Feedback stabilization of incompressible flows using quadratic constraints

Flow instabilities can be detrimental for engineered systems interacting with fluids, e.g., through increased drag, vibrations, or heating. Conventional feedback control approaches for stabilizing and suppressing these instabilities rely on linearized models of the fluid dynamics. As such, the resulting controllers tend to suppress instabilities over a narrow range of perturbation magnitudes. In this work, we propose a framework for synthesizing globally stabilizing feedback controllers that can stabilize the flow regardless of the perturbation magnitude. This is done using quadratic constraints that describe input-output properties of the nonlinear terms in the fluid dynamics. In particular, for incompressible flows, the nonlinearity is known to be lossless and energy conserving. The associated quadratic constraint can be used to yield a linear matrix inequality (LMI) for full-state feedback and static output-feedback controller synthesis. We demonstrate that controllers designed by this approach successfully stabilize a reduced-order model of plane Couette flow. We further show that the proposed controllers outperform controllers designed using prevailing linear control synthesis techniques on the same flow.