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Mathematical Modelling and Data Analysis

Department 8.4

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PhD thesis

Title: Analysis of Dynamic Measurements - Evaluation of dynamic measurement uncertainty
Author(s): S. Eichstädt
Work type: PhD Thesis
Year: 2012
Publisher address: Berlin
School: TU Berlin
File URL: fileadmin/internet/fachabteilungen/abteilung_8/8.4_mathematische_modellierung/Publikationen_8.4/842_Dynamik_Diss_Eichstaedt.pdf
Keywords: dynamic measurement, dynamic uncertainty, digital filter, deconvolution
Tags: 8.42, Dynamik
Abstract: Metrology is concerned with the establishment of measurement units and the transfer of measurement standards to industry. International comparability of measurement results requires internationally agreed guidelines for specific measurement tasks and a standardised treatment of measurement uncertain- ties. To this end, the Guide to the Expression of Uncertainty in Measurement (GUM) provides the framework for the evaluation and interpretation of mea- surement uncertainty in metrology. However, it does not address dynamic measurements, which are of growing importance for industry and metrology. Typical examples of dynamic measurements are in-cylinder measurements in the automotive industry (pressure), crash tests (e.g., acceleration and force) or assembly line measurements (e.g., torque and force). A reliable calibra- tion of the measurement systems employed, which can be related to national standards, requires a consistent evaluation of measurement uncertainty for dynamic measurements.The goal of this thesis is to develop a framework for the evaluation of uncer- tainty in dynamic measurements in metrology that are closely related to the treatment of static measurements. The measurement systems considered are those that can be modelled by a linear and time-invariant (LTI) system since such models cover a wide range of metrological applications. The measured values are the values of the system output signal, whereas the values of the quantity of interest serve as the system input signal. Estimation of the in- put signal is considered to be carried out by means of digital filtering in the discrete time domain from which inference of the continuous-time signal is sought.This requires the design of digital filters, an uncertainty evaluation for regu- larised deconvolution and a framework for the definition and propagation of the uncertainty of a continuous function. The design of digital filters for de- convolution is well-established in the signal processing literature. The same holds true for the propagation of variances through LTI systems. However, propagation of variances through uncertain LTI systems for evaluation of uncertainty in the sense of GUM has only recently been considered. The methods developed so far focus on the evaluation of uncertainties and do not address regularisation errors. Moreover, the relation of the discrete-time es- timate to the actual continuous-time measurand has not yet been addressed.We extend the available results for the evaluation of uncertainties to the propagation of associated probability density functions and propose efficient calculation schemes. Moreover, the ill-posed deconvolution problem requires regularisation. We develop a reliable quantitative evaluation of the uncer- tainty contribution due to regularisation assuming a particular type of prior knowledge. We present a framework for the evaluation of uncertainty for con- tinuous measurements, which addresses the definition, assignment and prop- agation of uncertainty. Finally, we develop a technique for the calculation of uncertainty associated with a continuous-time estimate of the measurand from a discrete-time estimate.The proposed techniques provide a complete framework for the consistent and reliable evaluation of uncertainty in the analysis of a dynamic measurement.

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