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
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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
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
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
"Non-parametric identification of hi..." refers methods in this paper
...For non-line ar systems which can be approximated by Volterra series the Related Linear Dynamic System can be i dentified using multi-sine based signals [2, 4, 5, 6, 7]....
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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
"Non-parametric identification of hi..." refers background in this paper
...The GF RFs have their limitations: systems with non-fading memory, like non-local memory hysteresis as see n in friction, can not be described with GFRFs and GFRFs are difficult to interpret due to their multid imensional nature [17, 18]....
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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
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
"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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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