Nonparametric Control-Koopman Operator Learning: Flexible and Scalable Models for Prediction and Control
<|reference_start|>Nonparametric Control-Koopman Operator Learning: Flexible and Scalable Models for Prediction and Control: Linearity of Koopman operators and simplicity of their estimators coupled with model-reduction capabilities has lead to their great popularity in applications for learning dynamical systems. While nonparametric Koopman operator learning in infinite-dimensional reproducing kernel Hilbert spaces is well understood for autonomous systems, its control system analogues are largely unexplored. Addressing systems with control inputs in a principled manner is crucial for fully data-driven learning of controllers, especially since existing approaches commonly resort to representational heuristics or parametric models of limited expressiveness and scalability. We address the aforementioned challenge by proposing a universal framework via control-affine reproducing kernels that enables direct estimation of a single operator even for control systems. The proposed approach, called control-Koopman operator regression (cKOR), is thus completely analogous to Koopman operator regression of the autonomous case. First in the literature, we present a nonparametric framework for learning Koopman operator representations of nonlinear control-affine systems that does not suffer from the curse of control input dimensionality. This allows for reformulating the infinite-dimensional learning problem in a finite-dimensional space based solely on data without apriori loss of precision due to a restriction to a finite span of functions or inputs as in other approaches. For enabling applications to large-scale control systems, we also enhance the scalability of control-Koopman operator estimators by leveraging random projections (sketching). The efficacy of our novel cKOR approach is demonstrated on both forecasting and control tasks.<|reference_end|>
@article{bevanda2024nonparametric,
    title={Nonparametric Control-Koopman Operator Learning: Flexible and Scalable
  Models for Prediction and Control},
    author={Petar Bevanda, Bas Driessen, Lucian Cristian Iacob, Roland Toth,
  Stefan Sosnowski and Sandra Hirche},
    journal={arXiv preprint arXiv:2405.07312},
    year={2024},
    archivePrefix={arXiv},
    eprint={2405.07312},
    primaryClass={eess.SY cs.LG cs.SY}
}