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Distributed State-Estimation Using Quantized Measurement Data from

Wireless Sensor Networks

Li Chai with Bocheng Hu

Professor College of Information Science and EngineeringWuhan University of Science and TechnologyWuhan, 430081, ChinaEmail: [email protected]

www.info.wust.edu.cn

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Outline

Introduction of WUST and College of ISE Motivation and related works Problem statements State estimator design Simulation Conclusion


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Introduction of WUST and College of ISE

LocationWuhan, besides the Yangtze river and very near to Three Gorges Dam

20 colleges, about 1,500s academic staff Feature: tight link with metallurgical company

(Wuhan Iron & Steel Co., Ltd, Panzhihua Iron & Steel Co., Ltd, Handan Iron & Steel Co., Ltd, Baoshan Iron & Steel Co., Ltd)


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Introduction of WUST and College of ISE

College of Information Science and Engineering 75 Academic staff including 16 professors, 15 AP and 8

professional engineer Two Departments:

Dept. of Automatic Control, Dept. of Electrical Engineering About 200 PG students and 1,200 UG

Feature: metallurgical automationEngineering Research Center for Metallurgical Automation and Measurement Technology, Ministry of Education, China


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Motivation and related works

A typical sensor network consists of a large number of nodes deployed in an environment being sensed and/or controlled.

The sensors collaborate to perform certain high level task: detection, estimation …

The sensors’ dynamic range, resolution, power and wireless communication capability can be severely limited.

Local data quantization/compression is not only a necessity, but also an integral part of the design of sensor networks.


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Sensor network applications– Environmental monitoring – Habitat monitoring – Acoustic detection – Seismic Detection – Military surveillance – Inventory tracking – Medical monitoring – Smart spaces – Process Monitoring


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The highly decentralized network architecture and severely limited communication constraints presents significant challenges in the design of signal processing algorithms.

In this talk, we will focus on a general state estimation problem

Will not consider Details of communication protocol / network topology Channel fading and uncertainty Location and routing issues


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Static decentralized estimation problemXiao and Luo (2005, 2006) and Riberiro and Giannakis (2006)


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Static decentralized estimation problem

Methods to design local message functions and final fusion function

Methods of estimation if one-bit sensor is assumed.

Analysis of the MSE.

Tradeoff between network size K and MSE under bandwidth constraint.

)( kk xm).,,( 1 Kmmf


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Problem statements

Dynamic decentralized estimation

1S 2S NS

Fusion Center

1( )y k 2 ( )y k ( )Ny k

1m 2m Nm

)()(

)()()1(

kCxky

kkAxkx


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Problem statements

In the figure


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Problem statements

To design the state estimator

such that is “close” to x(k).

Here, “close” means is small, where

0

1

)0(ˆ

))(,),(()(ˆ

)(ˆ)(ˆ)1(ˆ

xx

kmkmgky

kyBkxAkx

N

ff

)(ˆ kx

)()(ˆ)( kxkxke

pke )(


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Problem statements

Power spectral density

where

Power norm of the error is defined as

)]()([)( * kekeERe

1

0

*2

)}({)}0({

)]()([)(

dffStraceRtrace

kekeEke

ee

p

fjee eRfS 2)()(


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State estimator design

The augment system

)1(ˆ

)1()(

)(

)(

0

0

)(ˆ

)(0

)1(ˆ

)1(

kx

kxIIke

k

k

B

I

kx

kx

ACB

A

kx

kx

nn

f

n

ff

G

e


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The power norm of error

An upper bound

The above bound is tight in the sense that it can be achieved if is arbitrary.

1

0

2*2

1

0

2*22

)()()(

)()()()(

dfeGfSeGtrace

dfeGfSeGtraceke

fje

fje

fjeww

fjewp

222

2

2)(

peewpGGke


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To design the state estimator

such that is minimized.

0

1

)0(ˆ

))(,),(()(ˆ

)(ˆ)(ˆ)1(ˆ

xx

kmkmgky

kyBkxAkx

N

ff

222

2 peew GG


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Step 1, find g, and upper bound of

Step 2, find such that is

minimized.

Remark: Step 2 is a typical mixed optimization filtering problem, for which various efficient algorithms exist.

ff BA ,

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2))(),(( kmkmgyE Np

22

2 eew GG


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Numerical example

Consider the following LTI system

Let

0 0.5( 1) ( ) ( )

1 1

( ) 100 10 ( ) ( ), 1,...,i i

x k x k k

y k x k v k i N

0.4233 0.4457 0.0044,

0.9394 0.9851 0.0003f fA B

TxN 32,100,2 0


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Conclusion

Distributed state estimator is designed.

The power norm of the error is minimal in worst-case.

The idea applies to other cases, such as different types of sensors are used.


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Basic multirate elements in digital signal processing

M-fold decimator

M yD[n]x[n] ][][ MnxnyD

-2

-1 0 1 2 3 n

0 1 n

x[n]

yD[n]-1

M=2

Multirate signal processing


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L-fold expander

n40 1 2 3 865 7

x[n]

n0 1 2

yE[n]

3 4 5 6 7 8

Vaidyanathan 93

Multirate signal processing


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Multirate Signal Processing in WSNs

(a) Direct high sampling rate measurement x(n)

(b) Low sampling rate measurements vi(n)

(c) Relation between x(n) and vi(n)


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To estimate the power spectral density of x(n) using statistics

of the low-rate observable signals vi(n).

O. S. Jahromi, B. A. Francis, and R. H. Kwong, Relative information of multi-rate sensors,

Information Fusion, 5, pp. 119-129, 2004.


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Our research: Is it possible to achieve other goals using low-rate sampling

data? If yes, how to design suitable algorithms and how to evaluate those algorithms?

How to deal with quantization and channel uncertainty? Does the dual-rate assumption make sense? For arbitrary

sampling-rate data, what shall we do? Key

Distributed (multirate) signal processing


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Thank you!

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