Distributed event-triggered control for multi-agent formation stabilization and tracking

Transcription

1 strbuted event-trggered control for mult-agent formaton stablzaton and trackng Chrstophe Vel, Sylvan Bertrand, Mchel Keffer, Hélène Pet-Lahaner o cte ths verson: Chrstophe Vel, Sylvan Bertrand, Mchel Keffer, Hélène Pet-Lahaner. strbuted event-trggered control for mult-agent formaton stablzaton and trackng. 17. <hal > HAL Id: hal Submtted on Sep 17 HAL s a mult-dscplnary open access archve for the depost and dssemnaton of scentfc research documents, whether they are publshed or not. he documents may come from teachng and research nsttutons n France or abroad, or from publc or prvate research centers. L archve ouverte plurdscplnare HAL, est destnée au dépôt et à la dffuson de documents scentfques de nveau recherche, publés ou non, émanant des établssements d ensegnement et de recherche franças ou étrangers, des laboratores publcs ou prvés.

2 strbuted event-trggered control for mult-agent formaton stablzaton and trackng Chrstophe Vel a Sylvan Bertrand a Mchel Keffer b Hélène Pet-Lahaner a a OERA - he French Aerospace Lab, F-911 Palaseau e-mal: frstname.lastname@onera.fr. b LS, Unv Pars-Sud, CRS, CentraleSupelec, F-9119 Gf-sur-Yvette e-mal: frstname.lastname@lss.supelec.fr Abstract hs paper addresses the problem of formaton control and trackng a of desred trajectory by an Euler-Lagrange mult-agent systems. It s nspred by recent results by Qngka et al. and adopts an event-trggered control strategy to reduce the number of communcatons between agents. For that purpose, to evaluate ts control nput, each agent mantans estmators of the states of the other agents. Communcaton s trggered when the dscrepancy between the actual state of an agent and the correspondng estmate reaches some threshold. he mpact of addtve state perturbatons on the formaton control s studed. A condton for the convergence of the mult-agent system to a stable formaton s studed. Smulatons show the effectveness of the proposed approach. Key words: Communcaton constrants, event-trggered control, formaton stablzaton, mult-agent system MAS. 1 Introducton strbuted cooperatve control of a mult-agent system MAS usually requres sgnfcant exchange of nformaton between agents. In early contrbutons, see, e.g., 6, 4, communcaton s consdered permanent. Recently, more practcal approaches have been proposed. For example, n 4, 44, 45, communcaton s ntermttent, alternatng phases of permanent communcaton and of absence of communcaton. Alternatvely, communcaton may only occur at dscrete tme nstants, ether perodcally as n 1, or trggered by some event, as n 9, 11, 4, 48. hs paper proposes a strategy to reduce the number of communcatons for dsplacement-based formaton control whle followng a desred reference trajectory. Agent dynamcs are descrbed by Euler-Lagrange models and nclude perturbatons. hs work extends results presented n 7 by ntroducng an event-trggered strategy, and results of, 6, 7 by addressng systems wth more complex dynamcs than a smple ntegrator. o obtan effcent dstrbuted control laws, each agent uses an estmator of the state of the other agents. he proposed dstrbuted communcaton trggerng condton CC nvolves the nter-agent dsplacements and the relatve dscrepancy between actual and estmated agent states. A sngle a pror trajectory has to be evaluated to follow the desred path. Effects of state perturbatons on the formaton and on the communcatons are analyzed. Condtons for the Lyapunov stablty of the MAS have been ntroduced. he absence of Zeno behavor s proved. hs paper s organzed as follows. Related work s detaled n Secton. Some assumptons are ntroduced n Secton and the formaton parametrzaton s descrbed n Secton 4. As the problem consdered here s to drve a formaton of agents along a desred reference trajectory, the desgned dstrbuted control law conssts of two parts. he frst part see Secton 4 drves the agents to some target formaton and mantans the formaton, despte the presence of perturbatons. It s based on estmates of the states of the agents descrbed n Secton 4.. he second part see Secton 5 s dedcated to the trackng of the desred trajectory. Communcaton nstants are chosen locally by each agent usng an event-trggered approach ntroduced n Secton 6. A smulaton example s consdered n Secton 7 to llustrate the reducton of the communcatons obtaned by the proposed approach. Fnally, conclusons are drawn n Secton 8. Preprnt submtted to Automatca 19 September 17

3 Related work Event-trggered communcaton s a promsng approach to save energy. It s well-suted to applcatons where communcatons should be mnmzed, e.g., to mprove furtvty, reduce energy consumpton, or lmt collsons between transmtted data packets. Applcaton examples wth such constrants are exposed, e.g., n 18, 19 for the case of a fleet of vehcles, or n 4 where agents am at mergng local feature-based maps. he man dffculty conssts n determnng the CC that wll ensure the completon of the task assgned to the MAS, e.g., reachng some consensus, mantanng a formaton, etc. In a dstrbuted strategy, the states of the other agents are not permanently avalable, thus each agent usually mantans estmators of the state of ts neghbours to evaluate ther control laws. evertheless, wthout permanent communcaton, the qualty of the state estmates s dffcult to evaluate. o address ths ssue, each agent mantans an estmate of ts own state usng only the nformaton t has shared wth ts neghbours. When the dscrepancy between ths own state estmate and ts actual state reaches some threshold, the agent trggers a communcaton. hs s the approach consdered, e.g., n 9, 1, 14, 4, 9, 4, 49. hese works dffer by the complexty of the agents dynamcs 1, 4, 49, the structure of the state estmator 9, 14, 9, 4, and the determnaton of the threshold for the CC 4, 9. Most of the event-trggered approaches have been appled n the context of consensus n MAS 9, 14, 4. hs paper focuses on dstrbuted formaton control, whch has been consdered n, 6, 7. Formaton control conssts n drvng and mantanng all agents of a MAS to some reference, possbly tme-varyng confguraton, defnng, e.g., ther relatve postons, orentatons, and speeds. Varous approaches have been consdered, such as behavor-based flockng 6, 5, 1,, 8, or formaton trackng 5, 8, 1,,. Behavor-based flockng 6, 5, 1,, 8 mposes several behavor rules attracton, repulson, mtaton to each agent. her combnaton leads the MAS to follow some desred behavor. Such approach requres the avalablty to each agent of observatons of the state of ts neghbours. hese observatons may be deduced from measurements provded by sensors embedded n each agent or from nformaton communcated by ts neghbours. In all cases, these observatons are assumed permanently avalable. In addton, f a satsfyng global behavor may be obtaned by the MAS, behavor-based flockng cannot mpose a precse confguraton between agents. fferent formaton-trackng methods have been consdered. In leader-follower technques 5, 8, 1,, based on msson goals, a trajectory s desgned only for some leader agent. he other follower agents, am at trackng the leader as well as mantanng some target formaton defned wth respect to the leader. A vrtual leader has been consdered n 7, 8, to gan robustness to leader falure. hs requres a good synchronzaton among agents of the state of the vrtual leader. Vrtual structures have been ntroduced n, 41, where the agent control s desgned to satsfy constrants between neghbours. Such approaches also address the problem of leader falure. In dstancebased control, the constrants are dstances between agents. In dsplacement-based control, relatve coordnate or speed vectors between agents are mposed. In tensegrty structures 4, 7 addtonal flexblty n the structure s consdered by consderng attracton and repulson terms between agents, as formalzed by. In addton to constrants on the structure of the MAS, 5 mposes some reference trajectory to each agent. In most of these works, permanent communcaton between agents s assumed. Some recent works combne event-trggered approaches wth dstance-based or dsplacement-based formaton control, 6, 7. In these works, the dynamcs of the agents are descrbed by a smple ntegrator, wth control nput consdered constant between two communcatons. he proposed CCs consder dfferent threshold formulatons and requre each agent to have access to the state of all other agents. A constant threshold s consdered n 6. A tme-varyng threshold s ntroduced n, 7. he CC depends then on the relatve postons between agents and the relatve dscrepancy between actual and estmated agent states. hese CCs reduce the number of trggered communcatons when the system converges to the desred formaton. A mnmal tme between two communcatons, named nter-event tme, s also defned. Fnally, n all these works, no perturbatons are consdered. Logc-based control LBC technques have been ntroduced n, 9, 46, 47 to reduce the number of communcatons n trajectory trackng problems. MAS wth decoupled nonlnear agent dynamcs are consdered n, 9. Agents have to follow parametrzed paths, desgned n a centralzed way. CCs ntroduced by LBC lead all agents to follow the paths n a synchronzed way to set up a desred formaton. Communcaton delays, as well as packet losses are consdered. evertheless, f nput-to-state stablty condtons are establshed, absence of Zeno behavor s not analyzed.

4 q q x vector of coordnates of Agent n some global fxed reference frame R vector q1 q... q R.n, confguraton of the MAS state vector q, q of Agent ˆq j estmate of q performed by Agent j. ˆq j estmate of q performed by Agent j. ˆx j estmate of x performed by Agent j. e j estmaton error between q and ˆq j. r j relatve coordnate vector r j = q q j between agents and j. r j q q ε desred value for r j. reference trajectory reference trajectory for Agent trajectory error for Agent, ε = q q t j,k tme at whch the k-th message s sent by Agent j. t j,k tme at whch the k-th message sent by Agent j s receved by Agent. able 1 Man notatons otatons and hypotheses able 1 summarzes the man notatons used n ths paper. Consder a MAS consstng of a network of agents whch topology s descrbed by an undrected graph G =, E. = {1,,..., } s the set of nodes and E the set of edges of the network. he set of neghbours of Agent s = {j, j E, j}. s the cardnal number of. For some vector x = x 1 x... x n R n, we defne x = x 1 x... x n where x s the absolute value of the -th component of x. Smlarly, the notaton x wll be used to ndcate that each component x of x s non negatve,.e., x {1... n}. A contnuous functon β r, s :, a,, s sad to belong to class KL f for each fxed s, the functon β., s s strctly ncreasng and β, s =, and for each fxed r, the functon β r,. s decreasng and lm s β r, s =. Let q R n be the vector of coordnates of Agent n some global fxed reference frame R and let q = q1 q... q R.n be the confguraton of the MAS. he dynamcs of each agent s descrbed by the Euler-Lagrange model M q q + C q, q q + G = τ + d, 1 where τ R n s some control nput descrbed n Secton 4., M q R n n s the nerta matrx of Agent, C q, q R n n s the matrx of the Corols and centrpetal term on Agent, G accounts for gravtatonal acceleraton supposed to be known and constant, and d s a tme-varyng state perturbaton satsfyng d t < max. he state vector of Agent s x = q, q. Assume that the dynamcs satsfy the followng assumptons: A1 M q s symmetrc postve and there exsts k M > satsfyng x, x M q x k M x x. A Ṁ q C q, q s skew symmetrc or negatve defnte and there exsts k C > satsfyng x, x C q, q x k C q x x. A here exsts q max R n + and q max R n + such that q q max and q q max. A4 he left-hand sde of 1 can be lnearly parametrzed as M q x 1 + C q, q x = Y q, q, x 1, x θ for all vectors x 1, x R n, where Y q, q, x 1, x s a regressor matrx wth known structure and θ s a vector of unknown but constant parameters assocated wth the -th agent. A5 For each = 1,...,, θ s such that θ mn, < θ < θ max,, wth known θ mn, and θ max,.

5 Assumptons A1, A, A and A4 have been prevously consdered, e.g., n 1,,. Moreover, one assumes that A6 each Agent s able to measure wthout error ts own state x, A7 there s no packet losses or communcaton delay between agents. In what follows, the notatons M and C are used to replace M q and C q, q. 4 Formaton control problem hs secton ams at desgnng a decentralzed control strategy to drve a MAS to a desred target formaton n some global reference frame R, whle reducng as much as possble the communcatons between agents. he target formaton s frst descrbed n Secton 4.1. he potental energy of a MAS wth respect to the target formaton s ntroduced to quantfy the dscrepancy between the target and current formatons. he proposed dstrbuted control, ntroduced n Secton 4., tres to mnmze the potental energy. o evaluate the control nput of each agent despte the communcatons at dscrete tme nstants only, estmators of the coordnate vectors of all agents are managed by each agent, as presented n Secton 4.. he presence of perturbatons ncreases the dscrepancy between the state vector and ther estmates. A CC s desgned to lmt ths dscrepancy by updatng the estmators as descrbed n Secton Formaton parametrzaton Consder the relatve coordnate vector r j = q q j between two agents and j and the target relatve coordnate vector r j for all, j. A target formaton s defned by the set { r j,, j }. he potental energy P q, t of the formaton represents the dsagreement between r j and r j P q, t = 1 k rj j rj where the k j = k j are some sprng coeffcents, whch can be postve or null, and where k =. P q, t has been ntroduced for tensegrety formatons n 4, 7. he mnmum number of non-zero coeffcents k j, j to properly defne a target formaton s 1. Indeed, for a gven r, all target relatve coordnate vectors rj between any par of agents and j can be expressed from components of r. evertheless, a number of non-zero k j larger than 1 ntroduces robustness n the formaton, n partcular wth respect to the loss of an agent. he values of the k j s that make a gven r an equlbrum formaton may be chosen usng the method developed n 7. efnton 1 he MAS asymptotcally converges to the target formaton wth a bounded error ff there exsts some ɛ 1 > such as lm P q, t ɛ 1. 4 t A control law desgned to reduce the potental energy P q, t allows a bounded convergence of the MAS. o descrbe the evoluton of P q, t, one ntroduces as n 7 g = ġ = P q, t q = s = q + k p g k j ṙj ṙj k j rj rj where g and ġ characterze the evoluton of the dscrepancy between the current and target formatons and k p s a postve scalar desgn parameter. 4

6 4. strbuted control he control law proposed n 7 s defned as τ = τ q, q, q and ams at reducng P q, t, thus makng the MAS converge to the target formaton n case of permanent communcaton. In ths approach, each agent evaluates ts control nput usng the state vectors of ts neghbours obtaned va permanent communcaton. Here, n a dstrbuted context wth lmted communcatons between agents, agents cannot have permanent access to q. hus, one ntroduces the estmate ˆq j of q j performed by Agent to replace the mssng nformaton n the control law. he MAS confguraton estmated by Agent s denoted as ˆq = ˆq 1... ˆq R.n. he way ˆq j s evaluated s descrbed n Secton 4.. In a dstrbuted context wth lmted communcatons, wth the help of ˆq, Agent s able to evaluate ḡ = s = q + k p ḡ k j rj rj 8 9 wth r j = q ˆq j and r j = q ˆq j. Usng ḡ and s, Agent s able to evaluate the followng adaptve dstrbuted control nput to be used n 1 τ q, q, ˆq, ˆq = k s s k g ḡ + G Y q, q, k p ḡ, k p ḡ θ, 1 θ = Γ Y q, q, k p ḡ, k p ḡ s 11 wth k g >, k s 1 + k p k M + 1 a desgn parameter and Γ an arbtrary symmetrc postve defnte matrx. Secton 4. ntroduces the estmator ˆq j of q j needed n Communcaton protocol and estmator dynamcs In what follows, the tme nstant at whch the k-th message s sent by Agent j s denoted t j,k. Let t j,k be the tme at whch the k-th message sent by Agent j s receved by Agent. In ths paper, we assume that there s no communcaton delay between agents. herefore, t j,k = t j,k for all j. When a communcaton s trggered at t,k for Agent, t broadcasts a message contanng t,k, q t,k, q t,k and ts estmated matrx θ t,k. Once a message s receved by neghbours of Agent, ts content s used to update ther estmate of the state of Agent as presented n the next secton. Agent 1 t 1, 1 1 t, 1 t 1, Agent t 1, 1 t, 1 t 1, Agent t 1, 1 t, 1 t 1, Fg. 1. Example of transmsson tmes t,k by Agent of k-th message and recepton tmes t j,k of k-th message by Agent j. 5

7 4..1 Estmator dynamcs Followng the dea of 9, 4, the estmate ˆq j of q j made by Agent s evaluated consderng ˆM j where ˆM j ˆq j ˆq j + Ĉ j ˆq j, ˆq j ˆq j + G = ˆτ j, t t j,k, t j,k+1 t j,k = qj t j,k ˆq j ˆq j t j,k = qj t j,k, 14 ˆq j and Ĉj ˆq j, ˆq j are estmates of M j and C j computed from Y j ˆq j, ˆq j, x, y and θ j t j,k usng ˆM j 1 1 ˆq j x + Ĉj ˆq j, ˆq j y = Y j ˆq j, ˆq j, x, y θj t j,k. 15 he estmator 1 managed by Agent requres an estmate ˆτ j of τ j evaluated by Agent j. hs estmate, used by Agent, s evaluated as ˆθ j ˆτ j = k s ŝ j k g ĝj + G Y j ˆq j, ˆq j, k p ĝ j, k p ĝj ˆθ j 16 ˆθ j = Γ j Y j ˆq j, ˆq j, k p ĝ j, k p ĝj ŝ j 17 t j,k = θj t j,k 18 where ŝ j = ˆq j + k pĝj, ĝ j = k=1 k jk ˆr jk r jk, ĝ j = k=1 k jk ˆr jk ṙjk, ˆr jk = ˆq j ˆq k, and ˆθ j s the estmate of θ j. Errors appear between q and ts estmate ˆq j obtaned by an other Agent j due to the presence of state perturbatons, the non-permanent communcaton, and the msmatch between θ, θ, and ˆθ. he errors for the estmates performed by Agent j are expressed as e j = ˆqj q, j e j = ˆq j q. 19 hese errors are used n Secton 6 to trgger communcatons when e and ė become too large. Fgure summarzes the overall structure of the estmator and controller. Remark he structure of the estmator for ˆτ j s chosen so as to get an accurate estmate for q n order to keep the e s and ė s small. In absence of perturbatons,.e., when max = and f θ s perfectly known,.e., θ = ˆθ = θ, the estmaton error e ntroduced n 19 vanshes. he prce to be pad for the use of ths estmator structure for s that every agent needs to mantan an estmator of the state of all other agents. ˆτ j 4.. Communcaton protocol When a communcaton s trggered at t,k for Agent, t broadcasts a message contanng t,k, q t,k, q t,k and ts estmated θ t,k. We assume that ths message s receved by all other agents, ether drectly when the network s fully connected, or after several hops when the network s connected. he latter case requres the use of a floodng protocol 15, 8. Snce communcatons have been assumed wthout delay, one has ˆq t = ˆqj t for all, j. hs smplfes the stablty study n Appendx

8 d Estmatons made by Agent Estmate of Agent 1 q 1, q 1 Agent Θ jt j,k Control g, g, s, Y, Θ Estmator of other agents' state Receve from Agent j τ Measure q and q, q j t j,k, q j t j,k Agent dynamcs M q + C q + G = τ + d q, q Communcaton If CCe, e, s, g ransmsson of q, q Yes Θ t j,k, q t j,k, q t j,k Estmate of Agent j Control g j, g j, s j,y j, Θ j Estmate of Agent Update q j and q j τ j Agent dynamcs M j q j + C j q j = τ j Update M j and C j q j, q j q, q Receve from Agent j Θ jt j,k, q j t j,k, q jt j,k q,q Fg.. Formaton control system archtecture 5 me-varyng formaton and trackng Consder, wthout loss of generalty, the frst agent as a reference agent 1 and ntroduce the target relatve confguraton vector r = r11... r1 whch may be tme-varyng. In ths secton, the MAS has to follow some reference trajectory q1 t, whle remanng n a desred formaton. Agent 1, taken as the reference agent, ams at followng q1 t. It s assumed that all agents have access to q1 t. Moreover, assume that the target formaton can be tme-varyng and s represented by the relatve confguraton vector r t. herefore the reference trajectory of each agent can be expressed as q t = q 1 t + r1 t. o guarantee that ndvdual reference trajectores can be tracked by each agent, t s assumed that for = 1,...,, q < q max q < q max. 1 efnton he MAS reaches ts trackng objectve ff there exsts ɛ 1 > and ɛ > such that 4 s satsfed and lm q 1 t q1 t ɛ, t.e., ff the reference agent asymptotcally converges to the reference trajectory, and the MAS asymptotcally converges to the target formaton wth bounded errors. A dstrbuted control law s desgned to satsfy ths target. Introduce the trajectory error terms ε = q q ˆε j = ˆqj q. he terms g, ḡ, ĝ j, s and ŝ j ntroduced n Sectons 4 are now redefned as follows to address the trajectory trackng 1 a vrtual agent may also be consdered. 7

9 problem g = ḡ = k j rj rj + k ε k j rj rj + k ε ĝ j = k j ˆr j j r j + k ˆε j s = q q + k p g s = q q + k p ḡ ŝ j = ˆq j q + k p ĝ j where k s a postve desgn parameter whch may be used to control the trackng error wth respect to the reference trajectory. When no reference trajectory s consdered, k =. From these terms, a new dstrbuted control nput to be used n 1 s defned for Agent as τ = k s s k g ḡ + G Y q, q, p, p θ θ = Γ Y q, q, p, p s 1 where p = k p ḡ q and p = k p ḡ q. he estmators mantaned by Agent are defned wth the same dynamcs as 1 but the evaluaton of the estmate ˆτ j of τ j s now evaluated as ˆτ j = k s ŝ j k g ĝj + G Y j ˆq j, ˆq j, ˆp j, ˆp j ˆθ j ˆθ j = Γ j Y j ˆq j, ˆq j, ˆp j, ˆp j ŝ j where ŝ j = ˆq j q j + k pĝ j, ˆp j = k pĝ j q j and ˆp j = k p ĝ j q j. he communcaton protocol ntroduced n Secton 4.. remans the same. he way the estmator 1-14 for the state of all agents s defned wth the control nput, and the absence of communcaton delays ensure that ˆx = ˆxj for all par of agents and j n the network. 6 Event-trggered communcatons heorem 4 ntroduces a CC used to trgger communcatons to ensure a bounded asymptotc convergence of the MAS to the target formaton. he ntal value of the state vectors are consdered to be known by all agents. In practce, ths condton can be satsfed by trggerng a communcaton from all agents at tme t = to ntalze the estmates of the state of the neghbours of all agents. Let k max = max l = 1... k lj and k mn = mn l = 1... k lj, α = k j, α mn = mn =1,..., α and j = 1... j = 1... α max = max =1,..., α. efne also for θ R p and θ = θ,1,..., θ,p max { θ,1 θ mn,,1, } θ,1 θ max,,1 θ,max =. max { θ,p θ mn,,p, } θ,p θ max,,p 4 8

10 and θ = θ θ. heorem 4 Consder a MAS wth agent dynamcs gven by 1 and the control law. Consder some desgn parameters η, η >, < b < ks k sk p+k g, c = } mn {1, k 1, k p, k, k k + αmnkmn k max max {1, k M } and k 1 = k s 1 + k p k M + 1. In absence of communcaton delays, the system 1 s nput-to-state practcally stable ISpS and the agents can be drven to some target formaton such that wth lm t =1 ξ = k ε + 1 P q, t ξ 5 k g c max + η + c max where max = max =1: supt> θ Γ 1 θ, f the communcatons are trggered when one of the followng condtons s satsfed k s s s + k p k g ḡ ḡ + η α M ke e e + k p k M ė ė + α M kck p e k j ˆq j + η + kg b q q + k p e α M 1 + Y θ,max + 6 Y θ,max 1 + Y θ,max 7 q ˆq + η 8 wth k e = k s k p + k g k p + kg b, and Y = Y q, q, p, p. he proof of heorem 4 s gven n Appendx 9.1. Corollary 5 Consder a MAS wth agent dynamcs gven by 1 and the control law. For any Agent, let t,k and t,k+1 be two consecutve communcaton nstants at whch the CC of heorem 4 have been satsfed. hen t,k+1 t,k >. he proof of Corollary 5 s provded n Appendx 9.. he CCs proposed n heorem 4 are analyzed assumng that the estmators of the state of the agents and the communcaton protocol s such that, j, ˆx t =ˆx j t 9 ˆx t,k =x t,k, 4 where 9 s called the estmate synchronzaton condton and 4 the estmator reset condton. heorem 4 s vald ndependently of the way the estmate ˆx of x s evaluated provded that 9 and 4 are satsfed. From 5 and 7, one sees that η can be used to adjust the trade-off between the bound ξ on the formaton and trackng errors and the amount of trggered communcatons. If η =, there s no perturbaton and θ s perfectly known, the system converges asymptotcally. 9

11 he CC 8 s related to the dscrepancy between q and ˆq. Choosng a small value of η may lead to frequent communcatons. On the contrary, when η s large, 7 s more lkely to be satsfed. A value of η that corresponds to a trade-off between the two CCs 7 and 8 has thus to be found to mnmze the amount of communcatons. he CCs 7 and 8 manly depend on e and ė. A communcaton s trggered by Agent when the state estmate ˆx of ts own state vector x s not satsfyng,.e., when e and ė becomes large. o reduce the number of trggered communcatons, one has to keep e and ė as small as possble. hs may be acheved by ncreasng the accuracy of the estmator, as proposed n Secton 4., but possbly at the prce of a more complex structure for the estmator. he perturbatons have a drect mpact on e and ė, and, as a consequence, on the frequency of communcatons. 6 shows the mpact of max and η on the formaton and trackng errors: n presence of perturbatons, the formaton and trackng errors cannot reach a value below a mnmum value due to the perturbatons. At the cost of a larger formaton and trackng errors, η can reduce the number of trggered communcatons and so can reduce the nfluence of perturbatons on the CC 7. he dscrepancy between the actual values of M and C and of ther estmates ˆM and Ĉ determnes the accuracy of θ, so θ,max, and the estmaton errors. Even n absence of state perturbatons, due to the lnear parametrzaton, t s lkely that ˆM M, Ĉ C and θ,max >, whch leads to the satsfacton of the CCs at some tme nstants. hus, the CC 7 leads to more communcatons when the model of the agent dynamcs s not accurate, requrng thus more frequent updates of the estmate of the states of agents. he choce of the parameters α M, k g, k p and b also determnes the number of broadcast messages. Choosng the sprng coeffcents k j such that α = k j s small leads to a reducton n the number of communcaton trggered due to the satsfacton of 7. 7 Smulaton results he performance of the proposed algorthm s evaluated consderng a set of = 6 agents. wo models wll be consdered to descrbe the dynamcs of the agents. 7.1 Models of the agent dynamcs and estmator ouble ntegrator wth Corols term I he frst model conssts n the dynamcal system M q q + C q, q q = τ + d wth q R and where 1.1 M = C q = q hen the vectors θ = ˆθ j, = 1,..., are obtaned usng. In place of the estmator n Secton 4. a frst less accurate estmate of x j made by Agent, s evaluated as ˆq j t = q j t j,k 4 ˆq j t = q j t j,k. 4 hs estmator allows one to better observe the tradeoff between the potental energy of the formaton and the communcaton requrements. For ths dynamcal model, the parameters of the control law and the CC 7 have been selected as: k M = M = 1, k C = C =.1, k p = 1, k g = 15, k s = 1 + k p k M + 1, b = 1 k g, and k =. 1

12 7.1. Surface shp SS he second model consders surface shps wth coordnate vectors q = x y ψ R, = 1..., n a local earth-fxed frame. For Agent, x, y represents ts poston and ψ ts headng angle. he dynamcs of the agents s descrbed by the surface shp dynamcal model taken from 17, assumed dentcal for all agents, and expressed n the body frame as where v = and M b, v + C b, v v + b, v = τ b, + d b,, 44 u v r s the velocty vector n the body frame, τb, s the control nput, d b, s the perturbaton, 5.8 M b, = v 1.115r C b, v = 5.8u.8v r 5.8u.7 b, = At t =, one assumes that Agent has access to estmates ˆM b, of M b,, Ĉ b, of C b,, and ˆ b, of b, descrbed as ˆM b, = 1 +.1Ξ M Mb, Ĉb, = 1 +.1Ξ C Cb, ˆ b, = 1 +.1Ξ b,, where 1 s the matrx of ones, Ξ M, ΞC, and Ξ are matrces whch components are ndependent and dentcally Bernoull random varables wth values n { 1, 1}, and s the Hadamard product. hese estmates are transmtted at t = to all other agents. As a consequence, the estmates of M b, and C b, made by all agents at t = are all dentcal. he model 44 s expressed wth the coordnate vectors q n the local earth-fxed frame usng the transform q = J ψ v cos ψ sn ψ J ψ = sn ψ cos ψ 1 where J ψ s a smple rotaton around the z-axs n the earth-fxed coordnate. efne J can be rewrtten as J M b, J 1 q + J C b, v M b, J 1 J + b, J 1 q = J τ b + J d b, = J 1. hen, 44 and so 11

13 where M q q + C q, q q = τ + d C q, q = J M q = J M b J 1, C b, J 1 q Mb, J 1 J + b, J 1, and τ s the control nput n earth-fxed coordnates as defned n. he vectors θ = ˆθ j, = 1,..., are obtaned usng. he estmator descrbed n Secton 4. s employed. For ths dynamcal model, the parameters of the control law and the CC 7 have been selected as: k M = M =.8, k C = C v 1 = 4.96, k p = 6, k g =, k s = 1 + k p k M + 1, b = 1 k g, and k = Smulaton parameters One chooses the components of the ntal value x of the state vector as.5 q = , and q = n 1. he vector of relatve target confguratons corresponds to a hexagonal formaton r = 1. Usng the approach developed n 7, the followng matrx K = k j = 1... can be computed from r j = K = and α = k j =.46, for all = 1,..., and α M =.46. A fully-connected communcaton graph s consdered. he smulaton duraton s = s. Matlab s ode45 ntegrator s used wth a step sze t =.1 s. Snce tme has been dscretzed, the mnmum delay between the transmsson of two messages by the same agent s set to t. he perturbaton d t s assumed of constant value over each nterval of the form k t, k + 1 t. he components of d t are ndependent realzatons of zero-mean unformly dstrbuted 1

14 nose U max, max and are thus such that d t max. Let m be the total number of messages broadcast durng a smulaton. he performance of the proposed approach s evaluated comparng m to the maxmum number of messages that can be broadcast m = / t m. he percentage of resdual communcatons s defned as R com = 1 m m. R com ndcates the percentage of tme slots durng whch a communcaton has been trggered. When a trackng has to be performed, one consders the target trajectory of the frst agent 4 sn.4t q 1 t = 4 cos.4t,.4t the other agents havng to reman n formaton. efne the trackng error ε = q 1 q Formaton control wth I Fgure shows the evoluton of the communcaton rato R com and of the potental energy at t =. For all smulatons, one has P q, ξ for the dfferent values of max and η. In Fgure a, the number of communcatons obtaned once the system has converged ncreases as the level of perturbatons becomes more mportant, as expected. Increasng η n the CC 7 helps reducng R com. evertheless, ncreasng η also ncreases the potental energy P q, of the formaton, as can be seen n Fgure b. In Fgure b, when η, one observes that the potental energy starts to decrease wth the level of perturbaton max to ncrease agan when max gets large. o explan ths surprsng behavor, Fgure c shows that there exsts a threshold R com =.5 below whch the potental energy sgnfcantly ncreases to ensure proper convergence. herefore η should be chosen such that R com remans above ths threshold. Even large values of max can be tolerated provded that η s chosen large enough to provde a suffcent amount of communcatons. 7. Formaton control wth shp dynamcal model Fgure 4 shows the trajectores of the agents when the control s appled and the communcatons are trggered accordng to the CC of heorem 4. Fgure 4 a llustrates the results obtaned usng the accurate estmator 1, Fgure 4b llustrates results obtaned usng the smple estmator 4. he agents converge to the desred formaton wth a lmted number of communcatons, even n presence of perturbatons. Fgure 5 shows the evoluton of R com and of P q, parametrzed by η for dfferent values of max. For all smulatons, one has P q, ξ for the dfferent values of max and η. As expected and shown n Secton 7., the potental energy obtaned once the system has converged ncreases wth max. It can also be observed that ncreasng η reduces the number of messages broadcast, wthout a sgnfcant mpact on P q,, contrary to what was observed wth the I wth smple estmator. 7.4 rackng control wth I he smulaton duraton s =.5 s. Fgures 6 and 7 show the evoluton of the communcaton rato R com, the potental energy and the trackng error at t =. In Fgure 6 a, the number of communcatons obtaned once the system has converged decreases as the level of perturbaton becomes more mportant, especally when η s small, whch was not excepted. Such behavor s not observed wth the accurate estmator 1, where R com ncreases when the perturbatons become more mportant, as llustrated n Fgure 9 a wth the shp model. hs behavor can be explaned by the fact a large max makes ḡ and s larger, whch reduces the number of tmes the CC 7 s satsfed, even f the error e s also affected. fference wth accurate estmator s the error e s keepng small by the estmator, so the nfluence of perturbatons s more sgnfcant on e than on ḡ or s, whch leads to a larger number of communcatons trggered. 1

15 Rcom η =. η = 1. η =. η = 5. η = 7. η = 9. η = 11. lm t= Pq,t η =. η = 1. η =. η = 5. η = 7. η = 9. η = max a me s b 1 max =. max =. max = 4. max = 6. max = 8. max = 1. max = 1. lm t= Pq,t 1 1 η = 11 η = 9 η = 7 η = 5 η = η = 1 η = me s c { } { } Fg.. Evoluton of R com and P q, t for dfferent values of max,, 4, 6, 8, 1, 1, η, 1,, 5, 7, 9, 11, and η = 7.5. he I model and the smple estmator 4-4 are consdered. Fgure 6 a llustrates that the parameter η n the CC 7 can help reducng R com. It can be seen that there exsts for R com a threshold R com = 7 whch R com cannot reach : we can deduce a mnmal number of communcatons s requred for system converge wth the constant estmator 4-4. Fgures 6 b and c show that the potental energy of the formaton P q, t and the trackng error ε ncrease when the perturbaton level ncreases. he nfluence of parameter η s also llustrated: Fgure 7 shows that a larger value of η leads to an ncrease of P q, t, but reduces ε. Indeed, the less communcatons, the more dffcult t s for some Agent to be synchronzed wth the others agents to reach the target formaton. However, be less synchronzed wth the other agents allows Agent to be more synchronzed wth ts target trajectory q, nducng a small trackng error ε. hus, a trade off between the P q, t and ε has to be reached. 14

16 4 4 q, q, q,1 4 6 q,1 Agent ndex me s Agent ndex me s a Accurate estmator 1. b Constant estmator 4. Fg. 4. Hexagonal formaton wth max =, η = and η = 7.5. Agents are represented by crcles. In a, R com =.61% and P q, =.1. In b R com = 18.5% and P q, =.1. lmt= Pq,t..5. max =. =. max = 4. max = 5. max = 6. max = 7. max 65 η = η = η = R com Fg. 5. Evoluton of R com and P q, t for dfferent values of max η = 7.5. Model 44 and accurate estmator 1 are consderate. 7.5 rackng wth surface shp model he smulaton duraton s =.5 s. { } { },,..., 7, η, 5, 1, and Fgures 9 and 1 show the evoluton of the communcaton rato R com, the potental energy and the trackng error at t =. In Fgure 9 a, the number of communcatons obtaned once the system has converged ncreases as the level of 15

17 Rcom η =. η = 1. η =. η = 5. η = 7. η = 9. η = 11. lmt= Pq,t η =. η = 1. η =. η = 5. η = 7. η = 9. η = max a max b η =. η = 1. η =. η = 5. η = 7. η = 9. η = 11. lmt= ε max c { } { } Fg. 6. Evoluton of R com, P q, t and ε for dfferent values of max,, 4, 6, 8, 1, 1, η, 1,, 5, 7, 9, 11 and η = 7.5. Model 41 and constant estmator 4-4 are consderate. perturbatons becomes more mportant. he parameter η n the CC 7 can help to reduce R com. Fgure 9 b and c show that the potental energy of the formaton P q, t and the trackng error ε also ncrease when the perturbaton level ncreases. Influence of parameter η s also llustrated : Fgure 9 c shows that ncreasng η results n make ε decrease when max >. Influence of η on P q, t s less clearly detectable than n the case of the I model. In Fgure 1, t can be observed that R com cannot be reduced below the value of 1: a mnmum number of communcatons s ndeed requred to converge wth the accurate estmator 1. 16

18 lmt= Pq,t η = 9 η = 11 η = 7 η = 5 η = η = 1 η = max =. max =. max = 4. max = 6. max = 8. max = 1. max = 1. lmt= ε η = 7 η = 9 η = 11 η = η = 5 η = 1 η = max =. max =. max = 4. max = 6. max = 8. max = 1. max = R com R com a b { } { } Fg. 7. Evoluton of R com, P q, t and ε for dfferent values of max,, 4, 6, 8, 1, 1, η, 1,, 5, 7, 9, 11 and η = 7.5. Model 41 and constant estmator 4-4 are consderate q, q,1 6 Agent ndex me s Fg. 8. Hexagonal formaton and trackng problem wth max = 5, η = 5, and η = 7.5. Crcles represents agents top fgure and communcaton events bottom fgure. R com = 5%, P q, =.1 and ε =.1. = 6 s. 8 Concluson hs paper presents an adaptve control and event-trggered communcaton strategy to reach a target formaton for mult-agent systems wth perturbed Euler-Lagrange dynamcs. From estmate nformaton of agents dynamcs, an estmator has been proposed to provde the mssng nformaton requred by the control. Convergence to a desred formaton and nfluence of state perturbatons on the convergence and on the amount of requred communcatons have been studed. rackng control to follow an desre trajectory has been consderate and added to the formaton control. A dstrbuted event-trggered condton to converge to a desred formaton and follow the reference trajectory whle reduce the number of communcatons have been studed. Smulatons have shown the effectveness of the proposed method n presence of state perturbatons when ther level remans moderate. he tme nterval between consecutve communcatons has been shown to be strctly postve. 17

19 R com η =. 8 η = 1 η = 6 η = η = 4 4 η = 5 η = 6 η = 7 η = 8 4 max 6 8 lm t= ε η =. η = 1 η = η = η = 4 η = 5 η = 6 η = 7 η = max a b lmt= Pq,t x 1 η =. η = 1 η = η = η = 4 η = 5 η = 6 η = 7 η = max c { } { } Fg. 9. Evoluton of R com, P q, t and ε for dfferent values of max, 1,,... 7, η, 1,,... 8 and η = 7.5. he SS model 44 and accurate estmator 1 are consdered. In future work, the consdered problem wll be extended to communcaton delay and package drop. Acknowledgments We thanks recton Generale de l Armement GA and ICOE for a fnancal support to ths study. 18

20 x 1 max =. 6.5 max = 1. max =. 6 max =... max =. max = 1. max =. max =. lm t= Pq,t max = 4. max = 5. max = 6. max = 7. 4 η = η = lm t= ε max = 4. max = 5. max = 6. max = 7. η = η = R com R com a b { } { } Fg. 1. Evoluton of R com, P q, t and ε for dfferent values of max, 1,,... 7, η, 1,,... 8 and η = 7.5. he SS model 44 and accurate estmator 1 are consdered. 9 Appendx 9.1 Proof of heorem 4 Consder a gven value of max and η, one shows frst that the MAS s nput-to-state practcally stable. One then evaluates the nfluence of max and η on the behavor of the MAS Proof of the nput-to-state practcal stablty of the MAS Consder the contnuous postve-defnte canddate Lyapunov functon V = 1 =1 s M s + θ Γ 1 k g θ + 1 P q, t + k q q =1 45 where θ = θ θ. he tme dervatve of V s V = 1 =1 s Ṁ s + s M ṡ + θ Γ 1 θ + k g d 1 dt P q, t + k q q =1 46 where, from 7, one has ṡ = q q + k pġ. Injectng 11 n 46 one obtans 1 V = s Ṁ s + s M ṡ + θ Y q, q, p, p s =1 + k g d 1 dt P q, t + k q q. 47 =1 19

21 he last term n 47 may be wrtten as 1 d dt 1 P q, t + k q q =1 = 1 d k rj j r j + 1 d k q q 4 dt dt =1 = 1 k j ṙj ṙ j rj rj + k q q q q =1 = 1 +k q q q q = 1 +k q q ε. k j q q r j rj qj q j rj rj k j q q r j rj q q r j rj 48 Snce r j = r j, one gets 1 d dt P q, t + k q q = q q 1 =1 Combnng 47 and 49, one obtans V = =1 = =1 k j rj rj + k ε q q g. 49 =1 1 s Ṁ s + s M ṡ + θ Y q, q, p, p s + k g q q g. 5 One focuses now on the term M ṡ. Usng agan 7, one may wrte Usng 1, one gets M ṡ + C s = M q q + k p ġ + C q q + k p g 51 M ṡ + C s = τ + d G + M k p ġ q + C k p g q, 5 where one used 1. ow, ntroducng, one gets M ṡ + C s = k s s k g ḡ Y q, q, k p ḡ q, k p ḡ q θ +M k p ġ q + C k p g q + d 5 In what follows, one uses Y n place of Y q, q, k p ḡ q, k pḡ q to lghten notatons. Snce θ = θ θ, one obtans s M ṡ = k s s s k g s ḡ s C s + s M k p ġ q + C k p g q s Y θ + θ + s d. 54

22 Usng n 54 leads to s Y θ + θ = s Y θ s M k p ḡ q + C k p ḡ q. 55 Consderng and 54 n 5, one gets V = =1 1 s Ṁ s k s s s k g s ḡ s C s + s M k p ġ q + C k p g q s M k p ḡ q + C k p ḡ q s Y θ + s Y θ +k g q q g + s d. 56 ow, ntroduce 4 n 7 to get s = q q + k p =1 k j q q j rj + k ε. 57 Snce e j = ˆq j q j, one gets s = q q + k p =1 = q q + k p = s + k p E j, =1 k j q ˆq j + e j rj + k ε k j rj rj + k ε + k p j = 1 j k j e j 58 wth snce k = Usng smlar dervatons, one may show that Ej = k j e j. 59 =1 g = ḡ + E j. 6 Replacng 58 and 6 n 56, one gets V = =1 s +k p s 1 C s k s s Ṁ s k g q q + k p g ḡ M Ėj + C Ej + k p E j Y θ + k g q q g + s d. 61 Let V 1 = k p s =1 M Ėj + C Ej 1

23 and V = k p Snce 1 Ṁ C s skew symmetrc or defnte negatve, s of smlar sze, one has x y 1 Usng 6 wth b = 1, one deduces that d s 1 =1 E j Y θ. 1 Ṁ C s. For all b > and all vectors x and y bx x + 1 b y y. 6 max + s s and that V =1 k s s s k g k p g ḡ + 1 s s + 1 max +k g q q g ḡ + 1 V1 + V 6 One notces that r j = q q j = q ˆq j + e j = r j + e j, thus s s = s s s s + s s kp Ej = s s s s + s s s s = 1 k p Ej + 1 s s + 1 s s 64 In the same way, from 64, one shows that Injectng 65 n 6, g ḡ = 1 ks V kp E j s s s 1 s + k p k g =1 +k g q q g ḡ + 1 V1 + V k s 1 s s k s s s + k skp + k g k p =1 +k g q q g ḡ + 1 E j + 1 g g + 1 ḡ ḡ. 65 E j g g ḡ ḡ + 1 s s + 1 max E j 1 k pk g g g + ḡ 1 ḡ + max V1 + V. 66 Usng 6 wth b = b >, one shows that q g ḡ b q + 1 b E j. Usng ths result n 66, one gets V 1 k s 1 s s k s s s + k s kp + k g k p + k g E + b k g q q =1 k p k g g g + ḡ ḡ + max + 1 b j V1 + V. 67 Consder now V 1. Usng 6 wth b = 1, the fact that M s symmetrc postve defnte, and that x M x < k M x x,

24 one obtans k p s =1 M Ėj + C Ej =1 =1 k p s M s + s Ė s + j M Ėj + Ej C C Ej k p k M + 1 s s + k M Ė j Ėj + Ej C C Ej 68 Focus now on the terms Ej C C Ej =1 E j C C E j = k j e j =1 l=1 Usng 6 wth b = 1, one gets =1 E j C C E j 1 l=1 l=1 α C C k l e l l=1 k l k j C e j e l. 69 k l k j C e j k l k j C e j e j e j + e l e l k j C e j e j. 7 Snce one has assumed that 4 and 9 are satsfed, one has ˆq j = ˆqj j, e j = ej j. As a consequence, and snce k j = k j, =1 Ej C C Ej α M =1 α M =1 hen, the second CC 8 leads to =1 Ej C C Ej α M kc =1 k j e j = e j k j = k j e C j j k j e. 71 k j e k C q j. 7 e k j ˆq j + η. 7 Smlarly, one shows that =1 Ej Ej =1 α M e

25 and =1 Ėj Ėj =1 α M ė. Consder now V V = k p =1 Ej Y θ = k p k j e j =1 Y θ. 74 Snce e j = ej j, one gets V = k p =1 = k p = k p k j e j j =1 =1 e Y θ kj e Yj θ j k j Y j θ j. 75 Let n =,... R n be the all-zero vector. If e = n, one has k p e k jy j θ j =. Consderng now the case e n. Usng 6 wth b = b >, one obtans Snce =1 E j E j =1 α M e V = k p k p, one gets V =1 =1 =1 =1 E j Y θ 76 b E j E j + 1 b Y θ. 77 k p αmb e 1 + Y θ b k p α Mb e + 1 Y θ,max, 78 b where θ,max s gven by 4. 4

26 Snce e n, choosng b = 1+ Y θ,max, one obtans e V V wth V = = =1 k p α M k p e =1 α M 1 + Y θ,max e 1 + Y θ,max + e e + Y θ,max 1 + Y θ,max Y θ,max 1 + Y θ,max. 79 Injectng 68, 7, and 79 n 67, one gets ks 1 k p k M + 1 s s k s s s + max V 1 =1 k p k g g g k p k g ḡ ḡ + k g b q q + k p k M αm ė +αm k s kp + k g k p + k g e + α M k p kc e b k j ˆq j + η + 1 V. 8 he CC 7 leads to V 1 V 1 ks 1 k p k M + 1 s s k g k p g g + max + η =1 k1 s s k g k p g g + max + η 81 =1 wth k 1 = k s 1 k p k M + 1. Followng the steps gven n Appendx 9..1 from 16 to 11, one shows that V c V + max + η + c =1 θ Γ 1 θ, 8 where c > s a postve constant. Introducng max = max =1: supt> θ Γ 1 θ, one has V c V + c max + max + η. 8 efne the functon W such that W = V and Ẇ = c W + max + η + c max. 84 Usng the ntal condton W = V, the soluton of 84 s W t = exp c t V + 1 exp c t c max + η + c max. 85 hen, usng the Lemma.4 n 1 Comparson lemma, one has V t W t and so V t exp c t V + 1 exp c t c max + η + c max 86 5

27 Snce M and Γ are symmetrc, there exsts matrces S M and S Γ such that M = S M S M and Γ = S Γ S Γ. Introduce now y M = S M1 s 1... S M s... S M s 87 S y Γ = 1 Γ 1 θ 1... S 1 Γ θ y q = q 1 q1... q q... q q z = ym y Γ kg k yq... S 1 Γ θ k g P x, t 9 hen, V t can be rewrtten as Usng 91 n 86, one has t V z = 1 z z. 91 z t exp c t z + 1 exp c t c max + η + c max z t exp c t z + 1 exp c t max c + η + c max z t exp c t z + 1 exp c t max c + η + c max z t exp c t z + c max + η + c max 9 and so z t β z, t + ρ 9 wth ρ = c max + η + c max, β z, t = exp c t z, and β KL. Usng efnton.1 from 16, 9 mples that the MAS s nput-to-state practcally stable Convergence of V From 9, we know the system s ISpS. Moreover, from 8, one has V c V + max + η + c =1 θ Γ 1 θ 94 hen, f ntally c V + max + η + c =1 θ Γ 1 θ < 95 6

28 one has V and V s decreasng. hen, one has from 86 1 lm t =1 s M s + θ Γ 1 k g θ + k g lm t =1 lm V t t c max + η + c max k ε + 1 P q, t c max + η + c max k ε + 1 P q, t c max + η + c max lm =1 t =1 Asymptotcally, the formaton and trackng error are bounded. k ε + 1 P q, t 1 lm t =1 s M s + θ Γ 1 θ k g c max + η + c max Proof of t,k+1 t,k > From the CC 7, a communcaton s trggered at t = t,k when k s s s + k p k g ḡ ḡ + η = α M k e e + kp k M ė + α M kck p e k j ˆq j + η + kg b q q + k p e α M 1 + Y θ,max + Y θ,max 1 + Y θ,max 97 wth k e = k s kp + k g k p + kg b. hen, the estmaton errors e and ė are reset and one has e t +,k = and ė t +,k =. As a consequence, the CC 7 n heorem 4 s not satsfed at t = t +,k ff k s s s + k p k g ḡ ḡ + η > k g b q q. 98 o prove the absence of Zeno behavor,.e., that t,k+1 > t,k, one has to show that 98 s satsfed. Usng the property x y 1 b x x + 1 b y y for some b >, one deduces that s s = kpḡ ḡ + q q + k p ḡ q q kp k p b ḡ ḡ + 1 k p q q. 99 b Usng 99, a suffcent condton for 98 to be satsfed s 7

29 k s k p k p b ḡ ḡ + k s 1 k p b q q + k p k g ḡ ḡ + η > k g b q q k s 1 k p q q + k p k g + k s k b p k p b ḡ ḡ + η > k g b q q k 1 ḡ ḡ + η > k q q 1 where k 1 = k p k g + k s k p k p b and k = k g b k s 1 kp b. o ensure that the nequalty 1 s satsfed ndependently of the values of ḡ and q, t s suffcent to fnd b and b such that k 1 > and k <. Consder frst k 1. k p k g + k s k p k p b > k g k s > k p + b k s k p + k g k s > b. 11 Focus now on k Snce b >, one has kgb k s k g b k s 1 k p < 1 and so b < ks k g. hen b < k g b < 1 k p k s 1 b < 1 k p b 1 k gb k s. 1 k s k p k s k g b < b. 1 Fnally, one has to fnd a condton on b such that 11 and 1 can be satsfed smultaneously One may fnd b f k s k p + k g k s > b > k sk p k s k g b k g k s whch also ensures that b < ks k g. hus, once b < consequence t,k+1 t,k >. k sk p k s k g b > k s k p + k g k sk p > b k s k p + k g b < k s k s k p + k g. 15 k s k sk p+k g, there exsts some b such that 14 s satsfed. As a 8

30 9. Complementary proof elements 9..1 fferental equaton satsfed by V From 81, one gets V 1 km s s k g g g + max + η 16 =1 where k m = mn {k 1, k p }. Usng 117, one may wrte where hen g g k ε + k + α mnk mn P q, t k =1 =1 max k k ε + 1 P q, t k = =1 { k + αmnkmn k max f 1 else. k + αmnkmn k max < 1 g g k k ε + 1 P q, t =1 =1 k k ε + 1 P q, t = where k = k k f k < 1, k = 1 else. hen V 1 1 k M k 4 k M km s k k g s k ε + 1 P q, t + max + η =1 1 k m km s k k g s + k ε + 1 P q, t + max + η =1 =1 1 km s k g s + k ε + 1 P q, t + max + η 19 =1 =1 wth k M = 1 f k M < 1 and k M = k M else, and k 4 = mn k m, k. Let c = k4 k M V c 1 + =1 max + η + c V c V + =1 s M s + θ Γ 1 k g θ + =1 max + η + c he evaluaton of c s descrbed n Appendx θ Γ 1 θ =1 and one gets k ε + 1 P q, t =1 θ Γ 1 θ. 11 9

31 9.. Upper-bound on =1 g g From 4, one may wrte g g = k j rj rj + k ε =1 = =1 k j rj rj =1 k j rj rj + k ε + k ε + k ε k j rj rj. 111 Let P 1 = ε =1 k j rj rj. 11 Snce r j r j = q q j q q j = ε ε j, P 1 = = k j ε ε ε j Usng the fact that a b = a a + b b a b a b, one gets P 1 = k j ε ε ε ε j. 11 k j ε 1 ε + ε j ε ε j. 114 Snce k j = k j and ε ε j = r j r j P 1 = 1 = 1 = P q, t. k j ε 1 k j ε 1 k j ε j + 1 k j ε + P q, t k rj j rj 115 Injectng P 1 n 111, one gets g g = k j rj rj =1 =1 + k ε + k P q, t 116 and usng 15, one gets

32 g g k ε + k + α mnk mn P q, t 117 k =1 =1 max 9.. Upper-bound on =1 k j rj rj One may wrte k j rj rj = k j rj rj k l r l rl =1 = =1 Usng the fact a b = a a + b b a b a b k j rj rj =1 = 1 =1 l=1 =1 l=1 l=1 k l k j rj rj rl rl 118 rj k l k j rj + r l rl rj rj r l rl 119 One has rj rj rl rl = r j r l rj rl 1 = r lj rlj 11 Injectng ths result n 119 leads to k j rj rj =1 = 1 =1 l=1 k l k j r j rj + r l rl r lj rlj 1 wth k max = max l = 1... k lj k max =1 k max k max k max j = 1... k j rj rj k j rj rj k j rj rj k j rj rj =1 l=1 =1 l=1 rj k l k j k lj rj + r l rl rlj rlj =1 l=1 =1 l=1 =1 l=1 =1 l=1 k l k j k rj lj rj + 1 k l k j k rlj lj rlj k l k j k lj r j rj + 1 k l k j k lj r j rj =1 l=1 =1 l=1 k l k j k lj r l rl k l k j k lj r j rj k l k j k rj lj rj. 1 1

33 Let k mn = mn k lj and α mn = mn =1,..., α. One may wrte l = 1... j = 1... =1 l=1 k l k j k rj lj rj = k l =1 l=1 =1 l=1 k l k mn α k mn =1 α mn k mn k j k rj lj rj k rj j rj k j r j rj k rj j rj α mn k mn P q, t 14 Injectng 14 n 1 one gets k max =1 k j rj rj k j rj rj =1 α mn k mn P q, t α mnk mn k max P q, t Evaluaton of c One has c = k 4 k M = mn k m, k max {1, k M } = mn {mn {k 1, k p }, mn {k k, 1}} max {1, k M } = mn {k 1, k p, 1, k k } mn = mn = max {1, k M } { k 1, k p, 1, k mn { }} k + αmnkmn k max, 1 max {1, k M } } {k 1, k p, 1, k, k k + αmnkmn k max max {1, k M } 16 where k 1 = k s 1 k p k M + 1, α mn = mn =1,..., α, k max = max l = 1... j = 1... k lj and k mn = mn l = 1... j = 1... k lj

34 References 1 onlnear Systems, rd ed.. A. P. Aguar and A. M. Pascoal. Coordnated path-followng control for nonlnear systems wth logc-based communcaton. In Proc. IEEE ecson and Control Conference, pages , 7. A.Y. Alfakh and V.H. guyen. On affne motons and unversal rgdty of tensegrty frameworks. In Lnear Algebra and ts Applcatons, volume 49, pages Elsever, 1. 4 R. Aragues, J. Cortes, and C. Sagues. strbuted consensus algorthms for mergng feature-based maps wth lmted communcaton. In Robotcs and Autonomous Systems, volume 59, pages 16 18, F.J.M Arboleda, J. Alberto G. Luna, S.A.G. Aras, et al. Identfcaton of v-formatons, crcular, and doughnut formatons n a set of movng enttes wth outlers. In Abstract and Appled Analyss. Hndaw Publshng Corporaton, 1. 6 M Arboleda, F. Javer, G. Luna, J. Alberto, G. Aras, and S. Alonso. Identfcaton of v-formatons and crcular and doughnut formatons n a set of movng enttes wth outlers. Abstract and Appled Analyss, 1, Z. Chao, L. Mng, Z.. Shaole, and Z. Wenguang. Collson-free uav formaton flght control based on nonlnear mpc. In Proc. IEEE Electroncs, Communcatons and Control ICECC, pages , Z. Chao, S.-L. Zhou, L. Mng, and W-G Zhang. Uav formaton flght based on nonlnear model predctve control. In Mathematcal Problems n Engneerng, volume 1, pages Hndaw Publshng Corporaton, V. marogonas, E. Frazzol, and K. H. Johansson. strbuted event-trggered control for mult-agent systems. In IEEE ransactons on Automatc Control, volume 57, pages , 1. 1 K- o. Formaton trackng control of uncycle-type moble robots wth lmted sensng ranges. In IEEE ransactons on Control Systems echnology, volume 16, pages 57 58, Y. Fan, G. Feng, and Y. Wang. strbuted event-trggered control of mult-agent systems wth combnatonal measurements. In Automatca, volume 49, pages , 1. 1 E. Garca, Y. Cao, X. Wang, and. W. Casbeer. Cooperatve control wth general lnear dynamcs and lmted communcaton: Centralzed and decentralzed event-trggered control strateges. In Amercan Control Conference ACC, volume 1, pages , E Garca, Y Cao, X Wang, and. W. Casbeer. Cooperatve control wth general lnear dynamcs and lmted communcaton: Perodc updates. In Amercan Control Conference ACC, volume 1, pages 195, E. Garca, Y. Cao, X. Wang, and. W. Casbeer. ecentralzed event-trggered consensus of lnear mult-agent systems under drected graphs. In Amercan Control Conference ACC, volume 1, pages , W. R. Henzelman, J. Kulk, and H. Balakrshnan. Adaptve protocols for nformaton dssemnaton n wreless sensor networks. In Proceedngs of the 5th Annual ACM/IEEE Internatonal Conference on Moble Computng and etworkng, pages , Z.-P. Jang, I. MY Mareels, and Y. Wang. A lyapunov formulaton of the nonlnear small-gan theorem for nterconnected ss systems. Automatca, 8: , E Kyrkjeb, K.Y. Pettersen, M. Wondergem, and H. jmejer. Output synchronzaton control of shp replenshment operatons: heory and experments. 156: , S. Lnsenmayer. Event-trggered control of mult-agent systems wth double-ntegrator dynamcs: Applcaton to vehcle platoonng and flockng algorthms. Ph thess, KH Royal Insttute of echnology, Sweden, S. Lnsenmayer and. V. marogonas. Event-trggered control for vehcle platoonng. In Amercan Control Conference ACC, pages 11 16, 15. Q. Lu, Z. Sun, J. Qn, and C. Yu. stance-based formaton shape stablsaton va event-trggered control. In Proceedngs Chnese Control Conference CCC, pages , Z Lu, W Chen, J Lu, H Wang, and J Wang. Formaton control of moble robots usng dstrbuted controller wth sampled-data and communcaton delays. In IEEE ransactons on Control Systems echnology, volume 4, pages 15 1, 16. C. Makkar, G. Hu, W. G. Sawyer, and W.E. xon. Lyapunov-based trackng control n the presence of uncertan nonlnear parameterzable frcton. IEEE ransactons on Automatc Control, 51: , 7. J Me, W Ren, and G. Ma. strbuted coordnated trackng wth a dynamc leader for multple euler-lagrange systems. IEEE ransactons on Automatc Control, 566: , B. abet and -E Leonard. ensegrty models and shape control of vehcle formatons. In arxv preprnt arxv:9.71, volume 1, pages 1 1, 9. 5 R. Olfat-Saber. Flockng for mult-agent dynamc systems: Algorthms and theory. In Proc. IEEE ransactons on Automatc Control, volume 51, pages IEEE, 6. 6 R. Olfat-Saber, A. J. Fax, and R. M. Murray. Consensus and cooperaton n networked mult-agent systems. In Proceedngs of the IEEE, volume 95, pages 15, 7. 7 Y. Qngka, C Mng, F. Hao, C. Je, and H. Je. strbuted formaton stablzaton for moble agents usng