GRAPH EFFECTIVE RESISTANCE AND DISTRIBUTED CONTROL: SPECTRAL PROPERTIES AND APPLICATIONS

Transcription

1 GRAPH EFFECTIVE RESISTANCE AND DISTRIBUTED CONTROL: SPECTRAL PROPERTIES AND APPLICATIONS Prabir Barah Jã P. Hespanha Abstract We intrduce the cncept f matrix-valued effective resistance fr undirected matrix-weighted graphs. Effective resistances are defined t be the square blcks that appear in the diagnal f the inverse f the matrix-weighted Dirichlet graph Laplacian matrix. Hwever, they can als be btained frm a generalized electrical netwrk that is cnstructed frm the graph, and fr which currents, vltages and resistances take matrix values. Effective resistances play an imprtant rle in several prblems related t distributed cntrl and estimatin. They appear in least-squares estimatin prblems in which ne attempts t recnstruct glbal infrmatin frm relative nisy measurements; as well as in mtin cntrl prblems in which agents attempt t cntrl their psitins twards a desired frmatin, based n nisy lcal measurements. We shw that in either f these prblems, the effective resistances have a direct physical interpretatin. We als shw that effective resistances prvide bunds n the spectrum f the graph Laplacian matrix and the Dirichlet graph Laplacian. These bunds can be used t characterize the stability and cnvergence rate f several distributed algrithms that appeared in the literature. I. INTRODUCTION This paper cnsiders undirected graphs with a weight assciated with each ne f its edges. The edge-weights are symmetric psitive definite matrices. Fr such graphs we intrduce the cncept f effective resistances. The effective resistance f a nde is defined t be a square matrix blck that appears in the diagnal f the inverse f the matrix-weighted Dirichlet graph Laplacian matrix (cf. Sectin II). The terminlgy effective resistance is mtivated by the fact that these matrices als define a linear map frm currents t vltages in a generalized electrical netwrk that can be cnstructed frm the undirected matrix-weighted graph. Hwever, the vltages, currents, and resistances in this generalized electrical netwrk take matrix values ]. Effective resistances in regular electrical netwrks, where currents vltages and resistances are scalar valued, have been knwn t have far reaching implicatins in a variety f prblems. Recurrence and transience in randm walks in infinite netwrks 2] and the cverage and cmmute times f randm walks in graphs 3] can be determined by the effective resistance. There is a als a strng cnnectin between effective resistances and estimatin errr variances that arise in the estimatin f scalar-valued variables using This material is based upn wrk supprted by the Institute fr Cllabrative Bitechnlgies thrugh grant DAAD9-03-D-0004 frm the U.S. Army Research Office and by the Natinal Science Fundatin under Grant N. CCR Bth authrs are with the Dept. f Electrical and Cmputer Engineering and the Center fr Cntrl, Dynamical-Systems, and Cmputatin at Univ. f Califrnia, Santa Barbara, CA relative measurements 4]. It was later shwn by Barah and Hespanha 5] that this analgy can be extended t vectr measurements with matrix-valued cvariances, by intrducing generalized electrical netwrks with matrixvalued currents, vltages, and resistances. Crrespndingly, the effective resistance in a generalized electrical netwrk is matrix-valued. Every graph with psitive definite matrix weights can be thught f as a generalized electrical netwrk where the edge weights are the inverse-resistances. The effective resistances f matrix-weighted graphs play an imprtant rle in several prblems related t distributed cntrl and estimatin. We justify this statement in Sectin IV by presenting several such prblems. The first f these prblems is frmulated in Sectin IV-A and cnsists f estimating a certain number f variables, based n nisy relative measurements. The adjective relative refers t the fact that the measurements nly prvide infrmatin abut pairwise differences between the unknwn variables. This prblem was cnsidered in 4 7] and has multiple applicatins t sensr netwrks, including time synchrnizatin and sensr lcalizatin. It turns ut that the cvariance matrices f the estimatin errrs are given by the effective resistances f a graph that describes the measurement mdel. A secnd prblem is frmulated in Sectin IV-B and cnsists f cntrlling a grup f mbile agents twards a frmatin defined by the desired relative psitins between the agents. Each agent has available fr cntrl nisy measurements f its relative psitin with respect t a small set f neighbrs. We cnsider a prprtinal negativefeedback cntrl law that drives each agent t the current best estimate f where it shuld be, based n the current psitins f its neighbrs. Fr this law, the steady-state cvariance matrices f the vehicles psitins are given by the effective resistances f a graph that describes the available measurements. In Sectin IV-C we cnsider the cntrl f a swarm f vehicles based n relative measurements. We cnsider the setup intrduced by Fax and Murray 8] and shw that stability f the grup f vehicles is guaranteed when the Nyquist plt f the pen-lp system des nt encircle any pint in a line segment between the symmetric f the sum f the traces f the effective resistances and the rigin f the cmplex plane. A similar cnditin culd be develped fr the frmatin cntrl prblem in Yadlapalli et al. 9]. The stability cnditin in terms f effective resistances is typically mre cnservative than the ne that appeared in 8], but it is useful because effective resistances

2 have a physical interpretatin that can be used t build intuitin. Mre imprtantly, it is pssible t cnstruct upperand lw-bunds n the effective resistances fr large-scale graphs ]. II. GRAPH EFFECTIVE RESISTANCES An undirected matrix-weighted graph is a triple G = (V, E, W), where V is a set f n vertices; E V V a set f m edges; and W := {W u,v R k k : (u, v) E} a set f symmetric psitive definite matrix-valued weights fr the edges f G. Since we are dealing with undirected graphs, the pairs (u, v) and (v, u) dente the same edge. Fr simplicity f ntatin, we exclude the existence f multiple edges between the same pair f ndes and als edges frm a nde t itself. The matrix-weighted Laplacian f G is a nk nk matrix L with k rws and k clumns per nde such that the k k blck f L crrespnding t the k rws assciated with nde u V and the k clumns assciated with nde v V is equal t W u,v u = v v N u W u,v (u, v) E 0 (u, v) / E. where N u V dentes the set f neighbrs f u, i.e., the set f ndes that have an edge in cmmn with u. The matrix-weighted Dirichlet r Grunded Laplacian is btained frm the Laplacian by remval f rws and clumns. In particular, given a subset V V cnsisting f n n ndes, the matrix-weighted Dirichlet Laplacian fr the bundary V is a (n n )k (n n )k matrix L btained frm the matrix-weighted Laplacian f G by remving all rws and clumns crrespnding t the ndes in V. The usual graph Laplacian is a special case f the matrixweighted Laplacian when k = and all the weights are equal t ne. The submatrix L f L in the special case f k = is called the Dirichlet Laplacian, since it arises in the numerical slutin f PDE s with Dirichlet bundary cnditins 0]. It is als called a Grunded Laplacian since it arises in the cmputatin f nde ptentials in a electrical netwrk. Our terminlgy is derived frm this histry, but as we shall see shrtly, the matrix-weighted Dirichlet Laplacian als arises in several distributed cntrl/estimatin prblems. We say that a graph G is cnnected t V if there is a path frm every nde in the graph t at least ne f the bundary ndes in V. Lemma at the end f this sectin shws that the Dirichlet Laplacian L is invertible if and nly if G is cnnected t V. We nw frmally define effective resistance f a nde in a cnnected graph G: nde u s effective resistance t V, dented by Ru eff(v ), is the k k blck in the main diagnal f L crrespnding t the k rws/clumns assciated with the nde u V. This terminlgy is justified by the fact that these matrices als express a map frm (matrix-valued) currents t (matrix-valued) vltages in an apprpriately defined electrical netwrk (cf. ]). Hwever, fr nw we are mstly interested in the fact that these effective resistances have a direct physical relevance in many distributed cntrl/estimatin prblems. Mrever, they als allw us t deduce prperties f the spectrum f the matrix-weighted Dirichlet Laplacian and even prperties f the spectrum f the riginal matrix-weighted Laplacian. It shuld be nted that these effective resistances are matrixvalued. The previus definitins relied n the nn-singularity f L, which is established by the fllwing lemma. Lemma (Invertibility). The matrices L and L are bth psitive semi-definite. Mrever, the matrix L is psitive definite if and nly if G is cnnected t V. T prceed with the prf f this lemma as well as t build further insight int the structure f the matrices L and L, we nte that the matrix-weighted Laplacian can als be defined cmpactly using the incidence matrix f the graph. T d s, we cnsider a directed graph G btained by assigning arbitrary directins t the edges f G. The incidence matrix f the directed graph G with n ndes and m edges is an n m matrix with ne rw per nde and ne clumn per edge defined by A := a ue ], where a ue is nnzer if and nly if the edge e E is incident n the nde u V. When nnzer, a ue = if the edge e is directed twards u and a ue = therwise. Define W = diag(w,..., W m ). Figure shws a graph G, a pssible crrespnding directed graph G and its assciated incidence matrix. The matrix-weighted Laplacian f the graph G can be expressed as L := AWA T, A = A I k () where dentes the Krnecker prduct and I k is a k k identity matrix. Nte that the matrix-weighted Laplacian des nt depend n the directins f the edges chsen t define the incidence matrix. The Dirichlet Laplacian can als be expressed cmpactly as L := A WA T, A := A I k, (2) where the generalized basis incidence matrix A is a matrix btained by remving all the rws frm the incidence matrix A that crrespnd t ndes in V. Figure als shws a graph G and its assciated matrix-weighted Laplacian L when every edge weight is equal t the identity matrix. Prf f Lemma. The incidence matrix A R n m has a rw rank less than r equal t n, Therem 8.3.], and since W is psitive definite, L is psitive semi-definite. The matrix A is full-rw rank if and nly if G is cnnected t V ], which establishes the cnditin fr L t be psitive definite.

3 PSfrag replacements e e 4 3 e 4 e 2 PSfrag2 replacements 2 e 4 e 5 G 3» A = A = I 2I 0 0 2I 4I I I L = 0 I 2I I I I 2I 7 {z } L e 3 e 4 e 5 3 G e 2 h I I I I 0 i I I 0 0 I 0 I 5, D = " 4I 0 # 0 0 2I I Fig.. A graph G, a directed versin G and its incidence matrix A (rw and clumn indices f A crrespnd t nde and edge indices, respectively). The matrix A is fr the bundary with the single nde V = {}. The matrix-weighted Laplacian is shwn fr the case when all the edge weights are equal t the identity matrix. III. EFFECTIVE RESISTANCE VS. SPECTRAL PROPERTIES The effective resistances f a graph can be uses t study several spectral prperties f a graph. The fllwing result prvides the starting pint t establish these results. Lemma 2 (Spectrum f L and L ). Assume that G is cnnected t V and dente by λ (L) λ 2 (L) λ nk (L) the srted eigenvalues f the L and by λ (L ) λ 2 (L ) λ (n n)k(l ) the srted eigenvalues f L. Fr every i {, 2..., (n n )k} and λ i (L ) trace Reff u (V ) (3) λ i (L) λ i (L ) λ i+kn (L). (4) Prf. The inequality (3) is a cnsequence f the fact that any eigenvalue f the psitive definite matrix L can be upper-bunded by its trace, which can be btained by adding up all the traces f its diagnal blcks Ru eff(v ), u V. This means that every eigenvalue f L satisfies: λ i (L ) trace L = trace Ru eff (V ), frm which (3) fllws since the eigenvalues f L and L are reciprcals f each ther. The inequality (4) is a direct applicatin f the Interlacing Eigenvalues Therem 2, Therem 4.3.5]. By cnstructin, the matrix-weighted Laplacian L has the prperty that L(I k,..., I k ] T ) nk k = 0, where I k dentes the k k identity matrix. This means that L has k eigenvalues equal t zer. The effective resistances can be used t btain a lwer bund n the smallest nn-zer eigenvalue f L. In particular, when n =, we cnclude frm (3) and (4) that λ +k (L) λ (L ) trace Reff u (V ), which prvides a lwer-bund n the smallest strictly psitive eigenvalue f L. The fllwing lemma summarizes the discussin abve: Lemma 3 (Albegraic Cnnectivity). Fr any single-nde bundary set V, fr which G is cnnected, the matrix L has exactly k zer eigenvalues and all remaining eigenvalues satisfy λ i (L) trace Reff u (V ) > 0. When k = and all the edge-weights are equal t ne, the matrix L is the usual graph Laplacian and the smallest nnzer eigenvalue is knwn as the algebraic cnnectivity f the graph. The algebraic cnnectivity is a measure f perfrmance/speed f cnsensus algrithms 3]. The cnvergence f several discrete-time distributed algrithms is determined by the spectrum f the matrix J := I D L, (0, ], where D is a blck diagnal matrix that cntains the k k blcks that appear in the diagnal f the matrix-weighted Dirichlet Laplacian L (see Fig. fr an example). The next lemma establishes the stability f J and prvides a bund n the spectrum f this matrix. Lemma 4 (Discrete-time cnvergence). Assume that G is cnnected t V. Every eigenvalue f J is real and satisfies < λ i (J ) λ max (D ) trace Reff u (V ), (5) where λ max (D ) dentes the largest eigenvalue f D. Mrever, when the weighting matrices W u,v, (u, v) E are all diagnal r all equal (but nt necessarily diagnal), we actually have λ i (J ) λ max (D ) trace Reff u (V ). Prf. Since J = D 2 ( I D 2 L D 2 ) D 2, we cnclude that J can be transfrmed by a similarity transfrmatin int the symmetric matrix I D 2 L D 2 and therefre all eigenvalues f J must be real. Mrever, all eigenvalues f J have abslute value strictly smaller than ne because the discrete-time Lyapunv equatin J P J P = Q has a psitive definite slutin P := D L D > fr Q := (2D L ) + ( )L (2D L ) > 0. T verify that this matrix is psitive definite, we nte that 2D L is very similar t the matrix-weighted Laplacian L, and in fact, 2D L = A W A T, where A is a matrix whse entries are the abslute values f the entries f A. The psitive definiteness f A W A T can be prved in a manner similar t that used in prving the psitive definiteness f L = A WA T (Lemma ).

4 Denting by λ (, ) an eigenvalue f J and by x the crrespnding eigenvectr, we have that (I D L )x = λx L D x = λ x. Therefre λ max (D 0 ) x L D x = λ min (L ) λ x, frm which we cnclude using Lemma 2 that λ λ min(l ) λ max (D 0 ) λ max (D 0 ) trace Reff u (V ), which prves (5). When all the weighting matrices are diagnal, the matrix J is nn-negative fr every. Therefre by Perrn s Therem 2, Therem 8.3.] we knw that its eigenvalue with the largest abslute value must be a psitive number. The last inequality in the Lemma then fllws frm (5). When all weighting matrices are equal t sme W > 0, it fllws frm the definitin (2) that J = J I k, where J := I ( D L ) R (n n) (n n), L is the Dirichlet Laplacian fr scalar weights equal t, and D its main diagnal. Thus, the weights W play n rle. The matrix D L is nnnegative, and we cnclude frm the reasning abve that its eigenvalue with largest abslute value is psitive. Since J = J I k, the distinct eigenvalues f J are just the distinct eigenvalues f J, s we cnclude that the eigenvalue f J with largest nrm is als psitive. The last inequality again fllws frm (5). The cnvergence f several cntinuus-time distributed algrithms is determined by the spectrum f the matrix G := D L, > 0. where D is a blck diagnal matrix that cntains the k k blcks n the diagnal f the matrix-weighted Dirichlet Laplacian L. Lemma 5 (Cntinuus-time cnvergence). Assume that G is cnnected t V. Every eigenvalue f G is real and satisfies λ i (G ) λ max (D ) trace Reff u (V ), (6) where λ max (D ) dentes the largest eigenvalue f D. Prf. Since G = D 2 ( D 2 L D 2 ) D 2, we cnclude that G can be transfrmed by a similarity transfrmatin int the negative definite matrix D 2 L D 2 and therefre all eigenvalues f G must be real and negative. Mrever, defining P := 2 L we cnclude that (αi + G )P + P (αi + G ) = (αl D ). The matrix n the right hand side is then negative definite as lng as λ min (D ) > α λ max (L ) α < λ min(l ) λ max (D ). Frm Lyapunv s Therem, we thus cnclude that αi + G is Hurwitz fr every α < λmin(l) λ max(d ), which implies that the eigenvalue f G must be mre negative than λmin(l) λ. max(d ) The inequality (6) fllws frm this and Lemma 2. IV. APPLICATIONS TO DISTRIBUTED CONTROL AND ESTIMATION We nw describe a few prblems in distributed cntrl and estimatin fr which the matrix-weighted Dirichlet Laplacian and the effective resistances play a key rle. A. Graph estimatin Cnsider the prblem f estimating the values f n nde variables x, x 2,... x n R k, k frm nisy relative measurements f the frm y u,v = x u x v + ɛ u,v, (u, v) E V V, (7) where the ɛ u,v s are uncrrelated zer-mean nise vectrs with assciated cvariance matrices P u,v = Eɛ u,v ɛ u,v ]. This estimatin prblem is relevant fr such wide ranging applicatins such as lcatin estimatin and time synchrnizatin in sensr netwrks 4, 6, 7] This estimatin prblem can be assciated with a matrixweighted graph G = (V, E, W) with n ndes and m edges, with nde set V := {, 2,..., n}, edge set E cnsisting f all the pairs f ndes (u, v) fr which a nisy measurement f the frm (7) is available; and weight set W cnsisting f the inverses f the cvariance matrices W u,v := Pu,v, (u, v) E. Just with relative measurements, determining the x u s is nly pssible up t an additive cnstant. T avid this ambiguity, we assume that at least ne f the ndes is used as a reference and therefre its nde variable can be assumed knwn. In general, several nde variables may be knwn and therefre we may have several references. We dente by V the set f reference ndes. This prblem was intrduced in 5] fr a single reference nde. By stacking tgether all the measurements int a single vectr y R km, all nde variables (knwn and unknwn) int ne vectr X R kn, and all the measurement errrs int a vectr ɛ R km, we can express all the measurement equatins (7) in the fllwing cmpact frm: y = A T X + ɛ, (8) where A is as defined in (). By partitining X int a vectr x cntaining all the unknwn nde variables and anther vectr r cntaining all the knwn reference nde variables, we can re-write (8) as y = A T r r + AT x + ɛ, (9) where A r cntains the rws f A crrespnding t the reference ndes and A cntains the rws f A crrespnding t the unknwn nde variables (see Figure fr an example).

5 The estimatin f the unknwn x based n the linear measurement mdel (9) is a classical estimatin prblem. The Best Linear Unbiased Estimatr (BLUE) f x is given by ˆx := L b, L := A WA T, b := A W(y A T r r), (0) where A is as defined in (2) and W R km km is a blckdiagnal matrix with inverse-cvariances in the diagnal W = diag(p,..., Pm ); Amng all linear estimatrs f x, (0) has the smallest variance fr the estimatin errr x ˆx and the inverse f L prvides the cvariance matrix f the estimatin errr 4]: E(x ˆx)(x ˆx) ] = L. () As we saw befre, the inverse f L exists as lng as the graph G is cnnected t V. Frm (), we als cnclude that the cvariance matrix Σ u f the estimatin errr fr the variable x u appears in the crrespnding k k diagnal blck f L, which is precisely nde u s effective resistance t V defined in Sectin II. Fr large graphs ne is interested in hw the cvariance f the BLU estimate grws as a functin f distance frm the reference nde. This questin is answered in ] by determining hw the matrix-effective resistance scales with distance fr a large class f graphs. A distributed algrithms fr the cmputatin f the estimates have been prpsed in 7]. A similar algrithm is als prpsed in 6]. This algrithm is based n the bservatin that the ptimal estimate ˆx in (0) is a slutin t the fllwing equatin ˆx = J ˆx + D ˆb (2) where J := I D L, (0, ]. Since all eigenvalues f J have abslute value smaller than ne (cf. Lemma 4), the discrete-time system ˆx(t + ) = J ˆx(t) + D ˆb will cnverge precisely t the slutin t (2), which is the ptimal estimate ˆx. It turns ut that the recursin in (2) can be cmputed in a distributed fashin. In practice, each nde nly cmputes the k elements f ˆx(t) that crrespnd t its nde variable. Since the k rws f J that crrespnds t the nde u nly have nnzer k k ff-diagnal blcks in the psitins crrespnding t the neighbrs f u, all that u needs are the current estimates f its -hp neighbrs. The algrithm prpsed in 7] is actually a special case f this ne fr =. Lemma 4 relates the speed f cnvergence f this algrithm with the effective resistances. B. Frmatin cntrl with nisy measurements Cnsider a grup f n mbile agents mving in k- dimensinal space that ne desires t cntrl t a given frmatin defined by their relative psitins. In particular, denting by x u R k, u V := {, 2,..., n} the psitin f the uth agent, the cntrl bjective is t make the psitins cnverge t values fr which x u x v = r u,v, (u, v) V V, (3) where r u,v dentes the desired relative psitin f agent u with respect t agent v. One f the agents V will be called the leader and it will mve independently f the remaining nes. The remaining agents attempt t maintain the frmatin specified by (3). The leader may actually nt be a physical agent. Instead, it may be a reference that is knwn t at least ne f the physical agents. Nt all agents are able t measure their relative psitins with respect t all ther agents and therefre each agent is cnstrained t use nly a few relative psitin measurements t cmpute its cntrl signal. We dente by E V V the set f pairs f agents that can measure their relative psitins. In particular, the existence f a pair (u, v) in E signifies that agent u can measure its psitin with respect t v and similarly, v can measure its psitin with respect t u, althugh bth measurements will be crrupted with nise. Since the nise crrupting the measurement f x u x v available t u will be in general different frm the nise n the measurement f x v x u available t v, we need t distinguish these tw measurements. T this end, we intrduce a directed edge set E cntaining the tw rdered pairs (u, v), (v, u) whenever (u, v) E. We assume that a nisy measurement y u,v f the fllwing frm is available t agent u if (u, v) E: y u,v = x u x v + ɛ u,v (4) where ɛ u,v is a white randm nise prcess with autcrrelatin matrix given by Eɛ u,v (t )ɛ T u,v (t 2)] = δ(t t 2 )R u,v. Nte that by assumptin, if a measurement y u,v is available t u, then the measurement y v,u is available t v. The nise prcesses ver different edges are assumed independent f each ther. In particular, e u,v (t) is independent f e v,u (t) fr all t. In case x is a reference and nt a physical agent, an edge between the nde u and the leader means that the physical agent u is able t measure its psitin with respect t the reference. The prblem abve is nw assciated with a matrixweighted directed graph G = (V, E, W) with nde set V = {, 2,..., n}; directed edge set E cnsisting f all rdered pairs f ndes (u, v) fr which a nisy measurement f the frm (4) is available; and weight set W cnsisting f the inverses f the autcrrelatin matrices W u,v := Ru,v, (u, v) E. We assume that even thugh the measurement errrs n the tw edges (u, v) and (v, u) cnnecting the ndes u and v are uncrrelated, they have the same autcrrelatin matrix; i.e., R u,v = R v,u. We will refer t this assumptin, tgether with the assumptin that the directed edge (u, v) exists iff (v, u) exists, as bidirectinality. Fig. 2 shws an example f a bidirectinal directed graph and its assciated undirected graph. We are interested in cntrl laws fr which each agent uses all its measurements t cnstruct an ptimal estimate f the difference between its currently psitin and what this shuld be, in view f what it knw abut its neighbrs psitins. The measurements available t an arbitrary agent u V are y u,v = x u x v + ɛ u,v, v N u,

6 where N u V dentes set f ndes v such that (u, v) E. If agent u assumes that all its neighbrs are crrectly psitined then, accrding t (3), the desired psitin f u is given by any ne f the fllwing equatins x d u = x v + r u,v, v N u. Cmbining the tw previus sets f equatins, we btain y u,v = x u x d u + r u,v + ɛ u,v, v N u, frm which agents u estimates its psitin errr x u x d u. It is straightfrward t shw that the Best Linear Unbiased estimate ) f x u x d u is given by Du v N u Ru,v( yu,v r u,v, where Du := v N u Ru,v. This mtivates the fllwing negative prprtinal cntrl law fr the agents ẋ u = Du u,v (y u,v r u,v ), u V \ {}, v N u R where dentes sme psitive number. Fr analysis purpses it is cnvenient t describe the system dynamics in term f psitins with respect t the leader. Defining x u = x u x, ne cncludes that x u = D u v N u R u,v( x u x v r u,v + ɛ u,v ) ẋ, fr every u V \ {}. By stacking all the psitins x u, u V \ {} in a clumn vectr x, the abve systems can be written as fllws: x = D L x + D B W (r ɛ) ẋ, (5) where r is a clumn vectr btained by stacking all the r u,v n tp f each ther; ɛ is a clumn vectr btained by stacking all the ɛ u,v ; is a n clumn vectr f all s; W > 0 is a blck-diagnal matrix with k rws/clumns fr each edge in E, with the weights W u,v := Ru,v, (u, v) E in the diagnal; D > 0 is a blck-diagnal matrix with k rws/clumns fr each nde in V \ V, with D u, u V \ V as defined earlier in the diagnal; L = 2 A W A T where A is the generalized incidence matrix fr the directed graph (V, E) with V = {} (cf. sectin II); and B is a matrix with k rws fr each vertex in V\V and k clumns fr each edge in E, cnstructed as fllws: the k clumns crrespnding t edge (u, v) E are all equal t zer except fr the blck crrespnding t the nde u, which is equal t I k. The white nise prcess ɛ has blck diagnal autcrrelatin matrix given by Eɛ(t )ɛ T (t 2 )] = δ(t t 2 )W. Fig. 2 shws an example f the matrices defined abve. L is exactly the matrix-weighted Dirichlet Laplacian fr the matrix-weighted undirected graph (V, E, W) with bundary V := {} and with weight W u,v n every undirected edge (u, v) assigned as the weight n the crrespnding directed edge (u, v) E. Nte that we get the undirected Laplacian in the system dynamic equatins (5) due t the bidirectinality assumptin. Frm Lemma 5, we cnclude that (5) is an asympttically stable system and Lemma 5 actually relates the speed f its slwest ple with the effective resistances. It turns ut that the effective resistances play an even mre interesting A 0 = G G h I 0 I I I 0 I 0 I 0 i 0 I 0 I 0 I I I 0 I 0 B = I 0 0 I I I h I I I i I I I I I Fig. 2. A (bidirectinal) directed graph G and the assciated undirected graph G. The matrices A and B shwn are fr the graph G. rle fr this system. T see this, we further re-write the mdel (5) as x = D L x + w + b, where b := D B W r ẋ and w := D B W ɛ is a white nise randm prcess with autcrrelatin matrix given by Ew(t )w T (t 2 )] = 2 D B W Eɛ(t )ɛ T (t 2 )]W B T D = 2 δ(t t 2 )D B W B T D = 2 δ(t t 2 )D, where we used the fact that B W B T Lyapunv equatin = D. Since the D L Σ Σ L D + 2 D = 0 has a psitive definite slutin Σ = 2 L, it is straightfrward t shw that the cvariance matrix f x cnverges t Σ. In particular, the steady-state cvariance matrix f the relative psitin x u := x u x is given by k k diagnal blck f Σ, which is given by /2 times nde u s effective resistance Ru, eff t V := {} defined in Sectin II. The scaling f matrix-valued effective resistance Ru, eff as a functin f distance d u, f u frm the leader (addressed in ]) determine hw the structure f the graph G affect the grwth f the effective resistance, and therefre tracking errr cvariance. C. Stability f multi-vehicle swarms Cnsider a grup f n single-input/single-utput vehicles all with the same transfer functin P (s) frm their cntrl inputs u i, i V := {, 2,..., n} t their psitins x i. Each vehicle is able t determine its relative psitin with respect t a few ther vehicles. We dente by E V V the set f pairs f vehicles that can measure each ther relative psitins. In particular, the existence f a pair (i, j) in E signifies that the measurement y i,j := x i x j, (i, j) E is available t bth vehicles i and j. The cntrl bjective is fr all vehicles t rendezvus at a pint. T this effect each vehicle cmputes a weighted average e i := w i,j (x i x j ), i V j N i

7 f the relative psitins that it can measure and uses it fr feedback accrding t U i (s) = K(s)E i (s), where U i (s) and E i (s) dente the Laplace transfrms f u i and e i, respectively; and K(s) is a suitably selected cntrller transfer functin used by every vehicle. The errr equatin can be assciated with an undirected scalar-weighted graph G = (V, E, W), with weights W := {w i,j R : (i, j) E}. It was shwn by Fax and Murray 8] that the resulting clsed-lp system is stable if and nly if fr every nnzer eigenvalue λ i (L) f the scalar-weighted Laplacian matrix L f G there is n net encirclements f /λ i (L) by the Nyquist plt f K(s)P (s). We cnclude frm (3) that /λ i (L) trace Reff u (V ). Mrever, it can be easily seen that 2D L = A W A T 0 (see the prf f the discrete time cnvergence Lemma 4), s λ i (L) < 2λ max (D). We cnclude that fr all nnzer λ i (L), /λ i (L) lies in the range λ 2 (L), λ max (L) ] trace R eff u (V ), 2λ max (D) We therefre cnclude that the clsed-lp system is stable, as lng as the the Nyquist plt f K(s)P (s) des nt encircle any pint in the line segment frm trace Reff u (V ) t /2λ max (D). Large effective resistances will generally lead t lnger line segments and therefre mre stringent stability cnditins. Similar cnclusins can be drawn fr the type f frmatins cnsidered in 9] and 5], where every vehicle tries t maintain a cnstant separatin between itself and its neighbrs, and there is a leader whse cntrl actin des nt depend n the ther vehicles. In this case the quantities f interest are the eigenvalues f the Dirichlet Laplacian, λ i (L ). In particular, the frmatin will be stable if and nly if fr every eigenvalue λ i (L ) f the scalarweighted Dirichlet Laplacian matrix L f G there is n net encirclements f /λ i (L ) by the Nyquist plt f K(s)P (s). We can derive in a manner similar t that abve that all /λ i (L ) lies in the range λ min (L ), λ max (L ) ] trace Ru eff (V ), V. CONCLUSION ]. ]. 2λ max (D ) We intrduced the cncept f matrix-valued effective resistance fr undirected matrix-weighted graphs and shwed that effective resistances have a direct physical interpretatin in several prblems related t distributed cntrl and estimatin. We als shwed that the effective resistances can be used t cnstruct bunds n the spectrum f the Laplacian, Dirichlet Laplacian and ther matrices that arise in distributed cntrl and estimatin. We examined a few nvel and existing algrithms fr such prblems and shwed that their fundamental perfrmance limits, including stability and speed f cnvergence, can be determined by examining the effective resistances in the underlying netwrk. Our cverage f prblems fr which effective resistances are relevant was nt exhaustive and we encurage the readers t lk fr ther prblems where this tl can be f use. 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