Data Fusion Using Robust Estimator for Uncertain Noisy Systems Over Sensor Networks

Copyright © 2013 IJECCE, All right reserved

Data Fusion Using Robust Estimator for Uncertain

Noisy Systems Over Sensor Networks

Bijan Moaveni, Fatemeh Ebrahimi

Abstract – This paper, verifies the problem of designing

robust state estimator for multiple sensor networks with uncertain model and noisy measurements. Multi sensor data fusion by measurement fusion and state vector fusion structure using the Kalman filter were introduced. The standard Kalman filter requires an accurate system model. In order to get accurate information in modelling signals and sensors, the information form of robust Kalman filter by using the Krein space approach is proposed. Also, a new technique is presented by developing robust Kalman filter algorithm for fused state estimates. Simulation results demonstrate that the performance of data fusion with robust Kalman filter, as compared to data fusion with standard Kalman filter is improved.

Keywords – State Estimation, Uncertain Systems, Robust

Kalman Filter, Multi Sensor Data Fusion.

I.

I

The estimation of state variables of dynamic systems through the noisy measurements plays an important role in control engineering and signal processing. In most cases, the measured data available by sensors is incomplete, inconsistent or imprecise and contains errors, which are due to noise in the system components, the error resolution, inaccurate model used for sensors, mismatch between the static and dynamic characteristics of individual sensors and the influence of environmental conditions on the performance and characteristics of the sensors [1]. Hence, a single sensor alone cannot provide a complete assessment of actual performance of a system.

Multi sensor data fusion (MSDF) is the process of redundant and/or complementary sensors to generate comprehensive and specific information about an entity [4]. There are several papers that study on sensor networks with different objective functions including reducing the cost and increasing the reliability and accuracy [7,8,9]. Various techniques are available for MSDF, which is based on the minimization of the variance of the estimation error. Among these techniques, Kalman filtering-based approach is used for the present case, as it proves to be an efficient recursive algorithm suitable for real-time application using digital computers. It is well known that the preconditions of using the Kalman filter are the precise knowledge of dynamic model and covariance of the noises [2]. It has motivated several studies on robust Kalman filtering design for state estimation in uncertain systems [5]. Petersen and McFarlane gave a steady-state optimal guaranteed cost state estimator for linear time-invariant systems with uncertainties in the state matrix [2]. Another useful method of robust estimation is the set-valued estimation approach: given the output measurements from a linear system with noise inputs subject to a norm bound

constraint, one attempt to find the set of all possible states consistent with the uncertainty is described by an energy-type constraint or sum quadratic constraint (SQC) for a discrete-time system [1]. A commonly used approach to solve the robust Kalman filter problem is that the uncertainty is replaced with a scaling parameter, which guarantees an upper bound on the filtering error covariance [10].

In this paper, we first derive the information form of robust Kalman filter for discrete-time system with norm bounded uncertainty in the state space model by using the Krein space approach. Then we apply this filter to estimate the states, and obtaining a new method to improve the estimation in sensor networks with uncertain model.

II.

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Multi-sensor data fusion is a synergistic process, pertain to the mechanism of fusing uncertain, incomplete and sometimes conflicting data from a variety of disparate sensors in real time to extract a single incorporation of the overall system status for monitoring, control and decision-making purposes[4].

A.

Kalman filter

Consider a linear discrete-time system[11]

xk+1= A xk+ wk (1)

yk = Hkxk+ vk (2)

The Kalman filtering equation is given by Time update:

x k+1− = A x k (3)

Pk+1= A PkAT+ B Q BT (4)

Measurement update:

x k = x k−+ Kk yk− Hkx k (5)

Pk = I − KkHk Pk−1 (6)

Where the gain matrix Kk is defined by

Kk = Pk−HkT HkPk−HkT+ R

−1 ₍₇₎

Among different approaches for Kalman filter-based sensor fusion, two commonly employed techniques are presented.

B.

Measurement fusion

obtained at time step k into single augmented measurement vector as follows

y k + 1 = R−1i k + 1

I

i=1

−1

Ri−1 k + 1 yi k + 1

I

i=1

(8)

H k + 1 = Ri−1 k + 1 I

i=1

−1

R−1i k + 1 Hi k + 1 I

i=1

(9)

R k + 1 = Ri−1 (k + 1)

I

i=1

−1

(10)

The information combining is done as is shown in Fig.1.

Fig.1. Measurement fusion

In this method, the observation vector remains unchanged and the computation is reduced.

C.

State vector fusion based on Federated Kalman

filter

Data fusion method based on FKF-filter consist of a number of local Kalman filters, as are given for each set of observations, meaning that the algorithm is applied independently for each sensor and generates state estimates. Next, the state vectors by the total weighted state estimates, according to equation (11) combined by a master filter, as is shown in Fig.2. Finally, a non-biased common state estimation can be improved, provided that the local filter state estimation errors are independent [13].

x k+1 k+1,FKF =

P_{k+1 k+1,i} −1

I

i=1

−1

P_{k+1 k+1,i} −1 I

i=1

x k+1 K+1,i

(11)

The error covariance is as follows

Pk+1 k+1,FKF = Pk+1 k+1,i −1 I

i=1

−1

(12)

In this way, any number of sensors is to estimate the state. Simultaneously, it is possible to combine information.

Fig.2. FKF-State vector fusion

This approach is very suitable for the data from the outputs of non-commensurate sensors, since the data cannot be directly combined as it can in direct measurement data fusion process. Also, this method is superior to other methods of state vector fusion, reducing computation load.

III.

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Kalman filtering algorithm requires the knowledge of a perfect dynamic model for the signal generating system. Thus, the conventional Kalman filter may not be robust against modelling uncertainty. For a particular industrial process application, there might be abundant of associated sensor measurements located at different operational levels and having various accuracy and reliability specifications. One of the key issues in developing a MSDF system is the question of how can the multi-sensor measurements be fused or combine to overcome uncertainty [4].

In this section, we introduce approaches to MSDF by employing the robust estimators to deal with problem of the presence of uncertainty in sensors model.

A.

A robust Kalman information filter

Now we consider the following class of linear discrete-time uncertain process models defined onk ∈ [0, N]:

xk+1= Ak+ Fk∆kEk xk + wk (13)

yk = Hkxk+ vk (14)

Where, the uncertainty is additive uncertainty, k is the time iteration, xk is the state vector, yk is the measured

output vector, wk is the process noise, vk is the

measurement noise, and Ak, Bk, Hk, Ek and Fk are known

as real time-varying matrices with appropriate dimensions. The noises defined as zero mean Gaussian white noise processes which are reciprocally uncorrelated, and the covariance of wk is Qk, also the covariance of vk is Rk. The

initial state x0 has the covariance P0, and is uncorrelated

with wk and vk.

The matrix Δk represents the deterministic

norm-bounded time-varying uncertainty, and is said to be admissible if it satisfies the following condition:

] , 0 [

, k N

I

T k

k   

 ₍₁₅₎

The problem is solved by deriving a robust filter formula based on the Krein space Kalman filter, where the model and equations are an extension of the celebrated Kalman filter [2].

The robust Kalman information filtering (RIKF) equation is given by

Time update:

x k+1 k = Akx k k (16)

Pk+1 k = AkPk k ATk+ BkQkBkT+ FkFkT (17)

Measurement update:

x k k = Pk k P k k−1−1 x k k−1+ HkTRk−1yk (18)

P _{k k}−1 _{= P}

k k−1−1 + HkTR−1k Hk− EkTEk (19)

Copyright © 2013 IJECCE, All right reserved State estimation problem arises naturally due to the

noisy and uncertain observations obtained by different sensors. To overcome this problem in the next sections, we will expand the robust Kalman filter (RKF) equations for integrating information in sensor networks.

B.

Measurement fusion by a robust Kalman

information filter

This method, uses the Krein space-RKIF introduced in the previous section is performed. We applied RKF to estimate states in measurement fusion algorithm. In robust filter equations, the covariance of measurement noise Rt is

used instead of R(k+1) and noisy uncertain measurements, was fused as weighted additive measurements. Also, the measurement matrix Hk is replaced with Hk.

yu k + 1 = Ri−1(k + 1) I

i=1

−1

R−1_{i (k + 1)} I

i=1

yu,i(k + 1)

(20)

R k + 1 = R−1i (k + 1)

I

i=1

−1

(21)

Time update:

x k k= Pk k (Pk k−1 −1 x k k−1 + HkTRk−1yu,k) (22)

P_{k k} −1 _{= P}

k k−1 −1 + HtTRk−1Ht− EkTEk (23)

Measurement update:

x _{k k}= Pk k (Pk k−1 −1 x k k−1 + HkTRk−1yu,k) (24)

Pk k −1 = Pk k−1 −1 + HtTRk−1Ht− EkTEk (25)

C.

State vector fusion by the robust Kalman

information filter

In this algorithm, we use the Krein space-RKIFas local Kalman filters where is given by

Time update:

x k+1 k ,i= Akx k k ,i (26)

Pk+1 k = AkPk k ATk+ BkQkBkT+ FkFkT (27)

Measurement update:

Kk,i= Pk k,i HkRk−1 (28)

x k k,i= Pk k,i Pk k−1,i −1 x k k−1,i + HkT Rk,iyk,i (29)

Pk k,i −1 = Pk k−1,i −1 + HkTR−1k,iHk− EkTEk (30)

Finally, the sum of weighted state estimations combined via a master filter based on (11), (12) equations.

IV.

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In order to evaluate and compare the performance of approaches derived in section 3, we consider the following uncertain discrete-time invariant system presented in [1,2,9] with three sensors:

xk+1=

0 −0.5

1 1 +ζ xk+ wk

yk,i= 0 1 xk+ vk,i (31)

Where, i =1,2,3. The uncertain parameter, ζ, satisfies

𝜁 ≤ 0.3. The covariance of measurement noises by the

sensors are R1= 1, R2= 2.5, R3= 3and the covariance of

process noise is Q = 1. Also, the sensors assumed to besimilar, independent and assumed to have Competitive operation. This system can be written in the form of (13),(14) with

A = 0 −0.5

1 1 , Fk =

0 10

Ek = 0 0.03 , Hk = 0 1 , ∆k= 1

Rk = 1_{0 −1}0 , Hk = HkTEkT

′ ₍₃₂₎

We perform simulation in the sample period 0.5s. The

initial state vector x0= 0₀, and the initial state estimation

error covariance P0= 1 0_{0 1} are assumed.

Now we present the implementation results of Krein space robust Kalman filter and compare with Kalman filter for the mentioned example. Figures 3, 4 show the estimation improvement and the error reduction by applying robust Kalman filter.

Fig.3: Comparison the state estimation via Kalman filter and Krein space robust Kalman filter by using a single sensor (solid line: The actual state having noise and uncertainty. Dashed line: Estimated state produced by a Kalman filter. Dashed-dot line: Estimated state produced by a Krein space Robust Kalman filter)

Fig.4. Comparison the mean square error via Kalman filter and Krein space robust Kalman filter by using a single sensor.(Dashed line: MSE produced by a Kalman filter. Solid line:MSE produced by a Krein space Robust Kalman filter)

450 455 460 465 470 475 480 -4

-3 -2 -1 0 1 2 3 4 5

No. of samples

x

x - R K F x - K F x - u

0 50 100 150 200 250 300 350 400 450 500

0 1 2 3 4 5 6 7 8 9

X: 488 Y: 2.288

N0. of sample

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According the figure 4, by using the Krein space robust Kalman filter instead of Kalman filter the mean square error is reduced from 2.52 to 2.28.

In this section, at first, we performed data fusion algorithms for three sensors.

In state estimation includes uncertainty by using the Krein space robust Kalman filter instead of Kalman filter in measurement fusion, the mean square error reduces from 2.104 to 2.026as is shown in Fig.5 and the trace of covariance matrix reduces from 3.607 to 1.716 as is shown in Fig.6.

Fig.5 Comparison the mean square error via Kalman filter and Krein space robust Kalman filter by using measurement fusion technique. (Dashed line: MSE in measurement fusion by a Kalman filter. Solid line:MSE in measurement fusion by a Krein space Robust Kalman filter)

Fig.6. Comparison the trace of covariance matrix via Kalman filter and Krein space robust Kalman filter by using measurement fusion technique. (Dashed line: Trace (P) in measurement fusion by a Kalman filter. Solid line:Trace (P) in measurement fusion by a Krein space Robust Kalman filter)

In state estimation includes uncertainty by using the Krein space robust Kalman filter instead of Kalman filter in FKF state vector fusion, the mean square error reduces from 2.001 to 1.714 as is shown in Fig.7. Also, the trace of covariance matrix reduces from 4.252 to 1.065 as is shown in Fig.8.

We will have higher estimation error when using only the single sensor. Estimation error significantly is reduc by fusing the data of three sensors.

Fig.7. Comparison the mean square error via Kalman filter and Krein space robust Kalman filter by using state vector fusion technique. (Dashed line: MSE in state vector fusion by a Kalman filter. Solid line:MSE in state vector fusion by a Krein space Robust Kalman filter)

Fig.8. Comparison the trace of covariance matrix via Kalman filter and Krein space robust Kalman filter by using state vector fusion technique. (Dashed line: Trace (P) in state vector fusion by a Kalman filter. Solid line:Trace (P) in state vector fusion by a Krein space Robust Kalman filter)

The state estimation accuracy evaluated by sum of diagonal components of estimation error covariance matrix in figures 6 and 8. Also, the mean square error (MSE) curves are compared in Figures 5 and 7.

It is shown that the accuracy of the state vector fusionis higher than that of the measurement fusion with robust Kalman filtering. Therefore, from the view point of application, using this technique is suitable for this example.

V.

C

In this paper, we have proposed new design techniques for data fusion in sensor networks with noisy observations and uncertain models. These techniques allow us to effectively treat sensor networks with norm-bounded uncertainty in the state matrix. The corresponding robust estimation problem can be solved via a Krein space-robust 0 50 100 150 200 250 300 350 400 450

1 1.5 2 2.5 3

X: 489 Y: 2.026

No. of samples

M

S

E

mse - K F mse - R K F

0 50 100 150 200 250 300 350 400 450 500 1

1.5 2 2.5 3 3.5 4

X: 485 Y: 1.716

No. of samples

t

ra

ce

(

p

)

t r (p) - K F t r (p) - R K F

0 50 100 150 200 250 300 350 400 450 500 0

0.5 1 1.5 2 2.5 3 3.5

X: 491 Y: 1.714

M

S

E

No. of samples

mse - K F mse - R K F

50 100 150 200 250 300 350 400 450 500 1

1.5 2 2.5 3 3.5 4 4.5 5

X: 489 Y: 1.065

No. of samples

t

r

a

ce

(p

)

Copyright © 2013 IJECCE, All right reserved Kalman filter. Comparing with the previous works for

robust Kalman filtering, the simple structure of the proposed filter makes the filter easy to design online applications. Then, we have represented, it is mostly suitable to multi-sensory applications with undefined model and white noises. The accuracy of data fusion techniques is higher than that of each local Kalman filters. The proposed techniques give less estimation errors in comparison with data fusion via standard Kalman filtering. This property has been illustrated using an example.

R

[1] T.H. Lee, W.S. Ra, T.S. Yoon and J.B. Park ," Robust Kalman

filtering via Krein space estimation", IEE Proc.,Control Theory Appl.Vol. 151, No.1, January2004.

[2] Zhe Dong ,Zheng You, " Finite-Horizon Robust Kalman filtering

for Uncertain Discrete time-Varying Systems with Uncertain-Covariance White Noises", IEEE signal processing letters, Vol.13, No.8, August 2006.

[3] Minhas, R., Shin, V. I. and Jonathan We, Q.M.,"A decentralized

data fusion algorithm for Kalman estimates in multisensory environment ", IEEE International Conference on Automation and Logistics, China, September 2008.

[4] Salahshoor, K., Mosallaei ,M. ,Buyat ,M. , "Centralized and

decentralized process and sensor fault monitoring using data fusion based on adaptive extended Kalman filter algorithm", Measurement Vol. 41, PP.1059-1076, 2008.

[5] Adrees Ahmad, Mahbub Gani, Fuwen Yang,"Decentralized

robust Kalman filtering for uncertain stochastic systems over

heterogeneous sensor networks",Elsevier B.V., Signal

processingVol. 88, PP.1919-1928, 2008.

[6] Gandhi, M. A. "Robust Kalman filters using generalized

maximum likelihood-type estimators",[dissertation],UMI

dissertation publishing, united states,2009.

[7] I. Petersen and D. C. McFarlane. "Optimal guaranteed cost

filtering for uncertain discrete-time linear systems." Int. J Robust Nonlinear Control , vol .6,pp. 267-280, 1996.

[8] Ian R. Petersen ,Andrey V. Savkin ,"Robust Kalman filtering for

signals and systems with large uncertainties",Australia,1999.

[9] Uri Shaked, Lihua Xie and Yeng Chai Soh,"New Approach to

Robust Minimum Variance Filter Design"IEEETransactions on signal processing, Vol.49,no.11,November 2001.

[10] Minyue Fu, Carlos E. de Souza, Zhi-Quan Luo, "Finite-Horizon

Robust Kalman filter Design",IEEE Transactions on signal processing, Vol.49,no.9,September2001.

[11] E. W. Kamen , J. K. Su. "Introduction to optimal estimation ",

springer,1999.

[12] Gan, Q. and Harris, C. J.,"Comparison of Two Measurement

Fusion Methods for Kalman –Filter- Based Multi –Sensor Data Fusion", IEEE Transaction on Aerospace and Electronic systems , vol.37, January 2001,No. 1, pp.273-280.

[13] I. F. Akyildiz, S. Weilian, Y. Sanskaarasubramaniam, E.

Gayirci." A survey on sensor networks". IEEE Comm. Mag. 40, Vol. 4, 2002, pp.102-114.

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Bijan Moaveni

Received his Ph.D. degree in systems and control from K. N. Toosi University of Technology, Tehran, Iran.

Department of Railway Engineering Iran University of Science and Technology, Tehran, Iran.

E-mail: [email protected]

Fatemeh Ebrahimi

Received her M.Sc. degree in Control engineering from Department of Electrical Engineering, South-Tehran Branch, Islamic Azad University, Tehran, Iran.