Lemma 4: Th paper take a fxe, urecte, coecte etwork topologcal graph G whch combe wth mult-aget compoto to coerato, there at th Chee Cotrol a Deco Coferece (CCDC)

4 leat ext oe path betwee the aget a the vrtual leaer, a the matrx H = D+ L wll be efte, that the value of the mallet feature matrx λ m ( H ) > 0. Lemma 5: [16]aumg that aget of the commucato etwork topology G compoe fxe, urecte a coecte, at leat oe aget ha the acce to the vrtual leaer, a the tme-varyg referece tate of the vrtual leaer atfe 0 x ( kt + T) x ( kt)/ T ξ < T > 0 (16) where k = 0,1,, N. The the mult-aget ytem (10), applyg the coeu trackg protocol (1), achevg the boue coeu trackg f a oly f all the root of the equato m+ m+ 1 ( T ελ ) ( H) ελ( H) 0, I + + = (17) are wth the ut crcle. Obvouly, Lemma 1 ca be regare a a pecal cae of Lemma 5. Whe m = 0, the formula (17) reuce to T τ λ H τ λ H I + [( ) ( ) 1] + ( ) = 0 ( ) (18) Lemma 3 how that: f a oly f all root of the equato (19) are ope the left flat a all root of the equato (18) are locate e the ut crcle. The equato (19) ca be expree a: Tλ ( H) t + [1 τ λ ( H)] t+ [ + ( τ T) λ ( H)] = 0( I) (19) Becaue the ytem etwork topology G apparetly atfe the coto of Lemma 4, a λ m ( H ) > 0. Therefore, f a oly f the followg two equalty, all root of the equato (18) are ope the left half plae, amely: 1 τλ ( H ) > 0 (0) + ( τ T) λ ( H) > 0 (1) S Smplfy equalte (0) (1), a t ca be obtae coto (14) (15). 5 ILLUSTRATIVE EXAMPLE I th ecto we prove the umercal mulato to llutrate the effectvee of the above reult. It eay to verfy that the complete formato flow graph G cota a pag tree wth the leaer beg t root. The commucato amog follow urecte, whch mea the formato flow brectoal. We aopt 0-1 weghtg, a the Laplaca for follow L = () a D = ag(1,1,1,1,1) whch repreet the formato flow from leaer to the follower. The leaer tate choe a x ( t) = (0.1 t), t [0,100] (3) The tal coto for follower are x (0) = 3.5, x (0) = 3, x (0) = 1.5, 1 3 x (0) = 1.1, x (0) = Obvouly, tal tate error are ozero. We apply the ILC rule (4) wth learg ga 0.3 Q =. The egevalue of H are: λ = , λ = , λ = 1.000, 1 3 λ = 1.003, λ = (4) (5) By calculatg formula (14), the tme-elay (0,0.5). So choogτ = 0.4, by (15)we ca get the amplg pero T the rage of (0.71,1.8). Choog τ = 0.4, T = 1.5, the tme-elay τ a amplg pero T atfy the Lemma 1 coto, each mult-aget tate how Fg 1. The tate of each aget graually le wth the leaer traectory. A how Fg.1, uerlyg atfy coto of the Lemma 1, the covergece get fater whe the tme-elay a amplg pero get maller. Fg 1: τ = 0.4, T = 1.5 tate of leaer a each aget I Fg a Fg 3, we ca f that whe τ = 0.4, T = 1.8 a τ = 0.4, T = 1.9, whch tme-elay τ atfe the coto of Lemma 1, whle ample pero 016 8th Chee Cotrol a Deco Coferece (CCDC) 6819

5 T oe ot atfy the coto of the Lemma 1. Therefore, the tate of each aget graually verge a uable to trace the leaer traectory. Fg : τ = 0.4, T = 1.8 tate of leaer a each aget Fg 3: τ = 0.4, T = 1.9 tate of leaer a each aget Alo, whe τ = 0.5, T = 1.5 a τ = 0.6, T = 1.5, ample pero atfy the coto of Lemma 1, but the tme-elay oe ot atfy the coto of the Lemma 1. A how Fg 4 a 5, the tate of each aget graually verge, caot track the leaer traectory. Fg 5: τ = 0.6, T = 1. 5 tate of leaer a each aget Alo, orer to coer tme-elay by ILC rule (4), that coerg the calculato tme bae o the ILC rule (4), covergece tme a calculato tme are how below table. Calculato tme the learg tme bae o the ILC rule (4) here. Table 1: Covergece a calculato tme () terato covergece tme() calculato tme () 10th th th th th th th A how Table 1, a greater the umber of terato, the covergece tme horter a the calculato tme loger. I other wor, the calculato tme a tme-elay the crete ytem. Coerg commucato elay, the total tme-elay loger. We chooe the umber of terato that atfe the calculato tme, whch et to 5th. Fg 4: τ = 0.5, T = 1. 5 tate of leaer a each aget Fg 6: State at the 5th terato A how Fg 6, covergece tme wa gfcatly reuce. Thu, whe ILC rule apple to crete ytem, be ure to aalyze the covergece of the th Chee Cotrol a Deco Coferece (CCDC)

6 ytem that eterme the ample pero a umber of terato. Therefore, practcal applcato houl aalyze the covergece properte of the ytem, t houl eterme the umber of terato. 6 CONCLUSION I th paper, we aalyze the covergece of mult-aget ytem by teratve learg cotrol wth cotrae tme-elay. Alo, th paper aalyze the covergece properte of each aget the crete ytem a eterme the umber of reaoable terato bae o ILC rule. REFERENCES [1] J.-X. Xu a Y. Ta, Lear a Nolear Iteratve Learg Cotrol. Germay: Sprger-Verlag, 003. I ere of Lecture Note Cotrol a Iformato Scece. [] D.A. Brtow, M. Tharayl, a A. G. Alleye, A urvey of teratve learg cotrol a learg-bae metho for hgh-performace trackg cotrol, IEEE Cotrol Sytem Magaze, Vol. 6, pp , 006. [3] H.-S. Ah, Y. Che, a K. L. Moore, Iteratve learg cotrol: Bref urvey a categorzato, IEEE Traacto o Sytem, Ma, a Cyberetc - Part C: Applcato a Revew, Vol. 37, No. 6,pp , 007. [4] H.-S. Ah a Y. Che, Iteratve learg cotrol for mult-aget formato, ICROS-SICE Iteratoal Jot Coferece, (Fukuoka Iteratoal Cogre Ceter, Japa), pp , 18-1 Augut 009. [5] Y. Lu a Y. Ja, A teratve learg approach to formato cotrol of mult-aget ytem, Sytem & Cotrol Letter, Vol. 61, No. 1, pp , 01. [6] D. Meg a Y. Ja, Formato cotrol for mult-aget ytem through a teratve learg eg approach, Iteratoal Joural of Robut a Nolear Cotrol, 01. [7] J.-X. Xu, S. Zhag, a S. Yag, A hom-bae teratve learg cotrol cheme for mult-aget formato, 011 IEEE Iteratoal Sympoum o Itellget Cotrol, (Dever, CO, USA), pp ,8-30 September 011. [8] S. Yag, J.-X. Xu, a D. Huag, Iteratve learg cotrol for mult-aget ytem coeu trackg, The 51t IEEE Coferece o Deco a Cotrol, (Mau Hawa USA), pp , December 01. [9] S. Yag a J.-X. Xu, Aaptve teratve learg cotrol for mult-aget ytem coeu trackg, IEEE Iteratoal Coferece o Sytem, Ma, a Cyberetc, (COEX, Seoul, Korea), pp , October 01. [10] J. L a J. L Aaptve teratve learg cotrol for coorato of eco-orer mult-aget ytem, Iteratoal Joural of Robut a Nolear Cotrol, 013. [11] Y. Che, C. We, Z. Gog, a M. Su, A teratve learg cotroller wth tal tate learg, IEEE Traacto o Automatc Cotrol, Vol. 44, No., pp , [1] M. Su a D. Wag, Aaptve cooperatve trackg cotrol of hgher-orer olear ytem wth ukow yamc, Automatc, Vol. 48, No. 7, pp , 01. [13] Re W, Cao Y. Covergece of ample-ata coeu algorthm for ouble-tegrator yamc[c]//proceeg of the 47th IEEE Coferece o Deco a Cotrol. Mexco; IEEE, 008: [14] Xe U, Lu H, Wag L, et al. Coeu etworke mult-aget ytem va ample cotrol: fxe topology cae[c]// Amerca Cotrol Coferece. at Lou:Hyatt Regccy Rverfrot, 009: [15] Y. C. Cao, W. Re, a Y. L Dtrbute crete-tme coorate trackg wth a tme-varyg referece tate a lmte commucato, Automatca, 45(5): , 009. [16] R. Olfat-Saber, a R. M. Murray, Coeu problem etwork of aget wth wtchg topology a tme-elay, IEEE Tra. Automatc Cotrol, 49(9): , th Chee Cotrol a Deco Coferece (CCDC) 681