Sum-of-squares approach to feedback control of laminar wake flows

Davide Lasagna,Sergei Chernyshenko,Deqing Huang,Owen R Tutty

doi:10.1017/jfm.2016.688

Abstract

In this paper a novel nonlinear feedback control design methodology for incompressible fluid flows aiming at the optimisation of long-time averages of flow quantities is presented. It applies to reduced-order finite-dimensional models of fluid flows, expressed as a set of first-order nonlinear ordinary differential equations with the right-hand side being a polynomial function in the state variables and in the controls. The key idea, first discussed in Chernyshenkoet al.(Phil. Trans. R. Soc. Lond. A, vol. 372, 2014, 20130350), is that the difficulties of treating and optimising long-time averages of a cost are relaxed by using the upper/lower bounds of such averages as the objective function. In this setting, control design reduces to finding a feedback controller that optimises the bound, subject to a polynomial inequality constraint involving the cost function, the nonlinear system, the controller itself and a tunable polynomial function. A numerically tractable and efficient approach to the solution of such optimisation problems, based on sum-of-squares techniques and semidefinite programming, is proposed. To showcase the methodology, the mitigation of the fluctuation kinetic energy in the unsteady wake behind a circular cylinder in the laminar regime at$Re=100$, via controlled angular motions of the surface, is numerically investigated. A compact reduced-order model that resolves the long-term behaviour of the fluid flow and the effects of actuation, is first derived using proper orthogonal decomposition and Galerkin projection. In a full-information setting, feedback controllers are then designed to reduce the long-time average of the resolved kinetic energy associated with the limit cycle. These controllers are then implemented in direct numerical simulations of the actuated flow. Control performance, total energy efficiency and the physical control mechanisms identified are analysed in detail. Key elements of the methodology, implications and future work are finally discussed.

Highlights

In the past decades, the coordinated efforts of laboratory experiments using high-resolution flow diagnostics and large-scale direct numerical simulations (DNS) have considerably progressed our understanding of key physical processes and mechanisms in turbulent flows
In this paper a novel nonlinear feedback control design methodology for incompressible fluid flows aiming at the optimisation of long-time averages of flow quantities is presented. It applies to reduced-order finite-dimensional models of fluid flows, expressed as a set of first-order nonlinear ordinary differential equations with the right-hand side being a polynomial function in the state variables and in the controls
Additional tests have been performed with dV = 6 with no difference, except for additional computational costs, as all bounds are tight to the actual average from simulation with dV = 4

Summary

Introduction

The coordinated efforts of laboratory experiments using high-resolution flow diagnostics and large-scale direct numerical simulations (DNS) have considerably progressed our understanding of key physical processes and mechanisms in turbulent flows. Despite these new discoveries, progress in the ability to effectively control their spatiotemporal evolution in complex geometries has remained more elusive, owing to the nonlinear, multiscale nature of turbulent motion. The development of active control strategies is commonly regarded as one of the key enablers for future advances in efficient transportation, energy generation and distribution, and in many other technologically relevant industrial sectors. A variety of actuation/sensing strategies and control design methods have been proposed, as recently reviewed by Choi, Jeon & Kim (2008)

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Conclusion