On commutation of reduction and control: Linear feedback control of a von Kármán street

Design of feedback controls for distributed parameter systems in fluid flows remains a formidable task for researchers. One popular approach is "reduce-then-contol" which has been successfully implemented by first developing a reduced-order model and then applying the control theory. However, this approach has several drawbacks, such as ensuring the unknown feedback functional gains are well represented in the reduced-basis. Control of vortex shedding past a circular cylinder has become a canonical problem to test or validate a flow control design. Control of the von Karman vortex street has many applications, e.g. when designing ocean platforms-as the vortex shedding induces an oscillatory fatigue load on structural components. In this paper, we follow "control-then-reduce" approach to control von Kármán vortex street (periodic shedding) in a channel 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. 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. In this study, we map the functional gains computed for the flow past a cylinder in a channel onto an exterior flow past a cylinder to analyze the performance of the controller. We present numerical results for various flow conditions to test the robustness of the control mechanism.