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Proceedings ArticleDOI

Bias Removal in Higher Order Sinusoidal Input Describing Functions

TL;DR: A novel method is presented for the reduction of bias caused by harmonic excitation in the identification of higher order sinusoidal input describing functions (HOSIDF) and is demonstrated with real measurements on a mechanical system with friction.
Abstract: In this paper a novel method is presented for the reduction of bias caused by harmonic excitation in the identification of higher order sinusoidal input describing functions (HOSIDF). HOSIDF are a recently introduced generalization of the theory of the describing function. HOSIDF describe the magnitude and phase relations between the individual harmonic components in the output signal of a non-linear system and the sinusoidal excitation signal. In the presented method, the output signal of a non-linear system subjected to harmonic excitation is numerically split up into a fraction caused by the non-linear response due to the fundamental input signal component and the fraction caused by the quasi-linear response due to the harmonic input signal components. This separation is based on the assumption that the non-linear effects of intermodulation can be neglected, compared to the the effects caused by the generation of harmonics and gain compression/expansion. The method is demonstrated with real measurements on a mechanical system with friction.
Citations
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27 Aug 2003
TL;DR: A theoretical framework is proposed that extends the linear system description to include the impact of nonlinear distortions: the nonlinear system is replaced by a linear model plus a 'nonlinear noise source'.
Abstract: This paper studies the impact of nonlinear distortions on linear system identification. It collects a number of previously published methods in a fully integrated approach to measure and model these systems from experimental data. First a theoretical framework is proposed that extends the linear system description to include the impact of nonlinear distortions: the nonlinear system is replaced by a linear model plus a 'nonlinear noise source'. The class of nonlinear systems covered by this approach is described and the properties of the extended linear representation are studied. These results are used to design the experiments; to detect the level of the nonlinear distortions; to measure efficiently the 'best' linear approximation; to reveal the even or odd nature of the nonlinearity; to identify a parametric linear model; and to improve the model selection procedures in the presence of nonlinear distortions.

119 citations

Journal ArticleDOI
TL;DR: In this article, the authors proposed a frequency domain-based method for detection and optimal compensation of performance degrading nonlinear effects in Lur'e-type systems, where a sinusoidal response is necessary and sufficient to show the existence of an equivalent linear and time invariant dynamical model that fully captures the system dynamics for a well defined set of input signals and initial conditions.
Abstract: SUMMARY Nonlinearities often lead to performance degradation in controlled dynamical systems. This paper provides a new, frequency domain-based method, for detection and optimal compensation of performance degrading nonlinear effects in Lur'e-type systems. It is shown that for such systems a sinusoidal response to a sinusoidal input is necessary and sufficient to show the existence of an equivalent linear and time invariant dynamical model that fully captures the systems’ dynamics for a well-defined set of input signals and initial conditions. This allows to quantify nonlinear effects by using a frequency domain performance measure and yields a novel method to design optimized static compensator structures that minimize performance degrading nonlinear effects. Moverover, the methods discussed in this paper allow to quantify the performance of nonlinear systems on the basis of output measurements only while requiring little knowledge about the nonlinearity and other system dynamics, which yields a useful tool to optimize performance in practice without requiring advanced nonlinear modeling or identification techniques. Finally, the theoretical results are accompanied by examples that illustrate their application in practice.Copyright © 2013 John Wiley & Sons, Ltd.

11 citations

References
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Journal ArticleDOI
TL;DR: The quick method to measure directly the points of the Volterra kernels which are located along one or more axes in the frequency domain is developed, thus making the method complete and more accurate than that presented in Reference 1.
Abstract: Accurate measurement of Volterra kernels is an essential step in the modelling of weakly non-linear physical systems via the block box approach, and in model validation. This paper is a sequel to an earlier paper,1 where practical methods for measuring Volterra kernels were presented along with a quick method for measuring the second-order Volterra kernels. This paper extends that quick method for measuring higher-order frequency-domain Volterra kernels of weakly non-linear systems. We further developed the quick method to measure directly the points of the Volterra kernels which are located along one or more axes in the frequency domain, thus making our method complete and more accurate than that presented in Reference 1. We illustrate our method by actually measuring a weakly non-linear circuit whose Volterra kernels can be accurately calculated. the experimentally measured and the theoretically predicted results agree remarkably well.

96 citations


"Bias Removal in Higher Order Sinuso..." refers methods in this paper

  • ...The high numerical cost however limits these methods to the identification of GFRFs up to a maximum order of three [14], [15]....

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Journal ArticleDOI
TL;DR: A class of broadband pilot test signals is proposed, termed sparse odd multisines, which can be used to establish the system bandwidth and detect nonlinearities, and signals are defined within this class which allow the measurement of the best linear approximation of a nonlinear system.
Abstract: This paper examines the effects of nonlinearities on frequency response function measurements using periodic multifrequency signals. A class of broadband pilot test signals is proposed, termed sparse odd multisines, which can be used to establish the system bandwidth and detect nonlinearities. Signals are then defined within this class which allow the measurement of the best linear approximation of a nonlinear system. A comparison is made with related work in this area.

63 citations


"Bias Removal in Higher Order Sinuso..." refers methods in this paper

  • ...For non-linear systems which can be approximated by Volterra series the Related Linear Dynamic System can be identified using multi-sine based signals [2], [4], [5], [6], [7]....

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Journal ArticleDOI
TL;DR: In this article, analytical and graphical methods of interpreting generalised frequency response functions for non-linear systems are derived and it is shown that nonlinear phenomena can be classified into intra-kernel and inter-kernel interference and that worstcase responses can be computed.
Abstract: Analytical and graphical methods of interpreting generalised frequency response functions for non-linear systems are derived. It is shown that non-linear phenomena can be classified into intra-kernel and inter-kernel interference and that worst-case responses can be computed. The results are illustrated using several discrete- and continuous-time non-linear systems.

54 citations


"Bias Removal in Higher Order Sinuso..." refers background in this paper

  • ...The GFRFs have their limitations: systems with non-fading memory, like non-local memory hysteresis as seen in friction, can not be described with GFRFs and GFRFs are difficult to interpret due to their multidimensional nature [17], [18]....

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Journal ArticleDOI
TL;DR: In this paper, the characteristics of generalized frequency response functions (GFRFs) of non-linear systems in higher dimensional space are investigated using a combination of graphical and symbolic decomposition techniques.
Abstract: The characteristics of generalized frequency response functions (GFRFs) of non-linear systems in higher dimensional space are investigated using a combination of graphical and symbolic decomposition techniques It is shown how a systematic analysis can be achieved for a wide class of non-linear systems in the frequency domain using the proposed methods The paper is divided into two parts In Part 1, the concepts of input and output frequency subdomains are introduced to give insight into the relationship between one dimensional and multi-dimensional frequency spaces The visualization of both magnitude and phase responses of third order generalized frequency response functions is presented for the first time In Part 2 symbolic expansion techniques are introduced and new methods are developed to analyse the properties of generalized frequency response functions of non-linear systems described by the NARMAX class of models Case studies are included in Part 2 to illustrate the application of the new methods

42 citations

Journal ArticleDOI
TL;DR: This paper examines the effects of nonlinear distortions on frequency response functions estimated using multisine test signals, to minimize the distortion introduced by the nonlinearity, for a given input power constraint.
Abstract: For Pt. I see ibid. vol. 49, pp. 602-609, 2000. This paper examines the effects of nonlinear distortions on frequency response functions estimated using multisine test signals. The aim is to minimize the distortion introduced by the nonlinearity, for a given input power constraint. A number of different multisine signals are compared for this purpose, with zero, random and low crest factor harmonic phases. The results are compared with those of other authors in this field.

38 citations


"Bias Removal in Higher Order Sinuso..." refers methods in this paper

  • ...The frequency domain based techniques for the analysis of nonlinear systems mentioned in literature can roughly be classified in three groups: Identification of the linear system In weakly non-linear systems the true linear system can be identified by minimizing the influence of the non-linear distortion using odd multisine excitation signals with minimized crest factor and with an amplitude kept as small as possible [1], [2], [3]....

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