Deadbeat model predictive control (DB-MPC) is one of the advanced promising control methods for power converters thanks to its simplicity, high steady-state performance and fast dynamic response. However, the high sensitivity to parameter mismatch and the difficulty of handling multiple control targets are problematic issues in DB-MPC. This work presents an improved robust DB-MPC for a new nine-level ANPC-based inverter. This inverter requires a low number of power devices compared to other single dc-source inverters. Only nine active switches and two discrete diodes are utilized to obtain a nine-level waveform. Without the need for weighting factors, the proposed DB-MPC method tackles three control goals; current control, flying capacitors (FCs) stabilization and dc-link balance, which saves the laborious effort of adjusting the weighting factors in the traditional finite control set MPC (FCS-MPC) method. Moreover, an effective dc-link balancing scheme based on power flow control is proposed and integrated into the FCs control objective. To enhance the control robustness, an EKF-based estimator is designed to identify the system parameters online. In addition, the proposed DB-MPC scheme allows the considered inverter to continue operating with the generation of five levels in the failure condition of the four-quadrant switch, improving the fault tolerance of the inverter. The developed DB-MPC method is experimentally verified in steady-state and transient operation. To demonstrate the excellent performance of the presented DB-MPC scheme, experimental comparisons with other popular MPC methods are performed.