Nonparametric Control-Koopman Operator Learning: Flexible and Scalable Models for Prediction and Control

• URL: http://arxiv.org/abs/2405.07312v1
• Date: Sun, 12 May 2024 15:46:52 GMT
• Title: Nonparametric Control-Koopman Operator Learning: Flexible and Scalable Models for Prediction and Control
• Authors: Petar Bevanda, Bas Driessen, Lucian Cristian Iacob, Roland Toth, Stefan Sosnowski, Sandra Hirche,
• Abstract summary: We present a nonparametric framework for learning Koopman operator representations of nonlinear control-affine systems. We also enhance the scalability of control-Koopman operator estimators by leveraging random projections. The efficacy of our novel cKOR approach is demonstrated on both forecasting and control tasks.
• Score: 2.7784144651669704
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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