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In this paper, the application of model predictive control (MPC) for high-performance speed control and torsional vibration suppression in the drive system with flexible coupling is demonstrated. The control methodology presented in this paper relies on incorporating the drive's safety and physical limitations directly into the control problem formulation so that future constraint violations are anticipated and prevented. In order to reduce the computational complexity, the standard MPC controller is replaced by its explicit form. The resulting explicit controller achieves the same level of performance as the conventional MPC, but requires only a fraction of the real-time computational machinery, thus leading to fast and reliable implementation. The simulation results are confirmed by laboratory experiments.