# Uncertainty analysis of a subspace-based input estimation method for dynamic measurements

A measurement is an experimental procedure to determine the value of a physical magnitude. The true value of the to-be-measured quantity is unknown and the measurement result is an estimation of the true value. The difference between the true value and its estimate cannot be absolutely determined. Therefore, there is an uncertainty associated with the result of any measurement. Moreover, the accuracy and availability of the estimation depends on the sensor dynamics. A data-driven method that estimates the value of an unknown input by processing the sensor transient response suggested that a measurement can be performed in less time than classical model-based approaches.

The data-driven estimation method estimates the true value of a step input of level  by processing the transient response  of a sensor of order  and dc-gain . The perturbation noise is additive normally distributed so that the measured transient response is . The estimation of the step input level is obtained as the solution of the minimization problem

We study the statistical properties of the data-driven estimation method (1) to know the estimate uncertainty. The aim of the statistical analysis is to provide confidence bounds on the uncertainty associated with the estimate in practical measurements. We obtain analytical expressions for the prediction of the estimate bias and variance. The analytical solution to the problem (1) is approximated by a Taylor expansion. The first two moments of the estimate  are analyzed. The resulting expressions are functions of the measurement noise variance and of the exact transient response data and predict the bias and variance of the estimate.

Figure 12. . Relative errors of the estimate bias (left) and variance (right) observed in the Monte Carlo simulation. The Monte Carlo averages (MC) of the estimate bias and variance parameters is accurately predicted (p) by the formulas. The relative error between the empirical (e) and the predicted (p) bias and variance parameters increases as it is expected for decreasing signal-to-noise ratio

We validate the prediction formulas using Monte Carlo simulation and using real data measured from a weighing sensor in an experimental setup. The bias and variance of the estimate give insight on how these parameters are related with the signal-to-noise ratio and the processed number of samples of the sensor transient response.

## References

1. Markovsky, I., An application of system identification in metrology, Control Eng. Prac., 43:85-93, 2015.
2. Stewart, G., Stochastic Perturbation Theory, SIAM Review, 32:579-610, 1990.