Distributed Estimation of Dynamic Fields over Multi-agent Networks

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by  Subhro Das

DISTRIBUTEDESTIMATIONOFDYNAMICFIELDSOVERMULTI-AGENTNETWORKS
SubhroDas†,Advisor:Jose´ M.F.Moura‡
† IBMT.J.WatsonResearchCenter,YorktownHeights,NY10598,USA
‡ ECEDepartment,CarnegieMellonUniversity,Pittsburgh,PA15213,USA
Email: [email protected],[email protected]
ABSTRACT Dynamic Averaging Field Estimation
017 Tvahriysinwgorrakndporemsefinetslddsiostvreibrumteudltia-laggoernitth/smenssoforrneetswtimoraktsi.onAonfettwimorek- (((abc)))PPPssseeeuuudddooo---ioonbbnssoeevrravvtaaittoiinoosnnss E(((abcs)))timaaPvvsaeeetrrueaad:ggoee- sootffa PPtesseeuuddoo--ionbnsoevravtaitoinosns Locaolf esstatitmeate
of sensors makes sparse and noisy local measurements of the dy-
2
namic field. Each sensor aims to obtain unbiased distributed esti- Observations F(aro)mPs neeuidgoh-binonrso vreactieoinves :estimate (sata) tRee ecsetiivme ates
n mates of the entire field with bounded mean-squared error (MSE) ((bc))PPsseeuuddoo--ostbasteer veasttiiomnast eestimate from neighbors
a based on its own local observations and its neighbors’ estimates.
J
This work develops three novel distributed estimators: Pseudo-
Fig.1:Structureofdistributedestimators:(a)PIKF,(b)DIKF,and,(c)CIKF.
0 InnovationsKalmanFilter(PIKF),DistributedInformationKalman
1 Filter (DIKF) and Consensus+Innovations Kalman Filter (CIKF). singletimescaleestimator[6]extendstotimevaryingdynamicsthe
We design the gain matrices such that the estimators achieve un- consensus+innovations distributed estimator introduced in [7] for
] biased estimates with bounded MSE under minimal assumptions parameterestimation. Here,weconsiderthesingletimescalesetup
T
onthelocalobservationandnetworkcommunicationmodels. This because it reduces the onerous communication rounds required by
I
. work establishes trade-offs between these three distributed estima- theconsensusstepinthetwotimescaleestimators.
s
torsanddemonstrateshowtheyoutperformexistingsolutions. We
c
[ validateourresultsthroughextensivenumericalevaluations.
2. PROBLEMFORMULATION

1
1. INTRODUCTION WeconsideranetworkofN sensorsobservinganunderlyingtime-
v
varying random field x ∈ RM, where i is the time index. The
0 Motivation: In the era of large-scale systems and big data, dis- i
evolutionofthefieldx is:
1 tributed estimators, yielding robust and reliable state estimates by i
7 running local parallel inference algorithms, are capable of signifi- xi+1 =Axi+vi (1)
2 cantlyreducingthelargecomputationandcommunicationloadre-
where, A ∈ RM×M is the field matrix and v ∈ RM is the
0 quiredbyoptimalcentralizedestimators,namelytheKalman-Bucy i
fieldnoise.Thenoisev iszero-meanGaussian,i.e.,v ∼N(0,V).
. filter [2]. Distributed estimators have applications in estimation, i i
1 Theinitialcondition,x ∼N(x¯ ,Σ ),isnormallydistributedwith
0 for example, of temperature, rainfall, or wind-speed over a large meanx¯ andcovariance0Σ . 0 0
7 geographical area; dynamic states of a power grid; locations of a Let0zn ∈ RM denote0theobservationofthefieldx bythenth
1 groupofcooperatingvehicles;orbeliefsinsocialnetworks. sensorattiimei.Thelinearobservationsznfollow i
: Related work: Reference [3] provides a bibliographic review of i
v different approaches to distributed Kalman filtering and also dis- zin =Hnxi+rin (2)
i
X cussestheirapplications. Weclassifythepriorworkondistributed where,H ∈RMn×M isasparseobservationmatrixandrn∈RMn
dynamicfieldestimationintotwogroups,thesecondbeingfurther n i
r is the observation noise. In practice, the observation dimen-
a sub-divided in two sub-groups: i) Two timescale: Fast commu- sions M , ∀n are much smaller than the field dimension M,
nication – slow dynamics and sensing; and ii) Single timescale: n
i.e.,M (cid:28)M.Thenoisev iswhiteGaussian,i.e.,v ∼N(0,V).
a)GossipKalmanfilters;andb)Consensus+innovationsestimators. Thenonisesequencesv ,rniandthefieldx areindeipendent. The
The two timescale distributed estimators [4] reproduce a Kalman i i 0
fieldisgloballyobservable,butnotnecessarilylocallyobservable.
filter locally at each sensor with a consensus step on the observa-
Thesensornetworkisrepresentedbyanundirected,connected
tions. The sensing and the local filter updates occur at the same
graphG =(V,E),whereVandEdenotethesetofsensorsandlocal
slow time scale of the process dynamics, while the consensus it-
communicationchannelsrespectively.Theneighborhoodofagentn
erations happen at the fast time scale of communication among
isΩ ={n}∪{l|(n,l)∈E}.TheLaplacianmatrixofGisdenoted
sensors. In single timescale estimators, sensing and communica- n
byL,whoseeigenvaluesare0=λ (L)<λ (L)≤...≤λ (L).
tion occur once at every time step; the local estimators operate at 1 2 N
the same time scale of the process dynamics. In gossip Kalman
filters [5], the sensors exchange their field estimates and their er- 3. DISTRIBUTEDESTIMATORS
rorcovariancematricesfollowingthegossipprotocol, whereasthe
Incontradistinctionwiththecentralizedsolutions, weidentifythat
Thisdissertationwork[1]hasbeensubmittedforPhDdegreeinElectricaland a key component in the distributed estimators that we develop is
ComputerEngineeringatCarnegieMellonUniversityinJune2016.Thisworkhasbeen
a global average step of a linearly transformed version of one of
supportedbyNSFgrantsCCF1513936,CCF1011903andCCF1018509. Simulations
wereruninAzurecloudsupportedbyMicrosoftAzureforResearchGrant. thethreequantities: innovations,observations,orstate. Wereferto
Algorithm:Consensus+InnovationsKalmanFilter(CIKF)[13]
MSE
 (in
 dB)
 with
 time
Input:ModelparametersA,V,H,R,G,L,x ,Σ .
0 0 0
Initialize:xn =x ,yn =Gx . -­‐1 0 5 10 15 20 25 30
(cid:98)0|−1 0 (cid:98)0|−1 0 -­‐2
Pre-compute:GainmatricesBiandKiusingAlgo.2in[13]. -­‐3
whilei≥0do -­‐4
-­‐5
Communications: -­‐6
Broadcastyn toneighborsΩ ;receive{yl }froml∈Ω . -­‐7
(cid:98)i|i−1 n (cid:98)i|i−1 n -­‐8
Observation: -­‐9
Makemeasurementzn;transformintopseudo-observationzn, Centralized CIKF DIKF
i (cid:101)i
z(cid:101)in=HnTRn−1zin, (3) Fig.2:ComparisonofMSEperformanceoftheCIKFwithCKFandDIKF.
Filterupdates:Computeynandxn andpseudo-state[13]inthecontextofdistributedfieldestimation,
(cid:98)i|i (cid:98)i|i
and, developed a Consensus+Innovations distributed dynamic av-
y(cid:98)i|ni=y(cid:98)i|ni−1+(cid:88)Bnil(cid:16)y(cid:98)i|li−1−y(cid:98)i|ni−1(cid:17)+Bnin(cid:16)z(cid:101)in−(cid:16)H(cid:101)ny(cid:98)i|ni−1+Hˇnx(cid:98)in|i−1(cid:17)(cid:17) (4) eraging algorithm as a key part of field estimation; (c) Designed
l∈Ωn the optimized filter gain matrices [13] using the Gauss-Markov
x(cid:98)in|i=x(cid:98)in|i−1+Kin(cid:16)y(cid:98)i|ni−Gx(cid:98)in|i−1(cid:17) (5) tthheeodreismtr,ibsuotetdhavtetrhseionestoimf athtieonalMgeSbEraiics Rmiicnciamtiizeeqdu,aatniodn, dfoerrivtehde
Predictionupdates:Predictyn andxn DIKF and the CIKF [12,13]; (d) Expressed the tracking capacity
(cid:98)i+1|i (cid:98)i+1|i ofthedistributedestimators[8–13]intermsofthenetworkLapla-
y(cid:98)i+n1|i =A(cid:101)y(cid:98)i|ni +Aˇx(cid:98)in|i, x(cid:98)in+1|i =Ax(cid:98)in|i. (6) cian and the observation matrices; (e) Validated the convergence
results [8–13] through numerical simulations and evaluated exper-
endwhile imentallythesensitivityoftheperformanceoftheDIKF[12]with
respecttomodelparameters,noisestatistics,andnetworkmodels.
Futurework: Thegenericnatureoftheresultsinthisthesismakes
theselinearlytransformedandnormalizedversionsofthelocalinno- them applicable in a variety of problems in distributed inference.
vations,localobservations,andstateaspseudo-innovations,pseudo- Interestingextensionsincludedevelopingadistributedestimatorthat
observations,andpseudo-state,respectively.Apriori,itisnotknown isresilienttorandomsensorlinkandnodefailures.Thisisimportant
whichofthesethreepseudo-quantitieswillyieldbetterperformance, inpractice,becausethecommunicationlinksbetweensensorsorthe
or what are the trade-offs in the computation complexity, or what sensors themselves can fail. For distributed parameter estimation,
are the communications constraints of each algorithm. Hence, we thishasbeenconsideredin[7].
developthreedistinctdistributedestimators:(i)Pseudo-Innovations
Kalman Filter (PIKF in [8–11]) using pseudo-innovations; (ii)
5. REFERENCES
DistributedInformationKalmanFilter(DIKFin[12])usingpseudo-
observations; and (iii) Consensus+Innovations Kalman Filter [1] S.Das, DistributedLinearFilteringandPredictionofTime-varying
(CIKFin[13])usingpseudo-state.Thestructureoftheseestimators RandomFields,Ph.D.thesis,CarnegieMellonUniversity,2016.
involvestwosteps–adynamicaveragingstepandafieldestimation [2] R.E.KalmanandR.S.Bucy, “Newresultsinlinearfilteringandpre-
step,asshowninFig.1.Thethreeversionsofdistributedestimators dictiontheory,”JournalofbasicEngineering,1961.
estimatetheaverageofoneoftheselocalpseudo-quantitiesthrough [3] M.S. Mahmoud and H.M. Khalid, “Distributed Kalman filtering: a
adynamicconsensusstep. Thedistributedfieldestimatorusesthis bibliographicreview,”IETControlTheoryAppl.,2013.
averageestimateinthefilteringsteptocomputethefieldestimates. [4] R.Olfati-Saber, “Kalman-consensusfilter: Optimality, stability, and
Here, inAlgorithm1westateonlytheCIKF[13], thereadersare performance,”inIEEECDC,2009.
referredto[8–12]forthePIKFandtheDIKF. [5] D.Li,S.Kar,J.M.F.Moura,H.V.Poor,andS.Cui, “DistributedKal-
manfilteringovermassivedatasets:analysisthroughlargedeviations
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ofrandomRiccatiequations,”IEEETrans.Inf.Theory,2015.
diffusesthroughthenetwork,thefielditselfevolves.Thislagcauses
[6] U.A.Khan,S.Kar,A.Jadbabaie,andJ.M.F.Moura,“Onconnectivity,
agapinMSEperformancebetweenthedistributedandcentralized
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field estimators. Numerical simulations show that the CIKF per-
[7] S.Kar,J.M.F.Moura,andK.Ramanan,“Distributedparameterestima-
forms better among the three in terms of MSE performance. The
tioninsensornetworks: Nonlinearobservationmodelsandimperfect
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CIKFimprovestheperformanceby3dBovertheDIKF,reducingby
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halfthegaptothecentralized(optimal)Kalmanfilter,whileshowing randomfields,”inAllertonConf.onComm.,Control,&Comp.,2013.
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trackingcapacity,”inAsilomarConf.onSig.,Sys.,&Comp.,2013.
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4. CONTRIBUTIONS
SignalProcessingConference,2013.
The contributions of this work are: (a) Proposed novel Consen- [11] S.DasandJ.M.F.Moura, “Distributedstateestimationinmulti-agent
networks,”inIEEEICASSP,2013.
sus + Innovations distributed estimators [8–13], PIKF, DIKF, and
CIKF of dynamic fields over multi-agent networks and proved [12] S.DasandJ.M.F.Moura, “DistributedKalmanfilteringwithdynamic
observationsconsensus,”IEEETrans.SignalProcess.,2015.
their convergence under minimal assumptions on the local obser-
vation and network communication models; (b) Introduced the [13] S.DasandJ.M.F.Moura,“Consensus+innovationsdistributedKalman
filterwithoptimizedgains,”IEEETrans.SignalProcess,2016.
relevance of pseudo-innovations [8–11], pseudo-observations [12],