Bias Removal in Higher Order Sinusoidal Input Describing Functions
12 May 2008-pp 1653-1658
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
More filters
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
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
More filters
TL;DR: This paper examines the output properties of static power-series nonlinearities driven by periodic multiharmonic signals with emphasis given to their effect on linear frequency response function (FRF) measurements, based on the classification of nonlinear distortions into harmonic and interharmonic contributions.
Abstract: This paper examines the output properties of static power-series nonlinearities driven by periodic multiharmonic signals with emphasis given to their effect on linear frequency response function (FRF) measurements. The analysis is based on the classification of nonlinear distortions into harmonic and interharmonic contributions. The properties of harmonic contributions are examined in detail and explicit formulae are derived, by which the number of harmonic contributions generated at the test frequencies can be calculated for odd-order nonlinearities up to, and including, the ninth order. Although an analytic solution for any odd-order nonlinearity is still under investigation, a heuristic methodology is developed that solves this problem. It is shown that the derived formulae provide a useful tool in the examination of the behavior of FRF measurements in the presence of nonlinear distortions. Based on these formulae, different approaches in classifying nonlinear distortions are then compared with respect to their suitability in assessing the influence of system nonlinearities on linear FRF measurements.
29 citations
TL;DR: In this article, two measuring techniques are presented for measuring the higher order sinusoidal input describing functions (HOSIDF) of a non-linear plant operating in feedback.
26 citations
TL;DR: In this article, a non-parametric frequency domain based measurement technique is introduced that enables capturing the stick to gross sliding transition of a mechanical system with dry friction, which is an extension of the Sinusoidal Input Describing Function theory (SIDF) to higher order describing functions (HOSIDF).
19 citations
21 May 2001
TL;DR: In this article, the authors examined the minimisation of signal crest factor using a frequency domain method and found that the technique does not produce the lowest possible crest factors but that the signals do minimise the nonlinear distortion introduced by a cubic nonlinearity.
Abstract: This paper examines the minimisation of signal crest factor using a frequency domain method. Multisine signals are designed with their relative harmonic phases selected to minimise the distortion introduced by a cubic nonlinearity on the measurement of a frequency response function. This approach is based on the observation that low crest factor signals possess properties that reduce the nonlinear effect. The resulting signals are compared with other types of low crest factor signals. It is found that the technique does not produce the lowest possible crest factors but that the signals do minimise the nonlinear distortion introduced by a cubic nonlinearity. The signals are thus termed minimum distortion multisines.
9 citations
01 Jan 1995
TL;DR: In this paper, it was shown that an irrational pencil Σ of curves cannot be contained in a linear system of dimension r, where r is necessarily ≧ 2, because the curves of the pencil are of virtual degree zero (II, 1).
Abstract: In II, 1 we have accepted for temporary purposes a definition of an algebraic system Σ of curves C on a surface F, as a system cut out on F by an algebraic system Σ′ of hypersurfaces. It will serve as a preliminary clarifying remark if we point out immediately why that definition is not sufficiently general. The base loci of the given system Σ′ of hypersurfaces determine a complete linear system of hypersurfaces of the same order as the hypersurfaces of Σ′. This linear system cuts out on F a linear system of curves of the same order as the curves C and containing Σ. Hence Σ is totally contained in a linear system of curves. However—and this is a fundamental point of the theory, which will be discussed in section 3 of this chapter—there exist surfaces (notably, irregular surfaces) which carry algebraic systems of curves not contained in linear systems. The simplest example is given by surfaces carrying an irrational pencil Σ of curves. Obviously, such a pencil (supposing for simplicity that the curves of the pencil are irreducible) cannot be contained in a linear system of dimension r, where r is necessarily ≧2, because the curves of the pencil are of virtual degree zero (II, 1). If the pencil is reducible, it is seen immediately that its curves are composed of the curves of another irrational pencil, and the statement that Σ is not contained in a linear system is essentially equivalent to the statement to the statement that an irrational involution of sets of points on an algebraic curve cannot be contained in a linear series.
9 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]....
[...]
...Then-dimensional Fourier transform of then-th order Volterra kernel yields the n-th-order FRF, the Generalized Frequency Response Function (GFRF) [12], [13]....
[...]
...GFRFs can be estimated with nonparametric methods if the system can be described by a low order (truncated) Volterra kernel....
[...]
...Parametric identification techniques ease the numerical requirements and allow analytical expressions for GFRFs up to any order [16]....
[...]
...The n-dimensional Fourier transform of the n-th order Volterra kernel yields the n-th-order FRF, the Generalized Frequency Response Function (GFRF) [12], [13]....
[...]