Distributed event-triggered coordination for average consensus on weight-balanced digraphs

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1 Dstrbuted event-trggered coordnaton for average consensus on weght-balanced dgraphs Cameron Nowzar a Jorge Cortés b a Department of Electrcal and Systems Engneerng, Unversty of Pennsylvana, Phladelpha, PA, 9, USA b Department of Mechancal and Aerospace Engneerng, Unversty of Calforna, San Dego, CA, 993, USA Abstract Ths paper proposes a novel dstrbuted event-trggered algorthmc soluton to the mult-agent average consensus problem for networks whose communcaton topology s descrbed by weght-balanced, strongly connected dgraphs. The proposed event-trggered communcaton and control strategy does not rely on ndvdual agents havng contnuous or perodc access to nformaton about the state of ther neghbors. In addton, t does not requre the agents to have a pror knowledge of any global parameter to execute the algorthm. We show that, under the proposed law, events cannot be trggered an nfnte number of tmes n any fnte perod (.e., no Zeno behavor), and that the resultng network executons provably converge to the average of the ntal agents states exponentally fast. We also provde weaker condtons on connectvty under whch convergence s guaranteed when the communcaton topology s swtchng. Fnally, we also propose and analyze a perodc mplementaton of our algorthm where the relevant trggerng functons do not need to be evaluated contnuously. Smulatons llustrate our results and provde comparsons wth other exstng algorthms. Key words: dscrete event systems, event-trggered control, average consensus, mult-agent systems, weght-balanced dgraphs Introducton Ths paper studes the mult-agent average consensus problem, where a group of agents seek to agree on the average of ther ntal states. Due to ts numerous applcatons n networked systems, many algorthmc solutons exst to ths problem; however, a majorty of them rely on agents havng contnuous or perodc avalablty of nformaton from other agents. Unfortunately, ths assumpton leads to neffcent mplementatons n terms of energy consumpton, communcaton bandwdth, congeston, and processor usage. Motvated by these observatons, our man goal here s the desgn of a provably correct dstrbuted event-trggered strategy that prescrbes when communcaton and control updates should occur so that the resultng asynchronous network executons stll acheve average consensus. Lterature revew: Trggered control seeks to understand the trade-offs between computaton, communcaton, sensng, and actuator effort n achevng a desred task wth a guaranteed level of performance. Early A prelmnary verson was presented as [Nowzar and Cortés, ] at the Amercan Control Conference. Emal addresses: cnowzar@seas.upenn.edu (Cameron Nowzar), cortes@ucsd.edu (Jorge Cortés). works [Åström and Bernhardsson., ] consder tunng controller executons to the state evoluton of a gven system, but the deas have snce then been extended to consder other tasks, see [Heemels et al., ] and references theren for a recent overvew. Among the many references n the context of mult-agent systems, [Mazo Jr. and Tabuada, ] specfes the responsblty of each agent n updatng the control sgnals, [Wang and Lemmon, ] consders network scenaros wth dsturbances, communcaton delays, and packet drops, and [Stöker et al., 3] studes decentralzed eventbased control that ncorporates estmators of the nterconnecton sgnals among agents. Several works have explored the applcaton of event-trggered deas to the acquston of nformaton by the agents. To ths end, Xe et al. [9], Heemels and Donkers [3], Meng and Chen [3] combne event-trggered controller updates wth sampled data that allows for the perodc evaluaton of the trggers. Zhong and Cassandras [] drop the need for perodc access to nformaton by consderng event-based broadcasts, where agents decde wth local nformaton only when to obtan further nformaton about neghbors. Self-trggered control [Anta and Tabuada,, Wang and Lemmon, 9] relaxes the need for local nformaton by decdng when a future sample of the state should be taken based on the aval- Preprnt submtted to Automatca January 6

2 able nformaton from the last sampled state. Teamtrggered coordnaton [Nowzar and Cortés, 6] combnes the strengths of event- and self-trggered control nto a unfed approach for networked systems. The lterature on mult-agent average consensus s vast, see e.g., [Olfat-Saber et al., 7, Ren and Beard, 8, Mesbah and Egerstedt, ] and references theren. Olfat-Saber and Murray [] ntroduce a contnuoustme algorthm that acheves asymptotc convergence to average consensus for both undrected and weghtbalanced drected graphs. Dmarogonas et al. [] buld on ths algorthm to propose a Lyapunov-based event-trggered strategy that dctates when agents should update ther control sgnals but ts mplementaton reles on each agent havng perfect nformaton about ther neghbors at all tmes. The work [Seybotha et al., 3] uses event-trggered broadcastng wth tmedependent trggerng functons to provde an algorthm where each agent only requres exact nformaton about tself, rather than ts neghbors. However, ts mplementaton requres knowledge of the algebrac connectvty of the network. In addton, the strctly tme-dependent nature of the thresholds makes the network executons decoupled from the actual state of the agents. Closer to our treatment here, Garca et al. [3] propose an event-trggered broadcastng law wth state-dependent trggerng functons where agents do not rely on the avalablty of contnuous nformaton about ther neghbors (under the assumpton that all agents have ntal access to a common parameter). Ths algorthm works for networks wth undrected communcaton topologes, tolerates quantzed communcaton, and guarantees that all nter-event tmes are strctly postve, but does not dscard the possblty of an nfnte number of events happenng n a fnte tme perod. The work [Fan et al., 5] proposes a self-trggered algorthm for ths problem that s guaranteed to avod Zeno executons. We consder here a more general class of communcaton topologes descrbed by weght-balanced, drected graphs. The works [Gharesfard and Cortés,, Rkos et al., ] present provably correct dstrbuted strateges that, gven a drected communcaton topology, allow a network of agents to fnd such weght edge assgnments. Statement of contrbutons: Our man contrbuton s the desgn and analyss of novel event-trggered broadcastng and controller update strateges to solve the multagent average consensus problem over weght-balanced dgraphs. Wth respect to the conference verson of ths work [Nowzar and Cortés, ], the present manuscrpt ntroduces new trgger desgns, extends the treatment from undrected graphs to weght-balanced dgraphs, and provdes a comprehensve techncal treatment. Our proposed law does not requre ndvdual agents to have contnuous access to nformaton about the state of ther neghbors and s fully dstrbuted n the sense that t does not requre any a pror knowledge by agents of global network parameters to execute the algorthm. Our Lyapunov-based desgn bulds on the evoluton of the network dsagreement to synthesze trggers that agents can evaluate usng locally avalable nformaton to make decsons about when to broadcast ther current state to neghbors. In our desgn, we carefully take nto account the dscontnutes n the nformaton avalable to the agents caused by broadcasts receved from neghbors and ther effect on the feasblty of the resultng mplementaton. Our analyss shows that the resultng asynchronous network executons are free from Zeno behavor,.e., only a fnte number of events are trggered n any fnte tme perod, and exponentally converge to agreement on the average of all agents ntal states over weght-balanced, strongly connected dgraphs. We also provde a lower bound on the exponental convergence rate and characterze the asymptotc convergence of the network under swtchng topologes that reman weght-balanced and are jontly strongly connected. Lastly, we propose a perodc mplementaton of our event-trggered desgn that has agents check the trggers perodcally and characterze the samplng perod that guarantees correctness. Varous smulatons llustrate our results. Prelmnares Ths secton ntroduces some notatonal conventons and notons on graph theory. Let R, R >, R, and Z > denote the set of real, postve real, nonnegatve real, and postve nteger numbers, respectvely. We denote by N and N R N the column vectors wth entres all equal to one and zero, respectvely. We let denote the Eucldean norm on R N. We let dag(r N ) = {x R N x = = x N } R N be the agreement subspace n R N. For a fnte set S, we let S denote ts cardnalty. Gven x, y R, Young s nequalty [Hardy et al., 95] states that, for any ε R >, xy x ε + εy. () A weghted drected graph (or weghted dgraph) G = (V, E, W ) s comprsed of a set of vertces V = {,..., N}, drected edges E V V and weghted adjacency matrx W R N N. Gven an edge (, j) E, we refer to j as an out-neghbor of and as an n-neghbor of j. The sets of out- and n-neghbors of a gven node are N out and N n, respectvely. The weghted adjacency matrx W R N N satsfes w j > f (, j) E and w j = otherwse. A path from vertex to j s an ordered sequence of vertces such that each ntermedate par of vertces s an edge. A dgraph G s strongly connected f there exsts a path from all V to all j V. The out- and n-degree matrces D out and D n are dagonal matrces where d out = w j, d n = j N n w j,

3 respectvely. A dgraph s weght-balanced f D out = D n. The (weghted) Laplacan matrx s L = D out W. Based on the structure of L, at least one of ts egenvalues s zero and the rest of them have nonnegatve real parts. If the dgraph G s strongly connected, s smple wth assocated egenvector N. The dgraph G s weght-balanced f and only f T N L = N f and only f L s = (L + LT ) s postve semdefnte. For a strongly connected and weght-balanced dgraph, zero s a smple egenvalue of L s. In ths case, we order ts egenvalues as λ = < λ λ N, and note the nequalty x T Lx λ (L s ) x N (T N x) N, () for all x R N. The followng property wll also be of use later, λ (L s )x T Lx x T L sx λ N (L s )x T Lx. (3) Ths can be seen by notng that L s s dagonalzable and rewrtng L s = S DS, where D s a dagonal matrx contanng the egenvalues of L s. 3 Problem statement We consder the mult-agent average consensus problem for a network of N agents. We let G denote the weght-balanced, strongly connected dgraph descrbng the communcaton topology of the network. Wthout loss of generalty, we use the conventon that an agent s able to receve nformaton from neghbors n N out and send nformaton to neghbors n N n. All nter-agent communcatons are assumed nstantaneous and of nfnte precson. We denote by x R the state of agent {,..., N}. We consder sngle-ntegrator dynamcs ẋ (t) = u (t), () for all {,..., N}. It s well known [Olfat-Saber and Murray, ] that the dstrbuted contnuous control law u (t) = w j (x (t) x j (t)), (5) drves each agent of the system to asymptotcally converge to the average of the agents ntal condtons. In compact form, ths can be expressed by ẋ(t) = Lx(t), where x(t) = (x (t),..., x N (t)) s the column vector of all agent states and L s the Laplacan of G. However, n order to be mplemented, ths control law requres each agent to contnuously access state nformaton about ts neghbors and contnuously update ts control law. Here, we are nterested n controller mplementatons that relax both of these requrements by havng agents decde n an opportunstc fashon when to perform these actons. Under ths framework, neghbors of a gven agent only receve state nformaton from t when ths agent decdes to broadcast ts state to them. Equpped wth ths nformaton, the neghbors update ther respectve control laws. We denote by x (t) the last broadcast state of agent {,..., N} at any gven tme t R. We assume that each agent has contnuous access to ts own state. We then utlze an event-trggered mplementaton of the controller (5) gven by u (t) = w j ( x (t) x j (t)). (6) Lettng u(t) = (u (t),..., u N (t)) R N and x = ( x,..., x N ) R N, we wrte (6) as u(t) = L x. Note that although agent has access to ts own state x (t), the controller (6) uses the last broadcast state x (t). Ths s to ensure that the average of the agents ntal states s preserved throughout the evoluton of the system. More specfcally, usng ths controller, one has d dt (T Nx(t)) = T Nẋ(t) = T NL x(t) =, (7) where we have used the fact that G s weght-balanced. Our am s to dentfy trggers that prescrbe n an opportunstc fashon when agents should broadcast ther state to ther neghbors so that the network converges to the average of the ntal agents states. Gven that the average s conserved by (6), all the trggers should enforce s that the agents states ultmately agree. Dstrbuted trgger desgn In ths secton we synthesze a dstrbuted trggerng strategy that prescrbes when agents should broadcast state nformaton and update ther control sgnals. Our desgn bulds on the analyss of the evoluton of the network dsagreement characterzed by the followng canddate Lyapunov functon, V (x) = (x x)t (x x), (8) where x = N (T N x) N corresponds to agreement at the average of the states of all agents. The next result characterzes a local condton for all agents n the network such that ths canddate Lyapunov functon s monotoncally nonncreasng. Proposton. (Evoluton of network dsagreement) For {,..., N}, let a R > and denote by 3

4 e (t) = x (t) x (t) the error between agent s last broadcast state and ts current state at tme t R. Then, V (t) N = [ ] w j ( a )( x x j ) e. a PROOF. Note that, snce the average s preserved, cf. (7), under the control law (6), x = N (T N x()) N. The functon t V (x(t)) s contnuous and pecewse contnuously dfferentable, wth ponts of dscontnuty of V correspondng to nstants of tme where an agent broadcasts ts state. Whenever defned, ths dervatve takes the form V = x T ẋ x T ẋ = x T L x x T L x = x T L x, where we have used that the graph s weght-balanced n the last equalty. Let e = (e,..., e N ) R N be the vector of errors of all agents. We can then rewrte V as Expandng ths out yelds V = N = V = x T L x + e T L x. [ ] w j( x x j ) e w j ( x x j ). Usng Young s nequalty () for each product e ( x x j ) wth ε = a > yelds, V = N = N = whch concludes the proof. [ w j ( x x j ) e a ( x x j ) ] a [ ] w j ( a )( x x j ) e, a From Proposton., a suffcent condton to ensure that the proposed canddate Lyapunov functon V s monotoncally decreasng s to mantan [ ] w j ( a )( x x j ) e, a for all {,..., N} at all tmes. Ths s accomplshed by ensurng e a ( a ) d out w j ( x x j ), for all {,..., N}. The maxmum of the functon a ( a ) n the doman (, ) s attaned at a =, so we have each agent select ths value to optmze the trgger desgn. As a consequence of the above dscusson, we have the followng result. Corollary. For each {,..., N}, let σ (, ) and defne f (e ) = e σ d out w j ( x x j ). (9) If each agent enforces the condton f (e (t)) at all tmes, then V (t) N = σ w j ( x x j ). (Note that the latter quantty s strctly negatve for all x / dag(r N ) because the graph s strongly connected). For each {,..., N}, we refer to the functon f defned n Corollary. as the trggerng functon and to the condton f (e ) = as the trgger. Note that the desgn parameter σ affects how flexble the trgger s: as the value of σ s selected closer to, the trgger s enabled less frequently at the cost of agent contrbutng less to the decrease of the Lyapunov functon. An mportant observaton s that, snce the trggerng functon f depends on the last broadcast states x, a broadcast from a neghbor of mght cause a dscontnuty n the evaluaton of f(e ), where just before the update was receved, f (e ) <, and mmedately after, f (e ) >. Such event would make agent mss the trgger. Thus, rather than prescrbng agent {,..., N} to broadcast ts state when f (e ) =, we nstead defne an event by ether f (e ) > or () f (e ) = and φ () where for convenence, we use the shorthand notaton φ = w j ( x x j ) R. We note the useful equalty N = φ = x T L x. The reasonng behnd these trggers s the followng. The nequalty () makes sure that the dscontnutes of φ do not make the agent mss an event. The trgger () makes sure that the agent s not requred to contnuously broadcast ts state to neghbors when ts last broadcast state s n agreement wth the states receved from them. The trggers () and () are a generalzaton of the ones proposed n [Garca et al., 3]. However, t s unknown

5 whether they are suffcent to exclude the possblty of Zeno behavor n the resultng executons. To address ths ssue, we prescrbe the followng addtonal trgger. Let t last be the last tme at whch agent broadcast ts nformaton to ts neghbors N n. If at some tme t t last, agent receves new nformaton from a neghbor, then mmedately broadcasts ts state f j N out t (t last, t last + ε ). () Here, ε R > s a desgn parameter selected so that σ ε < w max N out, (3) d out where w max = max w j. Our analyss n Secton 5 wll expand on the role of ths bound and the addtonal trgger n preventng the occurrence of Zeno behavor. In concluson, the trggers ()-() form the bass of the event-trggered communcaton and control law, whch s formally presented n Table. At all tmes t agent {,..., N} performs: : f f (e (t)) > or (f (e (t)) = and φ (t) ) then : broadcast state nformaton x (t) and update control sgnal 3: end f : f new nformaton x j(t) s receved from some then 5: f agent has broadcast ts state at any tme t (t ε, t) then 6: broadcast state nformaton x (t) 7: end f 8: update control sgnal 9: end f neghbor(s) j N out Table event-trggered communcaton and control law. Each tme an event s trggered by an agent, say {,..., N}, that agent broadcasts ts current state to ts out-neghbors and updates ts control sgnal, whle ts n-neghbors j N n update ther control sgnal. Ths s n contrast to other event-trggered desgns, see e.g., [Zhongxn and Zengqang,, Dmarogonas et al., ], where events only correspond to updates of control sgnals because exact nformaton s avalable to the agents at all tmes. As a fnal observaton, we note that f we drop our assumpton on nfnte precson of the transmtted messages, step would reduce to f (e (t)) >. Dgtal platforms operate wth quantzed sgnals, makng t easy to determne when a functon has crossed some threshold at the cost of ntroducng errors assocated wth fnte precson. 5 Analyss of the event-trggered communcaton and control law Here we analyze the propertes of the control law (6) n conjuncton wth the event-trggered communcaton and control law of Secton. Our frst result shows that the network executons are guaranteed not to exhbt Zeno behavor. Its proof llustrates the role played by the addtonal trgger () n facltatng the analyss to establsh ths property. Proposton 5. (No Zeno behavor) Gven the system () wth control law (6) executng the eventtrggered communcaton and control law over a weght-balanced, strongly connected dgraph, the agents wll not be requred to communcate an nfnte number of tmes n any fnte tme perod. PROOF. We are nterested n showng here that no agent wll broadcast ts state an nfnte number of tmes n any fnte tme perod. Our frst step conssts of showng that, f an agent does not receve new nformaton from neghbors, ts nter-event tmes are lower bounded by a postve constant. Assume agent {,..., N} has just broadcast ts state at tme t, and thus e (t ) =. For t t, whle no new nformaton s receved, x (t) and x j (t) reman constant. Gven that ė = ẋ, the evoluton of the error s smply e (t) = (t t )ẑ, () where, for convenence, we use the shorthand notaton ẑ = j N w out j ( x j x ). Snce we are consderng the case when no neghbors of broadcast nformaton, the trgger () s rrelevant. We are then nterested n fndng the tme t when f (e ) = occurs, trggerng a broadcast of agent s state. If ẑ =, no broadcasts wll ever happen (t = ) because e (t) = for all t t. Hence, consder the case when ẑ, whch n turn mples φ. Usng (), the trgger () prescrbes a broadcast at the tme t t satsfyng (t t ) ẑ σ d out or, equvalently, (t t ) = σ j N out d out w j ( x x j ) =, w j ( x x j ) ( ). j N w out j ( x x j ) Usng the fact that ( p k= y k) p p k= y k for any y,..., y p R and p Z > (whch readly follows from the Cauchy-Schwarz nequalty), we obtan ( ) w j ( x x j ) N out N out w max w j( x x j ) w j ( x x j ). (5) 5

6 Therefore, we can lower bound the nter-event tme by τ = t σ t w max N out >, d out (ncdentally, ths explans our choce n (3)). Our second step bulds on ths fact to show that messages cannot be sent an nfnte number of tmes between agents n a fnte tme perod. Let t be the tme at whch agent has broadcast ts nformaton to neghbors and thus e (t ) =. If no nformaton s receved by tme t + ε < t + τ, there s no problem snce ε >, so we now consder the case that at least one neghbor of broadcasts ts nformaton at some tme t (t, t +ε ). In ths case, at least one neghbor j N out has broadcast new nformaton, thus agent would also rebroadcast ts nformaton at tme t due to trgger (). Let I denote the set of all agents who have broadcast nformaton at tme t. Ths means that, as long as no agent k / I sends new nformaton to any agent n I, the agents n I wll not broadcast new nformaton for at least mn j I τ j seconds, whch ncludes the orgnal agent. As before, f no new nformaton s receved by any agent n I by tme t + mn j I ε j there s no problem, so we now consder the case that at least one agent k sends new nformaton to some agent j I at tme t (t, t + mn j I ε j ). By trgger (), ths would requre all agents n I to also broadcast ther state nformaton at tme t and agent k wll now be added to I. Reasonng repeatedly n ths way, the only way for nfnte communcatons to occur n a fnte tme perod s for an nfnte number of agents to be added to I, whch s not possble gven the fnte number of agents. Remark 5. (Condtons for Zeno) The ntroducton of the trgger () s suffcent to rule out Zeno behavor but we do not know whether t s also necessary. The desgn n [Garca et al., 3, Corollary ] has trggers of a nature smlar to ()-() for undrected graphs and guarantees that no agent undergoes an nfnte number of updates at any gven nstant, but does not dscard the possblty of an nfnte number of updates n a fnte tme perod, as Proposton 5. does. Next, we establsh global exponental convergence. Theorem 5.3 (Exponental convergence to average consensus) Gven the system () wth control law (6) executng the event-trggered communcaton and control law over a weght-balanced strongly connected dgraph, all agents exponentally converge to the average of the ntal states,.e., lm t x(t) = x. PROOF. By desgn, we know that the event-trggers ()- () ensure that, cf. Corollary., V N = σ φ. (6) We show that convergence s exponental by establshng that the evoluton of V towards s exponental. Defne σ max = max {,...,N} σ to further boundng (6) by V σ max N = φ = σ max x T L x. Gven ths nequalty, our next step s to relate the value of V (x) wth x T L x. Note that V (x) λ (L s ) xt Lx = λ (L s ) ( x e)t L( x e) = ( x T L x x T L s e + e T Le ), λ (L s ) where we have used () n the nequalty. Now, e T Le λ N (L s ) e λ N (L s ) σ max d out x T L x, mn where d out mn = mn {,...,N} d out and we have used f (e ) n the second nequalty. On the other hand, x T σmax L s e L s x e λ N (L s ) x T L x d out x T L x mn = λ N (L s ) σ max d out x T L x, mn where we have used (3) n the second nequalty. Puttng these bounds together, we obtan V (x) A x T L x, ( ). wth A = λ (L s) + λn (L s ) σmax d Usng ths expresson n the bound for the Le dervatve, we out mn get V σ max x T L x σ max A V (x(t)). Ths, together wth the fact that t V (x(t)) s contnuous and pecewse dfferentable mples, usng the Comparson Lemma, cf. [Khall, ], that V (x(t)) V (x()) exp( σmax A t) and hence the exponental convergence of the network trajectores to the average state. The Lyapunov functon used n the proof of Theorem 5.3 does not depend on the specfc network topology. Therefore, when the communcaton dgraph s tme-varyng, ths functon can be used as a common Lyapunov functon to establsh asymptotc convergence to average consensus. Ths observaton s key to establsh the next result, whose proof we omt for reasons of space. 6

7 Proposton 5. (Convergence under swtchng topologes) Let Ξ N be the set of weght-balanced dgraphs over N vertces. Denote the communcaton dgraph at tme t by G(t). Consder the system () wth control law (6) executng the event-trggered communcaton and control law over a swtchng dgraph, where t G(t) Ξ N s pecewse constant and such that there exsts an nfnte sequence of contguous, nonempty, unformly bounded tme ntervals over whch the unon of communcaton graphs s strongly connected. Then, assumng all agents are aware of who ts neghbors are at each tme and agents broadcast ther state f ther neghbors change, all agents asymptotcally converge to the average of the ntal states. 6 Perodcally checked event-trggered coordnaton Here we propose an alternatve strategy, termed perodc event-trggered communcaton and control law, where agents only evaluate trggers () and () perodcally, nstead of contnuously. Specfcally, gven a samplng perod h R >, we let {t l } l Z, where t l+ = t l + h, denote the sequence of tmes at whch agents evaluate the decson of whether to broadcast ther state to ther neghbors. Ths type of desgn s more n lne wth the constrants mposed by real-tme mplementatons, where ndvdual components work at some gven frequency, rather than contnuously. An nherent and convenent feature of ths strategy s the lack of Zeno behavor (snce nter-event tmes are naturally lower bounded by h), makng the need for the addtonal trgger () superfluous. The strategy s formally presented n Table. At tmes t {, h, h,... }, agent {,..., N} performs: : f f (e (t)) > or (f (e (t)) = and φ (t) ) then : broadcast state nformaton x (t) and update control sgnal 3: end f : f new nformaton x j(t) s receved from some neghbor(s) j N out 5: update control sgnal 6: end f then Table perodc event-trggered communcaton and control law. Each tme an agent {,..., N} broadcasts, ths resets the error to zero, e =. However, because trggers are not evaluated contnuously, we no longer have the guarantee f (e (t)) at all tmes t but, nstead, have f (e (t l )), (7) for l Z. The next result provdes a suffcent condton on h that guarantees the correctness of our desgn. Theorem 6. (Exponental convergence under perodc event-trggered communcaton and control law) Let h R > be such that σ max + hw max N out max <, (8) where w max = max {,...,N} w max and Nmax out = max {,...,N} N out. Then, gven the system () wth control law (6) executng the perodc eventtrggered communcaton and control law over a weght-balanced strongly connected dgraph, all agents exponentally converge to the average of the ntal states. PROOF. Snce (7) s only guaranteed at the samplng tmes under the perodc event-trggered communcaton and control law, we analyze what happens to the Lyapunov functon V n between them. For t [t l, t l+ ), note that e(t) = e(t l ) + (t t l )L x(t l ). Substtutng ths expresson nto V (t) = x T (t)l x(t) + e T (t)l x(t), we obtan V (t) = x T (t l )L x(t l ) + e T (t l )L x(t l ) + (t t l ) x T (t l )L T L x(t l ), for all t [t l, t l+ ). For a smpler exposton, we drop all arguments referrng to tme t l n the sequel. Followng the same lne of reasonng as n Proposton. yelds V (t) N = Usng (5), we bound x T L T L x = σ φ + (t t l ) x T L T L x. N ( = N = N out = N out max w max Hence, for t [t l, t l+ ), w j ( x x j ) ) w max w j ( x x j ) N φ. (9) = N ( σ ) V (t) + hw max N out max φ = ( + σ ) max + hw max N out max x T L x. 7

8 Under (8), a reasonng smlar to the proof of Theorem 5.3 usng (9) leads to fndng B > such that V (t) ( ) σ max + hw max N out B max V (x(t)), whch mples the result. Note that checkng the suffcent condton (8) requres knowledge of the global quanttes σ max, w max, and Nmax. out Ensurng that ths condton s met can ether be enforced a pror by the desgner or, alternatvely, the network can execute a dstrbuted ntalzaton procedure, e.g., [Lynch, 997, Ren and Beard, 8], to compute these quanttes n fnte tme. Once known, agents can compute h by nstantatng a specfc formula to select t that s guaranteed to satsfy (8). 7 Smulatons Ths secton llustrates the performance of the proposed algorthms n smulaton. Fgure shows a comparson of the event-trggered communcaton and control law wth the algorthm proposed n [Garca et al., 3] for undrected graphs over a network of 5 agents. Both algorthms operate under the dynamcs () wth control law (6), and dffer n the way events are trggered. The algorthm n [Garca et al., 3] requres all network agents to have knowledge of an a pror chosen common parameter a R >, whch we set here to a =.. Fgure (a) shows the evoluton of the Lyapunov functon V and Fgure (b) shows the number of events trggered over tme by each strategy. Fgure shows an V Tme (a) N E 5 5 Tme (b) Fg.. Plots of (a) the evoluton of the Lyapunov functon V and (b) the total number N E of events of the event-trggered communcaton and control law wth σ =.999 for all (sold blue) and the algorthm proposed n [Garca et al., 3] wth a =. (dashed black). The network conssts of 5 agents wth communcaton topology descrbed by the undrected graph ({,..., 5}, {(, ), (, 3), (, ), (, 5)}). The ntal condton s x() = [,,,, ] T. executon of event-trggered communcaton and control law over a network of 5 agents whose communcaton topology s descrbed by a weght-balanced dgraph. We do not compare t aganst the algorthm V Tme (a) N E Tme (b) Fg.. Plots of (a) the evoluton of the Lyapunov functon V and (b) the total number N E of events of the event-trggered communcaton and control law wth σ =.999 for all. The network conssts of 5 agents wth communcaton topology descrbed by the weght-balanced dgraph ({,..., 5}, {(, ), (, 3), (, ), (3, ), (, 5), (5, ), (5, )}) wth weghts (,,.5,,.5,,.5). The ntal condton s x() = [,,,, ] T. n Garca et al. [3] because the latter s only desgned to work for undrected graphs. We have also compared the perodc event-trggered communcaton and control law wth a perodc mplementaton of Laplacan consensus, cf. [Olfat- Saber et al., 7]. For the latter, trajectores are guaranteed to converge f the perodc s less than /d max, where d max s the maxmum out-degree of the graph G. Fgure 3 shows ths comparson usng h =. and also demonstrates the effect of {σ } N = on the executons of the perodc event-trggered communcaton and control law. Ths s compared aganst the standard perodc mplementatons wth perods. and.3. For smplcty, we have used σ = σ to be the same for all agents n each executon. One can observe the trade-off between communcaton and convergence rate for varyng σ: hgher σ results n less communcaton but slower convergence compared to smaller values of σ. It should be noted that, although usng a perod of.3 n the standard consensus algorthm yelds a smlar performance n terms of convergence speed and requres a comparable amount of communcaton, there s no systematc way of selectng the perod a pror, whch n general depends on the ntal condton. Instead, for each executon, the perodc event-trggered communcaton and control law naturally tunes the communcatons to occur only when necessary for convergence. 8 Conclusons We have proposed novel event-trggered communcaton and control strateges for the mult-agent average consensus problem. Among the noveltes of our frst desgn, we hghlght that t works over weght-balanced drected communcaton topologes, does not requre ndvdual agents to contnuously access nformaton about the states of ther neghbors, and does not necesstate a pror agent knowledge of global network parameters to execute the algorthm. We have shown that our al- 8

9 V Tme (a) N E Tme (b) σ =. σ =.5 σ =.8 Fg. 3. Plots of (a) the evoluton of the Lyapunov functon V and (b) the total number N E of events of the perodc event-trggered communcaton and control law (wth varyng σ =.,.5,.8 and h =., sold blue) and a standard perodc Laplacan consensus algorthm (wth perod. and.3, dashed red). Network and ntal condton are as n Fgure. gorthms exclude the possblty of Zeno behavor and dentfed condtons such that the network state exponentally converges to agreement on the ntal average of the agents state. We have also provded a lower bound on the convergence rate and characterzed the network convergence when the topology s swtchng under a weaker form of connectvty. Fnally, we have developed a perodc mplementaton of our event-trggered law that relaxes the need for agents to evaluate the relevant trggerng functons contnuously and provded a suffcent condton on the samplng perod that guarantee ts asymptotc correctness. 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