The European Centre for
Mathematics and Statistics in Metrology

On model-free methods for dynamic measurements

We review a recent publication by Ivan Markovsky on input estimation in dynamic measurements with respect to its suitability for metrological applications.

Author: Sascha Eichstädt (PTB)

The analysis of dynamic measurements requires to estimate the dynamic input signal $x(t)$ from knowledge about the system output signal $y(t)$ and knowledge about the dynamic system itself. In a typical setup, the measurement system is calibrated in order to assess and quantify its dynamic behaviour (e.g. in terms of its frequency response). As a result, a (parametric or non-parametric) model of the dynamic system is obtained, see for instance Link et al.
From the model of the measurement system a deconvolution filter can be determined then in order to carry out estimation of the system input signal, cf. Eichstädt et al. Such a deconvolution filter can be obtained either from a parametric model of the system or from its frequency response. Especially in the latter case the estimation of the input signal does not require a system model in the classical sense.

However, carrying out dynamic calibration can turn out to be very challenging for some measurement systems. To this end, Markovsky suggests a so called "model-free" approach in his recently published "Comparison of adaptive and model-free methods for dynamic measurements" (IEEE Signal Processing Letters 2015). He considers use cases where the knowledge of the system dynamics is "unrealistic" (his words). Thus, a dynamic calibration of the measurement system is considered unavailable.
He focuses on step input signals $u(t)=\bar{u}s(t)$ with $s(t)$ the unit step signal and with unknown step height $\bar{u}$. The system is decomposed into a static part with gain $G$ and a dynamic part with unknown model. Therefore, the system output signal is written as $$y=\bar{y}s + \Delta y$$ where $\bar{y}=G\bar{u}$.
The input estimation proposed by Tarkovsky is carried out by forming a block-Hankel matrix from $\Delta y$ and solving a linear system of equations (in least-squares sense for noisy measurements). Markovsky considers dynamic weighing as an example and presents estimation results for simulated measurements. The proposed method outperforms naive least-squares estimation and adaptive filtering in terms of speed of convergence and estimation quality for the considered examples. The corresponding MATLAB software can be downloaded from Markovsky's website.

Although Markovsky's method appears to work quite well for simulated dynamic weighing, its applicability for metrology remains questionable. First of all, its method is in fact not model-free as it requires a specific parametric model of the input signal. The simple step signal example from the paper can be replaced by some multivariate model, but the requirement for a signal model remains. Hence, the proposed method is rather a regression problem than a non-parametric input estimation. In addition, Markovsky admits that his method is statistically inefficient when the noisy measurements are considered. Overall, the method is interesting from a mathematical point of view, but appears to be of limited use for metrologists.

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