The use of multi-phase electric machines exhibits specific advantages such as increased power comparing to the three-phase machines and robustness to failures. In this paper, the dynamic model of the 6-phase synchronous electric machine undergoes first an approximate linearisation, through Taylor series expansion. The linearization is performed round local operating points which are defined at each time instant by the present value of the system's state vector and the last value of the control input that was exerted on it. The linearisation procedure requires the computation of Jacobian matrices at the aforementioned operating points. The modelling error, which is due to the truncation of higher order terms in the Taylor series expansion is perceived as a perturbation that should be compensated by the robustness of the control loop. Next, for the linearized equivalent model of the 6-phase synchronous electric machine, an H-infinity feedback control loop is designed. This approach, is based on the concept of a differential game that takes place between the control input (which tries to minimize the deviation of the state vector from the reference setpoints) and the disturbance input (that tries to maximize it). In such a case, the computation of the optimal control input requires the solution of an algebraic Riccati equation at each iteration of the control algorithm. The known robustness properties of H-infinity control enable compensation of model uncertainty and rejection of the perturbation terms that affect the 6-phase synchronous machine. The stability of the control loop is proven through Lyapunov analysis. Actually, it is shown that H-infinity tracking performance is succeeded, while conditionally the asymptotic stability of the control loop is also assured. The efficiency of the proposed control scheme for the 6-phase synchronous machine is further confirmed through simulation experiments.