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

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

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