Higher order sinusoidal input describing functions : extending linear techniques towards non-linear systems analysis

P.W.J.M. Nuij

Research output: ThesisPhd Thesis 1 (Research TU/e / Graduation TU/e)

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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 essentiële 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
Original languageEnglish
QualificationDoctor of Philosophy
Awarding Institution
  • Mechanical Engineering
  • Steinbuch, Maarten, Promotor
  • Bosgra, Okko, Promotor
Award date29 Aug 2007
Place of PublicationEindhoven
Print ISBNs978-90-386-1066-5
Publication statusPublished - 2007


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