Controlling fluid flows with positive polynomials

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.