Kernel-based Koopman approximants for control: Flexible sampling, error analysis, and stability
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866909852817686528 |
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| author | Bold, Lea Philipp, Friedrich M. Schaller, Manuel Worthmann, Karl |
| author_facet | Bold, Lea Philipp, Friedrich M. Schaller, Manuel Worthmann, Karl |
| contents | Data-driven techniques for analysis, modeling, and control of complex dynamical systems are on the uptake. Koopman theory provides the theoretical foundation for the popular kernel extended dynamic mode decomposition (kEDMD). In this work, we propose a novel kEDMD scheme to approximate nonlinear control systems accompanied by an in-depth error analysis. Key features are regularization-based robustness and an adroit decomposition into micro and macro grids enabling flexible sampling. But foremost, we prove proportionality, i.e., explicit dependence on the distance to the (controlled) equilibrium, of the derived bound on the full approximation error. Leveraging this key property, we rigorously show that asymptotic stability of the data-driven surrogate (control) system implies asymptotic stability of the original (control) system and vice versa. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_02811 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Kernel-based Koopman approximants for control: Flexible sampling, error analysis, and stability Bold, Lea Philipp, Friedrich M. Schaller, Manuel Worthmann, Karl Optimization and Control Systems and Control Data-driven techniques for analysis, modeling, and control of complex dynamical systems are on the uptake. Koopman theory provides the theoretical foundation for the popular kernel extended dynamic mode decomposition (kEDMD). In this work, we propose a novel kEDMD scheme to approximate nonlinear control systems accompanied by an in-depth error analysis. Key features are regularization-based robustness and an adroit decomposition into micro and macro grids enabling flexible sampling. But foremost, we prove proportionality, i.e., explicit dependence on the distance to the (controlled) equilibrium, of the derived bound on the full approximation error. Leveraging this key property, we rigorously show that asymptotic stability of the data-driven surrogate (control) system implies asymptotic stability of the original (control) system and vice versa. |
| title | Kernel-based Koopman approximants for control: Flexible sampling, error analysis, and stability |
| topic | Optimization and Control Systems and Control |
| url | https://arxiv.org/abs/2412.02811 |