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We present an estimator-based control design procedure for flow control,using reduced-order models of the governing equations, linearized about apossibly unstable steady state. The reduced models are obtained using anapproximate balanced truncation method that retains the most controllable andobservable modes of the system. The original method is valid only for stablelinear systems, and we present an extension to unstable linear systems. Thedynamics on the unstable subspace are represented by projecting the originalequations onto the global unstable eigenmodes, assumed to be small in number. Asnapshot-based algorithm is developed, using approximate balanced truncation,for obtaining a reduced-order model of the dynamics on the stable subspace. Theproposed algorithm is used to study feedback control of 2-D flow over a flatplate at a low Reynolds number and at large angles of attack, where the naturalflow is vortex shedding, though there also exists an unstable steady state. Forcontrol design, we derive reduced-order models valid in the neighborhood ofthis unstable steady state. The actuation is modeled as a localized body forcenear the leading edge of the flat plate, and the sensors are two velocitymeasurements in the near-wake of the plate. A reduced-order Kalman filter isdeveloped based on these models and is shown to accurately reconstruct the flowfield from the sensor measurements, and the resulting estimator-based controlis shown to stabilize the unstable steady state. For small perturbations of thesteady state, the model accurately predicts the response of the fullsimulation. Furthermore, the resulting controller is even able to suppress thestable periodic vortex shedding, where the nonlinear effects are strong, thusimplying a large domain of attraction of the stabilized steady state.

Feedback control of unstable steady states of flow past a flat plate using reduced-order estimators