Time domain identification, frequency domain identification.Equivalencies! Differences?

J. Schoukens, R. Pintelon, and Y. RolainVrije Universiteit Brussel, Department ELEC, Pleinlaan 2, B1050 Brussels, Belgium

email: johan.schoukens@vub.ac.be

Abstract

- In the first part of this paper, the full equiva-lence between time and frequency domain identification isestablished. Next the differences that show up in the practi-cal applications are discussed. Finally, an illustration on theidentification of a servo-system in feedback is given.

I. INTRODUCTION

For a long time, frequency domain identification and timedomain identification were considered as competingmethods to solve the same problem: building a model for alinear time-invariant dynamic system. In the end, thefrequency domain approach got a bad reputation because thetransformation of the data from the time domain to thefrequency domain is prone to leakage errors: noiseless datain the time domain resulted in noisy frequency responsefunction (FRF) measurements. This is illustrated in thesimulation below. A system is excited with a random input.The input and the output are sampled in 256points with . No disturbingnoise is added. Starting from these measurements, the FRF

is measured and compared to the true FRF (FIG.1.). It can be seen that is strongly disturbed. This was themajor reason to drop the frequency domain approach. Thestatement: Why would we move from the time to thefrequency domain? The only thing we buy for it areproblems! expresses very well the feeling that lived in theidentification society.

This problem is further analysed in Section III where itis shown that i) exactly the same problem is present in thetime domain; ii) by extending the models, a full equivalenceexists between both domains.

Once this equivalence between both domains is estab-lished, one can wonder why to bother about it? Are thereany differences at all? The answer is definitely yes.Although both domains carry exactly the same information,it may be more easy to access this information in one

u0 t( ) y0 t( )u0 k( ) y0 k( ), k 1 2 … 256, , ,=

Ĝ jωl( ) G0

-40

-20

0

20

0 0.1 0.2 0.3 0.4 0.5

Am

plitu

de (

dB)

Frequency

FIG. 1. Comparison of the true FRF (__)with the estimatedFRF (+)

G0Ĝ

Ĝ

domain than in the other because the same information isrepresented differently.

When discussing the differences between time- and fre-quency domain methods, a clear distinction should be madebetween the aspects that are intrinsically due to the fre-quency domain formulation, and on the other hand the addi-tional signal processing possibilities that are opened byusing periodic excitations. The latter might also be usefulfor time domain identification methods.

The practical aspects that take advantage of the fre-quency domain formulation are: arbitrary selection of theactive frequencies where the model is matched to the meas-urements; continuous-time modelling; and identification ofunstable models. By including also periodic excitations, wewill be able to address in addition: the use of nonparametricnoise models; separation of plant and noise model estima-tion; errors-in-variables identification and identificationunder feedback; separation of nonlinear distortions and(process) noise. These aspects are discussed in Section IVand V. A comprehensive discussion of other aspects can befound in (Pintelon and Schoukens, 2001; Ljung, 2004).

In Section VI, an extensive case study on the closedloop identification of a compact disc servo-system is pre-sented. A frequency domain approach using periodic excita-tions is made. Many of the aspects mentioned before areillustrated.

II. SETUP

The discussion in this paper is completely focused onsingle-input-single-output linear systems (SISO), but thereader should be aware that many results can be directlyextended and generalized to multiple-input-multiple-output(MIMO) systems, and even a nonlinear behaviour can beincluded in the framework (Schoukens

et al.

, 2003). Consider the discrete time system

, (1)

with , and . The aim ofsystem identification is to extract the best model for the plant , and at times the disturbing noise powerspectrum .

III. TRANSIENTS: THE KEY TO THE EQUIVALENCE OF TIME AND FREQUENCY DOMAIN

In practice the identification should be done from a finite setof measurements

, . (2)

The description (1) should be extended to include theimpact of reducing the observation window from

y t( ) G0 q( )u0 t( ) H0 q( )e t( )+ y0 t( ) v t( )+= =

t : ∞– ∞→ x t 1–( ) q 1– x t( )=G q θ,( )

G q( )H q θ,( ) 2

u0 t( ) y t( ), t 0 1 … N 1–, , ,=

to . This is illustrated in FIG.2. The records are split in three parts: the observation win-dow, the preceding and following unobserved signals. Twoeffects can be seen:

i) The past excitation (before the start of the observation) contributes to the output in the observation window:

a tail is added to the observed output (FIG. 2.a). Theseeffects are well known in the time domain, a transient term

has to be added to describe the contributions of theinitial conditions to the output.

ii) When the data are processed in the frequency domain,it is implicitly assumed that the observed signals are periodi-cally repeated. At that moment not only the tail (a) is addedto the output, also the tail (b) will be missing. Although thislooks more complicated than the previous situation it leadseventually to expressions that are completely equivalent tothe time domain description. Both extended models aregiven below.

A. Extended time domain description

The reduction from to addsinitial conditions effects on the output of the dynamicsystems (plant and noise filter). These are described by theplant and the noise filter transients . Both decayexponentially to zero.

, . (3)

B. Extended frequency domain description

The finite data records aretransferred to the frequency domain using the discreteFourier transform (DFT):

, (4)

with , and . Although the spectra are affected by leakage, their relation remains

remarkably simple: it is again sufficient to add a transientterm to the system equations, completely similar to what is

done in the time domain (Pintelon

et al.

, 1997; Pintelon andSchoukens, 2001). This term describes the impact of the tails(a) and (b) on the input-output relation.

. (5)

The relation between and is:

,

with . (1)

The reader should be aware that this expression is valid forarbitrary signals, no periodic excitation is required.

Loosely spoken, the impact of the transients disappearsat a rate of or faster. In practice the noise transient

is always omitted. For simplicity we do the same in thispaper.

C. Equivalence between time- and frequency domain

During the identification step, parametric plant- and noisemodels are identified by minimizing the squared predictionerrors. This can be done in the time- or in the frequencydomain (Ljung, 1999; S derstr m, and Stoica, 1989):

( stands for the data). (6)

Both expressions result in exactly the same value for thecost function, and this for arbitrary excitations. This estab-lishes the full equivalence between the time- and frequencydomain formulation of the prediction error framework.

Remark: The transfer function model and the transientmodel have very similar expressions:

, (7)

Hence the inclusion of the transient term does not increasethe complexity of the cost function because both rationalforms have the same denominator.

D. Differences?

Although we established a full equivalence between thetime- and frequency domain formulation, there are someremarkable differences in the practical use of theseexpressions.

1) Unstable plant models: If an unstable plant model isidentified, the time domain calculations are tedious and oftenimpossible because the calculation noise explodes. FromForsell and Ljung (2000) it is known that the predictorshould be stable . In the frequency domain, this prob-lem is not a problem because only multiplications of thefinite DFT-spectra are made, and the stability of the models

t : ∞– ∞→ t 0 … N 1–, ,=

-3

0

3

0 200 400-3

0

3

0 200 400

-3

0

3

0 200 400-3

0

3

0 200 400

-3

0

3

0 200 400-3

0

3

0 200 400

-3

0

3

0 200 400-3

0

3

0 200 400

u(t)

y(t)

u(t)

u(t)

u(t)

y(t)

y(t)

y(t)

observation window

FIG. 2. : Illustration of the effect of the finite observation window.

(a)

(b)

t 0=

TG t( )

t : ∞– ∞→ t 0 … N 1–, ,=

TG t( ) TH t( ),

y t( ) G0 q( )u0 t( ) H0 q( )e t( ) TG t( ) TH t( )+ + += t 0≥

{u0 t( ) y t( ), : t 0 … N 1}–, ,=

X k( ) 1

N-------- x t( )e

j2πN------ tk–

t 0=N 1–

∑= k 0 1 … N 1–, , ,=

x u y e, ,= X U Y E, ,=U0 Y0,

Y0 k( ) G0 zk1–( )U0 k( ) TG k( )+=

U0 E, Y

Y k( ) G0 zk1–( )U0 k( ) H0 zk

1–( )E k( ) TG k( ) TH k( )+ + +=

zk ej2πk N⁄=

N 1 2/–

TH

VPE θ Z,( ) H1– q θ,( ) y t( ) G q θ,( )u0 t( )– TG q θ,( )–( )

2

t 0=

N 1–

∑=

Y k( ) G zk1– θ,( )U0 k( )– TG zk

1– θ,( )– 2

H zk1– θ,( ) 2

-----------------------------------------------------------------------------------------k 0=

N 1–

∑=

Z

G zk1– θ,( )

bnzn–

n 0=

nb∑

anzn–

n 0=

na∑------------------------------= TG zk

1– θ,( )inz

n–n 0=

max na nb,( )∑

anzn–

n 0=

na∑----------------------------------------=

H 1– G

is not at the order, even not for the intermediate models dur-ing the iteration process (The zeros of should be kept inthe unite circle!).

2) It is very easy in the frequency domain to restrict thesum to a selected set of frequencies, for example the fre-quency band of interest; to those frequencies with a goodsignal-to-noise ratio; or to eliminate spurious components.Although this should not affect the prediction error result (itcan be interpreted as a maximum likelihood estimator, andthrowing away information does never improve the result), ithas a strong impact on the calculation of initial estimateswith sub optimal (linear) methods like ARX or subspaceidentification.

IV. PERIODIC SIGNALS: A FREE ACCESS TO THE NONPARAMETRIC NOISE MODEL

A. Introduction

The prediction error method can be considered as thesolution of a weighted least squares problem:

(8)

with , , the covariance matrix of . The major problem is

that is a large dense matrix that should be inverted whichis impractical. This is nicely circumvented in the predictionerror method by whitening the residuals with a parametricnoise model , leading to

. (9)

This is exactly the time domain form of in (5). Inthe frequency domain the matrix inversion does not comeinto play. The covariance matrix of the frequencydomain noise is asymptotically ( ) a diagonal matrix:

, (10)

with . This gives an alternative fre-quency domain representation of the cost function:

.(11)

The full details and the proof of the equivalence of (5) and(11) are given in Schoukens et al. (1999).

B. Taking advantage of periodic excitations

In the classical prediction error framework, the plantmodel and the noise model are identifiedsimultaneously because this is the only possibility to sepa-rate the signal and the noise . However, if the exci-tation is periodical, , it is possible to collect

successive periods, and to average the measurementsover these repeated periods. This process is exemplified forthe output measurement in FIG. 3.:

,

. (12)

The sample variance is a nonparametric estimate of. Substituting it in the frequency domain expression of

(11) gives:

. (13)

This results in a two step procedure: i) The nonparametricnoise model is determined in the pre-processing step, ii) the plant model is estimated inthe 2nd step, keeping the noise model fixed.

C. Discussion

This approach has many advantages: i) It is no longerrequired to estimate plant and noise model simultaneously;ii) Even before starting the identification process, it ispossible to verify the quality of the raw data as is illustratedin the example of Section VI; iii) The estimated noise modelis no longer influenced by the plant model errors; iv) Thismethod can be extended to the errors-in-variables problem.This includes identification in feedback as a special case.

The price to be paid for all these advantages is therestriction to periodic signals and the need for multiple peri-ods to be measured, resulting in a frequency resolution loss.However, the required number of periods can be small,for example is enough for consistency, and reduces the efficiency loss to less than 33% in variance(Schoukens et al., 1997).

D. Extension: the errors-in-variables problem

In some applications, there is not only process noise on theoutput. Also the input measurements can be disturbed bynoise, as is for example the case for identification in

feedback. The nonparametric noise model is extended withthe input noise variance and the input-outputcovariance ( denotes the complex conjugate):

H

V v̂T

Cv1– v̂=

v̂T

v̂ 1( ) … v̂ N( )= v̂ k( ) y t( ) G q θG,( )u0 t( )–=Cv N N× v

Cv

ε t( ) H 1– q θ,( )v t( )=

VPE θ Z,( ) εTε=

VPE θ Z,( )

CVN ∞→

CV diag σY2 1( ) … σY

2 N( )( )=

σY2 k( ) H ejωk( ) 2=

VPE θ Z,( )Y k( ) G zk

1– θ,( )U0 k( )– TG zk1– θ,( )– 2

σY2 k( )

-----------------------------------------------------------------------------------------k 0=

N 1–

∑=

G q θ,( ) H q θ,( )

y0 t( ) v t( )u t T+( ) u t( )=

M

Ŷ k( ) 1M----- Y l[ ] k( )

l 1=M

∑=

σ̂Ŷ2 k( ) 1

M----- 1

M 1–-------------- Y l[ ] k( ) Ŷ k( )–

2

l 1=M

∑=

u t( )

y 1[ ] t( ) y 2[ ] t( ) y l[ ] t( )t

FIG. 3. Processing periodic excitations: is the period.y l[ ] t( ) lthFIG. 3. Processing periodic excitations: is the period.y l[ ] t( ) lth

… …

σ̂Y2 k( )

σY2 k( )

VPE

VPE θ Z,( )Ŷ k( ) G zk

1– θ,( )Û k( )– TG zk1– θ,( )–

2

σ̂Y2 k( )

--------------------------------------------------------------------------------------k 0=

N 1–

∑=

σ̂Y2 k( ) var Y k( )( )=

G zk1– θ,( )

MM 4= M 6=

u0 t( ) y0 t( ) y t( )

u0 t( ) y0 t( ) y t( )

ny t( )

v t( )

u t( )

nu t( )

FIG. 4. : Standard and alternative noise assumption.

+

+

+

plant

plantAlternative

Standard

FIG. 4. : Standard and alternative noise assumption.

σ̂U2 k( )

σ̂YU2 k( ) x

(14)

The corresponding cost function to be minimized is thesample maximum likelihood (Pintelon and Schoukens,2001):

(15)

The use of nonparametric noise models combined with iden-tification in feedback is illustrated in SectionVI.

V. FREQUENCY DOMAIN IDENTIFICATION: A HIGHWAY TO CONTINUOUS TIME

IDENTIFICATION

A. Arbitrary excitation signals

In the last theory section of the paper, we combine theadvantages of the time- and frequency domain identificationapproaches. A continuous-time modelling procedure isproposed using arbitrary excitations. The relation betweenthe spectra of band limited measurements is the continuous-time model(see FIG. 5):

. (16)

By combining this with a discrete time noise model, a mixedBox-Jenkins continuous time identification method is foundthat is based on the minimization of the cost function (Pinte-lon et al., 2000):

. (17)

The major advantages of this method compared to clas-sical continuous time identification techniques are that: i)The need for approximate differentiation or integration iscompletely removed; ii) There is no need for huge oversam-pling rates, the full bandwidth can be used; iii) A noisemodel is included which increases the efficiency; iv) It is alogical extension of the well known Box-Jenkins identifica-tion method.

The major drawback compared with the discrete timeBox-Jenkins method is the loss of consistency in feedbackwhich is due to the mixture of a discrete- and continuous-time model.

B. Periodic excitation signals

For periodic excitation signals the sample maximumlikelihood method (15) can be used without any modifica-tion to identify a continuous-time model.

VI. CASE STUDY: IDENTIFICATION OF A SERVO- SYSTEM IN CLOSED LOOP OPERATION

In this example we illustrate all the aspects that werediscussed before:

- use of periodic signals - extraction of a non parametric noise model - use of a selected set of frequencies - removal of large spurious components- separation of the signals, the noise, and the nonlinear

distortions- identification of a continuous or discrete time model in

feedback using the errors-in-variables frame work- identification of an unstable model (due to nonlinear

distortions);

A. Experimental set-up

The open loop transfer function of the radial posi-tion servo-system of a CD player is identified. Figure 6shows a simplified block diagram of the compact disc (CD)

player measurement setup. The block stands for the cas-cade of a power amplifier, a lowpass filter, the actuator sys-tem and, finally, the optical position detection system. Theblock stands for the parallel implementation of a lead-lagcontroller with some additional integrating action, that stabi-lizes the unstable actuator characteristics and takes care forthe position control. In order to excite and to measure theopen loop transfer function, two operational amplifiers havebeen inserted in between the lead-lag controller and thepower amplifier at the input of the process .

B. Need for closed loop identification

The actuator transfer function represents the dynamics of thearm moving over the compact disc, and is, in a firstapproximation, proportional to . In practice, due to thefriction, the double pole at the origin moves into the left halfplane to a highly underdamped position. This explains whythe characteristics of the position mechanism of a CD-playeris very hard to measure in open loop.

C. Use of the errors-in-variables framework

An external reference signal is injected in the loop, theresulting signals are measured (the input is not exactlyknown). Moreover, the loop is also disturbed by the processnoise , mainly induced by tracking irregularities due topotato shaped spirals; non eccentric spinning of the disc;

σ̂YU2 k( ) 1

M----- 1

M 1–-------------- Y l[ ] k( ) Ŷ k( )–( ) U l[ ] k( ) Û k( )–( )

l 1=M

∑=

VML θ Z,( )

Ŷ k( ) G Ωk θ,( )Û k( )–2

σ̂Y2 k( ) σ̂

U2 k( ) G Ωk θ,( )

2 2Re σ̂YU2 k( )G Ωk θ,( )( )–+

------------------------------------------------------------------------------------------------------------------------k 1=

F

∑=

ZOH plant plant

AAAA

ZOH- BL-

u t( )

u t( )

y t( )

y t( )

ucyc yc

FIG. 5. : Ideal ZOH- and BL-setup. AA: an ideal anti-alias filter.

G s θ,( ) bnsn

n 0=

nb∑ ansnn 0=na∑⁄=

VPE θ Z,( )Y k( ) G s θ,( )U k( )– TG zk

1– θ,( )– 2

H zk1– θ,( ) 2

--------------------------------------------------------------------------------k 0=

N 1–

∑=

GC

FIG. 6. Setup of the CD measurements.

r u G C y-

++

d

G

C

CG

1 s2⁄

ru y,

d

dirt, stains and scratches on the disc surface. The followingrelations exist between the Fourier spectra (assuming thatthey all exist):

(18)

In the absence of disturbances, the open loop transfer func-tion between and is and it is the aim of thissection to provide a parametric model for it. A periodic refer-ence signal will be applied, the contributions

are considered as noise.Hence the input and output measurements are disturbed bymutually correlated noise which fits perfectly in the errors-in-variables approach.

D. Design of a periodic excitation, use of a well selected set

of frequencies

For the sake of control, a 582.5 Hz sinusoidal wobble signalis internally injected in the feedback loop. It is measured atdifferent points in the electronic circuit, and serves as aninput signal for an automatic gain controller (AGC), tocompensate, amongst other things, for the effect that thedisplacement of the arm is not perpendicular to the trackover the whole disc, resulting in a variable gain of theprocess. The wobble signal complicates the measurementprocess significantly, since it is more than 20 dB above thenormal signal levels in the loop. For this reason, we had tomake a careful experiment design to eliminate its impact onthe measurements.

As an external reference signal a multisine with ,

, , and, is used (Schoukens and Pintelon, 2001).

The frequencies are selected to allow for the detection ofnonlinear distortions at the non-excited frequencies. Themultisine is generated with a clock frequency of

and points in one period.

E. Preprocessing: extraction of the signals and noise levels

In the first experiment, 256 K points are measured. The longrecord is broken in 16 blocks of 4 basic periods each( ). This is done in order to reduce the leakage effectof the wobble signal on the rest of the spectrum. Themeasurement window does not contain an integer number ofperiods of the wobble signal since its frequency is notsynchronized to the measurement system (Figure 7). In thisfigure, it can be seen that the contribution of the referencesignal is clearly above the disturbances level. Also thewobble signal (with its leakage) is clearly visible, itsamplitude is more than 20 dB above the signals of interest.

Starting from the 16 repeated spectra, are esti-mated, together with the (co-)variances andshown in Figure 8. It turns out that there is an extremely highcorrelation ( ) between the noise on and . From (18), it is seen that this indicates that the proc-ess noise dominates completely the measurement noise

F. Quantifying the nonlinear distortions

Checking the non-excited odd frequencies in Figure 7 seemsto indicate the presence of odd nonlinear distortions(Schoukens et al., 1998; Pintelon and Schoukens, 2001;Schoukens et al. 2003), but they are almost completelyhidden under the noise level of the test. Hence, a secondexperiment with a reduced set of frequencies:( with ) is made. This does not affectthe relative level of the nonlinearities with respect to thelinear contributions, but it increases the SNR (Pintelon andSchoukens, 2001). From the results in Figure 9 it is seen thatthe odd distortions are now well above the noise level of thetest. Especially at the lower frequencies a very highdistortion can be seen, indicating that the linearised models

U jω( ) 11 G jω( )C jω( )+--------------------------------------R jω( ) C jω( )

1 G jω( )C jω( )+--------------------------------------D jω( )–=

Y jω( ) G jω( )C jω( )1 G jω( )C jω( )+--------------------------------------R jω( ) C jω( )

1 G jω( )C jω( )+--------------------------------------D jω( )+=

u y G jω( )C jω( )

r t( )C jω( )D jω( )( ) 1 G jω( )C jω( )+( )⁄

r t( ) Ak 2πf0lkt ϕk+( )sink 1=F∑= F 305=

f0 2.3842 Hz= lk 1 3 9 11 17 19 25 27 …, , , , , , , ,=Ak constant=

10 MHz 210⁄ N 4096=

M 16=

Û Ŷ,σ̂U

2 σ̂y2 σ̂YU

2, ,

σYU2 σU

2 σy2⁄ 1–≈ U

Y

Input Output

-100

-80

-60

-40

-20

0 2500 5000

Am

plitu

de (

dB)

Frequency (Hz)

-100

-80

-60

-40

-20

0 2500 5000

Am

plitu

de (

dB)

Frequency (Hz)

Signal

Signal

Disturbances Disturbances

Wobble Wobble

FIG. 7. Pilot test with a special odd multisine signal.

-100

-80

-60

-40

0 1000 2000 3000

Am

plitu

de (

dB)

Frequency (Hz)

signal

noise

FIG. 8. SNR of the output measurements after processing the rawdata.

Y

σ̂Y

f0 19.07 Hz= F 39=

-100

-80

-60

-40

-20

0 1000 2000 3000A

mpl

itude

(dB

)Frequency (Hz)

Linear

Cubic

FIG. 9. nonlinearity test: power on the FRF measurementfrequencies (linear), detected distortions after compensationfor the linear feed through (cubic), --- .σY

will have poor value in this frequency band.

G. Identification of a continuous or discrete time model

After these non-parametric tests, we have already a goodidea about the limiting quality of the model. For the giveninput power, the nonlinearities are for sure less than 30 dBbelow the linear output. Next a linear model is identified thatapproximates the system as good as possible. A 24th orderdiscrete-time or continuous time model ( )was identified (both give very similar results). The measuredFRF is compared to the parametric model in Figure 10. Ascan be seen, a very good fit is obtained. The residuals arebelow the noise level. Only at the low frequencies, where thenonlinearities detected in the nonlinearity test are very large,the fit is poor. Because we knew in advance that in this bandthe data are of poor quality, the frequencies below 230 Hzwere not considered during the fit. The cost function of thefit is 842.6, while a theoretical value of 256.5 is expected.This points to model errors. However, the auto-correlation ofthe residuals is white, therefore we can conclude that the bestlinear approximation is extracted. The remaining errors aredue to the nonlinear behaviour of the process.

A stability analysis showed that two poles of the modelwere unstable ( ), but the correspond-ing closed loop model is stable and hence the model is valu-able for a closed loop analysis. This instability is due to thefact that the system has 2 poles, almost equal to one (doubleintegration in -domain), that are very difficult to estimatedue to the presence of the nonlinearities in this band.

VII. CONCLUSIONS

In this paper the equivalencies and differences of time- andfrequency domain identification are discussed. Thisdiscussion should not be mixed up with the use of periodicexcitations.

The major conclusion is that there exist a full equiva-lence between both approaches from theoretic and informa-tion point of view. Transforming data from the time to the

frequency domain does neither create or delete information.Hence, what can be done in one domain can also be done inthe other domain. However, in practice it might be easier toaccess the information in one of both domains. Arbitraryselection of active frequency bands, continuous time model-ling, and identification of unstable models are typical exam-ples.

It is also shown that the restriction to periodic inputsopens a number of practical possibilities and this for time-and frequency domain identification. It does not only allowto extract nonparametric noise models, it also simplifies sig-nificantly the identification under feedback conditions andgives access to a nonlinear distortion analysis.

VIII. ACKNOWLEDGEMENT

This work was supported by the Flemish government (GOA-ILiNos), and the Belgian government as a part of the Belgianprogram on Interunivrsity Poles of Attraction (IUAP V/22).

REFERENCES

[1] Forsell, U. and L. Ljung (2000). Identification of unstablesystems using output error and Box-Jenkins model structures.IEEE Transactions on Automatic Control, 45, 131-147, 2000.

[2] Ljung, L. (1987 and 1999). System Identification: Theory forthe User (second edition). Prentice Hall, Upper Saddle River.

[3] Ljung, L. (2004). State of the art in linear systemidentification: time and frequency domain methods. ACC2004.

[4] Pintelon, R., J. Schoukens and G. Vandersteen (1997).Frequency domain system identification using arbitrarysignals. IEEE Trans. on Automatic Control, 42, 1717-1720.

[5] Schoukens, J., R. Pintelon and Y. Rolain (1999). Study ofConditional ML estimators in time and frequency-domainsystem identification. Automatica, 35, 91—100.

[6] Pintelon R., J. Schoukens, and Y. Rolain. Box-Jenkinscontinuous-time modeling (2000). Automatica, 36, 983-991.

[7] Pintelon R. and J. Schoukens (2001). System Identification. Afrequency domain approach. IEEE-press, Piscataway.

[8] Schoukens, J., R. Pintelon, G. Vandersteen and P. Guillaume(1997). Frequency-domain system identification using non-parametric noise models estimated from a small number ofdata sets. Automatica, 1073-1086.

[9] Schoukens J. , T. Dobrowiecki, and R. Pintelon (1998).Identification of linear systems in the presence of nonlineardistortions. A frequency domain approach. IEEE Trans.Autom. Contr., Vol. 43, No.2, pp. 176-190.

[10] Schoukens J., R. Pintelon, T. Dobrowiecki, and Y. Rolain(2003). Identification of linear systems with nonlineardistortions. 13th IFAC Symposium on System Identification.Rotterdam, The Netherlands.

[11] S derstr m, T. and P. Stoica (1989). System Identification.Prentice-Hall, Englewood Cliffs

na nb 24= =

-60

-20

20

0 1000 2000 3000

Am

plitu

de (

dB)

Frequency (Hz)

0

90

180

270

0 1000 2000 3000

Phas

e (d

eg.)

Frequency (Hz)

FIG. 10. : Comparison of the estimated transfer function (full line)with the measured FRF (dots). The residuals (+) are compared tothe 95% noise level (thin full line).

z 1.021 j0.00344±=

z

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Header: Proceeding of the 2004 American Control ConferenceBoston, Massachusetts June 30 - July 2, 2004Footer: 0-7803-8335-4/04/$17.00 ©2004 AACCSession: WeM01.2Page0: 661Page1: 662Page2: 663Page3: 664Page4: 665Page5: 666