Data-Driven Analytic Differentiation via High Gain Observers and Gaussian Process Priors

TL;DR

This paper introduces a robust, data-driven method combining high-gain observers and Gaussian process regressors to accurately model unknown functions and derivatives from limited, scattered data, with applications in control safety.

Contribution

It proposes a novel cascade of high-gain observers and Gaussian process regressors that work effectively with limited data, improving robustness and flexibility over existing methods.

Findings

Outperforms previous approaches in numerical simulations.

Provides performance bounds on regression error.

Effective with small, sliding data windows.

Abstract

The presented paper tackles the problem of modeling an unknown function, and its first $r-1$ derivatives, out of scattered and poor-quality data. The considered setting embraces a large number of use cases addressed in the literature and fits especially well in the context of control barrier functions, where high-order derivatives of the safe set are required to preserve the safety of the controlled system. The approach builds on a cascade of high-gain observers and a set of Gaussian process regressors trained on the observers' data. The proposed structure allows for high robustness against measurement noise and flexibility with respect to the employed sampling law. Unlike previous approaches in the field, where a large number of samples are required to fit correctly the unknown function derivatives, here we suppose to have access only to a small window of samples, sliding in time. The paper presents performance bounds on the attained regression error and numerical simulations showing how the proposed method outperforms previous approaches.