Higher-order sinusoidal input describing functions for the analysis of non-linear systems with harmonic responses
01 Nov 2006-Mechanical Systems and Signal Processing (Academic Press Inc.)-Vol. 20, Iss: 8, pp 1883-1904
TL;DR: In this paper, an extension to higher-order describing functions is realized by introducing the concept of the harmonics generator, which relates the magnitude and phase of the higher harmonics of the periodic response of the system to the magnitude of a sinusoidal excitation.
About: This article is published in Mechanical Systems and Signal Processing.The article was published on 2006-11-01. It has received 100 citations till now. The article focuses on the topics: Describing function & Harmonic (mathematics).
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12 May 2008
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.
2 citations
TL;DR: In this article, the authors investigated the statistical distributions of the reradiated spectrum from an ensemble of nonlinear devices subjected to radiofrequency interference measured in a reverberation chamber and showed that the Monte Carlo simulations based on the statistical models can properly reconstruct the measured statistical distributions.
Abstract: The statistical distributions of the reradiated spectrum from an ensemble of nonlinear devices subjected to radio-frequency interference measured in a reverberation chamber are investigated. This phenomenon can be used to probe the interaction between threatening radio-frequency energy and digital hardware. Previously, a statistical model was developed to explain the statistics of the measured spectrum from a single nonlinear element. Here, the model is extended to the more realistic situation with multiple reradiating sources. It is observed in the measurement results that two parameters, the power level of the interference and the spatial distribution of the reradiating sources, have effects on the statistical distribution. It is shown that the Monte Carlo simulations based on the statistical models can properly reconstruct the measured statistical distributions. Understanding the signature of the onset of hard nonlinearity in the observed spectrum may lead to the development of an immunity diagnostic tool that does not depend on monitoring the functional behavior of the equipment-under-test.
2 citations
Additional excerpts
...input describing functions (HOSIDFs) [5], [7]....
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Posted Content•
08 May 2020TL;DR: This paper analyzes the CgLp behaviour from the perspective of first as well as higher-order harmonics and presents simple relations between the tuning parameters and the gain-phase behaviour of all the harmonics, which can be used for better tuning of these elements.
Abstract: This paper presents the tuning of a reset-based element called "Constant in gain and Lead in phase" (CgLp) in order to achieve desired precision performance in tracking and steady state. CgLp has been recently introduced to overcome the inherent linear control limitation - the waterbed effect. The analysis of reset controllers including ones based on CgLp is mainly carried out in the frequency domain using describing function with the assumption that the relatively large magnitude of the first harmonic provides a good approximation. While this is true for several cases, the existence of higher-order harmonics in the output of these elements complicates their analysis and tuning in the control design process for high precision motion applications, where they cannot be neglected. While some numerical observation-based approaches have been considered in literature for the tuning of CgLp elements, a systematic approach based on the analysis of higher-order harmonics is found to be lacking. This paper analyzes the CgLp behaviour from the perspective of first as well as higher-order harmonics and presents simple relations between the tuning parameters and the gain-phase behaviour of all the harmonics, which can be used for better tuning of these elements. The presented relations are used for tuning a controller for a high-precision positioning stage and results used for validation.
2 citations
Cites background or methods from "Higher-order sinusoidal input descr..."
...Hence the idea of higher-order sinusoidal input describing function (HOSIDF) presented in Nuij et al. (2006) was extended for reset controllers by Heinen (2018)....
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...Hence the idea of describing functions was extended to include the higher-order harmonics by Nuij et al. (2006) for nonlinear systems in general as higherorder sinusoidal input describing functions (HOSIDFs) and was applied for reset controllers by Heinen (2018)....
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TL;DR: In this paper, a relation between different frequency response functions has been established and sufficient conditions for this relation to exist and results on uniqueness and equivalence of the HOSIDF and GFRF are provided.
2 citations
Cites methods from "Higher-order sinusoidal input descr..."
...An example of such method are the higher order sinusoidal input describing functions [Nuij et al., 2006, Rijlaarsdam et al., 2011a], which are recently shown to be especially suitable for optimal pre-compensator design for nonlinear systems [Rijlaarsdam et al., 2011b]....
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...The HOSIDFs are introduced in [Nuij et al., 2006] and further discussed in [Rijlaarsdam et al., 2011a]....
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TL;DR: The authors constructs approximate analytical expressions of periodic disequilibrium fluctuations business cycles occurring in connection with Hopf bifurcations in nonlinear problems of economic interactions described by four-dimensional continuous time dynamical systems.
Abstract: Paper constructs approximate analytical expressions of periodic disequilibrium fluctuations business cycles occurring in connection with Hopf bifurcations in nonlinear problems of economic interactions described by four-dimensional continuous time dynamical systems. Two nonlinear macrodynamic models are employed as tests models. The region of equilibrium stability in parameter space is obtained in each case and a Hopf bifurcation curve is identified as a boundary of the region. Validity of the analytical approximations obtained for the cycles generated by the loss of equilibrium stability on this curve is confirmed by comparison to numerically determined cycles. Explicit analytical description of such limit cycles is of particular interest in the case of subcritical bifurcation, due to the difficulty of the numerical determination of the generated unstable cycles. Mathematics Subject Classification: 34A34, 34C07, 34C23, 37G10
2 citations
Cites methods from "Higher-order sinusoidal input descr..."
...In fact the HB method is the Describing Function (DF) Technique introduced by Krylov and Bogolybov [9] (see also [6], [22]) and has been widely used in classical Nonlinear Feedback Control Theory (see for example [21], [17] and [11])....
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References
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01 Jan 1991
TL;DR: Covers in a progressive fashion a number of analysis tools and design techniques directly applicable to nonlinear control problems in high performance systems (in aerospace, robotics and automotive areas).
Abstract: Covers in a progressive fashion a number of analysis tools and design techniques directly applicable to nonlinear control problems in high performance systems (in aerospace, robotics and automotive areas).
15,545 citations
TL;DR: This survey is the first to bring to the attention of the controls community the important contributions from the tribology, lubrication and physics literatures, and provides a set of models and tools for friction compensation which will be of value to both research and application engineers.
2,658 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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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 …
1,244 citations
Additional excerpts
...Some approaches addressed the describing function analysis [5,6] to replace a non-linear element with a quasi-linear descriptor which gain is a function of input amplitude....
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TL;DR: In this article, an iterative method is proposed for the identification of nonlinear systems from samples of inputs and outputs in the presence of noise, which consists of a no-memory gain (of an assumed polynomial form) followed by a linear discrete system.
Abstract: An iterative method is proposed for the identification of nonlinear systems from samples of inputs and outputs in the presence of noise. The model used for the identification consists of a no-memory gain (of an assumed polynomial form) followed by a linear discrete system. The parameters of the pulse transfer function of the linear system and the coefficients of the polynomial non-linearity are alternately adjusted to minimize a mean square error criterion. Digital computer simulations are included to demonstrate the feasibility of the technique.
707 citations
"Higher-order sinusoidal input descr..." refers background in this paper
...This structure, however, is not a Hammerstein model since the second block is not necessarily linear [14]....
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