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Stabilizing a flow around an unstable equilibrium is a typical problem in flow  control. Model-based designed of modern controllers like LQR/LQG or $H_\infty$  compensators is often limited by the large-scale of the discretized flow  models. Therefore, model reduction is usually needed before designing such a  controller. Here we suggest an approach based on applying balanced truncation  for unstable systems to the linearized flow equations usually used for  compensator design. For this purpose, we modify the ADI iteration for Lyapunov  equations to deal with the index-2 structure of the underlying descriptor  system efficiently in an implicit way. The resulting algorithm is tested for  model reduction and control design of a linearized Navier-Stokes system  describing von Kármán vortex shedding.

Balancing based model reduction for structured index-2 unstable descriptor systems with application to flow control