Content
Overview
Dynamic measurements can be found in many areas of metrology and industry such as, for instance, in the measurement of time-dependent forces or accelerations. Methods from signal processing are often applied in the analysis of dynamic measurements. In many applications linear time-invariant systems are appropriate to model dynamic measurements, where the output signal is obtained as a convolution of the input signal and the measurement system’s impulse response. Input and output signal are not proportional to each other, and estimation of the system’s input signal from its output signal constitutes one important task in the analysis of dynamic measurements. Often digital filters are employed for this purpose. The evaluation of the uncertainty associated with the estimated input signal is particularly important from a metrological perspective.
Typical examples are measurements of mechanical quantities as, for example, force, torque and pressure. Further examples are oscilloscope measurements for the characterization of high speed electronics, hydrophone measurements for the characterization of medical ultrasound devices, the spectral characterization of radiation sources, spectral color measurements and camera-aided temperature measurements.
Research
One focus of PTB‘s Working Group 8.42 is the development of methods for the estimation of the input signal from the output signal when the dynamic behavior of the measurement system has been characterized. This includes the development of procedures for the evaluation of the uncertainty associated with the estimated input signal. Another focus is the development of methods for the analysis of dynamic calibration measurements aimed at determining the dynamic behavior of a measurement system.
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Publications
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Article
| Title: | On the evaluation of uncertainties for state estimation with the Kalman filter |
|---|---|
| Author(s): | S. Eichstädt, N. Makarava and C. Elster |
| Journal: | Measurement Science and Technology |
| Year: | 2016 |
| Volume: | 27 |
| Issue: | 12 |
| Pages: | 125009 |
| DOI: | 10.1088/0957-0233/27/12/125009 |
| Keywords: | Kalman filter, uncertainty, dynamic measurement, state-space system, state estimation |
| Tags: | 8.4, 8.42, Dynamik |
| Abstract: | The Kalman filter is an established tool for the analysis of dynamic systems with normally distributed noise, and it has been successfully applied in numerous areas. It provides sequentially calculated estimates of the system states along with a corresponding covariance matrix. For nonlinear systems, the extended Kalman filter is often used. This is derived from the Kalman filter by linearization around the current estimate. A key issue in metrology is the evaluation of the uncertainty associated with the Kalman filter state estimates. The “Guide to the Expression of Uncertainty in Measurement” (GUM) and its supplements serve as the de facto standard for uncertainty evaluation in metrology. We explore the relationship between the covariance matrix produced by the Kalman filter and a GUM-compliant uncertainty analysis. In addition, the results of a Bayesian analysis are considered. For the case of linear systems with known system matrices, we show that all three approaches are compatible. When the system matrices are not precisely known, however, or when the system is nonlinear, this equivalence breaks down and different results can then be reached. For precisely known nonlinear systems, though, the result of the extended Kalman filter still corresponds to the linearized uncertainty propagation of the GUM. The extended Kalman filter can suffer from linearization and convergence errors. These disadvantages can be avoided to some extent by applying Monte Carlo procedures, and we propose such a method which is GUM-compliant and can also be applied online during the estimation. We illustrate all procedures in terms of a two-dimensional dynamic system and compare the results with those obtained by particle filtering, which has been proposed for the approximate calculation of a Bayesian solution. Finally, we give some recommendations based on our findings. |