Low-dimensional models for feedback stabilization of unstable steady states

When the angle of attack of an airfoil is gradually increased at low Reynolds numbers, the flow undergoes a Hopf bifurcation wherein the steady state loses stability in a transition to periodic vortex shedding. The goal of this work is to stabilize these unstable steady states using feedback control. For that purpose, we derive reduced order models valid in the neighborhood of these unstable steady states, using an approximate balanced truncation method valid for large dimensional systems. The original method is valid only for stable systems, and here we present a modification to derive models for unstable sytems, and use the same along with standard linear control techniques such as Linear Quadratic Regulator (LQR) to obtain stabilizing control laws. We use full-state feedback to stabilize an unstable steady state of a two-dimensional flow past a past flat plate at an angle of attack of 35 degrees. We also present observers that reconstruct the entire flow-field based on the flat-plate lift and drag measurements.