Feedback flow control is an emerging discipline which aims at applying the methods of closed loop feedback control to problems in fluid dynamics. A challenging area for application, which has drawn considerable academic research, concerns the control of absolute instabilities associated with the von Kármán vortex street that develops in the wake of a circular cylinder. The need for meaningful real-time control leads to the requirement of developing an appropriate low order model. A two dimensional, high resolution simulation is used to develop a neural network based low-dimensional model for controller design. Actuation is implemented as displacement of the cylinder normal to the flow. This control approach uses a low dimensional model based on a 15 mode Double Proper Orthogonal Decomposition (DPOD) applied to the velocity field. The truncated DPOD modes, used to construct a dynamic mathematical model of the wake, accurately predict the wake flow dynamics within the lock-in region at low forcing amplitudes. This low-dimensional model, derived using non-linear artificial neural network based system identification methods, is robust and accurate and can be used to simulate the dynamic behaviour of the wake flow. The control strategy involves the estimation of the DPOD modes and a respective feedback command based on these estimates. The control approach is based on both the Kármán fluctuating DPOD mode as well as the Kármán shift mode. The shift mode provides corrections for spatial variations in the POD Eigenfunctions which are caused by the feedback control induced modification of the wake. Using the reduced-order model, numerical investigations are conducted to determine effective controller gains. High resolution simulation having 63,700 nodes show that the developed control law stabilizes all 15 modes, the controller succeeds in substantially reducing drag and fluctuating lift. While the reduced order model is shown to be a very useful tool for controller development, major issues are identified and suggested remedies are discussed.