High-Order Lie Derivatives from Taylor Series in the ADTAYL Package

TL;DR

This paper introduces a fast numerical method for computing high-order Lie derivatives using Taylor series in MATLAB, significantly outperforming symbolic approaches in efficiency.

Contribution

The paper presents a novel numerical approach leveraging Taylor series to compute high-order Lie derivatives efficiently within the ADTAYL package.

Findings

Achieves orders of magnitude speedup over symbolic methods

Applicable to scalar, vector, and covector fields

Demonstrated on a gantry crane model

Abstract

High-order Lie derivatives are essential in nonlinear systems analysis. If done symbolically, their evaluation becomes increasingly expensive as the order increases. We present a compact and efficient numerical approach for computing Lie derivatives of scalar, vector, and covector fields using the MATLAB ADTAYL package. The method exploits a fact noted by R\"obenack: that these derivatives coincide, up to factorial scaling, with the Taylor coefficients of expressions built from a Taylor expansion about a trajectory point and, when required, the associated variational matrix. Computational results for a gantry crane model demonstrate orders of magnitude speedups over symbolic evaluation using the MATLAB Symbolic Math Toolbox.