Linear feedback control of a von Kármán street by cylinder rotation

This paper considers the problem of controlling a von Kármán vortex street (periodic shedding) behind a circular cylinder using cylinder rotation as the actuation. The approach is to linearize the Navier-Stokes equations about the desired (unstable) steady-state flow and design the control for the regulator problem using distributed parameter control theory. The Oseen equations are discretized using finite element methods and the resulting LQR control problem requires the solution to algebraic Riccati equations with very high rank. The feedback gains are computed using model reduction in a "control-then-reduce" framework. Model reduction is used to efficiently solve both Chandrasekhar and Lyapunov equations. The reduced Chandrasekhar equations are used to produce a stable initial guess for a Kleinman-Newton iteration. The high-rank Lyapunov equations associated with Kleinman-Newton iterations are solved by applying a novel model reduction strategy. This "control-then-reduce" methodology has a significant computational cost, but does not suffer many of the "reduce-then- control" setbacks, such as ensuring the unknown feedback functional gains are well represented in the reduced-basis. Numerical results for a 2-D cylinder wake problem at a Reynolds number of 100 demonstrate that this approach works when perturbations from the steady-state solution are small enough. When this feedback control is applied to a flow where vortex shedding has already occurred, the feedback control in the nonlinear problem stabilizes a nontrivial limit cycle. This limit cycle does have reduced lift forces and showcases the promise of the linear feedback control approach.