Non-parametric identification of higher order sinusoidal output describing functions
01 Jan 2008-pp 2361-2371
TL;DR: The HOSODF can be defined for the class of causal, stable, time invariant non-linear systems which give a sinusoidal response to a specific harmonic excitation of non- linear systems.
Abstract: In this paper the concept of the Higher Order Sinusoidal Output Describing Functions (HOSODF) is presented. HOSODF can be defined for the class of causal, stable, time invariant non-linear systems which give a sinusoidal response to a specific harmonic excitation. The HOSODF relate the magnitude and phase of the individual harmonics, which together compose that specific input signal, to the sinusoidal output signal of such a system. HOSODF are the dual of the Higher Order Sinusoidal Input Describing Functions (HOSIDF). Like the HOSIDF, the HOSODF are the results of an extension of linear techniques towards non-linear systems analysis. Using the HOSODF, the non-linear systems under investigation can be modeled as a cascade of the HOSODF and a Virtual Harmonics Compressor (VHC). The VHC is defined as a non-linear component which transforms a harmonic input signal y(t) into a sinusoidal output signal y(t) with frequency ω, amplitude â and phase φ. This input signal y(t) consists of an infinite amount of harmonics of the output signal y(t) with frequency nω, amplitude â and phase nω with n = 0, 1, ...∞. Special attention is paid to the non-parametric identification of the HOSODF. The identification requires control of the frequency and amplitude of the sinusoidal output of the system within its domain of possible sinusoidal output signals. This specific state of these non-linear systems can be reached by incorporating the system under test in a feedback loop. In this loop the desired sinusoidal output is defined as the control objective of a dedicated repetitive controller consisting of a memory loop with positive feedback. The design of the learning filter required for stability is also addressed. As a spinoff of the identification technique, the authors see opportunities for advanced non-linear control of shaker systems aimed at sinusoidal excitation of non-linear systems.
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: A mapping from the parameters defining the nonlinear and LTI dynamics to the output spectrum is derived, which allows analytic description and analysis of the corresponding higher order sinusoidal input describing functions.
29 citations
TL;DR: This paper provides a comparative overview of four classes of frequency domain methods for nonlinear systems: Volterra based models, nonlinear frequency response functions / Bode plots, describing functions and linear approximations in the presence of nonlinearities.
26 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
Cites methods from "Non-parametric identification of hi..."
...This is done as well in [16] by application of a describing function approach....
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TL;DR: In this paper, the authors present a book on multivariable system theory and design for postgraduate students with a focus on pole assignment, frequency domain, design techniques and robust servomechanism problem.
Abstract: Multivariable System Theory and Design: RAJNIKANT V. PATEL and NEIL MUNRO (Pergamon Press, 1982, 374 pp., £19.50 hardback, £9.50 paperback) There are some books whose presentation and content one immediately takes a liking to, and this book falls into that category. The book seems very appropriate for use by M.Sc. and Ph.D. students and it is also very up-to-date. I particularly liked the chapters on poles and zeros of multivariable systems, pole assignment, frequency domain, design techniques and the robust servomechanism problem. It is also nice to see the method of inequalities due to Zakian included in a student text (for the first time, I believe).The book contains an extensive list of references which should be helpful to research students. The subject is very mathematical but the treatment does not rely on advanced mathematics. The book should be of value to control engineers working in industry, but it does not address itself to the practical aspects of engineering problems. However, the description of the INA design technique which was developed at UMIST and is readily available on CAD computer packages will be valuable to engineers. The characteristic locus frequency domain design method (developed at Cambridge) which has also found wide application is described and examples given. It is a credit to the authors that most of the modern multi variable design techniques are considered and not only those that they have developed. I would have preferred more material on optimal control and a chapter on Kalman filtering but there is, of course, a limit on the size of such a text. In all, the book is well balanced and should be high on the list of recommended texts for postgraduate students. M. J. GRIMBLE, Professor ofElectrical Engineering. University ofStrathclyde. Glasgow
6 citations
References
More filters
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
"Non-parametric identification of hi..." refers methods in this paper
...The frequency domain based techniques for the analysis of non-linear systems mentioned in literature can roughly be c lassified in three groups: • Identification of the linear system In weakly non-linear systems the true linear system can be id entified by minimizing the influence of the non-linear distortion using odd multisine excitation s ignals with minimized crest factor and with an amplitude kept as small as possible [1, 2, 3]....
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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
TL;DR: In this paper, the authors present a book on multivariable system theory and design for postgraduate students with a focus on pole assignment, frequency domain, design techniques and robust servomechanism problem.
Abstract: Multivariable System Theory and Design: RAJNIKANT V. PATEL and NEIL MUNRO (Pergamon Press, 1982, 374 pp., £19.50 hardback, £9.50 paperback) There are some books whose presentation and content one immediately takes a liking to, and this book falls into that category. The book seems very appropriate for use by M.Sc. and Ph.D. students and it is also very up-to-date. I particularly liked the chapters on poles and zeros of multivariable systems, pole assignment, frequency domain, design techniques and the robust servomechanism problem. It is also nice to see the method of inequalities due to Zakian included in a student text (for the first time, I believe).The book contains an extensive list of references which should be helpful to research students. The subject is very mathematical but the treatment does not rely on advanced mathematics. The book should be of value to control engineers working in industry, but it does not address itself to the practical aspects of engineering problems. However, the description of the INA design technique which was developed at UMIST and is readily available on CAD computer packages will be valuable to engineers. The characteristic locus frequency domain design method (developed at Cambridge) which has also found wide application is described and examples given. It is a credit to the authors that most of the modern multi variable design techniques are considered and not only those that they have developed. I would have preferred more material on optimal control and a chapter on Kalman filtering but there is, of course, a limit on the size of such a text. In all, the book is well balanced and should be high on the list of recommended texts for postgraduate students. M. J. GRIMBLE, Professor ofElectrical Engineering. University ofStrathclyde. Glasgow
6 citations
"Non-parametric identification of hi..." refers background in this paper
...• Identification of the Generalized Frequency Response Funct ion For the class of causal, stable, time-invariant, non-linea r systems with fading memory 1, the convolution integral description of the linear system can be generalize d to an infinite series called the Volterra series [9, 10, 11]....
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