Under modeling and Border effect of Local Polynomial method

Research output: Chapter in Book/Report/Conference proceedingMeeting abstract (Book)


A non-parametric system identification method, the Local Polynomial (LP) method, was recently developed to remove the leakage errors, determine a non-parametric model of the linear part of the system, and estimate the covariance matrix of the additive noise. The transient response in time domain translates to leakage in the frequency domain, which causes errors in the frequency response function (FRF) estimation. The LP method assumes that the transient and the FRF are smooth functions of frequency, while the excitation is assumed to be a rough function of the frequency (e.g. random noise or random phase multisine). This enables the separation of the transient and the FRF by locally approximating both by a polynomial function of the frequencies. In this way, a significant part of the leakage can be removed. At the same time, interpolation errors are introduced by both the transient and the FRF polynomials. The interpolation errors are significant where the transient and/or FRF fluctuate fast, and hence demand a high order polynomial approximation. The order of the polynomial and the considered frequency interval, on which the polynomials are estimated, are controlled by the user. In most cases, this frequency interval can be chosen symmetrically around the frequency line at which the transient and/or FRF are estimated. At the borders, e.g. around DC, there are not enough frequency lines available, and hence the frequency interval becomes asymmetric. This poster discusses the non-parametric FRF estimate obtained using the LP method. The focus is put on the increase of the estimated variance of the FRF estimate at the border. This increase is caused by the under modeling of the transient/FRF at the border. The estimated variance does not only contain the true variance but also a bias error due to under modeling. This bias error, and thus the estimated variance, can be decreased by increasing the order of the polynomial estimate. Another solution exists in decreasing the amount of frequency lines used for the estimation of the polynomial. However, the amount of frequency lines always needs to be at least as large as the amount of parameters for the polynomial estimate. That is why we propose to use all the frequency lines (excited as well as non-excited) and estimate the FRF and transient at the same time. In this way the total amount of frequency lines remains large enough for estimating the parameters while the amount of excited frequency lines decreases. That is why the bias of the FRF estimate vanishes and thus the estimated variance decreases at the border in the latter case. The variance increases slightly at all the frequency lines. This increase in variability is, however, negligible compared to the decrease of the bias at the border. A simple simulated SISO system with added white noise is used to illustrate this problem.
Original languageEnglish
Title of host publicationPresentation of poster at the European Research Network on System Identification workshop (ERNSI 2011), Nice (France), 25-28 September 2011
Publication statusPublished - 25 Sep 2011
EventUnknown -
Duration: 25 Sep 2011 → …


Period25/09/11 → …


  • non-parametric system identification method
  • Local Polynomial (LP) method
  • frequency response function (FRF) estimation


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