Asymptotically stable data-driven koopman operator approximation with inputs using total extended DMD
The Koopman operator framework can be used to identify a data-driven model of a nonlinear system. Unfortunately, when the data is corrupted by noise, the identified model can be biased. Additionally, depending on the choice of lifting functions, the identified model can be unstable, even when the un...
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IOP Publishing
2025-01-01
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| Series: | Machine Learning: Science and Technology |
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| Online Access: | https://doi.org/10.1088/2632-2153/ada33b |
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| author | Louis Lortie James Richard Forbes |
| author_facet | Louis Lortie James Richard Forbes |
| author_sort | Louis Lortie |
| collection | DOAJ |
| description | The Koopman operator framework can be used to identify a data-driven model of a nonlinear system. Unfortunately, when the data is corrupted by noise, the identified model can be biased. Additionally, depending on the choice of lifting functions, the identified model can be unstable, even when the underlying system is asymptotically stable. This paper presents an approach to reduce the bias in an approximate Koopman model, and simultaneously ensure asymptotic stability, when using noisy data. Additionally, the proposed data-driven modeling approach is applicable to systems with inputs, such as a known forcing function or a control input. Specifically, bias is reduced by using a total least-squares, modified to accommodate inputs in addition to lifted inputs. To enforce asymptotic stability of the approximate Koopman model, linear matrix inequality constraints are augmented to the identification problem. The performance of the proposed method is then compared to the well-known extended dynamic mode decomposition (DMD) method and to the newly introduced forward–backward extended DMD method using a simulated Duffing oscillator dataset and experimental soft robot arm dataset. |
| format | Article |
| id | doaj-art-322d1f1b091d4fa3a6ad88de9fc2c985 |
| institution | Kabale University |
| issn | 2632-2153 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | IOP Publishing |
| record_format | Article |
| series | Machine Learning: Science and Technology |
| spelling | doaj-art-322d1f1b091d4fa3a6ad88de9fc2c9852025-01-13T07:27:51ZengIOP PublishingMachine Learning: Science and Technology2632-21532025-01-016101500310.1088/2632-2153/ada33bAsymptotically stable data-driven koopman operator approximation with inputs using total extended DMDLouis Lortie0https://orcid.org/0009-0006-8990-3805James Richard Forbes1https://orcid.org/0000-0002-1987-9268Department of Mechanical Engineering, McGill University , 817 Sherbrooke Street West, Montreal, QC H3A 0C3, CanadaDepartment of Mechanical Engineering, McGill University , 817 Sherbrooke Street West, Montreal, QC H3A 0C3, CanadaThe Koopman operator framework can be used to identify a data-driven model of a nonlinear system. Unfortunately, when the data is corrupted by noise, the identified model can be biased. Additionally, depending on the choice of lifting functions, the identified model can be unstable, even when the underlying system is asymptotically stable. This paper presents an approach to reduce the bias in an approximate Koopman model, and simultaneously ensure asymptotic stability, when using noisy data. Additionally, the proposed data-driven modeling approach is applicable to systems with inputs, such as a known forcing function or a control input. Specifically, bias is reduced by using a total least-squares, modified to accommodate inputs in addition to lifted inputs. To enforce asymptotic stability of the approximate Koopman model, linear matrix inequality constraints are augmented to the identification problem. The performance of the proposed method is then compared to the well-known extended dynamic mode decomposition (DMD) method and to the newly introduced forward–backward extended DMD method using a simulated Duffing oscillator dataset and experimental soft robot arm dataset.https://doi.org/10.1088/2632-2153/ada33bdynamical systemsKoopman operator theorylinear matrix inequalitiesnoisy datalinear systems theoryasymptotic stability |
| spellingShingle | Louis Lortie James Richard Forbes Asymptotically stable data-driven koopman operator approximation with inputs using total extended DMD Machine Learning: Science and Technology dynamical systems Koopman operator theory linear matrix inequalities noisy data linear systems theory asymptotic stability |
| title | Asymptotically stable data-driven koopman operator approximation with inputs using total extended DMD |
| title_full | Asymptotically stable data-driven koopman operator approximation with inputs using total extended DMD |
| title_fullStr | Asymptotically stable data-driven koopman operator approximation with inputs using total extended DMD |
| title_full_unstemmed | Asymptotically stable data-driven koopman operator approximation with inputs using total extended DMD |
| title_short | Asymptotically stable data-driven koopman operator approximation with inputs using total extended DMD |
| title_sort | asymptotically stable data driven koopman operator approximation with inputs using total extended dmd |
| topic | dynamical systems Koopman operator theory linear matrix inequalities noisy data linear systems theory asymptotic stability |
| url | https://doi.org/10.1088/2632-2153/ada33b |
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