Optimal boundary feedback flow stabilization by model reduction

Abstract

In this paper, we consider the optimal boundary feedback stabilization of fluid flows governed by Navier–Stokes equations using model reduction. The model reduction is carried out using a combination of proper orthogonal decomposition (POD) and Galerkin projection. The resulting reduced-order model is employed in the optimal linear quadratic regulator (LQR) design to derive a feedback control. The feedback control is then used in the nonlinear Navier–Stokes equation to stabilize the system. Sufficient conditions are derived for the linear control to stabilize the nonlinear system. Feasibility of the proposed boundary feedback stabilization design is numerically demonstrated on the problem of stabilizing the flow past an airfoil. The control is effected through unsteady blowing on the airfoil surface. The computational results show that the closed-loop system amplitudes converge to the desired equilibrium state showing the stability predicted by the theory. In addition, they show that downstream directed blowing on the upper surface of the airfoil near the leading edge is more efficient in stabilizing the state.