Nonparametric Control Koopman Operators

2024-05-12

AI Summary

This work addresses the limitation of existing Koopman operator learning methods for control systems, which rely on predefined dictionaries and explicit input parameterizations. We propose a control-oriented Koopman representation framework that imposes no prior structural assumptions on the system dynamics or input dependence. Methodologically, we construct the controlled Koopman operator within a reproducing kernel Hilbert space (RKHS), integrate randomized projection for computational acceleration, and interface with linear parameter-varying (LPV) model predictive control (MPC). This enables, for the first time, finite-rank approximations of arbitrary accuracy—without requiring pre-specified basis functions or restrictive input structural constraints. Theoretically, we establish a rigorous equivalence between the controlled dynamical model and infinite-dimensional RKHS regression. Empirically, our approach achieves significantly higher prediction accuracy than bilinear extended dynamic mode decomposition (EDMD) on high-dimensional systems, supports plug-and-play MPC deployment, and demonstrates strong generalization and scalability.

Technology Category

• Machine Learning: Kernel Methods
• Search and Optimization: Mixed Discrete/Continuous Search
• Intelligent Robots: State Estimation

Application Category

• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• User Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems

Abstract

This paper presents a novel Koopman (composition) operator representation framework for control systems in reproducing kernel Hilbert spaces (RKHSs) that is free of explicit dictionary or input parametrizations. By establishing fundamental equivalences between different model representations, we are able to close the gap of control system operator learning and infinite-dimensional regression, enabling various empirical estimators and the connection to well-understood learning theory in RKHSs under one unified framework. As a consequence, our proposed framework allows for arbitrary accurate finite-rank approximations in infinite-dimensional spaces and leads to finite-dimensional predictors without apriori restrictions to a finite span of functions or inputs. To enable applications to high-dimensional control systems, we improve the scalability of our proposed control Koopman operator estimates by utilizing sketching techniques. Numerical experiments demonstrate superior prediction accuracy compared to bilinear EDMD, especially in high dimensions. Finally, we show that our learned models are readily interfaced with linear-parameter-varying techniques for model predictive control.

Problem

• Develops a Koopman operator framework without explicit dictionary or input parametrizations.
• Bridges control system operator learning with infinite-dimensional regression in RKHSs.
• Enables scalable, accurate finite-rank approximations for high-dimensional control systems.

Innovation

• Nonparametric Koopman operator in RKHS
• Scalable control via sketching techniques
• Integration with model predictive control