Gain-Preserving Data-Driven Approximation of the Koopman Operator and Its Application in Robust Controller Design
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 <inline-for...
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MDPI AG
2021-04-01
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| Online Access: | https://www.mdpi.com/2227-7390/9/9/949 |
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| author | Keita Hara Masaki Inoue |
| author_facet | Keita Hara Masaki Inoue |
| author_sort | Keita Hara |
| collection | DOAJ |
| description | 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="script">L</mi><mn>2</mn></msub></semantics></math></inline-formula> gain of the system is known, the data-driven finite-dimensional approximation of the operator while preserving information about the gain, namely <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="script">L</mi><mn>2</mn></msub></semantics></math></inline-formula> 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: <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="script">L</mi><mn>2</mn></msub></semantics></math></inline-formula> 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. |
| first_indexed | 2024-03-10T12:01:37Z |
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| language | English |
| last_indexed | 2024-03-10T12:01:37Z |
| publishDate | 2021-04-01 |
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| spelling | doaj.art-2f39fbc8a93b49f2b3c9c174e7c13a432023-11-21T16:53:28ZengMDPI AGMathematics2227-73902021-04-019994910.3390/math9090949Gain-Preserving Data-Driven Approximation of the Koopman Operator and Its Application in Robust Controller DesignKeita Hara0Masaki Inoue1Department of Applied Physics and Physico-Informatics, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama, Kanagawa 223-8522, JapanDepartment of Applied Physics and Physico-Informatics, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama, Kanagawa 223-8522, JapanIn 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="script">L</mi><mn>2</mn></msub></semantics></math></inline-formula> gain of the system is known, the data-driven finite-dimensional approximation of the operator while preserving information about the gain, namely <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="script">L</mi><mn>2</mn></msub></semantics></math></inline-formula> 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: <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="script">L</mi><mn>2</mn></msub></semantics></math></inline-formula> 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.https://www.mdpi.com/2227-7390/9/9/949Koopman operatordata-driven modelingdata-driven controlrobust controlinternal model control |
| spellingShingle | Keita Hara Masaki Inoue Gain-Preserving Data-Driven Approximation of the Koopman Operator and Its Application in Robust Controller Design Mathematics Koopman operator data-driven modeling data-driven control robust control internal model control |
| title | Gain-Preserving Data-Driven Approximation of the Koopman Operator and Its Application in Robust Controller Design |
| title_full | Gain-Preserving Data-Driven Approximation of the Koopman Operator and Its Application in Robust Controller Design |
| title_fullStr | Gain-Preserving Data-Driven Approximation of the Koopman Operator and Its Application in Robust Controller Design |
| title_full_unstemmed | Gain-Preserving Data-Driven Approximation of the Koopman Operator and Its Application in Robust Controller Design |
| title_short | Gain-Preserving Data-Driven Approximation of the Koopman Operator and Its Application in Robust Controller Design |
| title_sort | gain preserving data driven approximation of the koopman operator and its application in robust controller design |
| topic | Koopman operator data-driven modeling data-driven control robust control internal model control |
| url | https://www.mdpi.com/2227-7390/9/9/949 |
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