Sequential Fusion Estimation for Clustered Sensor Networks

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by  Wen-An Zhang

SequentialFusionEstimationfor ClusteredSensorNetworks
Wen-An Zhanga,LingShib,
aDepartment ofAutomation, ZhejiangUniversity of Technology, Hangzhou 310023, China
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1 bElectronic and Computer Engineering, Hong KongUniversity of Science andTechnology, Clear Water Bay,Kowloon, Hong
0 Kong
2

n
a
J Abstract

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1 We consider multi-sensor fusion estimation for clustered sensor networks. Both sequential measurement fusion and state
fusion estimation methods are presented. It is shown that the proposed sequential fusion estimation methods achieve the
] same performance as the batch fusion one, but are more convenient to deal with asynchronous or delayed data since they
C are able to handle the data that is available sequentially. Moreover, the sequential measurement fusion method has lower
O computationalcomplexitythantheconventionalsequentialKalmanestimationandthemeasurementaugmentationmethods,
while the sequential state fusion method is shown to have lower computational complexity than the batch state fusion one.
h. Simulationsof a target trackingsystem are presentedtodemonstrate theeffectiveness of theproposed results.
t
a
m
Keywords: Multi-sensor information fusion; optimal estimation; sensornetworks; networkedsystems.
[

1
v 1 Introduction sensor networks have been available in the litera-
4 ture, including centralized fusion and distributed fu-
9 sion, as well as measurement fusion and state fusion
Fusion estimation for sensor networks has attracted
6
[9,10,11,12,13,14,15,16,17,18]. However, most of the re-
4 much research interest during the last decade, and has
sults are based on the idea of batch fusion, that is,
0 foundapplicationsinavarietyofareas[1,2,3,4,5].Com-
measurements or localestimates are fused all at a time
. pared with the centralized structure, the distributed
1 at the fusion instant until all of them are available at
structureismorepreferableforsensornetworksbecause
0 the estimator, as illustrated in Fig.1(a). Such a batch
ofitsreliability,robustnessandlowrequirementonnet-
7
fusion estimation may induce long computation delay,
1 workbandwidth [4,6,7]. When the number of sensorsis
thus it is not appropriate for real-time applications. A
: large, it is wasteful to embed in each sensor an estima-
v possible improvementis to adopt the idea of sequential
tor and the communication burden is high. Moreover,
i fusion, by which the measurements or local estimates
X for long-distancedeployedsensors,it may notbe possi-
are fused one by one accordingto the time order of the
ble to allocate communication channels for all sensors.
r dataarrivingattheestimator,asillustratedinFig.1(b).
a An improvement is to adopt the hierarchical structure
In this way, the fusion and the state estimation could
for distributed estimation [8], by which all the sensors
becarriedoutovertheentireestimationinterval,which
in the networkare divided into severalclusters and the
help reduce computation burdens at the estimation
sensorswithinthe sameclusterareconnectedtoaclus-
instant and ultimately reduce the computation delay.
terhead(CH)whichactsasalocalestimator.Then,the
Moreover, asynchronous or delayed data can be eas-
distributed estimation is carried out in two stages. In
ily handled. Some relevant results on sequential fusion
the first stage, the local estimator in each cluster fuses
estimation have been presented in [19] and [20]. The
the measurements from its cluster to generate a local
idea in [19] is similar to the conventional sequential
estimate. Then, the local estimators exchange and fuse
Kalman filtering, where the state estimate is updated
localestimatestoproduceafusedestimatetoeliminate
several times by sequentially fusing the various mea-
anydisagreementsamongthemselves.
surements, and both the procedures of state prediction
and measurement updating are involved over each step
Various results on multi-sensor fusion estimation for
oftheestimateupdating,whichincurssignificantmuch
computation cost. An alternative approach is to fuse
Emailaddresses: [email protected] (Wen-An all the measurements first, and then generate the state
Zhang),[email protected] (LingShi).
Preprintsubmitted toarXiv.org 18 January 2017
2 ProblemStatement
(a)batchfusionestimation (b)sequentialfusionestimation
Considerthehierarchicalfusionestimationforclustered
sensor networks as shown in Fig.2, where the plant,
t t t t
k-1 + k k-1 + + + * k whose state is to be estimated, is described by the fol-
*
lowingdiscrete-time state-spacemodel
data + fusioninstant * estimationinstant
x(k+1)=A(k)x(k)+B(k)ω(k) (1)
Fig. 1. Examples of batch fusion estimation and sequential
fusion estimation.
where x(k) ∈ ℜnx is the system state, and ω(k) ∈ ℜnω
is a zero-mean white Gaussian noise with variance Q .
ω
Asensornetworkwithmclustersisdeployedtomonitor
Clustera thestateofsystem(1).Thesetoftheclustersisdenoted
s
sa2 3 c2 cblyusΦte=r i{n1,t.h.e.,smen}s.oLrenteNtwso=rk{,1w,.h.e.r,enss}∈deΦnoatendthenstihs
s
sa1 1 s tahreencounmnbeecrteodfsteonasocrlsuisntetrhehecaludst(eCrHN)se.Tsheervnisngseanssoarns
s c1 s
a3 Clusterc estimator. The measurement equation of each sensor is
sb1 2 sb2 givenby
s
b3 Sensor ys,i(k)=C(k)x(k)+υs,i(k), i∈Ns, s∈Φ (2)
Clusterb Estimator
where y (k) ∈ ℜq, υ (k) is a zero-mean white Gaus-
s,i s,i
sian noise with variance R , and υ (k) are mutually
Fig.2.Astructureofhierarchicalfusionestimationforclus- s,i s,i
uncorrelatedandareuncorrelatedwithω(k).
tered sensornetworks.
AsshowninFig.2,thefusionestimationiscarriedoutin
estimate based on the fused measurement. This is the twostages.Atthefirststage,eachCHcollectsandfuses
novel method introduced in this paper. In [20], the se- measurementssequentiallyfromits cluster,thengener-
quentialcovarianceintersection(CI)fusionmethodwas ates a local estimate using the fused measurement. At
presented for state fusion estimation. However, the CI the second stage,eachCH collects local estimates from
fusion is not optimal since the cross-covariancesamong itself and the other CHs to produce a fused state esti-
the variouslocalestimatesareignored. mateusing the SSF methodto improveestimationper-
formance and eliminate any disagreements among the
estimators.
In this paper, both sequential measurement fusion
(SMF) estimation and state fusion estimation (SSF)
methods are developed for clustered sensor networks,
where the SMF is presented for local estimation, while 3 Designofthe SMFEstimators
the SSF is presented for state fusion estimation among
all the local estimators. The main contributions of the
ThissectionisdevotedtothedesignoftheSMFestima-
paperaresummarizedasfollows:
torsforeachcluster.ConsiderclusterN ,s∈Φ.Forno-
1) We present a design method for the SMF estima- s
tationalconvenience,thesubscriptsinthenotationswill
tors. We show that the SMF estimator is equivalent
bedroppedintheremainingofthissection,forexample,
to the conventional sequential Kalman (SK) and the
y isdenotedasy andn isdenotedasn.Denoteys as
batch measurement fusion (BMF) estimators, and is s,i i s f
the fused measurement and Y(k) = {y (k),...,y (k)}
equivalent to the one designed based on measurement 1 n
asthesetofmeasurementsforfusion.Thenitcanbeseen
augmentation (MA). We also show that the SMF es-
from Fig.1(b) that ys is obtained by sequentially fus-
timator has lower computational complexity than the f
estimatorsbasedonSK andMA. ingthenmeasurements.Thefusedmeasurementandits
2)WepresentadesignmethodfortheSSFestimators noisevarianceofthejthfusionovertheinterval(k−1,k]
with matrix weights. We further show that the SSF isdenotedbyy(j)(k)andR(j−1)(k),respectively,where
estimator is equivalent to the batch state fusion (BSF) j ∈ {1,2,...,n−1}. Denote the measurement noise of
estimators with matrix weights but has much lower y(j)(k)asυ(j)(k),theny(j)(k)=C(k)x(k)+υ(j)(k),and
computationalcomplexity. R(j−1)(k) = Cov{υ(j)(k)}. We now introduce the first
mainresultonSMF estimator.
2
Theorem 1. For the measurements in Y(k), the SMF Since υ(j−1) is uncorrelated with υj+1, it follows from
estimatorisgivenby the followingequations (10)thatR =Cov{υ }=[R−1 +R−1 ]−1,which
(j) (j) (j−1) j+1
is just the equation (3). Moreover, it follows from the
−1
R(j)(k)= R(−j1−1)(k)+Rj−+11(k) (3) equationR(−j1) = R(−j1−1)+Rj−+11 that R(j) ≤ R(j−1) and
y(j)(k)=hR(j)(k) R(−j1−1)(k)y(j−i1)(k) R{1(,j)..≤.,nR}.j+T1h,ewphriocohfilseatdhsustocoRmfsp(lke)ted≤. Ri(k), ∀ i ∈
+R−1 (hk)y (k) (4)
j+1 j+1 Whenthe fusedmeasurementys(k) isavailable,the es-
f
(cid:3) timatorisableto produceanoptimalstateestimateby
wherej =1,...,n−1,y (k)=y (k),R (k)=R (k),
(0) 1 (0) 1 using ys(k) and applying a standard Kalman filter. An
andthefusedmeasurementys(k)anditsnoisevariance f
f alternativeapproachtoobtainthefusedmeasurementis
Rfs(k) are given by yfs(k) = y(n−1)(k) and Rfs(k) = toapplytheBMFmethod,whichhasbeenpresentedin
R(n−1)(k), respectively. Moreover, one has R(j)(k) ≤ [12].IntheBMFmethod,allthemeasurementsinY(k)
R(j−1)(k)andRfs(k)≤Ri(k), i∈{1,...,n}. are fused all at a time and the fused measurement can
be obtainedby the WLSmethod, andit isgivenby
Proof. For brevity, the notation k will be dropped
in the following developments. Denote f as the se- Rb(k)=[Σn R−1(k]−1 (11)
m f i=1 i
quential measurement fusion operator, then y(j) = yb(k)=Rb(k)[Σn R−1(k)y (k)] (12)
fm{y(j−1),yj+1}.Augmenty(j−1) andyj+1 to get f f i=1 i i
y(j−1) Remark 1. If the measurement equations in (2) have
z(j) = =eCx+υ¯(j) (5) different measurement matrices Cs,i(k), i ∈ Ns, s ∈
" yj+1 # Φ, and C (k) can be decomposed as C = M C,
s,i s,i s,i
∀ i ∈ N , s ∈ Φ, where C ∈ ℜq×nx, M ∈ ℜq×q and
s s,i
where e = [I I]T and υ¯ = [υT υT ]T. Let MTR−1M is non-singular, then a similar SMF
(j) (j−1) j+1 s,i i s,i
R¯ = Cov{υ¯ }. The term z can be regarded as a i∈Ns
(j) (j) (j) ruPleasgiveninTheorem1canbeobtainedbyfollowing
measurementofCxwiththemeasurementnoiseυ¯ and
(j) somesimilarlines asin(5)-(10).
the measurement matrix e. Then by the weighted least
square(WLS)estimationmethod,aleastnormestimate
Ithasbeenshownin[12]thattheBMFisoptimalinthe
ofCxis givenby
sensethatthenoisevarianceofthefusedmeasurementis
minimalamongallthefusionruleswithmatrixweights.
zˆ =[eTR¯−1e]−1eTR¯−1z (6)
(j) (j) (j) (j) ThefollowingtheoremshowsthattheproposedSMFis
equivalentto the BMF.
Sinceυ(j−1) isrelatedto{υ1,...,υj},itisuncorrelated
withυ .Thus,onehas Theorem 2. The SMF is equivalent to the BMF, i.e.,
j+1
ys(k)=yb(k)andRs(k)=Rb(k).
f f f f
R¯(j) =diag{R(j−1),Rj+1} (7)
Proof.For j =n−1,onehas by(3)and(4)that
Substituting (5)and(7)into (6)yields
Rfs =R(n−1) =[R(−n1−2)+Rn−1]−1 (13)
zˆ(j) =[R(−j1−1)+Rj−+11]−1[R(−j1−1)y(j−1)+Rj−+11yj+1] (8) yfs =y(n−1) =R(−n1−1)[R(−n1−2)y(n−2)+Rn−1yn] (14)
Itcanbeseenfrom(8)thatzˆ(j) isa linearcombination Substituting the expressions of R(n−2) and y(n−2) into
of y(j−1) and yj+1, thus it canbe regardedas the fused (13)and(14)yields
measurement y , that is, y = zˆ , which leads to
(j) (j) (j)
equation(4).Sincey(j) isaWLSestimateofCx,itcan R(n−1)=[R(−n1−3)+Rn−−11+Rn−1]−1 (15)
sbuerewmrietntetnnoaissey.(jT)h=enCoxne+haυs(j), where υ(j) is the mea- y(n−1)=R(−n1−1)[R(−n1−3)y(n−3)+Rn−−11yn−1
+R−1y ] (16)
n n
υ =y −Cx (9)
(j) (j)
Followingthe similarproceduresas in(15)and(16)for
Substituting (5)into(6)andnotingy =zˆ ,onehas j =n−3,n−4,...,n,onefinally obtains
(j) (j)
by(9)that
Rs = Σn R−1 −1 =Rb (17)
f i=1 i f
υ(j) =[eTR¯(−j1)e]−1eTR¯(−j1)υ¯(j) (10) yfs =((cid:2)Rfs)−1 Σni(cid:3)=1Ri−1yi =yfb (18)
(cid:2) (cid:3)
3
The proofis thus completed. tionerrorvariancematrix of xˆ . Then, the SSF estima-
i
torwithmatrixweightsis givenby
Remark 2. Conventional approaches for the local es-
timation with multiple measurements either use the xˆ(j)=∆1,(j)xˆ(j−1)+∆2,(j)xˆj+1 (19)
measurementaugmentation(MA) oruse the sequential P =[eTΩ−1e]−1 (20)
Kalman (SK) estimation. It can be seen from Theo- (j) (j)
rem 1 and equation (12) that the fused measurement
obtained by the SMF or BMF has the same dimen- where j = 1,2,...,m−1, P is the estimation error
(j)
sion as each local measurement. Therefore, the SMF variance matrix of xˆ , xˆ = xˆ , P = P and the
(j) (0) 1 (0) 1
and BMF methods are more computationally efficient optimal matrix weights ∆ and ∆ are computed
1,(j) 2,(j)
than the MA method. Moreover, in the SMF or BMF as
method, the estimate is obtained using the fused mea-
surement and applying one step state update. How-
∆
ever, in the SK estimation method, the estimate is 1,(j) =Ω−1e[eTΩ−1e]−1 (21)
(j) (j)
obtained by sequentially applying a set of updates. "∆2,(j)#
Therefore, the SMF and BMF methods have lower
computation complexity than the SK method. Specif- P(j−1) P(j−1),j+1
Ω = (22)
ically, Define the computational complexity of the es- (j) " ∗ Pj+1 #
timation method as the number of multiplications and
divisions in the algorithm, and let δ , δ , δ , and e=[I I]T (23)
sm bm ma
δ denote the computational complexity of the SMF,
sk
BMF,MAandSKmethods,respectively.Then,onehas where P(j−1),j+1 = E{x˜(j−1)x˜Tj+1} is the cross covari-
δsm =(n2x+5nx+8n−7)q2+(4n2x+nx)q+2n3x+n2x, ancematrixofxˆ(j−1) andxˆj+1,andis computedas
δ =(n2+5n +3n+3)q2+(4n2+n )q+2n3+n2,
bm x x x x x x
δ =(n2+5n +8n−7)n2q2+(4n2+n )nq+2n3+n2,
ma x x x x x x j j−d+1
δ =(n2+5n +8n−7)nq2+(4n2+n )nq+n(2n3+n2).
Itskcan bexseenxthat both δ andxδ xare of magxnituxde P(j−1),j+1(k)= ∆1,(j−l)∆2,(d−1)Pd,j+1(k()24)
sm bm
O(nq2), while δma and δsk are of magnitudes O(n3q2) Xd=1 Yl=1
andO(n2q2), respectively. Pj,d(k)=[I−Kj(k)C(k)][A(k−1)Pj,d(k−1)
×AT(k−1)+B(k−1)Q BT(k−1)]
ω
×[I −K (k)C(k)]T (25)
d
4 DesignoftheSSF Estimators
where ∆ = ∆ = I and P = P . The fused
1,(0) 2,(0) (0),2 1,2
This section is devoted to the design of the state fu- stateestimateanditserrorvariancearefinallygivenby
sion estimator for each cluster. Suppose that m local xˆsf =xˆ(m−1)andPfs =P(m−1),andonehasPfs ≤Pi,i=
estimates xˆ , i = 1,2,...,m are available for fusion at
i 1,2,...,m,thatis,theprecisionoftheSSFestimatoris
the cluster N , s ∈ Φ over each estimation interval. To
s higherthaneachlocalestimator.
fusethemlocalestimates,onemayapplysomewellde-
veloped batch state fusion (BSF) methods, such as the
optimal fusion rule with matrix weights given in [11]. Proof. According to the SSF rule, xˆ(j−1) and xˆj+1 are
fusedinthejthfusion.Then,byapplyingthefusionrule
However,intheBSFmethod,allthelocalestimatesare
with matrix weights as presented in Theorem 1 of [11],
fused once at a time which usually involves computing
theoptimalfusioninthelinearminimumvariancesense
the inverse of a high dimensional matrix, and may not
is given by (19) and (20), where ∆ +∆ = I. In
besuitableforreal-timeapplications.Inwhatfollows,a 1,(j) 2,(j)
whatfollows,it will be shownthat the cross-covariance
SSFmethodwith matrixweightswillbe developed.
P(j−1),j+1 satisfies the equation(24).Note that for j =
1, one has P = P . For j ≥ 2 and t ∈ {j−1,j −
Without lossofgenerality,supposethat the m locales- (0),2 1,2
2,...,1}one has
timates arrive at the head of cluster N in time order
s
as xˆ , xˆ , ..., xˆ . In the SSF method, the cluster head
1 2 m
fusesthelocalestimatesonebyoneaccordingtothetime x˜ =x−xˆ (26)
(t) (t)
order. Denote the jth fused estimate as xˆ(j), then j ∈ xˆ(t) =∆1,(t)xˆ(t−1)+∆2,(t)xˆt+1 (27)
{1,2,...,m−1},xˆ(0) =xˆ1 andxˆ(j) =fs(xˆ(j−1),xˆj+1),
wheref isthestatefusionruletobedesigned.TheSSF
s Substituting (27) into (26) and taking the relation
with matrix weights are presented in the following the-
∆ +∆ =I intoconsiderationyields
orem. 1,(t) 2,(t)
Theorem 3. Let xˆi, i = 1,2,...,m be unbiased esti- x˜(t)=(∆1,(t)+∆2,(t))x−∆1,(t)xˆ(t−1)−∆2,(t)xˆt+1
mates of the state of system (1) and Pi be the estima- =∆1,(t)x˜(t−1)+∆2,(t)x˜t+1 (28)
4
Itfollowsfrom(28)that wherexˆ=[xˆT ··· xˆT]T,I =[I ··· I]T and
1 m o
m
P =E{x˜ x˜T }
(t),j+1 (t) j+1 | {z }
=∆1,(t)P(t−1),j+1+∆2,(t)Pt+1,j+1 (29) m−1 m−2
∆= ∆1,(m−l) ∆1,(m−l)∆2,(1) ···
"
l=1 l=1
Applying(29)recursivelyfort=j−1,j−2,...,1leads Y Y
to equation (24). Denote the fused measurement in the ∆1,(m−1)∆2,(m−2) ∆2,(m−1)
jthclusteras ys ,then ys canbe writtenas
f,j f,j (cid:3)
ys =Cx+υs , j ∈Z (30) Proof. Note that (19) is a recursive equation for com-
f,j f,j s
puting xˆ with respect to the variable j. Therefore,
(j)
whereυs isthefusedmeasurementnoise.Then,by(1), substitutingtheexpressionofxˆ(j−1) intothatofxˆ(j) for
(30)andf,tjhe standardKalmanfilter,onehas j = 1,2,...,m−1 yields xˆsf = xˆ(m−1) = ∆xˆ. By the
definitionofI ,onehas
o
x˜ (k)=x(k)−xˆ (k)
j j m m−j
=[I −K (k)C(k)][A(k−1)x˜ (k−1)
+B(k−j 1)ω(k−1)]−K (kj)υs (k) (31) ∆Io = ∆1,(m−l)∆2,(j−1) (35)
j f,j j=1 l=1
X Y
Bydefinition, the crosscovarianceis givenby
m−j
Denote Dm−j+1 = ∆1,(m−l)∆2,(j−1), j = 1,...,m.
P (k)=E{x˜ (k)x˜T(k)},j 6=d, j,d∈Z (32) l=1
j,d j d s Then,onehasby (35Q)that
Substituting (31) into (32) and taking into account the
m
relations x˜ (k − 1) ⊥ ω(k − 1), x˜ (k − 1) ⊥ υs (k),
j j f,j ∆I = D (36)
ω(k−1) ⊥ υs (k) and υs (k) ⊥ υs (k), one obtains o j
f,j f,j f,d j=1
equation(25). X
Since ∆ +∆ =I, j =1,2,...,m−1,one has
1,(j) 2,(j)
Moreover,bythe fusionrulegivenin[11],one has
D1+D2=∆1,(m−1)∆2,(m−2)+∆2,(m−1)
P(j) ≤P(j−1), P(j) ≤Pj+1 (33) =∆1,(m−1)∆2,(m−2)+I −∆1,(m−1)
Byapplying(33)recursivelyforj =1,2,...,m−1,one =I+∆1,(m−1)[∆2,(m−2)−I]
obtains Pfs = P(m−1) ≤ Pi, i ∈ Zs. The proof is thus =I−∆1,(m−1)∆1,(m−2) (37)
completed.
Then,itfollowsfrom(37)that
Remark3.Itcanbeseenfrom(22)thatthematrixΩ
(j)
hasdimension2nx×2nx,whileithasdimensionmnx× D1+D2+D3
mn intheBSF.Therefore,theSSFismuchmorecom-
x =I −∆1,(m−1)∆1,(m−2)+∆1,(m−1)∆1,(m−2)∆2,(m−3)
putational efficient than BSF. Specifically, let δ and
bs
δ denotethecomputationalcomplexityoftheBSFand =I +∆1,(m−1)∆1,(m−2)[∆2,(m−3)−I]
ss
SSFmethods,respectively.Then,onehasδbs =5n2xm2+ =I −∆1,(m−1)∆1,(m−2)∆1,(m−3) (38)
(n3 +n2)m and δ = (2n3 +22n2)m −2n3 −22n2.
x x ss x x x x
Therefore,themagnitudesofδbs andδss areO(m2)and Following the similar lines as in (37) and (38), one ob-
O(m), respectively. tains
The fused estimate xˆs given by Theorem 3 has the fol- m−1 m−1
f
lowingproperty. Dj =I− ∆1,(m−l) (39)
j=1 l=1
X Y
Theorem4.Thefusedestimatexˆs isanunbiasedesti-
f
mate ofthe systemstate,andsatisfies Since ∆ =I,one has
2,(0)
xˆs = ∆xˆ m−1 m−1
f (34) Dm = ∆1,(m−l)∆2,(0) = ∆1,(m−l) (40)
(∆Io = I l=1 l=1
Y Y
5
Then,itfollowsfrom(36),(39)and(40)that
1
E SMF
m m−1 MS0.5 BMF
R
∆I = D = D +D =I (41)
o j j m
0 5 10 15 20 25 30
Xj=1 Xj=1 time/steps
1
Sincexˆ ,i=1,2,...,mareunbiasedestimatesofx,one E SMF
has byithe fact ∆Io = I that E{x−xˆsf} = E{∆Iox− RMS0.5 SK
∆xˆ} = ∆E{Iox − xˆ} = 0, that is, E{x} = E{xˆsf}. 0 5 10 15 20 25 30
Therefore,xˆs isanunbiasedestimateofthestatex.The time/steps
f 1
proofis thus completed. E SMF
MS0.5 MA
R
Remark4.ItcanbeseenfromTheorem4thatthepro-
posedestimatexˆs isalsoalinearcombinationofallthe 0 5 10 15 20 25 30
f time/steps
localestimateswithmatrixweights.Thus,theestimator
givenin Theorem4 can be regardedas another formof
BSF estimator. However,the weighting matrices in the Fig.3.EquivalenceoftheBMFestimatorandtheSMF,SK
andMA estimator
BSFestimatorgiveninTheorem4 havemuchlowerdi-
mensions thanthose in the BSF estimatoras presented
1
in[11]. local estimator 1
0.8 local estimator 2
local estimator 3
5 Simulations SE0.6 SBSSFF eessttiimmaattoorr
M
R0.4
Consider a maneuvering target tracking system, where
the target moves in one direction, and its position and 0.2
velocity evolve according to the state-space model (1)
0
with 0 5 10 15 20 25 30
time/steps
1 h h2/2
A(k)= ,B(k)= (42) Fig. 4. Equivalence of the BSF estimator and the SSF esti-
"0 1# " h # mator
from the other two CHs and generates the fused state
where h is the sampling period. The state is x(k) = estimatesusingtheproposedBSFandSSFmethods.It
[xTp(k) xTv(k)]T,wherexp(k)andxv(k)aretheposition canbeseenfromFig.4thattheestimationperformance
andvelocityofthemaneuveringtargetattimek,respec- is improved by fusing the local estimates, and the pro-
tively. The variance of the process noise ω(k) is q = 1. posed SSF method is equivalent to the BSF method in
The position is measured by the sensors, and the mea- achievingthe same estimationprecision.
surementmatrixisC =[10].Theinitialstateofsystem
(1) is x = [1 0.5]T, and we take h = 0.5s in the sim-
0
6 Conclusions
ulation.Agroupofsensorsaredeployedtomonitorthe
target,andthesensornetworkisdividedintothreeclus-
ters,namely,N1with10sensors,N2with8sensors,and Some sequential fusion estimators have been developed
N with6sensors.ThereisaCHineachcluster,andthe in this paper for distributed estimation in clustered
3
CHcollectsmeasurementsfromitsclustertogeneratea sensor networks. It is shown that the sequential fu-
localestimate ofthe systemstate.Monte Carlosimula- sion methods have the same estimation performance
tionswillbecarriedoutandtherootmean-squareerror as the batch fusion one but have lower computational
RMSE= 1 L (xi −xˆi)2isusedtoevaluatethees- complexity than conventional approaches, such as the
L i=1 p p batch fusion estimation, sequential Kalman estimation
timationpqerforPmanceoftheestimators,whereL=1000 and that based on measurement augmentation. There-
is the number of Monte Carlo simulation runs. The es-
fore, the proposed methods are more appropriate for
timate ofthe initialstate issetasxˆ =[2 1]T.
0 real-time applications and are convenient to handle
asynchronousanddelayedinformation.
The local estimates in cluster N using the proposed
1
SMFmethod,theBMF,SKandMAmethodsareshown
inFig.3.ItcanbeseenfromFig.3thattheSMF,BMF, References
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