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.
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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.
115 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.
Abstract: When analyzing and modeling dynamical systems in the frequency domain, the effects of nonlinearities need to be taken into account. This paper contributes to the analysis of the effects of nonlinearities in the frequency domain by supplying new analytical tools and results that allow spectral analysis of the output of a class of nonlinear systems. 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. The theoretical results are illustrated by examples that show both the use and efficiency of the proposed algorithms.
27 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.
Abstract: The widespread acceptance of frequency domain techniques for linear and time invariant systems has been an impetus for the extension of these methodologies toward nonlinear systems. However, differences and equivalences between alternative methods have been less addressed. 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. Each method is introduced using consistent nomenclature and terminology, which allows for comparison in terms of system and signal classes for which the methods are valid as well as the type of (nonlinear) effects captured by each model. Summarizing, the paper aims to connect, and make different frequency domain methods for nonlinear systems accessible, by providing a comparative overview of such methodologies, accompanied by illustrative (experimental) examples.
20 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.
10 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
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Book•
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
2,278 citations
TL;DR: In this article, a digital feed-forward control algorithm for tracking desired time varying signals is presented, which is particularly suited to the general motion control problems including robotic arms and positioning tables.
Abstract: A digital feedforward control algorithm for tracking desired time varying signals is presented. The feedforward controller cancels all the closed-loop poles and cancellable closed-loop zeros. For uncancellable zeros, which include zeros outside the unit circle, the feedforward controller cancels the phase shift induced by them. The phase cancellation assures that the frequency response between the desired output and actual output exhibits zero phase shift for all the frequencies. The algorithm is particularly suited to the general motion control problems including robotic arms and positioning tables. A typical motion control problem is used to show the effectiveness of the proposed feedforward controller.
1,397 citations
"Non-parametric identification of hi..." refers methods in this paper
...The learning filterL can be designed with the ZPETC algorithm [31] and the resulti ng phase delay ofl samples is absorbed in the two delay blocks....
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Book•
01 Jan 1968
TL;DR: The theory of automatic control has been advanced in important ways during recent years, particularly with respect to stability and optimal control, but these theories do not, however, lay to rest all questions of importance to the control engineer.
Abstract: ABRAMSON Information theory and coding BATTIN Astronautical guidance BLACHMAN Noise and its effect on communication BREMER Superconductive devices BROXMEYER Inertial navigation systems GELB AND VANDER VELDE Multiple-input describing functions and nonlinear system design GILL Introduction to the theory of finite-state machines HANCOCK AND WINTZ Signal detection theory HUELSMAN Circuits, matrices, and linear vector spaces KELSO Radio ray propagation in the ionosphere MERRIAM Optimization theory and the design of feedback control systems MUUM Biological control systems analysis NEWCOMB Linear multiport synthesis PAPOULIS The fourier integral and its applications R. N. BRACEWELL) STEINBERG AND LEQUEUX (TRANSLATOR Radio astronomy WEEKS Antenna engineering PREFACE The theory of automatic control has been advanced in important ways during recent years, particularly with respect to stability and optimal control. These are significant contributions which appeal to many workers, including the writers, because they answer important questions and are both theoretically elegant and practically useful. These theories do not, however, lay to rest all questions of importance to the control engineer. The designer of the attitude control system for a space vehicle booster which, for simplicity, utilizes a rate-switched engine gimbal drive, must know the characteristics of the limit cycle oscillation that the system will sustain and must have some idea of how the system will respond to attitude commands while continuing to limit-cycle. The designer of a chemical process control system must be able to predict the transient oscillations the process may experience during start-up due to the limited magnitudes of important variables in the system. The designer of a radar antenna pointing system with limited torque capability must be able to predict the rms pointing error due to random wind disturbances on the antenna, and must understand how these random disturbances will influence the behavior of the system in its response to command inputs. But more important than just being able to evaluate how a given system will behave in a postulated situation is the fact that these control engineers must design their systems to meet specifications on important characteristics. Thus a complicated exact analytical tool, if one existed, would be of less value to the designer than an approximate tool which is simple enough in application to give insight into the trends in system behavior as a function of system parameter values or possible compensations, hence providing the basis for system design. As an analytical tool to answer questions such as these in a way …
1,227 citations
"Non-parametric identification of hi..." refers background in this paper
...• Describing Functions The Describing Function concept extends the FRF in the way th t identification of amplitude dependency becomes possible [19, 20]....
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Book•
27 Jan 2005
1,120 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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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.
844 citations