High-Order Lie Derivatives from Taylor Series in the ADTAYL Package
Nedialko S. Nedialkov, John D. Pryce
TL;DR
High-order Lie derivatives are essential for nonlinear analysis but become intractable symbolically at large orders. The paper introduces a purely numerical approach based on Taylor series arithmetic implemented in the MATLAB ADTAYL package, exploiting Röbenack identities that link $L_f^{k}$ to Taylor coefficients of the trajectory $x(t)$ and the variational matrix $J(t)$. By combining the adtayl class with the odets solver and a taylcoeffs interface, the method propagates Taylor coefficients through BAOs and sub-ODEs to compute $L_f^{k} X(x_0)$ for scalar, vector, and covector fields, including their families. Experiments on a gantry crane model demonstrate orders-of-magnitude speedups over symbolic evaluation with the MATLAB Symbolic Math Toolbox, highlighting a scalable, differentiable-programming-friendly workflow for high-order Lie-derivative computation in nonlinear control and observability analysis.
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 Röbenack: 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.