Stabilization of unsteady flows by reduced-order control with optimally time-dependent modes
In dynamical systems theory, suppression of instabilities around a fixed point is generally achieved by controlling the linearized dynamics of infinitesimal perturbations, because considering small-amplitude disturbances allows for application of a range of celebrated techniques from linear control...
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| Format: | Article |
| Language: | English |
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American Physical Society (APS)
2021
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| Online Access: | https://hdl.handle.net/1721.1/133541 |
| _version_ | 1826193834232512512 |
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| author | Blanchard, Antoine Sapsis, Themistoklis P |
| author2 | Massachusetts Institute of Technology. Department of Mechanical Engineering |
| author_facet | Massachusetts Institute of Technology. Department of Mechanical Engineering Blanchard, Antoine Sapsis, Themistoklis P |
| author_sort | Blanchard, Antoine |
| collection | MIT |
| description | In dynamical systems theory, suppression of instabilities around a fixed point is generally achieved by controlling the linearized dynamics of infinitesimal perturbations, because considering small-amplitude disturbances allows for application of a range of celebrated techniques from linear control theory. In this paper, we consider the problem of design and implementation of a controller for fully nonlinear, high-dimensional, dynamical systems with the goal of steering trajectories to an unstable fixed point of the governing equations. Our control strategy is based on our previous work [A. Blanchard, S. Mowlavi, and T. P. Sapsis, Nonlinear Dynam. 95, 2745 (2019)10.1007/s11071-018-4720-1] and takes advantage of the unique properties of the optimally time-dependent (OTD) modes, a set of global, time-evolving, orthonormal modes that track directions in phase space associated with transient growth and persistent instabilities. We show that the OTD control strategy introduced previously is robust with respect to perturbation amplitude even in cases in which the trajectory initially evolves on an attractor that lies far away from the target fixed point. In recognition of the fact that actuation capabilities are generally limited in practice, we also formulate a localized control strategy in which the OTD modes are computed in a spatially localized subdomain of the physical domain of interest. We suggest a strategy for selecting the optimal control domain based on a quantitative criterion derived from the OTD modes. We show that even when the range of the controller is reduced, OTD control is able to steer trajectories toward the target fixed point. |
| first_indexed | 2024-09-23T09:46:08Z |
| format | Article |
| id | mit-1721.1/133541 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T09:46:08Z |
| publishDate | 2021 |
| publisher | American Physical Society (APS) |
| record_format | dspace |
| spelling | mit-1721.1/1335412023-09-26T20:03:45Z Stabilization of unsteady flows by reduced-order control with optimally time-dependent modes Blanchard, Antoine Sapsis, Themistoklis P Massachusetts Institute of Technology. Department of Mechanical Engineering In dynamical systems theory, suppression of instabilities around a fixed point is generally achieved by controlling the linearized dynamics of infinitesimal perturbations, because considering small-amplitude disturbances allows for application of a range of celebrated techniques from linear control theory. In this paper, we consider the problem of design and implementation of a controller for fully nonlinear, high-dimensional, dynamical systems with the goal of steering trajectories to an unstable fixed point of the governing equations. Our control strategy is based on our previous work [A. Blanchard, S. Mowlavi, and T. P. Sapsis, Nonlinear Dynam. 95, 2745 (2019)10.1007/s11071-018-4720-1] and takes advantage of the unique properties of the optimally time-dependent (OTD) modes, a set of global, time-evolving, orthonormal modes that track directions in phase space associated with transient growth and persistent instabilities. We show that the OTD control strategy introduced previously is robust with respect to perturbation amplitude even in cases in which the trajectory initially evolves on an attractor that lies far away from the target fixed point. In recognition of the fact that actuation capabilities are generally limited in practice, we also formulate a localized control strategy in which the OTD modes are computed in a spatially localized subdomain of the physical domain of interest. We suggest a strategy for selecting the optimal control domain based on a quantitative criterion derived from the OTD modes. We show that even when the range of the controller is reduced, OTD control is able to steer trajectories toward the target fixed point. 2021-10-27T19:53:26Z 2021-10-27T19:53:26Z 2019 2020-08-04T17:09:04Z Article http://purl.org/eprint/type/JournalArticle https://hdl.handle.net/1721.1/133541 en 10.1103/PHYSREVFLUIDS.4.053902 Physical Review Fluids Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. application/pdf American Physical Society (APS) APS |
| spellingShingle | Blanchard, Antoine Sapsis, Themistoklis P Stabilization of unsteady flows by reduced-order control with optimally time-dependent modes |
| title | Stabilization of unsteady flows by reduced-order control with optimally time-dependent modes |
| title_full | Stabilization of unsteady flows by reduced-order control with optimally time-dependent modes |
| title_fullStr | Stabilization of unsteady flows by reduced-order control with optimally time-dependent modes |
| title_full_unstemmed | Stabilization of unsteady flows by reduced-order control with optimally time-dependent modes |
| title_short | Stabilization of unsteady flows by reduced-order control with optimally time-dependent modes |
| title_sort | stabilization of unsteady flows by reduced order control with optimally time dependent modes |
| url | https://hdl.handle.net/1721.1/133541 |
| work_keys_str_mv | AT blanchardantoine stabilizationofunsteadyflowsbyreducedordercontrolwithoptimallytimedependentmodes AT sapsisthemistoklisp stabilizationofunsteadyflowsbyreducedordercontrolwithoptimallytimedependentmodes |