An overview of non-centralized Kalman filters

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1 An overvew of non-centralzed Kalman flters Sjs, J; Lazar, M; van den Bosch, PPJ; Papp, Z Publshed n: Control Applcatons, 2008 CCA 2008 IEEE Internatonal Conference, San Antono, Texas, USA, September 3-5, 2008 DOI: /CCA Publshed: 01/01/2008 Document Verson Publsher s PDF, also known as Verson of Record (ncludes fnal page, ssue and volume numbers) Please check the document verson of ths publcaton: A submtted manuscrpt s the author's verson of the artcle upon submsson and before peer-revew There can be mportant dfferences between the submtted verson and the offcal publshed verson of record People nterested n the research are advsed to contact the author for the fnal verson of the publcaton, or vst the DOI to the publsher's webste The fnal author verson and the galley proof are versons of the publcaton after peer revew The fnal publshed verson features the fnal layout of the paper ncludng the volume, ssue and page numbers Lnk to publcaton Ctaton for publshed verson (APA): Sjs, J, Lazar, M, Bosch, van den, P P J, & Papp, Z (2008) An overvew of non-centralzed Kalman flters In Control Applcatons, 2008 CCA 2008 IEEE Internatonal Conference, San Antono, Texas, USA, September 3-5, 2008 (pp ) San Antono, Texas: Insttute of Electrcal and Electroncs Engneers (IEEE) DOI: /CCA General rghts Copyrght and moral rghts for the publcatons made accessble n the publc portal are retaned by the authors and/or other copyrght owners and t s a condton of accessng publcatons that users recognse and abde by the legal requrements assocated wth these rghts Users may download and prnt one copy of any publcaton from the publc portal for the purpose of prvate study or research You may not further dstrbute the materal or use t for any proft-makng actvty or commercal gan You may freely dstrbute the URL dentfyng the publcaton n the publc portal? Take down polcy If you beleve that ths document breaches copyrght please contact us provdng detals, and we wll remove access to the work mmedately and nvestgate your clam Download date: 03 Jan 2019

2 17th IEEE Internatonal Conference on Control Applcatons Part of 2008 IEEE Mult-conference on Systems and Control San Antono, Texas, USA, September 3-5, 2008 ThB023 An overvew of non-centralzed Kalman flters J Sjs, Student Member, IEEE M Lazar PPJ van den Bosch Z Papp Abstract The usage of Wreless Sensor etworks (WSs) for state-estmaton has recently ganed ncreasng attenton due to ts cost effectveness and feasblty One of the major challenges of state-estmaton va WSs s the dstrbuton of the centralzed state-estmator among the nodes n the network Sgnfcant emphass has been on developng non-centralzed stateestmators consderng communcaton, processng-demand and estmaton-error Ths survey paper presents dfferent methodologes to obtan non-centralzed state-estmators and focuses on the estmaton algorthms and ther mplementaton The temperature dstrbuton of a bar s used as a benchmark to assess the non-centralzed state-estmators n terms of estmaton-error and communcaton requrements Index Terms Wreless Sensor etworks, dstrbuted stateestmaton, Kalman flter I ITRODUCTIO State-estmaton s a wdely used technque n montorng and control applcatons An mportant state-estmator stll wdely used today s the Kalman flter formally presented n [1] The method requres that all process-measurements are sent to a central system whch estmates the global state-vector of the process The nterest for usng WSs to retreve the measurements has recently grown [2], due to mproved performance and feasblty n new applcaton areas However, for WSs consstng of a large amount of nodes a central state-estmator becomes mpractcable due to hgh processng demand and energy consumpton As a result, the dstrbuton of the centralzed Kalman flter, n whch each node estmates ts own state-vector, has become a challengng and actve research area Wthn ths research area two dfferent drectons can be notced In one drecton each node estmates the global statevector and a central system s used to fuse the nformaton of all the nodes together Examples of such methods, also called sensor fuson (SF) can be found n [3] [8] In the second drecton the central estmaton s absent Instead each node estmates a part of the global state-vector usng nformaton from other nodes n ts local regon, preferably ts drect neghbors Artcles that descrbe these dstrbuted Kalman flters (DKFs) are [9] [14] The purpose of ths paper s to provde a crtcal overvew of exstng non-centralzed Kalman flters, whch would help n choosng a partcular method for a partcular applcaton J Sjs and Z Papp are wth TO Scence and Industry, PO Box 155, 2600 AD Delft, The etherlands, E-mal: jorssjs@tnonl, zoltanpapp@tnonl M Lazar and PPJ van den Bosch are wth the Department of Electrcal Engneerng, Endhoven Unversty of Technology, PO Box 513, 5600 MB Endhoven, The etherlands, E-mal: mlazar@tuenl, ppjvdbosch@tuenl For each method we present ts characterstcs, algorthm and amount of decentralzaton n terms of processng demand and communcaton requrements per node Fnally all methods are assessed n a benchmark problem on ther performance n estmaton, communcaton and robustness to data loss or node break down The remander of the paper has the followng structure Some basc notaton and the prncples of the centralzed Kalman flter are descrbed n Secton II The ntal SF/DKF presented n [3] s then descrbed n Secton III Ths method was later used to desgn DKFs, as shown n Secton IV Secton V presents a herarchcal Kalman flter, whle a DKF wth weghted averagng s gven n Secton VI Secton VII dscusses a DKF wth consensus flters and a DKF wth bpartte fuson graphs s presented n Secton VIII Fnally, the dfferent non-centralzed Kalman flters are assessed n Secton IX usng the benchmark example of estmatng the temperature of a bar va a wreless sensor network Conclusons and recommendatons are formulated n Secton X II CETRALIZED KALMA FILTER Suppose a WS s used n combnaton wth a centralzed Kalman flter to estmate the states of a global process All nodes send ther measurements to one system where the centralzed Kalman flter estmates the global state-vector The measurements of the k th sample nstant are combned n the measurement-vector y[k] wth measurement-nose v[k] The global state-vector of the process s defned as x[k] wth process-nose w[k] Wth ths, the dscretsed process-model becomes: x[k] = Ax[k 1]+w[k 1], y[k] = Cx[k]+v[k] (1a) (1b) The probablty densty functon (PDF) of both w[k] and v[k] are descrbed by a Gaussan-dstrbuton, e E(w[k]) = 0 and E(w[k]w T [k]) = Q[k], E(v[k]) = 0 and E(v[k]v T [k]) = R[k] The centralzed Kalman flter [1] estmates the global state-vector ˆx[k] and the global error-covarance matrx P[k] Let E(α) represent the expectaton of the stochastc varable α Then, ˆx[k] and P[k] are defned as: ˆx[k] = E(x[k]), P[k] = E((x[k] ˆx[k])(x[k] ˆx[k]) T ) (3) The centralzed Kalman flter conssts of two stages that are performed each sample nstant k: the predctonstep and the measurement-update Frst the predctonstep computes the predcted state-vector ˆx[k k 1] and error- (2) /08/$ IEEE 739 Authorzed lcensed use lmted to: Endhoven Unversty of Technology Downloaded on March 18, 2009 at 05:55 from IEEE Xplore Restrctons apply

3 covarance P[k k 1] Second, the measurement-update calculates the estmated state-vector ˆx[k k] and error-covarance P[k k] The centralzed Kalman flter, wth ntal values ˆx[0 0] = x 0 and P[0 0] = P 0, s formally descrbed by the followng set of equatons: predcton-step ˆx[k k 1] = A ˆx[k 1 k 1], P[k k 1] = AP[k 1 k 1]A T + Q[k 1], measurement-update K[k] = P[k k 1]C T (CP[k k 1]C T + R[k]) 1, ˆx[k k] = ˆx[k k 1]+K[k](y[k] C ˆx[k k 1]), P[k k] = (I K[k]C)P[k k 1] (4a) (4b) For large scale WSs the centralzed mplementaton of (4) results n hgh processng demand, communcaton requrements and energy consumpton, whch prevents the usage of a centralzed Kalman flter To overcome ths ssue, a number of methodologes to mplement the Kalman flter n a dstrbuted fashon were desgned However, untl now there has been no comparson or evaluaton of the obtaned results n ths drecton The purpose of ths paper s to provde a crtcal overvew of exstng methods for desgnng noncentralzed Kalman flters The performance of the dfferent DKFs s llustrated usng a benchmark applcaton example n Secton IX Before explanng the dfferent methods of ths overvew n detal, we present three assumptons If not ndcated otherwse, these assumptons hold for the presented method Frstly, the exstence of a WS consstng of nodes s assumed n whch each node has ts own measurement-vector y wth correspondng measurement-nose v The global measurement-vector y, observaton-matrx C and equaton (1b) are rewrtten as follows: y [k] = C x[k]+v [k] { y = (y 1,y 2,,y ) T C = (C 1,C 2,,C ) T Secondly, the measurement-noses of two dfferent nodes are uncorrelated, e R (, j) = E(v v T j ) = 0, f j Resultng n an R-matrx of the form: (5) R = blockdag ( R (1,1),R (2,2),,R (,) ) (6) Thrdly, all nodes j that are drectly connected to a node are collected n the set, whch also ncludes the node Ths means that f node j s connected to node, then j Usually, s contanng only drect neghbors of node However, t s also possble that contans other nodes besdes the drect neghbors and n the case of global communcaton = Ths wll be made clear for each estmaton method III PARALLEL IFORMATIO FILTER Ths secton descrbes a parallel mplementaton of the Kalman flter [3] Each node has ts own Kalman flter calculatng the global state-estmates ˆx and P of node usng only ts measurement-vector y In the algorthm an nformaton-matrx I and an nformaton-vector are computed from the y and R (,) Each node sends ts stateestmates to a central system whch calculates the global state-estmates of the whole WS, e ˆx and P The sets of equatons of the parallel nformaton flter(pif) for node are: node predcton-step ˆx [k k 1] = A ˆx [k 1 k 1], P [k k 1] = AP [k 1 k 1]A T + Q[k 1], node nformaton-update I [k] = C T R 1 (,) [k]c, [k] = C T R 1 (,) [k]y [k], node measurement-update P 1 [k k] = P 1 [k k 1]+I [k], ˆx [k k] = P [k k](p 1 [k k 1] ˆx [k k 1]+ [k]) (7a) (7b) (7c) The global state-estmates ˆx and P are calculated takng the covarance ntersecton nto account [15]: α [k] = (tr(p [k k])) 1 =1 (tr(p [k k])) 1, P 1 [k] = ˆx[k] = =1 α [k]p[k]p 1 =1 α [k]p 1 [k k], [k k] ˆx [k k] (8) The calculaton of ˆx[k] and P[k] s done n a central system, whch can be located n one node only or even n every node A drawback of ths method s that every node estmates a global state-vector leadng to a hgh processng-demand A second drawback s global communcaton, for every node needs to send nformaton to at least one central system Ths method was mproved n the decentralzed nformaton flter presented n the next secton IV DECETRALIZED IFORMATIO FILTER In [9] the decentralzed nformaton flter (DIF) was proposed to overcome some drawbacks of the PIF Agan each node has ts own global state-estmates ˆx and P However, the central estmaton s decentralzed among the nodes and a node s only connected to ts neghborng nodes n These nodes exchange ther nformaton-matrx I and nformatonvector Meanng that node receves I j and j from the nodes j wth j, j The receved I j and j are added to I and respectvely The sets of equatons of the DIF for node are: node predcton-step ˆx [k k 1] = A ˆx [k 1 k 1], P [k k 1] = AP [k 1 k 1]A T + Q[k 1], node nformaton-update I [k] = C T R 1 (,) [k]c, [k] = C T R 1 (,) [k]y [k], local measurement-update P 1 [k k] = P 1 [k k 1]+ I j [k], ˆx [k k] = P [k k](p 1 [k k 1] ˆx [k k 1]+ j [k]) (9a) (9b) (9c) 740 Authorzed lcensed use lmted to: Endhoven Unversty of Technology Downloaded on March 18, 2009 at 05:55 from IEEE Xplore Restrctons apply

4 An mportant aspect of ths DKF s that f node s connected to all other nodes and assumptons (5) and (6) are vald, ts state-estmates ˆx and P are exactly the same as the estmates of a centralzed Kalman flter [1] An advantage s that only local communcaton s requred A drawback however, s that each node estmates the global state-vector V DECOUPLED HIERARCHICAL KALMA FILTER In [10] [12] decoupled herarchcal Kalman flters (DHKFs) are presented The common feature of ths method s that the global state-vector x and the process-model are dvded n parts Each node estmates one of the parts and exchanges ts state-estmates wth all other nodes n the WS The process-model s descrbed as: x 1 [k] = A x [k] y 1 [k] = C y [k] x 1 [k 1] + w 1 [k 1] x [k 1] w [k 1] x 1 [k] x [k] v 1 [k] +, v [k], (10) where, A (1,1) A (1,) C (1,1) C (1,) A =,C = A (,1) A (,) C (,1) C (,) Just as R, also the matrces Q and P are both assumed to be block-dagonal matrces Therefore we defne Q = E(w w T ) and P = E((x ˆx )(x ˆx ) T ) ode estmates ˆx [k] and P [k] The algorthm for each node s: node predcton-step ˆx [k k 1] = P [k k 1] = j=1 j=1 A (, j) ˆx j [k 1 k 1], node measurement-update (A (, j) P j [k 1 k 1]A T (, j) )+Q [k 1], (11a) K [k] = P [k k 1]C T (,) ( j=1(c (, j) P j [k k 1]C T (, j) )+R (,)[k]) 1, ˆx [k k] = ˆx [k k 1]+K [k](y [k] P [k k] = (I K [k]c (,) )P [k k 1] j=1 C (, j) ˆx j [k k 1]), (11b) otce that ths method s better compared to the PIF and DIF n terms of processng-demand and the amount of data transfer requred A drawback however, s that global communcaton s stll requred VI DISTRIBUTED KALMA FILTER WITH WEIGHTED AVERAGIG In prevous methods each node sends a vector wth ts correspondng covarance-matrx to the other nodes, e wth I or ˆx wth P In the dstrbuted Kalman flter wth weghted averagng (DKF-WA) [13] a node only sends ts state-vector, wthout covarance-matrx, to ts neghborng nodes n the set The weghted average of all receved state-vectors forms the node s estmated global state-vector ˆx One remark should be made: n ths case the matrx R s not necessarly block-dagonal, e R (, j) 0, j The algorthm of the DKF-WA s dvded nto an on-lne and an off-lne part In the on-lne part each node has ts own estmate of the global state-vector ˆx whch s partly calculated usng the equatons of the centralzed Kalman flter In ths method a node has a fxed, pre-calculated Kalman gan K After the measurement-update the nodes exchange ther estmated state-vector A node receves the state-vectors ˆx j ( j ) whch are then weghted wth a fxed, pre-calculated matrx W (, j) The weghted average s chosen as the new estmated global state-vector of node, e ˆx The on-lne algorthm s: node predcton-step (on-lne) ˆx [k k 1] = A ˆx [k 1 k 1], node measurement-update (on-lne) ˆx [k k] = ˆx [k k 1]+K (y [k] C ˆx [k k 1]), local weghted average (on-lne) ˆx [k k] = W (, j) ˆx j [k k] (12a) (12b) (12c) ext, we explan the off-lne algorthm whch s used to calculate K and W (, j) For that, the error-covarance between the estmated global state-vectors of node and j s: P (, j) [k] = E((x[k] ˆx [k])(x[k] ˆx j [k]) T ) (13) The off-lne algorthm uses the same stages for P (, j) as the on-lne algorthm for ˆx n (12) Frst the predcton-step (12a) and measurement-update (12b) of a node are gven to calculate P (, j) [k k 1] and P (, j) [k k], wth j : node predcton-step (off-lne) P (, j) [k k 1] = AP (, j) [k 1 k 1]A T + Q[k 1], node measurement-update (off-lne) K [k] = P (,) [k k 1]C T (C P (,) [k k 1]C T + R (,) [k]) 1, P (, j) [k k] = (I K [k]c )P (, j) [k k 1](I K j [k]c j ) T (14a) + K [k]r (, j) K T j [k] (14b) otce R (, j) and the calculaton of K n (14b) The next step s calculatng W (, j) of the weghted average as n (12c) To keep the state-estmaton unbased the followng constrant s ntroduced: W (, j) [k] = I n n (15) From (12c) and (15) we can derve: x[k] ˆx [k k] = W (, j) x[k] W (, j) ˆx j [k k], = W (, j) (x[k] ˆx j [k k]) (16) 741 Authorzed lcensed use lmted to: Endhoven Unversty of Technology Downloaded on March 18, 2009 at 05:55 from IEEE Xplore Restrctons apply

5 Usng (13) the weghted average of P (, j) [k k] results n: P (, j) [k k] = W (,p) [k]p (p,q) [k k]w T q j p ( j,q) [k] (17) Equaton (17) can also be wrtten n matrx form If = ( 1, 2,, ) and we defne W = (W (,1 ),,W (, )), equaton (17) becomes: P (1, j 1 )[k k] P (1, j j )[k k] P (, j) [k k] = W W j T P (, j 1 )[k k] P (, j j )[k k] (18) The last step n ths off-lne algorthm s to mnmze P (,) [k k] wth respect to W = (W (,1 ),,W (, )) takng constrant (15) nto account For further detals we refer the nterested reader to [13] The off-lne algorthm runs untl the values K and W (, j) reman constant These values are then used n the on-lne algorthm An mportant aspect n the performance of ths method s that each node estmates the global state-vector, but due to the fxed matrces K and W (, j) ts processng-demand remans low It was already notced that the DKF-WA has low communcaton requrements However, t s not robust aganst lost data or nodes breakng down For n that case the weghted averagng of (12c) wll not be accurate VII DISTRIBUTED KALMA FILTER WITH COSESUS FILTERS In [14], [16] the dstrbuted Kalman flter wth consensus flter (DKF-CF) was proposed In ths method a node has ts own estmate of the global state-vector ˆx and the node can only communcate wth ts neghborng nodes collected n Instead of averagng the receved state-vectors, a node tres to reach consensus on them usng a correcton-factor ε Bascally the algorthm of the DKF-CF adds an extra stage to the algorthm of the DIF n (9), e the local-consensus - stage Hence, every node has ts own global state-estmates ˆx and P We defne ˆx c to be the estmated global state-vector of node before the consensus-stage The algorthm s: node predcton-step ˆx [k k 1] = A ˆx [k 1 k 1], P [k k 1] = AP [k 1 k 1]A T + Q[k 1], node nformaton-update I [k] = C T R 1 (,) [k]c, [k] = C T R 1 (,) [k]y [k], local measurement-update P 1 [k k] = P 1 [k k 1]+ I j [k], ˆx c [k k] = P [k k](p 1 [k k 1] ˆx [k k 1]+ j [k]), local consensus ˆx [k k] = ˆx c [k k]+ε ( ˆx c j[k k] ˆx c [k k]) (19a) (19b) (19c) (19d) Due to the local-consensus -stage ths method requres more communcaton then the DIF, but t does not necessarly lead to an mproved estmaton-error A drawback s that each node estmates the global state-vector, meanng hgh processng-demand and data transfer per node A DKF that overcomes ths problem s the dstrbuted Kalman flter wth bpartte fuson graphs VIII DISTRIBUTED KALMA FILTER WITH BIPARTITE FUSIO GRAPHS Orgnally, the usage of graphs to show how sensors are related to state estmates n DKFs was employed n [17] More recently, DKFs wth bpartte fuson graphs (DKF- BFG) were presented n [18] The method assumes that each node s connected only to ts neghborng nodes collected n Furthermore, a node has ts own state-estmate whch s only a part of the global state-vector Ths means that the global state-vector at node, e x global, s dvded nto two parts: a part that s estmated, e x, and a part that s not estmated, e d The vectors x and d are defned usng some transformaton-matrces Γ and S as follows: ( ) ( ) x [k] Γ = x global d [k] [k] (20) S Preferably, the states of x are determned by takng those states of x global that have a drect relaton wth the measurement-vector y Meanng that Γ and S are defned by observaton-matrx C Assume I s the dentty matrx wth sze equal to the number of states n x global If the j th column of C contans non-zero elements, the j th row of I s put nto Γ If not, the j th row of I s put nto S An example of C wth ts correspondng Γ and S s: ( ) c11 c C = c 15 0 c Γ = ( ) (21) ,S = Due to the fact that a node estmates a part of the global state-vector, the node also has ts own process-model derved from the global one Ths s done by usng Γ and S on the global process-model The followng matrces are defned: A = Γ AΓ T, D = Γ AS T, H = C Γ T and w [k] = Γ w[k] Wth ths, the process-model of node becomes: x [k] = A x [k 1]+D d [k 1]+w [k 1], y [k] = H x[k]+v [k] (22) The method assumes that the state-vector x s estmated by node as ˆx, state-vector d s sent by other nodes and s represented by node as dˆ What remans s the matrx Q [k] = E(w [k]w T [k]) ow that the characterstcs of the DKF-BFG are presented, we proceed wth the estmaton algorthm otce that the algorthmc procedure s actually based on the the DIF algorthm n (9) Each node shares ts local nformatonmatrx I and nformaton-vector wth ts neghbors n But because the state-vectors n dfferent nodes are not necessarly equal, n contrast wth the DIF, the structure of 742 Authorzed lcensed use lmted to: Endhoven Unversty of Technology Downloaded on March 18, 2009 at 05:55 from IEEE Xplore Restrctons apply

6 I and dffers per node Ths means that I cannot be added to I j, as s the case n (9c) Ths s solved by usng Γ and S as shown n the algorthm: node predcton-step ˆx [k k 1] = A ˆx [k 1 k 1]+D ˆ d [k 1], P [k k 1] = A P [k 1 k 1]A T + Q [k 1], node nformaton-update I [k] = H T R 1 (,) H, [k] = H T R 1 (,) y [k], local measurement-update P 1 [k k] = P 1 [k k 1]+ (Γ Γ T j )I j [k](γ Γ T j ) T, ˆx [k k] = P [k k]p 1 [k k 1] ˆx [k k 1] + P [k k] (Γ Γ T j ) j [k](γ Γ T j ) T (23a) (23b) (23c) An mportant ssue n the performance of ths method s whether the global process-model s sparse and localzed so that the node s process-model can be derved wthout loss of generalty If ths s ndeed the case, ts performance should be equal to the DIF A drawback s that although only local communcaton s assumed n [18], t s also assumed that the states of dˆ are sent by other nodes Ths means that extended or even global communcaton may stll be needed A beneft of ths method s that a node only estmates a part of the global state-vector so that ts processng-demand per node s low IX APPLICATIO EXAMPLE Ths secton assess the non-centralzed Kalman flters presented n ths paper n terms of state-estmaton error, communcaton requrements and robustness aganst data loss or node break down The benchmark process s the heat transfer of a bar The bar s dvded nto 100 segments and the temperature T n of each segment n s estmated The state-vector of the the global process s therefore x = (T 1,T 2,,T 100 ) T The bar s heated at the 48 th segment The WS conssts of 5 nodes, placed at segment 11, 31, 51, 71 and 91 Each node measures the temperature of ts own specfc segment Several of the DKFs are used to estmate the temperature at all 100 segments A graphcal descrpton of ths system s shown n Fgure 1 state-vector and error-covarance together wth Q and R are the same for all methods Ths concludes the desgn of the PIF and the DHKF Communcaton s only allowed wth the neghborng nodes For example, node 3 receves from and sends data to node 2 and 4 In ths way the desgn of the DIF s also completed For the DKF-CF the value of ε s set 01, whch gave good smulaton-results The desgn of ths parameter s crtcal, for f too bg the estmaton algorthm becomes unstable, whle f too lttle the method has no mprovements over the DIF algorthm Matrces Γ and S of the DKF-BFG are constructed n such a way that node 1 estmates state 1 to 21, node 2 state 1 to 41, node 3 state 21 to 61, node 4 state 41 to 81 and node 5 state 61 to 100 Fgure 2 and Fgure 3 show the real temperature of all the states together wth the measurements (wth nose) both at 10,000 seconds Also the estmated states of the dfferent methods are plotted The estmaton of a state s nearest node s plotted, e the plotted states 41 to 60 were estmated by node 3 In case of Fgure 2 no data loss was smulated In Fgure 3 however, we smulated a 5% loss of the communcated data-packages temperature [K] segment Fg 2 temperature [K] real measurement PIF DIF DHKF DKF WA DKF CF DKF BFG State-estmaton at tme 10,000 seconds wthout data loss real measurement PIF DIF DHKF DKF WA DKF CF DKF BFG Fg 1 Bar wth Wreless Sensor etwork The DKFs are frst ntalzed The samplng tme s 10 seconds and the model runs for 10,000 seconds The ntal segment Fg 3 State-estmaton at tme 10,000 seconds wth 5% data loss Besde state-estmaton, communcaton s also an m- 743 Authorzed lcensed use lmted to: Endhoven Unversty of Technology Downloaded on March 18, 2009 at 05:55 from IEEE Xplore Restrctons apply

7 portant aspect Table I shows whch varables need to be transmtted and whether they are transmtted locally (e to node n ) or globally (e to all nodes n ) The total number of sent tems s shown n the fourth column Take for example DIF; has 100 tems and I 10,000 tems odes 2, 3 and 4 send ths data to 2 other nodes whch leads to 20,200 tems to be sent per node odes 1 and 5 send to 1 other node, resultng n 10,100 sent tems per node TABLE I REQUIRED COMMUICATIO DKF varables nodes send tems per node PIF ˆx [k k] P [k k] 40,400 DIF [k] I [k] 10,100 or 20,200 DHKF ˆx [k k 1] P [k k 1] 3360 ˆx [k k] P [k k] DKF-WA ˆx [k] 200 DKF-CF [k] I [k] ˆx c[k k] 20,400 DKF-BFG [k] I [k]/ ˆx [k k / 500 or 1800 or 3440 Fgure 2 and Fgure 3 together wth Table I show the performance, robustness to data loss and the communcaton requrement, respectvely, for each method Unfortunately, the methods that requre the least data transfer, e DKF- WA and DHKF, suffer the most from data loss ote that the estmated temperature values obtaned wth these two methods do not even appear n Fgure 3 (they are around 100K) Furthermore, also the DKF-BFG estmator, although t needs much less communcaton than the DIF estmator, n the presence of data loss s not robust, as can be observed n Fgure 3 On the overall, the least estmaton error was obtaned for the DIF estmator, whch s also the most robust aganst data loss Another aspect that can be observed s that the process-model s almost localzed and sparse, as the results of the DKF-BFG closely resemble the ones obtaned wth the DIF, when no data loss occurs X COCLUSIOS In ths paper we presented an overvew of dfferent methodologes for desgnng non-centralzed Kalman flters that can be used n WSs Each method was descrbed and analyzed n terms of communcaton requrements, robustness and estmaton-error It turned out that the DKF- WA requres the least communcaton and provdes a low state-estmaton error However, t lacks robustness for ts estmaton error ncreases sgnfcantly when data s lost or nodes break down, whch s usually the case n WSs For ths reason t s not sutable for most WSs A method that can deal wth unrelable data transfer and node loss, but stll has a low state-estmaton error s the DIF It also has average requrements regardng the amount of data transfer needed compared to other methods The amount of computatons and communcaton per node can be decreased when the DKF-BFG s used However, ths approach s vald only for processes that have a localzed and sparse structure, and assumng that there s no data-loss Hence, the DKF-BFG s not sutable for usage n WSs An extenson on ths survey paper s to take mathematcal models for communcaton nto account Meanng that both communcaton topology as well as the ntroduced errors and noses due to wreless communcaton lnks are used n the nose- and stablty analyss, as descrbed n [19] Based on the above conclusons, future work on noncentralzed estmators, sutable for WSs, needs to fnd new methods for reducng the communcaton and computaton requrements, wthout loosng robustness to data loss Improvng the robustness of the DKF-BFG seems to be a possble soluton REFERECES [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] I Akyldz, W Su, Y Sankarasubramanam, and E Cayrc, Wreless Sensor etworks: a survey, Elsever, Computer etworks, vol 38, pp , 2002 [3] J Speyer, Computaton and transmsson requrements for a decentralzed Lnear-Quadratc-Gaussan control problem, IEEE Transactons on Automatc Control, vol 24, no 2, pp , 1979 [4] S Felter, An overvew of decentralzed Kalman flters, n IEEE 1990 Southern Ter Techncal Conference, Brmngham, Y, USA, 1990, pp [5] S Roy, R Hashem, and A Laub, Square root parallel Kalman flterng usng reduced order local flters, IEEE Transactons on Aerospace and Electronc Systems, vol 27, no 2, pp , 1991 [6] S Roy and R Ilts, Decentralzed lnear estmaton n correlated measurement nose, 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