A benchmark problem in active aerodynamic flow control, suppression of strong pressure oscillations induced by flow over a shallow cavity, is addressed in this paper. Proper orthogonal decomposition and Galerkin projection techniques are used to obtain a reduced-order model of the flow dynamics from experimental data. The model is made amenable to control design by means of a control separation technique, which makes the control input appear explicitly in the equations. A prediction model based on quadratic stochastic estimation correlates flow field data with surface pressure measurements, so that the latter can be used to reconstruct the state of the model in real time. The focus of this paper is on the controller design and implementation. A linear-quadratic optimal controller is designed on the basis of the reduced-order model to suppress the cavity flow resonance. To account for the limitation on the magnitude of the control signal imposed by the actuator, the control action is modified by a scaling factor, which plays the role of a bifurcation parameter for the closed-loop system. Experimental results, in qualitative agreement with the theoretical analysis, show that the controller achieves a significant attenuation of the resonant tone with a redistribution of the energy into other frequencies, and exhibits a certain degree of robustness when operating in off-design conditions.