Extension list

In this list we present extensions to the material presented in the book, R. Pintelon, and J. Schoukens (2001). System Identification - A Frequency Domain Approach. IEEE Press, Piscataway.

E.1 Probability density function of FRF measurements using periodic excitation signals

Section 2.5.1 (p. 48) discusses the stochastic properties of FRF measurements using periodic excitation signals. The mean and variance of the FRF are calculated assuming that the signal- to-noise ratio (SNR) of the input measurements is sufficiently large. In Pintelon et al. 2003 the exact probability density function of the FRF measurement is given. It allows to calculate exact confidence regions for any input SNR.

Pintelon, R., Y. Rolain and W. Van Moer (2003). Probability density function for frequency response function measurements using periodic signals, IEEE Trans. Instrum. Meas., vol. 52, no. 1, pp. 61-68.

The material of the paper is covered by the technical report.

E.2 Study of the properties of FRF measurements in the presence of nonlinear distortions

The asymptotic behavior of the related linear dynamic system (Theorem 3.12 on page 80) and of the variance of the stochastic nonlinear distortions (Theorem 3.16 on page 82) are illustrated on simulation and real measurement examples in Pintelon and Schoukens (2002). Two methods for measuring the related linear dynamic system, the disturbing noise variance and the variance of the stochastic nonlinear distortions are presented in Pintelon et al. (2004), Part I and II.

Pintelon, R. and J. Schoukens (2002). Measurement and modelling of linear systems in the presence of non-linear distortions, Mechanical Systems and Signal Processing, vol. 16, no. 5, pp. 785-801.

The material of the paper is covered by the technical report.

Pintelon, R., G. Vandersteen, L. De Locht, Y. Rolain, and J. Schoukens (2004). Experimental characterization of operational amplifiers: a system identification approach - Part I: theory and simulations, IEEE Trans. Instrum. Meas., vol. 53, no. 3, pp. 854-862.

The material of the paper is covered by the technical report.

Pintelon, R., Y. Rolain, G. Vandersteen, and J. Schoukens (2004). Experimental characterization of operational amplifiers: a system identification approach - Part II: calibration and measurements, IEEE Trans. Instrum. Meas., vol. 53, no. 3, pp. 863-876.

The material of the paper is covered by the technical report.

E.3 Uncertainty calculation in the presence of model errors

Expressions (7-93) p. 216 (maximumum likelihood estimate) and (8-16) p. 286 (sample maximum likelihood estimate) of the covariance matrix of the estimated model parameters are accurate in the absence of model errors only. In Pintelon et al. (2003) explicit expressions are derived which are valid for "small" model errors and "large" signal-to-noise ratios. These analytic expressions establish a clear link between the parameter uncertainty, the type of model errors (unmodelled dynamics and nonlinear distortions), the type of excitation signal (deterministic, random phase, and random), and the noise level. Although the predicted uncertainty can be several dB off the actual uncertainty, the improvement w.r.t. the classical formula (7-93) and (8-16) is significant and good enough for most applications.

Pintelon, R., J. Schoukens and Y. Rolain (2003). Uncertainty of transfer function modeling using prior estimated noise models, Automatica, vol. 39, no. 10, pp. 1721-1733.

The material of the paper is covered by the technical report.

E.4 Uncertainty calculation in the absence of model errors

In the absence of model errors the expressions for the covariance matrix of the estimated model parameters (see eq. (7-93) p. 216, and eq. (8-16) p. 286) are accurate for sufficiently large signal-to-noise ratios only. Indeed these expressions are obtained by neglecting the second term in the asymptotic covariance matrix (see eq. (17-32) p. 543). In Pintelon (2006) exact expressions for the asymptotic covariance matrix are derived which are valid for circular complex normally distributed input-output noise. It follows that the second term in the asymptotic covariance matrix is important only if the input and output signal-to-noise ratios are both very low, and if the input-output errors are weakly correlated.

Pintelon, R (2006). Asymptotic covariance matrix of the frequency domain maximum likelihood estimator using non-parametric noise models, Internal Report no. RP070306, Vrije Universiteit Brussel, dept. ELEC, Pleinlaan 2, 1050 Brussel, Belgium.

The material of the paper is covered by the technical reportUP

E.5 Frequency scaling in continuous-time modeling

To avoid bad numerical conditioning, frequency scaling is indispensable in continuous-time modeling. Section 7.4 (p. 188) advices to scale the frequency axis with the arithmetic mean of the maximal and minimal frequency. In Pintelon and Kollar (2004) it is shown that the median of the frequency set is a better compromise for reducing the condition number.

Pintelon, R. and I. Kollar (2004). On the frequency scaling in continuous-time modeling, IEEE Trans. Instrum. Meas., vol. 53, no. 5.

The material of the paper is covered by the technical report.

E.6 Asymptotic properties of frequency domain subspace algorithms

Sections 7.14 (p. 221) and 8.6 (p. 292) describe frequency domain subspace algorithms using respectively the true and the sample covariance matrix of the disturbing noise. The strong convergence and the strong consistency of both algorithms are shown. Pintelon (2002) proves also the convergence rate and the asymptotic normality, and quantifies the loss in efficiency of the sample subspace algorithms (Section 8.6) w.r.t the subspace algorithms (Section 7.14).

Pintelon, R. (2002). Frequency domain subspace system identification using non-parametric noise models, Automatica, vol. 38, no. 8, pp. 1295-1311.

The material of the paper is covered by the technical report.

E.7 Modified AIC and MDL rules

Sections 9.3.2 (p. 329) and 17.7 (p. 550) give the Akaike information criterion (AIC) and maximum description lenght (MDL) rules for detecting overmodeling. The rules as presented in these sections are unable to handle undermodeling and short data records. In Schoukens et al. (2002) a modification of the AIC and MDL rules is presented that allows model selection in the presence of model errors, for example, nonlinear distortions and/or unmodeled dynamics. De Ridder et al. (2004) gives AIC and MDL extensions for handling short data records where the number of parameters is not much smaller than the amount of data.

Notes

(i) Although the modified AIC rule has been derived within a prediction error framework it is also valid for frequency domain identification within an errors-in-variables framework. It is sufficient to replace N by 2*F in the multiplicative correction term.

(ii) In the absence of model errors the modified AIC rule (multiplicative correction term) reduces to the original AIC rule (additive correction term).

Schoukens, J., Y. Rolain, and R. Pintelon (2002). Modified AIC rule for model selection in combination with prior estimated noise models, Automatica, vol. 38, no. 5, pp. 903-906.

The material of the paper is covered by the technical report.

De Ridder F., R. Pintelon, J. Schoukens, and D.P. Gillikin (2002). Modified AIC and MDL model selection criteria for short data records, IEEE Trans. Instrum. Meas., vol. 53, no. 5.

The material of the paper is covered by the technical report.

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