Koopman Bilinearization of Nonlinear Control Systems Article Swipe
Yaoguang Zhang
,
Jr-Shin Li
·
YOU?
·
· 2022
· Open Access
·
· DOI: https://doi.org/10.48550/arxiv.2211.07112
YOU?
·
· 2022
· Open Access
·
· DOI: https://doi.org/10.48550/arxiv.2211.07112
Koopman operators, since introduced by the French-born American mathematician Bernard Koopman in 1931, have been employed as a powerful tool for research in various scientific domains, such as ergodic theory, probability theory, geometry, and topology. The current use of Koopman operators mainly focuses on the characterization of spectral properties of ergodic dynamical systems. In this paper, we step forward from unforced dynamical systems to control systems and establish a systematic Koopman control framework. Specifically, we rigorously derive a differential equation system governing the dynamics of the Koopman operator associated with a control system, and show that the resulting system is a bilinear system evolving on an infinite-dimensional Lie group, which directly leads to a global bilinearization of control-affine systems. Then, by integrating techniques in geometric control theory with infinite-dimensional differential geometry, this further offers a two-fold benefit to controllability analysis: the characterization of controllability for control-affine systems in terms of de Rham differential operators and the extension of the Lie algebra rank condition to infinite-dimensional systems. To demonstrate the applicability, we further adopt the established framework to develop the Koopman feedback linearization technique, which, as one of the advantages, waives the controllability requirement for the systems to be linearized by using the classical feedback linearization technique. In addition, it is worth mentioning that a distinctive feature of this work is the maximum utilization of the intrinsic algebraic and geometric properties of Koopman operators, instead of spectral methods and the ergodicity assumption of systems, which further demonstrate a significant advantage of our work and a substantial difference between the presented and existing research into Koopman operators.
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- Type
- preprint
- Language
- en
- Landing Page
- http://arxiv.org/abs/2211.07112
- https://arxiv.org/pdf/2211.07112
- OA Status
- green
- Cited By
- 1
- Related Works
- 10
- OpenAlex ID
- https://openalex.org/W4309133721
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https://openalex.org/W4309133721Canonical identifier for this work in OpenAlex
- DOI
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https://doi.org/10.48550/arxiv.2211.07112Digital Object Identifier
- Title
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Koopman Bilinearization of Nonlinear Control SystemsWork title
- Type
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preprintOpenAlex work type
- Language
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enPrimary language
- Publication year
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2022Year of publication
- Publication date
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2022-11-14Full publication date if available
- Authors
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Yaoguang Zhang, Jr-Shin LiList of authors in order
- Landing page
-
https://arxiv.org/abs/2211.07112Publisher landing page
- PDF URL
-
https://arxiv.org/pdf/2211.07112Direct link to full text PDF
- Open access
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YesWhether a free full text is available
- OA status
-
greenOpen access status per OpenAlex
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https://arxiv.org/pdf/2211.07112Direct OA link when available
- Concepts
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Controllability, Mathematics, Dynamical systems theory, Affine transformation, Ergodic theory, Linearization, Lie group, Nonlinear system, Ergodicity, Differential geometry, Operator theory, Algebra over a field, Pure mathematics, Applied mathematics, Statistics, Physics, Quantum mechanicsTop concepts (fields/topics) attached by OpenAlex
- Cited by
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1Total citation count in OpenAlex
- Citations by year (recent)
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2024: 1Per-year citation counts (last 5 years)
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10Other works algorithmically related by OpenAlex
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| abstract_inverted_index.condition | 162 |
| abstract_inverted_index.dynamical | 51, 61 |
| abstract_inverted_index.establish | 67 |
| abstract_inverted_index.extension | 156 |
| abstract_inverted_index.framework | 175 |
| abstract_inverted_index.geometric | 124, 228 |
| abstract_inverted_index.geometry, | 32, 130 |
| abstract_inverted_index.governing | 81 |
| abstract_inverted_index.intrinsic | 225 |
| abstract_inverted_index.operators | 40, 153 |
| abstract_inverted_index.presented | 258 |
| abstract_inverted_index.resulting | 97 |
| abstract_inverted_index.topology. | 34 |
| abstract_inverted_index.associated | 88 |
| abstract_inverted_index.assumption | 240 |
| abstract_inverted_index.difference | 255 |
| abstract_inverted_index.ergodicity | 239 |
| abstract_inverted_index.framework. | 72 |
| abstract_inverted_index.introduced | 3 |
| abstract_inverted_index.linearized | 198 |
| abstract_inverted_index.mentioning | 211 |
| abstract_inverted_index.operators, | 1, 232 |
| abstract_inverted_index.operators. | 264 |
| abstract_inverted_index.properties | 48, 229 |
| abstract_inverted_index.rigorously | 75 |
| abstract_inverted_index.scientific | 24 |
| abstract_inverted_index.systematic | 69 |
| abstract_inverted_index.technique, | 182 |
| abstract_inverted_index.technique. | 205 |
| abstract_inverted_index.techniques | 122 |
| abstract_inverted_index.French-born | 6 |
| abstract_inverted_index.advantages, | 188 |
| abstract_inverted_index.demonstrate | 167, 245 |
| abstract_inverted_index.distinctive | 214 |
| abstract_inverted_index.established | 174 |
| abstract_inverted_index.integrating | 121 |
| abstract_inverted_index.probability | 30 |
| abstract_inverted_index.requirement | 192 |
| abstract_inverted_index.significant | 247 |
| abstract_inverted_index.substantial | 254 |
| abstract_inverted_index.utilization | 222 |
| abstract_inverted_index.differential | 78, 129, 152 |
| abstract_inverted_index.Specifically, | 73 |
| abstract_inverted_index.linearization | 181, 204 |
| abstract_inverted_index.mathematician | 8 |
| abstract_inverted_index.applicability, | 169 |
| abstract_inverted_index.control-affine | 117, 145 |
| abstract_inverted_index.bilinearization | 115 |
| abstract_inverted_index.controllability | 138, 143, 191 |
| abstract_inverted_index.characterization | 45, 141 |
| abstract_inverted_index.infinite-dimensional | 106, 128, 164 |
| cited_by_percentile_year | |
| countries_distinct_count | 0 |
| institutions_distinct_count | 2 |
| citation_normalized_percentile |