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Resolvent-based tools for optimal estimation and control via the Wiener–Hopf formalism

Published online by Cambridge University Press:  28 February 2022

Eduardo Martini Open the ORCID record for Eduardo Martini [Opens in a new window] ,
Junoh Jung Open the ORCID record for Junoh Jung [Opens in a new window] ,
André V.G. Cavalieri Open the ORCID record for André V.G. Cavalieri [Opens in a new window] ,
Peter Jordan Open the ORCID record for Peter Jordan [Opens in a new window]  and
Aaron Towne Open the ORCID record for Aaron Towne [Opens in a new window]
Eduardo Martini*
Affiliation:
Instituto Tecnológico de Aeronáutica, 12228-900 São José dos Campos/SP, Brazil Département Fluides, Thermique et Combustion, Institut Pprime, CNRS, Université de Poitiers, ENSMA, 86000 Poitiers, France
Junoh Jung
Affiliation:
University of Michigan, Ann Arbor, MI 48109, USA*
André V.G. Cavalieri
Affiliation:
Instituto Tecnológico de Aeronáutica, 12228-900 São José dos Campos/SP, Brazil
Peter Jordan
Affiliation:
Département Fluides, Thermique et Combustion, Institut Pprime, CNRS, Université de Poitiers, ENSMA, 86000 Poitiers, France
Aaron Towne
Affiliation:
University of Michigan, Ann Arbor, MI 48109, USA*
*
†Email address for correspondence: [email protected]
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Abstract

The application of control tools to complex flows frequently requires approximations, such as reduced-order models and/or simplified forcing assumptions, where these may be considered low rank or defined in terms of simplified statistics (e.g. white noise). In this work we propose a resolvent-based control methodology with causality imposed via a Wiener–Hopf formalism. Linear optimal causal estimation and control laws are obtained directly from full-rank, globally stable systems with arbitrary disturbance statistics, circumventing many drawbacks of alternative methods. We use efficient, matrix-free methods to construct the matrix Wiener–Hopf problem, and we implement a tailored method to solve the problem numerically. The approach naturally handles forcing terms with space–time colour; it allows inexpensive parametric investigation of sensor/actuator placement in scenarios where disturbances/targets are low rank; it is directly applicable to complex flows disturbed by high-rank forcing; it has lower cost in comparison to standard methods; it can be used in scenarios where an adjoint solver is not available; or it can be based exclusively on experimental data. The method is particularly well suited for the control of amplifier flows, for which optimal control approaches are typically robust. Validation of the approach is performed using the linearized Ginzburg–Landau equation. Flow over a backward-facing step perturbed by high-rank forcing is then considered. Sensor and actuator placement are investigated for this case, and we show that while the flow response downstream of the step is dominated by the Kelvin–Helmholtz mechanism, it has a complex, high-rank receptivity to incoming upstream perturbations, requiring multiple sensors for control.

JFM classification

Flow Control: Control theory Mathematical Foundations: Computational methods
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 937 , 25 April 2022 , A19
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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Footnotes

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The online version of this article has been updated since original publication. A notice detailing the change has also been published.

References

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