Differentiator for Noisy Sampled Signals with Best Worst-Case Accuracy

TL;DR

This paper introduces an optimal causal differentiator for noisy sampled signals with bounded noise and derivatives, providing the best worst-case accuracy and requiring no tuning beyond known bounds.

Contribution

It presents a linear program-based differentiator that achieves the tightest worst-case error bounds for signals with bounded noise and second derivatives, outperforming existing methods.

Findings

Achieves the best worst-case differentiation accuracy among causal differentiators.

Provides a fixed-step method for optimal differentiation in noisy environments.

Demonstrates superior performance compared to high-gain and sliding-mode differentiators.

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 sliding-mode differentiators illustrate the obtained results.