Feedback control of transitional flows: A framework for controller verification using quadratic constraints

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

The dynamics of incompressible fluid flows are governed by a non-normal linear dynamical system in feedback with a static energy-conserving nonlinearity. These dynamics can be altered using feedback control but verifying performance of a given control law can be challenging. The conventional approach is to perform a campaign of high-fidelity direct numerical simulations to assess performance over a wide range of parameters and disturbance scenarios. In this paper, we propose an alternative simulation-free approach for controller verification. The incompressible Navier-Stokes equations are modeled as a linear system in feedback with a static and quadratic nonlinearity. The energy conserving property of this nonlinearity can be expressed as a set of quadratic constraints on the system, which allows us to perform a nonlinear stability analysis of the fluid dynamics with minimal complexity. In addition, the Reynolds number variations only influence the linear dynamics in the Navier-Stokes equations. Therefore, the fluid flow can be modeled as a parameter-varying linear system (with Reynolds number as the parameter) in feedback with a quadratic nonlinearity. The quadratic constraint framework is used to determine the range of Reynolds numbers over which a given flow will be stable, without resorting to numerical simulations. We demonstrate the framework on a reduced-order model of plane Couette flow. We show that our proposed method allows us to determine the critical Reynolds number, largest initial disturbance, and a range of parameter variations over which a given controller will stabilize the nonlinear dynamics.

Original languageEnglish (US)
Title of host publicationAIAA Aviation and Aeronautics Forum and Exposition, AIAA AVIATION Forum 2021
PublisherAmerican Institute of Aeronautics and Astronautics Inc, AIAA
ISBN (Print)9781624106101
DOIs
StatePublished - 2021
EventAIAA Aviation and Aeronautics Forum and Exposition, AIAA AVIATION Forum 2021 - Virtual, Online
Duration: Aug 2 2021Aug 6 2021

Publication series

NameAIAA Aviation and Aeronautics Forum and Exposition, AIAA AVIATION Forum 2021

Conference

ConferenceAIAA Aviation and Aeronautics Forum and Exposition, AIAA AVIATION Forum 2021
CityVirtual, Online
Period8/2/218/6/21

Bibliographical note

Funding Information:
This material is based upon the work supported by the Army Research Office under Grant Number W911NF-20-1-0156 and the National Science Foundation under award number CBET-1943988. Maziar S. Hemati acknowledges support from the Air Force Office of Scientific Research under grant number FA9550-19-1-0034.

Publisher Copyright:
© 2021, American Institute of Aeronautics and Astronautics Inc.. All rights reserved.

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