Multiphase machines have recently gained importance in the research community for their use in applications where high power density, wide speed range and fault-tolerant capabilities are needed. The optimal control of such drives requires to consider voltage and current constraints imposed by the power converter and the machine itself. If classical three-phase drives have been optimally controlled under such limits for a long time, the same cannot be said in the case of multiphase drives. This paper deals with this issue, where an optimal control technique based on Cascaded Model Predictive Controls (MPC) is presented for a five-phase permanent magnet synchronous machine (PMSM). A Continuous-Control-Set MPC (CCS-MPC) numerically computes optimal current references in real-time in order to exploit the maximum performance for given DC bus voltage and current limits. Then, a Finite-Control-Set MPC (FCS-MPC) is used to carry out the current control in the machine, directly applying the switching state that minimizes a cost function related to the current tracking. Obtained mixed microprocessor-based and FPGA-based real-time simulations prove the interest of the proposal, which ensures the optimal control of the multiphase drive operating under current and voltage constraints.