Closed-Loop Koopman Operator Approximation
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866913345496416256 |
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| author | Dahdah, Steven Forbes, James Richard |
| author_facet | Dahdah, Steven Forbes, James Richard |
| contents | This paper proposes a method to identify a Koopman model of a feedback-controlled system given a known controller. The Koopman operator allows a nonlinear system to be rewritten as an infinite-dimensional linear system by viewing it in terms of an infinite set of lifting functions. A finite-dimensional approximation of the Koopman operator can be identified from data by choosing a finite subset of lifting functions and solving a regression problem in the lifted space. Existing methods are designed to identify open-loop systems. However, it is impractical or impossible to run experiments on some systems, such as unstable systems, in an open-loop fashion. The proposed method leverages the linearity of the Koopman operator, along with knowledge of the controller and the structure of the closed-loop system, to simultaneously identify the closed-loop and plant systems. The advantages of the proposed closed-loop Koopman operator approximation method are demonstrated in simulation using a Duffing oscillator and experimentally using a rotary inverted pendulum system. An open-source software implementation of the proposed method is publicly available, along with the experimental dataset generated for this paper. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_15318 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Closed-Loop Koopman Operator Approximation Dahdah, Steven Forbes, James Richard Systems and Control Machine Learning Dynamical Systems This paper proposes a method to identify a Koopman model of a feedback-controlled system given a known controller. The Koopman operator allows a nonlinear system to be rewritten as an infinite-dimensional linear system by viewing it in terms of an infinite set of lifting functions. A finite-dimensional approximation of the Koopman operator can be identified from data by choosing a finite subset of lifting functions and solving a regression problem in the lifted space. Existing methods are designed to identify open-loop systems. However, it is impractical or impossible to run experiments on some systems, such as unstable systems, in an open-loop fashion. The proposed method leverages the linearity of the Koopman operator, along with knowledge of the controller and the structure of the closed-loop system, to simultaneously identify the closed-loop and plant systems. The advantages of the proposed closed-loop Koopman operator approximation method are demonstrated in simulation using a Duffing oscillator and experimentally using a rotary inverted pendulum system. An open-source software implementation of the proposed method is publicly available, along with the experimental dataset generated for this paper. |
| title | Closed-Loop Koopman Operator Approximation |
| topic | Systems and Control Machine Learning Dynamical Systems |
| url | https://arxiv.org/abs/2303.15318 |