This paper reports the optimal speed control of a permanent magnet synchronous motor (PMSM) system. The predictive control method is an effective strategy for a fast dynamic response. The undesirable performance in the presence of system disturbances, including internal model uncertainties and external load disturbances, is analyzed. To achieve a fast response and ensure stronger robustness and improved disturbance rejection performance simultaneously, a robust generalized predictive controller (GPC) with a high-order terminal sliding-mode observer (HOTSMO) is proposed for a PMSM control system. The proposed observer can estimate the unknown disturbances with chattering elimination. A feed-forward compensation based on the estimated disturbances is provided to the predictive speed controller. The simulation and experimental results show that the proposed control method can achieve a better speed dynamic response and a stronger robustness.INDEX TERMS Permanent magnet synchronous motor (PMSM), generalized predictive control, high-order terminal sliding mode observe, speed control.