Abstract
AbstractThe chain rule is a standard tool in differential calculus to find derivatives of composite functions. Faà di Bruno’s formula is a generalization of the chain rule and states a method to find high-order derivatives. In this contribution, we propose an algorithm based on Faà di Bruno’s formula and Bell polynomials (Bell in Ann Math 29:38–46, 1927; Parks and Krantz in A primer of real analytic functions, 2012) to compute the structure of derivatives of function compositions. The application of our method is showcased using trajectory planning for the heat equation (Laroche et al. in Int J Robust Nonlinear Control 10(8):629–643, 2000).
Funder
Hochschule Ravensburg-Weingarten
Publisher
Springer Science and Business Media LLC
Reference15 articles.
1. Parks HR, Krantz SG. A primer of real analytic functions. Basel: Birkhäuser; 2012.
2. Laroche B, Martin P, Rouchon P. Motion planning for the heat equation. Int J Robust Nonlinear Control. 2000;10(8):629–43.
3. Meurer T. Control of higher–dimensional PDEs: flatness and backstepping designs. Berlin: Springer; 2012.
4. Utz T, Graichen K, Kugi A. Trajectory planning and receding horizon tracking control of a quasilinear diffusion-convection-reaction system. IFAC Proc Vol. 2010;43(14):587–92.
5. Scholz S, Berger L, Lebiedz D. Benchmarking of flatness-based control of the heat equation. 2023. arXiv preprint. arXiv:2307.16764.