A quantitative and constructive proof of Willems' Fundamental Lemma and its implications
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arXiv
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| Hauptverfasser: | , , , , , |
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| Format: | Preprint |
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2022
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| _version_ | 1866910724569169920 |
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| author | Berberich, Julian Iannelli, Andrea Padoan, Alberto Coulson, Jeremy Dörfler, Florian Allgöwer, Frank |
| author_facet | Berberich, Julian Iannelli, Andrea Padoan, Alberto Coulson, Jeremy Dörfler, Florian Allgöwer, Frank |
| contents | Willems' Fundamental Lemma provides a powerful data-driven parametrization of all trajectories of a controllable linear time-invariant system based on one trajectory with persistently exciting (PE) input. In this paper, we present a novel proof of this result which is inspired by the classical adaptive control literature and differs from existing proofs in multiple aspects. The proof involves a quantitative and directional PE notion, allowing to characterize robust PE properties via singular value bounds, as opposed to binary rank-based PE conditions. Further, the proof is constructive, i.e., we derive an explicit PE lower bound for the generated data. As a contribution of independent interest, we generalize existing PE results from the adaptive control literature and reveal a crucial role of the system's zeros. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_00905 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A quantitative and constructive proof of Willems' Fundamental Lemma and its implications Berberich, Julian Iannelli, Andrea Padoan, Alberto Coulson, Jeremy Dörfler, Florian Allgöwer, Frank Optimization and Control Systems and Control Willems' Fundamental Lemma provides a powerful data-driven parametrization of all trajectories of a controllable linear time-invariant system based on one trajectory with persistently exciting (PE) input. In this paper, we present a novel proof of this result which is inspired by the classical adaptive control literature and differs from existing proofs in multiple aspects. The proof involves a quantitative and directional PE notion, allowing to characterize robust PE properties via singular value bounds, as opposed to binary rank-based PE conditions. Further, the proof is constructive, i.e., we derive an explicit PE lower bound for the generated data. As a contribution of independent interest, we generalize existing PE results from the adaptive control literature and reveal a crucial role of the system's zeros. |
| title | A quantitative and constructive proof of Willems' Fundamental Lemma and its implications |
| topic | Optimization and Control Systems and Control |
| url | https://arxiv.org/abs/2208.00905 |