Optimal Decentralized Kalman Filter

Transcription

1 17th Medterranean Conference on Control & Automaton Makedona Palace, Thessalonk, Greece June 24-26, 2009 Optmal Decentralzed Kalman Flter S Oruç, J Sjs, PPJ van den Bosch Abstract The Kalman flter s a powerful state estmaton algorthm whch ncorporates nose models, process model and measurements to obtan an accurate estmate of the states of a process Implementaton of conventonal Kalman flter algorthm requres a central processor that harvests measurements from all the sensors n the feld Central algorthms have some drawbacks such as relablty, robustness and hgh computaton whch result n a need for non-central algorthms Ths study takes optmalty n Decentralzed Kalman Flter (DKF) as ts focus and derves the Optmal Decentralzed Kalman Flter (ODKF) algorthm, n case the network topology s provded to every node n the network, by ntroducng Global Kalman Equatons ODKF sets a lower bound of estmaton error n least squares sense for DKF Index Terms State estmaton, sensor network, decentralzed Kalman flter I INTRODUCTION The Kalman flter [1] combnes nformaton from the model and the real tme measurements of a process to estmate the states of the process In ts orgnal form, the Kalman flter requres all the measurements to be sent to a central processor for estmaton Later on research nterest n ths feld has been drected to decentralze the Kalman flter to apply t n dstrbuted systems lke Dstrbuted Sensor Networks (DSN) consstng of large amount of nodes The early papers suggested mult sensor Kalman flterng schemes wth herarchcal but decentralzed structures [2] [4] Later on the Informaton flter, whch s a nonherarchcal decentralzed Kalman flter, s ntroduced where all sensors work n parallel to obtan local estmaton based on ther own and neghbors nformaton [5] [9] These algorthms are optmal gven the constrant of sharng only measurements We refer to such optmal algorthms, where only measurements are shared but not state estmatons, as Local Kalman Flter (LKF) n ths paper Contrary to LKF, Decentralzed Kalman Flter (DKF) does share state estmatons as well as measurements whch create more complex problems does not let the nodes share state estmatons whch creates more complex problems By transmttng state estmaton, the past measurements of each node can dffuse throughout the network The mportance of transmttng state estmaton s llustrated n Temperature Dstrbuton example S Oruç s wth the Department of Electrcal Engneerng, Endhoven Unversty of Technology, PO Box 513, 5600 MB Endhoven, The Netherlands, E-mal: sertaco@gmalcom J Sjs s wth TNO Scence and Industry, PO Box 155, 2600 AD Delft, The Netherlands, E-mal: jorssjs@tnonl PPJ van den Bosch s wth the Department of Electrcal Engneerng, Endhoven Unversty of Technology, PO Box 513, 5600 MB Endhoven, The Netherlands, E-mal: ppjvdbosch@tuenl Ths paper ams to ntroduce an optmal formulaton and nfer an optmal algorthm for DKF whch we call ODKF The more recent papers n ths feld menton the agreement protocols between the nodes based on estmated states [10] [12] These algorthms provde non-optmal but practcally acceptable solutons II KEY CONCEPTS AND NOTATION Node Unt whch has a sensor to make measurements, a processor to run an algorthm and a network connecton to neghborng nodes Neghbor The node that s communcaton-wse adjacent to the node under consderaton Neghborhood The set of nodes that s adjacent to the node under consderaton and the node tself Optmalty Mnmzaton of the sum of squared errors between state estmate, ˆx and the real state x accordng to the model Data Incest The stuaton n whch the same data s used more than once n the estmaton x[k] The real state at the k th teraton y [k] The measurement of node at the k th teraton ˆx The estmate of state x of node ˆx The predcted estmate of state x of node P j [k] E [ (x[k] ˆx [k])(x[k] ˆx j [k]) T ] M j Kalman gan for state from j th node on th node K j Kalman gan for measurement from j th node on th node N The set of all nodes n the network N Neghborhood of th node N Neghborhood of th node w/o tself e N {} v Measurement-nose of the th sensor w Process-nose R E [ v v T ] Q E [ ww T ] *For clarty tme ndces are dropped when there s no rsk of confuson III PROBLEM DESCRIPTION The system we consder n ths paper s a network of nodes wth ther own sensor, hence own measurements The network topology s arbtrary but known by each node An example of a certan network topology and process dynamcs s shown n fgure 1 There s only one process and each /09/$ IEEE 803 Authorzed lcensed use lmted to: Endhoven Unversty of Technology Downloaded on December 24, 2009 at 05:47 from IEEE Xplore Restrctons apply

2 Fg 1 System node estmates the entre state-vector of the process based on ts own and neghborng nodes measurement and prevous state estmaton Therefore the dscrete state-space model of a local node becomes: x[k] = Ax[k 1] + w[k 1], y [k] = C x[k] + v [k] (1a) (1b) The probablty densty functons (PDFs) of w and v are modeled as zero mean Gaussan noses wth an errorcovarance matrx of Q and R respectvely The sensor measurements are assumed to be uncorrelated Wth ths network of sensors we want to estmate the state-vector x of the process The optmal estmator for the descrbed process of (1) s the Kalman flter [1] for whch all measurements are gathered by a central algorthm Therefore, f we defne y := [y T 1,yT 2,,yT n ]T and v := [v T 1,vT 2,,vT n ]T, then the process model of the Kalman flter becomes: x[k] = Ax[k 1] + w[k 1] (2) y[k] = Cx[k] + v[k] (3) The Kalman flter s update equatons, wth R := E[vv T ] = dag(r 1,R 2,,R n ), yelds: ˆx [k] = A ˆx[k 1] (4) P [k] = AP[k 1]A T + Q (5) K[k] = P [k]c T (CP [k]c T + R) 1 (6) ˆx[k] = ˆx [k] + K[k](y[k] C ˆx [k]) (7) P[k] = (I K[k]C)P [k] (8) Although ths estmator s optmal t requres that every measurement s sent to one central processor, mplyng that t cannot be used n the system of Fgure 1 As a result of that, t does not make sense to compare an estmator that s desgned for a sensor network wth the central Kalman flter snce the latter one does not take the communcaton topology of the network nto account Nevertheless, a lot of DKFs make ths comparson Therefore, the goal of ths paper s to present an optmal state-estmator, e ODKF, whch takes explctly the network topology nto account Then future DKFs can compare ther results wth the ODKF to assess the performance of ther estmator For a DKF the most mportant parameter s the estmatonerror e x[k] ˆx [k] In case of the Kalman flter, ts PDF s modeled as a Gaussan functon It can be proven that wth the equatons of the Kalman flter ths PDF has a zero mean, e E[x[k] ˆx [k]] = 0, and a covarance modeled as P [k] The am of any DKF s to mnmze the total value of all P [k] gven the communcaton requrements Therefore, the ODKF mnmzes trace of n =1 P [k] under the standng assumpton that the network topology s known and that any parameter of the node s local state-estmator can be sent to ts neghborng nodes The desgned ODKF s compared wth the local Kalman flter (LKF) n an applcaton example on the temperature dstrbuton of a bar Therefore, before gong nto the detals of the ODKF, let us frst present the LKF IV LOCAL KALMAN FILTER The local Kalman flter of [5], [8] s based on the Kalman flter n nformaton form In ths form, each measurement of node, e y, and ts covarance R are transformed nto an nformaton-vector and nformaton-matrx I : [k] := C T R 1 y [k], (9) I := C T R 1 C (10) The LKF does not take the communcaton topology nto account, t just assumes that there are neghborng nodes Each node shares ts local nformaton-vector and correspondng nformaton-matrx I wth ts neghbors Therefore node uses the receved nformaton of all the nodes n the set N to update ts local state-vector ˆx and error-covarance matrx P For that node apples the followng equatons: P [k] = AP [k 1]A T + Q, (11) ˆx [k] = A ˆx [k 1], (12) ( ) 1 (P P [k] = [k] ) 1 + I j, (13) ) (P ˆx [k] = P [k]( [k] ) 1 ˆx [k] + j [k] (14) Ths algorthm s proven to gve optmal estmate for th node gven the constrant that only the measurements are communcated to neghborng nodes and are not forwarded to other nodes V OPTIMAL DECENTRALIZED STATE-ESTIMATOR Contrary to the LKF, n the ODKF each node transmts both ts local state estmate ˆx and measurement y to ts neghbors Note that the measurements are fresh data whch are not correlated wth the current estmates or other measurements made n the neghborhood The estmates however are found by ncorporaton of hstorcal nformaton, therefore they are correlated wth the estmates from the other nodes Furthermore, for the LKF holds that measurements made at a second order neghbor of node can never have an effect on the estmaton of x However, f the nodes communcate the state estmates as well as the measurements, as n the ODKF Then nherently the nformaton of measurements made at a second order neghbor of node does have effect on the estmaton of x through the estmated state of a frst order neghbor 804 Authorzed lcensed use lmted to: Endhoven Unversty of Technology Downloaded on December 24, 2009 at 05:47 from IEEE Xplore Restrctons apply

3 The transmsson of estmated states to neghbors brngs n the rsk of data ncest snce the estmated states of dfferent nodes can be correlated To prevent data ncest, we assume that the network topology s known by each node and taken nto account n the dervaton of ODKF As stated n Secton III, the am of the ODKF s to mnmze the trace of n =1 P [k] However, calculaton of P [k] s not straght forward snce ˆx [k] nvolves the ncorporaton of ˆx j [k 1], for all j N Therefore to calculate P [k] we need P j [k 1], snce P j [k 1] s defned as P j [k 1] = E[(x[k 1] ˆx [k 1])(x[k 1] ˆx j [k 1]) T ] Calculaton of these P j terms s the key challenge n fndng the ODKF algorthm Our approach n the desgn of the ODKF s a smple yet effectve one The cross-covarances are specfcally taken nto account by consderng the network as a whole For ths purpose we ntroduce a new representaton, called Global System Representaton, as presented n Secton V-A Ths representaton enables us to calculate all cross covarances P j [k] n the network, from whch a cost-functon can be derved to mnmze the trace of n =1 P [k] as presented n Secton V-B A Global Kalman Equatons In ths paper we am to take the cross correlatons, e P j, between the estmates of each node explctly nto account as well as the auto correlatons, e P For clarty let us start wth the state space model of the process at node : x[k] = Ax[k 1] + w[k 1], (15) y [k] = C x[k] + v [k] (16) Then the process for the entre network can be expressed as; wth the followng defntons: x[k] = Ax[k 1] + w[k 1], (17) Y[k] = Γx[k] +V[k] (18) y 1 y 2 C 2 v 2 Y :=,Γ :=,V := y n In the dervaton of the Kalman flter [1] t s defned that the update of the state-vector s x[k] = x [k] + K[k](y[k] Cx [k]) Then the optmal soluton of the estmaton was derved by substtuton of that K[k] whch solves the equaton δ Tr(P[k]) δ K[k] C 1 C n = 0 Applyng the same dervaton here, gves: ˆx [k] = ˆx [k] + K j [k](y j [k] C j ˆx [k]) where + j N v 1 v n M j [k]( ˆx j [k] ˆx [k]), (19) = M j [k] ˆx j [k] + K j [k]y j [k], (20) M := I j N M j K j C j (21) Here the predcted state ˆx s corrected wth the measurement and state resdue terms y j C j ˆx and ˆx j [k] ˆx [k] respectvely These resdue terms are weghed wth ther correspondng Kalman gans K j and M j Then; y = C x + v, e = x ˆx = M j e j K j v j, (22) j N e 1 v 1 v 2 v 3 e 2 e = µ e 3 κ, (23) P11 P 12 R 1 0 P = µ P21 P22 µ T + κ 0 R 2 κ T (24) Makng the followng defntons; ˆx 1 A 0 0 Q Q Q ˆx 2 0 A 0 Q Q Q ˆX :=, := 0 0 A,Φ := Q Q Q ˆx n P 11 P 12 P 13 K 11 K 12 K 13 P 21 P 22 P 23 K 21 K 22 K 23 Π := P 31 P 32 P 33,κ := K 31 K 32 K 33, M 11 M 12 M 13 R M 21 M 22 M 23 0 R 2 0 µ := M 31 M 32 M 33,Ω := 0 0 R 3 The Global Kalman Equatons wthout optmal Kalman Gans κ, µ s found as: predct: ˆX [k] = ˆX[k 1], (25) Π [k] = Π[k 1] T + Φ, (26) update: ˆX[k] = µ[k] ˆX [k] + κ[k]y[k], (27) Π[k] = µ[k]π[k] µ[k] T + κ[k]ωκ[k] T (28) The trace of Π[k] equals the cost-functon n our optmzaton problem Meanng that our am s to fnd those κ and µ whch mnmze the trace of equaton (28) The network connectons are explctly taken nto account n dervng equaton (28) such that K j and M j are defned as the zero-matrx n case there s no communcaton between th and j th node Note that M j, for all j, and K j are ndependent varables whereas M depends on these varables as n equaton (21) B Optmal Decentralzed Kalman flter In order to fnd the ODKF equatons we need to mnmze Tr(Π) for the Kalman Gans K j and M l wth N, j N and l N If for clarty P s replaced wth P mm, then wth 805 Authorzed lcensed use lmted to: Endhoven Unversty of Technology Downloaded on December 24, 2009 at 05:47 from IEEE Xplore Restrctons apply

4 the equalty Tr(Π[k]) = N m=1 Tr(P mm), ths cost-functon becomes: N K j,m l m=1 mn (Tr(Π)) = mn K j,m l Tr(P mm ) (29) On the other hand we have for P mm, after substtuton of the defntons of κ, µ nto (24), P mm = j N m M ml Pl j MT m j + K m j R j Km T j (30) l N m j N m Due to the fact that all Pl j s are constants comng from prevous teraton, t follows from (30) that only the P mm for whch = m depends only on K j and M l All the other ones, e P mm for all m N\ do not depend on K j and M l As a result mnmzaton of the cost-functon as shown n (29) becomes: mn (Tr(Π)) = mn Tr(P ) (31) K j,m l K j,m l So to mnmze the global cost-functon t s enough to mnmze ndvdual P s wth respect to ther correspondng K j,m l The values for K j and M l, whch mnmze the cost-functon, are those values at whch the dervatve of P equals 0, e: δ Tr(P ) = 2 δk j M l Pl CT j + 2K j R j = 0, j N, (32) δ Tr(P ) = 2 δm j M l (Pl Pl j ) = 0, j N (33) Rewrtng equatons (32) and (33), by replacng M = I l N as n equaton (21), gves: P CT j = l N (P P j ) = l N M l (P P l )CT j M l K l C l + K l C l P CT j + K j R j, where j N, l N M l (P P j ) + K l C l (P P j ) (34) M l (P l P l j ), where j N (35) Ths set of equatons gves 2n 1 matrx equatons from whch the Kalman gans are calculated, where n s the number of nodes n N Suppose we defne: κ := [ K 1 K 2 K 3 K j ], where j N, µ := [ M 1 M 2 M j ], where j N, Γ := [ C T 1,CT 2,,CT j, ]T, where j N Then (34) and (35), for all nodes, can be wrtten n matrx form as; [ ] [ ] A κ µ B B T = [ P ] C ΓT P (36) where A = Γ P + Ω, (37) B = Γ P, (38) C = P P P P 1 P 2 P k, (39) P j = [ P j P j P j ] [ P j1 P j2 P jk ], (40) where j N and k N At ths pont t could be helpful for the reader to note that to wrte equatons (39) and (40), we assume node 1 and 2 are neghbors of th node for the sake of notaton Hence the optmal DKF at th node yelds: predct: ˆx [k] = A ˆx [k 1] (41) P j [k] = AP j[k 1]A T + Q (42) Determnaton of Kalman gans κ, µ [ ] [ κ µ = P Γ T ] [ ] 1 A B B B T (43) C update: ˆx [k] = ˆx [k] + K j [k](y j [k] C j ˆx [k]) + M j [k]( ˆx j [k] ˆx [k]) (44) Π[k] = µ[k]π[k] µ[k] T + κ[k]ωκ[k] T (45) Notce that the covarance matrces, whch are requred for A, B and C, are calculated wth ther governng equaton (45) It nvolves global varables whch are all the predcted auto and cross covarance matrces Π as well as all the Kalman gans κ, µ n the network The covarance matrces and the Kalman gans depend on the measurement s covarance R and not on the actual measurements y snce they are calculated wth (42), (43) and (45) Therefore Π, κ and µ can be calculated n each local node f the nodes know the network topology To buld κ and µ each node frst calculates κ and µ for all N usng (43) whch gves non zero entres for κ and µ matrces And then the rest of the elements n κ and µ are assgned to zero, whch corresponds to the unconnected nodes n the network Remark: Dependng on the network topology, some or all states n the dfferent nodes converge to the same value As a result all cross covarance matrces, e P j, wll also converge to the same value makng B and C sngular and thus makng the nverson of (43) mpossble Ths s an nherent pont that stems from the formulaton of Π and ˆX when the augmented state was defned As the states of dfferent nodes converge to each other, ˆx j ˆx goes to zero and M j of (45) mght have any value Occurrence of sngularty depends on the precson of the processor 806 Authorzed lcensed use lmted to: Endhoven Unversty of Technology Downloaded on December 24, 2009 at 05:47 from IEEE Xplore Restrctons apply

5 Next, an example of the temperature dstrbuton of a bar s used to compare the ODKF, the LKF and the central Kalman flter (CKF) as well as revealng the sngularty ssue 1 sec 5000 sec VI EXPERIMENT: TEMPERATURE DISTRIBUTION To llustrate the performance of the ODKF algorthm, as proposed n Secton V-B, we smulate an experment n whch a bar s connected to two temperature reservors on both ends and heated from the mddle Our purpose s comparng the ODKF wth both the CKF and LKF The CKF s done by a central processor whch accesses all four measurements drectly Both the LKF and DKF are run n the local processors at the nodes sec sec Real Meas CKF LKF(N1) ODKF(N1) Fg 3 Temperature profle of the bar over tme Fg 2 Setup of the measurement system The bar s dvded nto 11 segments and therefore modeled wth 11 states each representng the temperature of one segment of the bar, as shown n fgure 2 The reservors provde boundary condtons for the experment and keep the temperature of the end ponts at K The bar s heated from the 6 th segment and the temperatures of only the 3 rd,5 th,7 th and 9 th segments are measured Each measurement s done by a node and the nodes can communcate wth ther neghbors to collaborate data Intally the bar s kept at K whch changes n tme due to the heat at the 6 th segment The temperature profle of the bar over tme s shown n fgure 3 wth sold lne All three state-estmators use the same state-space model of the system as n fgure 1, e x[k] = Ax[k 1]+w[k 1] and y [k] = C x[k]+v [k] The state-vector x contans 11 elements, each representng the temperature of one segment and w also contans 11 elements, each representng the unknown heatng/coolng at one of the segments Also C denotes measurement matrx and v measurement nose Notce that n ths example the frst node has a measurement matrx C 1 = [ ] The nodes do not know about the heatng/coolng gven to the bar, nor about the heat-reservors at the bar s ends and models those as Gaussan nose Therefore the covarance of the process-nose, e Q, s qute hgh wth respect to R, meanng that the measurements are more relable then the process-model The reason for usng ths setup s that snce the uncertanty n the process model s very hgh, the measurements are most valuable for state-estmaton In our system, although we have 11 segments we only measure 4 of them whle the RMS error (Kelvn) RMS error per node n LKF CKF LKF(Node 1) LKF(Node 2) tme (sec) Fg 4 Local Kalman Flter rest s to be estmated from the model That s to say, ths setup contans a lot of states that are not measured at all but are yet to be estmated The comparson between the LKF and the DKF llustrates the mportance of sharng states For example n the LKF the 1 st node can only use measurements from the 1 st and the 2 nd node Therefore ts estmaton-error of states on the rght sde of the bar s hgh However n the ODKF sharng states results n the dffuson of measurements from the 3 rd and 4 th node nherently, whch results n a much better estmaton as llustrated n fgure 3 The performance of the LKF and the ODKF are compared wth the CKF, as the CKF s assumed to be state-estmator wth the best overall performance, e the least estmatonerror The result of the LKF are presented n Fgure 4 and of the ODKF n Fgure 5 Ths fgure suggest that the ODKF may converge to the CKF, whch s left as a future work Fgure 4 shows the root-mean-squared (RMS) error n tme 807 Authorzed lcensed use lmted to: Endhoven Unversty of Technology Downloaded on December 24, 2009 at 05:47 from IEEE Xplore Restrctons apply

6 RMS error (Kelvn) RMS error per node n ODKF CKF ODKF(Node 1) ODKF(Node 2) tme (sec) Fg 5 Optmal Decentralzed Kalman Flter of the LKF between the real states and estmated ones at 1 st and 2 nd node Snce the process and the node s poston are symmetrc from the center of the bar, the error of the 3 rd and 4 th nodes follow more or less the same trajectores as the ones of the 2 nd and 1 st node respectvely For ths reason these two nodes are not drawn for clarty The LKF of the 1 st node has access to the measurements from the 2 nd node and tself Because of ths reason ts estmaton s worse than 2 nd node, for that node has access to the measurements from the 1 st and 3 rd node In our experment for the estmaton of the temperature dstrbuton by the ODKF, shown n Fgure 5, a hybrd algorthm s used Here both states and measurements are sent and the estmator swtches between the ODKF and LKF When the matrx Π, as denoted n Secton V-A, s full rank ODKF algorthm s used In case Π s sngular the LKF s appled When the LKF algorthm runs, each node makes ther own estmaton resultng n dfferent errorcovarance matrces, P j Therefore after a whle Π matrx agan becomes full rank whch trggers the use of ODKF algorthm Ths dynamcs, whch depends on the sngularty of Π matrx causes the saw tooth graph of Fgure 5 Ths behavor also proves that the ODKF decreases the estmatonerror sgnfcantly and gves a clue that ODKF may ndeed converge to CKF In concluson ths experment s a clear example to show the mportance of sendng states rather then sendng only measurements (as s done n most other state-estmators dedcated to sensor networks) Many network-topology specfc solutons [13] can be suggested for the problem However the am of ths experment s to compare the ODKF wth both the LKF and CKF wth respect to already mentoned performance measure A smple structure s chosen for llustraton purposes although t s possble to use an arbtrary and connected graph wth ODKF VII CONCLUSION In ths study we addressed some optmalty problems arsng from the decentralzaton of the Kalman flter Consderng the network-topology, we stated a problem formulaton for optmalty of any state-estmator Solvng ths problem formulaton, we obtaned the ODKF n whch the network topology s known Ths result proposes an algorthm whch makes the best estmate that can be obtaned, takng the network topology nto account So far n the lterature the proposed algorthms for decentralzed Kalman flters have been compared wth the central KF However the central KF does not take the network topology nto account whereas our ODKF does For fully connected network topologes t could turn out that the ODKF converges to the central KF, as t s the case for the experment,whch stll needs to be proven In ths paper we examned the optmalty condton constraned by the network topology We dd not take ts processng demand or data transfer nto account Although ths soluton of the ODKF s proven to be optmal, snce t depends on the network topology, scalablty s stll an ssue to be solved Meanng that addng or removng a node n the network wll affect the algorthms on all nodes snce ths affects network topology Therefore future work s on ncorporatng scalablty condton nto the problem formulaton The scalablty s a key condton to make the algorthm feasble for use n Dstrbuted Sensor Networks REFERENCES [1] R Kalman, A new approach to lnear flterng and predcton problems, Transacton of the ASME Journal of Basc Engneerng, vol 82, no D, pp 35 42, 1960 [2] D Wllner, C Chang, and K Dunn, Kalman flter algorthms for a mult-sensor system, IEEE Conference on Decson and Control Vol15, Part 1, vol 15, no 1, pp , 1976 [3] M Hassan, G Salut, M Sgh, and A Ttl, A decentralzed algorthm for the global Kalman flter, IEEE Transactons on Automatc Control, vol 23, no 2, pp , 1978 [4] H Hashmpour, S Roy, and A Laub, Decentralzed structures for parallel Kalman flterng, IEEE Transactons on Automatc Control, vol 33, no 1, pp 88 93, 1988 [5] H Durant-Whyte, B Rao, and H Hu, Towards a fully decentralzed archtecture for mult-sensor data fuson, n 1990 IEEE Int Conf on Robotcs and Automaton, Cncnnat, Oho, USA, 1990, pp [6] B Rao and H Durrant-Whyte, Fully decentralzed algorthm for multsensor Kalman flterng, IEEE Proceedngs, vol 138, no 5, pp , 1991 [7] S Roy and R Ilts, Decentralzed lnear estmaton n correlated measurement nose, IEEE Transactons on Aerospace and Electronc Systems, vol 27, no 6, pp , 1991 [8] A Mutambara and H Durrant-Whyte, A formally verfed modular decentralzed robot control system, IEEE/RSJ Internatonal Conference on Intellgent Robots and Systems, 1993 [9] S Shu, Mult-sensor optmal nformaton fuson Kalman flters wth applcatons, Elsever, Aerospace Scence & Technology, vol 8, no 1, pp 57 62, 2004 [10] R Saber, Dstrbuted Kalman flters for sensor networks, n 46th IEEE Conf on Decson and Control, New Orleans, LA, USA, 2007 [11] R Olfat-Saber, Dstrbuted Kalman flterng for sensor networks, Proceedngs of the 46th IEEE Conference on Decson and Control, 2007 [12] S Roy, A Saber, and K Herlugson, A control-theoretc perspectve on the desgn of dstrbuted agreement protocols, 2005 [13] S Stankovc, M Stanojevc, and D Sljak, Decentralzed overlappng control of a platoon of vehcles, IEEE transactons on Control Systems Technology, Authorzed lcensed use lmted to: Endhoven Unversty of Technology Downloaded on December 24, 2009 at 05:47 from IEEE Xplore Restrctons apply