TY - JOUR

T1 - Gain-preserving data-driven approximation of the koopman operator and its application in robust controller design

AU - Hara, Keita

AU - Inoue, Masaki

N1 - Funding Information:
Funding: This work was supported by Grant-in-Aid for Scientific Research (B), No. 20H04473 from JSPS.
Publisher Copyright:
© 2021 by the authors. Licensee MDPI, Basel, Switzerland.

PY - 2021/5/1

Y1 - 2021/5/1

N2 - 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 L2 gain of the system is known, the data-driven finite-dimensional approximation of the operator while preserving information about the gain, namely L2 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: L2 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.

AB - 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 L2 gain of the system is known, the data-driven finite-dimensional approximation of the operator while preserving information about the gain, namely L2 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: L2 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.

KW - Data-driven control

KW - Data-driven modeling

KW - Internal model control

KW - Koopman operator

KW - Robust control

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U2 - 10.3390/math9090949

DO - 10.3390/math9090949

M3 - Article

AN - SCOPUS:85105326212

SN - 2227-7390

VL - 9

JO - Mathematics

JF - Mathematics

IS - 9

M1 - 949

ER -