Extended Dynamic Mode Decomposition, embedded in the Koopman framework, is a widely-applied technique to predict the evolution of an observable along the flow of a dynamical control system. However, despite its popularity, the error analysis is still fragmentary. We provide a complete and rigorous analysis for control-affine systems by splitting up the approximation error into the projection and estimation error resulting from the finite dictionary size [3] and the finite amount of i.i.d. data used to generate the surrogate model [1]. If time permits, an extension towards reproducible kernel Hilbert spaces is indicated, see [2] for details. Then, the usefulness of the derived surrogate model for predictive control of non-holonomic systems is demonstrated based on experimental data.
References
[1] Feliks Nüske, Sebastian Peitz, Friedrich Philipp, Manuel Schaller, and Karl Worthmann. Finite-data error bounds for Koopman-based prediction and control. Journal of Nonlinear Science, 33:14, 2023.
[2] Friedrich Philipp, Manuel Schaller, Karl Worthmann, Sebastian Peitz, and Feliks Nüske. Error bounds for kernel-based approximations of the Koopman operator. 2023. arXiv preprint https://arxiv.org/abs/2301.08637.
[3] Manuel Schaller, Karl Worthmann, Friedrich Philipp, Sebastian Peitz, and Feliks Nüske. Towards reliable data-based optimal and predictive control using extended DMD. In Proc. 12th IFAC Symposium on Nonlinear Control
Systems, 2022. Accepted for publication, arXiv preprint arXiv:2202.09084.