Distributed Kalman Filtering for Sensor Networks

Transcription

1 Proceedngs of the 46th IEEE Conference on Decson and Control New Orleans, LA, USA, Dec , 2007 Dstrbuted Kalman Flterng for Sensor Networks R. Olfat-Saber Abstract In ths paper, we ntroduce three novel dstrbuted Kalman flterng (DKF) algorthms for sensor networks. The frst algorthm s a modfcaton of a prevous DKF algorthm presented by the author n CDC-ECC 05. The prevous algorthm was only applcable to sensors wth dentcal observaton matrces whch meant the process had to be observable by every sensor. The modfed DKF algorthm uses two dentcal consensus flters for fuson of the sensor data and covarance nformaton and s applcable to sensor networks wth dfferent observaton matrces. Ths enables the sensor network to act as a collectve observer for the processes occurrng n an envronment. Then, we ntroduce a contnuous-tme dstrbuted Kalman flter that uses local aggregaton of the sensor data but attempts to reach a consensus on estmates wth other nodes n the network. Ths peer-to-peer dstrbuted estmaton method gves rse to two teratve dstrbuted Kalman flterng algorthms wth dfferent consensus strateges on estmates. Communcaton complexty and packet-loss ssues are dscussed. The performance and effectveness of these dstrbuted Kalman flterng algorthms are compared and demonstrated on a target trackng task. Index Terms sensor networks, dstrbuted Kalman flterng, consensus flterng, sensor fuson I. INTRODUCTION Dstrbuted estmaton and trackng s one of the most fundamental collaboratve nformaton processng problems n wreless sensor networks (WSN). Mult-sensor fuson and trackng problems have a long hstory n sgnal processng, control theory, and robotcs [1], [2], [3], [6], [16]. Moreover, estmaton ssues n wreless networks wth packet-loss have been the center of much attenton lately [19], [7], [18]. Decentralzed Kalman flterng [22], [15] nvolves state estmaton usng a set of local Kalman flters that communcate wth all other nodes. The nformaton flow s all-to-all wth communcaton complexty of O(n 2 ) whch s not scalable for WSNs. Here, we focus on scalable or dstrbuted Kalman Flterng algorthms n whch each node only communcates messages wth ts neghbors on a network. Control-theoretc consensus algorthms have proven to be effectve tools for performng network-wde dstrbuted computaton tasks such as computng aggregate quanttes and functons over networks [13], [12], [17], [8], [23], [24], [21], [5]. These algorthms are closely related to gossp-based algorthms n computer scence lterature [9], [4]. Recently n [10], the author ntroduced a dstrbuted Kalman Flterng (DKF) algorthm that uses dynamc consensus algorthms [14], [20]. The DKF algorthm conssts of a network of mcro-kalman Flters (MKFs) each embedded wth a low-pass and a band-pass consensus flter. The role Reza Olfat-Saber s an Assstant Professor at Thayer School of Engneerng, Dartmouth College, Hanover, NH 03755, emal: olfat@dartmouth.edu. of consensus flters s fuson of sensor and covarance data obtaned at each node. The exstng DKF algorthm of the author suffers from a key weakness: the algorthm s only vald for sensors wth dentcal sensng models. In other words, t s not applcable to heterogeneous mult-sensor fuson. To be more precse, let z (k) = H (k)x(k) + v (k) be the sensng model of node n a sensor network. Here, x(k) denotes the state of a dynamc process x(k + 1) = A k x(k) + B k w(k) drven by zero-mean whte Gaussan nose w(k). Then, the DKF algorthm n [10] s applcable to sensors wth dentcal H s. The reason s that under the assumpton of dentcal observaton matrces, or H = H,, we get z (k) = H(k)x(k) + v (k) = s(k) + v (k) and z s possess the structure necessary for the dstrbuted averagng feature of the low-pass consensus flter n [14]. Ths lmtaton of the exstng DKF algorthm motvates us to develop novel dstrbuted Kalman flterng algorthms for sensor networks that have broader range of applcatons. In partcular, we are nterested n DKF algorthms capable of performng the followng trackng and estmaton tasks: 1) The H s are dfferent across the entre network 1 and the process wth state x(k) s collectvely observable by all the sensors. 2) Extended Kalman flterng (EKF) for sensors wth dfferent nonlnear sensng models z (k) = h (x(k)) + v (k) whch after lnearzaton leads to the case above (up to trval terms) wth H s beng the Jacoban of the partal dervatves of h (x) w.r.t. x. We refer to each node of the dstrbuted Kalman flter that provdes a state estmate as a Mcroflter. A DKF s a Networked System (or Swarm) of nteractng mcroflters wth dentcal archtectures. Each mcroflter s computatonally mplemented as an embedded module n a sensor. The man results of ths paper are as follows: ) resolvng the lmtaton of the exstng DKF by ntroducng a novel mcroflter archtecture wth dentcal hgh-pass consensus flters. The revsed DKF algorthm s applcable to sensors wth dfferent H s, ) presentng an alternatve dstrbuted Kalman flterng strategy whch does not nvolve consensus 1 Some of the H s could be equal, but all of them are not the same /07/$ IEEE. 5492

2 flterng and nstead uses consensus on state estmates, ) ntroducng a contnuous-tme dstrbuted Kalman flter, and v) provdng a base performance standard for dstrbuted estmaton and comparson between DKF algorthms on target trackng. The outlne of the paper s as follows: a DKF algorthm wth dentcal consensus flters s ntroduced n Secton II. Our man results ncludng the DKF algorthms wth consensus on state estmates are presented n Secton III. Smulaton results are gven n Secton IV. Fnally, concludng remarks are made n Secton V. II. DISTRIBUTED KALMAN FILTER: TYPE I: CONSENSUS-BASED FUSION OF SENSORY DATA Frst, we present a modfed verson of the DKF algorthm n [10]. The key modfcaton s to replace the low-pass and band-pass consensus flters n the archtecture of the mcroflter by hgh-gan versons of the hgh-pass consensus flter n [20]. The resultng mcroflter s shown n Fg. 1. In the followng, we provde the equatons of the mcro-kalman flter (MKF) and the hgh-pass consensus flters (CFs). Fg. 1. Node Sensor Data Covarance Data Mcroflter Archtecture Hgh-Pass Consensus Flter Hgh-Pass Consensus Flter Mcro Kalman Flter Iteratons The archtecture of the mcroflter of the type-i DKF algorthm. Smlar to the problem set up n [10], consder a sensor network wth an ad hoc topology G = (V, E) and n nodes. The graph G s undrected, V = {1, 2,..., n}, and E V V. The objectve s to perform dstrbuted state estmaton (or trackng) for a process/target that evolves accordng to x(k + 1) = A k x(k) + B k w(k); x(0) N ( x(0), P 0 ). (1) The sensng model of the th sensor s z (k) = H (k)x(k) + v (k), z R p (2) and we assume the H s are dfferent (see Secton I). Both w k and v k are zero-mean whte Gaussan nose (WGN) and x(0) R m s the ntal state of the target. The statstcs of the measurement nose s gven by E[w(k)w(l) T ] = Q(k)δ kl, (3) E[v (k)v j (l) T ] = R (k)δ kl δ j. (4) where δ kl = 1 f k = l, and δ kl = 0, otherwse. ˆx Let z(k) = col(z 1 (k),..., z n (k)) R np be the collectve sensor data of the entre sensor network at tme k. Gven the nformaton Z k = {z(0), z(1),..., z(k)}, the estmates of the state of the process can be expressed as ˆx k = E(x k Z k ), x k = E(x k Z k 1 ), (5) P k = Σ k k 1, M k = Σ k k (6) where Σ k k 1 and Σ k k are the estmaton error covarance matrces and Σ 0 1 = P 0. Defnng an output matrx H = col(h 1, H 2,..., H n ), one can defne a central estmate ˆx(k) assocated wth the data z(k) gven by ˆx(k) = x(k) + K k (z(k) H x(k)). (7) Assumng that the measurement nose of the sensors are uncorrelated, the covarance matrx of the nose v = col(v 1,..., v n ) s R = dag(r 1,..., R n ) (the tme ndces are dropped). Let us defne two network-wde aggregate quanttes: the fused nverse-covarance matrces S(k) = 1 n n =1 =1 H T (k)r 1 (k)h (k) (8) and the fused sensor data y(k) = 1 n H T (k)r 1 (k)z (k). (9) n Theorem 1. (Mcro-KF Iteratons, [10]) Suppose every node of the network apples the followng teratons M (k) = (P (k) 1 + S(k)) 1, ˆx(k) = x(k) + M (k)[y(k) S(k) x(k)], P (k + 1) = A k M (k)a T k + B k Q (k)b T k, x(k + 1) = A k ˆx(k). (10) where Q (k) = nq(k) and P (0) = np 0. Then, the local and central state estmates for all nodes are the same,.e. ˆx (k) = ˆx(k) for all. The above theorem holds regardless of how the networkedwde fuson task necessary to compute y(k) and S(k) s performed. Now, f one can (approxmately) compute the averages y(k) and S(k), a dstrbuted Kalman flterng algorthm emerges. We propose to use a hgh-gan verson of the dynamc consensus algorthm of Spanos et al. [20] to perform dstrbuted averagng hence, the mcroflter archtecture n Fg. 1. Let N = {j : (, j) E} be the set of neghbors of node on graph G. Moreover, let L = D A be the Laplacan matrx of G and λ 2 = λ 2 (L) denote ts algebrac connectvty. The hgh-pass consensus flter s a lnear system n the form q = β (q j q ) + β (u j u ); β > 0 (11) y = q + u where u s the nput of node, q s the state of the consensus flter, and y s ts output. The gan β > 0 s relatvely large (β O(1/λ 2 )) for randomly generated ad hoc topologes 5493

3 that are rather sparse. The collectve dynamcs of ths CF s gven by { q = β ˆLq β ˆLu (12) p = q + u where ˆL = L I m s the m-dmensonal graph Laplacan. For a connected network, p (t) asymptotcally converges to 1/n u (t) as t (see [20]). Expressng ths flter n dscrete-tme s trval (see [12] for hnts on the rght choce of the step-sze). Remark 1. In [20], the equaton of the dynamc consensus algorthm s gven as ṗ = Lp + u whch reduces to (12) for m = 1 by defnng q = p u and assumng the graph s weghted wth weghts n {0, β}. Algorthm 1 Dstrbuted Kalman Flterng Algorthm wth hgh-pass consensus flterng of the sensed data. 1: Intalzaton: q = 0, X = 0 m m, P = np 0, x = x(0) 2: whle new data exsts do 3: Update the state of the data CF: u j = Hj T R 1 j z j, j N {} q q + ɛβ [(q j q ) + (u j u )] y = q + u 4: Update the state of the covarance CF: U j = Hj T R 1 j H j, j N {} X X + ɛβ S = X + U [(X j X ) + (U j U )] 5: Estmate the target state usng Mcro-KF: M = (P 1 + S ) 1 ˆx = x + M (y S x ) 6: Update the state of the Mcro-KF: 7: end whle P AM A T + nbqb T x Aˆx To compute y(k) and S(k) n a dstrbuted way, we use the dscrete-tme versons of both consensus flters. Each node uses the nputs u (k) = H T (k)r 1 (k)z (k) and U (k) = H T (k)r 1 (k)h (k) wth zero ntal states q (0) = 0 and X (0) = 0. The outputs y (k) and S (k) of the hgh-pass consensus flters asymptotcally converge to y(k) and S(k) (up to mnor condtons gven n [20]). Algorthm 1 s the new type-i DKF algorthm wth dentcal consensus flters. Accordng to Algorthm 1, node sends the message msg = (q (k), X (k), u, U ) to all of ts neghbors. The message conssts of the state and nput of ts consensus flters. Ths communcaton scheme s fully compatble wth packet-based communcaton n broadcast mode n real-world wreless sensor networks. The nformaton n each message can be contaned n one or multple packets (n case the message sze s large). The message sze s O(m(m + 1)) wth m beng the dmenson of the state x of the process/target. Packet-loss can be treated as loss of a communcaton lnk. Under mld connectvty condtons, packet-loss wll not affect consensus algorthms. III. DISTRIBUTED KALMAN FILTER: TYPE II: CONSENSUS ON ESTIMATES In ths secton, we dscuss an alternatve approach to dstrbuted Kalman flterng that reles on communcatng state estmates between neghborng nodes. We refer to ths second class of the DKF algorthms as type-ii algorthms. Before presentng the type-ii DKF algorthms, we frst need to dscuss a more prmtve DKF algorthm that nvolves local Kalman flterng and forms the bass of our man results. A. Local Kalman Flterng Assume that each node of the sensor network can communcate ts measurement z, covarance nformaton R, and output matrx H wth ts neghbors N. Should one avod usng any form of consensus (.e. wthout further nformaton exchange regardng states/estmates), what s the optmal state estmate by each node? The answer to ths queston s rather smple. In fact, one can use local Kalman flterng. The resultng algorthm can act as a base (or mnmum) performance standard for any current (or future) DKF algorthms for sensor networks. Intutvely, local Kalman flterng (LKF) does not perform well for a mnorty of nodes and ther neghbors that make relatvely poor observatons due to envronmental or geometrc condtons. In local Kalman flterng, node can assume that no nodes other than ts neghbors N exst as the nformaton flow from non-neghborng nodes to node s prohbted n ths case. Therefore, node can use a central Kalman flter that only utlzes the observatons and output matrces of the nodes n J = N {}. Ths leads to the followng prmtve DKF algorthm wth no consensus on data/states/estmates. Proposton 1. (LKF Iteratons) Let N c = V \ J be the set of non-neghborng nodes of node. Moreover, assume node receves no nformaton from ts non-neghbors. Then, the local Kalman flterng teratons for node are n the form S (k) = Hj T (k)r 1 j (k)h j (k), y (k) = Hj T (k)r 1 j (k)z j (k), M (k) = (P (k) 1 + S (k)) 1, ˆx (k) = x (k) + M (k)[y (k) S (k) x (k)], P (k + 1) = A k M (k)a T k + B k Q(k)B T k, x (k + 1) = A k ˆx (k). where node locally computes y (k) and S (k). (13) 5494

4 Proof: The proof follows from algebrac manpulaton of the equatons of the nformaton form of the Kalman flter wth restrcted nformaton to the observatons of the nclusve neghbors J of node. Accordng to ths LKF algorthm, there s no guarantee that the state estmates reman cohesve (or close to each other). Ths form of group dsagreement regardng the state estmates s hghly undesrable for a peer-to-peer network of estmators. Let ˆL = L I m be the m-dmensonal Laplacan of G. The dsagreement potental [13] of the state estmates s defned as Ψ G (ˆx) = ˆx T 1 ˆLˆx = ˆx j ˆx 2 2 (,j) E where ˆx = col(ˆx 1,..., ˆx n ). Algorthm 2 s a type-ii DKF algorthm that attempts to reduce the dsagreement regardng the state estmates n local Kalman flterng usng an ad hoc approach by mplementng a consensus step rght after the estmaton step. Later, n Algorthm 3, we provde a rgorous way of performng ths consensus on estmates. Algorthm 2 Dstrbuted Kalman Flterng Algorthm wth an ad hoc consensus step on estmates. 1: Intalzaton: P = P 0, ξ = x(0) 2: whle new data exsts do 3: Locally aggregate data and covarance matrces: J = N {} u j = Hj T R 1 j z j, j J, y = u j U j = Hj T R 1 j H j, j J, S = U j 4: Compute the ntermedate Kalman estmate of the target state: M = (P 1 + S ) 1 ϕ = ξ + M (y S ξ ) 5: Estmate the target state after a Consensus step: ˆx = ϕ + ɛ (ϕ j ϕ ) {Ths s equvalent to movng towards the average ntermedate estmate of the neghbors. } 6: Update the state of the local Kalman flter: 7: end whle P AM A T + BQB T ξ Aˆx Let us refer to ϕ as the ntermedate estmate of the state of the target. Each node sends the followng message to ts neghbors. msg = (u, U, ϕ ) In the followng, we provde a more rgorous dervaton of a type-ii DKF algorthm that ams at reducng dsagreement of estmates due to local Kalman flterng over the set of nclusve neghbors J. Our algorthm desgn method reles on solvng the problem n contnuous-tme and then nferrng the dscrete-tme verson of the dstrbuted Kalman flter wth cohesve estmates. B. Contnuous-Tme Dstrbuted Kalman Flter Consder a contnuous-tme (CT) lnear system representng a target ẋ = A(t)x + B(t)w and a sensng model z = H(t)x + v. The nose statstcs s rather analogous to the dscrete-tme problem setup. The contnuous-tme Kalman flter s n the form 2 : ˆx = Aˆx + K(z H ˆx) K = P H T R 1 P = AP + P A T + BQB T P H T R 1 HP (14) The followng lemma s the key n dervaton of the DKF n contnuous-tme: Lemma 1. Let η = x ˆx denote the estmaton error of the Kalman flter n contnuous-tme. The estmaton error dynamcs s n the form η = (A KH)η + w e (15) wth the nput nose w e = Bw + Kv. The error dynamcs wthout nose s a stable lnear system wth a Lyapunov functon V (η) = η T P (t) 1 η (16) Proof: By drect dfferentaton, we have V = η T P 1 η + η T P 1 η η T P 1 P P 1 η = η T [(A KH) T P 1 + P 1 (A KH)]η η T [P 1 A + A T P 1 + P 1 BQB T P 1 H T R 1 H]η = η T [H T R 1 H + P 1 BQB T P 1 ]η < 0 for all η 0. Thus, η = 0 s globally asymptotcally stable and V (η) s a vald Lyapunov functon for the error dynamcs. Consder a network wth n sensors wth the followng sensng model: z (t) = H (t)x + v (17) wth E[v (t)v T (s)] = R δ(t s). Assume the par (A, H) wth H = col(h 1,..., H n ) s observable. Here s our man result: 2 For smplcty of notatons, the tme-dependence of many tme-varyng matrces s dropped. 5495

5 Proposton 2. (Kalman-Consensus flter) Consder a sensor network wth contnuous-tme sensng model n (17). Suppose each node apples the followng dstrbuted estmaton algorthm ˆx = Aˆx + K (z H ˆx ) + γp (ˆx j ˆx ) K = P H T R 1, γ > 0 P = AP + P A T + BQB T K R K T (18) wth a Kalman-Consensus estmator and ntal condtons P (0) = P 0 and ˆx (0) = x(0). Then, the collectve dynamcs of the estmaton errors η = x ˆx (wthout nose) s a stable lnear system wth a Lyapunov functon V (η) = n =1 ηt P 1 η. Furthermore, V 2ΨG (η) 0 and asymptotcally all estmators agree ˆx 1 = = ˆx n = x. Proof: Defne vectors η = col(η 1,..., η n ), x = col(x,..., x) R mn and let ˆx = col(ˆx 1,..., ˆx n ) be the vector of all the estmates of the nodes. Note that ˆx j ˆx = η η j and thus, the estmator dynamcs can be wrtten as ˆx = Aˆx + K (z H ˆx )) γp (η j η ) whch gves the followng error dynamcs for the th node η = (A K H )η + γp (η j η ) or η = F η + γp (η j η ) wth F = A K H. By calculatng V (η), we get But V = η T P 1 η T P 1 η + η T P 1 η = η T P 1 F η + γη T η η T P 1 P P 1 η. (η j η ) and after transposton, the last term remans the same. Hence η T P 1 η = η T F T P 1 η + γη T (η j η ) Note that the evoluton of P s the same as the one for a standard Kalman flter (only the estmator s modfed). Addng the three terms n V (η) and usng Lemma 1 gves V = + 2γ η T [H T R 1 H T + P 1 BQB T P 1 ]η η T (η j η ) For undrected graphs, the second term s proportonal to the negatve of the dsagreement functon Ψ G (η) n consensus theory [13]. Thus V (η) = η T Λη 2γη T ˆLη 2γΨg (η) 0 (19) where Λ s a postve defnte block-dagonal matrx wth dagonal blocks H T R 1 H T + P 1 BQB T P 1. Due to the fact that V (η) = 0 mples all η s are equal and η = 0, asymptotcally ˆx = x, as t. Remark 2. All H s n Proposton 2 are, n fact, H,l s (l means local ) and H,l = col{h j } j J meanng that the aggregate observaton of all sensors n J are used as z. Interestngly, H can also be the observaton matrx of node and the result wll stll hold. C. Iteratve Kalman-Consensus Flter From the contnuous-tme DKF algorthm n Proposton 2, Algorthm 3 can be nferred. Ths s our man type-ii dstrbuted Kalman flterng algorthm. Algorthm 3 Kalman-Consensus flter: DKF Algorthm wth an estmator that has a rgorously derved consensus term. 1: Intalzaton: P = P 0, x = x(0) 2: whle new data exsts do 3: Locally aggregate data and covarance matrces: J = N {} u j = Hj T R 1 j z j, j J, y = u j U j = Hj T R 1 j H j, j J, S = U j 4: Compute the Kalman-Consensus estmate: M = (P 1 + S ) 1 ˆx = x + M (y S x ) + ɛm ( x j x ) 5: Update the state of the Kalman-Consensus flter: 6: end whle P AM A T + BQB T x Aˆx In Algorthm 3, the message broadcasted to all the neghbors by node s msg = (u, U, x ) and the nodes communcate ther predctons x s as well as ther sensed data. In smulaton results, we wll see that Algorthm 3 has the best performance on a trackng task. Algorthms 2 and 3 can be effectvely appled to moble sensor networks by assumng the tme-dependence of the set of neghbors N (t) as n [11]. wth IV. SIMULATION RESULTS Consder a target wth dynamcs ẋ = A 0 x + B 0 w [ 0 1 A 0 = ] 5496

6 and B 0 = c 2 wi 2 wth c w = 5 (.e. a pont movng on nosy crcular trajectores). We use the dscrete-tme model of ths target wth parameters x(k + 1) = Ax(k) + Bw(k) A = I 2 + ɛa 0 + ɛ2 2 A2 0 + ɛ3 6 A3 0, B = ɛb 0. The step-sze s ɛ = ( 70 Hz). The ntal condtons are x 0 = (15, 10) T, P 0 = 10I 2. A sensor network wth randomly located nodes s used n ths experment (see Fg. 2). The nodes make nosy measurement of the poston of the target ether along the x-axs, or along the y-axs,.e. (a) z = H x + v where ether H = H x = (1, 0), or H = H y = (0, 1). Moreover, R = c 2 v for = 1, 2,..., 50 wth cv = 30. Clearly, the target s not observable by ndvdual sensors, but s observable by all the sensors. Furthermore, we assume that each set of nclusve neghbors J of node contans nodes wth observaton matrces H x and H y. Fg. 2. A sensor network wth 50 nodes and 242 lnks. Half of the nodes sense along the x-axs and the other half sense along the y-axs. Let us refer to local Kalman flterng wth no exchange of state or estmates as Algorthm 0 (or A0). We compare four algorthms A0, A1, A2, and A3. For Algorthm 1, we set β = 7. Larger values of β force a smaller step-sze ɛ, or a hgher rate of nformaton exchange n the network. Ths s a crtcal weakness of the hgh-pass consensus flter and thus Algorthm 1. To measure the dsagreement of the estmates ndependent of the network topology, we use the followng measure ˆx. δ = ( n =1 δ2 )1/2 wth δ = ˆx µ and µ = 1 n One can also utlze Ψ g (ˆx) nstead. Fg. 3 demonstrates the comparson of the four dstrbuted Kalman flterng algorthms. A quck look at Fg. 3 reveals that the pars of algorthms (A0,A1) and (A2,A3) behave n a smlar manner (have comparable performances). Furthermore, both A2 and A3 perform sgnfcantly better than A0 and A1. Fnally, A3 performs better than all other algorthms on ths target trackng task. (b) Fg. 3. Comparson of the performance of dstrbuted Kalman flterng algorthms: (a) average estmaton error per node and (b) dsagreement of the estmates δ. Each curve s determned by averagng over 10 random runs of each algorthm). Estmaton error by tself s no longer an mportant measure of performance n dstrbuted estmaton n sensor networks. In a peer-to-peer estmaton archtecture, no partcular fuson centers exst and every node s supposed to know the estmate of the state. A relatvely hgh dsagreement n the estmates of the dfferent nodes goes aganst the purpose of performng dstrbuted estmaton wthout leaders. Therefore, only Kalman-Consensus flters (A2,A3) perform n a satsfactory manner. Ths result demonstrates the mportance of havng a base standard (A0) that can be employed to judge the usefulness of more complex dstrbuted estmaton algorthms that nvolve further nformaton exchange n terms of states/estmates. Fg. 4 compares the estmates of all nodes by A1 vs. A3. Clearly, A3 has a superor performance and provdes cohesve estmates. The estmates of A1 are somewhat dspersed for a relatvely long perod. 5497

7 (a) (b) (c) (d) (e) (f) (g) (h) Fg. 4. Comparson between the estmates of all nodes: (a)-(d) Algorthm 1 and (e)-(h) Algorthm 3 V. CONCLUSIONS Three novel dstrbuted Kalman flterng algorthms were ntroduced. Algorthm 1 s a modfcaton of a prevous DKF algorthm [10]. A contnuous-tme DKF algorthm s rgorously derved and analyzed. Two Kalman-Consensus flterng algorthms n dscrete-tme were nspred by ths contnuous-tme DKF algorthm. The objectve n type-ii DKF algorthms s to reduce dsagreement of the estmates by dfferent nodes. Ths led to the addton of a consensus term n the estmator of Algorthm 3 and an ad hoc consensus step n Algorthm 2. On a trackng task, both A2 and A3 perform better than A1 and a prmtve local Kalman flterng approach. Algorthm 3 has the best overall performance. ACKNOWLEDGMENT Many thanks goes to Jeff Shamma for several useful dscussons on dstrbuted estmaton and suggestng Lemma 1. REFERENCES [1] B. D. O. Anderson and J. B. Moore. Optmal Flterng. Prentce-Hall. Englewood Clffs. NJ., [2] Y. Bar-Shalom and T. E. Fortmann. Trackng and Data Assocaton. Academc Press, [3] Y. Bar-Shalom and X. R. L. Multtarget-Multsensor Trackng: Prncples and Technques. YBS Publshng, Storrs, CT, [4] S. Boyd, A. Ghosh, and D. Prabhakar, B. Shah. Gossp algorthms: desgn, analyss and applcatons. Proceedngs of the 24th Annual Jont Conference of the IEEE Computer and Communcatons Socetes (INFOCOM 05), pages , March [5] J. Cortés. Dstrbuted algorthms for reachng consensus on arbtrary functons. Automatca (submtted), Oct [6] D. Fox, J. Hghtower, L. Lao, D. Schulz, and G. Borrello. Bayesan flterng for Locaton estmaton. IEEE Pervasve Computng, 2(3):24 33, July Sept [7] V. Gupta, D. Spanos, B. Hassb, and R. M. Murray. On LQG control across a stocjastc packet-droppng lnk. Proc. of the 2005 Automatc Control Conference, pages , June [8] Y. Hatano and M. Mesbah. Agreement over random networks. IEEE Trans. on Automatc Control, 50(11): , Nov [9] D. Kempe, A. Dobra, and J. Gehrke. Gossp-based computaton of aggregate nformaton. Proc. of the 44th Annual IEEE Symposum on Foundatons of Computer Scence (FOCS 03), 8(3): , [10] R. Olfat-Saber. Dstrbuted Kalman flter wth embedded consensus flters. 44th IEEE Conference on Decson and Control, 2005 and 2005 European Control Conference (CDC-ECC 05), pages , Dec [11] R. Olfat-Saber. Dstrbuted trackng for moble sensor networks wth nformaton-drven moblty. Proc. of the 2006 Amercan Control Conference, July [12] R. Olfat-Saber, J. A. Fax, and R. M. Murray. Consensus and cooperaton n networked mult-agent systems. Proceedngs of the IEEE, 95, Jan [13] R. Olfat-Saber and R. M. Murray. Consensus problems n networks of agents wth swtchng topology and tme-delays. IEEE Trans. on Automatc Control, 49(9): , Sep [14] R. Olfat-Saber and J. S. Shamma. Consensus flters for sensor networks and dstrbuted sensor fuson. 44th IEEE Conference on Decson and Control, 2005 and 2005 European Control Conference (CDC-ECC 05), pages , Dec [15] B. S. Y. Rao, H. F. Durrant-Whyte, and J. A. Sheen. A fully decentralzed mult-sensor system for trackng and survellance. Int. Journal of Robotcs Research, 12(1):20 44, Feb [16] D. B. Red. An algorthm for trackng multple targets. IEEE Trans. on Automatc Control, 24(6): , Dec [17] W. Ren and R. W. Beard. Consensus seekng n multagent systems under dynamcally changng nteracton topologes. IEEE Trans. on Automatc Control, 50(5): , [18] L. Schenato, B. Snopol, M. Franceschett, K. Poola, and S. S. Sastry. Foundatons of control and estmaton over lossy networks. Proceedngs of the IEEE, 95(1): , Jan [19] B. Snopol, L. Schenato, M. Franceschett, K. Poola, M. I. Jordan, and S. S. Sastry. Kalman flterng wth ntermttent observatons. IEEE Trans. on Automatc Control, 49(9): , Sep [20] D. Spanos, R. Olfat-Saber, and R. M. Murray. Dynamc Consensus on Moble Networks. The 16th IFAC World Congress, Prague, Czech, [21] D. P. Spanos, R. Olfat-Saber, and R. M. Murray. Approxmate dstrbuted Kalman flterng n sensor networks wth quantfable performance. Fourth Internatonal Symposum on Informaton Processng n Sensor Networks, pages , Aprl [22] J. L. Speyer. Computaton and transmsson requrements for a decentralzed lnear-quadratc-gaussan control problem. IEEE Trans. on Automatc Control, 24(2): , Feb [23] J. N. Tstskls. Problems n Decentralzed Decson Makng and Computaton. PhD thess, Department of Electrcal Engneerng and Computer Scence, Laboratory for Informaton and Decson Systems, Massachusetts Insttute of Technology, Cambrdge, MA, Nov [24] J. N. Tstskls, D. P. Bertsekas, and M. Athans. Dstrbuted asynchronous determnstc and stochastc gradent optmzaton algorthms. IEEE Trans. on Automatc Control, 31(9): , Sep