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Journal ArticleDOI

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

01 Jan 2008-Control Engineering Practice (Elsevier)-Vol. 16, Iss: 1, pp 101-113
TL;DR: In this article, two measuring techniques are presented for measuring the higher order sinusoidal input describing functions (HOSIDF) of a non-linear plant operating in feedback.
About: This article is published in Control Engineering Practice.The article was published on 2008-01-01 and is currently open access. It has received 26 citations till now. The article focuses on the topics: Harmonic & Harmonics.
Citations
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Journal Article
TL;DR: In this paper, a modification of the integrated friction model structure proposed by Swevers et al. called the Leuven model is presented, which allows accurate modeling both in the presliding and the sliding regimes without the use of a switching function.
Abstract: This note presents a modification of the integrated friction model structure proposed by Swevers et al. (2000), called the Leuven model. The Leuven model structure allows accurate modeling both in the presliding and the sliding regimes without the use of a switching function. The model incorporates a hysteresis function with nonlocal memory and arbitrary transition curves. This note presents two modifications of the Leuven model. A first modification overcomes a recently detected shortcoming of the original Leuven model: a discontinuity in the friction force which occurs during certain transitions in presliding. A second modification, using the general Maxwell slip model to implement the hysteresis force, eliminates the problem of stack overflow, which can occur with the implementation of the hysteresis force.

288 citations

Journal ArticleDOI
TL;DR: In this paper, a linear dynamic time-invariant model is identified to describe the relationship between the reference signal and the output of the system, and the power spectrum of the unmodeled disturbances are identified to generate uncertainty bounds on the estimated model.
Abstract: 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.

83 citations

Journal ArticleDOI
TL;DR: A linear dynamic time-invariant model is identified to describe the relationship between the reference signal and the output of the system and the power spectrum of the unmodeled disturbances are identified to generate uncertainty bounds on the estimated model.
Abstract: This article addresses the following problems: 1) First, a nonlinearity analysis is made looking for the presence of nonlinearities in an early phase of the identification process. The level and the nature of the nonlinearities should be retrieved without a significant increase in the amount of measured data. 2) Next it is studied if it is safe to use a linear system identification approach, even if the presence of nonlinear distortions is detected. The properties of the linear system identification approach under these conditions are studied, and the reliability of the uncertainty bounds is checked. 3) Eventually, tools are provided to check how much can be gained if a nonlinear model were identified instead of a linear model. Addressing these three questions forms the outline of this article. The possibilities and pitfalls of using a linear identification framework in the presence of nonlinear distortions will be discussed and illustrated on lab-scale and industrial examples. In this article, the focus is on nonparametric and parametric black box identification methods, however the results might also be useful for physical modeling methods. Knowing the actual nonlinear distortion level can help to choose the required level of detail that is needed in the physical model. This will strongly influence the modeling effort. Also, in this case, significant time can be saved if it is known from experiments that the system behaves almost linearly. The converse is also true. If the experiments show that some (sub-)systems are highly nonlinear, it helps to focus the physical modeling effort on these critical elements.

61 citations

Journal ArticleDOI
TL;DR: In this paper, a frequency domain based method for controller design for nonlinear systems is presented, which is applied to optimally design a feed forward friction compensator for an industrial motion stage in a transmission electron microscope.

50 citations


Additional excerpts

  • ...Finally, the Higher Order Sinusoidal Input Describing Functions (HOSIDF) [16,18,26] are used to analyze nonlinear effects in the following....

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Journal ArticleDOI
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

References
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Proceedings ArticleDOI
07 Aug 2002
TL;DR: In this paper, a detailed analysis of the mechanisms at work on the generation of a class of nonlinear distortions, termed harmonic components, is presented, which leads to the derivation of formulae to calculate the number of harmonic components generated by any order of non-linearity at the excitation frequencies during frequency response function measurements.
Abstract: New insights on the frequency-domain output properties of nonlinear systems driven by multiharmonic signals will be discussed in this paper. These are based on a detailed analysis of the mechanisms at work on the generation of a class of nonlinear distortions, termed harmonic components. This leads to the derivation of formulae to calculate the number of harmonic components generated by any order of nonlinearity at the excitation frequencies during frequency response function measurements. Based on these formulae the classification of nonlinear distortions in harmonic and interharmonic components is compared against other methodologies on the classification of nonlinear distortions.

13 citations

Proceedings ArticleDOI
01 Dec 1999
TL;DR: In this paper, a phase advance compensator in the repetitive control loop is proposed to improve tracking performance for repetitive linear motion in a direct-drive linear motor with phase advance compensation.
Abstract: Direct drive linear motors have particularly good potential for use with the next generation of machine tool feed drives, such as high speed machining, since they can increase machining rates and improve servo accuracy by eliminating gear related mechanical problems. In the paper, repetitive control is employed to design a servo controller for reducing the steady-state error due to the periodic commands. A repetitive control scheme which has a phase advance compensator in the repetitive control loop is proposed to improve tracking performance for repetitive linear motion. The repetitive controller of an experimental direct feed drive system is implemented in real-time with a digital signal processor.

13 citations

Journal ArticleDOI
TL;DR: In this paper, a nonlinear hybrid parameter multibody dynamic system (HPMBS) model is proposed to estimate damaged structure parameters through significant nonlinear damage, such as loose bolted joints, dry frictional damping and large articulated motions.
Abstract: This research work is in the area of structural health monitoring and structural damage mitigation. It addresses and advances the technique in parameter identification of structures with significant nonlinear response dynamics. The method integrates a nonlinear hybrid parameter multibody dynamic system (HPMBS) modeling technique with a parameter identification scheme based on a polynomial interpolated Taylor series methodology. This work advances the model based structural health monitoring technique, by providing a tool to accurately estimate damaged structure parameters through significant nonlinear damage. The significant nonlinear damage implied includes effects from loose bolted joints, dry frictional damping, large articulated motions, etc. Note that currently most damage detection algorithms in structures are based on finding changed stiffness parameters and generally do not address other parameters such as mass, length, damping, and joint gaps. This work is the extension of damage detection practice from linear structure to nonlinear structures in civil and aerospace applications. To experimentally validate the developed methodology, we have built a nonlinear HPMBS structure. This structure is used as a test bed to fine-tune the modeling and parameter identification algorithms. It can be used to simulate bolted joints in aircraft wings, expansion joints of bridges, or the interlocking structures in a space frame also. The developed technique has the ability to identify unique damages, such as systematic isolated and noise-induced damage in group members and isolated elements. Using this approach, not just the damage parameters, such as Young's modulus, are identified, but other structural parameters, such as distributed mass, damping, and friction coefficients, can also be identified.

9 citations

DOI
01 Jan 2007
TL;DR: In this thesis an alternative concept for frequency based nonlinear system analysis is presented, the required measurement techniques are described and some application examples are shown.
Abstract: In modern positioning systems, accuracy and speed requirements have increased significantly. These accuracies can only be realized if account is given to nonlinear system behavior in both the mechanical and the control design. This requires additional tools for frequency based identification of nonlinear system behavior since existing tools either are either too limited to successfully describe nonlinear behavior or the results are very difficult to interpret and as such do not relate to the background of the intended user. In this thesis an alternative concept for frequency based nonlinear system analysis is presented, the required measurement techniques are described and some application examples are shown. The method is applicable for the class of causal, stable, time-invariant non-linear systems which have a harmonic response to a sinusoidal excitation. This new concept is the generalization of the Sinusoidal Input Describing Function to Higher Order Sinusoidal Input Describing Functions (HOSIDF) as it yields the magnitude and phase relations between the individual higher harmonics in the response signal and the sinusoidal excitation signal, both as function of magnitude and frequency of the excitation signal. An essential element in the HOSIDF theory is the concept of the Virtual Harmonics Expander (VHE). This nonlinear function describes the transformation of a single sinusoid into an infinite amount of harmonics, each with equal amplitude as the input signal and with a phase equal to the phase of the input signal times the harmonic number. Nonlinear systems belonging to the class can be modeled as a parallel connection of an (infinite) amount of HOSIDF describing quasi-linear subsystems in series with the VHE. Two measurement methods for nonparametric identification of HOSIDF are presented. The Fast Fourier Transform based method on fast fourier transforms shows ideal characteristics due to its perfect selectivity. The IQ (In phase-Quadrature phase) demodulation method has limited performance due to non perfect selectivity. The bias in the HOSIDF estimates caused by harmonic components in the input signal is analyzed and a compensation algorithm is presented to reduce this bias. Accept- ing harmonic distortion in the excitation signal allows the application of non-constant amplitude-time profiles for testing. It is demonstrated that a ramped amplitude-time signal reduces the required settling time of the digital filters used in the IQ methode. The capabilities of the HOSIDF technique are demonstrated in a real measurement in which the stick to gross sliding transition of a mechanical system with dry friction is captured as function of frequency. The odd HOSIDF clearly reveal this transition which is not possible with the Frequency Response Function technique. From the HOSIDF the pre-sliding displacement and the friction-induced stiffness are determined and the friction force which must be present in the stick-phase is calculated. Validation with force measurements shows excellent agreement. Special attention is paid to the determination of the HOSIDF of a nonlinear plant operating in feedback. In a controlled systemthe harmonics generated by the non-linear system will be fed back to the input, changing the sinusoidal excitation into an harmonic excitation. Two different solutions are presented to deal with this problem. The first method applies a numerical compensatie techniques to compensate the bias caused by the harmonic components in the excitation signal. The secondmethod uses amodified repetitive control scheme to suppress the harmonic components in the excitation signal. The effectiveness of both methods is tested in simulation experiments of a mass operating in feedback subjected to Coulomb friction, Stribeck-effect and hysteresis in the pre-sliding regime. The friction forces are modeled with the modified Leuven friction model. The results are compared with the HOSIDF measured under open loop condition and both methods yield correct results. It is shown that by rearranging the repetitive control loop, the output signal of a class of stable, time-invariant nonlinear systems becomes sinusoidal as response to an harmonic excitation. For this class of signals Higher Order Sinusoidal Output Describing Functions (HOSODF) can be defined as the dual of the HOSIDF. The HOSODF describe magnitude and phase relations between the individual higher harmonics in the input signal and the sinusoidal output signal, both as function of magnitude and frequency of the output signal. The required dual of the Virtual Harmonics Expander is defined as the Virtual Harmonics Compressor. This nonlinear function describes the transformation of an infinite amount of harmonics into a single sinusoid. Finally, an application example shows the extreme sensitivity of the HOSIDF technique for changes in friction characteristics, indicating interesting opportunities for application in the field of machine condition monitoring. De eisen die gesteld worden aan de snelheid en positioneringsnauwkeurigheid van moderne positioneringssystemen zijn significant toegenomen. Deze nauwkeurigheden kunnen alleen maar gerealiseerd worden als met niet-lineair systeemgedrag rekening wordt gehouden in zowel het mechanische als het regeltechnische ontwerp. In tegenstelling tot de tijddomein gebaseerde systeemidentificatie is de moderne regeltechniek op frequentiedomein technieken gebaseerd. Maar de transformatie van niet-lineaire tijddomeinmodellen naar het frequentiedomein is nietmogelijkmet alleen lineaire technieken. Dit vereist extra gereedschappen ten behoeve van de frequentiedomein gebaseerde identificatie van niet-linear systeemgedrag omdat de bestaande gereedschappen ofwel te beperkt zijn om met succes niet-linear gedrag te beschrijven ofwel resultaten leveren in een formaat dat moeilijk te interpreteren is en niet aansluit bij de achtergrond van de gebruiker. In dit proefschrift wordt een alternatief concept gepresenteerd voor een op frequentiedomeintechnieken gebaseerde niet-lineaire systeemanalyse. Eveneens worden de vereiste meetmethodes beschreven en enkele toepassingsvoorbeelden getoond. De methode is van toepassing op de klasse I gedefinieerd als de klasse van causale, stabiele, tijdsinvariante, niet-lineaire systemen welke een harmonische responsie hebben ten gevolge van een sinusvormige excitatie. Dit nieuwe concept is de generalisatie van de Sinusoidal Input Describing Function tot de Higher Order Sinusoidal Input Describing Functions (HOSIDF). De HOSIDF beschrijven de magnitude- en faserelaties die bestaan tussen de afzonderlijke hogere harmonische componenten in het responsiesignaal en de sinusvormige excitatie, allen als functie van amplitude en frequentie van dat excitatiesignaal. In de HOSIDF theorie wordt een essentiele plaats ingenomen door het begrip Virtual Harmonics Expander (VHE). Deze niet-lineaire functie beschrijft de transformatie van een zuiver sinusvormig signaal in een oneindige reeks harmonischen, elk met identieke amplitude gelijk aan de amplitude van het ingangssignaal en een fase gelijk aan de fase van het ingangssignaal maal het rangnummer van de harmonische component. Systemen die behoren tot de klasse I kunnen gemodelleerd worden als een parallel schakeling van een (oneindig) aantal HOSIDF in serie met de VHE. Twee meetmethodes voor de niet-parametrische identificatie van HOSIDF worden gepresenteerd. De op Fast Fourier

6 citations


"Measuring the higher order sinusoid..." refers background or methods in this paper

  • ...This conflicts with the sinusoidal excitation condition required [Slotine, & Weiping Li, 1991, Atherton, 1975, Nuij, Bosgra, & Steinbuch, 2006, Nuij, 2007]....

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  • ...They reveal information about non-linear system behavior like for example stick/sliding behavior, which is not visible in the classical Sinusoidal Input Describing Function [Nuij, 2007]....

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Journal ArticleDOI
TL;DR: In this article, a non-linear signal analysis technique was used to estimate the friction force using the vibrations signals, measured on a reciprocating wear testing machine, by the proposed nonlinear model.
Abstract: When the surfaces of two elastic bodies present relative motions under certain amount of contact pressure the mechanical system can be unstable. Experiments conducted on elastic bodies in contact shown that the dynamic system is self-excited by the non-linear behavior of the friction forces. The main objective of this paper is to estimate the friction force using the vibrations signals, measured on a reciprocating wear testing machine, by the proposed non-linear signal analysis formulation. In the proposed formulation the system global output is the sum of two outputs produced by a linear path associated in parallel with a non-linear path. This last path is a non-linear model that represents the friction force. Since the linear path can be identified by traditional signal analysis, the non-linear function can be evaluated by the global input/output relationships. Validation tests are conducted in a tribological system composed by a sphere in contact with and a prismatic body, which has an imposed harmonic motion. The global output force is simultaneously measured by a piezoelectric and by a piezoresistive load cells. The sphere and prismatic body vibrations are measured by a laser Doppler vibrometer and by an accelerometer respectively. All signals are digitalized at the same time base and the data is transferred to a microcomputer. The non-linear signal analysis technique uses this data to identify the friction force.

4 citations