Optimal robust exact first-order differentiators with Lipschitz continuous output

TL;DR

This paper introduces a novel first-order differentiator that is robust to noise, exact without noise, optimally accurate, and has a Lipschitz continuous output with a tunable constant, applicable in both continuous and discrete forms.

Contribution

It develops the first differentiator combining robustness, exactness, optimal error, and Lipschitz continuity, with theoretical guarantees and practical discretization.

Findings

Achieves optimal worst-case differentiation error

Ensures robustness to measurement noise

Provides Lipschitz continuous output with tunable constant

Abstract

The signal differentiation problem involves the development of algorithms that allow to recover a signal's derivatives from noisy measurements. This paper develops a first-order differentiator with the following combination of properties: robustness to measurement noise, exactness in the absence of noise, optimal worst-case differentiation error, and Lipschitz continuous output where the output's Lipschitz constant is a tunable parameter. This combination of advantageous properties is not shared by any existing differentiator. Both continuous-time and sample-based versions of the differentiator are developed and theoretical guarantees are established for both. The continuous-time version of the differentiator consists in a regularized and sliding-mode-filtered linear adaptive differentiator. The sample-based, implementable version is then obtained through appropriate discretization. An illustrative example is provided to highlight the features of the developed differentiator.