Measuring the higher order sinusoidal input describing functions of a non-linear plant operating in feedback

Nuij, Pwjm Pieter; Steinbuch, M Maarten; Bosgra, О.H.

doi:10.1016/j.conengprac.2007.04.003

Cited by 20 publications

(11 citation statements)

References 37 publications

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“…where a indicates the amplitude of the excitation signal. The method can be used under feedback conditions [62]. The HOSIDFs give a simple description of complex nonlinear behaviors of mechanical systems, for example, the transition from stick to sliding in precision mechatronic systems [63].…”

Section: Nonlinear Distortions Characterization: Alternative Methodsmentioning

confidence: 99%

Linear System Identification in a Nonlinear Setting: Nonparametric Analysis of the Nonlinear Distortions and Their Impact on the Best Linear Approximation

2016

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Linear system identification [1]-[4] is a basic step in modern control design approaches. Starting from experimental data, a linear dynamic time-invariant model is identified to describe the relationship between the reference signal and the output of the system. At the same time, the power spectrum of the unmodeled disturbances is identified to generate uncertainty bounds on the estimated model. Linear system identification is also used in other disciplines, for example vibrational analysis of mechanical systems, where it is called modal analysis [5], [6]. Because linear time-invariant models are a basic model structure, linear system identification is frequently used in electrical [7]-[10], electronic, chemical [11], civil [12], and also in biomedical applications [13]. It provides valuable information to the design engineers in all phases of the design process.Starting from the late 1960s, system identification tools have been developed to obtain parametric models to describe the dynamic behavior of systems. A formal framework is set up to study the theoretical properties of the system identification algorithms [1]- [3]. The consistency (does the estimated model converge to the true system as the amount of data grows?) and the efficiency (is the uncertainty of the estimated model as small as possible?) are analyzed in detail. Underlying all these results are the assumptions that the system to be modeled is linear and time invariant.It is clear that these assumptions are often (mostly?) not met in real-life applications. Most systems are only linear to a first approximation. Depending on the excitation level, the output is disturbed by nonlinear distortions so that the linearity assumption no longer holds. This immediately raises doubts about the validity of the results obtained and validated by the linear system identification framework. The term nonlinear distortions indicates that nonlinear systems with a (dominant) linear term are considered. The deviations from the linear behavior are called nonlinear distortions.

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Section: Nonlinear Distortions Characterization: Alternative Methodsmentioning

confidence: 99%

Linear System Identification in a Nonlinear Setting: Nonparametric Analysis of the Nonlinear Distortions and Their Impact on the Best Linear Approximation

2016

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“…In order to model systems with levels or types of nonlinearities that do not satisfy this assumption the concept of Higher Order Sinusoidal Input Describing Functions (HOSIDF) was developed [4], [5]. HOSIDFs describe the response of the system by describing not only the response of the system at the input frequency, but by describing the response at harmonics of the input frequency, generated by nonlinearities, as well.…”

Section: Higher Order Sinusoidal Input Describing Functionsmentioning

confidence: 99%

“…After the introduction of the industrial application tested in this paper, a method based on the results in [7], [8], [9] is presented, which uses multisine signals to compute a Best Linear Approximation (BLA) of the systems dynamics and detect and classify possible nonlinearities. The second method is based on [4], [5] Note that multisine method provides the best linear approximation in a stochastic sense for the class of Gaussian excitation signals. Part of the result of the second method is a Linear Approximation (LA) of the systems dynamics as well.…”

Section: Introductionmentioning

confidence: 99%

Nonlinearities in Industrial motion stages - detection and classification

Rijlaarsdam

Loon

Nuij

et al. 2010

Proceedings of the 2010 American Control Conference

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Abstract-Detection and classification of nonlinearities in motion systems becomes of increasing importance with high demands on (closed loop) performance. In this paper two methods are compared that aim to measure both the linearized dynamics and the influence of nonlinearities. First, a broadband signal is used to measure a linear approximation of the systems dynamics. This method uses multisine signals with identical amplitude spectrum, but randomly distributed phases. Averaging over multiple periodic responses to the same signal and over multiple realizations of the random phase multisine allows the computation of the level of nonlinearities and external disturbances separately. This yields both a linear approximation of the systems dynamics and the amount of nonlinear 'disturbance' as a function of frequency. Second, single sine based measurements are used to measure the Higher Order Sinusoidal Input Describing Functions (HOSIDF) of the system under test. HOSIDFs describe the response of the system by describing not only the 'direct' response (gain and phase shift) of the system at the input frequency, but by describing the response at higher harmonics of the input frequency as well. This yields a quantitative measure of the power generated by nonlinearities at harmonics of the input frequency as a function of this frequency and the signal amplitude. In the paper these methods are utilized to acquire a non-parametric model for an industrial high precision stage. The effects of and sources for nonlinear influences are discussed for this particular case as well.

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“…The results presented in the sequel rely on the quantification of nonlinear effects by measuring energy in the output spectrum at harmonics of the input frequency, when the system is subject to a sinusoidal input signal. This representation of nonlinear effects in the frequency domain is captured by the higher order sinusoidal input describing functions [2], [3], [4], [8], [10], which describe the systems response (gain and phase) at harmonics of the base frequency of a sinusoidal input signal. In [7], [9] recent results concerning frequency domain tuning of Coulomb friction feed forward are presented.…”

Section: Introductionmentioning

confidence: 99%

Frequency domain based friction compensation - Industrial application to transmission electron microscopes -

Rijlaarsdam

Nuij

Schoukens

et al. 2011

Proceedings of the 2011 American Control Conference

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Abstract-Friction is a performance limiting factor in many industrial motion systems. Correct compensation or control of friction and other nonlinearities is generally difficult. Apart from the complex nature of friction, compensation of even the most basic type of friction, Coulomb friction, is non trivial. Most available tuning methods rely on time domain data and are often unable to distinguish between nonlinear effects of friction and that of for example linear viscous damping. Furthermore, the sensitivity of time domain data to the influence of friction is too low for correct tuning in many of the high precision motion applications currently used in industry. In this paper a frequency domain method is introduced that allows fast and high accuracy tuning of controller parameters when the closed loop system is subject to nonlinear influences. This methodology is applied to optimally compensate friction in a high precision motion stage of a transmission electron microscope. Theoretical and experimental results are presented and related to time domain performance to illustrate the advantage of frequency domain tuning over time domain tuning.

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Measuring the higher order sinusoidal input describing functions of a non-linear plant operating in feedback

Cited by 20 publications

References 37 publications

Linear System Identification in a Nonlinear Setting: Nonparametric Analysis of the Nonlinear Distortions and Their Impact on the Best Linear Approximation

Linear System Identification in a Nonlinear Setting: Nonparametric Analysis of the Nonlinear Distortions and Their Impact on the Best Linear Approximation

Nonlinearities in Industrial motion stages - detection and classification

Frequency domain based friction compensation - Industrial application to transmission electron microscopes -

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