Automatica 43 (2007) 1590–1596www.elsevier.com/locate/automatica

Brief paper

Accuracy analysis of bias-eliminating least squares estimates forerrors-in-variables systems�

Mei Honga,∗, Torsten Söderströma, Wei Xing Zhengb

aDivision of Systems and Control, Department of Information Technology, Uppsala University, P.O. Box 337, SE-75105 Uppsala, SwedenbSchool of Computing and Mathematics, University of Western Sydney, Penrith South DC NSW 1797, Australia

Received 31 January 2006; received in revised form 4 October 2006; accepted 2 February 2007Available online 6 July 2007

Abstract

The bias-eliminating least squares (BELS) method is one of the consistent estimators for identifying dynamic errors-in-variables systems.In this paper, we investigate the accuracy properties of the BELS estimates. An explicit expression for the normalized asymptotic covariancematrix of the estimated parameters is derived and supported by some numerical examples.� 2007 Elsevier Ltd. All rights reserved.

Keywords: System identification; Parameter estimation; Errors-in-variables; Bias-eliminating least squares; Accuracy

1. Introduction

System identification and parameter estimation for stochas-tic errors-in-variables (EIV) systems, where the input as wellas the output measurements are noisy, have been a topic ofactive research for several decades. This class of system mod-els frequently appears in various problems of practical interest,such as time series econometric models, blind channel equal-ization in communications, multivariate calibration in analyti-cal chemistry, etc. See Van Huffel and Lemmerling (2002) formore descriptions.

Till now, many solutions to the EIV system identificationproblem have been proposed with different approaches. Forexample, the Koopmans–Levin (KL) method (Fernando andNicholson, 1985), the prediction error method (Söderström,1981), frequency domain methods (Pintelon and Schoukens,

� A preliminary version of this paper was presented at the 14th IFACSymposium on System Identification, March 2006, Newcastle, Australia. Thispaper was recommended for publication in revised form by Associate EditorTongwen Chen under the direction of Editor Ian Petersen. This researchwas partially supported by The Swedish Research Council, contract 621-2005-4207, The Australian Research Council and The University of WesternSydney.

∗ Corresponding author. Tel.: +46 18 4713398; fax: +46 18 511925.E-mail addresses: mhong@it.uu.se (M. Hong), ts@it.uu.se

(T. Söderström), w.zheng@uws.edu.au (W.X. Zheng).

0005-1098/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.automatica.2007.02.002

2001), and methods based on higher order cumulate statistics(Tugnait and Ye, 1995), etc. The focus of this paper is placed onthe bias-eliminating least squares (BELS) algorithms (Zheng,1998; Zheng and Feng, 1989), which are developed accord-ing to the bias compensation principle (Stoica and Söderström,1982). Since the BELS methods usually give quite good esti-mation accuracy (more accurate than standard IV methods andoften comparable to the use of a prediction error method) butwith a modest computational load, they seem to belong to theclass of the more interesting and efficient approaches for EIVidentification.

Noticeably, however, a statistical analysis of the accuracy ofthe BELS methods has been missing in the literature. Thereis no doubt that such an accuracy analysis can highly facili-tate evaluation of and comparison with different identificationapproaches. Besides, one may also get insight into importantissues like how different user choices in the algorithms can in-fluence the accuracy, and when the estimation problem is hardto solve and only a low accuracy can be expected.

In this paper, we will make an asymptotic accuracy analysisof the parameter estimates for the BELS method by followingthe analysis approach used for the Frisch method in Söderström(2005). First, the dynamic EIV problem is formulated in Sec-tion 2 and notations are described in Section 3. After a briefreview of the BELS methods in Section 4, we give linearization

M. Hong et al. / Automatica 43 (2007) 1590–1596 1591

results of the key equations utilized by the BELS algorithm forlarge data number N in Section 5. The asymptotic normalizedcovariance matrix of the BELS estimated parameters is pre-sented as a theorem in Section 6. Finally, an numerical exampleis studied in Section 7 before concluding in Section 8.

2. Problem formulation

Consider a linear and finite order system given by

A(q−1)y0(t) = B(q−1)u0(t), (1)

where u0(t) and y0(t) are the noise-free input and output, re-spectively, and A(q−1) and B(q−1) are polynomials describedas

A(q−1) = 1 + a1q−1 + · · · + anaq

−na , (2a)

B(q−1) = b1q−1 + · · · + bnb

q−nb . (2b)

Measurements of both the input and the output are contaminatedby additive noise, so the available measurements u(t) and y(t)

are

u(t) = u0(t) + u(t), y(t) = y0(t) + y(t). (3)

We assume that the system is asymptotically stable, observ-able and controllable; the system orders na and nb are a prioriknown; the noise-free input u0(t) is persistently exciting of suf-ficient order; the noise signals u(t) and y(t) are independent ofu0(t); and u(t) and y(t) are mutually independent white noisesequences of zero mean, and variances �u and �y , respectively.

The problem of identifying this error-in-variables system isconcerned with consistently estimating the parameter vector

ϑ = (a1 . . . ana b1 . . . bnb; �y �u)

T (4)

from the measured noisy data {u(t), y(t)}Nt=1. The task of thispaper is to derive the asymptotic covariance matrix of the esti-mated parameters for the BELS estimator:

P = limN→∞ E{N cov(ϑ − ϑ0)(ϑ − ϑ0)

T}, (5)

where ϑ is the estimate of ϑ, and ϑ0 denotes the true value.

3. Notations

We now introduce several notations. First, the regressor vec-tor is defined by

�(t)=(−y(t−1) . . . −y(t−na), u(t−1) . . . u(t−nb))T, (6)

�(t) = �0(t) + �(t), (7)

where �0(t) and �(t) denote the noise-free term and the noisecontribution to the regressor vector, respectively.

For convenience, the system parameter vector is expressedin partitioned form as

� =(

ab

), a =

⎛⎝ a1

...

ana

⎞⎠ , b =

⎛⎝ b1

...

bnb

⎞⎠ . (8)

We will use the conventions

� = � − �0, �y = �y − �0y, �u = �u − �0

u, (9)

where �0, �0y, �

0u denote the true value of �, �y, �u, respectively.

�, �y, �u are the corresponding estimates, and �, �y, �u are therelevant estimation errors.

Furthermore, �LS,N and VLS,N will be used to express theleast squares (LS) parameter estimate and the minimum valueof the LS loss function for a finite number of data N, and �LS,∞and VLS,∞ represent the quantities when N → ∞. The truevalues of the cross-covariance matrices and vectors and theirestimates are given by

R� = E�(t)�T(t), R� = 1

N

N∑t=1

�(t)�T(t), (10)

r�y = R��LS,∞ = E�(t)y(t),

r�y = R��LS,N = 1

N

N∑t=1

�(t)y(t). (11)

The covariance matrix of the noise terms and its estimate aredenoted as

R� = E�(t)�T(t) =(

�0yIna 0

0 �0uInb

), (12)

R� =(

�yIna 00 �uInb

). (13)

For R�, there exists the relation R� = R�0+ R�, where R�0

is the covariance matrix of the noise-free term, i.e., R�0=

E�0(t)�T0 (t).

4. Review of the BELS method

BELS algorithms are built upon the bias compensation prin-ciple:

�BELS = (R� − R�)−1r�y . (14)

From Eq. (12), we know that R� contains two unknown param-eters, i.e. the variances of the input and output noises �u and �y .So, in addition to the modified normal equation (14) (at least)two more relations for �u and �y are needed. One such relationcan be derived from the minimal value of the LS criterion:

VLS = min�

E(y(t) − �T(t)�)2 = �y + �T0 R��LS (15)

(see Söderström, Hong, and Zheng, 2005, Eq. (12)). To geta second relation for �u and �y , an extended model structureis considered in BELS. For this purpose introduce extendedversions of �(t), � and �0 as

� =(

��

), � =

(��

), �0 =

(�00

). (16)

1592 M. Hong et al. / Automatica 43 (2007) 1590–1596

The possible model extension includes, for example, appendingan additional A parameter, which yields

�(t) = −y(t − na − 1), � = ana+1 (17)

or, appending an additional B parameter, which gives

�(t) = u(t − nb − 1), � = bnb+1. (18)

In the extended model, by replacing �(t) with �(t), the least

squares estimate ˆ�LS, the covariance matrix R� and the covari-ance matrix of the noise R ˜� can be calculated in the same way

as that �LS, R� and R� in the normal model.Next consider LS estimation in the extended linear regression

model y(t)=�T(t)�+�(t) which leads to R�ˆ�LS=r�y . Similar

to (14), it holds that

R�ˆ�LS = r�0y0 + r ˜�y

= R�0�0 = (R� − R ˜�)�0. (19)

Set H = (0, . . . , 1) ∈ Rna+nb+1, J = (Ina+nb0), �0 =J�0, and

observe that H �0 = 0. Eq. (19) implies

Hˆ�LS = HR�

−1(R� − R ˜�)�0 = −HR−1� R ˜�J�0. (20)

See Zheng (1998) and Söderström et al. (2005) for details.To sum up, the BELS algorithm consists of the following

equations to determine the system parameter vector � and noisevariance vector �, where � = (�y �u)

T.

R��LS = (R� − R�(�))�, (21)

VLS = �y + �TLSR�(�)�, (22)

Hˆ�LS = −HR−1

� R�(�)J�. (23)

Eqs. (21)–(23) turn out to be bilinear in the unknowns � and �.There are different ways to solve these equations. In Söderströmet al. (2005), a variable projection algorithm has been proposedwhich has better convergence property than the classical BELSalgorithm.

5. Linearization

To analyze the estimation accuracy, we will examine how theestimates ϑ deviates from the true parameter vector ϑ0 for largedata sets (large N). The technique for doing so is to linearizeEqs. (21)–(23) by assuming that ϑ is close to ϑ0 (i.e. estimationerror is small) when N is large. Linearization results of Eqs.(21)–(23) are summarized in the following three lemmas, andtheir proofs are briefly given in Appendix A.

Lemma 5.1. Linearizing Eq. (21) leads to

R�0� +

(−a0

)�y +

(0

−b

)�u

= 1

N

N∑t=1

�(t)�(t) − E�(t)�(t), (24)

where

�(t) = y(t) − �T(t)�0 = A(q−1)y(t) − B(q−1)u(t). (25)

Lemma 5.2. Linearizing Eq. (22) leads to

rT�yR

−1� R�� +

(1 + rT

�yR−1�

(a0

))�y + rT

�yR−1�

(0b

)�u

= 1

N

N∑t=1

�LS(t)�(t) − E�LS(t)�(t), (26)

where

�LS(t) = y(t) − �T(t)�LS,∞

={

ALS(q−1)B(q−1) − BLS(q−1)A(q−1)

A(q−1)

}u0(t)

+ ALS(q−1)y(t) − BLS(q−1)u(t). (27)

Lemma 5.3. Linearizing Eq. (23) leads to

− HR−1� R�J � − HR−1

�

(a00

)�y − HR−1

�

(0b0

)�u

= HR−1�

(1

N

N∑t=1

�(t)�(t) − E�(t)�(t)

). (28)

As can be seen, each of the three equations (21)–(23) islinearized into the generic form

��� + ��y�y + ��u

�u ≈ �s , (29)

where the coefficients ��, ��y, ��u

are deterministic variables,while �s is a random term which has zero mean and a variancethat decreases when N increases. Moreover, we note that theterm �s in Eqs. (24) and (28) depends only on �(t), the distur-bances coming from the input and output noises. However, �s

in Eq. (26) is related not only to �(t) but also to �LS(t) whichdepends on the noise-free input u0(t).

6. Asymptotic covariance matrix

The main theoretical result of the paper is:

Theorem 6.1. Assume that the white noise y(t) has Ey(t)=0,Ey2(t)=�y and finite fourth-order moments Ey4(t)=�y , andsimilarly for u(t): Eu(t) = 0, Eu2(t) = �u and Eu4(t) = �u.Under the given assumptions in Section 2 and the central limittheorem in Ljung (1977), it follows that the BELS estimatedparameter ϑ is asymptotically Gaussian distributed:

√N(ϑ − ϑ0)

dist−→N(0, P ), (30)

where

P = limN→∞ E{N cov(ϑ − ϑ0)(ϑ − ϑ0)

T}= G−1 lim

N→∞ NE��TG−T

= G−1QG−T. (31)

M. Hong et al. / Automatica 43 (2007) 1590–1596 1593

The coefficients ��, ��y, ��u

, cf. (29), appear as block ele-ments of G. The block elements of Q are covariance matricesof the random terms �s . That is,

G =

⎛⎜⎜⎜⎜⎜⎜⎜⎝

R�0

(−a0

) (0

−b

)

rT�yR

−1� R� 1 + rT

�yR−1�

(a0

)rT�yR

−1�

(0b

)

−HR−1� R�J −HR−1

�

(a00

)−HR−1

�

(0b0

)

⎞⎟⎟⎟⎟⎟⎟⎟⎠

,(32)

Q = QG + QNG. (33)

In the case of Gaussian measurement noise, only the term QG

applies, and QNG vanishes. The two terms are given in parti-tioned form as

Q =⎛⎝QG

11 QG12 QG

13QG

21 QG22 QG

23QG

31 QG32 QG

33

⎞⎠ ,

⎛⎝QNG

11 QNG12 QNG

13QNG

21 QNG22 QNG

23QNG

31 QNG32 QNG

33

⎞⎠ .

The blocks in the symmetric matrix QG and QNG are as fol-lows:

QG11 =

∑

(r��()rT��(−) + R�()r�()),

QG12 =

∑

(r��LS()r�() + r��()r��LS()),

QG13 =

∑

(r��()rT��(−) + R��()r�())R

−1� HT,

QG22 =

∑

(r�LS()r�() + r�LS�()r��LS()),

QG23 =

∑

(r�LS�()rT��(−) + rT

�LS�()r�())R−1� HT,

QG33 = HR−1

�

(∑

(r��()rT��(−) + R�()r�())

)R−1

� HT,

QNG11 =

((�y − 3�2

y)aaT 0na × nb

0nb × na (�u − 3�2u)bbT

),

QNG12 =

(−(�y − 3�2y)a(aT

LSa)

(�u − 3�2u)b(bT

LSb)

),

QNG13 =

((�y − 3�2

y)aaT 0na×nb0na×1

0nb×na (�u − 3�2u)bbT 0nb×1

)R−1

� HT,

QNG22 = (�y − 3�2

y)(aTLSa)2 + (�u − 3�2

u)(bTLSb)2,

QNG23 = (−(�y − 3�2

y)(aTaLS)aT (�u − 3�2

u)(bTbLS)bT 0)

× R−1� HT,

QNG33 = HR−1

�

⎛⎝ (�y − 3�2

y)aaT 0na×nb0na×1

0nb×na (�u − 3�2u)bbT 0nb×1

01×na 01×nb0

⎞⎠

× R−1� HT.

Proof. See Appendix B.

The covariance elements in the blocks of QG satisfy

r�(k) ={

�y

∑iaiai+k + �u

∑ibibi+k,

0 |k| > max(na, nb − 1),(34)

r��LS(k) ={

�y

∑iaia(LS)i+k + �u

∑ibib(LS)i+k,

0 |k| > max(na, nb − 1),(35)

r��(k) = (−�y(a1−k . . . ana−k)T, −�u(b1−k . . . bnb−k)

T)T,

r��(k) = (−�y(a1−k . . . ana−k)T, −�u(b1−k . . . bnb−k)

T,

− �yana+1−k or �ubna+1−k)T, (36)

where the conventions

ai ={

1, i = 0,

0, i > na, i < 0,(37)

bi = 0, i > nb, i�0 (38)

are used. Note that the summations over in the elements ofQ are over all values making the terms nonzero. However, dueto the condition of r�(), r��LS(), and r��() and r��() willbe zero when || > max(na, nb − 1) or || > max(na + 1, nb),respectively. Thus, each sum is finite and will have only amodest number of nonzero terms.

7. Numerical example

We now present an example to illustrate the theoretical for-mulas derived in the preceding section. For simplicity, onlyGaussian data are considered here. Assume a second-order sys-tem given by

(1 − 1.5q−1 + 0.7q−2)y0(t) = (1.0q−1 + 0.5q−2)u0(t). (39)

The noise-free input u0(t) is an ARMA(1, 1) process

(1 − 0.5q−1)u0(t) = (1 + 0.7q−1)e(t), (40)

where e(t) is a zero-mean white noise with unit variance. Thewhite measurement noises at the input and output sides havevariances equal to 1 and 3, respectively. For each simulationresult, N = 10 000 data points are used and M = 1000 realiza-tions are made.

First, consider that an additional A parameter is appended,and in this case �(t) = −y(t − 3). The theoretical normalizedasymptotic covariance matrix of the estimation error is

P =

⎛⎜⎜⎜⎜⎝

0.842−0.644 0.513−2.031 1.191 40.00

4.378 −2.995 −38.17 45.97−1.720 1.409 −16.29 5.506 50.16

1.615 −1.444 24.12 −12.25 −23.82 37.88

⎞⎟⎟⎟⎟⎠ .

The corresponding normalized covariance matrix obtainedfrom the Monte-Carlo simulation is as follows:

Ps =

⎛⎜⎜⎜⎜⎝

0.846−0.636 0.500−1.893 1.024 44.45

4.372 −2.915 −40.13 47.08−1.617 1.374 −16.98 6.419 48.44

2.091 −1.826 31.56 −14.47 −25.19 53.12

⎞⎟⎟⎟⎟⎠ .

1594 M. Hong et al. / Automatica 43 (2007) 1590–1596

Second, consider the case of appending an additional B pa-rameter, i.e., �(t)=u(t−3). The theoretical normalized asymp-totic covariance matrix is

P =

⎛⎜⎜⎜⎜⎝

43.87−36.01 29.58102.9 −84.79 262.8

76.68 −62.69 164.2 147.5−106.4 87.35 −259.4 −179.3 298.9

313.7 −257.6 751.1 539.6 −770.0 2267

⎞⎟⎟⎟⎟⎠ ,

and the Monte-Carlo simulation result is

Ps =

⎛⎜⎜⎜⎜⎝

46.74−38.34 31.47111.3 −91.59 287.1

81.16 −66.32 176.8 154.8−109.9 90.17 −270.4 −185.1 300.2

333.4 −273.7 811.3 567.3 −793.2 2406

⎞⎟⎟⎟⎟⎠ .

As can be seen, the theoretical results are in good agreementwith the simulation results for both cases. It also shows thatadding an extra A parameter can result in much better estimationaccuracy than choosing an extra B parameter in the extendedmodels. For BELS, the user choice on the construction of theextended models, i.e. the extended regressor vector �(t), is veryimportant.

8. Conclusions

In this paper, we have analyzed the accuracy of the BELS es-timates for identifying the EIV systems. The normalized asymp-totic covariance matrix of the estimated system parameters andthe estimated noise variances has been derived. The numericalexamples demonstrate the correctness of the theoretical results.The analysis here suggests that, in the extended model of theBELS estimator, choosing to add the regressors which containmore information about the system and/or the noises would getbetter estimation result. The analysis of this paper can be ex-tended to the BELS algorithm under more general noise con-ditions.

Appendix A. Linearization of Eqs. (21)–(23)

Consider Eq. (21) for a finite N:

R�� = R��LS,N + R�� = r�y + R�� (A.1)

and for N → ∞:

R��0 = R��LS,∞ + R��0 = r�y + R��0. (A.2)

In Eq. (A.1), replace R�, r�y and � with R� + R�, r�y + r�y ,

and �0 + �, respectively. Assume that ϑ is close to ϑ0. ThenR�, r�y , �, and R� are all small. We can neglect the secondorder terms and use (A.2) to get

R�� + R��0 = r�y + R�� + R��0

⇒(R�0

−a 00 −b

)⎛⎝ ��y

�u

⎞⎠

= (r�y − r�y) − (R� − R�)�0.

Then have

�1= (r�y − r�y) − (R� − R�)�0

= 1

N

∑�(t)(y(t) − �T(t)�0) − E�(t)(y(t) − �T(t)�0)

= 1

N

∑�(t)�(t) − E�(t)�(t). (A.3)

Linearization of Eqs. (22) and (23) can be derived in a similarprocess. See Hong, Söderström, and Zheng (2006) for details.

Appendix B. Derivation of elements Qi,j

First, we have

Q11 = limN→∞ EN�1�

T1

= limN→∞ EN

(1

N

∑t

�(t)�(t) − E�(t)�(t)

)

×(

1

N

∑s

�(s)�T(s) − E�(s)�T(s)

)

= limN→∞

[1

N

∑t

∑s

E�(t)�(t)�(s)�T(s)

−E�(t)�(t)E�(s)�T(s)

]. (B.1)

We will first treat the case of Gaussian distributed measure-ment noise. Using the following property for jointly Gaussiandistributed variables:

Ex1x2x3x4 = (Ex1x2)(Ex3x4) + (Ex1x3)(Ex2x4)

+ (Ex1x4)(Ex2x3), (B.2)

we get

QG11 = lim

N→∞1

N

∑t

∑s

(E�(t)�T(s)E�(t)�(s)

+ E�(t)�(s)E�(t)�T(s)).

By changing variables, = t − s, the first double sum in Q11can be expressed as

limN→∞

1

N

∑t

∑s

(E�(t)�T(s)E�(t)�(s))

= limN→∞

(1

N

N−1∑=1−N

(N − ||)R�()r�()

)

=∞∑

=−∞R�()r�() − lim

N→∞1

N

N−1∑=1−N

||R�()r�().

Since the covariance functions R�() and r�() will decay ex-ponentially when || → ∞, the second term will converge tozero (Söderström, 2006, Appendix C). Hence,

limN→∞

1

N

∑t

∑s

(E�(t)�T(s)E�(t)�(s)) =∞∑

=−∞R�()r�().

M. Hong et al. / Automatica 43 (2007) 1590–1596 1595

Applying the same techniques to the second double sum in QG11

gives finally

QG11 =

∑

(r��()rT��(−) + R�()r�()).

A similar process is used to derive QG12, Q

G13, Q

G22, Q

G23, and

QG33. Next we consider the case of non-Gaussian noise. In this

case relation (B.2) cannot be used. Instead, we follow the tech-niques used in Söderström (2006), i.e. utilize that all xk termsare linear filters operating on a white noise source e(t) (beingeither y(t) or u(t)). Let the noise e(t) have zero mean, variance� and the fourth moment �. It holds that

xk(t) = Hk(q−1)e(t), Hk(q

−1) =∞∑

j=0

hkjq−j , k = 1, 2, 3, 4.

Ex1(t)x2(t)x3(t)x4(t)

=∞∑i=0

∞∑j=0

∞∑k=0

∞∑l=0

h1ih2j h1ih3kh4l

× Ee(t − i)e(t − j)e(t − k)e(t − l). (B.3)

As the white noise e(t) has zero mean and is uncorrelatedat different time points, the expectation in (B.3) is nonzeroonly when the time arguments are pairwise equal or all equal.Therefore,

Ee(t − i)e(t − j)e(t − k)e(t − l)

= �2[i,jk,l + i,kj,l + i,lj,k]+ (� − 3�2)i,jj,kk,l . (B.4)

Now we have

Ex1(t)x2(t)x3(t)x4(t) = [Ex1(t)x2(t)][Ex3(t)x4(t)]+ [Ex1(t)x3(t)][Ex2(t)x4(t)]+ [Ex1(t)x4(t)][Ex2(t)x3(t)]+ (� − 3�2)

∞∑i=0

h1ih2ih3ih4i . (B.5)

We see that using the first part of (B.5) leads precisely to theGaussian formula with QG. For the remaining terms of (B.5)that vanishes in the Gaussian case, we have, cf. (B.1), for i, j =1, . . . , na ,

(QNG11 )i,j =

∞∑=−∞

Ey(t − i)A(q−1)y(t)A(q−1)y(t + )

× y(t + − j) − (Q11)i,j

= (�y − 3�2y)

∞∑=−∞

aii,j−aj

= (�y − 3�2y)aiaj . (B.6)

The elements (QNG11 )i,j=(na+1):(na+nb) can be evaluated simi-

larly. The result can be summarized as

(QNG11 ) =

((�y − 3�2

y)aaT 0

0 (�u − 3�2u)bbT

). (B.7)

Similar processes apply for deriving the blocks QNG12 QNG

13 QNG22

QNG23 and QNG

33 . See Hong et al. (2006) for details.

References

Fernando, K. V., & Nicholson, H. (1985). Identification of linear systems withinput and output noise: The Koopmans–Levin method. IEE Proceedings,Part D, 132(1), 30–36.

Hong, M., Söderström, T., & Zheng, W. X. (2006). Asymptotic accuracyanalysis of bias-eliminating least squares estimates for identification oferrors in variables systems. Technical Report 2006-046, Department ofInformation Technology, Uppsala University.

Ljung, L. (1977). Some limit results for functionals of stochastic processes.Technical Report LiTH-ISY-I-0167, Department of Electrical Engineering,Linkping Univ., Linkping, Sweden.

Pintelon, R., & Schoukens, J. (2001). System identification. A frequencydomain approach. New York, NY, USA: IEEE Press.

Söderström, T. (1981). Identification of stochastic linear systems in presenceof input noise. Automatica, 17, 713–725.

Söderström, T. (2005). Accuracy analysis of the Frisch estimates foridentifying errors-in-variables systems. In Proceedings of 44th IEEECDC/ECC 2005, Seville, Spain, December 12–15.

Söderström, T. (2006). Statistical analysis of the Frisch scheme for identifyingerrors-in-variables systems. Technical Report No:2006-002, Departmentof Information Technology, Uppsala University, Uppsala, Sweden.

Söderström, T., Hong, M., & Zheng, W. X. (2005). Convergence propertiesof bias-eliminating algorithms for errors-in-variables identification.International Journal of Adaptive Control and Signal Processing, 19,703–722.

Stoica, P., & Söderström, T. (1982). Bias correction in least-squaresidentification. International Journal of Control, 35(3), 449–457.

Tugnait, J. K., & Ye, Y. (1995). Stochastic system identification withnoisy input–output measurement using polyspectra. IEEE Transactions onAutomatic Control, AC-40, 670–683.

Van Huffel, S., & Lemmerling, Ph. (Eds.). (2002). Total least squares anderrors-in-variables modelling. Analysis, algorithms and applications. TheNetherlands, Dordrecht: Kluwer.

Zheng, W. X. (1998). Transfer function estimation form noisy input and outputdata. International Journal of Adaptive Control and Signal Processing,12, 365–380.

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Mei Hong received her B.Sc. in Electronic En-gineering from Xidian University, China in 1994and M.Sc. from Chalmers University, Swedenin 2003. Now she is pursuing her Ph.D degreein Division of Systems and Control, Departmentof Information Technology, Uppsala University.Her current research interests are in systemidentification especially for errors-in-variablessystems.

Torsten Söderström received the M.Sc. degree(“civilingenjör”) in engineering physics in 1969and the Ph.D. degree in automatic control in1973, both from Lund Institute of Technology,Lund, Sweden. He is a Fellow of IEEE, and anIFAC Fellow.In the period 1967–1974 he held various teach-ing positions at the Lund Institute of Technol-ogy. Since 1974, he has been with Departmentof Systems and Control, Uppsala University,Uppsala, Sweden, where he is a professor ofautomatic control.

1596 M. Hong et al. / Automatica 43 (2007) 1590–1596

Dr. Söderström is the author or coauthor of many technical papers. His mainresearch interests are in the fields of system identification, signal process-ing, and control. He is the (co)author of four books: “Theory and Practiceof Recursive Identification”, MIT Press, 1983 (with L. Ljung), “The Instru-mental Variable Methods for System Identification”, Springer-Verlag, 1983(with P. Stoica), “System Identification”, Prentice-Hall, 1989 (with P. Stoica)and “Discrete-Time Stochastic Systems”, Prentice-Hall, 1994; second edition,Springer-Verlag, 2002. In 1981 he was, with coauthors, given an AutomaticaPaper Prize Award.Within IFAC he has served in several capacities including vice-chairman of theTC on Modelling, Identification and Signal Processing, 1993–1999, IPC chair-man of the IFAC SYSID’94 Symposium, Council member 1996–2002, Ex-ecutive Board member 1999–2002 and Awards Committee Chair 1999–2002.Within Automatica he was an associate editor 1984–1991, guest associate ed-itor or editor for four special issues, and is the editor for the area of SystemParameter Estimation since 1992.

Wei Xing Zheng was born in Nanjing, China.He received the B.Sc. degree in Applied Math-ematics and the M.Sc. and Ph.D. degrees inElectrical Engineering, in January 1982, July1984 and February 1989, respectively, all fromSoutheast University, Nanjing, China.From 1984 to 1991, he was with Institute ofAutomation at Southeast University, Nanjing,China, first as a Lecturer and later as an As-sociate Professor. From 1991 to 1994, he wasa Research Fellow in Department of Electrical

and Electronic Engineering at Imperial College of Science, Technology andMedicine, London, U.K.; in Department of Mathematics at University ofWestern Australia, Perth, Australia; and in Australian Telecommunications

Research Institute at Curtin University of Technology, Perth, Australia. Since1994, Dr. Zheng has been with University of Western Sydney, Sydney, Aus-tralia, where he is currently a Professor. He has also held various visitingpositions in Institute for Network Theory and Circuit Design at MunichUniversity of Technology, Munich, Germany; in Department of ElectricalEngineering at University of Virginia, Charlottesville, VA, USA; and in De-partment of Electrical and Computer Engineering at University of California,Davis, CA, USA. His research interests are in the areas of systems and con-trols, signal processing, and communications. He coauthored the book LinearMultivariable Systems: Theory and Design (SEU Press, Nanjing, 1991).Dr. Zheng has received several science prizes, including the Chinese Na-tional Natural Science Prize awarded by the Chinese Government in 1991.He is a Senior Member of IEEE. He has served on the technical programor organizing committee of numerous international conferences, includingthe 14th IFAC Symposium on System Identification (SYSID’2006), the 45thIEEE Conference on Decision and Control (CDC’2006), and the 46th IEEEConference on Decision and Control (CDC’2007). He has also served onseveral technical committees, including Member of IFAC Technical Com-mittee on Modelling, Identification and Signal Processing, Member of IEEEControl Systems Society’s Technical Committee on System Identification andAdaptive Control, Secretary of IEEE Circuits and Systems Society’s Techni-cal Committee on Blind Signal Processing, and Secretary of IEEE Circuitsand Systems Society’s Technical Committee on Neural Systems and Appli-cations. He served as an Associate Editor for IEEE Transactions on Circuitsand Systems—I: Fundamental Theory and Applications (2002–2004). He iscurrently an Associate Editor for IEEE Transactions on Automatic Control(2004–), an Associate Editor for IEEE Signal Processing Letters (2007–) andan Associate Editor of IEEE Control Systems Society’s Conference EditorialBoard (2000–).