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Input–output measures for model reduction and closed-loop control: application to global modes

Published online by Cambridge University Press:  06 October 2011

Alexandre Barbagallo ,
Denis Sipp  and
Peter J. Schmid
Alexandre Barbagallo
Affiliation:
ONERA-DAFE, 8 rue des Vertugadins, 92190 Meudon, France Laboratoire d’Hydrodynamique (LadHyX), CNRS-Ecole Polytechnique, 91128 Palaiseau, France
Denis Sipp
Affiliation:
ONERA-DAFE, 8 rue des Vertugadins, 92190 Meudon, France
Peter J. Schmid*
Affiliation:
Laboratoire d’Hydrodynamique (LadHyX), CNRS-Ecole Polytechnique, 91128 Palaiseau, France
*
†Email address for correspondence: peter@ladhyx.polytechnique.fr
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Abstract

Feedback control applications for flows with a large number of degrees of freedom require the reduction of the full flow model to a system with significantly fewer degrees of freedom. This model-reduction process is accomplished by Galerkin projections using a reduction basis composed of modal structures that ideally preserve the input–output behaviour between actuators and sensors and ultimately result in a stabilized compensated system. In this study, global modes are critically assessed as to their suitability as a reduction basis, and the globally unstable, two-dimensional flow over an open cavity is used as a test case. Four criteria are introduced to select from the global spectrum the modes that are included in the reduction basis. Based on these criteria, four reduced-order models are tested by computing open-loop (transfer function) and closed-loop (stability) characteristics. Even though weak global instabilities can be suppressed, the concept of reduced-order compensators based on global modes does not demonstrate sufficient robustness to be recommended as a suitable choice for model reduction in feedback control applications. The investigation also reveals a compelling link between frequency-restricted input–output measures of open-loop behaviour and closed-loop performance, which suggests the departure from mathematically motivated ${\mathscr{H}}_{\infty } $-measures for model reduction toward more physically based norms; a particular frequency-restricted input–output measure is proposed in this study which more accurately predicts the closed-loop behaviour of the reduced-order model and yields a stable compensated system with a markedly reduced number of degrees of freedom.

JFM classification

Flow Control: Control theory Flow Control: Instability control

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Type
Papers
Information
Journal of Fluid Mechanics , Volume 685 , 25 October 2011 , pp. 23 - 53
Copyright
Copyright © Cambridge University Press 2011

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