From Product Hilbert Spaces to the Generalized Koopman Operator and the Nonlinear Fundamental Lemma

2025-08-10

AI Summary

Addressing two open challenges in data-driven control of nonlinear systems—the difficulty of incorporating control inputs into the Koopman operator framework and the absence of a nonlinear fundamental lemma—this paper constructs a product Hilbert space spanned by tensor products of state- and input-dependent observable functions. Within this space, it proposes a generalized Koopman operator framework based on orthogonal expansion—the first application of such a space to system identification—and rigorously derives the nonlinear fundamental lemma leveraging its inherent bilinear structure. Furthermore, the paper develops scalable finite-dimensional approximation algorithms and accommodates multiple classes of observable function parametrizations. Experimental validation on the Van der Pol oscillator demonstrates high modeling and predictive accuracy, significantly enhancing the unification and practicality of data-driven design for nonlinear control systems.

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• Search and Optimization: Non-convex Optimization
• Machine Learning: Kernel Methods
• Intelligent Robots: State Estimation

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Abstract

The generalization of the Koopman operator to systems with control input and the derivation of a nonlinear fundamental lemma are two open problems that play a key role in the development of data-driven control methods for nonlinear systems. Both problems hinge on the construction of observable or basis functions and their corresponding Hilbert space that enable an infinite-dimensional, linear system representation. In this paper we derive a novel solution to these problems based on orthonormal expansion in a product Hilbert space constructed as the tensor product between the Hilbert spaces of the state and input observable functions, respectively. We prove that there exists an infinite-dimensional linear operator, i.e. the generalized Koopman operator, from the constructed product Hilbert space to the Hilbert space corresponding to the lifted state propagated forward in time. A scalable data-driven method for computing finite-dimensional approximations of generalized Koopman operators and several choices of observable functions are also presented. Moreover, we derive a nonlinear fundamental lemma by exploiting the bilinear structure of the infinite-dimensional generalized Koopman model. The effectiveness of the developed generalized Koopman embedding is illustrated on the Van der Pol oscillator.

Problem

• Generalizing Koopman operator for control input systems
• Deriving nonlinear fundamental lemma for data-driven control
• Constructing observable functions in product Hilbert spaces

Innovation

• Orthonormal expansion in product Hilbert space
• Generalized Koopman operator for nonlinear systems
• Data-driven computation of finite-dimensional approximations