Formation Control of Multi-agent Systems with Connectivity Preservation by Using both Event-driven and Time-driven Communication

Transcription

1 Foraion Conrol of Muli-agen Syses wih Conneciviy Preservaion by Using boh Even-driven and Tie-driven Counicaion Han Yu and Panos J. Ansaklis Absrac In his paper, disribued conrol algorihs and even-based counicaion sraegies are developed o achieve foraion conrol wih conneciviy preservaion aong a group of neworked obile agens. Each agen ransis is curren sae inforaion o is neighbors when is own riggering condiion is saisfied or when he ie elapsed fro is las even ie is going o exceed he agen s axial adissible iner-even ie. We have focused our sudies on wo ypes of syses dynaics: agens ha can be odeled as firs order inegraors and agens ha can be odeled as double inegraors. Siulaion resuls are provided o validae our resuls. I. Iɴʀɪɴ Exising resuls on disribued coordinaion conrol of uli-agen syses criically rely on ainaining a conneced counicaion nework aong he agens, eiher for all ie (i.e., [6],[5],[8]) or over sequence of bounded ie inervals (i.e., [],[9]). However, for a given se of iniial condiions, hose assupions on conneciviy of he neworks are difficul o verify. In paricular, conneciviy of he iniial deployen of he uli-agen syses canno guaranee conneciviy of he syses in fuure ies. Moivaed by he iporance of nework conneciviy in he conrol of uli-agen syses, any researchers have ephasized conneciviy preservaion in neworked dynaical syses. In [11], nework conneciviy is ainained by eans of poenial fields ha guaranee ha he second salles eigenvalue of he graph Laplacian arix is posiive definie; in [7], a easure of local conneciviy of a nework is inroduced and under cerain condiions i is also sufficien for global conneciviy; disribued ainenance of neares neighbor links by eans of unbounded edge ension funcions is addressed in [3], where a conrol hyseresis is inroduced o avoid infinie conrol inpus when new links are abou o be insered o he nework; siilarly, in [1], a syse of inerconneced unicycles is seered o a coon configuraion by eans of non-sooh, poenialbased conrol inpus ha urn unbounded when he disance beween adjacen agens approaches a cerain hreshold; in [4], a disribued coordinaion algorih ha allows he robos o decide wheher a desired collecive oion breaks conneciviy is proposed, and his procedure is used o design a second coordinaion algorih ha allows he robos o odify a desired collecive oion o guaranee ha conneciviy is preserved. Oher relaed recen work have been repored in [1]-[16]. The auhors are wih he Deparen of Elecrical Engineering, Universiy of ore Dae, ore Dae, I, 46556, USA, hyu@nd.edu, ansaklis.1@nd.edu. While conneciviy preservaion for coordinaed conrol of obile agens has been exensively sudied in he lieraure, one should noice ha coninuous or frequen counicaions beween coupled agens are sill required in os of hese works; oreover, he conrol acion updaes and he daa ransissions beween agens are usually assued o be ipleened in a synchronous fashion. oe ha uli-agen dynaic syses are disribued syses which usually ac in an asynchronous anner and in general, i is difficul o ipleen synchronous oions in he. However, analyzing he dynaics of asynchronous syses is ore difficul copared o heir synchronous counerpars. This paper sudies foraion conrol of uli-agen syses wih conneciviy preservaion by using boh evendriven and ie-driven counicaion. We have derived disribued riggering condiions and whenever an agen saisfies is riggering condiion, i will send is curren sae inforaion o is neighbors a ha ie. Moreover, here exiss an upper bound on he iner-even ie of each agen. Hence, an agen will ransi is curren sae inforaion o is neighbors whenever i saisfies is own riggering condiion or if he ie elapsed fro is las even ie is going o exceed he agen s axial adissible iner-even ie. We have derived disribued conrol acions o achieve boh foraion conrol and conneciviy preservaion under he proposed daa ransission sraegy provided ha he iniial deployen of he agens are wihin he counicaion radius of heir neighbors. oe ha he even-driven conrol approach has been exensively sudied in he area of neworked conrol syses, see [18]-[7]. However, o he bes of our knowledge, no uch work have been repored on he foraion conrol proble sudied in he presen paper. The res of his paper is organized as follows. Secion II provides soe background aerial. Secion III describes he probles sudied in his paper. The ain resuls are saed in Secion IV and Secion V. Siulaion sudies are included in Secion VI. Finally, concluding reakes are provided in Secion VII. II. Bɢʀɴ Mʀɪʟ The inforaion exchange opology beween agens can be odeled by a graph. In he following, we give soe basic erinologies and definiions fro graph heory [17]. A direced graph is a graph whose edges have direcion and are called arcs. A bi-direced graph is a graph in which each edge is given an independen orienaion a each end. Consider a finie weighed direced graph G := (V, E) wih no self-loops and adjacency arix A, where V denoes he se of

2 all verices, E denoes he se of all edges, and A := [a ij ] wih a ij > if here is a direced edge fro verex i ino verex j, and a ij = oherwise. The in-degree and ou-degree of verex k are given by d i (k) = j a jk and d o (k) = j a kj respecively. The Laplacian arix of a direced graph is defined as L = D A, where D is he diagonal arix of verex oudegrees. Definiion 1: A direced graph is srongly conneced if for any pair of disinc verices ν i and ν j, here is a direced pah fro ν i o ν j. Definiion : A verex is balanced if is in-degree is equal o is ou-degree. A direced graph is balanced if every verex is balanced. Definiion 3: A pah of lengh r in a direced graph is a sequence ν,...,ν r of r + 1 disinc verices such ha for every i {,...,r 1}, (ν i,ν i+1 ) is an edge. A weak pah is a sequence ν,...,ν r of r + 1 disinc verices such ha for each i {,...,r 1}, eiher (ν i,ν i+1 ) or (ν i+1,ν i ) is an edge. A direced graph is weakly conneced if any wo verices can be joined by a weak pah. Lea 1: Le G be a direced graph and assue i is balanced. Then G is srongly conneced if and only if i is weakly conneced. III. Pʀʙʟ Sɴ The evoluion of uli-agen syses depends fundaenally on heir inforaion exchange opology. In his paper, we have he following assupion wih respec o he underlying inforaion exchange graph: Assupion A. The underlying counicaion graph is bidirecional and balanced, and weakly conneced in ie. Definiion 4: Le p i () denoes he posiion of agen i a ie ; i denoes he se of agens sending inforaion o agen i; d ij R + denoes he desired disance beween agen i and agen j; d ij = d ji if boh i j and j i. For a group of agens, he agens are said o esablish a disance-based foraion if li p j() p i () = d ij, j i, for i = 1,...,. Consider a group of obile uli-agens, where he agens ay have differen counicaion capabiliies (i.e., counicaion radius) and differen liiaions on obiliy (i.e., axial allowable speed). The underlying counicaion nework is odeled by a graph Laplacian. Assue ha each agen has access o is curren sae inforaion (i.e., curren posiion or speed), and i can also exchange inforaion wih is neighbors (agens ha are wihin is counicaion radius are defined as neighbors in he counicaion graph). The proble invesigaed in he presen paper is o achieve disance-based foraion aong he neworked agens wih even-driven and/or ie-driven counicaion while preserving conneciviy of he underlying inforaion exchange graph. The fundaenal challenges regarding he proble sudied in he curren paper are he design of he disribued conrol laws and he disribued daa ransission sraegy o achieve boh foraion and conneciviy preservaion based on he local inforaion available o each agen. The disribued daa ransission sraegy will deerine he even-ie a which an agen ransis is curren sae inforaion o is neighbors. Since conneciviy preservaion is required, inuiively, one ay expec ha each agen should have soe sor of echanis o esiae he curren axial disance fro is neighbors based on he las sae inforaion i has received fro is neighbors. Moreover, one ay also expec ha each agen should be able o updae is conrol acions and schedule is daa ransissions based on his esiae in order o preserve conneciviy wih is neighbors(i.e., keep he disance fro is neighbors wihin is counicaion radius). The conneciviy preservaion conrol algorihs repored in he lieraure have been osly devoed o wo ypes of syses dynaics: agens ha can be odeled as firs order inegraors and agens ha can be odeled as double inegraors. In he following secions, we will also focus our sudies on hese wo ypes of uli-agen syses. IV. Fʀɪɴ Cɴʀʟ ɪʜ Cɴɴɪɪʏ Pʀʀɪɴ: Fɪʀ Oʀʀ Iɴɢʀʀ The foraion conrol proble sudied in he presen paper is focused in he D space. We firs consider he case when he dynaics of he agens can be odeled as firs order inegraors given by ṗ i () = u i (), p i (), u i () R, i = 1,,...,. (1) Define an edge-ension funcion beween agen i and agen j as pi () p j d ij V ij (δ i, p i ) = δ i p i () p j υτ j j, (i, j) E(G), () where p j = p j ( j k ), for [ j k, j k +1 ], { j k } k =,1,,... is he evenie of agen j; δ i R + /{} is he counicaion radius of agen i; υ j R + /{} is he axial allowable agniude of he velociy of agen j; τ j R + /{} is an upper bound on he adissible iner-even ie of agen j; d ij R + /{} is he desired disance beween agen i and agen j, d ij +υτ j j <δ i, j i ; if (i, j) E(G), hen d ij = d ji. Le l ij = p i () p j, hen one can verify ha Le i = lij d ij δi l ij υ j τ j d ij δi l ij υ j τ j l ij pi p j. (3) lij d ij δi l ij υτ j j d ij ij = δi l ij υτ j j, (4) l ij our designed conrol inpu for agen i is given by ij pi p j if ij pi p j υ i, u i () = υi sgn if ij pi p j >υ i, i (5)

3 where p i = p i (k i ), for [i k,i+1 ], { k k i } k=,1,,..., is he even-ie of agen i. Define h i = ij pi p j υ i, we can rewrie (5) as u i () = 1 sgn(h i) ij pi p j 1 + sgn(h i) υ i (6) sgn, i where 1, if hi > ; sgn(h i ) = 1, if h i. Reark 1: One can see ha he conrol law (6) requires ha each agen knows is own counicaion radius (δ i ), is curren posiion (p i ()), is las ransied sae inforaion (p i ), he laes received inforaion fro is neighbors (p j, j i ), he axial agniude of he velociy of is neighbors (υ j, j i ) and he axial adissible iner-even ie of is neighbors (τ j, j i ). Based on his inforaion, agen i can esiae he axial disance fro agen j (which is l ij + υ j τ j ) before agen i receives he nex sae inforaion fro agen j, j i. A riggering condiion o achieve disance-based foraion is saed in Theore 1. Theore 1: Consider a group of agens wih dynaics given by (1) and conrol laws given by (6). Assue ha a he iniial ie ( ), each agen broadcass he iniial sae o is neighboring agens and we have p i ( ) p j + υτ j j = pi ( ) p j ( ) + υτ j j <δ i, (7) (i, j) E(G). If each agen ransis is curren sae inforaion (p i ) o is neighboring agens whenever e pi () ij pi () p j >γ 1, i, (8) ij where e pi () = p i () p i, γ 1 (,1), or when i k = τi, (9) where k i is he las even-ie of agen i, hen under assupion A., he neworked agens will achieve disance-based foraion asypoically. Proof: The oal ension energy of he enire neworked syse can be defined as V(δ, p) = V ij (δ i, p i ), and we have T Vij T Vij V = ṗ i = u i i i = i i T 1 sgn hi ij pi p j T 1 + sgn hi υ i sgn i, (1) which furher yields 1 sgn h i T V = ij pi p j ij pi p j 1 + sgn h i υ i V T ij (11) sgn i i hus 1 sgn h i T V = ij pi p j ij pi p j 1 + sgn h i υ i. Wih e pi = p i p i, we can ge (1) ij pi p j = ij pi p j e pi = ij pi p j ij e pi, (13) and we can rewrie (1) as 1 + sgn(h i ) υ V i = 1 sgn h i T ij pi p j ij e pi ij pi p j j i 1 sgn h i = ij pi p j 1 sgn h i T + ij e pi ij pi p j 1 + sgn(h i ) υ i, (14) which furher yields 1 sgn h i V e pi ij ij pi p j 1 sgn(h i ) ij pi p j (15) 1 + sgn(h i ) υ i. So if e ij pi p j pi, i, (16) ij hen V,. oe ha he riggering condiion (8) guaranees ha (16) is saisfied. Under he riggering condiion (8), we have V(δ, p) V(δ, p ),. This indicaes ha p i p j + υτ j j will never approach δ i, (i, j) E(G), oherwise we igh have V(δ, p) V(δ, p ) since he iniial deployen of he agens (7) guaranees ha V(δ, p ) is finie. This furher indicaes ha if he iniial deployen of he agens are wihin he counicaion radius of neighboring agens, hen conneciviy is preserved over ie because p i p j p i p j + υτ j j <δ i, (i, j) E(G). (17) Moreover, since p i p j + υτ j j will never approach δ i, wih V(δ, p) and V, we can conclude ha li V(δ, p) exiss and is finie, and furherore li V(δ, p) =, hus in view of (1), we can ge li ij =, (i, j) E(G), which furher yields li p i p j = d ij, (i, j) E(G). (18) In view of he riggering condiion (8), li ij =, (i, j) E(G) furher indicaes ha

4 li p i p i = li e pi =, i. (19) (18) and (19) ogeher iply ha li p i p j = d ij, (i, j) E(G), () which coplees he proof. V. Fʀɪɴ Cɴʀʟ ɪʜ Cɴɴɪɪʏ Pʀʀɪɴ: Dʙʟ Iɴɢʀʀ We nex consider he case when he agens can be odeled as double inegraors wih consrains on he second order dynaics given by ṗ i () = q i () = u i () q i (), if υ i sgn(q i ()), if q i () υ i q i () >υ i (1) where q i (), p i (), u i () R, i = 1,,...,. We can also rewrie (1) as ṗ i = 1 sgn q i υ i q i sgn q i υ i υ i sgn(q i ) q i = u i, for i = 1,,...,. () We sill use an edge-ension funcion beween agen i and agen j as defined in (), he conrol inpu o agen i is given by 1 sgn qi υ i u i () = K p ij pi () p j 1 + sgn q i υ i υ i sgn(q i ) T sgn(q i ) (3) q i 1 + K d q j q i, where K p, K d > are designed conrol gains, q i = q i (k i ), for [k i,i k+1 ], and q j = q j ( j ), for [ j, j ]. A riggering condiion k k k +1 o achieve disance-based foraion is saed in Theore. Theore : Consider a group of agens wih dynaics given by () and conrol laws given by (3). Assue ha a he iniial ie ( ), each agen broadcass is iniial sae inforaion o he neighboring agens and he iniial deployen of he agens saisfies (7), (i, j) E(G). If each agen ransis is curren sae inforaion (q i () and p i ()) o is neighboring agens whenever e qi () q j q i >γ q j q, i, (4) i where e qi () = q i () q i, γ (,.5), or when i k = τi, (5) where k i is he las even-ie of agen i, hen under assupion A., he neworked agens will achieve disance-based foraion asypoically. Proof: Le he oal ension funcion for he enire neworked syse be defined as V (δ, p) = V ij. (6) Define he energy funcion for he enire neworked syse as 1 V(δ, p,q) = K p V + q i (), (7) hen we have V = K p V + q T i () q i() = K p V ij + q T i ()u i() T = K p ij pi () p j ṗi () + q T i ()u i() T 1 sgn qi () υ i = K p ij pi () p j q i () T 1 + sgn qi () υ i υ i + K p ij pi () p j sgn(q i ()) K p ij pi () p T 1 sgn qi () υ i j q i () T 1 + sgn qi () υ i υ i K p ij pi () p j sgn(q i ()) T T + K d q j q i qi () = K d q j q i qi (). T = K d q j q i eqi () +q i T Tqi = K d q j q i eqi () + K d q j q i, (8) since he underlying inforaion exchange graph is balanced, we have Tqi K d q j q i = K d q T j q K d i qt i q i (9) K d qt j q K d j = q j q i, replace (9) ino (8), we can ge so if T K d V = K d q j q i eqi () q j q i e qi () K d q j q i K d q j q i, e qi () (3).5 q i q j, i, (31) q j q i hen V,. oe ha he riggering condiion (4) will guaranee ha (31) holds. Under he riggering condiion (4), we have V(δ, p) V(δ, p ),. This indicaes ha p i () p j + υ j τ j will never approach δ i, (i, j) E(G), oherwise we igh have V(δ, p) V(δ, p ) since he iniial deployen of he agens (7) guaranees ha V(δ, p ) is finie. This furher indicaes ha if he iniial deployen of he agens are wihin he counicaion

5 radius of neighbors, hen conneciviy is preserved over ie since p i () p j () p i () p j + υτ j j <δ i, (i, j) E(G). (3) Moreover, since pi () p j + υ j τ j will never approach δ i, wih V(δ, p) and V, we can conclude ha li V(δ, p) exiss and is finie, and furherore li V(δ, p) =. Under he riggering condiion (4), we can ge = li V li K d (.5 γ ) q j q i, hus li K d (.5 γ ) q j q i =, which indicaes ha li q j = li q i, (i, j) E(G). (33) In view of (31), (33) furher yields li e q i () = li qi () q i =, i. (34) Based on (33) and (34), we can conclude ha li q i() = li q i = li q j = li q j (), (i, j) E(G). (35) Furherore, wih li V exiss, V,V and qi (), we can conclude ha li V and li qi () exis; wih li V =, in view of (8), we can furher conclude ha li V ij = and q T i u i =. Thus, he soluions of he dynaical syse should converge o he se S = {p i (),q i () R q i () = q j = q i ij =, (i, j) E(G)}, which furher iplies ha li q j() = li q i () =, and li p j p i () d ij =, (36) (i, j) E(G). Assue j is an even ie of agen j a ie, k f hen a ie j, based on (36), we have k f p j ( j k f ) p i ( j k f ) = d ij, where j i. (37) Since li p j () = p j ( j k f )+li j k f q j (τ)dτ and li p i () = p i ( j k f ) + li j k f q i (τ)dτ, j k f, we can furher ge li p j() li p i () = p j ( j ) p k f i ( j ) k f + li q j (τ)dτ li q i (τ)dτ. j k j f k f (38) Since li q j () = li q i () =, j i, wih j k f, we have li q j (τ)dτ = li q i (τ)dτ =, j k j f k f (39) hus li p j () p i () = p j ( j ) p k f i ( j ) = d k f ij, j i, (4) which coplees he proof. VI. Sɪʟɪɴ Sʏ Exaple: Consider a group of 3 agens rying o esablish a equilaeral riangle foraion in a D space, wih each side lengh equal o 1. Each agen can be odeled as a double inegraors wih consrains on he second order dynaics as described in Secion V. Le p ix () denoes agen i s posiion on x-axis and p iy () denoes agen i s posiion on y-axis; q ix () denoes agen i s velociy on x-axis and q iy () denoes agen i s velociy on y-axis, he dynaics of each agen are given by ṗ ix () = 1 sgn q ix υ i q ix sgn q ix υ i ṗ iy () = 1 sgn q iy υ i q iy sgn q iy υ i q ix () = u ix () q iy () = u iy (), i = 1,,3, The iniial condiions of agens are given by υ i sgn(q ix ) υ i sgn(q iy ) p 1 () = [, 3] T, q 1 () = [1, ] T, p () = [5, 1] T, q () = [, 4] T, p 3 () = [1, ] T, q 3 () = [3, ] T. (41) (4) The counicaion radius of agen 1 is δ 1 = 8, he axial allowable agniude of he velociy of agen 1 is υ 1 = 1/s, and he axial iner-even ie of agen 1 is τ 1 = 6s; he counicaion radius of agen is δ = 1, he axial allowable agniude of he velociy of agen is υ = 5/s, and he axial iner-even ie of agen is τ = 1s; he counicaion radius of agen 3 is δ 3 = 9, he axial allowable agniude of he velociy of agen 3 is υ 3 = 15/s, and he axial iner-even ie of agen 3 is τ 3 = 4s. The Laplacian arix of he underlying inforaion exchange graph is given by 1 1 L = 1 1, (43) 1 1 which saisfies assupion A. Choose γ 1 =.45, by applying he resuls in Theore, we ge he siulaion resuls shown in Fig.1- Fig.3. In Fig.1, he x-axis shows he even-ie of each agen (k i ) and he y-axis shows he evoluions of iner-even ie [k+1 i i ]; Fig. k shows he evoluion of he disances beween agen 1 and agen (d 1 ), agen and agen 3 (d 3 ), and agen 1 and agen 3 (d 13 ), and one can observe ha agens are kep wihin he counicaion radius of heir neighboring agens; Fig.3 shows he evoluion of he foraion aong he hree agens, where squares denoe he iniial posiions and circles denoe he final posiions. 1 k+1 1 k k+1 k 3 k+1 3 k 1 5 even ie of agen (s) even ie of agen (s) even ie of agen (s) Fig. 1: Even-ie of each agen

6 Y () d 1 d 3 d (s) Fig. : Disance Evoluion X() Fig. 3: Foraion Evoluion VII. Cɴʟɪɴ In his paper, disribued conrol algorihs and even-based counicaion sraegies o achieve foraion conrol wih conneciviy preservaion aong a group of neworked obile agens are proposed. Each agen ransis is curren sae inforaion o is neighbors whenever i saisfies is riggering condiion or whenever he ie elapsed fro is las even ie is going o exceed he agen s axial adissible iner-even ie. Disribued conrol algorihs and disribued riggering condiions for daa ransissions are derived o esablish foraion aong he obile agens wih conneciviy preservaion, provided ha he iniial deployen of he agens is wihin he counicaion radius of heir neighbors. VIII. Aɴʟɢɴ The suppor of he aional Science Foundaion under Gran o. CS is graefully acknowledged. Rʀɴ [1] D. V. Diarogonas and K. J. Kyriakopoulos, Conneciviy Preserving Sae Agreeen for Muliple Unicycles, in Proceedings of Aerican Conrol Conference, 7, pp [] A. Jadbabaie, J. Lin, and A. S. Morse, Coordinaion of groups of obile auonoous agens using neares neighbor rules, IEEE Transacions on Auoaic Conrol, vol. 48, no. 6, pp. 988C11, Jun. 3. [3] M. Ji and M. Egersed, Disribued Coordinaion Conrol of Muliagen Syses While Preserving Connecedness, IEEE Transacions on Roboics, vol. 3, no. 4, pp , Aug. 7. [4] M. Schuresko and J. Corés, Disribued Moion Consrains for Algebraic Conneciviy of Roboic eworks, Journal of Inelligen and Roboic Syses, vol. 56, no. 1-, pp , Apr. 9. [5] R. Olfai-Saber, Flocking for Muli-Agen Dynaic Syses: Algorihs and Theory, IEEE Transacions on Auoaic Conrol, vol. 51, no. 3, pp. 41-4, 6. [6] R. Olfai-Saber and R. M. Murray, Consensus Probles in eworks of Agens Wih Swiching Topology and Tie-Delays, IEEE Transacions on Auoaic Conrol, vol. 49, no. 9, pp , 4. [7] D. P. Spanos and R. M. Murray, Robus conneciviy of neworked vehicles, in Proceedings of IEEE Conference on Decision and Conrol (CDC), 4, pp , Vol.3. [8] H. G. Tanner, A. Jadbabaie, and G. J. Pappas, Flocking in Fixed and Swiching eworks, IEEE Transacions on Auoaic Conrol, vol. 5, no. 5, pp , May. 7. [9] W. Ren and R. Beard, Consensus seeking in uliagen syses under dynaically changing ineracion opologies, IEEE Transacions on Auoaic Conrol, vol. 5, no. 5, pp , 5. [1] M. M. Zavlanos, H. G. Tanner, A. Jadbabaie, and G. J. Pappas, Hybrid Conrol for Conneciviy Preserving Flocking, IEEE Transacions on Auoaic Conrol, vol. 54, no. 1, pp , Dec. 9. [11] M. M. Zavlanos and G. J. Pappas, Poenial Fields for Mainaining Conneciviy of Mobile eworks, IEEE Transacions on Roboics, vol. 3, no. 4, pp , Aug. 7. [1] H. Su and X. Wang, Coordinaed conrol of uliple obile agens wih conneciviy preserving, in Proceedings of The 17h IFAC World Congress, no. 1, pp , 8. [13] H. Su, X. Wang, and G. Chen, A conneciviy-preserving flocking algorih for uli-agen syses based only on posiion easureens, Inernaional Journal of Conrol, vol. 8, no. 7, pp , Jul. 9. [14] D. V. Diarogonas and K. H. Johansson, Even-riggered Conrol for Muli-Agen Syses, Join 48h IEEE. Conference on Decision and Conrol and 8h Chinese Conrol Conference, pp , 9. [15] A. Ajorlou, A. Moeni, and A. G. Aghda, A Class of Bounded Disribued Conrol Sraegies for Conneciviy Preservaion in Muli- Agen Syses, IEEE Transacions on Auoaic ConrolConrol, vol. 55, no. 1, pp , 1. [16] J. Dai, S. Zhu, and C. Chen, Conneciviy-Preserving Consensus Algorihs for Muli-agen Syses, in Proceedings of The 18h IFAC World CongressWorld Congress, 11, pp [17] C. Godsil and G. Royle. Algebraic Graph Theory. Springer Graduae Texs in Maheaics 7, 1. [18] K. J. Asrö, Even Based Conrol, Analysis and Design of onlinear Conrol Syses, Par 3, pp , 8. [19] W. P. M. H. Heeels, J. H. Sandeeb, P. P. J. Van Den Boscha, Analysis of even-driven conrollers for linear syses, Inernaional Journal of Conrol, Volue 81, Issue 4, pp , April 8. [] P. G. Oanez, J. R. Moyne, D. M. Tilbury, Using deadbands o reduce counicaion in neworked conrol syses, in Proceedings of he Aerican Conrol Conference, pp.315-3,. [1] E. Kofan, J. H. Braslavsky, Level Crossing Sapling in Feedback Sabilizaion under Daa-Rae Consrains, in Proceedings of he IEEE Conference on Decision and Conrol, pp , 6. [] P. Tabuada, Even-riggered real-ie scheduling of sabilizing conrol asks, IEEE Transacion on Auoaic Conrol, vol.5, no.9, pp , Sepeber 7. [3] X. Wang and M. Leon, Even-riggering in disribued neworked syses wih daa dropous and delays, in Hybrid Syses: Copuaion and Conrol, 9. [4] M.C.F. Donkers and W.P.M.H. Heeels, Oupu-Based Even- Triggered Conrol wih Guaraneed L -gain and Iproved Even- Triggering, 49h IEEE Conference on Decision and Conrol Deceber 15-17, 1. [5] H. Yu and P. J. Ansaklis, Even-Triggered Real-Tie Scheduling For Sabilizaion of Passive/Oupu Feedback Passive Syses, Proceedings of he 11 Aerican Conrol Conference, pp , San Francisco, CA, June 9-July 1, 11. [6] H. Yu and P. J. Ansaklis, Even-Triggered Oupu Feedback Conrol for eworked Conrol Syses using Passiviy: Triggering Condiion and Liiaions, Proceedings of he 5h IEEE Conference on Decision and Conrol (CDC 11) and ECC 11, pp.199-4, Orlando, Florida, Deceber 1-15, 11. [7] H. Yu and P.J. Ansaklis, Even-Triggered Oupu Feedback Conrol for eworked Conrol Syses using Passiviy: Tie-varying ework Induced Delays, Proceedings of he 5h IEEE Conference on Decision and Conrol (CDC 11) and ECC 11, pp.5-1, Orlando, Florida, Deceber 1-15, 11.