Synchronization in Networks of Identical Linear Systems

Transcription

1 Proceedings of he 47h IEEE Conference on Decision and Conrol Cancun, Meico, Dec. 9-11, 28 Synchronizaion in Neworks of Idenical Linear Sysems Luca Scardovi and Rodolphe Sepulchre Absrac The paper invesigaes he synchronizaion of a nework of idenical linear ime-invarian sae-space models under a possibly ime-varying and direced inerconnecion srucure. The main resul is he consrucion of a dynamic oupu feedback coupling ha achieves synchronizaion if he decoupled sysems have no eponenially unsable mode and if he communicaion graph is uniformly conneced. Sronger condiions are shown o be sufficien bu o some een, also necessary o ensure synchronizaion wih he diffusive saic oupu coupling ofen considered in he lieraure. I. INTRODUCTION In hese las years, consensus, coordinaion and synchronizaion problems have been popular subjecs in sysems and conrol, moivaed by many applicaions in physics, biology, and engineering. These problems arise in muli-agen sysems wih he collecive objecive of reaching agreemen abou some variables of ineres. In he consensus lieraure, he emphasis is on he communicaion consrains raher han on he individual dynamics: he agens echange informaion according o a communicaion graph ha is no necessarily complee, nor even symmeric or ime-invarian, bu, in he absence of communicaion, he agreemen variables usually have no dynamics. I is he echange of informaion only ha deermines he ime-evoluion of he variables, aiming a asympoic synchronizaion o a common value. The convergence of such consensus algorihms has araced a lo of ineres in he recen years and i only requires a weak form of conneciviy for he communicaion graph [1], [2], [3], [4], [5]. In he synchronizaion lieraure, he emphasis is on he individual dynamics raher han on he communicaion limiaions: he communicaion graph is ofen assumed o be complee (or all-o-all, bu in he absence of communicaion, he ime-evoluion of he agens variables can be oscillaory or even chaoic. The sysem dynamics can be modified hrough he informaion echange, and, as in he consensus problem, he goal of he inerconnecion is o reach synchronizaion o a common soluion of he individual dynamics [6], [7], [8], [9]. Coordinaion problems encounered in he engineering world, can ofen be rephrased as consensus or synchro- Luca Scardovi is wih he Deparmen of Mechanical and Aerospace Engineering, Princeon Universiy, USA. scardovi@princeon.edu. The work is suppored in par by ONR grans N and N Rodolphe Sepulchre is wih he Deparmen of Elecrical Engineering and Compuer Science, Universiy of Liège, Belgium. r.sepulchre@ulg.ac.be. This paper presens research resuls of he Belgian Nework DYSCO (Dynamical Sysems, Conrol, and Opimizaion, funded by he Ineruniversiy Aracion Poles Programme, iniiaed by he Belgian Sae, Science Policy Office. The scienific responsibiliy ress wih is auhor. nizaion problems in which boh he individual dynamics and he limied communicaion aspecs play an imporan role. Designing inerconnecion conrol laws ha can ensure synchronizaion of relevan variables is herefore a conrol problem ha has araced quie some aenion in he recen years [1], [11], [12], [13], [14], [15]. The presen paper deals wih a fairly general soluion of he synchronizaion problem in he linear case. Assuming N idenical individual agens dynamics each described by he linear sae-space model (A, B, C, he main resul is he consrucion of a dynamic oupu feedback conroller ha ensures eponenial synchronizaion o a soluion of he linear sysem ẋ = A under he following assumpions: (i A has no eponenially unsable mode, (ii (A, B is sabilizable and (A, C is observable, and (iii he (possibly ime-varying and direced communicaion graph is uniformly conneced. Uniform connecedness is a very mild condiion. I allows he communicaion from one sysem o anoher o be indirec, involving inermediae sysems. Also, he required communicaion does no need o occur insananeously bu may be spread over ime. The resul can be inerpreed as a generalizaion of classical consensus algorihms sudied in he recen years corresponding o he paricular case A =. The generalizaion includes he non-rivial eamples of synchronizing harmonic oscillaors or chains of inegraors. The dynamic conroller srucure proposed in his paper differs from he saic diffusive coupling ofen considered in he synchronizaion lieraure, which requires more sringen assumpions on he communicaion graph. The paper also provides sufficien condiions for synchronizaion by saic diffusive coupling and illusraes, on simple eamples, ha synchronizaion may fail under diffusive coupling when he sronger assumpions on he communicaion graph are no saisfied. The paper is organized as follows. In Secion II he noaion used hroughou he paper is summarized and some preliminary resuls are reviewed. In Secion III he synchronizaion problem is inroduced and defined. In Secion IV he linear case is sudied when sae coupling among he sysems is allowed while, in Secion V, he oupu coupling case is considered. Finally, in Secion VI, wo-dimensional eamples are repored o illusrae he role of he proposed dynamic conroller in siuaions where saic diffusive coupling fails o achieve synchronizaion. A. Noaion and Terminology II. PRELIMINARIES Throughou he paper we will use he following noaion. Given N vecors 1, 2,..., N we indicae wih he /8/$ IEEE 546

2 sacking of he vecors, i.e. = [ T 1, T 2,..., T N ]T. We denoe wih I N he diagonal mari of dimension N and we define 1 N [1, 1,...,1] T R N. Given wo marices A and B we denoe heir Kronecker produc wih A B, for he definiion and basic properies see e.g. [16]. For noaional convenience, we use he convenion ÃN = I N A and  N = A I N. B. Communicaion Graphs Given a se of inerconneced sysems he communicaion opology is encoded hrough a communicaion graph. The convenion is ha sysem j receives informaion from sysem i if and only if here is a direced link from node j o node i in he communicaion graph. Le G( = (V, E(, A d ( be a ime-varying weighed digraph (direced graph where V = {v 1,...,v N } is he se of nodes, E( V V is he se of edges, and A d ( is a weighed adjacency mari wih nonnegaive elemens a kj (. In he following we assume ha A d ( is piece-wise coninuous and bounded and a kj ( {} [η, γ], k, j, for some finie scalars < η γ and for all. Furhermore {v k, v j } E( if and only if a kj ( η. The se of neighbors of node v k a ime is denoed by N k ( {v j V : a kj ( η}. A pah is a sequence of verices such ha for each of is verices v k he ne vere in he sequence is a neighbor of v k. Assume ha here are no self-cycles i.e. a kk ( =, k = 1,...,N and for any. The graph Laplacian L( associaed o he graph G( is defined as { L kj ( = i a ki(, j = k a kj (, j k. The in-degree (respecively ou-degree of node v k is defined as d in k = N a kj (respecively d ou k = N a jk. The digraph G is said o be balanced if he in-degree and he ou-degree of each node are equal, ha is, a kj = a jk, k = 1,...,N. j j Balanced graphs have he paricular propery ha he symmeric par of heir Laplacian is nonnegaive: L + L T [17]. We recall some definiions ha characerize he concep of conneciviy for ime-varying graphs. Definiion 1: The digraph G( is conneced a ime if here eiss a node v k such ha all he oher nodes of he graph are conneced o v k via a pah ha follows he direcion of he edges of he digraph. Definiion 2: Consider a graph G(. A node v k is said o be conneced o node v j (v j v k in he inerval I = [ a, b ] if here is a pah from v k o v j which respecs he orienaion of he edges for he direced graph (V, I E(, I A d(τdτ. Definiion 3: G( is said o be uniformly conneced if here eiss a ime horizon T > and an inde k such ha for all all he nodes v j (j k are conneced o node v k across [, + T]. C. Convergence of consensus algorihms Consider he coninuous dynamics ẋ k = a kj (( j k, k = 1,..., N. (1 Using he Laplacian definiion, (1 can be equivalenly epressed as ẋ = ˆL n (, (2 Algorihm (2 has been widely sudied in he lieraure and asympoic convergence o a consensus value holds under mild assumpions on he communicaion opology. The following heorem summarizes he main resul in [2]. Theorem 1: Le k, k = 1, 2,...,N, belong o a finiedimensional Euclidean space W. Le G( be a uniformly conneced digraph and L( he corresponding Laplacian mari bounded and piecewise coninuous in ime. Then he equilibrium ses of consensus saes of (1 and (2 are uniformly eponenially sable. Furhermore he soluions of (1 and (2 asympoically converge o a consensus value 1 N β for some β W. III. THE SYNCHRONIZATION PROBLEM Consider N idenical dynamical sysems ẋ k = f( k, u k y k = h( k, k = 1,...,N where k R n is he sae of he sysem, u k R m is he conrol and y k R p is he oupu. We assume ha he coupling among he sysems involves only he oupu differences y k y j and he conroller sae differences ξ k ξ j. Following he erminology and noaion of Secion II, wo sysems are coupled a ime if here eiss an edge connecing hem in he associaed (ime-varying communicaion graph G( a ime. We will call a conrol law dynamic if i depends on an inernal (conroller sae, oherwise i is called saic. For he sysems o be synchronized, he conrol acion (ha will depend on he coupling mus vanish asympoically and mus force he soluions of he closed-loop sysems o asympoically converge o a common soluion of he individual sysems. This leads o he formulaion of he following problem: Synchronizaion Problem: Given N idenical sysems described by he model (3 and a communicaion graph G(, find a (disribued conrol law such ha he soluions of (3 asympoically synchronize o a soluion of he open-loop sysem ẋ = f(,,. In he presen paper we focus he aenion on synchronizaion of linear ime-invarian sysems. Generalizaions will be he subjec of fuure work. IV. SYNCHRONIZATION OF LINEAR SYSTEMS WITH STATE FEEDBACK Consider N idenical linear sysems, each described by he linear model (3 ẋ k = A k + Bu k, k = 1, 2,..., N, (4 547

3 where k R n is he sae of he sysem and u k R m is he conrol vecor. For noaional convenience i is possible o rewrie (4 in compac form as ẋ = ÃN + B N u. (5 Theorem 1 can be inerpreed as a synchronizaion resul for linear sysems wih A = and B = I. A sraighforward generalizaion is as follows. Theorem 2: Consider he linear sysems (4. Le B be a n n nonsingular mari and assume ha all he eigenvalues of A belong o he imaginary ais. Assume ha he communicaion graph G( is uniformly conneced and he corresponding Laplacian mari L( piecewise coninuous and bounded. Then, under he conrol law N u k = B 1 a kj (( j k, k = 1, 2,...,N, (6 all soluions of (4 eponenially synchronize o a soluion of he sysem ẋ = A. Proof: Consider he closed-loop sysem ẋ k = A k + The change of variable, leads o a kj (( j k. z k ( = e A( k (, k = 1, 2,...,N (7 ż k = Ae A( k + e A( A k +e A( N a kj(( j k = N a kj((z j z k or, in compac form, ż = ˆL n (z. (8 From Theorem 1 he soluions z k (, k = 1, 2,..., N eponenially converge o a common value R n as, ha is, here eis consans δ 1 > and δ 2 > such ha for all, z k ( δ 1 e δ2( z k (, >. (9 In he original coordinaes, his means k ( e A( δ 1 e A( e δ2( k (, (1 for every >. Because all he eigenvalues of he mari A lie on he imaginary ais, here eiss a consan δ 3 > such ha k ( e A( δ1 e δ3( k (, (11 for every >, which proves ha all soluions eponenially synchronize o a soluion of he open loop sysem. Remark 1: The resul is of course unchanged if A also possesses eigenvalues wih a negaive real par. Eponenially sable modes synchronize o zero, even in he absence of coupling. In conras, he siuaion of sysems wih some eigenvalues wih a posiive real par can be addressed in a similar way bu i requires ha he graph conneciviy is sufficienly srong o dominae he insabiliy of he sysem. This is clear from he las par of he proof of Theorem 2 where he eponenial synchronizaion in he z coordinaes mus dominae he divergence of he unsable modes of A. The assumpion of a square (nonsingular mari B in Theorem 2 can be weakened o a sabilizabiliy assumpion on he pair (A, B. For an arbirary sabilizing feedback mari K, consider he (dynamic conrol law ξ = N KN ξ + ˆL n (( ξ, u = K (12 N ξ which leads o he closed-loop sysem ẋ = ÃN + B N KN ξ (13a ξ = N KN ξ + ˆL n (( ξ. (13b Theorem 3: Consider he sysem (4. Assume ha all he eigenvalues of A belong o he closed lef-half comple plane. Assume ha he pair (A, B sabilizable and le K a sabilizing mari such ha A + BK is Hurwiz. Assume ha he graph is uniformly conneced and he Laplacian is piecewise coninuous and bounded. Then he soluions of (13 eponenially synchronize o a soluion of he open loop sysem ẋ = A. Proof: Wih he he change of variable s k = k ξ k we can rewrie (13b as ṡ = ÃNs ˆL n (s, and he closed-loop dynamics wrie ẋ = N KN + B N KN s (14a ṡ = ÃNs ˆL n (s. (14b Observe ha he wo sysems (14a and (14b are decoupled. Since he hypoheses of Theorem 2 are saisfied for he subsysem (14b, is soluions eponenially synchronize o a soluion of ṡ = As. The subsysem (13b is herefore an eponenially sable sysem driven by an inpu ˆL n (s( ha eponenially converges o zero. As a consequence, is soluion ξ( eponenially converges o zero, which implies ha he soluions of (13a eponenially synchronize o a soluion of ẋ = A. Remark 2: In he presen paper we focus on imeinvarian linear sysems in coninuous ime. However, he resuls here presened, are easily eendable o discree-ime sysems and periodic sysems. For he ineresed reader hese generalizaions are discussed in [18]. V. SYNCHRONIZATION OF LINEAR SYSTEMS WITH OUTPUT FEEDBACK Consider a group of N idenical linear sysems described by he linear model ẋ k = A k + Bu k, y k = C k k = 1, 2,...,N, (15 548

4 where k R n is he sae of he sysem, u k R m is he conrol vecor, and y k R p is he oupu. For noaional convenience i is possible o rewrie (15 in compac form as ẋ = ÃN + B N u y = C N y. (16 The sae feedback conroller of Theorem 3 is easily eended o an oupu feedback conroller if we assume observabiliy of he pair (A, C. Pick an observer mari H such ha A + HC is Hurwiz and consider he oupu feedback conroller ξ = N KN ξ + ˆL n ((ˆ ξ ˆ = à N ˆ + B N u + H N (ŷ y u = K (17 N ξ ŷ = C N ˆ, where observabiliy is assumed and H is a suiable observer mari. The convergence analysis is similar o he one for Theorem 2 and is mainly based on he observaion ha he esimaion error is decoupled from he consensus dynamics. Theorem 4: Assume ha he open-loop sysem (4 is sabilizable and observable and ha all he eigenvalues of A belong o he closed lef-half comple plane. Assume ha he communicaion graph is uniformly conneced and he Laplacian is piecewise coninuous and bounded. Then for any gain marices K and H such ha A+BK and A+HC are Hurwiz, he soluions of (5 wih he dynamic conroller (17 eponenially synchronize o a soluion of ẋ = A. Proof: Define s k = ˆ k ξ k and e k = k ˆ k, and rewrie he closed loop sysem as ẋ = N KN + B N KN (e + s ṡ = ÃNs ˆL n s ė = (ÃN + H N CN e. This sysem is he cascade of he closed-loop sysem analyzed in he proof of Theorem 3 wih an eponenially sable esimaion error dynamics, which proves he resul. Theorem 4 provides a general synchronizaion resul for linear sysems bu he soluion requires a dynamic conroller. For he sake of comparison, we provide a se of sufficien condiions o prove synchronizaion under a simple saic oupu feedback (diffusive inerconnecion. These sufficien condiions assume a passiviy propery for he sysem (A, B, C, ha is, he eisence of a symmeric posiive definie mari P > ha verifies PA + A T P, B T P = C. (18 Passiy condiions have been considered previously in [19] (where i is assumed ha he communicaion opology is bidirecional and srongly conneced and in [8] (where synchronizaion is sudied for a class of (nonlinear oscillaors assuming ha he communicaion opology is ime-invarian and balanced. Assumpions A1 and A2 below lead o a imevarying eension of he resuls in [8] and [19] in he special case of linear sysems. Theorem 5: Consider sysems (15 equipped wih he saic oupu feedback conrol laws u k = a kj ((y j y k. Le he graph Laplacian mari L( be piecewise coninuous and bounded. Then eponenial synchronizaion o a soluion of ẋ = A is achieved under eiher one of he following assumpions: A1. The sysem (A, B, C is passive and observable, he communicaion graph is conneced and balanced a each ime; A2. The sysem (A, B, C is passive and observable, he communicaion graph is symmeric, i.e. he Laplacian mari can be facorized as L = DD T (, and he pair (ÃN, ˆD p T C N is uniformly observable. Proof: Supppose firs ha assumpion A1 holds and consider he mari P soluion of (18. Consider he Lyapunov funcion V ( = 1 2 (ˆΠ n T PN (ˆΠ n, (19 he derivaive along he soluions of he closed loop sysem is V ( = 1 2ẋT ˆΠn PN ˆΠn à N T ˆΠn PN ˆΠn à N ẋ. (2 By using he commuaion propery of Kronecker producs (see e.g. [16] and he passiviy relaion (18 we obain V ( = 1 2 T ˆΠn ( P N à N + ÃT N P N ˆΠ n T CT N Π pˆlsym p (ˆΠ p y y T ˆΠpˆLsym p (ˆΠ p y. (21 Because he graph is balanced, he mari L sym ( (L(+ L T (/2 is posiive semi-definie for each and (ˆΠ p y T ˆLsym p (ˆΠ p y λ 2 ˆΠ p y 2, where λ 2 = inf λ 2 (, and λ 2 ( is he algebraic conneciviy of he graph a ime. Noe ha λ 2 > because he graph is conneced a each ime and he values of he adjacency mari relaed o he conneced componens are assumed o be bounded away from zero (see Secion II. This allows o rewrie (21 as V ( λ 2 ˆΠ p y 2, λ 2 >. (22 Inegraing (22 over he inerval [, + T] where T > is arbirary, we obain V d λ 2 ˆΠ p y 2 d γλ (23 2 ˆΠ n ( 2, γ >, for all (, where he las inequaliy follows from he observabiliy condiion of he pair (A, C. We conclude from a sandard Lyapunov argumen ha he soluions eponenially synchronize. 549

5 Assume ha assumpion A2 holds. Firs observe ha from he symmery of he communicaion graph he Laplacian mari can be facorized as L( = DD T (. Uniform observabiliy of he pair (ÃN, ˆD p T C N means ha for all > here eis posiive consans T and α (independen from such ha ΦN (, T CT N ˆDp ˆDT p (τ C N ΦN (, d αi nn, (24 where Φ(, τ is he ransiion mari. This implies ha he sysem ẋ = ÃN z = ˆD p T( C (25 N, is uniformly observable. Applying oupu injecion o sysem (25 we obain ẋ = ÃN K( D p T C N z = ˆD p T ( C N. (26 1 Choose K( P C N N T ˆD p T ( and observe ha, since L( is bounded, K( belongs o L 2 (, + T. Then oupu injecion preserves observabiliy (see [2] and references herein and he sysem ẋ = Ã B N ˆDT p ˆDp ( C N z = ˆD p T( C N (27 is sill uniformly observable (here we have also used he passiviy condiion C N = B N T P N. Therefore for all > here eis posiive consans T and β (independen from such ha for every ( z 2 d = y( T ˆDp ˆDT p (y(d β. (28 Consider he Lyapunov funcion (19. Inegraing is ime derivaive over he inerval [, + T] where T > is arbirary we obain V d ˆΠ p ˆDp ˆDT p (y 2 d σ ˆΠ n ( 2, σ >. We conclude from sandard Lyapunov resuls ha he soluions asympoically synchronize. VI. EXAMPLES The condiions of Theorem 5 are only sufficien condiions for eponenial synchronizaion under diffusive coupling. We provide wo simple eamples o illusrae ha hese condiions are no far from being necessary when considering ime-varying and direced graphs and ha he inernal model of he dynamic conroller (12 plays an imporan role in such siuaions. Eample 1: Synchronizaion of harmonic oscillaors Consider a group of N harmonic oscillaors ẋ 1k = 2k ẋ 2k = 1k + u k, (29 Fig. 1. The ime-varying communicaion opology used in Eample 1 and Eample Fig. 2. Firs componen of he soluions of he closed loop harmonic oscillaors by using he dynamic conrol law (o he lef and he saic conrol law (31 (o he righ. The dynamic conrol ensures eponenial synchronizaion. In conras, synchronizaion is no observed wih he diffusive inerconnecion. for k = 1, 2,...,N, which corresponds o sysem (4 wih ( ( 1 A =, B =. 1 1 The assumpions of Theorem 2 are saisfied: A is Lyapunov sable and (A, B is sabilizable. Choosing he sabilizing gain K = ( 1, he dynamic conrol law (12 yields he closed-loop sysem ẋ 1k = 2k ẋ 2k = 1k ξ 2k ξ 1k = ξ 2k + N a kj((ξ 1j ξ 1k + 1k 1j ξ 2k = ξ 1k ξ 2k + N a kj((ξ 2j ξ 2k + 2k 2j. (3 Theorem 3 ensures eponenial synchronizaion of he oscillaors o a soluion of he harmonic oscillaor if he graph is uniformly conneced. Fig. 2 illusraes he simulaion of a group of 4 oscillaors coupled according o he imevarying communicaion opology shown in Fig. 1 (he period T is se o 7 sec. The dynamic conrol ensures eponenial synchronizaion. In conras, synchronizaion is no observed wih he diffusive inerconnecion u k = a kj (( 2j 2k. (31 The sysem (A, B, C is neverheless passive, meaning ha sronger assumpions on he communicaion graph would ensure synchronizaion wih he diffusive coupling (31. We 55

6 Fig. 3. Firs componen of he soluions of he closed loop double inegraors by using he dynamic conrol law (o he lef and he saic conrol law (34 (o he righ. The dynamic conrol ensures eponenial synchronizaion. In conras synchronizaion is no observed wih he diffusive inerconnecion. menion he recen resul [15] ha proves (in discree-ime synchronizaion of harmonic oscillaors wih diffusive coupling under he assumpion ha he graph is ime-invarian and conneced. The following eample illusraes an analog scenario wih unsable dynamics. Eample 2: Consensus for double inegraors Consider a group of N double inegraors ẋ 1k = 2k ẋ 2k = u k, (32 for k = 1, 2,...,N, which corresponds o sysem (4 wih ( ( 1 A =, B =. 1 The assumpions of Theorem 2 are saisfied: he wo eigenvalues of A are a zero and (A, B is sabilizable. Choosing he sabilizing gain K = ( 1 1, he dynamic conrol law (12 yields closed-loop sysem ẋ 1k = 2k ẋ 2k = ξ 1k ξ 2k ξ 1k = ξ 2k + N a kj((ξ 1j ξ 1k + 1k 1j ξ 2k = ξ 1k ξ 2k + N a kj((ξ 2j ξ 2k + 2k 2j. (33 Theorem 3 ensures eponenial synchronizaion o a soluion of he double inegraor if he graph is uniformly conneced. Fig. 3 illusraes he simulaion of a group of 4 double inegraors coupled according o he ime-varying communicaion opology shown in Fig. 1 (he period T is se o 2 sec. The dynamic conrol ensures eponenial synchronizaion. In conras, synchronizaion is no observed wih he diffusive inerconnecion u k = a kj ((y j y k, y k = 1k + 2k. (34 The mari A αbc is neverheless sable for every α >, suggesing ha sronger assumpions on he communicaion graph would ensure synchronizaion. VII. CONCLUSION AND FUTURE WORK In his paper he problem of synchronizing a nework of idenical linear sysems described by he sae-space model (A, B, C under general inerconnecion opologies has been addressed. A dynamic conroller ensuring eponenial convergence of he soluions o a synchronized soluion of he decoupled sysems is provided assuming ha (i A has no eponenially unsable mode, (ii (A, B is sabilizable and (A, C is observable, and (iii he communicaion graph is uniformly conneced. Sronger condiions are shown o be sufficien (and, o some een, also necessary o ensure synchronizaion wih he ofen considered saic diffusive oupu coupling. The eension of he proposed echnique for synchronizaion of nonlinear sysems is he subjec of ongoing work. REFERENCES [1] L. Moreau, Sabiliy of muli-agen sysems wih ime-dependen communicaion links, IEEE Trans. on Auomaic Conrol, vol. 5, no. 2, pp , 25. [2], Sabiliy of coninuous-ime disribued consensus algorihms, in Proceedings of he 43rd IEEE Conference on Decision and Conrol, Paradise Island, Bahamas, 24, pp [3] V. D. Blondel, J. M. Hendrick, A. Olshevsky, and J. N. Tsisiklis, Convergence in muliagen coordinaion, consensus, and flocking, in Proceedings of he 44nd IEEE Conference on Decision and Conrol and European Conrol Conference, Seville, Spain, 25, pp [4] A. Jadbabaie, J. Lin, and S. Morse, Coordinaion of groups of mobile auonomous agens using neares neighbor rules, IEEE Tran. on Auomaic Conrol, vol. 48, pp , 23. [5] R. Olfai-Saber and R. Murray, Consensus problems in neworks of agens wih swiching opology and ime-delays, IEEE Trans. on Auomaic Conrol, vol. 49, no. 9, pp , 24. [6] J. K. Hale, Diffusive coupling, dissipaion, and synchronizaion, Journal of Dynamics and Differenial Equaions, vol. 9, pp. 1 52, [7] Q.-C. Pham and J.-J. Sloine, Sable concurren synchronizaion in dynamic sysem neworks, Neural Neworks, vol. 2, no. 1, pp , 27. [8] G.-B. San and R. Sepulchre, Analysis of inerconneced oscillaors by dissipaiviy heory, IEEE Transacions on Auomaic Conrol, vol. 52, no. 2, pp , 27. [9] A. Pogromsky, Passiviy based design of synchronizing sysems, In. J. Bifurcaion and Chaos, vol. 8, pp , [1] L. Scardovi, A. Sarlee, and R. Sepulchre, Synchronizaion and balancing on he N-orus, Sysems and conrol Leers, vol. 56, no. 5, pp , 27. [11] R. Sepulchre, D. Paley, and N. Leonard, Sabilizaion of planar collecive moion wih limied communicaion, IEEE Trans. on Auomaic Conrol, 27, acceped. [12] L. Scardovi, N. Leonard, and R. Sepulchre, Sabilizaion of collecive moion in hree dimensions: A consensus approach, in Proceedings of he 46h IEEE Conference on Decision and Conrol, New Orleans, La, 27, pp [13] S. Nair and N. Leonard, Sable synchronizaion of mechanical sysem neworks, SIAM J. Conrol and Opimizaion, vol. 47, pp , 28. [14] A. Sarlee, R. Sepulchre, and N. Leonard, Auonomous rigid body aiude synchronizaion, in Proceedings of he 46h IEEE Conference on Decision and Conrol, New Orleans, La, 27, pp [15] S. E. Tuna, Synchronizing linear sysems via parial-sae coupling, Auomaica, 28, in press. [16] R. A. Horn, Topics in mari analysis. New York, NY, USA: Cambridge Universiy Press, [17] J. C. Willems, Lyapunov funcions for diagonally dominan sysems, Auomaica J. IFAC, vol. 12, pp , [18] L. Scardovi and R. Sepulchre, Synchronizaion in neworks of idenical linear sysems, Auomaica, 28, submied. [Online]. Available: hp:// [19] M. Arcak, Passiviy as a design ool for group coordinaion, IEEE Transacions on Auomaic Conrol, vol. 52, pp , 27. [2] D. Aeyels, R. Sepulchre, and J. Peueman, Asympoic sabiliy condiions for ime-varian sysems and is relaion o observabiliy, Mahemaics of Conrol, Signals, and Sysems, pp. 1 27,