Differentiator for Noisy Sampled Signals With Best Worst-Case Accuracy

Abstract

This paper proposes a differentiator for sampled signals with bounded noise and bounded second derivative. It is based on a linear program derived from the available sample information and requires no further tuning beyond the noise and derivative bounds. A tight bound on the worst-case accuracy, i.e., the worst-case differentiation error, is derived, which is the best among all causal differentiators and is moreover shown to be obtained after a fixed number of sampling steps. Comparisons with the accuracy of existing high-gain and slidingmode differentiators illustrate the obtained results.

Optimal Robust Exact Differentiation via Linear Adaptive Techniques

“…For the continuous-time case, an interesting consequence of the results in (Haimovich et al, 2022) is that the best possible worst-case accuracy may also be achieved by a linear finite difference. However, this is true only if the noise magnitude is known to the differentiator; if the actual noise magnitude happens to be lower than the assumed bound, then performance can worsen greatly.…”

“…A radically different strategy for signal differentiation in a digital implementation setting is to directly consider the information carried by noisy sampled measurements accounting for noise magnitude and known derivative bound. If a suitable bound on the noise magnitude is known, then the differentiation problem can be performed through solving specific convex optimization problems (Haimovich et al, 2022) in the form of linear programs. This strategy is shown to achieve the best possible worst-case accuracy and to have explicitly computable fixed-time convergence and accuracy bounds for first-order differentiation, provided that the noise amplitude is known.…”