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This paper focuses on direct torque control (DTC) for three-phase AC electric drives. A novel model predictive control scheme is proposed that keeps the motor torque, the stator flux, and (if present) the inverter's neutral point potential within given hysteresis bounds while minimizing the switching frequency of the inverter. Based on an internal model of the drive, the controller predicts several future switch transitions, extrapolates the output trajectories, and chooses the sequence of inverter switch positions (voltage vectors) that minimizes the switching frequency. The advantages of the proposed controller are twofold. First, as underlined by the experimental results in the second part of this paper, it yields a superior performance with respect to the industrial state of the art. Specifically, the switching frequency is reduced by up to 50% while the torque and flux are kept more accurately within their bounds. Moreover, the fast dynamic torque response is inherited from standard DTC. Second, the scheme is applicable to a large class of (three-phase) AC electric machines driven by inverters.