Controllable rotor-gas bearings are popular offering adaptability, high speed operation, low friction and clean operation. Rotor-gas bearings are however highly sensitive to disturbances due to the low friction of the injected gas. These undesirable damping properties call for controllers, which can be designed from suitable models describing the relation from actuator input to measured shaft position. Current state of the art models of controllable gas bearings however do not provide such relation, which calls for alternative strategies. The present contribution discusses the challenges for feedback controller design using the state of the art method, and an alternative data driven modelling approach is pursued based on Grey- Box system identification. The method allows development of models of the rotor-gas bearing suitable for controller design, which can be identified from data over the range of operation and are shown to accurately describe the dynamical behaviour of the rotor-gas bearing. Design of a controller using the identified models is treated and experiments verify the improvement of the damping properties of the rotor-gas bearing.