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 H ∞ 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 H ∞ 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.

Dynamically correct formulations of the linearised Navier–Stokes equations

“…However, the question remains, what effect do these differences have when using the system formulations for feedback control design? We again stress at this point that models that are suitable for simulation are not necessarily suitable for control, and vice-versa [5,12]. In order to gain insight into this, actuation and sensing are now defined for the 2D channel flow in order to determine the system zeros, and compare the frequency responses from actuation to sensing.…”

“…As in many previous studies (see, e.g. [48,12,13]), we assume actuation in the form of wall transpiration, in this case at the upper wall. The actuator is modelled as a first-order system:…”

Feedback control of unstable flows: a direct modelling approach using the Eigensystem Realisation Algorithm

“…In this section, we show that robust control tools can quantify the model uncertainty associated with these choices. A related approach was used by Jones et al (2015) in a grid refinement study.…”

“…Maximising b(G w , K ∞ ) can be shown (Vinnicombe 2001;Jones et al 2015) to maximise the robust stability of the system with respect to perturbations to the normalised coprime factors of the model, which can be used to represent a large set of realistic model uncertainties. In fact, it can be shown (Vinnicombe 2001) that the norms of both sets of transfer functions are equal:…”

“…Hence, maximising b(G w , K ∞ ) minimises the influence of the worst-case disturbances entering the model from all positions in the feedback loop and can be seen as giving a measure of robust performance as well as robust stability (Jones et al 2015).…”