Controlling fluid flows with positive polynomials

Abstract

A novel nonlinear feedback control design methodology for incompressible fluid flows aiming at the optimisation of
long-time averages of key flow quantities is presented. The key idea, first outlined in Ref. [1], is that the difficulties of treating
and optimising long-time averages are relaxed by shifting the analysis to upper/lower bounds for minimisation/maximisation
problems, respectively. In this setting, control design reduces to finding the polynomial-type state-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 approach, based on Sum-of-Squares of polynomials techniques
and semidefinite programming, is proposed. As a prototypical example of control of separated flows, the mitigation of the
fluctuation kinetic energy in the unsteady two-dimensional wake past a circular cylinder at a Reynolds number equal to 100,
via controlled angular motions of the surface, is investigated. A compact control-oriented reduced-order model, resolving 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, linear state-feedback controllers are then designed to reduce the long-time
average of the resolved kinetic energy associated to the limit cycle of the system. Controller performance is then assessed in
direct numerical simulations.

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Identifiers

ORCID for Davide Lasagna: orcid.org/0000-0002-6501-6041

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