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Many research efforts have been dedicated to matrix converters for several years. As major technological issues are now solved, this structure will widespread in industrial applications, in particular with AC motors. Current control is a key issue for AC motor drives, so many control schemes have been proposed. Some of them proposed at first for inverters, were applied to matrix converters. Among algorithms used with inverters, predictive control shows very good performances. In this paper a new control scheme is proposed for a matrix converter- fed permanent magnet synchronous machine. Literature about matrix converter technology and control and about predictive control for inverter-fed AC machines is reviewed. The proposed predictive control principle, the model of the whole machine - converter and the cost-function are detailed. The method offers a trade-off between the quality of motor currents and input power factor. Finally experimental results are reported. The feasibility and the effectiveness of the proposed method is assessed.