A class of event-triggered coordination algorithms for multi-agent systems on weight-balanced digraphs

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1 2018 Annual Amercan Control Conference (ACC) June 27 29, Wsconsn Center, Mlwaukee, USA A class of event-trggered coordnaton algorthms for mult-agent systems on weght-balanced dgraphs Png Xu Cameron Nowzar Zh Tan Abstract Ths paper revsts the mult-agent average consensus problem on weght-balanced drected graphs. In order to reduce communcaton among the agents, many recent works have consdered event-trggered communcaton and control as a method to reduce communcaton whle stll ensurng that the entre network converges to the desred state. One common way to do ths s to desgn events such that a specfcally chosen Lyapunov functon s monotoncally decreasng; however, dependng on the chosen Lyapunov functon the transent behavors can be very dfferent. Consequently, we are nstead nterested n consderng a class of Lyapunov functons such that each Lyapunov functon produces a dfferent event-trggered coordnaton algorthm to solve the mult-agent average consensus problem. The proposed class of algorthms all guarantee exponental convergence of the resultng network and excluson of Zeno behavor. Ths allows us to easly consder the mplementaton of dfferent algorthms that all guarantee correctness to be able to meet varyng performance needs. Smulatons are provded to llustrate our fndngs. I. INTRODUCTION The dstrbuted coordnaton problem of dynamc multagent systems has been wdely studed due to ther broad applcatons n areas such as unmanned vehcles, moble robots, and wreless communcaton networks [1] [3]. In many of these applcatons, groups of agents are requred to agree upon certan quanttes of nterest, or n other words, to acheve a consensus state; typcal results can be found n [4] [9]. When consderng mplementaton of these deas, some of these algorthms requre agents to communcate and update control sgnals contnuously or wth a fxed samplng perod [4] [6], whch are neffcent and generally result n excessve consumpton of on-board energy resources. To reduce the amount of communcatons and controller updates whle mantanng the desred performance of the network, event-trggered algorthms have recently been ganng popularty [7] [9]. The man dea behnd event-trggered algorthms s to take actons only when necessary so that some desred property of the system can stll be mantaned effcently. There are many recent works on dstrbuted event-trggered control over mult-agent systems for both undrected and drected graphs [9] [20]. Among them, [9] proposes an algorthm usng a trggerng functon whose threshold s tme-dependent wth predefned constant parameters. In general, these tme-dependent thresholds are easy to desgn to guarantee deadlocks (or Zeno behavors, meanng an nfnte number of events trggered n a fnte number of tme perod) do not occur, but requre global nformaton to guarantee The authors are wth the Department of Electrcal and Computer Engneerng, George Mason Unversty, Farfax, VA 22030, USA, {pxu3@masonlve.gmu.edu} convergence to exactly a consensus state. Instead, some event-trggered algorthms use state-dependent thresholds to determne when actons should be taken [10], [11]; however, these trggers mght be rsky to mplement as Zeno behavors are harder to exclude. A combnaton of tme-dependent and state-dependent algorthms are gven n [12], [13], ether by ntroducng a tme-dependent nternal dynamc varable or a bounded convergent functon to the state-dependent threshold to rule out Zeno behavors. As Zeno behavors are mpossble n a gven physcal mplementaton, t s necessary and essental to exclude t n the event-trggered algorthm desgn to guarantee ts correctness. The event-trggered algorthms we propose n ths paper are state-dependent and Lyapunov functon-based. More specfcally, gven a Lyapunov functon for a certan system, an event-trggered controller can generally be developed to mantan stablty of the system whle reducng samplng or communcaton, usng the gven Lyapunov functon as a certfcate of correctness. In other words, all events are trggered based on how we want the gven Lyapunov functon to evolve n tme. However, t s known that a Lyapunov functon s not unque for a gven system, and each ndvdual functon may result n a totally dfferent, but equally vald/correct trggerng law. Consequently, there are many works that propose one such algorthm based on one functon that all have the same guarantee: asymptotc convergence to a consensus state. Smulatons then show that these deas are promsng when compared aganst perodc mplementatons n reducng communcaton whle mantanng stablty, but there are no formal guarantees on the ganed effcency. Moreover, ths means there s no establshed way to compare the performance of two dfferent event-trggered algorthms that solve the same problem. In partcular, gven two dfferent event-trggered algorthms that both guarantee convergence, ther trajectores and communcaton schedules may be wldly dfferent before ultmately convergng to the desred set of states. There are some new works that are addressng exactly ths topc [21] [23], whch set the bass for ths paper. More specfcally, once establshed methods of comparng the performance of event-trggered algorthms aganst one another are developed, currently avalable algorthms wll lkely be revsted to optmze dfferent types of performance metrcs. In partcular, we notce that dfferent algorthms are better than others n dfferent scenaros when consderng metrcs such as convergence speed or total energy consumpton. Therefore, nstead of tryng to desgn only one eventtrggered algorthm that smply guarantees convergence, we desgn an entre class of event-trggered algorthms that can be easly tuned to meet varyng performance needs /$ AACC 5988

2 Statement of contrbutons: The man contrbuton of ths paper s that we propose an entre class of event-trggered coordnaton algorthms that all guarantee exponental stablty whle excludng Zeno behavors. One such algorthm that solves the exact problem we consder here s gven n [11], whch s named as Algorthm 1 for smplcty. For our work, we frst desgn a dstrbuted event-trggered algorthm based on an alternatve Lyapunov functon and name t as Algorthm 2. Usng these two algorthms, we then parameterze an entre class of Lyapunov functons and show how each ndvdual functon can be used to develop a Combned Algorthm. More specfcally, choosng any parameter λ [0, 1] yelds an event-trggered algorthm that guarantees convergence whle usng a dfferent Lyapunov functon as a certfcate for correctness. Changng λ can then help acheve varyng performance goals whle always guaranteeng stablty. Wth the asymptotc convergence and excluson of Zeno behavor for both Algorthm 1 and Algorthm 2, we establsh that the Combned Algorthm also excludes Zeno behavor and guarantees convergence of the system. Varous smulatons llustrate the correctness and performance of dfferent algorthms we propose. Organzaton: The rest of the paper s organzed as follows. Secton II ntroduces the prelmnares and Secton III formulates the problem of nterest. Secton IV frst summarzes the related work n [11] and then proposes a novel strategy based on a new Lyapunov functon. The combned algorthm that based on the combned Lyapunov functon s proposed n Secton V. Secton VI presents the smulaton results, followed by the conclusons gven n Secton VII. Notatons: R denotes the set of real numbers. 1 N R N denotes the column vector wth each components beng one and dmenson N. denotes the Eucldean norm for vectors or nduced 2-norm for matrces. II. PRELIMINARIES Let G = {V, E, W} denote a weghted dgraph of N agents wth a vertces set V = {1,..., N}, drected edges E V V and a weghted adjacency matrx W R N N 0. 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 elements n W satsfes w j > 0 f (, j) E and w j = 0 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 = w j, j N n respectvely. A dgraph s weght-balanced f D out = D n and the weghed Laplacan matrx s L = D out W. Young s nequalty [24] states that gven x, y R, for any ε R >0, xy x2 2ε + εy2 2, (1) whch shall be used n the theoretcal analyss of ths paper. For a strongly connected and weght-balanced dgraph, zero s a smple egenvalue of L, therefore, we order ts egenvalues as λ 1 = 0 < λ 2... λ N. The followng property wll also be used: λ 2 (L)x T L T x x T L T Lx λ N (L)x T L T x. (2) III. PROBLEM STATEMENT Consder the mult-agent average consensus problem for a network of N agents over a weght-balanced and strongly connected dgraph. Let G denote the communcaton topology of ths network. Wthout loss of generalty, we say that an agent s able to receve nformaton from neghbors n N out and send nformaton to neghbors n N n. Assume that all nter-agent communcatons are nstantaneous and of nfnte precson. Let x denote the state of agent {1, 2,..., N} and consder the sngle-ntegrator dynamcs ẋ (t) = u (t). (3) As s well known, the dstrbuted contnuous control law u (t) = w j (x (t) x j (t)) (4) drves the states of all agents n the system asymptotcally converge to the average of the ntal condtons [25]. However, mplementng ths protocol requres all agents to contnuously access ther neghbors state nformaton and keep updatng ther own control sgnals, whch s unrealstc n practce n terms of both communcaton and control. Therefore, here we consder the stuaton where neghbors of a gven agent receve nformaton from t only when ths agent decdes to broadcast, and wth the nformaton receved, neghbors can update ther states accordngly. Let ˆx (t) denote the last broadcast state of agent {1,..., N} at tme t R 0 and assume that all agents have contnuous access to ther own states, then the dstrbuted event-trggered control law (4) s modfed nto [7] u (t) = w j (ˆx (t) ˆx j (t)). (5) Wth the above controller (5), each agent s equpped wth a trggerng functon f ( ) that takes values n R and depends on local nformaton only,.e., on the true state x (t) and the broadcast state ˆx (t). An event for agent s trggered as soon as the trggerng condton f (t, x (t), ˆx (t)) > 0 (6) s fulflled. The trggered event drves agent to broadcast ts state so that the neghbors of agent can update ther states. Therefore our purpose of ths paper s to dentfy event-based trggers that work effcently under the Lyapunov functonbased trggerng law wth state-dependent thresholds. A. Related work IV. DISTRIBUTED TRIGGER DESIGN The dstrbuted event-trggered coordnaton problem for mult-agent systems over weght-balanced dgraphs has been studed n [11]. As we study the same problem and ther 5989

3 results are essental for us to develop our algorthms, we summarze ther results frst and name ther algorthm as Algorthm 1. The event-trggered law proposed n [11] s Lyapunov functon-based, wth canddate Lyapunov functon be V 1 (x(t)) = 1 2 (x(t) x)t (x(t) x), (7) where x(t) = (x 1 (t),..., x N (t)) T, x = 1 N (1T N x(0))1 N s the agreement at the average of the states of all agents. The dervatve of V 1 (x(t)) s upper bounded by V 1 (x(t)) 1 2 N =1 [ w j (1 a )(ˆx (t) ˆx j (t)) 2 e2 (t) ], a (8) where a are arbtrary postve constants and e (t) = ˆx (t) x (t) s the measurement error between agent s last broadcast state and ts current state at tme t. The condton to ensure that the canddate Lyapunov functon V 1 (x(t)) s monotoncally decreasng s to mantan [ w j (1 a )(ˆx (t) ˆx j (t)) 2 e2 (t) ] 0, a for all agents {1,..., N} at all tmes, whch can be accomplshed by ensurng e 2 (t) a (1 a ) d out w j (ˆx (t) ˆx j (t)) 2. (9) As the trgger desgn s optmal when a = 0.5 for all agents {1,..., N} [11], ther trggerng functon s defned as f (e (t)) = e 2 (t) w j (ˆx (t) ˆx j (t)) 2, (10) where (0, 1) s a desgn parameter that affects how flexble the trgger s. Accordng to the trggerng functon, the event s trggered when f (e (t)) > 0 or when f (e (t)) = 0 and φ = w j (ˆx (t) ˆx j (t)) 2 0. Bascally, the trgger above makes sure that V1 (x(t)) s always negatve as long as the system has not converged, therefore, Algorthm 1 guarantees all agents to converge to the average of the ntal states,.e. lm t x(t) = x = 1 N (1T N x(0))1 N, nterested readers can refer to [11, Theorem 5.3] for more detals. B. Proposed new algorthm As we know, the Lyapunov functon s not unque for the stablty studyng of the same system and each ndvdual functon may result a totally dfferent trggerng law. Therefore, we propose a novel trggerng strategy based on an alternatve Lyapunov functon V 2 (x(t)) = 1 2 x(t)t L T x(t), (11) and name our algorthm as Algorthm 2. Proposton 4.1: For {1,..., N}, let b, c, c j > 0 for all, j {1,..., N} and e (t) = ˆx (t) x (t), then the 5990 dervatve of V 2 (x(t)) s upper bounded by where V 2 (x(t)) N =1 ( ( d δ u 2 out (t) 2b δ 1 dout b 2 ) ) + dout e 2 (t), 2c (12) w j c j, (13) 2 and u (t) s what defned n (5). The proof to Proposton 4.1 s omtted due to space lmt. Note that the coeffcent of e 2 s always postve. To ensure the coeffcent of u 2 s also postve, we requre b, c j < 1. d out From Proposton 4.1, a suffcent condton to guarantee that the proposed canddate Lyapunov functon V 2 (x(t)) s monotoncally decreasng s to ensure that δ ( ) 2 ( d out w j (ˆx (t) ˆx j (t)) 2b ) + dout e 2 (t) 0 2c for all agents {1,..., N} at all tmes, meanng that e 2 2δ b c ( 2. (t) (b + c )d out w j (ˆx (t) ˆx j (t))) (14) The trggerng functon for Algorthm 2 s therefore defned as f (e (t)) = e 2 (t) 2 δ b c ( 2, (b + c )d out w j (ˆx (t) ˆx j (t))) (15) where as before s a desgn parameter that affects how flexble the trgger s and controls the trade-off between communcaton and performance. Settng close to 0 s generally greedy, meanng that the trgger s enabled more frequently and the network requres more communcatons, whch makes agent contrbute more to the decrease of the Lyapunov functon V 2 (x(t)) and therefore the network converges faster whle settng the value of close to 1 acheves the opposte results. Corollary 4.2: For each agent {1,..., N} wth the trggerng functon defned n (15), f each agent enforces the condton f (e ) 0 at all tmes, then V 2 (x(t)) N (1 )δ ( =1 w j (ˆx (t) ˆx j (t))) 2. Smlar as [11], to avod the possblty that agent may mss any trggers, we defne an event ether by f (e (t)) > 0 or (16) f (e (t)) = 0 and φ 0 (17) where φ = ( j N w out j (ˆx (t) ˆx j (t))) 2. We also prescrbe the followng addtonal trgger as n [11] to address the non-zeno behavor. Let t last be the last tme at whch agent broadcasts ts nformaton to ts neghbors. If at some tme t t last, agent receves nformaton from a neghbor j N out, then agent mmedately

4 Algorthm 1 f (e ) e 2 j N w out j (ˆx ˆx j ) 2, ε < f (e ) e 2 w max N out 2δbc (b +c )d out ε < 2δ b c (b +c )d out Algorthm 2 ( TABLE I ) 2 w j (ˆx ˆx j ) DIFFERENCE BETWEEN ALGORITHM 1 AND ALGORITHM 2 broadcasts ts state f where t (t last, t last + ε ), (18) 2 δ b c ε < (b + c )d out (19) s a desgn parameter selected to ensure the excluson of Zeno. The reasonng s smlar as that n [11]. We summarze the dfferences between Algorthm 1 proposed n [11] and Algorthm 2 proposed here n Table I. Once the trggerng functon and parameters ɛ are chosen for each agent, ether algorthm can be mplemented usng the coordnaton algorthm provded n Table II. At all tmes t agent {1,..., N} performs: 1: f f (e (t)) > 0 or (f (e (t)) = 0 and φ 0) then 2: broadcast state nformaton x (t) and update control sgnal u (t) 3: end f 4: f new nformaton x j (t) s receved from some neghbor(s) j N out 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 u (t) 9: end f TABLE II EVENT-TRIGGERED COORDINATION ALGORITHM. For Algorthm 2, we have the followng proposton and theorem specfy ts non-zeno behavor and convergence. Proof s omtted due to space lmt. Proposton 4.3: (No Zeno Behavor) Consder the system (3) executng control law (5) wth the trggerng functon gven by (15). For the weght-balanced, strongly connected dgraph wth any ntal condtons, when executng Table II, the system wll not exhbt Zeno behavor. Theorem 4.4: (Asymptotc Convergence to Average Consensus). Gven the system (3) executng Table II over a weght-balanced, strongly connected dgraph, all agents asymptotcally converge to the average of the ntal states,.e. lm t x(t) = x, where x = 1 N (1T N x(0))1 N. V. A CLASS OF EVENT-TRIGGERED ALGORITHMS As stated n Secton I, gven a system and a Lyapunov functon, there are many works studyng event-trggered control to reach the goal of mantanng the stablty of the system whle ncreasng the effcency of the system. However, there s very lttle work currently avalable that mathematcally quantfes these benefts. Recently, some works began establshng results along ths lne [21] [23], however, ths area s stll n ts nfancy. In partcular, there are not yet establshed ways to compare the performance of an event-trggered algorthm wth another. Consequently, many dfferent algorthms can be proposed to ultmately solve the same problem, whle each algorthm s slghtly dfferent and can produce dfferent trajectores. Specfcally n our case, Algorthm 1 and Algorthm 2 solve the same problem, but what we can say about the two algorthms s only that they both exclude Zeno behavor and ensure asymptotcal convergence of the network. However, we have found that dependng on the ntal condton and network topology, each algorthm may out-perform the other n terms of dfferent evaluaton metrcs. In any case, once these performance metrcs become better researched, there wll lkely be more standard ways to mathematcally compare the two dfferent algorthms. Therefore, for now, nstead of desgnng only one event-trggered algorthm for the system that only works better n one stuaton, we am to desgn an entre class of algorthms that can easly be tuned to meet varyng performance needs. We do ths by parameterzng a set of Lyapunov functons rather than studyng only a specfc one. To the best of our knowledge, ths paper s then a frst study of how to desgn an entre class of algorthms that use dfferent Lyapunov functons to guarantee correctness, wth the ntenton of beng able to use the best one at all tmes. More specfcally, gven any λ [0, 1], we defne a combned canddate Lyapunov functon as V λ (x(t)) = λv 1 (x(t)) + (1 λ)v 2 (x(t)). (20) Accordngly, the dervatve of V λ (x(t)) takes the form V λ (x(t)) = λ V 1 (x(t)) + (1 λ) V 2 (x(t)). (21) Followng the steps of dervng the trggerng functons n Secton IV, the trggerng functon developed based on (20) s gven by f (e (t)) = e 2 (t) [ + (1 λ)2δ b c (b + c )d out λ ( ) 2 w j (ˆx (t) ˆx j (t) ) 2 ] w j (ˆx (t) ˆx j (t)) (22) We refer to ths as the Combned Algorthm that s parameterzed by λ [0, 1]. Smlarly, we set a parameter ε 5991

5 such that ε < λ w max N out + 2(1 λ) δ b c (b + c )d out. Then, wth ths trggerng functon (22) and ε, the Combned Algorthm can also be mplemented usng Table I. Note that λ = 0 n the Combned Algorthm recovers Algorthm 2 and λ = 1 recovers Algorthm 1. Corollary 5.1: Algorthm 1 and Algorthm 2 both ensure all agents to asymptotcally converge to the average of ther ntal states through provng that ther Lyapunov functons converge asymptotcally. Therefore, as a lnear combnaton of V 1 (x(t)) and V 2 (x(t)), V λ (x(t)) also converges exponentally, whch means that a network executng the Combned Algorthm shall converge asymptotcally to the average of ts ntal state. VI. SIMULATIONS We demonstrates the performance of the proposed algorthms through several smulatons. In partcular, we show how ether Algorthm 1 or Algorthm 2 could be argued to be better gven dfferent ntal condtons and network topologes, whch has set the bass for our ntroducton of the Combned Algorthm to easly go between the two. In all smulatons we consder a system of N = 5 agents wth dynamcs (3) and control law (5). The trggerng functons for Algorthm 1, Algorthm 2 and Combned Algorthm are defned n (10), (15) and (22), respectvely. Throughout the smulatons, we set b = c = 0.5 for all agents {1,..., N}. We mplement the λ = 0.5 verson of the Combned Algorthm. We adopt two dfferent networks wth dfferent ntal condtons for comparson. The ntal state of Network 1 s x 1 (0) = [1, 1, 0, 2, 0] T and ts weghted adjacency matrx s 1/4 1/4 0 1/3 1/ /2 1/6 1/3 W 1 = 1/2 1/3 1/ /4 1/6 0 1/3 1/4. 0 1/4 1/3 1/6 1/4 The ntal state of Network 2 s x 2 (0) = [0, 1, 1, 1, 1] T, wth an weghted adjacency matrx W 2 whose dagonal elements are 0 and the rest of the entres are 1/4. Fgure 1 shows the evolutons of the three Lyapunov functons for the two networks wth = 0.9 for all agents, whch corroborates our analyss that the proposed Algorthm 2 and Combned Algorthm ensure convergence for the resultng systems. In addton, Fgure 1(a) shows that Network 1 converges fastest when executng Algorthm 2 whle Fgure 1(b) shows that Network 2 converges fastest when executng Algorthm 1. A more drect comparson s gven n Fgure 2, on whch we have T con denote the tme needed for the two networks to reach a 99% convergence of the Lyapunov functon when executng all algorthms wth respect to varyng. It s clear that for Algorthm 1 and Algorthm 2, there exst stuatons when one outperforms the other n terms of convergence tme. In addton to convergence tme, other mportant metrcs may nclude power consumpton and total energy expendture. To compare these we use a smulaton step sze of h = second and we adopt the followng power calculaton model n unts of dbmw [26] : N P = 10 log 10, =1 j {1,...,N},j β10 0.1P j+α x xj where α > 0 and β > 0 depend on the characterstcs of the wreless medum and P j s the power of the sgnal transmtted from agent to agent j n unts of dbmw. Smlar as [27], we set α, β and P j to be 1. The total energy needed can be calculated by multplyng the power n unts of mllwatt (mw) wth the number of steps for convergence, whch s T con /h, whch n decbels (db) s E = P + 10 log 10 T cov h. Fgures 3(a) and 3(b) compare the average power consumpton for each algorthm and Fgures 4(a) and 4(b) show the total communcaton energy requred to reach a 99% consensus state. These fgures show that n Network 1, Algorthm 2 can always reach consensus usng less total communcaton energy for varyng. On the other hand, n Network 2, Algorthm 1 can complete the same task usng less total communcaton energy. Therefore, dependng on dfferent network topologes and ntal condtons and dependng on what performance metrcs are most mportant for the applcaton at hand, t may be desrable to mplement dfferent types of event-trggered algorthms. Note that the Combned Algorthm can easly be tuned to approach ether Algorthm 1 or Algorthm 2 or anythng n between to meet varyng system needs by settng values for λ. Ths also motvates our future work of adaptng λ onlne to further mprove performance. 1 V 0.5 Algorthm1 (IV.A) Algorthm2 (IV.B) Combned Algorthm (V) t Fg V Algorthm1 (IV.A) Algorthm2 (IV.B) Combned Algorthm (V) t The evoluton of the Lyapunov functons. VII. CONCLUSION Ths paper frst proposes a novel dstrbuted eventtrggered communcaton and control law based on a new Lyapunov functon that acheves consensus and excludes the possblty of Zeno behavor for mult-agent systems on weght-balanced dgraphs. We then show how the algorthm desgn can easly be extended by consderng a class of Lyapunov functons parameterzed by λ [0, 1] such that each λ defnes a new Lyapunov functon coupled wth a new eventtrggered coordnaton algorthm whch uses that partcular 5992

6 T con Algorthm1 (IV.A) Algorthm2 (IV.B) Combned Algorthm (V) T con 0.7 Algorthm1 (IV.A) Algorthm2 (IV.B) Combned Algorthm (V) 0.6 E Algorthm1 (IV.A) Algorthm2 (IV.B) Combned Algorthm (V) E Algorthm1 (IV.A) Algorthm2 (IV.B) Combned Algorthm (V) Fg. 2. Tme needed to reach 99% convergence. Fg. 4. Total energy expendture P Algorthm1 (IV.A) Algorthm2 (IV.B) Combned Algorthm (V) 59.5 Fg P Algorthm1 (IV.A) Algorthm2 (IV.B) Combned Algorthm (V) 56.8 Average communcaton power consumpton. functon to guarantee correctness. Although any λ [0, 1] produces an algorthm that guarantees Zeno-free asymptotc convergence to the desred state, the trajectores (or performance) can be very dfferent. Consequently, ths gves us an easy way to consder many event-trggered algorthms that all have the mnmum requrement of guaranteed asymptotc stablty. Future work wll be devoted to studyng how to adapt λ onlne to take full advantage of ths class of algorthms to meet varyng performance needs. REFERENCES [1] J. N. 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