Modelling for robust feedback control of fluid flows

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, 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 \(\mathcal{H}_\infty\) 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, a \(\mathcal{H}_\infty\) loop-shaping controller is designed and applied to the problem of reducing perturbation wall-shear stress in plane channel flow. DNS results demonstrate robust attenuation of the perturbation shear-stresses across a wide range of Reynolds numbers with a single, linear controller.

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ORCID for Ati S. Sharma: orcid.org/0000-0002-7170-1627

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