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.

Optimization With Affine Homogeneous Quadratic Integral Inequality Constraints

“…Other input-to-state/output properties such as passivity, reachability, and input-to-state stability can be studied in a similar way using dissipation inequalities for integral functionals of the state variable [2,4]. Finally, the computational cost of designing optimal control policies for systems with complex dynamics, such as turbulent flows, may be reduced by requiring the control law to minimize an upper bound on the objective function rather than the objective itself [20,21,23,24], and in the case of PDEs such upper bounds can be found by solving suitable integral inequalities [8,9,[11][12][13]19].…”