Feedback stabilization of fluids using reduced-order models for control and compensator design

In many cases, feedback control of fluids can effect large energetic changes in the state while using relatively small amounts of control input energy. One such example is the stabilization of the wake behind a circular cylinder by slight rotations of the cylinder itself. For a range of low-Reynolds number flows, this can have a large impact on the drag and lift forces. To design such a feedback control law from first principles leads to large discretized systems and requires model reduction methods. The same is true for compensator design. While proper orthogonal decomposition (POD) is the standard tool for controlling fluids, interpolatory model reduction methods (IMOR) have emerged as effective candidates for very large-scale linear problems due to their ability to produce high-fidelity (optimal in some cases) reduced models with modest computational cost. Furthermore, they don't require the expensive simulations needed for POD and are input-independent. In this paper, we will use the interpolation framework for model reduction of differential algebraic equations (DAEs) to develop a linear feedback control law and compensator design. We then compare IMOR with POD for producing a low-order compensator. We find that switching from IMOR to POD for this step can lead to a compensator that does not preserve the stabilizing properties of the linear feedback operator term. Numerical experiments using the two-dimensional Navier-Stokes equations and stabilizing the flow behind a pair of circular cylinders demonstrates the effectiveness of retaining the IMOR basis.