Uncertainty evaluation for dynamic measurements modelled by a linear time-invariant system

Abstract

Evaluation of measurement uncertainty is considered when the value of the measurand depends on the continuous variable time. A concept of dynamic measurement uncertainty is introduced by generalizing the GUM approach. The concept is applied to linear and time-invariant systems which are often appropriate to model dynamic measurements. Digital filtering is proposed for estimating the time-dependent value of the measurand and the design of an appropriate FIR filter is described. Dynamic uncertainty evaluation is then carried out for this analysis and conditions are specified for its proper use. The approach is illustrated for the particular example of a second-order model. It is shown in terms of simulations that the proposed analysis yields significantly improved results when compared with the sometimes applied quasi-static treatment.

Evaluation of measurement uncertainty for time-dependent quantities

“…Both are continuous functions and thus the estimation of the value of the measurand is related to continuous modelling [7]. Let us assume that there is a mapping ‫ܨ‬ so that the estimate ܻ ሺ‫ݐ‬ሻ of ܻሺ‫ݐ‬ሻ is calculated from the estimate ܺ ሺ‫ݐ‬ሻ by ܻ ሺ‫ݐ‬ሻ ൌ ‫ܨ‬൛ܺ ሺ‫ݐ‬ሻൟǤ Then the evaluation of uncertainties requires us to (i) assign an uncertainty to the continuous function ܺሺ‫ݐ‬ሻ and (ii) to propagate this uncertainty through the mapping ‫ܨ‬Ǥ The treatment of continuous functions is beyond the current GUM and requires the employment of a stochastic process as a model for the state of knowledge instead of a probability density function as in the GUM [7,9]. An extension of the GUM methodology to continuous functions using stochastic calculus has been proposed recently, but is beyond the scope of this paper [7].…”

“…In this paper we outline the design of such digital compensation filters as a means of estimating the quantity of interest. For the propagation of uncertainty in the application of such digital filters we present sequential formulae for finite impulse response (FIR) and infinite impulse response (IIR) filters [8,9]. We also demonstrate a recently developed efficient Monte Carlo method for uncertainty propagation in dynamic measurements which, in principle, allows calculating point-wise coverage intervals in real-time [10].…”

“…where ‫ܪ‬ሺ݆߱ሻ denotes the frequency response of the measurement device LTI system model and ߱ ଵ തതതത ൏ ߱ ଶ തതതത are chosen frequencies [9]. The inverse filter ݃ ௩ ሺ‫ݐ‬ሻ is designed from knowledge about the measurement device.…”

“…can be designed as an approximation to ෩ ି ൌ ቀ‫ܪ‬ ෩ ିଵ ሺ߱ ଵ ሻǡ ǥ ǡ ‫ܪ‬ ෩ ିଵ ሺ߱ ெ ሻቁ ் , the reciprocal of the shifted frequency response values, using linear least-squares [9]. Therefore, the coefficients (5) of the inverse filter are calculated as the solution of…”

“…As a consequence, the compensation filter has a linear phase. For instance, the time-delay of the compensation filter in Figure 3 is ݊ ௗ ଶ ൌ ͳͳ samples for ܶ ௦ ൌ ͲǤͲͲʹ ms, where ‫ܮ‬ denotes the order of the lowpass filter and ݊ ௗ the time delay introduced for the design of the inverse filter [9]. When the estimation error ઢ, due to an imperfect inverse filter and the chosen cut-off of the low-pass filter, is significant compared to the other sources of uncertainty, it has to be taken into account in the uncertainty budget.…”