zum Seiteninhalt

Physikalisch-Technische Bundesanstalt

StructureDiv. 8 Medical Physics and Metrological Information Technology8.4 Mathematical Modelling and Data Analysis8.42 Data Analysis and Measurement Uncertainty > Analysis of dynamic measurements
Analysis of dynamic measurements
Working Group 8.42



The goal of dynamic measurements is the determination of a physical quantity whose value shows a time-dependence. Examples are for instance the measurement of a mechanical force or acceleration, electrical high-voltage impulses or a time-dependent temperature.

Ideally, a dynamic measurement system is designed such that its output signal is proportional to the input signal, i.e. the time-dependent value of the measurand. However, often such an ideal behavior is achieved only for input signals with a low-frequency spectrum. For input signals with large bandwidth, the output signal of the measurement system is no longer proportional to its input signal. By taking the output signal of the measurement system as a time-dependent estimate of the value of the measurand, the so-called dynamic error is introduced. In order to compensate for the dynamic error, post-processing of the output of the measurement system is required for which digital filtering is an appropriate tool. In order to design a compensation filter, the dynamic behavior of the measurement system needs to be characterized, i.e. the measurement system has to be identified.

The determination of measurement uncertainty plays a key role in metrology. For a measurand with a constant value, agreed de-facto standards for the determination of measurement uncertainty are available. These standards are based on a Bayesian point of view and allow for a consistent treatment of random and systematic influences. However, these standards cannot be immediately applied to the case where the value of the measurand is time-dependent, and the concepts applied for the static case need to be extended.

Currently, we focus our work on the development and application of methods for

  • Identification of measurement systems
  • Design of digital compensation filters
  • Evaluation of dynamic measurement uncertainty

 

Figure 1 shows schematically a dynamic measurement system with subsequent post-processing. The time-dependent value  of the measurand passes through the measurement system and an analogue- to-digital conversion. Subsequent digital filtering compensates for the dynamic error and yields an estimate  of the measurand including the dynamic uncertainty  associated with this estimate.

 

Fig. 1:   Schematic set-up of dynamic measurement with subsequent compensation.

 

Actually, a compensation filter should have a frequency response which equals the inverse frequency response of the measurement system. However, this would imply a strong amplification of high frequency (noise) components. Therefore, the design of a compensation filter requires some tradeoff between sufficient noise suppression in the high frequency region and tolerable signal distortion due to non-ideal inverse filtering. Figure 2 shows the magnitude frequency response of such a compensation filter for a particular measurement system. The frequency response of the compensated measurement system is constant up to 60 kHz.  

 

 

Fig. 2:   Magnitude plots of the frequency responses of a measurement system (black), the compensation filter (blue) and the compensated measurement system (green).


The need and the benefit of applying a compensation filter to this measurement system is illustrated for a particular test signal in figure 3 where also the dynamic error is shown if no compensation is applied.

 

 

Fig. 3:  Input and output signals of a measurement system (top) as well as difference signal between the (time-shifted) output and input with (green) and without (red)  applying the  digital compensation filter of Fig. 2.

 

Figure 4 finally illustrates for simulated data the comparison of an uncertainty evaluation based on a static and a dynamic analysis for the above example. The static analysis treats the output of the measurement system as proportional to its input and ignores the dynamic behavior. The dynamic analysis is based on a subsequent application of the compensation filter. Figure 4 shows the frequency with which 95% credibility intervals determined by the analyses cover the underlying true value of the measurand at the time of the input signal’s maximum. The results show that as the bandwidth of the measurand increases, the coverage probability of 95% credibility intervals tends to zero for the static analysis in contrast to the dynamic analysis, and reliable uncertainties can in this case only be obtained by accounting for the dynamic behavior of the sensor system.

 

 

Fig. 4:   Estimated coverage probability for 95 % credibility intervals obtained for the peaks of a Gaussian-like measurand using the static and dynamic approaches in the uncertainty analysis.




Software

Monte Carlo  

The propagation of measurement uncertainty in dynamic measurements requires an efficient implementation in order to achieve high accuracies. The reason is that a standard implementation requires a large amount of computer memory.

  

S. Eichstädt, A. Link, P. Harris and C. Elster (2012). Efficient implementation of a Monte Carlo method for uncertainty evaluation in dynamic measurements. Metrologia 49, 401-410 (doi:10.1088/0026-1394/49/3/401).

In the PTB working group 8.42 a MATLAB software package has been developed, which implements these methods in an easy-to-use way. This package can be downloaded for free.

  

Software download

This software will be updated continuously. If you want to get informed about updates or have any questions or suggestions, please contact Sascha Eichstädt .

 
Richardson-Lucy

The correction of deviations in spectra measured with a spectrometer is often necessary in order to obtain accurate results. The classical approach for such a correction is based on a method from Stearns. However, we demonstrated in

   S. Eichstädt F. Schmähling G. Wübbeler, K. Anhalt, L. Bünger, K. Krüger and C. Elster (2013). Comparison of the Richardson-Lucy method and a classical approach for spectrometer bandpass correction. Metrologia 50, 107-118 (doi: 10.1088/0026-1394/50/2/107).

that the Richardson-Lucy method can result in much better results. Therefore, a software implementation of an adapted Richardson-Lucy method with automatic stopping rule has been written in PTB working group 8.42. This software includes MATLAB code as well as a graphical user interface written in Python. The software can be downloaded for free. 

  

Software download

This software will be updated continuously. If you want to get informed about updates or have any questions or suggestions, please contact Sascha Eichstädt .

 

 

 


References

  • S. Eichstädt, B. Arendacká, A. Link and C. Elster (2014). Evaluation of measurement uncertainties for time-dependent quantities. EPJ Web of Conferences 77. [DOI: 10.1051/epjconf/20147700003]. (Opens external link in new windowOpen Access).

  • C. Matthews, F. Pennecchi, S. Eichstädt, A. Malengo, T. Esward, I. Smith, C. Elster, A. Knott, F. Arrhén and A. Lakk (2014). Mathematical modelling to support tracable dynamic calibration of pressure sensors. Metrologia 51, 326-338. [DOI: 10.1088/0026-1394/51/3/326].
  • B. Arendacká, A. Täubner, S. Eichstädt, T. Bruns and C. Elster (2014). Linear mixed models: GUM and beyond. Meas. Sci. Rev. 14, 52-61. [DOI: 10.2478/msr-2014-0009], Opens external link in new window[online].
  • S. Eichstädt, F. Schmähling, G. Wübbeler, K. Anhalt, L. Bünger, U. Krüger and C. Elster (2013). Comparison of the Richardson-Lucy method and a classical approach for spectrometer bandpass correction. Metrologia 50, 107-118. DOI: 10.1088/0026-1394/50/2/107.
  • H. Füser, S. Eichstädt, K. Baaske, C. Elster, K. Kuhlmann, R. Judaschlke, K. Pierz and M. Bieler (2012). Optoelectronic time-domain characterization of a 100 GHz sampling oscilloscope. Meas. Sci. Technol. 23, 025201 (10pp).

  • S. Eichstädt, A. Link, P. Harris and C. Elster (2012). Efficient implementation of a Monte Carlo method for uncertainty evaluation in dynamic measurements. Metrologia 49, 401-410.

  • T. Esward, C. Matthews, S. Downes, A. Knott, S. Eichstädt and C. Elster (2012). Uncertainty evaluation for traceable dynamic measurement of mechanical quantities: A case study in dynamic pressure calibration, in "Advanced Mathematical & Computational Tools in Metrology and Testing IX" , Series on Advances in Mathematics for Applied Sciences vol. 84, eds. F. Pavese, M. Bär, J.-R. Filtz, A. B. Forbes, L. Pendrill, K. Shirono. World Scientific New Jersey.

  • S. Eichstädt (2012). Analysis of Dynamic Measurements - Evaluation of dynamic measurement uncertainty. PhD Thesis. [download pdf (1MB)]

  • S. Eichstädt and C. Elster (2012). Uncertainty evaluation for continuous-time measurements, in "Advanced Mathematical & Computational Tools in Metrology and Testing IX" , Series on Advances in Mathematics for Applied Sciences vol. 84, eds. F. Pavese, M. Bär, J.-R. Filtz, A. B. Forbes, L. Pendrill, K. Shirono. World Scientific New Jersey.

  •  Th. Bruns, A. Link and A. Täubner (2012). The influence of different vibration exciter systems on high frequency primary calibration of single-ended accelerometers: II. Metrologia 49, 27-31.

  • S. Eichstädt, C. Elster, T. J. Esward and J. P. Hessling (2010). Deconvolution filters for the analysis of dynamic measurement processes: a tutorial. Metrologia 47, 522-533.

  • S. Eichstädt, A. Link and C. Elster (2010). Dynamic Uncertainty for Compensated Second-Order Systems. Sensors 2010, 10, 7621-7631. [download pdf (450 KB)]
  • S. Eichstädt, A. Link, T. Bruns and C.Elster (2010). On-line dynamic error compensation of accelerometers by uncertainty-optimal filtering. Measurement 43,  708-713 .
  • C. Elster and A. Link (2009). Analysis of dynamic measurements: compensation of dynamic error and evaluation of uncertainty , in "Advanced Mathematical & Computational Tools in Metrology VIII" , Series on Advances in Mathematics for Applied Sciences vol. 78, eds. F. Pavese, M. Bär, J.M. Limares, C. Perruchet, N.F. Zhang, 80-89, World Scientific New Jersey.
  • G. Wübbeler, A. Link, T. Bruns and C. Elster (2009). Impact of correlation in the measured frequency response on the results of a dynamic calibration , in "Advanced Mathematical & Computational Tools in Metrology VIII" , Series on Advances in Mathematics for Applied Sciences vol. 78, eds. F. Pavese, M. Bär, J.M. Limares, C. Perruchet, N.F. Zhang, 369-374, World Scientific New Jersey.
  • A. Link and C. Elster (2009). Uncertainty evaluation for IIR (infinite impulse response) filtering using a state-space approach. Meas. Sci. Technol. 20, 055104 (5pp).
  • C. Elster and A. Link (2008). Uncertainty evaluation for dynamic measurements modelled by a linear time-invariant system. Metrologia 45, 464-473.
  • C. Elster, A. Link and T. Bruns  (2007). Analysis of dynamic measurements and determination of time-dependent measurement uncertainty using a second-order model. Meas. Sci. Technol. 18, 3682-3687.
  • A. Link, A. Täubner, W. Wabinski, T. Bruns and C. Elster (2007). Modelling accelerometers for transient signals using calibration measurements upon sinusoidal excitation. Measurement 40,  928-935 .
  • A. Link, M. Kobusch, T. Bruns and C. Elster (2006). Modellierung von Kraft- und Beschleunigungsaufnehmern für die Stoßkalibrierung. Tech. Mess. 73, 675-683.
  • A. Link, A. Täubner, W. Wabinski, T. Bruns and C. Elster (2006). Calibration of accelerometers: determination of amplitude and phase response upon shock excitation. Meas. Sci. Technol. 17, 1888-1894.
  • A. Link, W. Wabinski and H.-J. von Martens (2005). Identifikation von Beschleunigungsaufnehmern mit hochintensiven Stößen. Tech. Mess. 72, 153-160.
  • A. Link and H.-J. von Martens (2004). Accelerometer identification using shock excitation. Measurement 35, 191-199.
  • A. Link, W. Wabinski and H.-J. von Martens (2004). Accelerometer identification by high shock intensities using laser interferometry. Proc. SPIE 5503, 580-587.
  • A. Link and H.-J. von Martens (2000). Calibration of accelerometers by shock excitation and laser interferometry. Shock. Vib. 7, 101-112.
    • A. Link, W. Wabinski, A. Pohl and H.-J. von Martens (2000). Accelerometer identification using laser interferometry. Proc. SPIE 4072, 126-136.
    • J. Gerhardt and H.-J- Schlaak (2000). Zeitdiskrete Amplituden- und Nulllageregelung für sinusförmige Beschleunigungen bis 50 kHz. Tech. Mess. 6, 274-282.
    • H.-J. von Martens, A. Täubner, W. Wabinski, A. Link and H.-J. Schlaak (2000). Traceability of vibration and shock measurements by laser interferometry. Measurement 28, 3-20.

    To top




    Contact

    Physikalisch-Technische Bundesanstalt
    Working Group 8.42 Data Analysis and Measurement Uncertainty
    Abbestr. 2-12
    10587 Berlin
    Germany

    To top


    © Physikalisch-Technische Bundesanstalt, last update: 2014-09-22, Webmaster Abteilung 8 Seite drucken PrintviewPDF-Export PDF