Active Magnetic Bearing (AMB) technology is becoming attractive for several reasons such as friction-free suspension and high-speed operation, high reliability, and vibrations exemption. These desirable features come at the cost of an increased complexity of the system, which includes position sensors, a power electronic converter and a control system dedicated to the AMBs. This paper focus on the control system design of an AMB featuring a Wheatstone bridge winding configuration. To achieve a high-bandwidth current control able to generate the desired forces, a Finite Control Set Model Predictive Control (FCS-MPC) has been proposed in this paper. The proposed FCS-MPC includes an estimation of the AMB coils inductance, which can be used to improve the control system performance and, at the same time, provide a rotor position estimation without the need of additional sensing coils and possibly without the use of conventional position sensors. The AMB is modelled considering finite element simulation results in order to evaluate the relation between coil inductance variation and rotor position. A standard PI position control is also included in the system. Finally, the control system is validated through simulation.