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

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 modeled 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 linearised around these equilibria are 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 * Email address for correspondence: rnelson@ic.ac.uk 1 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.

Periodic Schwarz–Christoffel mappings with multiple boundaries per period

“…They can be used, for example, in fluid dynamics to study a wide class of two-dimensional ideal irrotational flows via a new calculus involving the prime function recently expounded by one of the authors [31]. Of particular interest could be the configurations of stationary vortices [35][36][37], which have been shown to enhance the lift on an aerofoil. An intriguing application is to periodic arrays of slits, aerofoils or blades relevant in aviation and turbomachinery applications.…”

Harnessing the Kelvin–Helmholtz instability: feedback stabilization of an inviscid vortex sheet

“…While numerical solution of this equation typically requires a suitable regularization, we focused here on an admittedly harder problem when there is no regularization. There have already been many successful applications of the methods of linear control theory to flow problems (Kim & Bewley, 2007;Bagheri et al, 2009), but only relatively few to inviscid vortex flows (Protas, 2008;Nelson et al, 2017), as they often give rise to a range of unique technical challenges, some of which are reiterated below.…”

“…In practical settings feedback strategies relying on some limited system output rather than full state information are more applicable. In such cases, feedback controllers are usually combined with state estimators to form "compensators" and in the context of inviscid vortex control problems such approaches were pursued by Protas (2004); Nelson et al (2017). However, since the present study in the first place aims to assess the fundamental opportunities and limitations inherent in the linear feedback control of the inviscid Kelvin-Helmholtz instability, we will focus here on the more basic problem with the full-state feedback.…”

The compact Green’s function for multiple bodies

“…In this case, analytic expressions for the conformal map are available. This conformal map is an example of a radial slit map and was derived by Nelson and Sakajo [20]. The mapping function is given by…”