Multiple Bits Distributed Moving Horizon State Estimation for Wireless Sensor Networks. Ji an Luo

Transcription

1 Multiple Bits Distributed Moving Horizon State Estimation for Wireless Sensor Networks Ji an Luo

2 Outline Background Problem Statement Main Results Simulation Study Conclusion

3 Background Wireless sensor network (WSN) could be used to perform advanced signal processing tasks such as distributed detection and estimation. Environment monitoring and target tracking. Under bandwidth constraints quantization, only quantized observations are communicated. Dynamic systems state estimation.

4 Deterministic parameter estimators Ribeiro and Giannakis etc. studied problems of maximum liklihood estimation under various noise assumption and signal and noise ratio (SNR) condition. Xiao and Luo etc. designed decentralized linear estimation schemes that do not require the knowledge of noise pdf, the estimator can be used in large scale WSN.

5 Distributed estimation framework Consider a generic distributed estimation problem using a WSN with an fusion center (FC) x = () + w, k = 1, L, K k k k : p where k is generally a nonlinear function and the noise terms wk, k = 1, L, Kare zero-mean independent random variables.

6 Distributed estimation setup.

7 If there is no quantization Consider the simple signal model: x = + w k = 1, L, K k The best linear unbiased estimator (BLUE): Mean-square error (MSE): ( ) 2 K ˆ 2 E BLUE 1 k k K K 2 2 BLUE = xk k k k= 1 k= 1 ˆ 1 { } = k = 1 K 2 2 Especially, when k =, ˆ BLUE = xk K, whose k = 1 MSE is known to be 2 K. 1

8 Example -estimator Consider a set of K distributed sensors with observations Let noise be uniform over U, U. k Local message functions are binary. m + where =. S k x = + w k = 1, L, K k w [ ] k ( x ) k k 1, = 0, x x k k S k S k

9 An -estimator of : ˆ = ( m,, 1 L m ) with K ( ˆ ) E We have P( mk = 1) =, P( mk = 0) =, 2 U + 2U U + E( mk ) =, k = 1, L, K 2U U 2U

10 Choose the final fusion function as K ˆ 2U = ( m1, L, mk) = U + m K k = 1 E ˆ =, ( ˆ ) 2 2 E U K () 2 Requiring a total of K = U sensors to compute an -estimator for k

11 Example : A decentralized estimation scheme (DES) for the known sensor variances case Assume that fusion center knows sensor noise variances. Let and write where and are the integer and decimal parts of. Construct sensor k s message function as where is a 0 1 binary random variable with

12 The final estimator of at the fusion center is: is an unbiased estimator of, and the MSE

13 Moving Horizon State Estimation Consider a joint density over N states: * * x ( n N + 1 ), L, x ( n) = arg max p( x( n N + 1 ), L, x() n y() 0, L, y() n ) { } n { xm ( )} m= n N+ 1 Equivalent optimization: n ( ) ( ) ( ),min n xn N { wm Q R ( )} m= n N m= n N subject to model: ( w m + v m ) + n N x( n N) ( ) () ( + 1) = ( ) + () = ( ()) + () (), (), () x n f x n w n y n h x n v n wn Wvn Vxn X ( ) prior information arrival cost

14 What can moving horizon state estimation do? Linear model nonlinear model extended Gaussian noise Kalman Gaussian noise Kalman Stability Linear model nonlinear model General noise MHE general noise MHE Inequality constraints inequality constraints stability stability online solution to quadratic/nonlinear program

15 Problem Statement Let us consider a WSN with a fusion center distributed sensors deployed with the objective of tracking a real random vector p 1 x() n

16 The underlying system and sensor nodes observing processes can be modeled as () ()( ) () x n = n x n 1 + w n n = 0,1,2, L ( ) ( ) ( ) ( ) y n, k = C n, k x n + v n, k k = 1, L, K where,,,,,,,,

17 Assumption This communication takes place over the shared wireless channel that we will assume can afford transmission of a single packet per time slot. Leading to a one-to-one correspondence between time n and sensor index t and allowing us to drop the sensor index t. Which sensor is active at time n depends on the underlying scheduling algorithm.

18 Quantization Observation model: y() n = C()() n x n + v() n Define a set of thresholds { i, i }, 1 y() n > i bi () n = 0 y() n = i i 1 Each is a Bernoulli random variable with parameter i( ()) { i() 1} ( i ()()) ( i ()) where q g n = r b n = = F C n x n = F g n i i

19 Sensor obtains total binary observation at time n: Without loss of generality, we will assume that, when, can only take on realizations when,

20 Main results If there is no quantization, we can obtain the estimate of state by Methods Batch filter Kalman filter Moving horizon estimate Particle filter etc.

21 State estimation using quantized observations Consider a joint density over N states: Approximate optimization: where Online solution to nonlinear program

22 If, then, the thresholds are, the binary observations are Approximate optimization is :

23 Solution steps of moving horizon state estimation for WSN Initialization At time n, by using state transfer equation, obtain. Only reserve as final estimate at time. At time n+1, obtain Only reserve as final estimate at time.

24 Matlab toolbox Unconstrained nonlinear programming halver method Newton method Choose functions of MATLAB optimization tool box One variable fminbnd, fminsearch, fminunc Multi-variable fminsearch, fminunc

25 Constrained nonlinear programming penalty function method Choose functions of MATLAB optimization tool box Multi-variable fmincon

26 The application of Multi-bit DMHE in target tracking Consider sensors randomly and uniformly deployed in a square region of 100*100 meters and suppose that sensor positions are known New observation equation obtained by DES method with distribution,

27 Suppose only one sensor node works at a time slot, then the observations are given by

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30 Conclusion An important problem of wireless sensor network signal processing is that bandwidth is limited, quantization of observations is necessary. A dynamic state estimation problem in the context of WSN has been considered. Develop a moving horizon state estimation method based on several quantized data for WSN.

31 Thanks!