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The benefits of model predictive control (MPC) have been well established; however its application to reference tracking on digital servo drives (DSDs), which typically have very fast update rates, is limited by the computational power of present-day processors. This paper presents a novel MPC formulation, which provides a mechanism to trade-off online computation effort with tracking performance, while maintaining stability. This is achieved by introducing a trajectory horizon, which is distinct from the prediction and control horizons typically encountered in MPC formulations. It is shown that increasing the trajectory horizon inherently leads to improved tracking; however larger horizon lengths also have the unwanted effect of increasing online computation. The proposed MPC formulation is compatible with recently developed explicit MPC solutions, and hence the burden of online optimization is avoided. The new approach is successfully implemented on an industrial machine tool DSD, and in terms of tracking accuracy, is shown to outperform the incumbent approach of cascaded PID control.