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Linear feedback stabilization of point-vortex equilibria near a Kasper wing

Published online by Cambridge University Press:  18 August 2017

R. Nelson Open the ORCID record for R. Nelson [Opens in a new window] ,
B. Protas Open the ORCID record for B. Protas [Opens in a new window]  and
T. Sakajo
R. Nelson*
Affiliation:
Department of Mathematics, Imperial College London, 180 Queen’s Gate, London SW7 2AZ, UK
B. Protas
Affiliation:
Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario L85 4K1, Canada
T. Sakajo
Affiliation:
Department of Mathematics, Kyoto University, Kitashirakawa Oiwake-cho, Sakyo-ku, Kyoto, 606-8502, Japan
*
†Email address for correspondence: [email protected]
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Abstract

This paper concerns feedback stabilization of point-vortex equilibria above an inclined thin plate and a three-plate configuration known as the Kasper wing in the presence of an oncoming uniform flow. The flow is assumed to be potential and is modelled by the two-dimensional incompressible Euler equations. Actuation has the form of blowing and suction localized on the main plate and is represented in terms of a sink–source singularity, whereas measurement of pressure across the plate serves as system output. We focus on point-vortex equilibria forming a one-parameter family with locus approaching the trailing edge of the main plate and show that these equilibria are either unstable or neutrally stable. Using methods of linear control theory we find that the system dynamics linearized around these equilibria is both controllable and observable for almost all actuator and sensor locations. The design of the feedback control is based on the linear–quadratic–Gaussian (LQG) compensator. Computational results demonstrate the effectiveness of this control and the key finding of this study is that Kasper wing configurations are in general not only more controllable than their single-plate counterparts, but also exhibit larger basins of attraction under LQG feedback control. The feedback control is then applied to systems with additional perturbations added to the flow in the form of random fluctuations of the angle of attack and a vorticity shedding mechanism. Another important observation is that, in the presence of these additional perturbations, the control remains robust, provided the system does not deviate too far from its original state. Furthermore, except in a few isolated cases, introducing a vorticity-shedding mechanism enhanced the effectiveness of the control. Physical interpretation is provided for the results of the controllability and observability analysis as well as the response of the feedback control to different perturbations.

JFM classification

Flow Control: Instability control Nonlinear Dynamical Systems: Low-dimensional models Vortex Flows: Vortex dynamics
Type
Papers
Information
Journal of Fluid Mechanics , Volume 827 , 25 September 2017 , pp. 121 - 154
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Copyright
© 2017 Cambridge University Press 

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