# 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.

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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

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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

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31 Dec 2003TL;DR: Focusing mainly on frequency domain techniques, System Identification: A Frequency Domain Approach, Second Edition also studies in detail the similarities and differences with the classical time domain approach.

Abstract: Preface to the First Edition Preface to the Second Edition Acknowledgments List of Operators and Notational Conventions List of Symbols List of Abbreviations Chapter 1 An Introduction to Identification Chapter 2 Measurement of Frequency Response Functions Standard Solutions Chapter 3 Frequency Response Function Measurements in the Presence of Nonlinear Distortions Chapter 4 Detection, Quantification, and Qualification of Nonlinear Distortions in FRF Measurements Chapter 5 Design of Excitation Signals Chapter 6 Models of Linear Time-Invariant Systems Chapter 7 Measurement of Frequency Response Functions The Local Polynomial Approach Chapter 8 An Intuitive Introduction to Frequency Domain Identification Chapter 9 Estimation with Know Noise Model Chapter 10 Estimation with Unknown Noise Model Standard Solutions Chapter 11 Model Selection and Validation Chapter 12 Estimation with Unknown Noise Model The Local Polynomial Approach Chapter 13 Basic Choices in System Identification Chapter 14 Guidelines for the User Chapter 15 Some Linear Algebra Fundamentals Chapter 16 Some Probability and Stochastic Convergence Fundamentals Chapter 17 Properties of Least Squares Estimators with Deterministic Weighting Chapter 18 Properties of Least Squares Estimators with Stochastic Weighting Chapter 19 Identification of Semilinear Models Chapter 20 Identification of Invariants of (Over) Parameterized Models References Subject Index Author Index About the Authors

2,379 citations

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27 Jan 2005

1,152 citations

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

...the convolution integral description of the linear system can be generalized to an infinite series called the Volterra series [9], [10], [11]....

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TL;DR: In this article, it was shown that any time-invariant continuous nonlinear operator with fading memory can be approximated by a Volterra series operator, and that the approximating operator can be realized as a finite-dimensional linear dynamical system with a nonlinear readout map.

Abstract: Using the notion of fading memory we prove very strong versions of two folk theorems. The first is that any time-invariant (TI) continuous nonlinear operator can be approximated by a Volterra series operator, and the second is that the approximating operator can be realized as a finite-dimensional linear dynamical system with a nonlinear readout map. While previous approximation results are valid over finite time intervals and for signals in compact sets, the approximations presented here hold for all time and for signals in useful (noncompact) sets. The discretetime analog of the second theorem asserts that any TI operator with fading memory can be approximated (in our strong sense) by a nonlinear moving- average operator. Some further discussion of the notion of fading memory is given.

923 citations

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RAND Corporation

^{1}TL;DR: Results, both old and new, which will aid the reader in applying Volterra-series-type analyses to systems driven by sine waves or Gaussian noise are presented.

Abstract: Troublesome distortions often occur in communication systems. For a wide class of systems such distortions can be computed with the help of Volterra series. Results, both old and new, which will aid the reader in applying Volterra-series-type analyses to systems driven by sine waves or Gaussian noise are presented. The n-fold Fourier transform G n of the nth Volterra kernel plays an important role in the analysis. Methods of computing G n from the system equations are described and several special systems are considered. When the G n are known, items of interest regarding the output can be obtained by substituting the G n in general formulas derived from the Volterra series representation. These items include expressions for the output harmonics, when the input is the sum of two or three sine waves, and the power spectrum and various moments, when the input is Gaussian. Special attention is paid to the case in which the Volterra series consists of only the linear and quadratic terms.

479 citations

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

...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]....

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TL;DR: A related linear dynamic system (RLDS) approximation to the nonlinear system (NLS) is defined, and it is shown that the differences between the NLS and the RLDS can be modeled as stochastic variables with known properties.

Abstract: This paper studies the asymptotic behavior of nonparametric and parametric frequency domain identification methods to model linear dynamic systems in the presence of nonlinear distortions under some general conditions for random multisine excitations. In the first part, a related linear dynamic system (RLDS) approximation to the nonlinear system (NLS) is defined, and it is shown that the differences between the NLS and the RLDS can be modeled as stochastic variables with known properties. In the second part a parametric model for the RLDS is identified. Convergence in probability of this model to the RLDS is proven. A function of dependency is defined to detect and separate the presence of unmodeled dynamics and nonlinear distortions and to bound the bias error on the transfer function estimate.

276 citations