Estimation over Communication Networks: Performance Bounds and Achievability Results

Transcription

1 Estimation ovr Communication Ntworks: Prformanc Bounds and Achivability Rsults A. F. Dana, V. Guta, J. P. Hsanha, B. Hassibi R. M. Murray Abstract This ar considrs th roblm of stimation ovr communication ntworks. Suos a snsor is taking masurmnts of a dynamic rocss. Howvr th rocss nds to b stimatd at a rmot location connctd to th snsor through a ntwork of communication links that dro ackts stochastically. W rovid a framwork for comuting th otimal rformanc in th sns of xctd rror covarianc. Using this framwork w charactriz th dndncy of th rformanc on th toology of th ntwork and th ackt droing rocss. For indndnt and mmorylss ackt droing rocsss w find th stady-stat rror for som classs of ntworks and obtain lowr and ur bounds for th rformanc of a gnral ntwork. W also illustrat how this framwork can b usd in th synthsis of ntworks for th uros of stimation. Finally w find a ncssary and sufficint condition for th stability of th stimat rror covarianc for gnral ntworks with satially corrlatd and Markov ty droing rocss. This intrsting condition has a max-cut intrrtation. I. INTRODUCTION AND MOTIVATION In rcnt yars, systms comrising of multil snsors coorating with ach othr hav rcivd wid-srad intrst (s,.g., [1], [2]). Although such systms admittdly hav a highr comlxity than th stratgy of using only on snsor, th incrasd accuracy oftn mak ths systms worthwhil. From an stimation and control rsctiv, such systms rsnt many nw challngs, such as daling with data dlay or data loss imosd by th communication links, fusion of data mrging from multil nods and so on. Most of ths issus aris bcaus of th tight couling btwn th stimation and control tasks that dnd on th snsd data and th communication channl ffcts that affct th transmission and rction of data. Communication links introduc many otntially dtrimntal hnomna, such as quantization rror, random dlays, data loss and data corrution to nam a fw. It is imrativ to undrstand and countract th ffcts of th communication channls. Motivatd by this, thr has bn a lot of work don on stimation and control ovr ntworks of communication links (s,.g., [3], [4] and th rfrncs thrin). Bginning with th sminal ar of Dlchams [5], quantization ffcts hav bn variously studid both in stimation and control contxt by Tatikonda [6], Nair and Evans [7], Hsanha t al [8] and many othrs. Th ffct of dlayd ackt dlivry using various modls for ntwork dlay has also bn considrd by many rsarchrs. In this work, w focus on stimation across a ntwork of communication links that dro ackts. W considr a dynamical rocss volving in tim that is bing obsrvd by a snsor. Th snsor nds to transmit th data ovr a ntwork to a dstination nod. Howvr th links in th ntwork stochastically dro ackts. Prior work in this ara has focusd on studying th ffct of ackt dros by a singl link in an stimation or control roblm. Assuming crtain statistical modls for th ackt dro rocss, stability of such systms was analyzd,.g., in [10], [11] and th control rformanc by Silr in [10] and by Ling and Lmmon in [12]. Aroachs to comnsat for th data loss wr roosd by Nilsson [9], Hadjicostis and Touri [13], Ling and Lmmon [12], [14], Azimi-Sadjadi [15], Sinooli t al. [16] and Imr t al. [17]. Sinooli t al. [18] also considrd th roblm of otimal stimation across a ackt-droing link that dros ackt in an i.i.d. fashion and obtaind bounds on th xctd rror covarianc. Most of th abov dsigns aimd at dsigning a ackt-loss comnsator. Th comnsator accts thos ackts that th link succssfully transmits and roducs an stimat for th tim sts whn data is lost. If th stimator is usd insid a control loo, th stimat is thn usd by th controllr. A mor gnral aroach is to dsign an ncodr and a dcodr for th communication link. This was considrd for th cas of a singl communication link in [19]. It was dmonstratd that using ncodrs and dcodrs can imrov both th stability margin and th rformanc of th systm. For gnral ntworks, th roblm is much mor comlicatd than th cas of a singl communication link sinc otntially thr ar multil aths from th sourc to th dstination. Rcnt work [20] idntifid otimal information rocssing schms that should b followd by th nods of th ntwork to allow th sink to calculat th otimal stimat at vry tim st. That work also idntifid th condition on th ntwork for th stimat rror covarianc to b stabl undr this algorithm. In this ar, w calculat th rformanc of such a stratgy. This rformanc also rovids a lowr bound on th rformanc that can b achivd by any othr schm (.g., transmitting masurmnts without any rocssing). W also gnraliz th condition for stability of th stimat rror covarianc from th indndnt and mmorylss ackt dro rocsss to ons that ar dscribd by Markov chains or ar satially corrlatd across th ntwork. W rovid a mathmatical framwork for valuating th rformanc for a gnral ntwork and rovid xrssions for ntworks containing links in sris and aralll. W also rovid lowr and ur bounds for th rformanc ovr gnral ntworks. As an xaml of how such rsults can b usd for synthsis of

2 ntworks to imrov stimation rformanc, w rovid a siml xaml in which th otimal numbr of rlay nods to b lacd is idntifid. Simulation rsults ar rovidd to illustrat th rsults. Th bttr stimation rformanc can also translat to bttr control rformanc if th stimat is usd for control uross [19], [16], [17]. Th ar is organizd as follows. In th nxt sction, w st u th roblm and stat th various assumtions. Thn, w rovid th mathmatical framwork ndd to calculat th stady-stat rformanc and introduc th conct of latncy. W show how to valuat latncy for sris and aralll ntworks. W thn rovid bounds for th rformanc for a gnral ntwork. W conclud with som rmarks and avnus for futur work. II. PROBLEM SETUP Considr a rocss volving in discrt-tim as x k+1 = Ax k + w k, (1) whr x k R n is th rocss stat and w k is th rocss nois modld whit and Gaussian with man zro and covarianc matrix Q. Th rocss is obsrvd using a snsor that gnrats masurmnts of th form y k = Cx k + v k, (2) whr v k R m is th masurmnt nois also assumd to b whit, Gaussian with man zro and covarianc R. Furthrmor, th noiss v k and w k ar assumd to b indndnt of ach othr. W considr th scnario in which th rocss nds to b stimatd in th minimum man squar rror (MMSE) sns at a rmot oint dnotd by a dstination nod d. W assum that th snsor (dnotd by s) and th dstination nod d ar connctd via a communication ntwork. Th communication grah is rrsntd by a dirctd grah G with nod st V (that contains, in articular, s and d) and link st E V V. Th link = (u, v) modls a communication channl btwn nod u and nod v. For any nod i V, th st of outgoing dgs corrsonds to th links along which th nod can transmit mssags whil th st of incoming dgs corrsonds to th links along which th nod rcivs mssags. W dnot th st of in-nighbors of nod v by N (v). Th communication links ar modld using a ackt rasur modl. Th links tak in as inut a finit vctor of ral numbrs. For vry link, at ach tim-st, a ackt is ithr drod or rcivd comltly at th outut nod. In this ar, w assum indndnt and mmorylss ackt dro rocsss, i.., th robability of droing a ackt on link E is givn by indndnt of othr links and tim. W ignor quantization issus, data corrution or random dlays. W also assum a global clock so that ach nod is synchronizd. W furthr assum that ach nod can listn to all th mssags ovr th diffrnt incoming links without intrfrnc from ach othr. 1 1 This rorty can b achivd by using a division multil accss schm lik FDMA, TDMA, CDMA tc. Th oration of diffrnt nods in th ntwork at vry tim-st k can b dscribd as follows: 1) Each nod comuts a function of all th information it has accss to at that tim. 2) It transmits th function on all th outgoing dgs. W allow som additional information in th mssag that tlls us th tim st j such that th function that th nod transmits corrsonds to th stat x j. Th dstination nod calculats th stimat of th currnt stat x k basd on th information it osssss. 3) Evry nod obsrvs th signals from all th incoming links and udats its information st for th nxt tim st. For th sourc nod, th mssag it rcivs at tim st k corrsonds to th obsrvation y k. Th timing squnc w hav scifid lads to strictly causal stimats. At tim st k, th function that th sourc nod transmits dnds on masurmnts y 0, y 1,, y k 1. Furthr vn if thr wr no ackt dros, if th dstination nod is l hos away from th sourc nod, its stimat for th stat x k at tim k can only dnd on masurmnts y 0, y 1,, y k l 1 till tim k l 1. In [20], th otimal information rocssing stratgy at ach nod in th ntwork that rsults in MMSE stimat at th dstination nod was idntifid. W rstat th algorithm in this ar for th sak of comltnss. W driv th ncssary and sufficint condition for stability of th xctd stimat rror covarianc for th algorithm using an altrnat mthod. Th chif contribution of this ar is to find th MMSE stady-stat rror at th dstination nod and to idntify its dndncy on th toology of th ntwork and th ackt dro robabilitis. Th framwork dvlod hr also allows us to gnraliz th condition for stability of th stimat rror covarianc to mor comlicatd ackt droing modls. III. OPTIMAL ENCODING AND DECODING W now dscrib an algorithm A, originally dvlod in [20], that achivs th otimal rformanc at th xns of constant mmory and transmission (modulo th transmission of th tim stam). At ach tim st k, vry nod v taks th following actions: 1) Calculat its stimat ˆx v k of th stat x k basd on any data rcivd at th rvious tim st k 1 and its rvious stimat. Th stimat can b comutd using a switchd linar filtr as follows. Th sourc nod imlmnts a Kalman filtr and udats its stimat at vry tim st with th nw masurmnt rcivd. Evry othr nod chcks th tim-stams on th data coming on th incoming dgs. Th timstams corrsond to th latst masurmnt usd in th calculation of th stimat bing transmittd. Lt th tim-stam for nod u V at tim k b t u (k). Also lt D uv (k) b th binary random variabl dscribing th ackt dro vnt on link (u, v) E at tim k. D uv (k) is 0 if th ackt is drod on link

3 (u, v) at tim k and 1 othrwis. For a ntwork with indndnt and mmorylss ackt dros, D uv (k) is distributd according to Brnoulli with aramtr uv. W dfin D uu (k) = 1. Nod v udats its timstam using th rlation t v (k) = max D uv(k)t u (k 1). (3) u N (v) {v} Not that for th sourc nod s, t s (k) = (k 1) for all k 1. Suos that th maximum of (3) is givn by nod n N (v) {v}. Th nod v udats its stimat as ˆx v k = Aˆxn k 1. 2) Affix a tim stam corrsonding to th last masurmnt usd in th calculation of its stimat and transmit th stimat on th outgoing dgs. 3) Rciv any data on th incoming dgs and stor it for th nxt tim st. Proosition 1: (Otimality of Algorithm A): Th algorithm A is otimal in th sns that it lads to th minimum ossibl rror covarianc at any nod at any tim st. Th roof of th abov thorm is rovidd in [20]. W should rmark that th abov rsult holds for any ackt dro squnc. Thus th algorithm A is otimal for any ackt dro attrn, i.., irrsctiv of whthr th ackt dros ar occurring in an i.i.d. fashion or ar corrlatd across tim or sac or if ackt dros ar tim-varying or vn advrsarial in natur. Th algorithm also dos not assum any knowldg of th statistics of th ackt dros at any of th nods. IV. STEADY-STATE ERROR COVARIANCE CALCULATION In this sction w calculat th stady-stat stimat rror covarianc at any nod using th algorithm A. For nod v and tim k, t v (k) dnots th tim-stam of th most rcnt obsrvation usd in stimating x k. This tim-stam volvs according to (3). Th xctd stimation rror covarianc at tim k at nod v can b writtn as = = E x k ˆx k tv(k) 2 k Pr (t v (k) = k l 1)E x k ˆx k k l 1 2 k [ Pr (l v (k) = l) A l P k l 1 A l + l 1 A j QA ],(4) j j=0 whr P k is th stimation rror covarianc of x k basd on {y 0, y 1,..., y k 1 } and l v (k) = k 1 t v (k) is th latncy for nod v at tim k. P k volvs according to a Riccati rcursion. Th abov quation givs th xctd stimation rror covarianc for a gnral ntwork with any ackt droing rocss. Th ffct of th ackt droing rocss aars in th distribution of th latncy l v (k). W considr th stady-stat rror covarianc in th limit as k gos to infinity, i.., P = lim k E x k ˆx k tv(k) 2. If P is boundd, w will say that th stimat rror is stabl; othrwis it is unstabl. As w can s from (4), th stability of th systm dnds on how fast th robability distribution of th latncy dcrass. For now w focus on an i.i.d. ackt dro modl. At any tim and for any link = (u, v), th ackt is drod with robability indndnt of tim k and of othr links of th ntwork. From (3), D uv (k) indicats th vnt of ackt dro for link = (u, v). For ach link and tim k lt Z (k) b th diffrnc btwn k and th most rcnt succssful transmission on link rcding tim k., i.., Z (k) = min{j 1 D uv (k + 1 j) = 1} Using th dfinition of Z (k), th last tim that any mssag is rcivd at nod v from link (u, v) is k Z uv (k)+1 and that mssag has tim-stam t u (k Z uv (k)). Thn (3) can b writtn in trms of Z (k) as t v (k) = max t u(k Z uv (k)) u N (v) Now Z (k) is distributd as a truncatd gomtric random variabl with Pr (Z (k) = i) = (1 ) i 1 k > i 1 and Pr (Z (k) = k) = 1 k 1 i=1 Pr (Z (k) = i). W can gt rid of th truncation by xtnding th dfinition of t u (k) for ngativ k s as wll. For k < 0 w dfin t u (k) = 0. Thn, for instanc, for th sourc nod s w hav t s (k) = (k 1) +, whr x + = max{0, x}. In gnral, w hav t v (k) = max t u(k Z uv ), u N (v) whr Z s ar indndnt random variabls distributd according to a gomtric distribution, i.., Pr (Z = i) = (1 ) i 1 i 1. Furthr not that Z s do not dnd anymor on k. Solving th abov rcursiv formula, w can writ t v (k) in trms of th tim-stam at th sourc nod (i.., (k 1) + ) as t v (k) = max (k 1 Z ) +, (5) P :an s-v ath P whr th maximum is takn ovr all aths P in th grah G from sourc s to th nod v. Thrfor th latncy at nod v can b writtn as l v (k) = k 1 t v (k) = min{k 1, min P :an s-v ath ( P Z )}. From th abov quation it can b sn that as k th distribution of l d (k) aroachs th distribution of l d dfind as l d = min ( Z ). (6) P :an s-d ath P W rfr to l d as th stady-stat latncy of th ntwork. Thrfor, th stady-stat rror covarianc can b writtn as l 1 P = Pr (l d = l) A l P A l + A j QA j, (7) j=0

4 whr P is th stady-stat stimation rror covarianc of x k basd on {y 0, y 1, y k 1 } and is th solution to th Discrt Algbraic Riccati Equation (DARE) P = AP A T + Q AP C T (CP C T + R) 1 CP A T. W assum that th systm {A, Q 1 2 } is stabilizabl. Hnc th rat of convrgnc of P k to P is xonntial [21] and th substitution of P for P k l in (4) dos not chang th stady-stat rror covarianc. Lt us dfin th gnrating function of th comlmntary dnsity function G(X) and th momnt gnrating function F (X) of th stady stat latncy l d G(X) = F (X) = Pr (l d l + 1)X l (8) Pr (l d = l)x l, whr X is a matrix. On vctorizing (7) w obtain vc (P ) = F (A A)vc (P ) + G(A A)vc (Q), whr A B is th Kronckr roduct of matrics A and B. Using th fact that F (X) = (X I)G(X) + I yilds vc (P ) = ((A A I)G(A A) + I) vc (P ) + G(A A)vc (Q). (9) W can s from (9) that th rformanc of th systm dnds on th valu of G(X) valuatd at X = A A. In articular, th systm is stabl if and only if G(X) is boundd at A A. Sinc G(X) is a owr sris, this is quivalnt to th bounddnss of G(x) (valuatd for a scalar x) at th squar of th norm of th ignvalu of A with th largst norm. W summariz th rsult of th abov argumnts in th following thorm. Thorm 1: Considr th systm modl dscribd in Sction II. Lt th ackt dros ar indndnt from on tim st to th nxt and across links. Thn th minimum xctd stady-stat stimation rror covarianc is givn by (9). Furthrmor, th rror covarianc is stabl, iff λ max (A) 2 lis in th rgion of convrgnc of G(x) whr λ max (A) is th maximum-norm ignvalu of A. Th abov thorm allows us to calculat th stady stat xctd rror covarianc for any ntwork as long as w can valuat th function G(X) for that ntwork. W now considr som scial ntworks and valuat th rformanc xlicitly. W start with a ntwork consisting of links in sris, or a lin ntwork. 1) Lin Ntworks: In this cas, th ntwork consists of only on ath from th sourc to th dstination. Sinc th dros across diffrnt links ar uncorrlatd, th variabls Z s ar indndnt. Thus F (X) = E [X l d ] = E [X P Z ] = E [X Z ]. Sinc Z is a gomtric random variabl, E [X Z ] = (1 )X(I X) 1 rovidd that λ max (X) < 1. Thrfor, F (X) = E [X l d ] = ] [(1 )X(I X) 1. Using artial fractions, w can asily show that G(X) = n 1 i=0 X i + X n c 1 (I X) 1, whr c = ( (1 )) 1. Thrfor th cost can b writtn as vc (P ) = [ (A A)( I ] (A A) ) 1 vc (P ) 1 + G(A A)vc (Q). (10) Rmark 1: W can s from th abov argumnt that th systm is stabl if for vry link w hav λ max (A) 2 < 1 or quivalntly max λ max (A) 2 < 1. This matchs with th condition in [20] Rmark 2: For th cas that som of s ar qual, a diffrnt artial fraction xansion alis. In articular for th cas whn thr ar n links all with th rasur robability, w obtain + + vc (P ) = (A A) n ( n 1 i=0 n 1 [ 1 (A A)n ( (A A) i vc (Q). i=0 I (A A) ) n vc (P ) 1 I (A A) ) ]vc i 1 (Q) 1 Whn thr is only on link btwn th sourc and th dstination th stady stat rror covarianc will b th solution to th Lyaunov quation P = AP A + (Q + (1 )AP A). This matchs with th xrssion drivd in [19] using Markov jum linar systm thory. 2) Ntwork of Paralll Links: Now considr a ntwork with on snsor connctd to a dstination nod through n links with robabilitis of ackt dro 1,..., n. In this cas th stady stat latncy is givn by l d = min 1 i n (Z i ). Sinc th minimum of indndnt gomtrically distributd random variabls with aramtrs { i } is itslf gomtrically distributd with aramtr q = i i, G(X) can b writtn as G(X) = (I i i X) 1 Thus th stady-stat rror can b valuatd using (9). Not that th rgion of convrgnc of G(X) nforcs i i λ max (A) 2 < 1 for stability which again matchs with th condition in [20].

5 Fig. 1. s Examl of a ntwork of combination of aralll and srial links 3) Arbitrary Ntwork of Paralll and Srial Links: Using similar argumnts as in rvious sctions, w can find th stady-stat rror covarianc of any ntwork of aralll and srial links. Ths ntworks ar drivd from th aralll and srial concatnations of sub-ntworks. Th following two siml ruls can giv th stady stat rror of any ntwork of aralll and sris links. Lt l d (G) dnot th stady-stat latncy function of ntwork G. Also givn two subntworks G 1 and G 2, dnot thir sris combination by G 1 G 2 and thir aralll combination by G 1 G 2. 1) For sris connction, w hav l d (G 1 G 2 ) = l d (G 1 )+ l d (G 2 ). Using th indndnc of latncy functions of th two sub-ntworks, th gnrating function of th ntwork is givn as G(X) = (X I)G 1 (X)G 2 (X) + G 1 (X) + G 2 (X). 2) For aralll connction, w hav l d (G 1 G 2 ) = min{l d (G 1 ), l d2 (G 2 )}. Using th indndnc of l d (G 1 ) and l d (G 2 ), th comlmntary distribution function of l d (G) can b writtn as th roduct of th functions for G 1 and G 2. As an xaml considr th ntwork dictd in Fig. 1. In this cas th ntwork G can b writtn as (((G 0 G 1 ) G 2 ) G 3 ) G 4 ) whr ach of th sub-ntworks G i is just a link with robability of ackt dro. Using th abov ruls and dnoting th momnt gnrating function of th aralll combination of any ntwork (with G(X)) and a link with robability of ackt dro by L (G)(X), th gnrating function of th ntwork can b writtn as G(X) = L (L (G 0 G 1 ) G 3 )(X) whr G i (X) = (I X) 1, i = 0, 1, 3 is th gnrating function for th i-th link and for ach function F ( ), L is an orator that such that L (F )(X) = F (X). Th stady stat rror covarianc can thus b valuatd. 4) Ntworks with Arbitrary Toology: Finding th distribution of th stady-stat latncy l d of a gnral ntwork is not an asy task. Howvr, w can rovid ur and lowr bounds on th rformanc. W first mntion th following intuitiv lmma without roof. Lmma 1: Lt P (G, {, E}) dnot th xctd stady-stat rror of a systm with communication ntwork rrsntd by grah G = (V, E) and robabilitis of ackt dro, E. Thn th xctd stady-stat rror is nonincrasing in s, i.., if q E P (G, {, E}) P (G, {q, E}), d whr A B mans that A B is ositiv smi-dfinit. Using th abov lmma w can lowr bound th stadystat rror by making a subst of links rasur fr. In articular, considr any sourc-dstination cut in th ntwork (which is simly a artition of th nods in two sts on containing th sourc nod (th sourc st) and th othr containing th dstination nod (th dstination st)). Stting th robability of rasur qual to zro for vry link xct thos crossing th cut givs a lowr bound on th rror. Thrfor, P (G, {, E}) P (G, {q, E}) whr q = iff is in th cut and zro othrwis. Now th lft sid of th abov quation can b calculatd asily using th rsults from Sction IV-.3. In articular, it can b shown that for stability w rquir that max ( ) λ max (A) 2 < 1 C:s-d cut C W rfr to mc (G) = max C:s-d cut ( C ) as th maxcut valu of th ntwork. On way to ur bound th stady-stat rror is by stting th robability of ackt dro of som of th dgs qual to on. In [20], it is shown that th rformanc of th ntwork G is lowr boundd by th rformanc of anothr ntwork G with th following rortis: G has th sam nod st. G is th combination of dg-disjoint aths from th sourc to dstination. Along ach ath th links hav th sam robability of droing ackts qual to th robability of ackt dro of on of links in th max-cut of th original ntwork G. Basd on th rvious rorty th valu of th maxcut in G is th sam as th original ntwork G. Now G is a ntwork with sris and aralll links only. Thus its rformanc can b comutd and rovids an ur bound on th stady-stat rror covarianc of G. In articular, sinc all th aths from s to d ar disjoint, Pr (l d (G ) l + 1) = i Pr (l d (P i ) l + 1) whr l d (P i ) is th stady-stat latncy of ath P i. But for any ath with n links, n 1 ( ) i + l n Pr (l d l + 1) = (1 ) i l n+1 l n i=0 Using th Stirling formula for larg l, w obtain c 1 Pr (l d l + 1) ( 1 1)n 1 (l 1) n 1 l c 2 (11) whr c 1, c 2 ar two ositiv constants indndnt of l. Thrfor, for larg l, Pr (l d (G ) l + 1) bhavs lik f(l)( i i ) l = f(l)( mc (G )) l = f(l)( mc (G)) l

6 whr f(l) grows olynomially in l. Thus it is asy to vrify that for ntwork G th systm is stabl if mc (G) satisfis mc (G) λ max (A) 2 < 1. Thrfor th abov condition is both ncssary and sufficint for stability. 5) Synthsis of a Ntwork: On can us th rsults on th rformanc of ntworks to dsign ntworks that rsult in minimal rror covarianc. To considr a siml xaml, considr a scalar systm obsrvd by snsor s. Assum that th dstination is locatd at distanc d 0 from th snsor. Th robability of droing a ackt on a link dnds on its hysical lngth. A rasonabl modl for robability of droing ackts is givn by 2 (d) = 1 x( βd α ), whr β, α ar ositiv constants. α dnots th xonnt of owr dcay in th wirlss nvironmnt. W ar intrstd in th otimal numbr n of rlay nods btwn snsor and th dstination so as to minimiz th xctd stady-stat rror covarianc. Assuming that th snsor ar uniformly lacd, P satisfis ( a P 2 (1 ) = 1 a 2 ) n+1 (P + Q a 2 1 ) Q a 2 1 Thus th otimal n (assuming that a 2 > 1) is th solution to th roblm ( a 2 (1 ( d0 n+1 min )) ) n+1 n 1 ( d0 n+1 )a2 If a 2 < 1 thn minimization is rlacd with maximization. V. EXAMPLES In this sction, w illustrat th abov rsults using a siml xaml. Considr a scalar rocss volving as x k+1 = 0.8x k + w k, that is bing obsrvd through a snsor of th form y k = x k + v k. Th noiss w k and v k ar assumd zro-man, whit and Gaussian with covariancs Q = 1 and R = 1 rsctivly. Furthr, th two noiss ar assumd indndnt of ach othr. To bgin with, suos that th sourc and th dstination nod ar connctd using two links in sris, ach with a robability of ackt rasur. Figur 2 shows th rformanc of our stratgy as th robability is varid. Th simulation rsults rfr to data gnratd by a random run avragd ovr tim sts whil th thortical valus rfr to th valu rdictd by using (9). W can s that th two sts of valus match quit closly. W also carrid out a similar xrcis for th sourc and dstination nods connctd by two links in aralll, with ackt rasur robability ach. Th rsults ar lottd in 2 This xrssion can b drivd by considring th robability of outag in a Rayligh fading nvironmnt. Stady stat stimat rror covarianc Simulatd valus Thortical valus Erasur Probability Fig. 2. Simulatd and thortical rsults for a lin ntwork. Figur 3. W can onc again s that th simulatd valus match quit closly with th thortical valus. Stady stat stimat rror covarianc Fig. 3. Simulatd valus Thortical valus Erasur Probability Simulatd and thortical rsults for a aralll ntwork. As a final xaml, w considr th sourc and dstination nods connctd by a bridg ntwork shown in Figur 4. W assum all th links in th ntwork to hav robability of rasur. This ntwork cannot b rducd to a sris of sris and aralll sub-ntworks. W can howvr, calculat th rformanc analytically in this articular cas and comar it to th ur and lowr bounds rsntd arlir. Th ntworks usd for calculating th bounds ar also shown in figur 4. Figur 5 shows a comarison of th analytical and simulatd valus with th lowr and ur bounds. Th simulatd valus do not fall blow th ur bound vry-tim bcaus of numrical issus; othrwis th bounds ar tight. W also calculatd th otimal numbr of nods to b lacd btwn th sourc and th dstination nod using our synthsis rsults. Th valus w usd ar d 0 = 5, α = 2, β = 1. In this cas, th otimal numbr of rlays turns out to b n = 4.

7 s s Bridg Ntwork Ur bound d d s Lowr Bound Fig. 4. Bridg ntwork and th ntworks usd for calculating lowr and ur bounds. Fig. 5. Exctd rror covarianc Simulatd valu Analytical valu Lowr bound Ur bound Probability of ackt dro Simulatd valus and thortical bounds for th bridg ntwork. VI. GENERALIZATIONS Corrlatd rasur vnts: Th analysis so far assumd that th rasur vnts ar mmorylss and indndnt across diffrnt links in th ntwork. W could thus formulat th rformanc in trms of a gnrating function of th stady-stat latncy distribution as dfind in (6). W now look at th ffct of droing ths assumtions. Markov vnts: If w assum that th dro vnts on ach link ar govrnd by a Markov chain (but ar still indndnt of othr links), w can obtain th rformanc as follows. Lt us assum that th ackt dro vnt on link (u, v), dnotd by D uv (k) volvs according to a Markov chain with transition matrix M uv. W furthr assum that M uv is irrducibl and rvrsibl. Lt us first considr th cas whr th initial distribution of ackt dro on ach link is th stationary distribution of th Markov chain on that link. Thn w can rwrit (3) as (5) as bfor, whr Z l is a gomtric random variabl with distribution { α uv M uv (1, 2)M uv (1, 1) l 2 l 2 Pr (Z uv = l) =, 1 α uv l = 1 with α uv as th robability of ackt dro basd on th stationary distribution of link = (u, v) and M uv (i, j) as th (i, j)-th lmnt of M uv. Thrfor, all th rvious d analysis gos through. In articular, th stability condition is ( max M (1, 1)) λ max (A) 2 < 1. c:s-d cut c Now, if th initial distribution is not th stationary distribution, th variabls Z uv (k) will not b tim-indndnt and th analysis dos not gos through. Howvr, sinc for larg k th chains aroach thir stationary distribution, th stability condition rmains unchangd. Satially corrlatd vnts: Suos that th ackt dro vnts ar corrlatd across th ntwork but mmorylss ovr tim. In othr words, at ach tim st k, th ackt dro vnts occur according to distribution Pr 0 (D uv, (u, v) E). Now Z (k) s ar not indndnt across th ntwork and hnc finding th stady-stat rror covarianc dos not sm to b tractabl. Howvr, w can find th condition for stability. For this, w dfin a gnralizd notion of quivalnt robability of ackt dro for corrlatd vnts. Considr a s d cut c, and lt B(c) dnot th st of dgs crossing this cut. Thn th quivalnt robability of ackt dro for this cut is dfind as q (c) = Pr (D uv = 0, (u, v) B(c)). Th valu of th max-cut for th ntwork is th maximum of q (c) ovr all th cuts, mc (G) = max c:s dcut q (c). W can show that th condition for stability of th systm is mc (G) λ max (A) 2 < 1. To s this, considr th scnario whn only on ackt is to b routd from th sourc to dstination starting at tim t 0. For ach tim-st t t 0 lt V r (t) dnot th st of nods that hav rcivd th ackt at tim t. Clarly V r (t 0 ) = {s}. W want to bound th robability that at tim t 0 + T, dstination nod has not yt rcivd th ackt. Not that for vry tim-st btwn t 0 and t 0 + T, V r (t) clarly forms a cut-st sinc it contains s and not d. Now th siz of V r (t+1) dos not incras with rsct to tim-st t iff all th links that cross th cut gnratd by V r (t) dro ackts. Howvr by th dfinition of mc (G) th robability of this vnt is at most mc (G). Thrfor, w hav { V r (t) + 1 with rob. at most mc (G) V r (t+1) = V r (t) with rob. at last 1 mc (G) Thus for larg T, th robability that at tim t 0 + T th dstination nod has not rcivd th ackt is ur boundd by n(1 mc (G)) n T n mc (G) T n, whr n is th numbr of nods in th ntwork. In th original scnario, a nw ackt is gnratd at th sourc at ach tim st. Howvr, sinc th imortanc of th ackts is incrasing with tim, w can ur bound th rror by considring that th ntwork is only routing ackt gnratd at tim k l. Th robability that th latncy is largr than l grows lik f(l) mc (G) l, whr f(l) is olynomial in l with boundd dgr and thus th sufficincy of th stability condition

8 follows. Th ncssity art involvs similar idas and is omittd. Unicast Ntworks: So far, w assumd that th toology of th ntwork was givn and any nod could transmit a mssag on all th out-going links. If th ntwork is unicast, ach nod chooss on link out of a st to transmit its mssag. Th roblm is to choos th otimal ath for th data to flow from th sourc to th dstination nod. Clarly onc th ath is chosn th otimal oration at ach nod on th ath is givn by algorithm A dscribd in Sction III. In ordr to choos th otimal ath, w nd to dfin a mtric for th cost of a ath. If th mtric is th condition for stability of th stimat rror covarianc, thn th roblm can b rcast as choosing th shortst ath in a grah with th lngth of a ath bing givn by its quivalnt robability of ackt dro, i., for ach ath P, q = max P. Thus th shortst ath roblm is to find th ath that has th minimum quivalnt robability of ackt dro among all th aths,i., min P :s-d ath max P. Th abov roblm is wll studid in th comutr scinc socity and can b solvd as a short-ath roblm ovr min-max smi-ring in a distributd fashion [24]. If th mtric is th stady-stat rror thn th roblm is mor comlicatd in gnral. For th scial cas of a scalar systm and no rocss nois, from (10), w hav for ath q, log P q = q Now th roblm is quivalnt to min q:s-d ath q log( (1 )a 2 1 a 2 ) log( (1 )a 2 1 a 2 ) This roblm can also b solvd in a distributd way [25]. VII. CONCLUSIONS AND FUTURE WORK In this ar, w considrd th roblm of otimal stimation across a ntwork. W modld th links as ackt rasur links. W rovidd a framwork for comuting th otimal stimat rror covarianc and gav ur and lowr bounds on th rformanc of gnral ntworks. W showd how to utiliz this framwork for th synthsis of ntworks for th uros of stimation. W also carrid out th stability analysis for arbitrary ntworks and for ackt rasur rocsss that ar ossibly corrlatd across tim or th ntwork. In this ar, w hav ignord issus of quantization. On intrsting and challnging roblm is to includ constraints of a limitd bit rat into th framwork. Th work of Sahai [22] and Ishwar t al [23] may b rlvant to this roblm. In th futur, w would lik to xlor ths connctions. REFERENCES [1] An Introduction to Multisnsor Data Fusion, D. L. Hall and J. Llinas, Procdings of th IEEE, 85(1), 1997, [2] Distributd Dtction with Multil Snsors: Part I - Fundamntals, R. Viswanathan and P. K. Varshny, Procdings of th IEEE, 85(1), 1997, [3] Scial Issu on Ntworks and Control, L. Bushnll (Gust Editor), IEEE Control Systms Magazin, 21(1), Fb [4] Scial Issu on Ntworkd Control Systms, P. Antsaklis and J. Bailliul (Gust Editors), IEEE Transactions on Automatic control, 49(9), St [5] Stabilizing a Linar Systm with Quantizd Stat Fdback, D. F. Dlchams, IEEE Transactions on Automatic Control, 35, 1990, [6] Control undr Communication Constraints, S. Tatikonda, PhD Thsis, MIT, Cambridg, MA [7] Stabilizability of Stochastic Linar Systms with Finit Fdback Data Rats, G. N. Nair and R. J. Evans, SIAM Journal on Control and Otimization, 43(2), July 2004, [8] Towards th Control of Linar Systms with Minimum Bit-rat, J. Hsanha, A. Ortga and L. Vasudvan, Procdings of th 15th Intrnational Symosium on th Mathmatical Thory of Ntworks, [9] Ral-tim Control Systms with Dlays, J. Nilsson, PhD Thsis, Dartmnt of Automatic Control, Lund Institut of Tchnology, [10], Coordinatd Control of Unmannd Arial Vhicla, P. Silr, PhD Thsis, Univrsity of California, Brkly, [11] Stability of Ntworkd Control Systms, W. Zhang, M. S. Branicky and S. M. Philis, IEEE Control Systm Magazin, 21(1), Fb 2001, [12] Robust Prformanc of Soft Ral-tim Ntworkd Control Systms with Data Droouts, Q. Ling and M. D. Lmmon, Procdings of th IEEE Confrnc on Dcision and Control, [13] Fdback Control Utilizing Packt Droing Ntwork Links, C. N. Hadjicostis and R. Touri, Procdings of th IEEE Confrnc on Dcision and Control, [14] Otimal Droout Comnsation in Ntworkd Control Systms, Q. Ling and M. D. Lmmon, Procdings of th IEEE Confrnc on Dcision and Control. [15] Stability of Ntworkd Control Systms in th Prsnc of Packt Losss, Procdings of IEEE Confrnc on Dcision and Control, [16] Tim Varying Otimal Control with Packt Losss, B. Sinooli, L. Schnato, M. Francschtti, K. Poolla and S. S. Sastry, Procdings of th IEEE Confrnc on Dcision and Control, [17] Otimal Control of Dynamical Systms ovr Unrliabl Communication Links, O. C. Imr, S. Yuskl and T. Basar, NOLCOS, [18] Kalman Filtring with Intrmittnt Obsrvations, B. Sinooli, L. Schnato, M. Francschtti, K. Poolla, M. Jordan and S. S. Sastry, IEEE Transactions on Automatic Control, 49(9), St. 2004, [19] On LQG Control Across Packt-Droing Links, V. Guta, D. Sanos, B. Hassibi and R. M. Murray, Systm and Control Lttrs, submittd July [20] Data Transmission ovr Ntworks for Estimation, V. Guta, A. F. Dana, J. P. Hsanha and R. M. Murray, 17th Intrnational Symosium on Mathmatical Thory of Ntworks and Systms, MTNS 2006, Submittd. [21] Linar Estimation, T. Kailath, A. H. Sayd and B. Hassibi, Prntic Hall, Nw Jrsy, [22] Anytim Information Thory, A. Sahai, PhD Thsis, MIT, Cambridg, MA [23] On Rat-constraind Distributd Estimation in Unrliabl Snsor Ntworks, P. Ishwar, R. Puri, K. Ramchandran and S. S. Pradhan IEEE Journal on Slctd Aras in Communications: Scial Issu on Slf-organizing Distributd Collaborativ Snsor Ntworks, 23(4), Ar. 2005, [24] Smiring framworks and algorithms for shortst-distanc roblms, M. Mohri Journal of Automata, Languags and Combinatorics, 2002, [25] Introduction to Algorithms, T. H. Cormn, C. E. Lisrson, R. L. Rivst, and C. Stin, MIT Prss and McGraw-Hill, 1990.