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Control of linear instabilities by dynamically consistent order reduction on optimally time-dependent modes

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Abstract

Identification and control of transient instabilities in high-dimensional dynamical systems remain a challenge because transient (non-normal) growth cannot be accurately captured by reduced-order modal analysis. Eigenvalue-based methods classify systems as stable or unstable on the sole basis of the asymptotic behavior of perturbations and therefore fail to predict any short-term characteristics of disturbances, including transient growth. In this paper, we leverage the power of the optimally time-dependent (OTD) modes, a set of time-evolving, orthonormal modes that capture directions in phase space associated with transient and persistent instabilities, to formulate a control law capable of suppressing transient and asymptotic growth around any fixed point of the governing equations. The control law is derived from a reduced-order system resulting from projecting the evolving linearized dynamics onto the OTD modes and enforces that the instantaneous growth of perturbations in the OTD-reduced tangent space be nil. We apply the proposed reduced-order control algorithm to several infinite-dimensional systems, including fluid flows dominated by normal and non-normal instabilities, and demonstrate unequivocal superiority of OTD control over classical modal control.

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Acknowledgements

The authors gratefully acknowledge insightful discussions with Dr. Mohammad Farazmand.

Funding

This study was supported by Army Research Office Grant W911NF-17-1-0306 and Air Force Office of Scientific Research Grant FA9550-16-1-0231.

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Authors and Affiliations

  1. Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA, 02139, USA

    Antoine Blanchard, Saviz Mowlavi & Themistoklis P. Sapsis

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  1. Antoine Blanchard
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  2. Saviz Mowlavi
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  3. Themistoklis P. Sapsis
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Correspondence to Antoine Blanchard, Saviz Mowlavi or Themistoklis P. Sapsis.

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Blanchard, A., Mowlavi, S. & Sapsis, T.P. Control of linear instabilities by dynamically consistent order reduction on optimally time-dependent modes. Nonlinear Dyn 95, 2745–2764 (2019). https://doi.org/10.1007/s11071-018-4720-1

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