Computation of Lie Brackets by Algorithmic Differentiation

Lie brackets are Lie derivatives of vector fields. They are usually computed using computer algebra software. This can result in very large expressions. We suggest an alternative approach using algorithmic of automatic differentiation.

Nonlinear Control Systems and Lie Brackets

Consider a nonlinear control system

\[\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x})+\mathbf{g}(\mathbf{x})\,u\]

with the state vector $\mathbf{x}$, the input $u$ and the smooth vector fields $\mathbf{f},\mathbf{g}:\mathbb{R}^{n}\to\mathbb{R}^{n}$. The Lie bracket of $\mathbf{f}$ and $\mathbf{g}$ is given by

\[[\mathbf{f},\mathbf{g}](\mathbf{x}) = L_{\mathbf{f}}h(\mathbf{x})=\frac{\partial h(\mathbf{x})}{\partial\mathbf{x}}\mathbf{f}(\mathbf{x}).\]

Publication

• Röbenack, K.: On the efficient computation of higher order maps $ad_kf g(x)$ using Taylor arithmetic and the Campbell-Baker-Hausdorff formula.
  In: Zinober, A.; Owens, D. (Hrsg.), Nonlinear and Adaptive Control, Springer, 2002, 281, 327-336.
• Röbenack, K.; Reinschke, K. J.: The Computation of Lie Derivatives and Lie Brackets based on Automatic Differentiation.
  Zeitschrift für Angewandte Mathematik und Mechnik (ZAMM), 2004, 84, 114-123
• Röbenack, K.: Computation of Lie Derivatives of Tensor Fields Required for Nonlinear Controller and Observer Design Employing Automatic Differentiation.
  Proc. in Applied Mathematics and Mechanics, 2005, 5, 181-184
• Röbenack, K.; Winkler, J.; Wang, S.: LIEDRIVERS - A Toolbox for the Efficient Computation of Lie Derivatives Based on the Object-Oriented Algorithmic Differentiation Package ADOL-C.
  In: Proc. of the 4th International Workshop on Equations-Based Object-Oriented Modeling Languages and Tools (EOOLT 2011). Zurich, Switzerland, September 5, 2011. ISSN: 1650-3686.