Kalman Filtering with Markovian Packet Losses and Stability Criteria

Transcription

1 Proceedings of he 45h IEEE Conference on Decision & Conrol Mancheser Grand Hya Hoel San Diego, CA, USA, December 3-5, 26 Kalman Filering wih Markovian Packe Losses and Sabiliy Crieria Minyi Huang and Subhrakani Dey Absrac We consider Kalman filering in a nework wih packe losses, and use a wo sae Markov chain o describe he normal operaing condiion of packe delivery and ransmission failure. We analyze he behavior of he esimaion error covariance marix and inroduce he noion of peak covariance, which describes he upper envelope of he sequence of error covariance marices {P, } for he case of an unsable scalar model. We give sufficien condiions for he sabiliy of he peak covariance process in he general vecor case; for he scalar case we obain a sufficien and necessary condiion, and derive upper and lower bounds for he ail disribuion of he peak variance. For pracically verifying he sabiliy condiion, we furher inroduce a subopimal esimaor and develop a numerical procedure o generae igher esimae for he consans involved in he sabiliy crierion. I. INTRODUCTION The problem of sae esimaion is of grea imporance in various applicaions ranging from racking, deecion and conrol, and in linear sochasic dynamical sysems, Kalman filering [2], [] plays an essenial role. Recenly here has been an increased research aenion for filering in disribued sysems where sensor measuremens and final signal processing ake place in geographically separae locaions and he usage of wireless or wireline communicaion channels is essenial for daa communicaion. In conras o radiional filering problems, an imporan feaure in hese neworked sysems is ha he delivery of measuremens o he esimaor is no always reliable and losses of daa may occur. In his paper, we consider he opimal filering of a linear sysem wih random packe losses. We focus on he n dimensional linear ime-invarian sysem x + = Ax + w,, where he iniial sae is x a =. The sensor measuremens are obained saring from in he form y = Cx + v, where C R m n,andheny is ransmied by a channel. Here {w, } and {v, } are wo muually independen sequences of i.i.d. Gaussian noises wih covariance marices Q and R>, respecively. The wo noise sequences Work parially suppored by Ausralian Research Council (ARC). M. Huang is wih Deparmen of Informaion Engineering, Research School of Informaion Sciences and Engineering, The Ausralian Naional Universiy, Canberra 2, ACT, Ausralia. minyi.huang@rsise.anu.edu.au S. Dey is wih Deparmen of Elecrical and Elecronic Engineering, The Universiy of Melbourne, Vicoria 3, Ausralia. s.dey@ee.unimelb.edu.au are also independen of x, which is a Gaussian random vecor wih mean x = Ex and covariance marix P x.the underlying probabiliy space is denoed as (Ω, F, P) where F is he σ-algebra of all evens. We consider a communicaion channel such ha y is exacly rerieved or he packe conaining y is los due o corruped daa or subsanial delay. When he packe is successfully received, one obains he observaion y = y, and if here is a packe loss, by our convenion, he observaion obained by he receiver is y. Under his assumpion, he underlying communicaion link may be looked a as an erasure channel a he packe level. We use γ {, } o indicae he arrival (wih value ) or loss (wih value ) of packes. Here γ may be inerpreed as resuling from he physical operaing condiion of a nework and is assumed o be known a he filer. Specifically, he sae for γ may correspond o channel error or nework congesion which causes a sraigh packe loss or long delay resuling in packe dropping a he receiver. For faciliaing he presenaion, and shall be called he failure sae and normal sae, respecively. To capure he emporal correlaion of he channel variaion (e.g, in bursy error condiions), γ is modelled by a wo sae Markov chain wih he ransiion marix [ ] q q α =, () p p where p and q, respecively, are called he failure rae and recovery rae and p, q >. For insance, p denoes he probabiliy of he channel remaining a he normal sae afer one sep ransiion if i sars wih sae. This is usually called he Gilber-Ellio channel model [5], [2]. Obviously, a small value (close o ) for p and large value (close o ) for q mean he channel is more reliable. Based on he hisory F = σ(y i,γ i,i ), which is he σ-algebra generaed by he available informaion up o ime (i.e., all evens ha can be generaed by hese random variables), one can wrie a se of filering and predicion equaions corresponding o he opimal esimae ˆx = E[x F ] and ˆx + = E[x + F ],, respecively, by he same mehod as in [6] which deal wih i.i.d. packe losses. The deails for he recursion of ˆx and ˆx + will no be repeaed here. In his paper we focus on he esimaion /6/$2. 26 IEEE. 562

2 45h IEEE CDC, San Diego, USA, Dec. 3-5, 26 error of ˆx + wih an associaed covariance marix P + = E(x+ ˆx + )(x + ˆx + ). We also wrie P + = P +. To characerize he filering covariance condiioned on he pas hisory, one can easily derive he following random Riccai equaion P + = AP A + Q γ AP C (CP C + R) CP A,, (2) where M denoes he ranspose of a vecor or marix M. The iniial condiion in (2) is P = Var(x )=AP x A +Q. Noe ha γ appears as a random coefficien in he recursion. Under a Bernoulli i.i.d. packe loss modelling, he filering sabiliy can be efficienly sudied by a modified algebraic Riccai equaion (MARE), which is obained by replacing γ in equaion (2) by he arrival rae λ. Subsequenly, he analysis amouns o idenifying a criical value λ c (as a hreshold) such ha sabiliy holds if and only if he arrival rae is greaer han λ c (see Secion IV for addiional discussion) [6]. In conras, when he channel model is given by he Markov chain γ, such a conversion ino a deerminisic MARE is no longer feasible, and since he channel is described by several independen parameers, he usual hreshold argumen is no applicable. A. Background and Relaed Work Nowadays, filering and esimaion consiue an imporan aspec in sensor nework deploymen for monioring, deecion or racking [], [2], [2], as well as muli-vehicle coordinaion [9], since in realiy sensors can only obain noisy informaion abou a physical aciviy in is viciniy. And for many linear sochasic models, a useful ool is he sandard Kalman filering heory which has been widely used in various esimaion and conrol scenarios. Recenly here is an increased aenion for is applicaion in disribued neworks while new heoreical quesions and implemenaion issues emerge. In close relaion o esimaion in lossy sensor neworks, here also has been a long hisory of research on filering wih missing signals a cerain poins of ime, i.e., he oupu does no necessarily conain he signal in quesion and i may be only a noise componen. Such models were referred o as sysems wih uncerain observaions [5], [], [7], [8], where a ypical mehod for sabiliy analysis is o consruc a deerminisic recursion uilizing he saisics of he uncerainy sequence indicaing he availabiliy of signals. In he more recen research on nework models, he work [7] and [3] considered sae esimaion wih lossy measuremens resuling from ime-varying channel condiions. In paricular, he auhors in [7] developed a subopimal jump linear esimaor for complexiy reducion in compuing he correcor gain using finie loss hisory where he loss process is modelled by a wo sae Markov chain. The work [3] inroduced a more general muliple sae Markov chain o model he loss and non-loss channel saes, and he asympoic mean square esimaion error for subopimal linear esimaors is analyzed and opimized by a linear marix inequaliy (LMI) approach. The MARE based analysis in [6] for i.i.d. packe losses was exended o he wo sensor siuaion in [4]. In hese resuls, he occurrence of packe losses is known a he esimaor and his leads o a random Riccai equaion involving he loss indicaor sequence. Conrol problems wih packe losses have been examined in [8], [3], [6]. B. Conribuions and Organizaion In his paper we consider a Markovian packe loss model which capures he emporal correlaion naure of pracical channels, and we develop new analyic echniques for filering sabiliy analysis. In Secion II, we inroduce he noion of peak covariance. The general sufficien condiion in Secion III was iniially obained in our earlier work [9]. In Secion IV we examine he sabiliy propery for he scalar model, and presen a ail disribuion analysis for he peak variance. To find pracically more verifiable sufficien condiions han in [9], in Secion V we inroduce an appropriaely parameerized subopimal esimaor, and opimize he parameer in he subopimal esimaor o produce a igher selecion of he consans in he sabiliy crieria. Secion VI presens some simulaion and compuaional examples. II. EVOLUTION OF THE COVARIANCE In order o simplify he analysis, in he following we assume he iniial sae for γ is γ =. Noe ha his assumpion imposes no essenial resricion and he oher case wih γ =may be reaed in he same manner. Based on equaion (2), we wrie wo separae equaions P + = AP A + Q AP C (CP C + R) CP A, γ = (3) P + = AP A + Q, γ = (4) depending on he value of γ. The covariance process P,asa random process, may be regarded as being governed by a bimodal hybrid sysem where he evoluion of he coninuum componen is driven by a wo sae Markov chain. Such a bi-modal srucure is especially useful and will be exploied in he sabiliy analysis. To make he model nonrivial, hroughou his paper we make he following assumpions: (H) The failure and recovery rae p, q are boh in (, ). (H2) The sysem [A, C] is observable, i.e, he rank of he marix [C,A C,, (A n ) C ] is n. For he reader s convenience, we inroduce he basic definiion of sopping imes alhough i is easily found in exbooks (see, e.g., [4]). A sopping ime τ (associaed wih he Markov chain γ, ) is a measurable map from Ω o he se {, 2,, } such ha {τ k} depends on γ up o ime k. In our filering conex, he wo sequences of sopping imes inroduced during he analysis simply describe he random swich ime of he filer, or equivalenly, he jump ime of he Markov chain γ. Given he iniial condiion γ =, we inroduce he following sopping ime: τ = inf{ >, γ = }. We make he usual convenion ha he infimum of an empy se 5622

3 45h IEEE CDC, San Diego, USA, Dec. 3-5, 26 is +. Thus τ is he firs ime when a packe loss occurs. Furhermore, we define β = inf{, > τ,γ = }. I is clear β is he firs ime he channel recovers from he firs failure. The above procedure is repeaed o define wo sequences τ, τ 2, τ 3,, β, β 2, β 3,, which gives he value of γ a he swich imes: {, if = τi <, γ =, if = β i <. Obviously he following order relaionship holds: (5) <τ <β < <τ k <β k <τ k+ <, (6) whenever each of he enries is finie on he associaed sample poin ω Ω. Lemma : Under condiion (H), wih probabiliy one, he wo sequences {τ i,i } and {β i,i } have finie values for each of heir enries. Lemma forms he basis for he peak covariance noion o be inroduced laer. Define τi = τ i β i, i βi = β i τ i, i where we adop he convenion β =.Hereτi and βi denoe he sojourn imes (i.e., he lengh of a coninuous say) a he success sae and failure sae, respecively. Lemma 2: Under (H), we have (i) he random variables {τi,i } are i.i.d., and τ i is geomerically disribued wih P (τi =k) =( p) k p, k. (ii) he random variables {βi,i } are i.i.d., and β i is geomerically disribued wih P (βi = k) = ( q) k q, k. (iii) The wo sequences of random variables {τi,i } and {βi,i } are independen of each oher. Now we define β k = β k. (7) In fac, β k is he las ime of visi of γ, o he failure sae sinceτ k. The ime β k is useful for analyzing he filering performance in ha i provides a basis for esimaing o wha exen he covariance process may deeriorae resuling from successive packe losses. Immediaely from β k,anew packe will arrive a he observer, and he sae predicion for he nex sep will sar o improve. The period [τ i,β i ] and [β i,τ i+ ] shall be called he loss cycle and normal cycle, respecively. Labelling a subsequence of he covariance process P k by he sequence of imes β k, we denoe M k = P βk. (8) M k denoes he predicion error covariance P βk β k compued by (4) a = β k. For an unsable scalar model, saring from τ k +, P monoonically increases o reach a maximum M k = P βk a ime β k before urning downward; he sequence {M k,k } gives he upper envelope of he covariance sequence. For his reason, we shall call M k he peak covariance process. In he muli-dimensional (vecor) case, P does no necessarily change monoonically before or afer reaching M k according o he packe arrival or loss; o faciliae our presenaion, however, we shall sill refer o M k as he peak covariance process. Definiion 3: We say he sequence {M k,k } is sable if sup k E M k <. Accordingly, we say he (filering) sysem saisfies peak covariance sabiliy. III. SUFFICIENT CONDITION FOR PEAK COVARIANCE STABILITY Le S n denoe he se of all n n nonnegaive definie real marices. Based on Kalman filering, define he map F (P )=AP A + Q AP C (CPC + R) CPA, (9) where P S n.iiseasyoshowf(p) S n.toanalyze he map F, we inroduce he following definiion. Definiion 4: For he observable linear sysem [A, C], he observabiliy index is he smalles ineger I o such ha [C,A C,, (A Io ) C ] has rank n. Under he observabiliy assumpion (H2), he ineger I o specified in Definiion 4 obviously exiss. For a deerminisic sysem, I o specifies he minimum number of observaions which are required in order o reconsruc he iniial condiion of an observable sysem. Define S n = {P : P A PA +Q, for some P }, which is a convex subse of S n. Lemma 5: Leing F be defined by (9), here exiss a consan K>such ha (i) for any P S n, F k ( P ) KI for all k I ; (ii) for any P S n, F k+ ( P ) KI for all k I,whereI is he n n ideniy marix. The sraegy o prove he lemma is o run an auxiliary Kalman filer; see [9] for deails. Remark: The observabiliy condiion may be relaxed o deecabiliy, and one can idenify an associaed index I o such ha Lemma 5 holds. Then he subsequen analysis in his paper can be exended o he deecable model in a sraighforward manner. We inroduce a few consans. For i (I o ), le C () i and C () i saisfy he following inequaliy F i (P ) C () i P + C () i, P S n, () where denoes he induced norm for marices. By he fac F (P ) AP A + Q, i is clear he above pair (C () i,c () i ) always exiss. For he case I =,wemayakec () =. Theorem 6: [9] The peak covariance process is sable if he following wo condiions hold: (i) λ A 2 ([ q) <, (ii) pqc () + ] I o i= C() i ( p) i j= Aj 2 ( q) j <, where λ A is an eigenvalue of A wih he larges absolue value. 5623

4 45h IEEE CDC, San Diego, USA, Dec. 3-5, 26 We give a brief discussion on condiion (ii). Noice ha under condiion (i), he infinie series in condiion (ii) converges. Now le he pair A and q be fixed such ha (i) holds. Then i is easy o check ha for he given pair (A, q), ifp is sufficienly small, condiion (ii) is always saisfied. Corollary 7: If C is inverible, condiion (ii) in Theorem 6 vanishes and he peak covariance sabiliy holds under condiion (i). IV. STABILITY OF THE SCALAR MODEL For he scalar case, condiion (ii) in Theorem 6 vanishes since in his case C () =. The reason is ha for he scalar Riccai equaion, once here is an arrival of one packe a, P + becomes bounded by a fixed consan, regardless of he value of P. Furhermore we can show ha condiion (i) in Theorem 6 is also necessary. This leads o a sufficien and necessary condiion. Noe ha his condiion only depends on he recovery rae of he Markov chain {γ, }. For he scalar case, we se he coefficiens A and C in he dynamics o heir lower case form, i.e., A = a and C = c. The erm covariance is also replaced by variance. A. The Sufficien and Necessary Condiion for Sabiliy Theorem 8: Leing r, wehavesup k E P βk r < if and only if a 2r ( q) <. I is clearly seen from Theorem 8 ha, wih a given a >, for obaining higher order sabiliy resuls, we need o pu a more sringen condiion on he recovery rae q. By aking r = in Theorem 8, we conclude ha in he scalar model a sufficien and necessary condiion for peak covariance sabiliy is a 2 ( q) <. In he following we esablish he sabiliy on he sandard variance process P. To simplify he esimaes, we only analyze he symmeric case wih p = q, in which he disribuion of he random variable τ k 2k + is he convoluion of 2k i.i.d. geomeric disribuions, and his subsanially simplifies he calculaions. For he general case wih p q, he calculaion is much more involved. Theorem 9: For he scalar model wih p = q, ifa 2 ( q) <, hen he variance process has he usual sabiliy propery, i.e., sup EP <. B. The Relaion beween Differen Sabiliy Noions For illuminaing he relaionship beween our peak covariance sabiliy wih oher exising sabiliy resuls in he lieraure, we specialize o he scalar model wih i.i.d. packe losses. In his case, [ he ransiion] marix of he channel given q q by () reduces o, wih an associaed packe q q loss probabiliy p = q. I is shown in [6] (Theorem 2 and Sec. IV) ha for he scalar model wih i.i.d. packe losses, sup E P < (we erm his as he usual sabiliy of P ) if and only if he packe arrival rae λ>λ c = /a 2,or equivalenly, q> /a 2. () Recalling Theorem 8, () is also a necessary and sufficien condiion for he peak variance sabiliy for he special case of i.i.d. packe losses. Then we can immediaely claim he following relaionship. Corollary : For he scalar model wih i.i.d. packe losses, he peak variance sabiliy is equivalen o he usual sabiliy (i.e., sup E P < ). For he scalar model wih i.i.d. packe losses, i is of ineres o noe ha he peak variance sabiliy is seemingly sronger han he usual sabiliy as he former characerizes a cerain boundedness propery along he upper envelope of he variance rajecories, bu acually i is no, as saed in Corollary. For he vecor case when P is a marix, he relaion beween he wo sabiliy noions as discussed above is much more complicaed as he sabiliy condiion is no jus reduced o he inequaliy (). C. Tail Disribuion of he Peak Variance Now we examine he ail disribuion of he peak variance when i is sable, i.e., a 2 ( q) <. We resric o he case a > and Q>. ForM>, define he ail disribuion of P βk, k, asp ail (M) =P {P βk M}. I is easy o show ha Q P τk a 2 R/c 2 + Q = P. Below we resric o M P (a 2 Q). I can be verified ha {P βk M} {βk ln(m/q) ln a }. Denoe by x he 2 smalles ineger no less han x. Hence P {P βk M} ( q) ln(m/q) ln a 2 ( q) ln(m/q) ln a 2. Denoe κ = ln( q) ln a >, where a >. We have 2 P ail (M) Q κ M κ. In a similar manner, we have {P βk M} = {F β k (Pτk ) M} {βk ln(m/ P ) ln a 2 + ln(a2 ) ln a 2 2}. Hence P {P βk M} ( q) ln(m/ P ) ln a 2 + ln(a2 ) ln a 2 2,which gives P ail (M) ζ P κ M κ,whereζ =( q) ln(a 2 ) ln a 2 2. For M P (a 2 Q), i follows ha Q κ M κ P ail (M) ζ P κ M κ which gives he lower and upper bound esimaes. I is seen ha he decaying rae of he ail disribuion depends on he raio κ which in urn is relaed o he sabiliy margin associaed wih he condiion a 2 ( q) < and a >. V. STABILITY CHECK BY LINEAR SUBOPTIMAL ESTIMATORS Theorem 6 gives a crierion for checking he sabiliy of he peak covariance process. In paricular, condiion (ii) depends on some consans relaed o he operaor F defined by (). In general, i is difficul o explicily compue hese consans, and a rough selecion may lead o very conservaive condiions for he pair (p, q). In his secion, we combine he analyic echnique in Secion III wih a numerical procedure o give a pracically useful selecion of hese consans. The basic idea is as follows. Firs we 5624

5 45h IEEE CDC, San Diego, USA, Dec. 3-5, 26 consruc a subopimal esimaor and inroduce is predicion error covariance Pk s, where s indicaes i is yielded by he subopimal esimaor. Nex, i is easy o esablish a dominance relaionship beween P k and Pk s, i.e., P k Pk s, Subsequenly, if he sysem dynamics ogeher wih he channel saisics ensure sabiliy of Pβ s k, hen he sabiliy of P βk naurally follows. Below we resric o linear subopimal esimaors where he filering correcor gain L for he case of a packe arrival is a consan marix o be seleced. We consruc he following subopimal esimaor: ˆx s k k =ˆxs k k + {γ k =}L(y k C ˆx s k k ) ˆx s k k = Aˆxs k k. The predicion error covariance P+ s = P+ s,, is described by P s + ={A(I LC)P s (I LC) A + ALRL A } {γ=} + AP s A {γ=} + Q. For he case γ =,wege P s + = A(I LC)P s (I LC) A + ALRL A + Q. (2) Based on (2) we inroduce he operaor F L (P )=A(I LC)P (I LC) A + ALRL A + Q. For he Kalman filer and he subopimal esimaor, we have he following dominance relaionship. Theorem : Le he common iniial condiion P x be given for he Kalman filer and he subopimal esimaor. Wih probabiliy one, we have P s k P k, k. (3) Proof. By using Kalman filering wihou packe losses, we can esablish he relaion F (P ) F L (P ) for all P. We assume P = Var(x ); hen for any marix H, H ˆx 2 is he minimum covariance esimae of Hx 2. Thus, H(x 2 ˆx 2 )(x 2 ˆx 2 ) H H(x 2 ˆx s 2 )(x 2 ˆx s 2 ) H which means HF(P )H = HP 2 H HP2 s H = HF L (P )H, and herefore F (P ) F L (P ). Nex, he Kalman filering covariance sequence P k (P ) as a funcion of P is monoone, i.e., P k (P ) P k ( P ) if P P.I is easy o check ha Ψ(P ), sanding for F L (P ) or AP A +Q, also has he same monoone propery. Finally, by applying hese monoone properies along he sequence γ, we see ha (3) holds wih probabiliy one. Corollary 2: If {Pβ s k,k } is sable, hen he peak covariance of P k is also sable. For he operaor F L, in analogy o Secion III, we also define he consans as follows. For i (I o ), le C L,() i and C L,() i saisfy he following inequaliy FL(P i ) C L,() i P + C L,() i, P S n. (4) I is obvious he above pair (C L,() i,c L,() i ) always exiss. The following sabiliy resul can be proved by he same mehod as in proving Theorem 6. Corollary 3: The covariance process {Pβ s k,k } is sable if he following wo condiions hold: (i) λ A 2 ( q) <, (ii) pqc L,() [ + ] I o i= CL,() i ( p) i j= Aj 2 ( q) j <, where λ A is an eigenvalue of A wih he larges absolue value. For a pracical applicaion o he sabiliy Theorem 6 for he Kalman filer, we may choose a suiable L o reduce he magniude of C L,() i. Then by he fac F (P ) F L (P ), we may se C () i = C L,() i, i I. VI. NUMERICAL EXAMPLES A. Simulaions for he Peak Covariance We firs consider a scalar sysem wih parameers [A, C] = [a, c] =[.4, ], Q = R =and P =. For his model, in order o guaranee sabiliy, he minimum recovery rae is q c = /a 2 = Fig. shows a ypical sample pah wih he parameer q =.6 >q c, which ensures sabiliy of he peak variance process. The horizonal axis in he figure is he discree ime. Along ha sample pah, we have τ =3, β =6, τ 2 =23, β 2 =25, ec. In Fig. -op, he curve displays he change of he variance along ha sample pah, and Fig. -boom shows he associaed channel sae jumping beween and. A high peak value for he variance is observed near =6, and his is due o he muliple successive packe losses. Fig. 2 shows a sample pah wih q =.32 <q c. Since in his case he recovery rae is low, he variance process has more chances o reach a high level. We coninue o examine a vecor example specified by A = [ ], C =[, ]. (5) The covariance of w is Q = I R 2 2, and he variance of v is R =.Wehave F (P ) AA P. I is easily checked ha he observabiliy index I o =2andwemayake C () =2.83, (6) since AA has wo eigenvalues λ =.2879 and λ 2 = By condiion (i) in Theorem 6, he recovery rae mus saisfy q > λ A 2 = From now on we ake q =.65. By numerical calculaion, we have j= Aj 2 ( q) j Then if p <.4, condiion (ii) holds. Fig. 3 shows a sample pah for his model wih parameers p =.3 and q =.65; P () and P 2 () are wo enries in he 2 2 marix P, and he channel sae is displayed beween = and = 2. Forhe associaed channel wih (p, q) =(.3,.65), he saionary disribuion of he failure sae is P (γ =)=.448. Thus he long erm packe loss rae is abou 4.4%. Unlike he scalar case, we only have a sufficien condiion for filering sabiliy, and condiion (ii) in Theorem 6 specifying he region for (p, q) may be conservaive. However, his crierion is sill useful since i covers some pracical models wih packe loss rae as high as several percens. 5625

6 45h IEEE CDC, San Diego, USA, Dec. 3-5, P P () P 2 () γ γ Fig.. The variance P and channel sae γ, q =.6. Fig. 3. P (),P 2 () and channel sae γ, q =.65 for he vecor case Fig. 2. The variance P and channel sae γ, q =.32. B. Subopimal Esimaor Aided Sabiliy Check For he wo dimensional example given by (5), wih he aid of he subopimal esimaor, we may ake C () = (A ALC)(A ALC) by recalling F (P ) F L (P ) and he srucure of F L (P ). NowheaskisofindL such ha A L = (A ALC)(A ALC) is minimized. Here we use a numerical mehod o search for an ideal value for L = [l,l 2 ] T. We compue he value A L for (l,l 2 ) onagrid[.6, 2] [.7, 2] wih a sep size.5 on boh edges, which gives a oal of poins. On he grid, he minimum of.22 for A L is aained by (l,l 2 )=(.5,.5). Now we may ake C () =.22 wih a significan improvemen from he rough esimae in (6). On he oher hand, he seady-sae ( soluion o he Riccai ) equaion P = F (P ) is P = Accordingly, he seady-sae correcor gain for he sae predicion equaion in a sandard Kalman filer is L = PC (CPC+R) =(.385,.292) T. I can be checked ha (A AL C)(A AL C) =4.46 (>.22), which only gives a very poor bound for C (). This suggess a careful search of L is imporan. REFERENCES [] C.-Y. Chong and S. P. Kumar. Sensor neworks: evoluion, opporuniies, and challenges, Proc. IEEE, vol. 9, pp , Aug. 23. P γ [2] E. O. Ellio. Esimaion of error raes for codes on burs-noise channels. Bell Sys. Tech. J., vol. 42, pp , Sep [3] A. K. Flecher, S. Rangan, and V. K. Goyal. Esimaion from lossy sensor daa: jump linear modeling and Kalman filering. Proc. he 3rd ACM/IEEE Inernaional Symposium on Informaion Processing in Sensor Neworks, Berkeley, CA, pp , Apr. 24. [4] D. Freedman. Markov Chains, Springer-Verlag, New York, 983. [5] E. N. Gilber. Capaciy of burs-noise channels. Bell Sys. Tech. J., vol. 39, pp , Sep., 96. [6] V. Gupa, D. Spanos, B. Hassibi, and R. M. Murray. Opimal LQG conrol across packe-dropping links. Proc. American Conrol Conference, Porland, OR, pp , June 25. [7] M. T. Hadidi and S. C. Schwarz. Linear recursive sae esimaion under uncerain observaions. IEEE Trans. Auoma. Conrol, vol. 24, no. 6, pp , Dec [8] C. N. Hadjicosis and R. Touri. Feedback conrol uilizing packe dropping nework links. Proc. IEEE Conf. Decision and Conrol, Las Vegas, NV, pp. 25-2, Dec. 22. [9] M. Huang and S. Dey. Sabiliy of Kalman filering wih Markovian packe losses. Auomaica (acceped), 26. [] A. G. Jaffer and S. C. Gupa. Recursive Bayesian esimaion wih uncerain observaion. IEEE Trans. Inform. Theory, vol. 7, pp , Sep. 97. [] T. Kailah, A. H. Sayed, and B. Hassibi. Linear Esimaion, Prenice- Hall, Upper Saddle River, NJ, 2. [2] R. E. Kalman. A new approach o linear filering and predicion problems. Trans. ASME J. Basic Eng., Ser. D., vol. 82, pp , Mar. 96. [3] Q. Ling and M. Lemmon. Sof real-ime scheduling of neworked conrol sysems wih dropous governed by a Markov chain. Proc. American Conrol Conf., Denver, CO, pp , June, 23. [4] X. Liu and A. J. Goldsmih. Kalman filering wih parial observaion losses. Proc. IEEE Conf. Decision Conr., Bahamas, pp , Dec. 24. [5] N. E. Nahi. Opimal recursive esimaion wih uncerain observaion. IEEE Trans. Inform. Theory, vol. 5, pp , July 969. [6] B. Sinopoli, L. Schenao, M. Franceschei, K. Poolla, M. I. Jordan, and S. S. Sasry. Kalman filering wih inermien observaions, IEEE Trans. Auoma. Conrol, vol. 49, no. 9, pp , Sep. 24. [7] S. C. Smih and P. Seiler. Esimaion wih lossy measuremens: jump esimaors for jump sysems. IEEE Trans. Auoma. Conr., vol. 48, pp , Dec. 23. [8] J. K. Tugnai. Asympoic sabiliy of he MMSE linear filer for sysems wih uncerain observaions. IEEE Trans. Inform. Theory, vol. 27, no. 2, pp , Mar 98. [9] P. Varaiya. Smar cars on smar roads: Problems of conrol. IEEE Trans. Auoma. Conrol, vol. 38, pp , Feb [2] H. Zhang, J. M. F. Moura and B. Krogh. Esimaion in sensor neworks: A graph approach. Proc. 4h Inerna. Symp. Inform. Process. in Sensor Neworks (IPSN), Los Angeles, pp , April, 25 [2] F. Zhao, J. Shin and J. Reich. Informaion-driven dynamic sensor collaboraion, IEEE Sig. Process. Mag., vol. 9, pp. 6-72, Mar