Optimal Robust Exact Differentiation via Linear Adaptive Techniques

TL;DR

This paper introduces a linear adaptive differentiator that is robust to noise, achieves optimal worst-case error bounds, and converges to the exact derivative instantly in noise-free conditions, outperforming existing methods.

Contribution

It develops a novel linear adaptive differentiator that is both robust and optimal, with a simple implementation achieving quasi-exactness after one sample.

Findings

Outperforms existing differentiators in robustness and accuracy.

Achieves the smallest possible worst-case differentiation error.

Converges to the continuous-time optimal error as sampling period decreases.

Abstract

The problem of differentiating a function with bounded second derivative in the presence of bounded measurement noise is considered in both continuous-time and sampled-data settings. Fundamental performance limitations of causal differentiators, in terms of the smallest achievable worst-case differentiation error, are shown. A robust exact differentiator is then constructed via the adaptation of a single parameter of a linear differentiator. It is demonstrated that the resulting differentiator exhibits a combination of properties that outperforms existing continuous-time differentiators: it is robust with respect to noise, it instantaneously converges to the exact derivative in the absence of noise, and it attains the smallest possible -- hence optimal -- upper bound on its differentiation error under noisy measurements. For sample-based differentiators, the concept of quasi-exactness is introduced to classify differentiators that achieve the lowest possible worst-case error based on sampled measurements in the absence of noise. A straightforward sample-based implementation of the proposed linear adaptive continuous-time differentiator is shown to achieve quasi-exactness after a single sampling step as well as a theoretically optimal differentiation error bound that, in addition, converges to the continuous-time optimal one as the sampling period becomes arbitrarily small. A numerical simulation illustrates the presented formal results.