Data-Driven Control of Linear Parabolic Systems using Koopman Eigenstructure Assignment
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866910508051857408 |
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| author | Deutscher, J. |
| author_facet | Deutscher, J. |
| contents | This paper considers the data-driven stabilization of linear boundary controlled parabolic PDEs by making use of the Koopman operator. For this, a Koopman eigenstructure assignment problem is solved, which amounts to determine a feedback of the Koopman open-loop eigenfunctionals assigning a desired finite set of closed-loop Koopman eigenvalues and eigenfunctionals to the closed-loop system. It is shown that the designed controller only needs a finite number of open-loop Koopman eigenvalues and modes of the state. They are determined by extending the classical Krylov-DMD to parabolic systems. For this, only a finite number of pointlike outputs and their temporal samples as well as temporal samples of the inputs are required resulting in a data-driven solution of the eigenstructure assignment problem. Exponential stability of the closed-loop system in the presence of small Krylov-DMD errors is verified. An unstable diffusion-reaction system demonstrates the new data-driven controller design technique for distributed-parameter systems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_00432 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Data-Driven Control of Linear Parabolic Systems using Koopman Eigenstructure Assignment Deutscher, J. Systems and Control This paper considers the data-driven stabilization of linear boundary controlled parabolic PDEs by making use of the Koopman operator. For this, a Koopman eigenstructure assignment problem is solved, which amounts to determine a feedback of the Koopman open-loop eigenfunctionals assigning a desired finite set of closed-loop Koopman eigenvalues and eigenfunctionals to the closed-loop system. It is shown that the designed controller only needs a finite number of open-loop Koopman eigenvalues and modes of the state. They are determined by extending the classical Krylov-DMD to parabolic systems. For this, only a finite number of pointlike outputs and their temporal samples as well as temporal samples of the inputs are required resulting in a data-driven solution of the eigenstructure assignment problem. Exponential stability of the closed-loop system in the presence of small Krylov-DMD errors is verified. An unstable diffusion-reaction system demonstrates the new data-driven controller design technique for distributed-parameter systems. |
| title | Data-Driven Control of Linear Parabolic Systems using Koopman Eigenstructure Assignment |
| topic | Systems and Control |
| url | https://arxiv.org/abs/2407.00432 |