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Gain-Preserving Data-Driven Approximation of the Koopman Operator and Its Application in Robust Controller Design

Author

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  • Keita Hara

    (Department of Applied Physics and Physico-Informatics, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama, Kanagawa 223-8522, Japan)

  • Masaki Inoue

    (Department of Applied Physics and Physico-Informatics, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama, Kanagawa 223-8522, Japan)

Abstract

In this paper, we address the data-driven modeling of a nonlinear dynamical system while incorporating a priori information. The nonlinear system is described using the Koopman operator, which is a linear operator defined on a lifted infinite-dimensional state-space. Assuming that the L 2 gain of the system is known, the data-driven finite-dimensional approximation of the operator while preserving information about the gain, namely L 2 gain-preserving data-driven modeling, is formulated. Then, its computationally efficient solution method is presented. An application of the modeling method to feedback controller design is also presented. Aiming for robust stabilization using data-driven control under a poor training dataset, we address the following two modeling problems: (1) Forward modeling: the data-driven modeling is applied to the operating data of a plant system to derive the plant model; (2) Backward modeling: L 2 gain-preserving data-driven modeling is applied to the same data to derive an inverse model of the plant system. Then, a feedback controller composed of the plant and inverse models is created based on internal model control, and it robustly stabilizes the plant system. A design demonstration of the data-driven controller is provided using a numerical experiment.

Suggested Citation

  • Keita Hara & Masaki Inoue, 2021. "Gain-Preserving Data-Driven Approximation of the Koopman Operator and Its Application in Robust Controller Design," Mathematics, MDPI, vol. 9(9), pages 1-18, April.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:9:p:949-:d:542244
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    References listed on IDEAS

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    1. Steven L Brunton & Bingni W Brunton & Joshua L Proctor & J Nathan Kutz, 2016. "Koopman Invariant Subspaces and Finite Linear Representations of Nonlinear Dynamical Systems for Control," PLOS ONE, Public Library of Science, vol. 11(2), pages 1-19, February.
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