In this paper, a Model Predictive Control (MPC) strategy is introduced for its application in a four-level quasi-nested topology, feeding an Interior Permanent Magnet Synchronous Machine (IPMSM) AC-drive. The proposed control strategy is capable to synthesize the required output space vectors to ensure perfect tracking of the AC-drive speed reference under different loading conditions, while also ensuring voltage balance between the dc-link capacitors. The proposed converter topology is based on a reduced number of components compared to other mature converter topologies, such as the neutral-point clamped converter (NPC) or the active neutral-point clamped converter (ANPC) topologies, when compared in terms of the number of output voltage levels, since this quasi-nested topology does not require passive clamping devices such as diodes or active switches. Moreover, no floating dc-link capacitors with asymmetrical voltage levels are employed, thus simplifying the dc-link capacitor voltage balance mechanism. This work presents the switching operation principles and MPC control law when supplying an IPMSM AC-drive load are addressed in detail. Simulation and validation results using a Hardware in the Loop (HIL) prototype under different operation conditions are presented in order to validate the proposed converter topology and control strategy.