Sensor Fusion in Centralized and Decentralized...

Vesa Hasu

Sensor Fusion in Centralized and Decentralized Networks

2.6.2006

2Hasu - Sensor Fusion in Centralized and Decentralized Networks TKK – Control Engineering Laboratory

Data Fusion Motivation

Sensing and measurements themselves are not always the end product of wireless sensor networks (WSN)Measurements may not be as accurate as desired

→ Data fusion for estimation and filtering is needed

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Fusion Topology Basics

Two main categories: centralized (star topology) and decentralized fusion (mesh or clusterized topology)Data fusion in WSN are examined in this presentation through:– Centralized Kalman filter (KF)

Some theoryKF in a weather station network

– Decentralized Kalman filter (DKF)Some theoryAn extension to DKFDKF implementation issues in sensor networks

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Centralized Data Fusion: Kalman Filter

Centralized KF application to WSN requires star topology – all measurements are sent to the fusion center– A good example on centralized data fusion

Kalman filter is a classic linear filter offering optimal linear estimation with certain assumptions on e.g. noise properties

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Centralized Kalman Filter

Kalman filter can be applied to a state space model:

Kalman filter noise assumptions (independent white Gaussian noises):

{ } ˆ(0) (0)E =X X

( )( ){ }ˆ ˆ(0) (0) (0) (0) (0)T

E − − =X X X X P

{ }( ) ( ) , ,TE k j k j= ∀w v 0

( ) (0, ( ))( ) (0, ( ))k N kk N k

w Qv R

∼∼

( 1) ( ) ( ) ( )( ) ( ) ( )

k k k kk k k+ = +⎧

⎨ = +⎩

X Φ X wx HX v

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Centralized Kalman Filter

Kalman filter equations– Prediction:

– Update:

ˆ ˆ( 1| ) ( ) ( | )k k k k k+ =X Φ X

( 1 | ) ( ) ( | ) ( ) ( )Tk k k k k k k+ = +P Φ P Φ Q

( ) 1( 1) ( 1 | ) ( 1 | ) ( )T Tk k k k k k

−+ = + + +K P H HP H R

( )( 1| 1) ( 1) ( 1| )k k k k k+ + = − + +P I K H P

( )ˆ ˆ ˆ( 1 | 1) ( 1 | ) ( 1) ( 1) ( 1 | )k k k k k k k k+ + = + + + + − +X X K x HX

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Kalman Filter Example in Wireless Sensor Network

Wireless sensor network: Helsinki Testbed– New type weather transmitter stations (Vaisala

WXT510) around Helsinki area

– Measurements: wind speed and direction, liquid precipitation, barometric pressure, temperature and relative humidity

– Measurements are sent through GPRS

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Kalman Filter Example in Wireless Sensor Network

Measurements are sent to a central database, where the centralized Kalman estimation can be doneIn this case, Kalman filter can be used e.g.estimation of missing values based on the neighbouring stations

5560 5580 5600 5620 5640 5660 5680 5700 5720 5740 5760 5780

14

16

18

20

22

24

26

28

Iteration (5 min)

T (o C

)

Suvisaaristo

Black = measured temperature

Red = measurement missing in estimation

Cyan = estimated temperature

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Decentralized Data Fusion

Decentralized in data fusion means doing the fusion distributedly in many equivalent nodes– Mesh-type communication

The application of decentralized data fusion to WSN brings up new problems, such as– Out-of-sequence-measurements problem (OOSM)– Application in clusterized network, when clusters

change

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Decentralized Data Fusion: Decentralized Kalman Filter

DKF is derived from the information filter form of the Kalman filterTotally decentralized and mathematically equivalent version to centralized KFMathematical starting point: partitioning the measurement equation of state model into i blocks, i.e.

Information filter basics for ith local model: – Information state vector– Information matrix

( 1) ( ) ( ) ( ) ( )k k k k k+ = +x F x G w

( ) ( ) ( ) ( )i i ik k k k= +z H x v

System equation – the same for all nodes i

Partitioned measurement equation for node i

1ˆ ˆ( | ) ( | ) ( | )i ik l k l k l−y P x1( | ) ( | )i ik l k l−Y P

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Decentralized Kalman Filter

Local filter update for node i

Communication between nodes – every node sends its variance error information Ii and state error information ii to all other nodesGlobal update– Information state vector– Information matrix

Estimated state:

( ) 11( 1 | ) ( 1) ( | ) ( 1) ( 1) ( 1) ( 1)T Ti i i i i i ik k k k k k k k k

−−+ = + + + + + +Y F Y F G Q G1ˆ ˆ( 1 | ) ( 1 | ) ( ) ( | ) ( | )i i i i ik k k k k k k k k−+ = +y Y F Y y

1( 1) ( 1) ( 1) ( 1)Ti i i ik k k k−+ = + + +I H R H 1( 1) ( 1) ( 1) ( 1)T

i i i ik k k k−+ = + + +i H R z

( 1| 1) ( 1| ) ( 1)i i ik k k k k+ + = + + +Y Y I ˆ( 1| 1) ( 1| ) ( 1)i i ik k k k k+ + = + + +y y i

1

ˆ ˆ( 1 | 1) ( 1| ) ( )m

i i jj

k k k k k=

+ + = + +∑y y i

1( 1 | 1) ( 1 | ) ( )

m

i i jj

k k k k k=

+ + = + +∑Y Y I1ˆ ˆ( 1| 1) ( 1| 1) ( 1| 1)i i ik k k k k k−+ + = + + + +x Y y

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Decentralized Kalman Filter

Global update requires communication between nodes – every node sends its variance error information Ii and state error information ii to all other nodes

→ Problem: every node sends one vector and one matrix to all other nodes – requires a lot of communication capacity, especially if the number of nodes is large

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An Extension to Decentralized Kalman Filter

An extension towards the best linear unbiased estimation (BLUE)Enables the process noise to be arbitrarily coloured, i.e.

The update equations in each node:

( )( )

cov ( ), ( ) ( , ), , ,

and cov (0), ( ) ( ),

i j i j i j

i i i

= ∀ ∈

= ∀ ∈

w w Q

x w B

( ( ) ) 11( | 1) ( 1) ( 1| 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) TT Ti i i ik k k k k k k k k k k k k

−−− = − − − − + − − − + − − + − −Y F Y F G Q G F Y F Y1| 1( 1) ( 1)k kk k− −− = −Ψ Ψ

0|0 ( 1) ( 1) (0)Tk k− = −B GΨ| 1 1| 1( 1) ( 1) ( 1) ( 1) ( 1, 1) ( 1)i i i i Tk i k i i k i− − −− = − − + − − − −F G Q GΨ Ψ

( )| 1 | 11( 1) ( | ) ( ) ( 1)Ni i i i

i jjk i i i k− −=

− = − −∑I Y IΨ Ψ

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Decentralized Kalman Filter Implementation Issues

If the number of nodes is large, the communcation issue prevents direct implementation in WSN:– Possible solution: clusterization– Local estimate and global estimate communication

done only in cluster heads

Other issues:– Out-of-sequence-measurements (OOSM) problem

due to wireless link– Changing cluster configuration when nodes are

moving

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Out-of-Sequence-Measurements Problem

Wireless radio link is more likely to cause random delays or lose data completely than traditional wired connections – hence the treatment of OOSM is more important than everIn DKF, the optimal solution to OOSM problem is well-known, but it requires a lot of system resources: memory and computational capacityMeanwhile, WSN nodes are desired to keep as simple as possible – hence suboptimal heuristic approaches must be considered

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Out-of-Sequence-Measurements Problem in DKF

The optimal solution to OOSM problem in DKF:

where

The above iteration must be done starting from the delayed measurement

( )( ) ( )

1

11

( 1) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

I Y T Y T Y

TT

k k k k k k k k k k

k k k k k k k k

+ −

−−

+ = + +

− +

I M M G G M G Q G M

M G G M G Q M G

( )( )( ) ( )

-1

1

ˆ( 1) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( | )

ˆ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( | ) ( )

T Y T T

T I T T

k k k k k k k k k k

k k k k k k k k k k k

+ − −

− −

+ = +

− + +

i F i M G G F y

M G G M G G F y i

Σ

Σ

1( ) ( ) ( | ) ( )Y Tk k k k k− −=M F Y F1( ) ( ) ( ) ( )I Tk k k k− −=M F I F

( ) ( ) ( )Y Ik k k= +M M M

The delayed

measurements

are i and I

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Suboptimal Out-Of-Sequence-Measurements Solutions in DKF

Simple and suboptimal OOSM solutions:– Use only the information gotten in the current time-step,

and ignore the delayed information.– Use the latest information gotten from each node, and

ignore the possible delays.– Use the optimal back propagation of old measurements,

but only if the new information is at most n steps delayed with n being a small positive integer.

Properties:– the n-step truncated iterative propagation has

substantially better performance than the others– the performance difference decreases, while the average

delay increases – the trade-off between accuracy and resources depends

on the statistical properties of variable delay

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Clustered Sensor Network: Mobility Problem for Data Fusion

Clusterization of network is suggested for reducing the communication burden in large sensor networksPossible configurations:– All cluster heads communicate their measurements

to the other cluster heads – making global estimates

– Cluster heads are satisfied to local estimates, using just data from own cluster

The second configuration leads to cluster change problem in networks with moving sensors– In KF: How to update error covariances in local KFs,

if clusterization changes?

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An Example: Cluster Change Problem in Kalman Filter

The state error covariance matrix P is required for KFThe biggest problem: if clusters are changed drastically, covariances between measurements should not be lostTrade-off: memory and communication requirements during cluster change for a better filter performance

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Conclusions on Decentralized Kalman Filter in Wireless Sensor Networks

Both of the OOSM and changing cluster problems have the same characteristic – The need for more accurate handling of OOSM and

cluster change problems are dependent on the filter accuracy requirements

The lighter protocols must be used, if the communication is the restricting bottleneck in the systemIf the wireless network size is not large and the filter must operate accurately, the use of accurate techniques is justified

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Conclusions on Data Fusion in Wireless Sensor Networks

Wireless sensor network needs not to mean “only for decentralized fusion”Data fusion must often (or always) be tailor-made according to the caseDifferent network configurations bring up different problemsTrade-offs between network resources and fusion accuracy have to considered carefully

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