TLDR: This paper introduces a novel approach to data-driven control for nonlinear systems by generalizing the Koopman operator and deriving a nonlinear fundamental lemma. It achieves this by using product Hilbert spaces for state and input observable functions, leading to an infinite-dimensional bilinear Koopman model that doesn’t require restrictive assumptions on system dynamics. The method, called GeKo, shows improved accuracy in approximating nonlinear system behavior from data, demonstrated on the Van der Pol oscillator.
Understanding and controlling complex, real-world systems is a significant challenge, especially when their behavior is nonlinear. Two prominent approaches in data-driven control, the Koopman operator and Willems’ fundamental lemma, have shown great promise but face hurdles when dealing with systems that have control inputs.
A recent research paper, “From Product Hilbert Spaces to the Generalized Koopman Operator and the Nonlinear Fundamental Lemma” by Mircea Lazar, introduces a novel solution to these long-standing problems. The core idea revolves around constructing a “product Hilbert space” by combining the mathematical spaces of state and input observable functions. This allows for a new way to represent nonlinear systems as infinite-dimensional, linear systems.
The Generalized Koopman Operator
The Koopman operator approach aims to transform a nonlinear system into a linear one in a higher-dimensional space. Traditionally, applying this to systems with control inputs has been difficult, often requiring specific assumptions about how the input affects the system. Lazar’s work proposes a “generalized Koopman operator” that operates on this newly defined product Hilbert space. Unlike previous methods that simply “stack” state and input information, this new approach uses a “tensor product,” which fundamentally changes how the system’s dynamics are represented, leading to a more complete and general framework.
This results in an infinite-dimensional “bilinear” Koopman model, meaning it involves products of lifted states and inputs. A key advantage is that this model doesn’t require restrictive assumptions about the system’s underlying structure, making it applicable to a wider range of nonlinear systems.
A Nonlinear Fundamental Lemma
Building on the generalized Koopman operator, the paper also derives a “nonlinear fundamental lemma.” Willems’ fundamental lemma is a powerful concept for linear systems, allowing their behavior to be described directly from data. Extending this to nonlinear systems has been a major challenge. By leveraging the bilinear structure of the generalized Koopman model, Lazar provides a data-driven representation of nonlinear systems based on their trajectories, without needing explicit knowledge of the system’s equations.
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Practical Applications and Approximations
While the theoretical framework involves infinite dimensions, the paper also provides practical methods for computing finite-dimensional approximations of these generalized Koopman operators from data. It discusses how existing techniques like Extended Dynamic Mode Decomposition (EDMD) can be adapted. Furthermore, it suggests several choices for “observable functions” – the mathematical functions used to “lift” the system’s state and input into the higher-dimensional spaces. These include universal kernels, deep neural networks, and Takens delay embeddings, all of which are shown to be valid choices for approximating the generalized Koopman operator.
The effectiveness of the proposed “GeKo” method was demonstrated using the Van der Pol oscillator, a classic example of a nonlinear system. The results showed that the GeKo method consistently improved prediction accuracy as the number of observable functions increased, outperforming a benchmark method (KIC) that struggled to improve with increased complexity.
This research opens up new avenues for data-driven control of nonlinear systems, offering a more complete and scalable framework for analysis and control design. Future work will explore its application in areas like predictive control and distributed control for interconnected systems.
