This article proposes an optimization-based control approach for the modular multilevel matrix converter (M3C), integrating energy storage to enable simultaneous variable-speed drive and grid-feeding operations. The proposal has two stages: 1) optimal power control, based on the instantaneous power theory, which generates optimal references for circulating current (CC) and common-mode voltage (CMV) and 2) a continuous-control-set model-predictive-control (CCS-MPC) for the motor drive. The proposed control approach ensures the converter entire operation, even with unbalanced inter-cluster powers at critical operating points, such as low speed, while simultaneously feeding the grid. Experimental results on an M3C prototype of 27 submodules validate the effectiveness and performance of the proposal.