Consensus-Based Distributed Linear Filtering

Transcription

1 49th IEEE Conference on Decson and Control December 5-7, 200 Hlton Atlanta Hotel, Atlanta, GA, USA Consensus-Based Dstrbuted Lnear Flterng Ion Mate and John S. Baras Abstract We address the consensus-based dstrbuted lnear flterng problem, where a dscrete tme, lnear stochastc process s observed by a network of sensors. We assume that the consensus weghts are known and we frst provde suffcent condtons under whch the stochastc process s detectable,.e. for a specfc choce of consensus weghts there exsts a set of flterng gans such that the dynamcs of the estmaton errors (wthout nose) s asymptotcally stable. ext, we provde a dstrbuted, sub-optmal flterng scheme based on mnmzng an upper bound on a quadratc flterng cost. In the statonary case, we provde suffcent condtons under whch ths scheme converges; condtons expressed n terms of the convergence propertes of a set of coupled Rccat equatons. We contnue wth presentng a connecton between the consensusbased dstrbuted lnear flter and the optmal lnear flter of a Markovan jump lnear system, approprately defned. More specfcally, we show that f the Markovan jump lnear system s (mean square) detectable, then the stochastc process s detectable under the consensus-based dstrbuted lnear flterng scheme. We also show that the optmal gans of a lnear flter for estmatng the state of a Markovan jump lnear system approprately defned can be used to approxmate the optmal gans of the consensus-based lnear flter. I. Introducton Sensor networks have broad applcatons n survellance and montorng of an envronment, collaboratve processng of nformaton, and gatherng scentfc data from spatally dstrbuted sources for envronmental modelng and protecton. A fundamental problem n sensor networks s developng dstrbuted algorthms for the state estmaton of a process of nterest. Genercally, a process s observed by a group of (moble) sensors organzed n a network. The goal of each sensor s to compute accurate state estmates. The dstrbuted flterng (estmaton) problem has receved a lot of attenton durng the past thrty years. An mportant contrbuton was brought by Borkar and Varaya [], who address the dstrbuted estmaton problem of a random varable by a group of sensors. The partcularty of ther formulaton s that both estmates and measurements are shared among neghborng sensors. The authors show that f the sensors form a communcaton rng, through whch nformaton s exchanged nfntely often, then the estmates converge asymptotcally to the same value,.e. they asymptotcally agree. An extenson of the results n reference [] s gven n []. The recent technologcal advances n moble sensor Ion Mate and John S. Baras are wth the Insttute for Systems Research and the Department of Electrcal and Computer Engneerng, Unversty of Maryland, College Park,mate, baras@umd.edu Ths materal s based upon work supported by the US Ar Force Offce of Scentfc Research MURI award FA , by the Defence Advanced Research Projects Agency (DARPA) award number to the Unversty of Calforna - Berkeley and by BAE Systems award number W9F networks have re-gnted the nterest for the dstrbuted estmaton problem. Most papers focusng on dstrbuted estmaton propose dfferent mechansms for combnng the Kalman flter wth a consensus flter n order to ensure that the estmates asymptotcally converge to the same value; these schemes wll be henceforth called consensus based dstrbuted flterng (estmaton) algorthms. Relevant results related to ths approach can be found n [7], [8], [9], [2], [0], [2]. In ths paper we address the consensus-based determnstc dstrbuted lnear flterng problem as well. We assume that each agent updates ts (local) estmate n two steps. In the frst step, an update s produced usng a Luenberger observer type of flter. In the second step, called consensus step, every sensor computes a convex combnaton between ts local update and the updates receved from the neghborng sensors. Our focus s not on desgnng the consensus weghts, but on desgnng the flter gans. For gven consensus weghts, we wll frst gve suffcent condtons for the exstence of flter gans such that the dynamcs of the estmaton errors (wthout nose) s asymptotcally stable. These suffcent condtons are also expressble n terms of the feasblty of a set of lnear matrx nequaltes. ext, we present a dstrbuted (n the sense that each sensor uses only nformaton avalable wthn ts neghborhood), sub-optmal flterng algorthm, vald for tme varyng topologes as well, resultng from mnmzng an upper bound on a quadratc cost expressed n terms of the covarances matrces of the estmaton errors. In the case where the matrces defnng the stochastc process and the consensus weghts are tme nvarant, we present suffcent condtons such that the aforementoned dstrbuted algorthm produces flter gans whch converge and ensure the stablty of the dynamcs of the covarances matrces of the estmaton errors. We wll also present a connecton between the consensus-based lnear flter and the lnear flterng of a Markovan jump lnear system approprately defned. More precsely, we show that f the aforementoned Markovan jump lnear system s (mean square) detectable then the stochastc process s detectable as well under the consensus-based dstrbuted lnear flterng scheme. Fnally we show that the optmal gans of a lnear flter for the state estmaton of the Markovan jump lnear system can be used to approxmate the optmal gans of the consensusbased dstrbuted lnear flterng strategy. We would lke to menton that, due to space lmtatons, not all the results are proved. The reader s nvted to consult the extended verson of ths paper represented by reference [6], whch contans all the mssng proofs /0/$ IEEE 7009

2 Paper structure: In Secton II we descrbe the problems addressed n ths note. Secton III ntroduces the suffcent condtons for detectablty under the consensus-based lnear flterng scheme together wth a test expressed n terms of the feasblty of a set of lnear matrx nequaltes. In Secton IV we present a sub-optmal dstrbuted consensus based lnear flterng scheme wth quantfable performance. Secton V makes a connecton between the consensus-based dstrbuted lnear flterng algorthm and the lnear flterng scheme for a Markovan jump lnear system. otatons and Abbrevatons: We represent the property of postve (sem-postve) defnteness of a symmetrc matrx A, by A 0 (A 0). By conventon, we say that a symmetrc matrx A s negatve defnte (sem-defnte) f A 0 ( A 0) and we denote ths by A 0 (A 0). Gven a set of square matrces {A }, by dag(a,...n) we understand the block dagonal matrx whch contans the matrces A s on the man dagonal. By A B we understand that A B s postve defnte. We use the abbrevatons CBDLF, MJLS and LMI for Consensus-Based Lnear Flter(ng), Markovan Jump Lnear System and Lnear Matrx Inequalty, respectvely. Remark.: Gven a postve nteger, a set of vectors {x }, a set of non-negatve scalars{p } summng up to one and a postve defnte matrx Q, the followng nequalty holds p x Q p x p x Qx. () II. Problemformulaton We consder a stochastc process modeled by a dscretetme lnear dynamc equaton x(k+ )=A(k)x(k)+w(k), x(0)= x 0, (2) where x(k) R n s the state vector and w(k) R n s a drvng nose, assumed Gaussan wth zero mean and (possbly tme varyng) covarance matrxσ w (k). The ntal condton x 0 s assumed to be Gaussan wth meanµ 0 and covarance matrx Σ 0. The state of the process s observed by a network of sensors ndexed by, whose sensng models are gven by y (k)=c (k)x(k)+v (k),,...,, (3) where y (k) R r s the observaton made by sensor and v (k) R r s the measurement nose, assumed Gaussan wth zero mean and (possbly tme varyng) covarance matrx Σ v (k). We assume that the matrces{σ v (k)} andσ w(k) are postve defnte for k 0 and that the ntal state x 0, the noses v (k) and w(k) are ndependent for all k 0. For later reference we also defneσv /2 (k),σw /2 (k), where Σ v (k) Σv /2 (k)σv /2 (k) andσ w (k) Σw /2 (k)σw /2 (k). The set of sensors form a communcaton network whose topology s modeled by a drected graph that descrbes the nformaton exchanged among agents. The goal of the agents s to (locally) compute estmates of the state of the process (2). Let ˆx (k) denote the state estmate computed by sensor and letǫ (k) denote the estmaton error,.e.ǫ (k) x(k) ˆx (k). The covarance matrx of the estmaton error of sensor s denoted byσ (k) E[ǫ (k)ǫ (k) ], wthσ (0)=Σ 0. The sensors update ther estmates n two steps. In the frst step, an ntermedate estmate, denoted byϕ (k), s produced usng a Luenberger observer flter ϕ (k)= A(k) ˆx (k)+ L (k)(y (k) C (k) ˆx (k)),,...,, (4) where L (k) s the flter gan. In the second step, the new state estmate of sensor s generated by a convex combnaton betweenϕ (k) and all other ntermedate estmates wthn ts communcaton neghborhood,.e. ˆx (k+ )= p j (k)ϕ j (k),,...,, (5) where p j (k) are non-negatve scalars summng up to one ( p j (k)=), and p j (k)=0 f no lnk from j to exsts at tme k. Havng p j (k) dependent on tme accounts for a possbly tme varyng communcaton topology. Combnng (4) and (5) we obtan the dynamc equatons for the consensus based dstrbuted flter: ˆx (k+ )= p j (k) [ A(k) ˆx j (k)+ L j (k) ( y j (k) C j (k) ˆx j (k) )], (6) for,...,. From (6) the estmaton errors evolve accordng to ǫ (k+ )= p j (k) [( A(k) L j (k)c j (k) ) ǫ j (k)+ +w(k) L j (k)v j (k) ],,...,. Defnton 2.: (dstrbuted detectablty) Assumng that A(k), C(k) {C (k)} and p(k) {p j(k)}, are tme nvarant, we say that the lnear system (2) s detectable usng the CBDLF scheme (6), f there exst a set of matrces L {L } such that the system (7), wthout the nose, s asymptotcally stable. We ntroduce the followng fnte horzon quadratc flterng cost functon J K (L( ))= (7) K E[ ǫ (k) 2 ], (8) k=0 where by L( ) we understand the set of matrces L( ) {L (k),k=0... K }. The optmal flterng gans represent the soluton of the followng optmzaton problem L ( )=argmn L( ) JK (L( )). (9) Assumng that A(k), C(k) {C (k)},σ w(k),σ v (k) {Σ v (k)} and p(k) {p j (k)}, are tme nvarant, we can also defne the nfnte horzon flterng cost functon J (L)= lm K K J K(L)= lm E[ ǫ (k) 2 ], (0) k where L {L } s the set of steady state flterng gans. By solvng the optmzaton problem L = argmn L J (L), () we obtan the optmal steady-state flter gans. In the next sectons we wll address the followng problems: 700

3 Problem 2.: (Detectablty condtons) Under the above setup, we want to fnd condtons under whch the system (2) s detectable n the sense of Defnton 2.. Problem 2.2: (Sub-optmal scheme for consensus based dstrbuted flterng) Ideally, we would lke to obtan the optmal flter gans by solvng the optmzaton problems (9) and (), respectvely. Due to the complexty of these problems, we wll not provde the optmal flterng gans but rather focus on provdng a sub-optmal scheme wth quantfable performance. Problem 2.3: (Connecton wth the lnear flterng of a Markovan jump lnear system) We make a parallel between the consensus-based dstrbuted lnear flterng scheme and the lnear flterng of a partcular Markovan jump lnear system. III. Dstrbuteddetectablty Let us assume that no sngle par (A,C ) s detectable n the classcal sense, but the par (A,C) s detectable, where C = (C,...,C ). In ths case, we can desgn a stable (centralzed) Luenberger observer flter. The queston s, can we obtan a stable consensus-based dstrbuted flter? As the Example 3. of [6] shows, ths s not true n general. That s why t s mportant to fnd condtons under whch the CBDLF can produce stable estmates. Proposton 3.: Consder the lnear dynamcs (2)-(3). Assume that n the CBDLF scheme (6), we have p j = and that ˆx (0)= x 0, for all,... If the par (A,C) s detectable, then the system (2) s detectable as well, n the sense of Defnton 2.. Proof: Rewrte the matrx C as C= C, where C = (O n r,...,o n r, C, O n r +,...,O n r ). Ignorng the nose, we defne the measurements ȳ (k)= C x(k), whch are equvalent to the ones n (3). Under the assumpton that p j = and ˆx =x 0 for all,..., t follows that the estmaton errors follow the dynamcs ǫ(k+ )= (A L C )ǫ(k). (2) Settng L = L for..., t follows that ǫ(k+ )=(A LC)ǫ(k). Snce the par (A,C) s detectable, there exsts a matrx L such that A LC has all egenvalues nsde the unt crcle and therefore the dynamcs (2) s asymptotcally stable, whch mples that (2) s detectable n the sense of Defnton 2.. The prevous proposton tells us that f we acheve (average) consensus between the state estmates at each tme nstant, and f the par (A,C) s detectable (n the classcal sense), then the system (2) s detectable n the sense of Defnton 2.. However, achevng consensus at each tme nstant can be tme and numercally costly and that s why t s mportant to fnd (testable) condtons under whch the CBDLF produces stable estmates. Lemma 3.: (suffcent condtons for dstrbuted detectablty) If there exsts a set of symmetrc, postve defnte matrces{q } and a set of matrces{l } such that Q = p j (A L j C j ) Q j (A L j C j )+S,..., (3) for some postve defnte matrces{s }, then the system (2) s detectable n the sense of Defnton 2.. Proof: The dynamcs of the estmaton error wthout nose s gven by ǫ (k+ )= p j (A L j C j )ǫ j (k),,...,. (4) In order to prove the stated result we have to show that (4) s asymptotcally stable. We defne the Lyapunov functon V(k)= x (k) Q x (k), and our goal s to show that V(k+) V(k)<0 for all k 0. The Lyapunov dfference can be upper bounded by V(k+ ) V(k) p j ǫ j (k) (A L j C j ) Q (A L j C j )ǫ j (k) ǫ (k) Q ǫ (k), (5) where the nequalty followed from Remark.. By changng the summaton order we can further wrte V(k+) V(k) ǫ (k) p j (A L j C j ) Q j (A L j C j ) Q ǫ (k). Usng (3) yelds V(k+ ) V(k) ǫ (k) S ǫ (k)<0, snce{s j } are postve defnte matrces and therefore asymptotc stablty follows. The followng result relates the exstence of the sets of matrces{q } and{l } such that (3) s satsfed, wth the feasblty of a set of lnear matrx nequaltes (LMI). Proposton 3.2: (dstrbuted detectablty test) The lnear system (2) s detectable n the sense of Defnton 2. f the followng lnear matrx nequaltes, n the varables{x } and{y }, are feasble ( X M M dag(x,...) ) > 0, (6) ( for =..., where M = p (A X C Y ),..., p (A X C Y )) and where {X } are symmetrc. Moreover, a stable CBDLF s obtaned by choosng the flter gans as L = X Y for... Proof: Gven n [6]. 70

4 IV. Sub-Optmal Consensus-Based Dstrbutedlnear Flterng Obtanng the closed form soluton of the optmzaton problem (9) s a challengng problem, whch s n the same sprt as the decentralzed optmal control problem. In ths secton we provde a sub-optmal algorthm for computng the flter gans of the CBDLF, wth quantfable performance n the sense that we compute a set of flterng gans whch guarantee a certan level of performance wth respect to the quadratc cost (8). A. Fnte Horzon Sub-Optmal Consensus-Based Dstrbuted Lnear Flterng The sub-optmal scheme for computng the CBDLF gans results from mnmzng an upper bound of the quadratc flterng cost (8). The followng proposton gves upperbounds for the covarance matrces of the estmaton errors. Proposton 4.: Consder the followng coupled dfference equatons Q (k+ )= p j (k) [( A(k) L j (k)c j (k) ) Q j (k) (A(k) L j (k)c j (k) ) + L j (k)σ v j (k)l j (k) ] +Σ w (k), (7) wth Q (0)=Σ (0), for,...,. Then, the followng nequalty holds Σ (k) Q (k), (8) for,..., and for all k 0. Proof: Gven n [6]. Defnng the fnte horzon quadratc cost functon J K (L( ))= K k= tr(q (k)), (9) the next Corollary follows mmedately. Corollary 4.: The followng nequaltes hold J K (L( )) J K (L( )), lmsup K K J K (L) lmsup K K J K (L). (20) Proof: Follows mmedately from Proposton 4.. In the prevous corollary we obtaned an upper bound on the flterng cost functon. Our sub-optmal consensus based dstrbuted flterng scheme wll result from mnmzng ths upper bound n terms of the flterng gans{l (k)},.e. mn L( ) J K (L( )). (2) Proposton 4.2: The optmal soluton for the optmzaton problem (2) s L (k)=a(k)q (k)c (k) [ Σ v (k)+c (k)q (k)c (k) ], (22) and the optmal value s gven by J K (L ( ))= K tr(q (k)), k= where Q (k) s computed usng Q (k+ )= p j (k) [ A(k)Q j (k)a(k) +Σ w (k) A(k)Q j (k)c j(k) (Σ v j (k)+c j (k)q j (k)c j(k) ) C j (k)q j (k)a(k) ], wth Q (0)=Σ (0) and for,...,. (23) Proof: Let J K (L( )) be the cost functon when an arbtrary set of flterng gans L( ) {L (k),k=0,..., K } s used n (7). We wll show that J K (L ( )) J K (L( )), whch n turn wll show that L ( ) {L (k),k=0,..., K } s the optmal soluton of the optmzaton problem (2). Let {Q (k)} and{q (k)} be the matrces obtaned when L ( ) and L( ), respectvely are substtuted n (7). In what follows we wll show by nducton that Q (k) Q (k) for k 0 and,...,, whch bascally prove that J K (L ( )) J K (L( )), for any L( ). For smplfyng the proof, we wll omt n what follows the tme ndex for some matrces and for the consensus weghts. Substtutng{L (k),k 0} n (7), after some matrx manpulatons we get Q (k+ )= p j [ AQ j (k)a +Σ w AQ j (k)c j (Σ v j + +C j Q j (k)c j ) C j Q j (k)a ], Q (0)=Σ (0),.... We can derve the followng matrx dentty (for smplcty we wll gve up the tme ndex): (A+ L C )Q (A + L C ) + L Σ v L = (A+ L C )Q (A + L C ) + +L Σ v L + (L L )(Σ v +C Q C )(L L ). (24) Assume that Q (k) Q (k) for... Usng dentty (24), the dynamcs of Q (k) becomes Q (k+ )= p j [ (A+ L j C j )Q j (k)(a+ L j C j ) + L j Σ v j L j (L j L j )(Σ v j +C j Q j (k)c j )(L j L j ) +Σ w ]. The dfference Q (k+) Q (k+) can be wrtten as Q (k+ ) Q (k+ )= p j [ (A+ L j C j )(Q j (k) Q j(k)) (A+ L j C j ) (L j L j )(Σ v j +C j Q j (k)c j )(L j L j ) ]. SnceΣ v +C Q (k)c s postve defnte for all k 0 and =,..., and snce we assumed that Q (k) Q (k), t follows that Q (k+) Q (k+). Hence we obtan that J K (L ( )) J K (L( )), for any set of flterng gans L( )={L (k),k=0,..., K }, whch concludes the proof. We summarze n the followng algorthm the suboptmal CBDLF scheme resulted from Proposton 4.2. Algorthm : Consensus Based Dstrbuted Lnear Flterng Algorthm Input:µ 0, P 0 Intalzaton: ˆx (0)=µ 0, Y (0)=Σ 0 2 whle new data exsts 3 Compute the flter gans: 4 5 Update the state estmates: Update the matrces Y : Y L AY C (Σ v +C Y C ) ϕ A ˆx + L (y C ˆx ) ˆx j p j ϕ j ( p j (A L j C j )Y j (A L j C j ) + L j Σ v j L j) +Σw 702

5 B. Infnte Horzon Consensus Based Dstrbuted Flterng We now assume that the matrces A(k), {C (k)}, {Σ v (k)} andσ w(k) and the weghts{p j (k), } are tme nvarant. We are nterested n fndng out under what condtons Algorthm converges and f the flterng gans produce stable estmates. From the prevous secton we note that the optmal nfnte horzon cost can be wrtten as J = lm k tr(q (k)), where the dynamcs of Q (k) s gven by Q (k+ )= [ p j AQ j (k)a +Σ w AQ j j( (k)c Σv j + ) +C j Q ] j (k)c j C j Q j (k)a, and the optmal flterng gans are gven by L ( (k)=aq (k)c Σv +C Q (k)c ), (25) for,...,. Assumng that (25), converges, the optmal value of the cost s gven by J where{ Q } satsfy Q = J = tr( Q ), [ p j A Q j A +Σ w A Q j C j (Σ v j +C j Q j C j ) C j Q j A ]. (26) Suffcent condtons under whch (26) has a unque soluton and (25) converges to ths unque soluton are provded by Proposton. n the Appendx secton. V. Connectonwththe Markovan Jump Lnear Systems stateestmaton In ths secton we present a connecton between the detectablty of (2) n the sense of Defnton 2. and the detectablty property of a MJLS, whch s gong to be defned n what follows. We also show that the optmal gans of a lnear flter for the state estmaton of the aforementoned MJLS can be used to approxmate the soluton of the optmzaton problem (9), whch gves the optmal CBDLF. We assume that the matrx P(k) descrbng the communcaton topology of the sensors s doubly stochastc and we assume, wthout loss of generalty, that the matrces{c (k),k 0} n the sensng model (3), have the same dmenson. We defne the followng Markovan jump lnear system ξ(k+ )=à θ(k) (k)ξ(k)+ B θ(k) (k) w(k) z(k)= C θ(k) (k)ξ(k)+ D θ(k) (k)ṽ(k),ξ(0)=ξ 0, (27) whereξ(k) s the state, z(k) s the output,θ(k) {,..., } s a Markov chan wth probablty transton matrx P(k), w(k) and ṽ(k) are ndependent Gaussan noses wth zero mean and dentty covarance matrces. Also,ξ 0 s a Gaussan nose wth mean µ 0 and covarance matrx Σ 0. We denote byπ (k) the probablty dstrbuton ofθ(k) (Pr(θ(k)= )=π (k)) and we assume that π (0)>0. We have that à θ(k) (k) {à (k)}, B θ(k) (k) { B (k)}, C θ(k) (k) { C (k)} and D θ(k) (k) { D (k)}, where the ndex refers to the state ofθ(k). We set à (k)=a(k), B (k)= π (0) π (k) Σ/2 w (k), C (k)= π (0) C (k), D (k)= (28) π (k) Σ/2 (k), v for all,k 0 (note that snce P(k) s assumed doubly stochastc andπ (0)>0, we have thatπ (k)>0 for all,k 0). In addton,ξ 0,θ(k), w(k) and ṽ(k) are assumed ndependent for all k 0. The random processθ(k) s also called mode. Assumng that the mode s drectly observed, a lnear flter for the state estmaton s gven by ˆξ(k+ )= à θ(k) (k)ˆξ(k)+ M θ(k) (k)(z(k) C θ(k) (k)ˆξ(k)), (29) where we assume that the flter gan M θ(k) depends only on the current mode. The dynamcs of the estmaton error e(k) ξ(k) ˆξ(k) s gven by e(k+)= ( à θk (k) M θ(k) (k) C θ(k) (k) ) e(k)+ + B θ(k) (k)w(k) M θ(k) (k) D θ(k) (k)v(k). (30) Letµ(k) and Y(k) denote the mean and the covarance matrx of e(k),.e.µ(k) E[e(k)] and Y(k) E[e(k)e(k) ], respectvely. We defne also the mean and the covarance matrx of e(k), when the system s n mode,.e. µ (k) E[e(k) {θ(k)=} ] and Y (k) E[e(k)e(k) {θ(k)=} ], where {θ(k)=} s the ndcator functon. It follows mmedately that µ(k)= µ (k) and Y(k)= Y (k). Defnton 5.: The optmal lnear flter (29) s obtan by mnmzng the followng quadratc fnte horzon cost functon J K (M( ))= K K tr(y(k))= tr(y (k)), (3) k= k= where M( ) {M (k),k=0,..., K } are the flter gans and where M (k) corresponds to M θ(k) (k) whenθ(k) s n mode. We can gve a smlar defnton for an optmal steady state flter usng the nfnte horzon quadratc cost functon. Defnton 5.2: Assume that the matrces à (k), C (k) and P(k) are constant for all k 0. We say that the Markovan jump lnear system (27) s mean square detectable f there exts {M } such that lm k E[ e(k) 2 ]=0, when the noses w(k) and ṽ(k) are set to zero. The next result makes the connecton between the detectablty of the MJLS defned above the dstrbuted detectablty of the process (2). Proposton 5.: If the Markovan jump lnear system (27) s mean square detectable, then the lnear stochastc system (2) s detectable n the sense of Defnton 2.. Proof: Gven n [6]. The next result establshes that the optmal gans of the flter (29) can be used to approxmate the soluton of the optmzaton problem (9). Proposton 5.2: Let M ( ) {M (k),k=0,..., K } be the optmal gans of the lnear flter (29). If we set L (k)= π (0) M (k) as flterng gans n the CBDLF scheme, then the flter cost functon (8) s guaranteed to be upper bounded by J K (L( )) K k=0 π (0) tr(y (k)), (32) where Y (k) are the covarance matrces resultng from mnmzng (3). Proof: 703

6 By Theorem 5.5 of [5], the flterng gans that mnmze (3) are gven by M (k)=ã (k)y (k) C (k) [ π (k) D j (k) D j (k) + C (k)y (k) C (k) ], (33) for..., where Y (k) satsfes Y (k+ )= p j (k) [ Ã j (k)y j (k)ã j (k) +π j (k) B j (k) B j (k) Ã j (k)y j (k) C j (k) ( π j (k) D j (k) D j (k) + C j (k)y j (k) C j (k) ) C j (k)y j (k)ã j (k) ]. (34) In what follows we wll show by nducton that Y (k)= π (0)Q (k) for all,k 0, where Q (k) satsfes (23). For k=0 we have Y (0)=π (0)Y (0)=π (0)Σ 0 =π (0)Q (0). Let us assume that Y (k)=π (0)Q (k). Then, from (28) we have π j (k) B j (k) B j (k) =π (0)Σ w (k), π j (k) D j (k) D j (k) =Σ v (k), π j (k) D j (k) D j (k) + C j (k)y j (k) C j (k) =Σ v j (k)+c j (k)q j (k)c j(k). Also, (35) M (k)= π (0)A(k)Q (k)c (k) [ Σ v j (k)+c j (k)q j (k)c j(k) ], (36) and from (22) we get that M (k)= π (0)L (k). From (34), (35), t can be easly argued that Y (k+ )=π (0)Q (k+ ). By Corollary 4. we have that J K (L( )) J K (L( )), for any set of flterng gans L( ) and n partcular for L (k)= π (0) M (k)= L (k), for all and k. But snce the result follows. J K (L ( ))= K k=0 VI. Conclusons π (0) Y (k), In ths paper we addressed three problems. Frst we provded (testable) suffcent condtons under whch stable consensus-based dstrbuted lnear flters can be obtaned. Second, we gave a sub-optmal, lnear flterng scheme, whch can be mplemented n a dstrbuted manner and s vald for tme varyng communcaton topologes as well, and whch guarantees a certan level of performance. Thrd, under the assumpton that the stochastc matrx used n the consensus step s doubly stochastc we showed that f an approprately defned Markovan jump lnear system s detectable, then the stochastc process of our nterest s detectable as well. We also showed that the optmal gans of the consensus-based dstrbuted lnear flter scheme can be approxmated by usng the optmal lnear flter for the state estmaton of a partcular Markovan jump lnear system. [3] O.L.V. Costa and M.D. Fragoso, Stablty results for dscrete-tme lnear systems wth Markovan jumpng parameters, Journal of Mathematcal Analyss and Applcaton, no. 79(993), pp [4] O.L.V. Costa, Dscrete-Tme Coupled Rccat Equatons for Systems wth Markov Swtchng Parameters, Journal of Mathematcal Analyss and Applcatons, no. 94(995), pp [5] O.L.V. Costa, M.D, Fragoso and R.P. Marques, Dscrete-Tme Markov Jump Lnear Systems, Sprnger [6] I. Mate and J.S. Baras, Consensus-Based Lnear Flterng, ISR Tecncal Report, TR-200-5, [7] R. Olfat-Saber, Dstrbuted Kalman Flterng for Sensor etworks, Proceedngs of the 46 th IEEE Conference on Decson and Control, pages , 2007 [8] R. Olfat-Saber, Dstrbuted Kalman Flter wth Embedded Consensus Flters, Proc. of the 44 th IEEE Conference on Decson and Control, Dec [9] R. Olfat-Saber, Dstrbuted Trackng for Moble Sensor etworks wth Informaton-Drven Moblty, Proc. of the 2007 Amercan Control Conference, pages , July -3, [0] A. Speranzon, C. Fschone, K. H. Johansson and A. Sangovann- Vncentell, A Dstrbuted Mnmum Varance Estmator for Sensor etworks, IEEE Journal on Select Areas n Communcaton, vol. 26, no. 4, pages , May [] D. Teneketzs and P. Varaya, Consensus n Dstrbuted Estmaton, Advances n Statstcal Sgnal Processng, JAI Press, H.V. Poor Ed., pp , January 988. [2] Y. Zhu, Z. You, J. Zhao, K. Zhang, X.R. L, The optmalty for the dstrbuted Kalman flterng fuson wth feedback, Automatca no. 37, pages , 200. Appendx Gven a postve ntegers, a sequence of postve numbers p={p j }, and a set of matrces F={F }, we consder the followng matrx dfference equatons W (k+ )= p j F j W j (k)f j, W (0)=W 0,,...,. (37) Defnton. ([4]): Gven a set of matrces C={C }, we say that (p,l,a) s detectable f there exsts a set of matrces L={L } such that the dynamcs (37) s asymptotcally stable, where F = A L C, for,...,. Defnton.2 ([4]): Gven a set of matrces C={C }, we say that (A,L,p) s stablzable, f there exsts a set of matrces L={L } such that the dynamcs (37) s asymptotcally stable, where F = A C L, for... Proposton.: LetΣ v ={Σv /2 }, whereσ v =Σv /2 Σ /2 v. Suppose that (p,c,a) s detectable and that (A,Σv /2,p) s stablzable n the sense of Defntons. and.2, respectvely. Then there exsts a unque set of symmetrc postve defnte matrces Q={ Q } satsfyng (26). Moreover, for any ntal condtons Q 0 0, we have that lm k Q (k)= Q, where the dynamcs of Q (k) s gven by (25). Proof: See [6]. References [] V. Borkar and P. Varaya, Asymptotc agreement n dstrbuted estmaton, IEEE Transacton on Automatc Control, vol. ac-27, no. 3, 982. [2] R. Carl, A. Chuso, L. Schenato and S. Zamper, Dstrbuted Kalman Flterng Based on Consensus Strateges, IEEE Journal on selected area n communcaton, vol. 26, no. 4, pages , May