Coordination of multi-agent systems via asynchronous cloud communication

Transcription

1 2016 IEEE 55h Conference on Decson and Conrol (CDC) ARIA Resor & Casno December 12-14, 2016, Las Vegas, USA Coordnaon of mul-agen sysems va asynchronous cloud communcaon Sean L. Bowman Cameron Nowzar George J. Pappas Absrac In hs work we sudy a mul-agen coordnaon problem n whch agens are only able o communcae wh each oher nermenly hrough a cloud server. To reduce he amoun of requred communcaon, we develop a self-rggered algorhm ha allows agens o communcae wh he cloud only when necessary raher han a some fxed perod. Unlke he vas majory of smlar works ha propose dsrbued evenand/or self-rggered conrol laws, hs work doesn assume agens can be lsenng connuously. In oher words, when an even s rggered by one agen, neghborng agens wll no be aware of hs unl he nex me hey esablsh communcaon wh he cloud hemselves. Usng a noon of promses abou fuure conrol npus, agens are able o keep rack of hgher qualy esmaes abou her neghbors allowng hem o say dsconneced from he cloud for longer perods of me whle sll guaraneeng a posve conrbuon o he ask. We show ha our self-rggered coordnaon algorhm guaranees ha he sysem asympocally reaches he desred sae. Smulaons llusrae our resuls. I. INTRODUCTION Ths paper consders a mul-agen coordnaon problem where agens can only communcae wh one anoher ndrecly hrough he use of a cenral base saon or cloud. Specfcally, we consder he problem of coordnang a number of submarnes ha only can communcae wh a base saon whle a he surface of he waer. Whle a majory of relaed works allow for an agen o push nformaon o s neghbors a any desred me, communcang wh he ousde world when underwaer s exremely expensve, f no mpossble [1], [2], and so a submarne mus perform all communcaon whle surfaced. Each me a submarne surfaces, mus deermne he nex me o surface and he conrol law o use whle underwaer n order o acheve some desred global ask based only on nformaon avalable on he server a ha momen. In hs paper we are neresed n desgnng a self-rggered coordnaon algorhm n whch agens auonomously schedule he nex me o communcae wh he cloud based on currenly avalable nformaon. Whle we movae our problem va an underwaer coordnaon problem n whch communcaon whle submerged s mpossble, s also drecly applcable o scenaros where wreless-capable agens canno be lsenng o any communcaon channels connuously. Leraure revew: In he conex of he mul-agen coordnaon problem n general, he leraure s exensve [3], [4], [5]. In our specfc problem of mul-agen consensus, Olfa-Saber and Murray [6] nroduce a connuous-me law ha guaranees consensus convergence on undreced as well as wegh-balanced dgraphs. However, he majory of hese Sean L. Bowman s wh he Compuer and Informaon Scence deparmen, and Cameron Nowzar and George J. Pappas are wh he Elecrcal and Sysems Engneerng Deparmen, Unversy of Pennsylvana, Phladelpha, {seanbow,cnowzar,pappasg}@seas.upenn.edu. works assume agens can connuously, or a leas perodcally, oban nformaon abou her neghbors. A useful ool for deermnng dscree communcaon mes n hs manner s even-rggered conrol, where an algorhm unes conroller execuons o he sae evoluon of a gven sysem, see e.g., [7], [8]. In parcular, even-rggered conrol has been successfully appled o mul-agen sysems o reduce overall communcaon, sensng, and/or acuaon effor of he agens. In [9], he auhors formulae a hreshold on sysem error o deermne when conrol sgnals need o be updaed. Even-rggered deas have also been appled o he acquson of nformaon raher han conrol. Several approaches [10], [11], [12] ulze perodcally sampled daa o reevaluae he conroller rgger. Even-rggered approaches generally requre he perssen monorng of some rggerng funcon as new nformaon s beng obaned. Unforunaely, hs s no drecly applcable o our seup because he submarnes only ge new nformaon when hey surface. Insead, self-rggered conrol [13], [14], [15] removes he need o connuously monor he rggerng funcon, nsead requrng each agen o compue s nex rgger me based solely on he nformaon avalable a he prevously rggered sample me. The frs o apply hese deas o consensus, Dmarogonas e al. [16], remove he need for connuous conrol by nroducng an even-rggered rule o deermne when an agen should updae s conrol sgnal, however sll requrng connuous nformaon abou her neghbors. In [17], he auhors furher remove he need for connuous neghbor sae nformaon, creang a me-dependen rggerng funcon o deermne when o broadcas nformaon. The auhors n [18] smlarly broadcas based on a sae-dependen rggerng funcon. Recenly, hese deas have been exended from undreced graphs o arbrary dreced ones [12], [19], [20]. A major drawback of all aforemenoned works s ha hey requre all agens o be lsenng, or avalable o receve nformaon, a all mes. Specfcally, when any agen decdes o broadcas nformaon o s neghbors, s assumed ha all neghborng agens n he communcaon graph are able o nsananeously receve ha nformaon. Insead, we are neresed n a suaon where n beween surfacngs when dsconneced from he cloud an agen has zero ably o communcae wh oher agens. In [21], he auhors sudy a very smlar problem o he one we consder here bu develop an even-rggered soluon n whch all Auonomous Underwaer Vehcles (AUVs) mus surface ogeher a he same me. Ths problem has very recenly been looked a n [22], [23] where he auhors ulze even- and self-rggered coordnaon sraeges o deermne when he AUVs should resurface. In [22], a medependen rggerng rule β(σ 0, σ 1, λ 0, ) s developed ha /16/$ IEEE 2215

2 ensures praccal convergence (n he presence of nose) of he whole sysem o he desred confguraon. Insead, he auhors n [23] develop a sae-dependen rggerng rule wh no explc dependence on me; however, he self-rggered algorhm developed here s no guaraneed o avod Zeno behavors whch makes an ncomplee soluon o he problem. In hs work we ncorporae deas of promses from eam-rggered conrol [24], [25] o develop a sae-dependen rggerng rule ha guaranees asympoc convergence o consensus whle ensurng ha Zeno behavor s avoded. Saemen of conrbuons: Our man conrbuon s he developmen of a novel dsrbued eam-rggered algorhm ha combnes deas from self-rggered conrol wh a noon of promses. These promses allow agens o make beer decsons snce hey have hgher qualy nformaon abou her neghbors n general. Our algorhm ncorporaes hese promses no he sae-dependen rgger o deermne when hey should communcae wh he cloud. In conras o [22], [23], our algorhm uses a sae-dependen rggerng rule wh no explc dependence on me, no global parameers, and no possbly of Zeno behavor. In general, dsrbued even- and self-rggered algorhms are desgned so ha agens are never conrbung negavely o he global ask, generally defned by he evoluon of a Lyapunov funcon V. Insead, we acually allow an agen o be conrbung negavely o he global ask emporarly as long as s accouned for by s ne conrbuon over me. Our algorhm guaranees he sysem converges asympocally o consensus whle ensurng ha Zeno execuons canno occur. Fnally, we llusrae our resuls hrough smulaons. II. PROBLEM STATEMENT We consder an N agen sysem wh sngle-negraor dynamcs ẋ () = u () for all {1,..., N}, where we are neresed n reachng a consensus confguraon,.e. where x () x j () 0 as for all, j {1,..., N}. For smplcy, we consder scalar saes x R, bu all resuls are exendable o arbrary dmensons. Gven a conneced communcaon graph G, s well known [6] ha he dsrbued connuous conrol law u () = (x () x j ()) (1) drves each agen of he sysem o asympocally converge o he average of he agens nal condons. However, n order o be mplemened, hs conrol law requres each agen o connuously have nformaon abou s neghbors and connuously updae s conrol law. Several recen works have been amed a relaxng hese requremens [12], [19], [20], [17]. However, hey all requre agens o be lsenng connuously o her neghbors,.e. when an even s rggered by one agen, s neghbors are mmedaely aware and can ake acon accordngly. Unforunaely, as we assume here ha agens are unable o perform any communcaon whle submerged, we canno connuously deec neghborng evens. Insead, we assume ha agens are only able o updae her conrol sgnals when her own evens are rggered (.e., when hey are surfaced). las Las me agen surfaced expre Conrol expraon me of agen nex Nex me agen wll surface x ( las ) Las updaed poson of agen u ( las ) Las rajecory of agen M ( las ) Mos recen conrol bound promse from agen TABLE I DATA STORED ON THE CLOUD FOR ALL AGENTS AT ANY TIME. Le { l } l Z 0 be he sequence of mes a whch agen surfaces. Then, our algorhm s based on a pecewse consan mplemenaon of he conroller (1) gven by u () = (x ( l ) x j ( l )), [ l, l+1 ). (2) Remark II.1 Laer we wll allow he conrol npu u () o change n a lmed way whle agen s submerged, bu for now we assume he conrol s pecewse consan on he nervals [ l, l+1 ). Movaon for and deals behnd changng he conrol whle submerged are dscussed n secon III-B. The purpose of hs paper s o develop a self-rggered algorhm ha deermnes how he he sequence of mes { l } and conrol npus u () can be chosen such ha he sysem converges o he desred consensus saemen. More specfcally, each agen a each surfacng me l mus deermne he nex surfacng me l+1 and conrol u () only usng nformaon avalable on he cloud a ha nsan. The closed loop sysem should hen have rajecores such ha x () x j () 0 as for all, j {1,..., N}. We descrbe he cloud communcaon model nex. A. Cloud communcaon model We assume ha here exss a base saon or cloud ha agens are able o upload daa o and download daa from when hey surface. A any me [ l, l+1 ), he cloud sores he followng nformaon abou agen : he las me las () = l ha agen surfaced, he nex me nex () = l+1 ha agen s scheduled o surface, he sae x ( las ) of agen when las surfaced, and he las conrol sgnal u ( las ) used by agen. The server also conans a conrol expraon me expre nex and a promse M for each agen whch wll be explaned n Secon III where we develop he self-rggered coordnaon algorhm. Ths nformaon s summarzed n Table I. For smplcy, we assume ha agens can download/upload nformaon o/from he cloud nsananeously. Whle he lnk s open, agen downloads all he nformaon n Table I for each neghbor j N. Agen hen (nsananeously) compues s conrol sgnal u ( l ) and nex surfacng me l+1 such ha knows wll make a ne posve conrbuon o he consensus over he nerval [ l, l+1 ). Fnally, before closng he communcaon lnk and dvng, agen calculaes a promse M boundng s fuure conrol npus and uploads all daa o he server. The goal of hs paper s o desgn a self-rggered algorhm pckng 2216

3 hese conrol npus and mes such ha as, x j () x () 0 (3) for all agens, j {1,..., N}. In he nex secon we descrbe hs algorhm n deal. III. DISTRIBUTED TRIGGER DESIGN Consder he objecve funcon V (x()) = 1 2 xt ()Lx(), (4) where L s he Laplacan of he conneced communcaon graph G. Noe ha V (x) 0 for all x R N and V (x) = 0 f and only f x = x j for all, j {1,..., N}. Thus, he funcon V (x) encodes he objecve of he problem and we are neresed n drvng V (x) 0. For smplcy, we drop he explc dependence on me when referrng o me. Takng he dervave of V wh respec o me, we have N V = ẋ T Lx = ẋ (x j x ). =1 Le us spl up V = N =1 V, where V ẋ (x j x ). (5) Noe ha we have essenally dsrbued V n a way ha clearly shows how each agen s moon conrbues o he global objecve, allowng us o wre N V (x()) = V (x(0)) + V (x(τ))dτ. =1 Ideally, we now wsh o desgn a self rggered algorhm such ha V (x()) 0 for all agens a all mes. Thus a surfacng me l, agen mus deermne l+1 and u () such ha V () 0 for all [ l, l+1 ). Whle n he full algorhm, we allow an agen o modfy s conrol by seng o 0 whle sll submerged, for clary n hs subsecon we assume he conrol s consan on he enre submerged nerval and gnore he conrol expraon me expre ; s movaon and (mnor) modfcaons o he algorhm wll be descrbed n secon III-B. Noe ha gven he nformaon agen downloaded from he server a me l, s able o exacly compue he sae of a neghborng agen j N up o he me resurfaces. For any l < nex nex j j, x j () = x j ( las j 0 ) + u j ( las j )( las j ). (6) Le T = mn j N nex j be he frs me a whch any neghborng agen j s scheduled o surface. Thus, agen can calculae he poson of s neghbors exacly unl me T. For mes wh l < T, we can hen wre he local objecve funcon conrbuon as V = u( l ) [ (xj ( l ) x ( l )) + ( l )(u j ( las j ) u ( l )) ] (7). Le be he smalles me l such ha V 0, assumng all neghbor agens connue o use a consan conrol, s no sasfed (hs me s easly compuable wh known nformaon from (7)). If T, hen agen smply ses nex = l+1 =, and by connuy of V s guaraneed ha V () 0 for all [ l, l+1 ). If > T, however, can no longer be guaraneed ha V () 0 for > T. For any > T, we can wre x j () = x j (T ) + and also, for l+1, T ẋ j (τ)dτ, (8) x () = x (T ) + u ( las )( T ). (9) We can hen wre V () as [ V () = u ( l ) (x j (T ) x (T )) + ] ẋ j (τ)dτ u ( l )( T ). T (10) Le us frs consder he case where u ( l ) < 0. We see ha V 0 ff he brackeed quany n (10) s 0. Whle agen s submerged, however, agen j s auonomously updang s conrol sgnal; hus, agen does no have access o ẋ j (), makng hard o deermne how o move whou surfacng. To remedy hs, we borrow an dea of promses from eamrggered conrol [25]. Suppose ha alhough we don know ẋ j () exacly for > T, we know some bound M j () 0 such ha ẋ j () M j (). Then, ẋ j (τ)dτ M j ()dτ = M j ()( T ). (11) T T Usng hs bound, we ge he followng suffcen condon for V 0: ( T ) ( u ( l ) M j () ) (x j (T ) x (T )). If u ( l ) > 0, we smlarly ge ( T ) ( u ( l ) + M j () ) (12) (x j (T ) x (T )). (13) Leng T be he smalles T T such ha (12) (or (13)) s no longer sasfed,.e. he smalles me such ha we can no longer guaranee ha V 0, we know ha V 0 on all of [ l, T ] by connuy of V. I s addonally possble o allow agen o reman submerged for longer by allowng V o emporarly become posve as long as we selec l+1 such ha he oal conrbuon o he objecve funcon on he nerval [ l, l+1 V l l+1 l ), V (τ)dτ, (14) s nonposve. Le B be he conrbuon from agen o he objecve 2217

4 funcon beween l and T,.e. B = T l V (τ)dτ. (15) Noe here we are specfcally neresed n he case when > T, and so B s guaraneed o be negave. Smlarly, le C () be he conrbuon from T o,.e. C () = T V (τ)dτ. (16) Usng he bounds M j (), s possble o bound C (): where and Le T oal C () C () = β 2 ( T ) 2 + γ ( T ) (17) β = u ( las ) M j () + u ( las ) 2 (18) γ = u ( las ) (x j (T ) x (T )). (19) be he smalles me T oal T such ha B + C (T oal ) < 0 (20) s no longer sasfed,.e., he frs me a whch agen can no longer guaranee s sll makng a posve conrbuon o he global objecve snce he las me surfaced. Seng l+1 = T oal hus ensures ha he oal conrbuon from agen o he global objecve funcon, V l = B +C ( l+1 ), s nonposve. Selecng l+1 = T ensures ha V < 0 over he submerged nerval, ensurng ha agen s makng progress owards he global objecve a all. Selecng l+1 = T oal s a rade-off; whle hs me allows he agen o reman submerged for longer, as allows some posve conrbuon o he objecve funcon, overall progress s slower. Thus, we propose a unng parameer σ [0, 1], selecng a me l+1 such ha T l+1 T oal : l+1 = (1 σ )T + σ T oal. (21) By connuy of V and he defnons of T, T oal, s guaraneed ha T l+1 T oal and ha we sll have B + C ( l+1 ) 0. Seng all σ near 0 allows faser convergence wh more frequen surfacng, whle σ near 1 resuls n slower convergence bu less frequen surfacng. A. Selecng promses M j As sn possble n general for agen o bound a neghbor agen j s fuure conrol npus from pas sae and conrol nformaon, nsead each agen makes a promse M abou s fuure conrol npus each me connecs o he server. Le M l be he promse made by agen a me l. From equaons (12), (13), (20), s clear ha he smaller M j s for any j N, he longer agen s able o say submerged. However, lmng he conrol oo much below he deal conrol (2) wll slow convergence. As a balance, a me l agen ses s promse o be exacly he magnude of s deal conrol npu: M l = u ( l ). Noe however ha hs does no mean agen can use s deal conrol law a all mes and sll abde by prevous promses has made; f he new desred npu and promse s greaer n magnude han s prevous promse, o reman ruhful o prevous promses agen mus wa unl he new promse has been receved by all of s neghbors when hey surface before can use s desred conrol npu. Le τj l be he me ha agen j sees agen s lh promse,.e. τj l = nex j ( l ). When submergng for an nerval [l, l+1 ), agen needs o guaranee ha all promses M currenly beleved by j N are abded by. Le p las j () be he ndex of he mos recen promse by agen ha agen j s aware of a me,.e. p las j () = arg max l : τ l j τ l j, (22) and le P l be he se of promse ndces ha agen mus abde by when submergng on [ l, l+1 ),.e. P l = { p las j () j N, [ l, l+1 ) }. (23) To abde by all promses ha agen s neghbors beleve abou s conrols, hen, smply needs o bound s conrol npu magnude by u max ( l ) = mn M k k P l. (24) Wh hs maxmum, he acual conrol law used and uploaded by agen on he nerval [ l, l+1 ) s gven by boundng he deal conrol magnude by u max ( l ), or u u ( l ) = (l ) u ( l ) u max ( l ), u max ( l ) u (l ) (25) u ) oherwse. (l B. Avodng Zeno behavor Whle he presened mehod of selecng surfacng mes guaranees convergence, s suscepble o Zeno behavor,.e. requrng some agen o surface an nfne number of mes n a fne me perod. To avod hs behavor, we nroduce a fxed dwell me T dwell > 0, and force each agen o reman submerged for a leas a duraon of T dwell. Unforunaely, hs means ha n general, here may be mes a whch an agen s forced o reman submerged even when does no know how o move o conrbue posvely o he global ask (or may no even be possble f s a a local mnmum). Remarkably, from he way have have dsrbued V usng (5), f agen ses u () = 0 s nsananeous conrbuon o he global objecve s exacly 0. Thus, we allow an agen s conrol o change whle s submerged and modfy he conrol law descrbed n he prevous secon as follows. If he chosen deal surfacng me deal = (1 σ )T + σ T oal s greaer han or equal o l + T dwell, hen nohng changes; agen ses s nex surfacng me l+1 = deal and uses he conrol law (25) on he enre submerged nerval. If, on he oher hand, deal < l + T dwell, we le agen use he usual conrol law unl deal, unl whch knows can make a posve conrbuon. Afer deal, we force agen o reman sll unl has been submerged for a dwell me 2218

5 Algorhm 1 : self-rggered coordnaon A surfacng me l, agen {1,..., N} performs: 1: download las j, expre j, nex j, x j ( las j ), u ( las j ), M j j N from cloud 2: compue neghbor posons x j ( l ) usng (27) 3: compue deal conrol u (l ) = (x ( l ) x j( l )) 4: compue u max ( l ) usng (24) and saved τ j daa 5: compue conrol u ( l ) wh (25) 6: compue T = mn j N nex j 7: compue usng (7) as frs me when V 0 s no longer sasfed 8: f < T hen 9: se deal = 10: else 11: f u ( l ) = 0 hen 12: se nex = T + T dwell 13: else 14: f u ( l ) < 0 hen 15: compue T as frs me when (12) s no longer sasfed 16: else 17: compue T as frs me when (13) s no longer sasfed 18: end f 19: compue B usng (15) 20: compue T oal as frs me when (20) s no longer sasfed 21: se deal = (1 σ )T + σ T oal 22: end f 23: end f 24: f deal 25: se expre < l + T dwell = deal hen 26: se l+1 = l + T dwell 27: else 28: se expre = l+1 = deal 29: end f 30: upload promse M = u ( l ) o cloud 31: upload las = l, nex = l+1, expre, u ( l ), x ( l ) o cloud 32: dve and se u () = u ( l ) for [l, expre ), u () = 0 for [ expre, l+1 ) duraon. In oher words, we se nex = l + T dwell, expre = deal and use conrol law (25) on he nerval [ l, expre ). For [ expre, l+1 ), we hen se u () = 0, and noe ha because V () = 0 on hs nerval we sll have he desred conrbuon o he global objecve l+1 l V (τ)dτ = expre l V (τ)dτ < 0. (26) Agen s hen sll able o calculae he poson of any neghbor j exacly for any < nex j usng nformaon avalable on he cloud by { x j ( las j ) + u j ( las j )( las j ) < expre j, x j () = x j ( las j ) + u j ( las j )( expre j las j ) oherwse. (27) An overvew of he fully synheszed self-rggered coordnaon algorhm s presened n Algorhm 1. Nex, we presen he man convergence resul of hs algorhm. The proof s omed for space consderaons. Theorem III.1 Gven he dynamcs ẋ () = u () and G conneced, f he sequence of updae mes { l } and conrol laws u ( l ) are deermned by Algorhm 1 for all {1,..., N}, hen for all, j {1,..., N} as. x () x j () 0 (28) Fg. 1. Smulaed communcaon nework IV. SIMULATION In hs secon we smulae a sysem of 5 agens wh nal condon x = [ ] T for a oal me of 6 seconds, and wh all σ = σ for hree dfferen values of σ: 0.01, 0.5, and The opology of he communcaon nework s shown n fgure 1. For comparson, we ran he smulaon from he same confguraon wh a smple perodc rggerng rule where each agen surfaces every T seconds and uses he consan conrol law u (l ) on each submerged nerval. We show resuls here for T = 0.2, 0.35, and 0.4 seconds. Noe ha for he undreced graph n fgure 1 he sysem wll converge as long as T < T = 2/λ max (L) = The cumulave number of surfacngs by all agens up o me, denoed N S (), s shown n fgure 2 for our algorhm and he perodc rgger. As expeced, as a hgher value of σ allows an agen o say submerged for longer, N S () s decreasng n σ. The evoluon of he objecve funcon V (x()) for he same sx confguraons descrbed above s dsplayed n fgure 3. Agan, as expeced, he lower he value of σ, he more quckly he objecve V (x()) decreases, and values of σ farher from 0 acually allow he objecve o momenarly ncrease. I s neresng o noe ha when comparng agans our self-rggerng algorhm agans he perodc algorhm wh a perod ha resuls n a smlar number of surfacngs (e.g. σ = 0.5 and T = 0.35 or σ = 0.01 and T = 2), our algorhm resuls n he objecve V (x()) decreasng more rapdly. Ths shows ha we are able o yeld beer performance wh a smlar amoun of communcaon. Addonally, deermnng he mnmum perod T = requres global nformaon makng even more undesrable. These benefs are n addon o he oher clear advanage of our algorhm beng naurally asynchronous and no requrng agens o be connuously lsenng. A sngle agen s conrol law (agen 5) from a run of our algorhm wh σ = 0.5 s shown n fgure 4, along wh s promse currenly on he cloud server. As can be seen, here exss a lag beween when he promsed conrol max M 5 () ncreases and when he acual conrol ncreases lkewse. Whle M 5 () represens he deal conrol ha he agen would use, s sll bound o a prevous promse unl he newer one propagaes o all neghbor agens. V. CONCLUSION We have presened a novel self-rggerng algorhm ha, gven only he ably o communcae asynchronously a dscree nervals hrough a cloud server, provably drves a se of agens o consensus whou Zeno behavor. Unlke mos prevous work, we do no requre an agen o be able o lsen connuously, nsead only beng able o receve nformaon 2219

6 N() σ = 0.01 σ = 0.5 σ = Fg. 2. Cumulave surfacngs up for σ Fg. 3. Objecve funcon V (x()) wh σ {0.01, 0.5, 0.99}, and for a perodc rggerng rule {0.01, 0.5, 0.99}, as well as for a perodc rggerng rule wh T {0.2, 0.35, 0.4}. Our runs are wh T {0.2, 0.35, 0.4}. Our surfacngs are sold lnes, perodc as dashed. Perodc plos are sacked sold lnes, perodc are dashed. Perodc resuls are vercally wh T boom (or n purple, green, and blue, respecvely, wh color). V (x()) 10 2 σ = σ = 0.5 σ = = 0.2, 0.35, 0.4 from op o sacked vercally wh T = 0.4, 0.35, 0.2 from op o boom (or n blue, green, and purple, respecvely, wh color). u() 4 2 conrol promsed conrol max Fg. 4. Magnude of conrol law n use by agen = 5, u 5 (), as well as s curren promse on he cloud server M 5 (). a s dscree surfacng mes. Through he use of conrol promses, we are able o bound he saes of neghborng agens, allowng an agen o reman submerged unl s oal conrbuon o he consensus would become dermenal. The gven algorhm requres no global parameers, and s fully dsrbued, requrng no compuaon o be done off of each local plaform. Smulaon resuls show he effecveness of he proposed algorhm. In he fuure, we are neresed n nvesgang conrol laws dfferen from (25) ha may be able o provde more nfrequen surfacngs or faser convergence. We are addonally neresed n mehods o reach approxmae consensus raher han rue asympoc consensus, and guaraneeng no Zeno behavor whou he nroducon of a dwell me. ACKNOWLEDGMENTS Ths work was suppored n par by he TerraSwarm Research Cener, one of sx ceners suppored by he STARne phase of he Focus Cener Research Program (FCRP) a Semconducor Research Corporaon program sponsored by MARCO and DARPA. REFERENCES [1] N. A. Cruz, B. M. Ferrera, O. Kebkal, A. C. Maos, C. Perol, R. Perocca, and D. Spaccn, Invesgaon of underwaer nerworkng enablng he cooperave operaon of mulple heerogeneous vehcles, Marne Technology Scence Journal, vol. 47, pp , [2] E. Forell, N. E. Leonard, P. Bhaa, D. 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