Robust parametric transfer function estimation using complex logarithmic frequency response data

IEEE TRANSACTTONS ON AUTOMATIC CONTROL, VOL. 40,NO. 7, JULY 1995

118(1

Robust Parametric Transfer Function Estimation Using Complex Logarithmic Frequency Response Data Patrick Guillaume, Member, IEEE, Rik Pintelon, Member, IEEE, and Johan Schoukens, Senior Member, IEEE

Abstract-The statistical properties of a logarithmic leastsquares frequency-domain estimator are analyzed in an "error in-variables" framework. It is shown that, contrary to most frequency-domain estimators, the logarithmic least-squares estimator remaim "practically" "&tent when the variEsnces of the frequency-ddn data are not U priori known. If available, the variance at each spectral b e of the logarithmic frequency response data CM be osed to weight the logarithmic least-squares estimator, resulting in parameter estimates with a d e r variability. An estimate of these variances can be derived from the its robustness residual errors at the excited spectral lines. -des to lack of prior nohe information, it is demonstrated that the logarithmic estimator behaves remarkably well in the presence of outliers.

I. INTRODUCTION HE purpose of an identification experiment is to determine the dynamics of a given device from its inputloutput (WO) observations and the available a priori information. The devices under consideration are assumed to be linear timeinvariant (LTI) single-input, single-output (SISO) systems. The WO observations are obtained by means of a band-limited measurement setup [22]. The signals z ( t ) and y ( t ) in Fig. 1 denotes the observed (continuous-time) YO signals. They are described by means of an errors-in-variables (EV) stochastic model U], VI, WI

T

an EV frequency-domain identification framework [21], [26], [29], [30]. Before being sampled, the signals z ( t )and y ( t ) pass through an anti-aliasing filter. The impulse responses a, (t) and a,(t) in Fig. 1 account for the acquisition channels (the antialiasing filters included), while T, is the sampling period. The impulse responses a, (t) and a, (t) are assumed to be identical (a,(t) = a y ( t ) = a ( t ) ) . If not, a relative calibration of the measurement setup is required [22]. The observed U 0 time series, { z [ n ]= ( U * z)(nT,),y[n] = ( U * y)(nT,)}f:i, are quasi-stationary processes [ 131

41.

=4 1.

+ d[n]

+I + 4 7 4

(2)

consisting of a deterministic part {U[.] = ave { ~ [ n ] }w[n] , = ave { y[n]}}f:i ("aye" denotes the expectation operator) and a (zero-mean) random part { d [ n ] ,e [ n ] } f g i If . ~ ( tand ) v(t) are periodic signals with period T such that NT,/T is an integer number, then the following relation holds exactly

VINIIIC]= G o ( ~ k ) U [ ~ ] [ k ]

(3)

with G o ( w k ) the transfer function of-the LTI system, W k = 27rk/NT,, IC = O,...,[N/2], and {UIN1[IC],V [ " ] [ k ] }the discrete Fourier transforms (DFT)

+

x ( t )= u(t) d ( t )

{Y ( t ) =

(1) * .)(t, + e ( t ) . time functions u ( t ) and w ( t ) (= (go * u ) ( t ) ,with "*" (90

The the convolution operator) represent the deterministic part of the VO signals, gO(t) is the impulse response function of the system under test, while d ( t ) and e ( t )are zero-mean stationary noise sources standing for all pbssible stochastic disturbances (e.g., process noise, ambient noise, and generator noise). The VO disturbances d ( t ) and e ( t ) are allowed to be correlated. Furthermore, it will not be assumed that their autopower and cross-power spectra are a pridri known as is usually the case in Manuscript received April 9, 1993; revised March 1, 1994 and November 11, 1994. Recommended by Past Associate FAtor, E. W. h e n . This work was supported in part by the Belgian National Fund for Scientific Research, the Flemish government ( G O A - M I ) , and the Belgian government as a part of the Belgian programme on Interuniversity Poles of Attraction (IIJAP50) initiated by the Belgian State, Prime Minister's Office, Science Policy Programming. The authors are with the Vrije Universiteit Brussel, Department ELEC, Pleinlaan 2, B-1050 Brussels, Belgium. IEEE Log Number 94 1 1874.

of the deterministic time series { u [ n ] ,v[n]}rgi. This results in the following frequency-domain EV equations

x["l[ k ] = U[Nl[IC] + D [ N l[ k ] , IC]

+

G o ( ~ k ) U [ ~ l [E["I[IC]. k]

(5)

The transfer function Go(w) in ( 5 ) is assumed to belong to the parameterized model set of all rational functions G(w, 0) with real coefficients 6' = { a [ i , ] , b[ia]:i a = 0, * ,n,; z b = 0,.

'

,n b }

For a continuous-time model, R(w) = iw, while for a discretetime model with sampling period T , R(w) = eiwT.

0018-9286/95$04.00 0 1995 IEEE

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GULLAUME er al.: ROBUST PARAMETRIC TRANSFER FUNCTION

To identify the transfer-function model (6), a frequencydomain approach is considered, which has the following advantages compared with a time-domain approach: Complex Normally Distributed Fourier Coeficients: Practical procedures for the estimation of dynamic system parameters in the frequency domain start with the discrete Fourier transformation (DFT) of the measured (periodic) U 0 signals [27]. The subsequent parameter estimation uses the (discrete) Fourier coefficient estimates thus obtained as primary observations. For a wide class of probability density functions (PDF) of the time-domain noise, the Fourier coefficients are asymptotically complex normally distributed and independent over the frequencies [ 5 ] . The importance of this rather detailed statistical knowledge is that it may be exploited to improve and assess the quality of the system-parameter estimates. Errors-in-Variables:Many system identification methods assume that the applied excitation is an exactly known or measurable (zero-order-hold) signal [22]. Hence, only the output signal is subjected to disturbing noise. Based on this assumption, it is possible to derive estimates for the system parameters as well as for a parametric noise model [13], [14]. When the input is disturbed by noise, however, these estimators are not consistent anymore. To preserve the consistency, one must take into account the errors in all variables. Thereto, an errors-in-variables (EV) stochastic model (2) is needed (see for instance [l], [7], and [24]). Examples of EV frequency-domain identification methods are the generalized total least squares (GTLS; see [26] and [30]) the bootstrapped total least squares (BTLS; [291), and the maximum likelihood estimator (MLE) called EliS (estimation of linear systems; [21]). These frequency-domain estimators are only consistent, however, when the “true” covariance matrix of the U 0 disturbances, { D [ k ] ,E [ k ] } ,is a priori known. Validation: A frequency-domain representation gives a better insight into the dynamics of a system and allows for an easy detection of modeling errors by comparing the estimated transfer function with the frequency response data. Lurge Dynamical Range: Transfer functions with a large dynamical range can be estimated [3]. Wide Frequency Bands: It is ,possible to combine measurements gathered at different sampling frequencies so that wide frequency bands can be covered [3]. Some other practical aspects of a direct frequency-domain approach that could be quite useful are listed in Ljung [I41 and in Schoukens and Pintelon [21]. The main drawbacks of a frequency-domain approach are: Periodic Excitations: The correct use of the DIT algorithm requires the ability to apply a periodic excitation. Off-lineZdentijcution: The measurements are processed off-line (or in batch mode). Covariance Matrix of the Fourier Coeflcients: To obtain consistent estimates within an EV stochastic framework, the covariance matrix of the observed Fourier coefficients has to be known a priori (see for instance [l]). If this

1181

I x l nA :l; }

IY [nl A;:}

Fig. 1. Band-limited “errors-in-variables”measurement setup.

matrix is not available, the consistency of the GTLS, the BTLS, and the ML estimator are lost. The purpose of this paper is to release this last drawback. One possible approach is given by Feng and Zheng [SI who constructed a modified least-squares (MLS)estimator. They assumed white input and output noise. Using a “known” firstorder prefilter, an estimate of the noise variances can be derived. Once the variances are known, the asymptotic bias of the least squares method can be determined and subtracted from the system parameter estimates. In this paper an alternative solution is given which does not require prefiltering and can handle colored VO noise as well. The idea consists of constructing a parametric frequencydomain estimator whose consistency is robust to lack of prior noise information. It turns out that minimizing a weighted sum of squared residuals involving complex logarithmic transforms of the transfer function model and the frequency response data yields an estimator that satisfies our objective. This logarithmic (nonlinear) least squares estimator (LOG-LSE) is shown to be “practically” consistent, even if the PDF and/or the (co)variances of the observed Fourier coefficients are unknown. The term “practically” will be defined more precisely in the sequel. Moreover, if available, noise information can be used to improve the efficiency of the estimator. Our motivation for using complex logarithmic frequency response data stems from the field of nonparametric frequency response function (FRF) estimation [20]. Averaging the logarithm of the empirical FRF measurements results in an estimator with a much smaller bias than the classical spectral analysis (SA) estimator. In [ 101, closed-form expressions are given for the asymptotic bias as a function of the signal-tonoise ratio (SNR) of the VO Fourier coefficients. Recently, a logarithmic least-squares estimator was proposed in [23] to ameliorate the numerical ill-conditioning of the nonlinear least-squares estimator in the low gain areas of an FRF with large dynamic range. Further, a weighting function is introduced which only depends on the frequency spacing of the measured FRF. In [2], a logarithmic leastsquares estimator is used to generate starting values because of its good convergence. In a final step, the classical nonlinear least-squares estimator is used to “refine” the solution. Both papers are based on a deterministic reasoning (e.g., the shape of the cost function is studied in [2]); no statistical analysis of the estimator is given (i.e., no attention is paid to the influence of the disturbing noise on the minimum of the cost function). This will be the main topic of the present paper.

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t

I182

7

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 40, NO. 7, JULY 1995

Based on a maximum likelihood approach, optimal weightings (resulting in a minimal variability of the parameter estimates) are deduced; they clearly differ from the ones given in [23]. Furthermore, the additional refinement step proposed in 121 will turn out to introduce a larger bias than the logarithmic estimator in many cases. In a recent survey paper [18], the LOG-LSE has been compared with other parameter frequency-domain estimators. The LOG-LSE and the Van den Enden (VDE) nonlinear leastsquares estimator [28] performed remarkably well on real measurement data, especially when no noise information is available. In the present paper, it will be shown why the LOG and the VDE estimator are more robust to lack of prior noise information than, for instance, the GTLS, the BTLS, and the ML estimator. The outline of the paper is as follows. Section I1 is concemed with the nonparametric estimation of the FRF when the U 0 Fourier coefficients are both disturbed by noise. The power of logarithmic averaging is illustrated by means of an example. In Section 111, the (parametric) LOG-LSE is introduced, and its statistical properties are discussed. These statistical properties are verified in Section IV using Monte-Carlo simulations. Moreover, it is demonstrated that the logarithmic estimator behaves remarkably well in the presence of outliers. A link with the robust identification theory [ll], [19] is established. Eventually, the conclusions are drawn in Section V. XI. NONPARAMETRICFRF ESTIMATION Before studying the parametric LOG-LSE, the use of logarithmic averaging as a tool to obtain "practically" consistent nonparametric estimates of the FRF is illustrated by means of a simple example. Consider a static system gO(t) = GoS(t) (see Fig. 2)

Y(t) = Goz(t)+

(7)

controlled by the regulator f ( t ) = H S ( t )

z ( t )= - H y ( t )

+ r(t)

(8)

where r ( t )is an external periodic reference signal. For the sake of simplicity, only process noise, p ( t ) , is considered, resulting in the following I/O disturbances (9)

Fig. 2. Band-limited "errors-in-variables"measurement setup for a system operating in a closed loop.

containing A4 data points corresponding with an integer number of periods (usually a single period) of the signals N = MP). For the considered example, the estimate G b z A ( w k ) converges in probability as P 4 cy, to ([13, p. 1641)

where n ( w k j is given by

and is independent of the considered batch p. Estimates of the FRF can also be obtained by averaging the empirical FRF'S, G ~ " I ( W ~ ) = Y ~ " ] [ ~ I / X ~ " p] [=~ ] , 1,. . , P, and W k = 2.rrlc/MT8.Arithmetic averaging yields ([lo], [13, p. 1571) 3

1'

G E 1 ( w k ) = -CGFI(wk).

Pp=i

(13)

One can show that the arithmetic frequency response function estimator (ARI-FWE) converges with probability one (w.P.1) to its expected value when the random variables { GIw1( w k ) } are independent and identically distributed (i.i.d.) [15]. For complex normally distributed I/O Fourier coefficients and the given setup, the expected value of (13) equals

G & ( u ) = ave (GLM1(wk)) = G O - [ $ + G ~ ] e x ap( w(k )- ~ ) (14) .

Using nonlinear averaging techniques, even better results Notice that the U 0 time series {z[n]}rg: and { ~ [ n ] } ~ ~ ,l are mutually correlated dye to the feedback loop, and, con- can be obtained [lo]. Consider, for instance, the logarithmic sequently, applying classical SA will yield biased estimates. frequency response function estimator (LOG-FRFE; log to The SA estimate of the (static) transfer function Go at angular base e) frequency w = W k (i.e., G!&l(wb) = 8$~1(wk)/@~Y(wk.; see [5, Chap. 61, [13, p. 1511 is usually evaluated as It converges w.p.1 to

Goexp($Ei(-w)) for periodic signals (Xi"' [k]stands for the complex conjugate of X p l [ k ] ) Thereto, . the data is split into P batches, each

GTOOG(Wk)

= exp

(4 Ei (-Gia(wb)))

with Ei (.) the exponential integral function.

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(16)

1183

GUILLAUME et a1 ROBUST PARAMETRIC TRANSFER FUNCTION

The amplitude of G s ( W k ) , G & ( W k ) . and G ~ O G ( U I as ~ ) a function of the reference-signal-to-process-noiseratio Q ( W k ) are shown in Fig. 3 for a deterministic reference signal with Fourier coefficients IRLY( W k ) l = 1. p = 1,. . , P ; a system with open-loop gain Go = 10; and a controller H = 1. When ~ ( w k= ) 5 % 7 dB (i.e., var{PaM1[k.]} = 0.2, p = l , . . . , P ) , the amplitude of G z ( W k ) is 18.24 dB, IG&(Uk>l equals 19.935 dB and l G & ( d k ) / 19.995 dB, which is rather close to the true amplitude of Go (=20 dB) compared with the SA estimate. Definition 2.1: The (power) SNR of a Fourier coefficient, Z [ k ] , is defined as the square of the absolute value of its expected value (deterministic part) over its variance

20 2

20 19 8

19 6

e

194 ‘p e)

s3

192 19

? 188 18 6

18 4 18 2 18

0

2 4 6 8 10 12 14 16 18 20 reference-signal-to-process-noiseratio (dB)

Fig. 3. Asymptotic estimates of the open-loop transfer function Go. Solid line: LOG; dashed line: ARI; dotted line: SA.

Definition 2.2: A nonparametric FRF estimator is called “practically” consistent if its asymptotic bias remains smaller Notice that &(e. W k ) = var {€(e, W k ) } , and consequently, than 1 mdB for mutually correlated and complex normally the “true” variances and covariances { c ~ $ ~ [ I c ].&[k], , &, distributed U 0 Fourier coefficients with SNR’s larger than 10 [ k ] } k g K of the observation { X [ k ] ,Y [ k ] } & K are needed in dB. theory (the complex covariance is defined as 0 5 ~ [ k ]= Remark 2. I : The LOG-FRFE is “practically” consistent: its ave { ( X [ k ]- a v e { X [ k ] } ) ( Y [ k-] a v e { Y [ k ] } ) } )In. practice, worst case asymptotic bias equals 18 pdB (< 1 mdB) for the these “true” (co)variances will be replaced by some estimates noise settings of Definition 2.2 (see [lo] for the analytical { & $ x [ k ] , b$y[Ic],& $ y [ k ] ) k E K . As aresult of this, a bias will expressions of the bias). The AH-FFWE satisfies Definition be introduced on the parameter estimates. In Appendix A it is 2.2 also (0.8 mdB < 1 mdB). shown that this bias is approximately a linear function of the To sum up, it has been shown that the LOG-as well following (second-order) noise-to-signal ratios as the AH-are able to give “practically” consistent nonparametric estimates of a system operating in a feedback loop, there where classical spectral analysis fails to do so. It should be noted that, for open-loop systems also, the LOGFRFE outperforms the classical SA estimator when disturbing The same conclusion is valid for the GTLS and the BTLS estimator (Appendix A). input noise is present (see [lo]). The LOG-LSE presented in the next section is based upon the LOG-FRFE, which partially explains its small bias in B. The Complex-Logarithmic Least-Squares Estimator presence of (uncorrelated as well as mutually correlated) U 0 Consider the following stochastic model disturbances. Z L O G = WLOG + f L O G (21)

111. PARAMETRIC TRANSFERFUNCTION ESTIMATION A. The Errors-in-Variables Maximum Likelihood Estimator Using the frequency-domainEV equations ( 5 ) and assuming the disturbances { D [ k ] ,E [ k ] } & K to be independent complex nonnals (where the set K contains the K spectral line numbers of interest), it is possible to construct a Gaussian ML estimator to extract the system-parameters from the observed YO Fourier coefficients [21]. The ML estimates are obtained by minimizing the followingcost function with respect to 0

where f ( B , w k ) = B ( 0 , # k ) X [ k ] - .4(0, W k ) Y [ k ] is the equation error function with B(0, W k ) and A(0, W k j the numerator and denominator polynomial of the transfer function model (6) and K(0. w k ) is a weighting function which is given by

+

Vf(0,W k ) =IB(O. W k ) 1 2 0 i - ~ [ k . ] IA(0, w k ) 1 2 0 $ > . [ k ] (19) - 2Re ( B ( 6 ,W k ) . 4 ( ( 9 , W k ) C & y [ k ] ) .

with Z L O G = { Z L O G ( W k ) } k E K a K x 1 vector having as entries the complex logarithmic frequency response data, Z L O G ( W k ) = log(L$Agl((wk)) for IC E K; K a set containing the K spectral line indexes of interest; WLOG the expected value of the vector ZLOG; and f LOG a zero-mean stochastic vector. Assuming f L o G to be complex normally distributed, V L O G[ )5 ] , with VLOG a diagonal covariance matrix, the following (negative) log-likelihood function is obtained (the constant term is omitted)

With LOG(^, W k ) = log ( G ( 0 ,u k ) ) (see (6)) and V L O G ( W k ) the kth diagonal entry of the covariance matrix V L O G . According to the central limit theorem [ 6 ] ,it is sensible to P assume that log ( G i $ ; ’ ( c d k ) ) E,,=, 10g(GkM1(Uk))/P) is complex normally distributed as P -+ 00. The independency over the frequency follows from the asymptotic properties of

(=

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?

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 40,NO. 7, JULY 1995

I184

,

the Fourier coefficients [51. Nonlinear least-squares estimators obtained by taking the inverse of the Fisher information matrix, like (22) are proven to be consistent and asymptotically M = 2 Re ( J ‘ J ) , where the Jacobian matrix, J, is evaluated efficient maximum likelihood estimators when the errors, in the parameter estimates, &G, using the estimated weightinstead of the (unknown) “true” ones {FLOG ( w k ) } . are independent complex normals with mean ing function, VLOG(W~), zero and a priori known variances, {VJ,OG(W~)} (see, for instance, [25]). Notice that VLOG(W~) = var {FLOG(WL)} Z l/snr { X ( w k ) } l/snr { Y ( w k ) } (for uncorrelated YO errors). This means that spectral lines k with large U 0 SNR’s ( V L O G ( Wsmall) ~ ) will have a more pronounced impact on From [21], it also follows that, in absence of modeling errors, the cost function (22) than ones with low SNR’s. Notice also the expected value of the minima of the logarithmic maximum that this “maximum likelihood” weighting function (which likelihood cost function (22) equals K+? with K the number minimizes the uncertainties on the parameter estimates) clearly of spectral lines and n, the number of coefficients to be differs from the one proposed in [23]. estimated (usually n, = n, n b 1; see (6)). In practice, the “true” variances, { V L O G ( W ~ ) }are , unknown so that estimates, { V L O G ( W ~have ) } , to be used instead. RESULTS IV. SIMULATION Contrary to the GTLS, the BTLS, and the ML estimator ELiS, it can be proven that the LOG-LSE remains consistent as A. Robustness to Lack of Prior Noise Information E Kzero-mean random long as the errors { F L o G ( w ~ ) } ~are The aim of this section is to illustrate the robustness of the variables with jointly uncorrelated and bounded moments LOG-LSE to lack of prior noise information. By means of up to and including the fourth-order moments. Thus, only the asymptotic efficiency of the LOG-LSE is lost when the Monte-Carlo simulations, the LOG-LSE is compared with the “true” variances are unknown, not its consistency. It should be ML and the Van den Enden (VDE) estimator. Consider thereto mentioned, however, that the assumption of zero-mean errors, the feedback setup of Fig. 2 with system transfer function { F I , o G ( w ~ )is} ,not strictly correct in general. As a result of Go(w) = 10/(1 t-i w ) and controller H ( w ) = 1. The Fourier this, there is a bias, but, as was illustrated in Section 11, the coefficients of the periodic reference signal, R[k],equals one mean values of the errors, { F L o G ( w ~are ) } , so small that the for k = 1,.. . ,100 and W k = 0.1k. The synthetic U 0 Fourier bias can be neglected. When the SNR’s of the U 0 Fourier coefficients are generated in the frequency domain using the coefficients at the spectral lines k E K are larger than 10 EV equations dB, the mean value of the complex normals { F L o G ( w ~ ) } ~ E K X [ k ]= U [ k ] D [ k ] , (24) is smaller than 2 x which corresponds with an error Y [ k ]= Go(wk)U[k] E[kI of less than 18 pdB on the FRF [lo]. Using (broadband) ), periodic excitation signals like multisines for instance, which where U [ k ]= R [ k ] / ( l G o ( w k ) H ( ~ k ) while concentrate their energy at a finite number of spectral lines in the frequency band of interest, it is rather common to get Fourier coefficients with SNR’s of 20 dB or even much more, without having to average the measurements [21]. In Appendix The coefficients P [ k ] ,k = 1,. ,100, are complex normals B it is shown that the bias on the parameter estimates is with mean zero and variance 0.2. During the estimations, approximately a linear function of the mean values of the the LOG and the VDE estimator are used without weights and hence, very small. When (i.e., weighting function equal to one), while the U 0 variances complex normals {FLoG(w~)}, are ) }not normally distributed, the bias occurring in the ML estimator are set equal to zero and 0.2, the errors { F L o G ( w ~ ] 0.2, is linearly related to the fourth-order noise-to-signal moments respectively (see Section 111-A, i?$x[k] = 0, i ? & - [ k= of the U 0 Fourier coefficients. The proof of this is given in i?$y[k] = 0; the ML estimates obtained with these settings Appendix B and is also applicable to the VDE estimator as will be denote by ML(0)). Notice that, if there were no feedwell as to all other estimators which are obtained by replacing back loop, the settings of the ML estimator would be correct. the complex logarithmic function by any analytic function, The parameter estimates averaged over 20 runs, ( 6 [ i ] ) ,their (sample) standard deviation, g ( b [ i l ) , and their (sample) root.9(.): c -+ 43: 2 + g ( z ) .

+

+ +

{

Remark 3.1: To avoid problems with the phase discontinuity of the complex logarithmic function along the negative real axis (e.g., loge’((“-‘) - logeZ(”+‘)z 27r instead of zero, with E an infinite small positive number), the subtraction of the logarithmic functions occurring in (22) can be implemented as log ( ~ ( 0wk)/GL:gl(wk)) , (log ( e L ( “ - E ) / e z ( “ + E ) ) E 0). Remark3.2: Many results derived for the EV maximum likelihood estimator ELiS in [21] remain valid for the logarithmic maximum likelihood estimator (22). So, a good approximate of the covariance matrix of the parameter vector estimated with the LOG-LSE (assuming the logarithmic frequency response data to be complex normally distributed) is

+

+

+

with S the number of runs (S =’ 20), and 00 the “true” parameter vector) are given in Table I. The LOG as well as the VDE estimator perform well while the ML(0) estimates are strongly biased (i.e., the true parameter values do not fall inside the 3u uncertainty interval around the estimates; they are far away from it). According to Appendix A [see (43)-(44)], the ML parametric transfer-function estimate will converge in probability to the SA-FRF estimate [see (1 l)]

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1185

GUILLAUME er al.: ROBUST PARAMETRIC TRANSFER FUNCTION

TABLE I PARAMETER E~TIMATES (THE COEFFICIENT a1 IS FIXEDTO ONE)

e,[i]

estmator

(6[iI)

50 40

30

RMSE

c~(g[,l)

h

LOG VDE

6, = 10

MUO) MUT)

LOG VDE

U,

= 1

MU01 MUT)

9.9022 9.3892 7.9915 9.9333

0.0886 0.1820 0.0748

0.1319 0.6373 2.0094 0.1002

0.9831 0.9464 0.8750 0.9785

0.0237 0.0337 0.0159 0.0221

0.0291 0.0633 0.1260 0.0308

0.0606

20

3 10

$

0

!-20 -I0

..... .....

-30

-50 0

TABLE 11 ML(0) PARAMETER JZSTIMATES FOR A MOD= ORDER1/1 (THE C O ~ I Wa1T IS FD(ED TO ONE)

(6[ i l )

OFA [;I

b, = -0.1667 -0.1557

0.0129

bo = 8.1667

8.1228

0.0620

1.oooO

0.9717

0.0209

=

when the input variance is set equal to zero, provided that the order of the parametric transfer-function model is large enough. To verify this, the order of the transfer-function model is increased to 1/1. The new ML(0) estimates given in Table II corroborate the theoretical results. Notice that when the “true” (co)variances are used 1171, the ML, estimator is consistent (see Table I, ML(T)). In short, it has been demonstrated that the LOG and the VDE estimator remain “practically” consistent, while the ML estimator can be strongly biased when incorrect noise information is used. When the ‘ h e ” (c0)variances of the U 0 disturbances are available, the ML estimator is consistent 1171. If no (or insufficient) prior noise information is available, however, the LOG-LSE is generally to be prefened.

B. Robustness to Outliers The robustness of the LOG-LSE to outliers is the topic of this section. The same simulation data is considered as in Section IV-A except for twdoutliers: one outlier around 3 rads is added to E ( w k ) ( E ( w k ) + E ( u ) 100630,k with 6 z , J the Kronecker delta) and another at 7 rad/s to D ( w k ) ( D ( w k ) +- D ( w k ) 100&0,k ) . As a result of this, the logarithmic frequency response data contains a large positive as well as negative peak (see Fig. 4). For example, the 5 0 4 0 Hz component of the mains (and its higher order harmonics) could be the cause of outliers in the frequency response data. If one knows that there are outliers present, these lines should be removed from the data prior to the estimation. If not, these outliers, which do not satisfy the noise assumptions .(they are not stationary stochastic processes), will introduce a bias. Fig. 4 illustrates the behavior of the LOG, the W E , and the ML estimator in presence of outliers. The LOG estimate coincides very welt with the “true” FRF (only at the lowest

+

+

1

2

3

4

5

6

7

8

9

10

angular hequency (rad/s)

Fig 4. Outliers. Diamonds: frequency response data; thin solid h e : the “true” FRF,solid line: LOG, dashed line: WE dotted Line: W O ) ; thsn dotted line: MLO.

qb1,l)

< U,

.....(....,

..a

-40

frequencies a minute discrepancy can be observed see inset of Fig. 4). Notice that the LOG-LSE is rather insensitive to the outliers, compared with the VDE and the ML estimator. The reason for this seems obvious: taking the logarithm, before squaring, attenuates the effect of large outliers. In Appendix C it is shown that there exists a link with Huber’s robust statistics [111. Moreover, the effect of outliers bringing some FRF measurements very close to the singularity point zero of the logarithmic function is studid.

C. Improving the Eficiency of the Parameter Estimates

In this section, different ways of deriving the variances {VLOG(W~)} are discussed. Using the EXK;-LSE with proper weights will result in smaller parameter uncertainties. To illustrate this, synthetic data is genera@ with

{

+’

= =

(27)

I+GO(w~)W(wk)

The random variates M [ k ] ,N [ k ] , and P[k] are independent complex normals with mean zerc). The variance of the measurement noise source M [ k ] and N [ k ] equals l x the variance of the process noise P[k] is chosen as follows var { P [ k ] }=

100 110(iW&)2

+ saw& + 20012

(28) *

R[k],Go(wk), and H ( w k ) remain as in Section IV-A. When several periods are available, the sample variance can be used to weight the LOG-LSE

By smoothing (29) (i.e., by averaging over neighboring frequencies), its mean-squareerror (MSE)can be further reduced. The following example illustrates that, even when only one period is measured, it is still possible to derive an estimate of V L Q G ( ~from ~ ) the residual equation errors at the excited spectral lines. Thereto, a two-step approach is considered. Firstly, the unweighted LOG-LSE as described above is used

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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 40, NO. 7, JULY 1995

118h

TABLE III ESTIMATES: (1): UNWEIGHTED, (2): WITH THE WEIGHTING FUNCTION OBTA~NED FOR 1 = 500, (T): USING THE ‘“kuE” WEIGHTING FUNCTION

PARAMEIER

LW1) LOG(2) bo = 10 MUT) Low) LOG(1) LOG(2)

U,

=

ML(T) 0

1

2

3

4

5

6

7

8

9

LOG(T)

10

1

10.033 10.012 10.005

0.0339 0.0097 0.0079

10.005

0.0080

0.0350 0.0351 .

1.0179 1.0102 1.0084 1.0085

0.0065

0.0346

0.0047

0.0230 0.0212 0.0217

0.0045 0.0046

0.1512 0.0440

angular frequency (rad/s)

Fig. 5. Smoothing the residual errors. Diamonds: the residual errors; thin solid line: the ‘’true’’variance;solid line: estimate of the variance for 7 = 100; dashed line: y = 500; dotted line: y = 1000.

to get a “practically” unbiased parametric estimate of the transfer function, ~ ( d f J ~w k, ) . The square absolute value of the residual errors are then smoothed by means of a frequency window W , ( W ~ ) I;iOG(Wk)

-[MP]

- C k ’ E K I ’ O R (G(d!.,%2! u k ) / G L o G ( w k ) ) l CktEKW-r(Wk’

w?’(wk’

-wk)

- wk)

(e.g., the bias of the GTLS, the BTLS,and the ML estimator is inversely proportional to the (power) SNR’s of the I/O data). Moreover, it is demonstrated that the LOG-LSE is robust to outliers. Eventually, it is shown that by estimating the variance of the residual errors at the excited spectral lines, the parameter estimates of the LOG-LSE can be improved. APPENDIX A In this section, an expression will be derived for the asymptotic bias on the parameter estimates of the ML, the GTLS, and the BTLS estimator. The (equivalent) cost functions of these estimators are given by, respectively, [18]

(30)

and the resulting weighting function is used during the second estimation step. The “smoothing” procedure is illustrated in Fig. 5 for different widths of a Parzen window. The width is controlled by a scalar 7 (a large value of 7 corresponds to a narrow window; see [13, p. 15SJ). For small values of 7 the bias error dominates; for large values the bias decreases but the variance becomes important. When model errors are present, the variance of the equation error will be overestimated. The parameters estimated for y = 500 are given in Table 111. Notice that the uncertainty on the parameter estimates of the weighted LOG-LSE decreased to a level that is close to the ML(T) uncertainties (the “true” autopower and cross-power spectra of the I/O noise were used during the ML estimation [17]). When the “true” weights of the LOG-LSE are used instead of the estimated ones, almost Qe same results are obtained as with the ML estimator.

with ~ ( 8w, k ) the equation error

40,Wk) = B(8, wlc)X[k]- 4 0 , W k ) Y [ k ]

B(8, W k ) and A(8, W k ) the numerator and the denominator polynomial of the transfer function model (6),and k(8,W k ) an estimate of the variance of the equation error

k(0,wk) =IB(@!W k ) l z 6 i x [ k ] + IA(e, W k ) I 2 3 $ y [ k ] - ‘2 Re

V. CONCLUSIONS

I

In the present paper the statistical properties of the (complex) LOG-LSE have been discussed. It is shown that the bias on the parameter estimates of the LOG-LSE decreases faster than exponentially as a function of the I/O SMR’s; for the nonlinear least-squares estimator (i.e., the VDE estimator [28]), it decreases exponentially (for Gaussian I/O disturbances in the frequency domain). Moreover it is shown that the LOGLSE as well as the VDE estimator are more robust to lack of prior noise information; they remain “practically” consistent when the second-order moments of the I/O disturbances are not a priori known. This is not the case for the EV estimators

(34)

(we,

W k ) A ( @ ,W k ) + i Y [ k I ) .

(35)

The parameter vector 8, in (33) represents a prior parameter estimate [IS]. To be consistent the “true” system-parameter vector 80 must be the minimizer of the expected value of the loss function 60

= arge minaveZ{L(0)/K}.

(36)

If (36) is satisfied and if the U 0 noise vectors f [ k ] = {D[k]E , [ k ] } ,k = 1.. . ,K , are zero-mean vector-valued random variables with jointly uncorrelated and bounded moments up to (and including) the fourth-order moments, then tbe minimizer of the cost function L(8) converges in probability to 80 in l / f i as the number of spectral lines K -+ 00 [9],

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1187

GULLAUME er al.: ROBUST PARAMETRIC TRANSFER FUNCTION

[16]. The division by K in (36) is introduced so as to have a finite limit value. The expected value of the four cost functions with respect to the Fourier coefficients are

where

= G(0, wk) - GGOA(uk)

(44)

G(8, W k ) = B(0, ifjk),/A(O, w k ) and p is a constant independent of B. Clearly, LkL(0) attains a minimum for G(B,w k ) -+ G?*(W~)which (together with the results of [SI)completes our proof. Let us now consider the more general case where 6$x [k]# 0. To assess the impact of the inaccurate knowledge of the-variances on the parameter estimate, the Taylor expansion of Lo(@) around the true system-parameter vector 80 is considered where cg(f3, W k ) equals the mean value of the equation error and V,(8, wf ) its “true” variance

1/E(e. wk) =IB(e3u k ) ( 2 0 : ~ [ k+] IA(6, wk)12&[k] - 2Re(B(O, ~ k ) A ( ewk)&-[k]). ,

(40)

These cost functions can be written-as the sum of two terms: one that is linear in*leo(e, w k ) I 2 , L’(B), and another that is linear in &(e, w k ) , L 2 ( 8 ) .For instance, for the ML estimator these two terms are

The parameter vector solving

minimizing (45) is obtained by

Substituting (45) into (46)yields

L1(O)

Notice that i l ( S ) is minimal in the “true” model parameter vector BO, so that $e first-order derivative occurring in (47) only depends on L2(8), which is a linear function of the “true” covariances of the U 0 Fourier coefficients. Working out the derivatives explicitly for the ML es:imator (assuming that the “true” system belongs to the model set, i.e., V [ k ]= G(@o,w k ) U [ k ] )gives

Notice that is minimal in the “true” model parameter vector BO. If the “true” variances are known &(e, w k ) = V e ( 8 ,w k ) ) , then i2(H)= 1, and 60 is the minimizer of LO(8). Consequently, the ML, the GTLS, and the BTLS estimator are (weakly) consistent. When the “ F e ” variances are unknown (i.e., 1/F(8, w k ) # VF(e,w k ) ) , L 2 ( 8 ) will no longer be independent of 6, and consequently, 00 will not be the minimizer of Lo(0).So, the consistency property will be The second-order derivative in (47) can be approximated by lost for all these estimators. In the sequel, we will derive some explicit expressions for the (scaled) Fisher information matrix evaluated in the wrong the bias of the ML estimator. It will turn out that the bias is variances (approximately)a linear function of the second-prder moments of the observed Fourier coefficients. Because L 2 ( e ) ,which is at the origin of the bias, is a line? function of the second-order moments for ail the considered estimator, it will become clear that the conclusions can be extended to the other estimators also. with b(B, w k ) = alog(G(0, w k ) ) / a O and In [9] it has been shown that, for uncorrelated YO errors, the ML parametric transfer-function estimate converges in probability (for K -+ m)to the asymptotic SA-FRF estimate (i.e., P -+ oc) when the input variance is set equal to zero, provided that the order of the parametric transfer-function model is large enough. This result will now be extended to the case with correlation. Thereto, it suffices to rewrite the The vector iiK2 in (48) stands for the first-order derivative cost function (37) with i?$x[k] L- 0 as and depends only on i 2 ( 0 ) (af,l(B)/aOle,= 0)

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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 40,NO. 7, JULY 1995

I188

where SML(WL) is given by 6ML(Wk)

- vY(Wk)K(Wk)

= c(Wk)vY(Wk) V Z ( 4

+ %(U)

(52)

with Vz(Wli)

4 x [kl 4 Y [kl -2 IWII wm4’

wg(Wk) = g ( G O ( W k ) )

~

(53)

For K + oc,the minimizer of the ML cost function (31) converges in probability to 6zL= limK-+w6’Ei [9], [16]. Hence, the asymptotic bias of the ML estimator is (approximately) a linear function of the noise-to-signal ratios of the U 0 Fourier coefficients. One can verify that the same conclusion holds for the GTLS and the BTLS estimator. The iterated quadratic maximum likelihood (IQML) estimator ([4], [MI) has not been considered here because it is an inconsistent estimator when the “true” variances are known. When wrong variances are used instead of the “true” ones, the additional bias introduced will be approximately a linear function of the noise-to-signal ratios also. APPENDIX

B

The asymptotic bias on the parameter estimates of the LOGLSE can be assessed in exactly the same way as followed in Appendix A. To make the results as general as possible, the complex-logarithmic function will be replaced by an arbitrary function g ( - ) . We will assume that g(.) is an analytic function in the neighborhood of the points G o ( w L ) , k = l , . . . , K. Consequently, the stochastic model (21) has to be replaced by zg

= wg

+ f,

-tMg(WL).

(59)

For complex normally distributed U 0 noise, and g ( . ) = log (.), [lo1 Ei(-snr{X[k]}) - Ei(-snr{Y[k]}) (60) MLOG(WL) = 2 while for g ( . ) the identity function, [lo] MARI(WL)

= - G o ( W k ) e x p (-snr {x[k]}).

(61)

Contrary to the estimators considered in Appendix A, the bias introduced here stems from the fact that the expected value of the primary data, w g ( W k ) , differs from the “true” one, g ( G o ( w e ) ) , and not from the fact that the noise information W k ) converges in is inaccurate. For K -, m, wg(eiKl, stands for the minimizer of (56); probability to W g ( w k )(t9iK1 the model order is assumed to be large enough so that (57) tends to zero when K + CO; [9]). As a result of this, the parametric “g” estimator is “practically” consistent when the underlying nonparametric estimator is “practically” consistent. Similarly to Appendix A, the bias on the parameter estimates can be assessed using

,9y1 Bo - (j$K1)-lh$~~.

(62)

The vector aiK] is given by

(54)

with zg = { Z g ( w k ) } a K x 1 vector containing z g ( W k ) = C ~ = l g ( G ~ M 1 ( ~ k ) )k/ PE , K; tug is the expected value of zg and f, is a zero-mean stochastic vector. Notice that, for g ( . ) the identity function, Zg(w,) reduces to the ARI-FRF estimator (13). The generalized (scaled) cost function of the resulting parametric estimator becomes

with Wg(8, W L ) = g ( G ( 8 , wk))and Vg(uk)an estimate of the variance of Zg(wk). Taking the expected value of the scale cost function with respect to the data zg yields

i;(e)= avez, {2,(0))= ii(8) + ii,

Thus, if w g ( W k ) = g ( G o ( w k ) ) (and G o ( w k ) = G ( @ O , W k ) , i.e., there are no model errors), then (56) is minimal for 6’ = 80, whatever weighting function is used. So, the estimator is consistent regardless of the weighting function used. As the U 0 Fourier coefficients are corrupted by noise, however, w g ( W k ) only approximately equals g ( G O ( w k ) )

(56)

Notice that the term L;, which was at the origin of the bias error in the previous appendix, is now parameter independent.

With b(8, W k ) = dlOg(G(8, W k ) ) / 8 8

and g ’ ( - ) the derivative of g ( . ) . For K + m, the minimizer [9], of (55) converges in probability to 8 7 = limK,,8r1 [16]. Hence, the asymptotic bias on the parameter estimates is a linear function of M g ( w k ) . When the PDF of the U 0 Fourier coefficients is known (complex Gaussian for instance), analytical expressions can be derived for h f g ( W k ) (see (60) and (61) as well as [lo]). Moreover, if the PDF is unknown, it can be shown that M g ( w k ) is a linear function of the fourthorder noise-to-signal ratios of the U 0 Fourier coefficients (see [lo] for all the details) and not of the second-order moments as is the case for the estimators examined in Appendix A. Thus, the (complex) logarithmic function, g ( z ) = logz, and the identity function, g ( z ) = z , are not the only ones leading to “practically” consistent parametric estimators (e.g., g ( z ) = 1/z is a good choice too). When one knows a priori that

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GUILLAUME el OZ.: ROBUST PARAMETRIC TRANSFER FlJh’CTION

1189

1 05

K = 100 104 1 03

3

K = 200

102

E

3 101

K = 500 K = loo00

c

.-. ? 099 s 8

n = 2

098

0 91 0 96

095

-2

3

-1

0

1

2

3

a

Fig. 6. LOG cost function versus 0 = a e x p ( i n / n ) (with dashed line represents the quadratic VDE cost function.

1 001

&

= 1). The

, , ,,,,,,; , ,,,,,,,/ ,

,,,,

01 1 10 standard deviation of alpha

Fig. 7. Theoretical mean value of

, , ,,,d 100

60, LOG.

TABLE IV PARAMETER ESTlMATEs (THE COeFncIENT a0 IS FIXED M ONE) only the output (input) measurements are disturbed by noise then g ( z ) = z ( g ( z ) = l / z ) is the best choice resulting in a consistent estimator. When both the input and the output data 0 ( i o . LOG) ‘J( A.&-) ave { 6 0 , ‘00 1 are noisy then g( z ) = log z can be used (the LOG-LSE is 0.86847 0.00102 1~10-~ 0.86848 consistent when snr { X [ k ] }= snr { Y [ k ] }Vk , E K;see (60)). 0.93058 0.00110 l ~ l O - ~ 0.93059 Thus, depending on the actual SNR’s (and PDF of the noise) 1 .oooo2 1.oooO3 0.00118 1.334568 one choice of g(.) will be better than another. Anyway, for 1.06846 0.00126 1.06845 lx103 all possible choices of g(.), the asymptotic bias is a linear 0.00135 1.14487 1x106 1.14487 function of the fourth-order noise-to-signal ratios as long as g(-) is analytic. Eventually, notice that v(Wk) in (63) and vz((wk) vy((wk) The mean value of &O,LOG as a function of (J and K is in (51) are both estimates of the noise-to-signal ratio of the given in Fig. 7. For large values of K, ave{6o,mG} M frequency response data, var {G(Wk)}/lGo(Wk)l*. 1 (with y Euler’s constant) and var {io, LOG} M Hence, for U M exp (7/2) the bias vanishes. These APPENDIX c results are confirmed by means of Monte-Carlo simulations. In this appendix we will study what happens when some The simulation results (averaged over 20 runs with K = 100) FRF measurements are very close to the singularity point zero are mentioned in Table IV. Notice that the bias remains quite of the logarithmic function. To keep the computations simple, small, even when the outliers are very close to the singularity. we consider a static system Go(wk) = 1. Assume there is an An outlier with a very small or a very large variance h a almost outlier that brings one of the FRF measured close to zero, e.g., the same impact on the bias (only the sign differs).

+

+

A.

zLOG(uk) =

k ( l -k &,ka)

(66)

with the Kronecker delta and cy E Ny(-1, 02).So, one spectral line in every K is affected by an outlier while at the other spectral lines, k = 2 , . . K, Z L O G ( Wequals ~) the ‘‘true’’ value. A zero-order model is used with a0 = 1. In Fig. 6 the cost function of the LOG estimator (evaluated in the “true” parameter bo = 1) is shown as a function of Q = aexp(iT/n). For n = 2, the LOG cost function is quadratic in the neighborhood6of the origin and almost linear outside this region, analogous to the robust norm proposed in [13, pp. 396-4011, and in [19], which is based on Huber’s robust statistics [ 111. The same is roughly true for larger values of n, while for n --+ 03 the cost function blows up around a = -1 (which corresponds with a = -1). LOG estimate of the unknown It can be shown that coefficient bo is given by bo, LOG = The expected value (with respect to a ) of BO, LOG equals ave {BO, LOG} = Kfil?( 1) and its variance var {io, LOG} = K’~(r’( +1) - r (& 1)) (I?(.) stands for the gamma function). e ,

vm.

4+

+

&

REFERENCES B. D.0. Anderson, “ldentification of scalar errors-in-variablesmodels with dynamics,”Automaticu, vol. 21, no. 6,pp. 709-716, 1985. J. R. F. Arruda, “Objective functions for the nonlinear curye fit of frequency response functions,”AIAA Journal, vol. 30, no. 3, pp. 855-857,

1992. K. Beya, R. Pintelon, J. Schoukens, P. Iataire, P. Guillaume, B. Mpanda-Mabwe, and M. Delhaye, “Identification of synchronous machines parameters using broadband excitations,” IEEE Trans. Energy Conversion, vol. 9,no. 2, pp. 270-280, 1994. Y. Bresler and A. Macovski, “Exact maximum likelihood estimation of superimposed exponential sign& in noise,” IEEE Trans. Acousr., Speech, Signal Processing, vol. ASSP-34, no. 10,pp. 1081-1089,1986. D. R. Brillinger, lime Series Data Analysis and Theory. New York: Holt, Reinhart, and Winston, 1975,pp. 88-115. W. Feller, An Introduction to Probability Theory and Its Applications. New York Wiley, 1971. L. J. Gleser, “Estimation in a multivariate “errors-in-variables”regression model: Large sample results. Annals Statistics, vol. 9,pp. 24-44.

1981. C. Feng and W. Zheng, “On-line modified least-squares parameter

estimation of linear systems with input-output data polluted by measurement noises,” in Proc. 1988 IFAC Symp. Identification and System Parameter Estimation, Beijing, China, 1988,vol. 3, pp. 1189-1194.

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#

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[91 P. Guillaume, J. Schoukens, and R. Pmtelon, “On the use of signals with a constant signal-to-noiseratio in the frequency domain,” IEEE Trans. Instrum. Meas., vol. 39, no. 6, pp. 835-842, 1990. P. Guillaume, R. Pintelon, and J. Schoukens, “Nonparametric frequency response function estimators based on nonlinear averaging techniques,” IEEE Trans. Instrum. Meas., vol. 41, no. 6, pp. 739-746, 1992. P. J. Huber, Robust Staristics. New York: Wiley, 1981. L. Ljung, “On the estimation of transfer functions,” in Proc. 7th MAC Symp. System Identification and Parameter Estimation, York, United Kmgdom, 1985, vol. 2, pp. 1653-1657. ~, System Identijcation: Theory for the User. Englewood Cliffs, N J Prentice-Hall, 1987. -, “Some results on identifying linear systems using frequency domain data,” in Proc. 32nd Conf. Decis. Conrr., San Antonio, TX, Dec. 1993, pp. 3534-3538. E. Lukacs, Stochastic Convergence. New York: Academic, 1975. R. Pintelon and J. Schoukens, “Robust identification of transfer functions in the s- and zdomains,” IEEE Trans. Instnun. Meas., vol. 39, no. 4, pp. 565-573, 1990. r171 R. Pintelon, P. Guillaume, Y.Rolain, and F. Verbeyst, “Identification of linear systems captured in a feedback loop,” IEEE Trans. Instrum. Meas., vol. 41, no. 6, pp. 747-754, 1992. R. Pintelon, P. Guillaume, Y.Rolain, J. Schoukens, and H. Van hamme, “Parametric identification of transfer functions in the frequency domain, a survey,” IEEE Trans. A u : m t . Contr., vol. 39, no. 11, 1994. B. T.Poljak and Ja. 2.Tsypkin, “Robust identification,” Automatica, vol. 16, pp. 53-63, 1980. J. Schoukens and R. Pintelon, “Measurement of frequency response functions in noisy environments,” IEEE Trans, Instrum. Meas., vol. 39, no. 6, pp. 905-909, 1990. J. Schoukens and R. Pintelon, Identijkatwn of Linear Systems. A Practical Guideline to Accurate Modeling. London: Pergamon, 1991. J. Schoukens, R. Pintelon, and H. Van hamme, “Identification of linear dynamic systems using piecewise constant excitations: Use,misuse and alternatives,”Automatica, vol. 39, no. 8, pp. 1733-1737, 1994. M. D. Sidman, F. E. Dehgelis, and G C. Verghtse, “PtuamCtric system identification on logarithmic frequency response data,” IEEE Trans. Automat. Contr., vol. 36, no. 9, pp. 1065-1070, 1991. T. SijdersWm and P. Stoica, System Identljcarion. Englewood Cliffs, NJ: Prentice-Hall, 1989, p. 256. H. W. Sorenson, Parameter Estimation: Principles and Problems. New York Marcel Dekker, 1980. J. Swevers, B. De Moor, and H. Van Brussel, “Stepped sine system identification, errors-in-variablesand the quotient singular value decomposition,” Mech. Syst. Signal Processing, vol. 6, no. 2, pp. 121-134, 1992. A. Van den Bos, “Estimation of Fourier coefficients,” IEEE Trans. Insrrum. Meas., vol. 38, no. 5, pp. 1005-1007, 1989. A. W. M. Van den Enden, G. C. Oroendael, and E. Van de Zee, “An improved complex-cure fitting method,” in Proc. C o f C o m p f e r Aided Design of Electronic, Microwave Circuits and Sysr. Hull, United Kingdom, 1977, pp. 53-58. H. Van hamme and R. Pintelon, “Application of the bootstrapped total least squares (BTLS) estimator in linear system identification,”

IEFE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 40,NO. 7, JULY 1995

Signal Processing VI: Theories and Applications. J. Vandewalle et al., Eds. New York: Elsevier Science, 1992, pp. 731-734. [30] S. Van Huffe and J. Vandewalle, n e Toto1Least Squares Pmblem: Computational Aspects and Analysis. Philadelphia, PA SIAM, Frontiers in Applied Mathematics, 1991.

Patrick GuiUsume (S’87-M’87) was born in Anderlecht, Belgium, on December ’7, 1963. He received the degree of civil electrotechnicalmechanical engineer (burgerlijk ingenieur) in 1987 and the degree of doctor in applied sciences in 1992, both from the Vrije Wniversiteit Bmsel (VUB), Bmsds, Belgium. He is premtly a Senior Research Assistant of the Belgian Mational Fund for Scientific Research (”0) and part-time Lecturer at the VUB in the Electrical Measurement Department (ELEC). His research interests are parameter estimttiodsystem identification and modal analysis.

Rik Pintelon (M’90) was born in Gent, Belgium, on December 4, 1959. He received the degree-of civil electrotechnical-mechanicalengineer murgezlijk ingenieur) in 1982 and the degree of doctor in applied science in 1988, both from the Vrije Universiteit Brussel (WB), Brussels, Belgium. He is presently a Senior Research Associate of the National Fund for Scientific Research (NFWO) and part-time Lecturer at the VUB in the Electrical Measurement Department. His main research interests are in the field or parameter estimatiodsystem identification and signal processing.

J&an Sdrollkens (M90-SM’92) was born in Belgium in 1957. He received the degree of engineer in 1980 anid the degree of doctor in applied sciences in 1985 from the Vrije Universiteit Bmasel (VUB). He is presently a Senior Research Asrociate of the National Fund for Scientific Research (”’0) and pert-time at the WB. The prime factors of his research IUO in the field of system identification for linear and nonlinear systems.

P

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