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Modelling for robust feedback control of fluid flows

Published online by Cambridge University Press:  25 March 2015

Bryn Ll. Jones ,
P. H. Heins ,
E. C. Kerrigan ,
J. F. Morrison  and
A. S. Sharma
Bryn Ll. Jones*
Affiliation:
Department of Automatic Control and Systems Engineering, University of Sheffield, Sheffield S1 3JD, UK
P. H. Heins
Affiliation:
Department of Automatic Control and Systems Engineering, University of Sheffield, Sheffield S1 3JD, UK
E. C. Kerrigan
Affiliation:
Department of Electrical and Electronic Engineering, Imperial College London, London SW7 2AZ, UK Department of Aeronautics, Imperial College London, London SW7 2AZ, UK
J. F. Morrison
Affiliation:
Department of Aeronautics, Imperial College London, London SW7 2AZ, UK
A. S. Sharma
Affiliation:
Engineering and the Environment, University of Southampton, Highfield, Southampton SO17 1BJ, UK
*
†Email address for correspondence: b.l.jones@sheffield.ac.uk
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Abstract

This paper addresses the problem of designing low-order and linear robust feedback controllers that provide a priori guarantees with respect to stability and performance when applied to a fluid flow. This is challenging, since whilst many flows are governed by a set of nonlinear, partial differential–algebraic equations (the Navier–Stokes equations), the majority of established control system design assumes models of much greater simplicity, in that they are: firstly, linear; secondly, described by ordinary differential equations (ODEs); and thirdly, finite-dimensional. With this in mind, we present a set of techniques that enables the disparity between such models and the underlying flow system to be quantified in a fashion that informs the subsequent design of feedback flow controllers, specifically those based on the $\mathscr{H}_{\infty }$ loop-shaping approach. Highlights include the application of a model refinement technique as a means of obtaining low-order models with an associated bound that quantifies the closed-loop degradation incurred by using such finite-dimensional approximations of the underlying flow. In addition, we demonstrate how the influence of the nonlinearity of the flow can be attenuated by a linear feedback controller that employs high loop gain over a select frequency range, and offer an explanation for this in terms of Landahl’s theory of sheared turbulence. To illustrate the application of these techniques, an $\mathscr{H}_{\infty }$ loop-shaping controller is designed and applied to the problem of reducing perturbation wall shear stress in plane channel flow. Direct numerical simulation (DNS) results demonstrate robust attenuation of the perturbation shear stresses across a wide range of Reynolds numbers with a single linear controller.

JFM classification

Flow Control: Control theory Flow Control: Drag reduction Nonlinear Dynamical Systems: Low-dimensional models
Type
Papers
Information
Journal of Fluid Mechanics , Volume 769 , 25 April 2015 , pp. 687 - 722
Copyright
© 2015 Cambridge University Press 

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