Estimation Over Wireless Sensor Networks: Tradeoff between Communication, Computation and Estimation Qualities

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1 Proceedngs of the 7th World Congress The Internatonal Federaton of Automatc Control Estmaton Over Wreless Sensor Networs: Tradeoff between Communcaton, Computaton and Estmaton Qualtes Lng Sh Karl Henr Johansson Rchard M. Murray Control and Dynamcal Systems, Calforna Insttute of Technology, Pasadena, CA 95 USA (emal: School of Electrcal Engneerng, Royal Insttute of Technology, Stocholm, Sweden (emal: Control and Dynamcal Systems, Calforna Insttute of Technology, Pasadena, CA 95 USA (emal: Abstract: In ths paper we consder a state estmaton problem over a wreless sensor networ. A fuson center dynamcally forms a local mult-hop tree of sensors and fuses the data nto a state estmate. It s shown that the optmal estmator over a sensor tree s gven by a Kalman flter of certan structure. Usng estmaton qualty as a metrc, two communcaton schemes are studed and compared. In scheme one, sensor nodes communcate measurement data to ther parent nodes, whle n scheme two, sensor nodes communcate ther local state estmates to therparentnodes. We showthatunderperfect communcatonlns, thetwoschemes produce thesame estmate atthefusoncenterwthunlmtedcomputatonateachsensornode; scheme one s always better than scheme two wth lmted computaton. When data pacet drops occur on the communcaton lns, we show that scheme two always outperforms scheme one wth unlmted computaton; wth lmted computaton, we show that there exsts a crtcal pacet arrval rate, above whch, scheme one outperforms scheme two. Smulatons are provded to demonstrate the two schemes under varous crcumstances.. INTRODUCTION Control over large resource-constraned nfrastructures requres new desgn paradgms beyond tradtonal sampleddata control. The man dffcultes arse from the constraned communcaton and computaton capabltes. Communcaton between networ nodes s lmted due to the lmted energy avalable, partcularly, f nodes are located physcally far way from each other. It taes tme to transfer nformaton from one node to another, and n many cases ths tme ncreases f the nformaton needs to be relably delvered. Node nterference, obstacles that bloc wreless sgnals, etc can sometmes cause datapacets dropped n the networ and hence lead to mperfect communcatons. Computaton capablty of each networ node may be lmted as well, for example, n a wreless sensor networ, the central processng unt of each sensor node can only perform smple calculatons. We study the problem of state estmaton over a wreless sensor networ n ths paper. The man contrbuton of ths paper s to provde a qualtatve and quanttatve The wor by L. Sh and R. M. Murray s supported n part by AFOSR grant FA The wor by K. H. Johansson was supported by the Swedsh Research Councl and the Swedsh Foundaton for Strategc Research. analyss of the tradeoff between estmaton qualtes and the communcaton and computaton capabltes of each networ node. In partcular, we study two communcaton schemes andcompare therperformances undercommuncaton and computaton constrants. We assume data are communcated over a mult-hop wreless networ, nstead of a sngle-hop networ to conserve networ energy usage and therefore enhance whole system lfe. The resultng local sensor topology has the structure of a tree for whch the fuson center s the root. The qualty of the state estmate depends not only on the sensor qualty but also on the communcaton delay,.e., the number of hops the sensor readng needs to tae untl t reaches the fuson center. There are several potental applcaton areas of the wor presented n ths paper, ncludng buldng automaton, envronmental montorng, ndustral automaton, power dstrbuton, and transportaton systems. Some wor related to ths paper s descrbed next. How to effcently encode control nformaton for event-trggered control over communcaton channels wth severe bandwdth lmtatons s dscussed n Bao et al. [6]. Kalman flterng under certan nformaton constrants, such as decentralzed mplementaton, has been exten /8/$. 8 IFAC 65.38/876-5-KR-.69

2 svely studed Slja [978]. Implementatons for whch the computatons are dstrbuted among networ nodes s consdered n Alrsson and Rantzer [6], Olfat-Saber [5], Spanos et al. [5]. Kalman flterng over lossy networs s consdered n Snopol et al. [4], Hespanha et al. [7]. The nteracton between Kalman flterng and how data s routed on a networ seems to be less studed. Routng of data pacets n networs are typcally done based on the dstance to the recever node as n Bertseas and Gallager [99]. Some recent wor addresses how to couple data routng wth the sensng tas usng nformaton theoretc measures was gven by Lu et al. [5]. For control over wreless sensor networs, the experenced delays and pacet losses are mportant parameters. Randomzed routng protocols that gves probablstc guarantees on delay and loss are proposed n Bonvento et al. [6], La and Paschalds [6]. A compensaton scheme n the controller for the varatons on the transport layer that such routng protocols gve rse s presented n Wtrant et al. [7]. A robust control approach to control over mult-hop networs s dscussed n Panousopoulou et al. [6]. A general crosslayer approach to control and data routng seems to be an open and rather dffcult topc due to many practcal constrants. The two schemes proposed n ths paper have been studed for a specal case (.e., a sensor networ consstng of only one node wth unlmted computaton capablty) n Gupta et al. [5], where the authors showed that preprocessng the measurement by the sensor before sendng t to the controller s better than post-processng the measurement after recevng t by the controller when the communcaton ln s mperfect. The rest of the paper s organzed as follows. The problem setup s descrbed n Secton. Some defntons and prelmnary facts on Kalman flter s provded at Secton 3. Optmal estmaton over a sensor tree usng a Kalman flter s dscussed n Secton 4 usng both estmate and measurement communcaton schemes. The two schemes are further studed and compared n Secton 5 under communcaton and computaton constrants. An example s presented n Secton 6 to llustrate the results obtaned n prevous few sectons. The paper s concluded n Secton 7 wth a dscusson on future wor.. Mathematcal Models. PROBLEM SET-UP Consder the feedbac control system n Fg.. The plant s gven by x = Ax + w, () where x IR n s the state, w s whte Gaussan nose wth zero-mean and covarance matrx Q. A wreless sensor networ s used to measure the state. The measurement equaton for sensor S s gven by y = H x + v, () where y IRm s the measurement, v s whte Gaussan nose wth zero-mean and covarance matrx Π >. Fg.. Structure of Closed-Loop Control System wth Measurements Gathered by a Wreless Sensor Tree Each sensor can potentally communcate va a sngle-hop connecton wth a subset of all the sensors by adjustng ts transmsson power. Let us ntroduce a sensor S, whch we denote the fuson center, and consder a tree T wth root S. Assume T = n +,.e., T contans n other sensor nodes. We suppose that there s a non-zero sngle-hop communcaton delay, whch s smaller than the samplng tme of the plant. All sensors are synchronzed n tme, so the data pacet transmtted from S to S s delayed one sample when compared wth the parent node of S.. Problems of Interest We focus on the followng estmaton problem n the paper, thus we assume the control law s computed and sent to the plant once the controller obtans the state estmate. Problem.. Gven a tree T representng sensor communcatons wth S, what s the optmal state estmate ˆx (T) generated at the fuson center? Apparently, the optmal estmaton procedure depends on whether the sensors communcate raw measurement or local estmates to ther parent nodes. Therefore we also dscuss the two sensor communcaton schemes n detal and try to answer: Problem.. What s the optmal sensor communcaton scheme when the communcaton lns ntroduce pacet drops and when each sensor has lmted computaton power? In the remanng sectons of the paper, we provde answers to these two problems. Remar.3. Although we consder ln falure n ths paper,.e., by studyng the effect of pacet drops when sensors transmt data, we assume no sensor node falure. In case node falure does occur n practce, a new sensor communcaton tree can be establshed based on remanng set of sensors, for example, by executng the sensor reconfguraton algorthm n Sh et al. [7]. 3. DEFINITIONS AND KALMAN FILTER PRELIMINARIES In ths secton, we provde some defntons whch are used frequently n later sectons. A bref ntroducton to Kalman Flter s also ncluded upon whch the soluton to the optmal estmaton problem s based. 66

3 Defne the followng terms for a gven a tree T representng sensor communcatons wth S. Fam T (S ): The subtree of T that s rooted at S. Par T (S ): The parent node of S n T. The depth of T s denoted h T. For all notatons, we drop the subscrpt T when the consdered tree follows from the context. Next, we formalze estmaton over a sensor tree under communcaton energy constrants, n whch the operatons above can be used to mprove the performance. Let us defne the followng state estmates and other quanttes at S : Fg.. Sensor Communcaton Schemes ˆx (T) E[x all measurements up to ], ˆx (T) E[x all measurements up to ], P (T) E[(x ˆx (T))(x ˆx (T)) ], P (T) E[(x ˆx (T))(x ˆx (T)) ], P (T) lm P (T),f the lmt exsts, P (T) lm P (T),f the lmt exsts. Consder the followng dscrete tme system x = Ax + w y = C x + v where w and v are whte Gaussan noses wth zeromean and covarances Q and R >, respectvely. The estmates ˆx and P can be computed as (ˆx, P ) = KF(ˆx, P, y, C, Q,R ), where KF denotes the Kalman flter gven by the followng update equatons: ˆx = Aˆx, (3) P = AP A + Q, (4) K = P C [C P C + R ], (5) ˆx = Aˆx + K (y C Aˆx ), (6) P = (I K C )P. (7) 4. OPTIMAL ESTIMATION OVER SENSOR TREES In ths secton, we consder two sensor communcaton schemes. Optmalestmatonovera sensor tree usngeach schemes s gven. Then the two schemes are compared n cases when communcaton lns ntroduce pacet drops and when sensor nodes have lmted computaton power. We provde condtons on whch scheme s better than the other n these cases. 4. Sensor Communcaton Schemes Scheme One Measurement Communcaton In ths scheme, the sensors only send ther raw measurement data to ts parent wthout processng whch has been used n Sh et al. [7] to construct mnmum energy sensor tree. The fuson center S T collects data from all sensors n T. The role of the fuson center s to compute an estmate of x and forward t to a controller node. Fg. 3. Kalman Flter Iteratons at Tme Scheme Two Estmate Communcaton In ths scheme each sensor S runs a local Kalman flter and computes ther own state estmate ˆx at tme. After ˆx s obtaned, S sends ˆx to Par(S ). Par(S ) then updates ts state estmate by fusng the estmates from ts chldren wth ts own. Snce the computaton s dstrbuted n ths scheme and each S mantans a local estmate of the state, ths scheme s more robustthanscheme one whenthere sdata pacet drops as shown n Theorem 5.. In prncple, after fusng the estmates from ts chld nodes, the parent node can also send the estmate bac to ts chld nodes and hence mprove ther estmaton qualty. The analyss n ths paper extends straghtforward to ths case. 4. Optmal Estmaton Over Sensor Trees: Scheme One Let the tree T wth depth h that represents the sensor communcatons wth the FC be gven. Defne Y + as all measurements avalable at the FC for tme + at tme, =,, h. Fgure 3 shows the overall estmaton scheme at tme, where Y l {y l+ : S s l hop away from S, l =,, h. Let S j be the node that s j hops away from S. Defne For X, defne Γ j [H j ;H j ; ], j =,, h C [Γ ; ; Γ ], =,, h Υ j dag{π j,π j,, j =,, h R dag{υ,,υ, =,, h g C (X) AXA + Q AXC [C XC + R ] C XA. 67

4 Theorem 4.. Consder a sensor tree T wth depth h. () ˆx and P can be computed from h parallel flter as (ˆx h+, P h+ ) = KF(ˆx h, P h, Y h+, C h, Q,R h ). (ˆx, P ) = KF(ˆx, P, Y, C, Q,R ) (ˆx, P ) = KF(ˆx, P, Y, C, Q,R ) () Furthermore P and P satsfes P = g C g Ch (P h ) (8) P = g C g Ch ( P ) (9) where P s the unque soluton to g Ch ( P ) = P. Proof: See Sh et al. [7]. 4.3 Optmal Estmaton Over Sensor Trees: Scheme Two In ths secton, we assume each sensor node has enough computaton power to run a local Kalman flter and hence generate ts own estmate of the state. It then communcates the estmate to ts parent node. Let x and ˆx be the local state estmate and P and P be the local error covarance of S before and after fusng wth ts chldren s estmates respectvely. After y s obtaned, S runs the Kalman flter n ts nformaton form as follows ˆx = Aˆx, () P where K = P H Π. = AP A + Q, () ( ) P ( ) = P + H Π H () x = ˆx + K [y H ˆx ] (3) Then S sends ˆx, P, x, P to ts parent node. Notce that for the leaf nodes, x and P are the same as ˆx and P. For ntermedate nodes, after they receve the local estmates from ther chldren, they update the state estmate and error covarance accordng to ( P ) = ( P ) + j:par(s j)=s ˆx = P { ( P ) x + [ ( P j) ( P j ) ] j:par(s j)=s (4) [( P j) x j ( P j ) ˆx j ] (5) Denote ˆx () and ˆx () as the state estmate at S usng scheme one and two respectvely. P () and P () etc, are denoted n the same way. Snce S s the root of subtree Fam(S ), f we defne Γ j, C, Υ j, R, g C smlarly as n the prevous secton for root S, we mmedately obtan the followng theorem. Theorem 4.. () If ˆx () = ˆx () and P () = P (), then ˆx () = ˆx () for all. () P ( ) = g C g Ch P (6) where h s the depth of Fam(S ) and P s the unque soluton to ( ) g Ch P = P. (7) Proof: () It s shown n Rao and Durrant-Whyte [99] that ˆx from Eqn (5) s the same as computed from a centralzed Kalman flter where the measurements from ts chldren are collected at S. Consequently, ˆx at S usng scheme two s the same as usng scheme one. () Drect result from Theorem COMPARISON OF THE TWO SCHEMES We assume when the computaton power of each sensor node s lmted, the local state estmate transmtted from a sensor node to ts parent node s delayed by one tme step usngscheme two. Snce usngscheme one, thesensor nodes smply forward the measurement data, hence there s no delay n transmttng ts data. Wth ths assumpton, the two schemes are compared n the followng four scenaros when communcaton lns ntroduce pacet drops and when sensor nodes have lmted computaton power. We provdecondtonsononescheme sbetterthantheother. 5. Perfect Communcaton wth Unlmted Computaton Wth no pacet drops and unlmted computaton, S has the same estmate usng ether scheme as we have stated n the prevous secton. 5. Imperfect Communcaton wth Unlmted Computaton Wth unlmted computaton at each sensor node, f pacet drops occur due to nterference, networ congeston etc, scheme twooutperformsscheme onentheexpected sense as shown n Theorem 5.. Furthermore, there exst cases that scheme one s arbtrarly worse than scheme two as shown n Theorem 5.. Let γ () and γ () be the ndcator functons of whether the pacet from S s successfully transmtted to Par(S ) ( = ) or dropped ( = ). To compare the performance of the two schemes, assume γ = γ () = γ () for all and. Then we have the followng result. Theorem 5.. Assume unlmtedcomputatonateachsensor node and P () = P (). Then E[P ()] E[P ()] for all. Proof: We gve a proof for the case where there s only one sensor (see Fg. 4). The dea s straghtforward to present n the general case. Snce we only consder one 68

5 Fg. 4. Comparson of Two Communcaton Schemes sensor here, we smply wrte γ as γ. Then by defnton of the expected value, we only need to show that P () P () (8) for a partcular realzaton of the pacet drop sequences γ. We use mathematcal nducton to prove Eqn (8). () P () P () holds by assumpton n the theorem. () Assume Pm() Pm() for m <. (3) At m +. (a) If γ m+ =, Pm+ () := gm+( P ()) Pm+ () = g( Pm() ) (b) If γ m+ =, g m+( P ()) g m+( P ()) = P m+ (). P m+ () = h( P m() ) h ( P m() ) = P m+ (). where the nequaltes are from propertes of h and g functons and can be found n Snopol et al. [4]. The three steps above complete the nducton. Theorem 5.. Consder the one sensor case (Fg. 4). Assume unlmted computaton at each sensor node and P () = P (). Further assume that A s unstable, (H, A) s detectable and (A, Q) s controllable. Then the followng are true. () For any and any pacet drop sequences γ l, l =,,, P () P (). (9) () If H does not exst, then there exst c >, c >, a postve nteger mn and a pacet drop sequence γ l, l =,,, such that P () c c P () () for all mn. (3) If H exsts, then for any pacet drop sequence, there does not exst c > such that Equ () holds. Proof: () See proof to Theorem 5.. () Assume H does not exst. We explctly construct a class of pacet drop sequences for whch c > and Eqn () holds. Let θ be a gven postve nteger. Consder the pacet drop sequence {{ {{ θ zeros θ zeros.e., γ l = for l mod (θ + ) = and γ l = otherwse. Let mn be such that g mn( P ()) ( + ǫ)p () where < ǫ <. Such mn exsts as lm g( P ()) = P (). Therefore P () hθ( ( + ǫ)p () ) for all mn. Wrte = θ + for mn, where < θ. Then P () = h g ( (h θ g) (P ())). Snce H does not exst and A s unstable, we can pc a θ > such that (h θ g) dverges. For ths θ, there exsts c > such that P () c P (). Let c = ( c) θ P, c = () ch θ( ( + ǫ)p () ) then P () c c P (). (3) If H exsts, then g(x) M where M = AH Π (H ) A + Q accordng to Sh et al. [5]. Hence there exsts mn > such that f mn and γ =, we have P () c P () where M c = ( ǫ)p (). If γ = and let = mn{s : γ s () = γ s () =, s then P () = h ( P () ) h ( c P () ) = c h ( P () ) = c P (). Remar 5.3. Theorem 5. only consders the one sensor case. Generalzaton to multple sensor case s straghtforward. Theorems 5. and 5. show that scheme one s at most as good as and could be arbtrarly worse than scheme two dependng on the pacet drops. The results suggest that n a networ where busty pacet drops can occur, t s always better to process the measurement before sendng t to ts parent node assumng unlmted computaton power at each sensor node. However, ths does not always hold when the computaton power at each sensor node s lmted as shown n the next two sectons. 5.3 Perfect Communcatons wth Lmted Computaton Wth lmted computaton at each sensor node, f there are nopacet drops, scheme onesalwaysbetter thanscheme 69

6 two. Ths s qute dfferent from the unlmted computaton case where scheme two s always better than scheme one. Theorem 5.4. Assume lmtedcomputatonateachsensor node and no pacet drops occur. Then f P () = P (), P () P (). Proof: We gve a proof for the case where there s only one sensor (Fg. 4). We have assumed that transmttng the estmate to ts parent node has one tme delay for a sensorwhenusngscheme two, andhasnotmedelaywhen usng scheme one, therefore n ths case, P () = g(p ()), P () = h(g(p ())). Assume P () P () for all m. Then we have P m() = g(g(p m ())) g(g(p m ()) h(g(p m ())) = P m() Ths completes the nducton step as P () = P (). 5.4 Imperfect Communcatons wth Lmted Computaton Wth lmted computaton at each sensor node, f the communcaton networ ntroduces pacet drops, scheme one s not always better than scheme two. As we show next, there exsts some crtcal pacet arrval rate, only above whch, scheme one s better. Fg. 8 n Secton 6 demonstrates ths phenomenon. The results s stated n the followng theorem. Theorem 5.5. Assume lmtedcomputatonateachsensor node and P () = P (). Further assume [H ;H ; ;H n ] does not exst. Let γ () = γ () = γ for all and denote γ = (γ,, γ, γ +,, γ n ). Then for fxed γ, there exsts γ such that f γ γ, E[P ()] E[P ()] and otherwse f γ > γ. Proof: We gve a proof for the case where there s only one sensor (Fg. 4). Agan, t s straghtforward to extend t n the general case. Followng the same lne as n the proof of Theorem 5. (.e., by constructng explct pacet arrval sequences), we can show that there exsts γ such that f γ γ, E[P ()] E[P ()]. We can mae the bound tghter by notcng that when γ = for all, from Theorem 5.4, E[P ()] = P () P () = E[P ()] Snce E[P ()] and E[P ()] are contnuous and decreasng functons wth respect to γ, from real analyss, we now there exsts γ γ such that f γ γ, E[P ()] E[P ()] and otherwse f γ > γ. 6. EXAMPLES We consder an ntegrator chan as an example n ths secton to demonstrate the performance of the two schemes 6 Fg. 5. Integrator Chan Example Tr(P ) γ = γ = wth unlmted computaton Centralzed KF Scheme Scheme Tr(P ) γ = γ = wth lmted computaton Centralzed KF Scheme Scheme 4 Fg. 6. Comparson of Two Schemes Under Perfect Communcaton under varous crcumstances. The dscrete tme system dynamcs s gven by Eqn () wth A = [ ].. wth process nose covarance Q =.3I. There are two sensors avalable. The measurement equatons are gven by y = [ ]x + v, y = [ ]x + v where v are whte Gaussan wth zero-mean and covarances Π =.5 and Π =.5. Assume sensor s hops away from S (Fg. 5). Notce that A s unstable n ths example. In Fg. 6, we plot the error covarance evoluton of the two schemes. We also plot the centralzed Kalman flter soluton. The fgure on the left hand sde show that the two schemes produce thesame errorcovaranceandtsbggerthanthesoluton from the centralzed Kalman flter, whch s as expected. The fgure on the rght sde shows the case wth lmted computaton at the sensor nodes. It shows n ths case that scheme two produces a worse result than scheme one, whch s stated n Theorem 5.4. In Fg. 7, we plot the dfference of the error covarance for the two schemes. The top fgure shows that scheme one s always worse than scheme two after the transent perod wth mperfect communcaton and unlmted computaton at each sensor node, whch s captured n Theorem 5.. The bottom fgure shows that when the pacet drop rates are low, the expected value of the error covarance from scheme one s smaller than that from scheme two, whch corresponds to Theorem 5.5. In Fg. 8, we fx γ at dfferent values and plot the average of Trace(P ()) and Trace(P ()) respectvely. It shows that when γ s relatvely bg,.e., γ =.8, scheme one starts to become worse than scheme two after γ becomes smaller than.3; when γ s relatvely small,.e., γ =.4, scheme one starts to become worse than scheme two after

7 γ =.7, γ =.8 wth unlmted computaton Trace(P () P ()) γ =.7, γ =.8 wth lmted computaton Trace(P () P ()) Fg. 7. Comparson of Two Schemes Under Imperfect Communcaton γ =.8 Average of Trace(P ()) Average of Trace(P ()) γ γ =.4 Average of Trace(P ()) Average of Trace(P ()) γ Fg. 8. Comparson of Two Schemes Wth Lmted Computaton γ becomes smaller than.63. The results agree wth Theorem CONCLUSION AND FUTURE WORK In ths paper, we consder an estmaton problem over wreless sensor networs. We derve optmal estmaton for a sensor tree and study two communcaton schemes. We show that the two schemes have dstnct performance wth/wthout stochastc pacet drops and wth/wthout unlmted computaton at each sensor node. We also provde condtons on whch scheme s better n dfferent scenaros. There are a few extensons of ths wor that wll be pursued n the near future, whch ncludes: closng the loop over the networ usng the estmaton schemes we have proposed; fndng the closed form soluton to the crtcal pacet arrval rate n Theorem 5.5; conductng experments usng the MVWT wreless testbed ( Jn et al. [4]) at Caltech to demonstrate the results n the paper. REFERENCES Peter Alrsson and Anders Rantzer. Dstrbuted alman flterng usng weghted averagng. Proc. 7th Internatonal Symposum on Mathematcal Theory of Networs and Systems, Kyoto, Japan, 6. L. Bao, M. Soglund, and K. H. Johansson. Encoder decoder desgn for event-trggered feedbac control over bandlmted channels. Amercan Control Conference, Mnneapols, Mnnesota, USA, 6. Dmtr P. Bertseas and R. Gallager. Data Networs. Prentce Hall, nd edton, 99. A. Bonvento, C. Fschone, and A. Sangovann- Vncentell. Randomzed protocol stac for ubqutous networs n ndoor envronment. IEEE CCNC,, Aprl 6. Vjay Gupta, Demetr Spanos, Baba Hassb, and Rchard M. Murray. On lqg control across a stochastc pacet-droppng ln. Proceedngs of the Amercan Control Conference, Portland, Oregon, 5. Joao Hespanha, Payam Naghshtabrz, and Yonggang Xu. Networed control systems: Analyss and desgn. To appear n the Proc. of IEEE, Specal Issue on Networed Control Systems, 7. Z. Jn, S. Waydo, E. B. Wldanger, M. Lammers, H. Scholze, P. Foley, D. Held, and Rchard M. Murray. Mvwt-: The second generaton caltech mult-vehcle wreless testbed. Proceedngs of the Amercan Control Conference, Boston, MA, 4. We La and Ioanns Ch. Paschalds. Routng through nose and sleepng nodes n sensor networs: latency vs. energy trade-offs. pages Proceedngs of the 45th IEEE Conference on Decson and Control, San Dego, CA, USA, Dec 6. Juan Lu, Feng Zhao, and Dragan Petrovc. Informatondrected routng n ad hoc sensor networs. IEEE Journal on Selected Areas n Communcatons, 3(4), Aprl 5. R. Olfat-Saber. Dstrbuted alman flter wth embedded consensus flters. Proceedngs of the 44th IEEE Conference on Decson and Control and European Control Conference, 5. A. Panousopoulou, G. Nolaopoulos, A. Tzes, and J. Lygeros. Expermental evaluaton of a moble adhoc networed (manet)controlled system. Proc. 7th Internatonal Symposum on Mathematcal Theory of Networs and Systems, Kyoto, Japan, 6. B. S. Rao and H. F. Durrant-Whyte. Fully decentralsed algorthm for multsensor alman flterng. In IEE Proceedngs-Control Theory and Applcatons, Sept 99. Lng Sh, Mchael Epsten, Ahbshe Twar, and Rchard.M.Murray. Estmaton wth nformaton loss: Asymptotc analyss and error bounds. Proceedngs of the IEEE Conference on Decson and Control and European Control Conference, 5. Lng Sh, Karl Henr Johansson, and Rchard M. Murray. Change sensor topology when needed: How to effcently use system resources n control and estmaton over wreless networs. Proceedngs of the IEEE Conference on Decson and Control, New Orleans, 7. Drago Slja. Large-Scale Dynamc Systems: Stablty and Structure. North-Holland, New Yor, 978. B. Snopol, L. Schenato, M. Franceschett, K. Poolla, M. Jordan, and S. Sastry. Kalman flterng wth ntermttent observatons. IEEE Transactons on Automatc Control, 49(9): , 4. D. Spanos, R. Olfat-Saber, and Rchard M. Murray. Dstrbuted alman flterng n sensor networs wth quantfable performance. Proceedngs of the 4th Internatonal Conference on Informaton Processng n Sensor Networs, 5. E. Wtrant, P. G. Par, M. Johansson, C. Fschone, and K. H. Johansson. Predctve control over wreless multhop networs. Proceedngs of the IEEE Conference on Control Applcatons, Sngapore, 7. 6