Singularly perturbed algorithms for dynamic average consensus

Transcription

1 Singularly perurbed algorihms for dynamic average consensus Solmaz S. Kia Jorge Corés Sonia Marínez Absrac This paper proposes wo coninuous-ime dynamic average consensus algorihms for neworks wih srongly conneced and weigh-balanced ineracion opologies. The proposed algorihms, ermed s-order-inpu Dynamic Consensus (FOI-DC) and nd-order-inpu Dynamic Consensus (SOI-DC), respecively, allow agens o rack he average of heir dynamic inpus wihin an O(ǫ)-neighborhood wih a pre-specified rae. The only requiremen on he se of reference inpus is having coninuous bounded derivaives, up o second order for FOI- DC and up o hird order for SOI-DC. The correcness analysis of he algorihms relies on singular perurbaion heory for non-auonomous dynamical sysems. When dynamic inpus are offse from one anoher by saic values, we show ha SOI-DC converges o he exac dynamic average wih no seady-sae error. Simulaions illusrae our resuls. I. ITRODUCTIO Given a muli-agen sysem and a se of ime-varying inpu signals, one per agen, he dynamic average consensus problem consiss of designing disribued algorihms ha allow individual agens o obain he average of he inpus. This problem has applicaions in numerous areas, including mulirobo coordinaion [], disribued esimaion [], sensor fusion [3], [4], and disribued racking []. In his paper, we employ a singular perurbaion approach o design provably correc dynamic consensus algorihms. Lieraure review: The work [3] generalizes he average saic consensus algorihm proposed in [6] o rack he average of inpus wih uniformly bounded rae which are differen from one anoher by zero-mean whie Gaussian noise. The algorihm acs as a low-pass filer which allows agens o rack he average of he agens dynamic inpus wih a nonzero seady sae error, which vanishes in he absence of noise. The work [7] proposes a dynamic average consensus algorihm ha, from a proper iniializaion, is able o rack wih zero seady-sae error he average of dynamic inpus whose Laplace ransfer funcion has all is poles in he lef half-plane, and has a mos one pole a origin. In [8], a proporional dynamic average consensus algorihm can rack, wih a bounded non-zero seady-sae error, he average of reference inpus whose weighed sum wih heir derivaives is bounded. For saic inpus, his algorihm converges wih zero-seady-sae error if i is iniialized properly. This work also proposes a proporional-inegral (PI) algorihm which achieves dynamic average consensus, wih a non-zero seadysae error, provided signals are slowly varying. The PI *This work was suppored by L3 Communicaions hrough he UCSD Cymer Cener for Conrol Sysems and Dynamics. The auhors are wih he Deparmen of Mechanical and Aerospace Engineering, Universiy of California San Diego, La Jolla, CA 993, USA {skia,cores,soniamd}@ucsd.edu. algorihm is generalized in [9] o achieve zero-error dynamic average consensus of special class of ime-varying inpus wih raional Laplace ransforms and no poles in he lef complex half-plane. The aforemenioned algorihms are all designed in coninuous ime. The work [] develops insead a discree-ime dynamic average consensus esimaor ha, wih a proper iniializaion, can rack wih bounded seadysae error he average of he ime-varying inpus whose nhorder difference is bounded. The soluions o he dynamic average consensus problem menioned above each suffer from a leas one of he following shorcomings: hey require proper iniializaions ha makes hem prone o iniializaion errors and no robus o changes o agens joining and leaving he nework; or hey relay on knowledge of he dynamics generaing he inpus a each agen herefore hey are ailored o specified classes of inpus which limis heir applicabiliy. In conras, he algorihms proposed in his paper do no suffer from hese shorcomings. Saemen of conribuions: The saring poin for our algorihm design is he following observaion: given a hypoheical saic average consensus algorihm able o converge infiniely fas, one could solve he dynamic average consensus problem by running i a each ime. In pracice, however, some ime is required for informaion o flow across he nework, and hence he resul of he repeaed applicaion of any saic average consensus algorihm will operae wih some error whose size depends on is speed of convergence and how fas inpus change. A follow-up observaion is ha, in some applicaions, he ask is no jus o obain he average of he dynamic inpus bu raher o physically rack his value, possibly wih limied conrol auhoriy. In hese cases, high rae algorihms migh no be implemenable. We propose wo dynamic average consensus algorihms (ermed s-order- Inpu Dynamic Consensus (FOI-DC) and nd-order-inpu Dynamic Consensus (SOI-DC)) whose design incorporaes wo ime scales, a fas and a slow one. The fas dynamics, which builds on he PI algorihm menioned above, acs in a similar way o a saic average consensus wih a high rae of convergence ha is able o compue he dynamic inpu average a each ime. The slow dynamics allows agens o rack his average a a feasible rae. The novely here is ha hese slow and fas dynamics are running simulaneously and, hus, here is no need o wai for convergence of he fas dynamics and hen ake slow seps owards he inpu average. Our echnical approach uses singular perurbaion heory o sudy he algorihm convergence. We show ha, when he derivaives of he agens inpus are coninuous and bounded (up o second order for FOI-DC and up o hird order for

2 FOI-DC), he algorihms converge o an O(ǫ)-neighborhood of he dynamic inpu average. Here, ǫ is a design parameer. Our algorihms do no require any specific iniializaion and do no relay on knowledge of he dynamics generaing he inpus. We also show how an appropriae variaion of our algorihms allows each agen o converge a heir own desired rae of convergence. Simulaions illusrae our resuls. Organizaion: Secion II inroduces basic noaion and conceps from graph heory, saic consensus, and singular perurbaion heory. Secion III presens he problem saemen. Secion IV moivaes he use of singular perurbaion heory o solve he dynamic average consensus problem. Secion V inroduces wo novel dynamic average consensus algorihms and analyzes heir correcness. Secion VI illusraes heir performance in simulaion. II. PRELIMIARIES This secion gahers basic preliminaries on noaion, graph heory, saic consensus and singularly perurbed dynamical sysems. A. oaion The vecor n represens a n-dimensional vecor wih all elemens equal o one, and I n represens he ideniy marix wih dimensionn n. We denoe bya he ranspose of marix A. We use Diag(A,,A ) o represen he blockdiagonal marix consruced from marices A,...,A. We le δ (ǫ) O(δ (ǫ)) denoe he fac ha here exis posiive consans c and k such ha δ (ǫ) k δ (ǫ), ǫ < c. For nework relaed variables, he local variables a each agen are disinguished by a superscrip i, e.g., u i is he local inpu of agen i. We denoe he aggregae vecor of local variables p i s by p = (p,...,p ) R. B. Graph heory Here, we briefly review some basic conceps from graph heory and linear algebra following []. A direced graph, or simply a digraph, is a pair G = (V, E), where V = {,...,} is he node se and E V V is he edge se. We make he convenion ha an edge from i o j, denoed by (i,j), models he fac ha agen j can send informaion o i. For an edge (i,j) E, i is called an in-neighbor of j and j is called an ou-neighbor of i. A direced pah is an ordered sequence of verices such ha any ordered pair of verices appearing consecuively is an edge of he digraph. A digraph is called srongly conneced if for every pair of verices here is a direced pah beween hem. A weighed digraph is a riple G = (V,E,A), where (V,E) is a digraph and A R is a weighed adjacency marix wih he propery ha a ij > if (i,j) E and a ij =, oherwise. A weighed digraph is undireced if a ij = a ji for all i,j V. The weighed ou-degree and weighed indegree of a node i, are respecively, d in (i) = j= a ji and d ou (i) = j= a ij. A digraph is weigh-balanced if a each node i V, he weighed ou-degree and weighed in-degree coincide (alhough hey migh be differen a differen nodes). The ou-degree marix D ou is he diagonal marix whose D ou ii = d ou (i), for i V. The (ou-) Laplacian maris L = D ou A. Based on he srucure of L, a leas one of he eigenvalues of L is zero and he res of hem have nonnegaive real pars. Also, L =. For a srongly conneced digraph, zero is a simple eigenvalue of L. A digraph G is weigh-balanced if and only if T L =. C. Saic consensus Here, we briefly review he soluion given in [8] o find in a disribued way he average of a se of saic inpus. Le G be a srongly conneced and weigh-balanced digraph. Assume each nodei {,...,} has access o a saic inpuu i R. Consider he dynamics ẋ i = ( u i ) j= L ijx j j= L ijν j, ν i = j= L jix j. The paper [8] esablishes ha, saring from any iniial condiion (),ν i () R, he variable converges o j= uj exponenially fas for all i {,...,}. oe ha he disribued implemenaion of his algorihm requires each agen o know he weighs of is in-neighbors. If he graph is undireced, his requiremen is rivially saisfied. D. Singularly perurbed dynamical sysems Here we give a shor accoun of he erminology employed in singularly perurbed dynamical sysems following [, Chaper ]. Le () ẋ = f(,x,z,ǫ), x( ) = η(ǫ), (a) ǫż = g(,x,z,ǫ), z( ) = ζ(ǫ). (b) The use of a small consan ǫ > creaes wo-ime scales in he sysem, resuling ino a fas and a slow dynamics. Singular perurbaion heory esablishes precise condiions under which he behavior of he sysem follows ha of he limiing sysem when ǫ goes o. We assume ha f and g are coninuously differeniable in heir argumens for (,x,z,ǫ) [, ) D x D z [,ǫ ], where D x R n and D z R m are open conneced ses. When we se ǫ = in (), he dimension of he sae equaion reduces from n+m o n because he differenial equaion (b) degeneraes ino he algebraic equaion = g(,x,z,). (3) We say ha he model () is in sandard form if (3) has k isolaed real roos z i = h i (,x), i {,...,k}, (4) for each (,x) [, ) D x. This assumpion assures ha a well-definedn dimensional reduced model (slow dynamics)

3 will correspond o each roo of (3). To obain he ih reduced model, we subsiue (4) ino (a), a ǫ =, o obain ẋ = f(,x,h(,x),), () where we have dropped he subscrip i from h. The boundary-layer sysem (fas dynamics) is dz dτ = g(,x,z,), τ = ǫ, (6) where x and are reaed as fixed parameers. The sabiliy and convergence properies of hese dynamical sysems can be esablished via [, Theorem.]. III. PROBLEM STATEMET We consider a nework of agens wih single-inegraor dynamics given by ẋ i = c i, i {,...,}, (7) where R is he agreemen sae andc i R is he driving command of agen i. The nework ineracion opology is modeled by a weighed digraph G. Agen i {,...,} has access o a ime-varying inpu signal u i : [, ) R. The problem we seek o solve is saed nex. Problem : (Dynamic average consensus wih pre-specified leas rae of convergence): Le G be srongly conneced and weigh-balanced. Design a disribued algorihm such ha each in (7) racks he average j= uj () of he inpus wih a convergence rae less han or equal o β >, i.e., κ > such ha () j= uj () κ () j= uj () e β for all. For vecor-valued inpus, one can apply he soluion of Problems in each dimension independenly. oe ha he algorihm () provides a soluion o Problem for saic inpus and no pre-specified rae of convergence. I is worh noicing he fac ha β in he problem saemen is an upper bound of he convergence rae, no a lower bound. This is moivaed by scenarios where agens have limied conrol auhoriy and canno implemen arbirary driving commands dicaed by he consensus algorihm. This is normally he case when (7) corresponds o a model of a physical process. IV. MOTIVATIO TO USE SIGULAR PERTURBATIO THEORY FOR THE DESIG OF COSESUS ALGORITHMS In his secion, we explain he raionale behind he design of our algorihmic soluions o solve Problem. The simples dynamics ha achieves for each agen () j= uj (), as, exponenially fas wih rae β, is he following ( ẋ i = β j= ) u j () + u i (). j= To decenralize his dynamics, we can make use of a mechanism ha generaes he average of he inpus and also he average of he derivaive of inpus, rapidly, in each agen in a disribued fashion. Then, he disribued dynamic consensus algorihm becomes a wo-ime scale operaion, a fas dynamics o generae each average and a slow dynamics o rack he inpu average. Such dynamics could be realized by means of he following mixed discree/coninuous-ime algorihm running synchronously a each node i {,...,}. : (Iniializaion) a k = iniialize () R n : while daa exiss do 3: Obain inpus u i (k) and u i (k) 4: Iniialize z i (),ν i () R : Solve he following dynamical equaion ż i () = (z i ()+βu i (k)+ u i (k)) i=j L ij(z j ()+ν j ()), ν i () = j= L jiz j (), 6: Le z i converge o equilibrium z i 7: Define: 8: k k + 9: end while (8) (k +) = (k) β ( (k)+ z i (k) ) (9) In he above algorihm is he sepsize. Building on he discussion in Secion II-C, a each imesep k, he dynamical sysem (8) acs as a saic consensus algorihm wih saic inpuβu i (k)+ u i (k). This algorihm converges exponenially o z i (k) = j= (βui (k) + u i (k)). Therefore, a any imesep k, for all i {,...,}, (9) becomes ( (k +)= (k) β (k) ) (u i (k)+ u i (k)). j= For small, he sabiliy and convergence of he above difference equaion can be sudied using he following coninuous-ime model where ẏ i = βy i, i {,...,}. () y i = u j, i {,...,}. () j= The dynamical sysem () is a sable linear sysem wih eigenvalue β. Therefore, i converges o zero exponenially fas wih rae β. As a resul, in (9) converges o j= uj () exponenially, for all i {,...,}. The aforemenioned algorihm solves Problem, provided (8) converges o is equilibrium in a ime inerval wih lengh. Hence, his algorihm is only concepual: he cos of solving (8) a each imesep makes i un-implemenable. Inspired by he muli-ime scale srucure observed above, we use singular perurbaion heory o weave ogeher seps 7 and devise a coninuous-ime dynamic average consensus algorihm. By doing so, we avoid solving he fas dynamics a each ieraion, i.e., he slow dynamics does no need o wai for he fas dynamics o converge.

4 V. DYAMIC COTIUOUS-TIME COSESUS ALGORITHMS VIA SIGULARLY PERTURBED DYAMICS In his secion, we presen novel coninuous-ime dynamic average consensus algorihms whose design is based on singular perurbaion heory. For β > and i {,...,}, consider he following dynamical sysems s-order-inpu Dynamic Consensus (FOI-DC): { ǫż i = (z i +βu i + u i ) i=j L ij(z j +ν j ), ǫ ν i = j= L jiz j, (a) ẋ i = β z i, (b) nd-order-inpu Dynamic Consensus (SOI-DC): ǫż i = (z i +βu i + u i ) j= L ij(z j +ν j ) ǫ(β u i +ü i ), ǫ ν i = j= L jiz j, (3a) ẋ i = β z i, (3b) The following resul esablishes in wha sense s-order- Inpu Dynamic Consensus solves Problem. Theorem. (Convergence of FOI-DC): Le G be a srongly conneced and weigh-balanced digraph. Assume ha he firs and he second derivaives of he inpu signal u i a each agen i {,...,} are coninuous and bounded for. Then, here exiss ǫ > such ha, for all ǫ (,ǫ ], saring from any iniial condiions x(),z(),ν() R, he sae, i {,...,}, of he algorihm () converges exponenially fas wih rae β o an O(ǫ)-neighborhood of j= uj (). Proof: We show he algorihm saisfies he condiions of [, Theorem.] globally (for he erminology used here we refer o Secion II-D). The boundary-layer (fas) dynamics of he algorihm () is, for i {,...,}, dz i dτ = (zi +βu i ()+ u i ()) i= L ij(z j +ν j ), dν i dτ = i= L jiz j. Invoking he discussion in Secion II-C, his fas dynamics globally exponenially converges o z i = (βu i + u i ), i {,...,}. (4) j= Subsiuing (4) ino (b), and using he change of variables (), we obain () as he reduced sysem (slow dynamics) model. For β >, () is a sable linear sysem wih sysem marix eigenvalue equal o β. Thus, for all i {,...,}, y i () converges globally exponenially fas o zero wih a rae of β, which is equivalen o () u j () () j= u j () e β. () j= Based on he required condiions for inpu signals, he algorihm FOI-DC saisfies he differeniabiliy and Lipschiz condiions of [, Theorem.] on any compac se of (x, z, ν). Thus, all he condiions of [, Theorem.] are saisfied globally. As a resul, for all i {,...,}, (,ǫ) () O(ǫ) where (,ǫ) is he soluion of he singularly perurbed sysem () and () is he soluion of he slow dynamics. Recall (), hen for all i {,...,} and all we have (,ǫ) u j () < O(ǫ)+ () j= which concludes our proof. u j () e β, j= The following resul esablishes in wha sense nd-order- Inpu Dynamic Consensus solves Problem. Theorem. (Convergence of SOI-DC): Le G be a srongly conneced and weigh-balanced digraph. Assume ha he firs, second, and hird derivaives of he inpu signal u i a each agen i {,...,} are coninuous and bounded for. Then, here exiss ǫ > such ha, for all ǫ (,ǫ ], saring from any iniial condiions x(),z(),ν() R, he sae, i {,...,}, of he algorihm (3) converges exponenially fas wih rae β o an O(ǫ)-neighborhood of j= uj (). Proof: The proof of his resul is very similar o he proof of Theorem. so we only provide a brief skech. oice ha he parallelism beween he slow and fas dynamics of SOI-DC and FOI-DC. As shown in he proof of Theorem., hese dynamics are boh globally exponenially sable. Based on he required condiions for he inpu signals, he algorihm SOI-DC also saisfies he differeniabiliy and Lipschiz condiions of [, Theorem.] on any compac se of (x,z,ν). Thus, all he condiions of [, Theorem.] are saisfied globally. I is worh noicing ha, wih respec o he algorihms available in he lieraure, boh () and (3) perform racking wih a pre-specified rae of convergence, can handle arbirary iniial condiions (and are herefore robus o iniializaion errors) and do no require any knowledge of he dynamics generaing he inpus. In he following, we show ha he algorihm SOI-DC has some advanages over FOI-DC a he expense of he exra condiion on he inpu signals. Lemma.: (SOI-DC for inpus offse by a saic value): Le G be a srongly conneced and weigh-balanced digraph. Assume ha he difference in he inpu signals is a saic offse, i.e., u i () = u c ()+ū i, where ū i is a consan scalar for each i {,...,}. Then, saring from any iniial condiions x(),z(),ν() R, for any ǫ > and β >, he algorihm SOI-DC converges exponenially fas o he exac inpu average, i.e., () j= uj () as for all i {,...,}. Proof: Consider he following change of variables: p = z +(βu c + u c ), q = x u c, η = [ r R ] ν,

5 where r = and R R saisfies R R = I and r R =. We le η = (η,η : ) where η R and η : R. Then, we can re-wrie (3) as follows ǫṗ = (p+βū) Lp LRη :, ǫ η : = R L p, ǫ η =, q = βq p. (6a) (6b) (6c) (6d) We can show ha he equilibrium poin of his sysem is ( p = ( β j=ūj ), η : = β(r LR) R ū, η = j= νj (), q = ( j=ūj ) ). Consider he following Lyapunov funcion where q = q q, p = p p and η : = η : η :. V = β q q + ǫ 8 p p+ ǫ 8 η : η :. The derivaive of his Lyapunov funcion along he rajecories of (6a), (6b) and (6d) is V = 8 p (L+L ) p ( p+β q) ( q), p+β which for srongly conneced digraph, i is negaive semidefinie. For a srongly conneced and weigh-balanced digraph, we have S = { p, q R, η : R V = } = { p, q R, η : R p = α, p = β q, α R}. ex, we show ha no soluion of (6a), (6b) and (6d) can say in S excep { p =, q =, η : = }. A rajecory (p(),q(),η : ()) belonging o S mus saisfy p() α() and p() β q(). Then, (6a), (6b) and (6d) become, respecively, ǫ α = α LR η :, ǫ η : =, α = βα. (7a) (7b) (7c) From (7b), η : () mus be consan. Recall ha for srongly conneced and weigh-balanced digraphs R LR is inverible. Therefore, muliplying (7a) by R from he lef, we conclude ha η : =. As a resul, from (7a) and (7c) we deduce ha α() =. In oher words, { p =, q =, η : = } is he only soluion of (6) ha idenically belongs o S. Invoking he LaSalle invarian principle [, Theorem 4.4 and Corollary 4.]), we conclude ha q and as a resul j= uj (), for all i {,...,}, globally asympoically as. The sysems (6a), (6b) and (6d) are linear ime-invarian, herefore he rae of convergence is exponenial. Remark.: (Relaionship beween he size of β and ǫ): As saed in Theorems., and., β is he convergence rae of o an O(ǫ)-neighborhood of j= uj (). One can increase he rae of convergence by choosing a largeβ. However, o keep he wo-ime scale srucure of he algorihms, one would hen be forced o use a smaller ǫ. Quanifying his rade-off, and specifically he range of admissible values of ǫ for a given β, is lef as fuure work. We conclude his secion by describing a variaion of he algorihms FOI-DC and SOI-DC ha does no require all agens o use he same parameer β o guaranee convergence. Consider he dynamical sysem { ǫż i = (z i +u i ) i=j L ij(z j +ν j ), ǫ ν i = j= L jiz j (8a), { ǫẏ i = (y i + u i ) i=j L ij(y j +µ j ), ǫ µ i = j= L jiy j (8b), ẋ i = β i β i z i y i, (8c) where β i > s for all i {,...,}. oe ha his algorihm has he benefi of each agen using is own local parameer. The drawback is he need for addiional disribued processing and communicaion. The following resul characerizes he convergence properies of his algorihm. Is proof is along he same lines of he proof of Theorem. and omied for breviy. Theorem.3 (Convergence of (8)): Le G be a srongly conneced and weigh-balanced digraph. Assume ha he firs and he second derivaives of he inpu signal u i a each agen i {,...,} are coninuous and bounded for. Then, here exiss ǫ > such ha, for all ǫ (,ǫ ] saring from any iniial condiions x(),z(),ν() R, he sae, i {,...,}, of he algorihm (8) converges exponenially fas wih rae β i o an O(ǫ)-neighborhood of j= uj (). VI. UMERICAL EXAMPLES Here, we presen hree numerical examples o demonsrae he performance of he algorihms FOI-DC, SOI-DC and (8). Firs, we consider a randomly generaed undireced nework (using Malab BGL package [3]) consising of = agens. The local inpu signals are u i () = a i sin(b i +c i ), i {,...,}, (9) where he inpu coefficiens are generaed randomly uniformly in he following ranges: a i U[,], b i U[,], c i U[,π/]. Figure shows he ime hisories of he local inernal saes generaed by he algorihm FOI-DC for differen values of ǫ and β. Figure 3 demonsraes he performance of he algorihms FOI-DC and SOI-DC when he difference in he inpu signals is a saic offse. Figure shows he nework and inpus employed (we se θ = in all he inpu signals). Figure 4 demonsraes he performance of he algorihm (8) when agens use differen β s. Figure shows he nework and inpus employed (here, we se θ = in all he inpu signals). VII. COCLUSIOS We have proposed wo coninuous-ime dynamic average consensus algorihms for neworks wih srongly conneced and weigh-balanced ineracion opologies. The proposed sraegies have a wo-ime scale srucure and do no require model informaion on he dynamic inpus. Using singular

6 (a) ǫ =. and β = (b) ǫ =. and β = (c) ǫ =. and β = 3 Fig. : Performance evaluaion of he algorihm FOI-DC wih respec o differen choices for ǫ and β using a random nework of = agens and inpus given in (9): Smallerǫresuls in smaller error and largerβ resuls in faser convergence. The solid blue line is he average of he inpus and he dashed lines are he agreemen saes of agens. 3 4 u ()=sin()+θ + +, u ()=sin()+θ., u 3 ()=sin()+θ cos(.)+4, u 4 ()=sin()+θlog(+)+, u ()=sin()+θ aan()+. Fig. : A digraph and he corresponding inpu a each agen. error error (a) ǫ = and β = (b) ǫ =. and β = Fig. 3: Performance evaluaion of he algorihms FOI-DC and SOI-DC wih respec o inpu signals which are differen from one anoher by saic values. Figure shows he nework and inpus employed (here, wih θ = ). Dashed (resp. solid) lines represen he error beween he agreemen saes of he algorihm FOI-DC (resp. SOI-DC) and he inpu average. As guaraneed by Lemma., he algorihm SOI- DC converges wih zero seady-sae error for arbirary values of ǫ. x x x 3 x 4 x Inpu Average Fig. 4: Execuion of he algorihm (8) over he neworked sysem of Figure, wih θ =, ǫ =. and β =., β =, β 3 =., β 4 =.4, β =.. Agen i has rae of convergence β i, hence, agen has he slowes rae of convergence. perurbaion analysis, we have shown ha he algorihms reach an O(ǫ)-neighborhood of he dynamic inpu average wih an exponenial rae irrespecive of he iniial condiions. Fuure work will be devoed o rigorously characerizing he O(ǫ)-convergence neighborhood and exending he resuls o neworks wih swiching opology. REFERECES [] P. Yang, R. Freeman, and K. Lynch, Muli-agen coordinaion by decenralized esimaion and conrol, IEEE Transacions on Auomaic Conrol, vol. 3, no., pp , 8. [] S. Meyn, Conrol Techniques for Complex eworks. Cambridge Universiy Press, 7. [3] R. Olfai-Saber and J. Shamma, Consensus filers for sensor neworks and disribued sensor fusion, in IEEE In. Conf. on Decision and Conrol and European Conrol Conference, (Seville, Spain), pp , December. [4] R. Olfai-Saber, Disribued Kalman filering for sensor neworks, in IEEE In. Conf. on Decision and Conrol, (ew Orleans, LA), pp , Dec. 7. [] P. Yang, R. Freeman, and K. Lynch, Disribued cooperaive acive sensing using consensus filers, in IEEE In. Conf. on Roboics and Auomaion, (Roma, Ialy), pp. 4 4, April 7. [6] R. Olfai-Saber and R. Murray, Consensus problems in neworks of agens wih swiching opology and ime-delays, IEEE Transacions on Auomaic Conrol, pp. 33, Sepember 4. [7] D. P. Spanos, R. Olfai-Saber, and R. M. Murray, Dynamic consensus on mobile neworks, in IFAC World Congress, (Prague, Czech Republic), July. [8] R. A. Freeman, P. Yang, and K. M. Lynch, Sabiliy and convergence properies of dynamic average consensus esimaors, in IEEE In. Conf. on Decision and Conrol, (San Diego, CA), pp , Dec. 6. [9] H. Bai, R. Freeman, and K. Lynch, Robus dynamic average consensus of ime-varying inpus, in IEEE In. Conf. on Decision and Conrol, (Alana, GA, USA), pp , December. [] M. Zhu and S. Marínez, Discree-ime dynamic average consensus, Auomaica, vol. 46, no., pp. 3 39,. [] F. Bullo, J. Corés, and S. Marínez, Disribued Conrol of Roboic eworks. American Mahemaical Sociey, Princeon Universiy Press, 9. Available a hp:// [] H. K. Khalil, onlinear Sysems. Prenice Hall, 3 ed.,. [3] D. Gleich, Models and Algorihms for PageRank Sensiiviy. PhD hesis, Sanford Universiy, Sepember 9. Chaper 7 on MalabBGL.