Robust HI. Decomposed-Composed (D-C) StateFusion Estimation for Multisensor System with

Uncertain Correlated Measurement Covariance

Xue-bo JIN 1, You-xian SUN 2

' College of Informatics and Electronics, Zhejiang Sci-Tech University, Hangzhou, 310018, China2 Institute ofModem Control Engineering, Zhejiang University, Hangzhou, 310027, China

E-MAIL: lysjxbgmail.hz.zj.cn

Abstract-It's very difficult to obtain the exact covariance matrixof correlated measurement noise in the practical multisensorsystem. This paper develops the robust filtering by decomposingthe measurement system to two parts, designing filtersrespectively to one part of the real state and composing the twoestimates to obtain the final estimate of the real state. Thenumerical example has shown that the robustHX decomposed-composed (D-C) state fusion methodoutperforms the general robust filter for this special kind ofuncertain system.

I. INTRODUCTION

In the practice, lots of signals are needed to constitute thecontrollable loop of automatic control. However, somerequired states can not be online-measured as to the limit ofsensor technique.Each kind of sensor has its sphere of application and exact

degree. So it's very difficult to know exactly and completelythe true conditions of the installation by single sensor. In thepractice, many sensors are employed simultaneously to theindustrial processing in order to meet someprecise-production-demands. Be subjected to the restraint ofsensor technique and the manufacturing cost, people oftenchoose the same sensor to obtain redundancy messages ofsignals, by which the more exact estimation can be obtained[I1 ] .However, as to the usage of same sensor, a close distance of

different sensors and an jam or uncertain source inmeasurement environment, the measurement noises fromdifferent sensors are ordinarily correlated. Moreover, theyoften has the same correlation covariance[ 2 ] .

The studies of [3,4] have shown that the performance ofestimation results will decease greatly if the correlation can'tbe correctly handled. [3, 4] use similarity transformation todecouple the correlated noise covariance and develop theoptimal state fusion estimation algorithm. But it is seemed socomplex because they didn't consider the peculiarity ofmultisensor system used for the production process conditionmonitoring, i.e., people often use the same sensor.

This paper studies a practical state monitoring system, inwhich the same sensors are employed. The multisensor system

has the same measurement matrix, the same covariance ofmeasurement noise and the same correlation covariance.

Ii. MODEL TRANSFORM AND D-C METHOD

A state -space model of the formx(k + 1) = A(k)x(k) + B(k)w(k)

yi (k) = Ci (k)x(k) + vi (k)(2. 1 a)(2.1b)

is considered, where x(k) ER' is the state of the system to beestimated and whose initial mean and covariance are known,Elx} =xo and E{(x(O)- xo) (x(O)- xo)T} =PO, w(k) ERP and vi(k)ER' are uncorrelated white noise with zero mean andindependent of the initial state x(O), the covariance of w(k) isQ and yi(k) ERm is the measurement vector of the ith sensor,i=l, 2, ...N.The measurement noises between sensors are correlated,

with the uncertain covariance matrixR(k) = E T(V (j)VT (j) ... VT (j)) (VT (j)VT (j) ... VT (j))(2.2)

Let the measurement vector yi be combined into N blocksy(k)=[yT(k),y2T(k),.. .,yjT(k)]T, then the measurement functionat fusion center is:

y(k) = C(k)x(k) + v(k)where C(k) [CT (k),CT (k),..., CT (k)]T

v(k) [vT (k), VT (k), . , VT (k)]T.

(2.3)and

Let R = DDT and ;(k) [WT(k) VT(k)4 where vi(k) is

Gaussian white noise with zero mean and unit covariance,

then the system function (2. la) and the measurement function

(2.3) can be rewrited as the following systemx(k + 1) = Ax(k) + B ((k)

y(k) = Cx(k) + D ;(k)(2.4a)

(2.4b)where B = [B Ol D = [O D].

Choose C and R DDT , which can constitute aPei-Radman system (Refer to [5] about the definition of thePei-Radman system), and denote C andD as

0-7803-9484-4/05/$20.00 ©2005 IEEE 1354

CC +AC

D D +ADwhere AD is a uncertain matrix and AD is a convexcombination of the vertex matrices D1, D2, ... Dk.We decompose the measurement system to the following

two measurement functionsy'(k) = Cx(k) + Db(k) (2.5a)

y2(k) = ACx(k) + AD;(k) (2.5b)Obviously, the relation between (2.4b), (2.5a) and (2.5b) is

given byy(k) = y'(k) + y2(k) (2.5c)

The D-C method developed here is to design two filtersFt and F2 based on the measurementoutput y'(k) and y2(k) , respectively, such that

<k+ = A 1 + B I (2.6a)

xk+1 = Af2' + Bf2 k (2.6b)

to estimate xl (k) and x2 (k) respectively, where

x(k)= x'(k) +x2(k), then Xk xi +Xj2 is the estimate of

the real state xk.

III. ROBUST H, D-C FusioN ESTIMATIoN ALGORITHM

A. The Design ofF1

Let xl (k) =x(k) and we can obtain x2(k)_ 0. Based onthe discussion in the previous section, we can conclude thatthe design of Flis a filter design problem for a Pei-Radmansystem. Applying the Pei-Radman fusion algorithm in [5] , wedevelop the following steady filter algorithm without proof.

Theorem 1 Given the Pei-Radman multisensor system (2.4a)and (2.5a). If there exists a positive-define solution P to theARE

PApAT - APCT(CPCT +R)- CPAT +BQBTThe filter Ft is given by

Afl = A - KCwhere K PCTS-l S= pCT +R.

Bf1 = K

B. The Design ofF2

For the subsystem (2.4a) and (2.5b), we define

T = B], where T varies with a fixed ploytopic of

m k k imatrix, i.e., T c Ya T,ai 20, Y a = I; Tl T̂2^ Tkare given vertex system:

K A BJT, = AC DI ' T2 A B1. A B]

LAC D2 LAC Dkj

In other words, T is a convex combination of the vertexmatrices TI 1,T2 ... Tk. Connecting the filter given by (2.6b)to (2.4a) and (2.5b), and considering x2(k) _0 yield theclosed-loop uncertain ploytopic system

x-(k + 1) = Ax-(k) + h,;(k) (a 1 I

e(k) = Cx-(k)

where x(k + 1) = xL(k +1)

13I. 1)

!Bf2AC Af21

B= B D C=[O -I]!BJ2ADI'U o -]The problem design F2 can be stated as follows:Find a robust Hr fusion filter of the form given by (2.6b)

that guarantees the following three conditions:1) The closed-loop matrix A is stable.2) The closed-loop transfer function matrix from ;(k) to

e(k) is bounded by|IT(z)II <

3) There exists a symmetric nonnegative definite matrixH such that

lim E{eT (k)e(k)} < Trace(H)k->co

(3.2)

for all admissible uncertainties, where E(.) denotes theexpectation.Theorem 2. Given y > 0, there exists a robust Hr fusionfilter F2 if and only if a symmetric matrix PX > 0 and amatrix H, such as

° y2I *

~A Pj3 * > 0 (3.3)PA PB P

LC 0 0 I

BTPB<H'IC

(3.4)

proof By applying the bound real lemma[6] , the condition

|IT(z)II < Y just as in the nominal case, can be shown to be

equivalent to the follow3P, > O, s.t. ATpA - P + ATPB[y2 - BTP B]-1

xBTPooA+CTC <O and 2/BTPB>By using the Schur complement formula, the following

block matrix inequality can be obtainedP -C Tc O AT

- 2I h > O

Then the LMI (3.3) is obtained by pre- and post-multiplying diag[I,I,P ] and using the Schur complementformula again.We know the estimation covariance

1355

satisfies limE{eT(k)e(k)} = Trace(BTPB) , where P > 0 is

the solution of the following Lyapunov function'~Tp' p TATPA-P+C C=CO (3.6)

While considering (3.5), we can obtain

ATPAP +CTC.<0 (3.7)Then by (3.7) and (3.6) and the fact A is stable, we

haveP<P_, i.e.,

lim E{eT (k)e(k)} = Trace(BTPB)

< Trace(BTPJ3) = Trace(H)Theorem 3 Given y > 0, there exists a robust Hr fusionfilter F2 if and only if the symmetric and positive definitematrices X, R and matrices H, S, T,Z, such as

R * * * * *

R X * * * *

0 0 y2I > 0 (3.8)RA RA RB R * *

XA+ZAC+S XA+ZAC XB+ZD. R X *

-T 0 0 0 0 I

H * *1RB R * .0 (3.9)

XB + ZD. R X

LI Y]>0 (3.10)

Then the filter F2 is given by

Af2 M-'SR-'NT, B2 M-1T (3.11)

N R-1TT M (I -XY)N-Tand the estimation covariance satisfied

lim El [X2 (k)]T x2 (k)} < Trace(H) (3.12)

Proof Partition PX and its inverse as

co -MT U-.0

NTVwhere X, Y E R"' and U, V e Rnfxnf are symmetric and

positive definite matrices.

Multiplying the first row of PX by the first column of

P'1 reveals that XY + MVNT= I and taking into account the

partition of PX and that of its inverse we get

y-I = X - MU-lMT. The key observation is that with n = nf

and for any given symmetric and positive definite matrices

such that X > Y-1 and M is nonsingular then N is

nonsingular too. Therefore we can obtain (3.10) by applying

schur complement.Introduce the following matrices JT =diag[JT, I, JT,I],

J=diag[J,I,J,I], where J= L T 0j. By substituting

the partitioned matrix PX, together with A, B, C, and pre-

and post-multiplying (3.3) by JT and J, the nonlinearmatrix inequality can be described as follows:

y * * * * *

I X * * * *

0 0 * ** >0AY A B Y**

XAY+MBf2ACY+MAr2NT XA + MBf2AC XB + MB2AD I X-NT 0 0 0 0 I

Then LMI (3.8) is obtained by pre- and post-multiplying theupper LMI by diag[Y-1, I,I,Y-1, I,I] and defining the changesof variables

Y =R, MBf =Z, MA4fNTy-I =S NTy-I T (3.13)Then we consider the estimation covariance. Applying the

Schur complement formula to (3.4) and pre- andpost-multiplying diag[I,PJ , we can obtain

H hTp1

Substituting B and P , and pre- and post-multiplying

diag[I, JT] and diag[I, J], we haveH eeB Y {j20 (3.14)

XB + MBfAD I X_Pre- and post-multiplying the upper LMI by diag[I, Y-1I]

and applying the changes of (3.13), (3.9) can be obtained. Bythe Theorem 3.2 and the fact x2 (k) _ 0, we can conclude theestimation covariance satisfies (3.12).

IV. ILLUSTRATIVE EXAMPLE

In this section, a numerical example is given todemonstrate the properties of the D-C algorithm. Considerthe following multisensor system

x(k + 1) = ]x(k) +H w(k)

y, (k) Ci x(k) + v, (k)

whereE (k) ((W T (j J ( ) 'k 1 0[v(k)) V011 R)k 1[ ]

- 2 1+A 21

C2 = [1 1] C3 = [0 1], q1 1+A 3 1

- 2 1 4]

1356

O<A<1.Based on the model decomposing method in Section 2, let

I- 0- ~2 1 1-I 1 2 1 and

-1 0- 1 1 2-0.4082 0.7071 1.1547

have D 0.4082 - 0.7071 1.1547

-0.8165 0 1.1547]-O 0-

Then we obtain AC 0 1

-1.1353 - 0.5896 0.0526AD= -0.2647 +fA - 0.6643 - 0.1065

1.2145 0.7090 0.6725-1 7S-5 < 1R < 0

we can

and

where

We ca,n_v%.%."I 6 _

2 1+A1 2R =DDT +ADADT = +A1 3+A2 1+A3 where

2 1+A3 4

0<A <1, 0<A2 < 1.4972, 0 < A3 < 0.5475. It's noted that

the matrix R has been introduced the conservatismcompared with R after the model decomposing process.

For the Pei-Radman system (C, D), Theorem 1 yields thefilter F I

0.3645 0.82221Afl - 0.1778 0.04091

0.1789 0.3567 - 0.17891Bf1 L-0.1406 0.3184 0.14061

The estimation covariances of two states are 1.4027 and1.2684, respectively. Lety = 1. Theorem 3.4 yields the filterF2

0.8662 0.11861Af2 - 0.3394 0.10801

- 0.2177 0.4807 - 0.0678B = lo6xXf2 0.1253 -0.1446 0.0974

The results of D-C fusion estimation are obtained bycombining the estimates of F1 and F2, which is shown inFig.1.

To illustrate the performance of D-C method, the robustfiltering algorithm developed by [7] is used to estimate thestates for the same system model. Fig.2 shows the estimateresults and the matrices of the filter obtained are

- 0.2806 2.1806 F1.7868 0 -1.1806Af

- 0.1050 0.6050 Bf L0.1590 0 0.1050jCompared the Fig.1 and Fig.2, it's clearly that D-C fusion

method have better estimation performance than the robustfiltering method[7] for all -1.3755<f8<0, especially, forthe bound uncertain parameter , = -1.3755.

Fig. 1 The results ofD-C fusion estimationThe 'thick' lines represent the real state and the 'thin' lines represent theresults of F1+2.

Fig.2 The results of robust filteringThe 'thick' lines represent the real state, the 'thin' lines represent the

results of robust filter with , = 0 and the 'dotted' lines represent the

results with ,O -1.3755 .

V. CONCLUSIONS

This paper presents the decomposed-composed (D-C)fusion estimation method. This approach can successfullysolve the robust filtering problem of the multisensor systemwith uncertainty correlated measurement covariance matrix.The principle of this method is to decompose themeasurement system to two parts and design filtersrespectively to one part of the real state. Finally, the twoestimates are combined to obtain the final estimate of the realstate. The numerical example has shown that the robust fusionfiltering here outperforms the general robust filter for thisspecial kind of uncertain system.

ACKNOWLEDGEMENTS

This research has been supported by Zhejiang ProvincialNatural Science Foundation of China grants No. M603174,PH.D Foundation of Zhejiang University of Sciences.

1357

the esitmation results of filter Fl and F210

0

0 tO 20 30 40 fl0 60 70 80

get

REFERENCES

[1] K. C., Chang, Saha, R. K., Bar-Shalom, Y., On optimal track-to-trackfusion, IEEE Transaction on Aerospace and Electronic Systems, Vol.33, pp. 1271-1276, 1997.

[2] Qiang Gan, Chis J. Harris, Comparison of Two Measurement FusionMethods for Kalman-Filter-Based Multisensor Data Fusion, IEEETransaction on Aerospace and Electronic Systems, Vol. 37, pp.273-280, 2001.

[3] Summit Roy, Ronald A. Iltis, Decentralized Linear Estimation inCorrelated Measurement Noise, IEEE Transaction on Aerospace andElectronic Systems. 1991, 27(6): 939-941.

[4] JIN Xue-bo, SUN You-xian, Optimal Centralized State FusionEstimation for Multi-sensor System with Correlated Measurementnoise, Proceedings of 2003 IEEE Conference on Control Applications,Vol. I, 2003, pp. 770-772

[5] JIN Xue-bo, SUN You-xian, Multisensor State Fusion Estimation withCorrelated Measurement Noise, Proceedings of the SecondInternational Conference on Machine Learning and Cybernetics,pp.1000-1003, 2003

[6] Carlos E.de Souza and Lihua Xie, On the discrete-time bounded reallemma with application in the characterization of static state feed backH. controller. System & Control Letters, 1992, 18:61-7 1.

[7] Reinaldo M. Palhares, LMI Approach to the Mixed H2 IH.Filtering Design for Discrete-Time Uncertain Systems, IEEE Trans.on AES, Vol. 37, No. 1, 2001, pp: 292-296

1358