Best Linear Approximation of Nonlinear Continuous-Time Systems Subject to Process Noise and Operating in Feedback

Rik Pintelon (Corresponding author), Maarten Schoukens, John Lataire

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In many engineering applications, the level of nonlinear distortions in frequency response function (FRF) measurements is quantified using specially designed periodic excitation signals called random phase multisines and periodic noise. The technique is based on the concept of the best linear approximation (BLA), and it allows one to check the validity of the linear framework with a simple experiment. Although the classical BLA theory can handle measurement noise only, in most applications, the noise generated by the system - called process noise - is the dominant noise source. Therefore, there is a need to extend the existing BLA theory to the process noise case. In this article, we study in detail the impact of the process noise on the BLA of nonlinear continuous-time systems operating in a closed loop. It is shown that the existing nonparametric estimation methods for detecting and quantifying the level of nonlinear distortions in FRF measurements are still applicable in the presence of process noise. All results are also valid for discrete-time systems and systems operating in an open loop.

Original languageEnglish
Article number9064602
Pages (from-to)8600-8612
Number of pages13
JournalIEEE Transactions on Instrumentation and Measurement
Issue number10
Publication statusPublished - Oct 2020


Manuscript received November 19, 2019; revised February 4, 2020; accepted April 4, 2020. Date of publication April 13, 2020; date of current version September 15, 2020. This work was supported in part by the Research Council of the Vrije Universiteit Brussel, in part by the Research Foundation Flanders (FWO-Vlaanderen), and in part by the Flemish Government through the Methusalem Fund METH1. The work of Maarten Schoukens was supported by the European Union’s Horizon 2020 Research and Innovation Program through the Marie Sklodowska-Curie Fellowship under Grant 798627. The Associate Editor coordinating the review process was Loredana Cristaldi. (Corresponding author: Rik Pintelon.) Rik Pintelon and John Lataire are with the Electrical Engineering Department, Vrije Universiteit Brussel, 1050 Brussel, Belgium (e-mail:;

FundersFunder number
European Union 's Horizon 2020 - Research and Innovation Framework Programme
Research Foundation Flanders
Marie Skłodowska‐Curie
Horizon 2020 Framework Programme798627
Fonds Wetenschappelijk Onderzoek
Vlaamse regering


    • Best linear approximation (BLA)
    • continuous-time
    • feedback
    • frequency response function (FRF)
    • nonlinear systems
    • nonparametric estimation
    • process noise


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