This paper proposes a new fuzzy speed controller for the precise speed control of a permanent magnet synchronous motor(PMSM). The proposed control system needs the information of the angular acceleration instead of the load torque, so the third-order fuzzy acceleration observer estimates it. Moreover, the LMI conditions are derived for the existence of the fuzzy acceleration observer and fuzzy speed controller, and the gain matrices of the observer and controller are obtained. It is analytically proven that the proposed observer-based fuzzy speed regulator is exponentially stable. To evaluate the performance of the proposed control algorithm, experimental results as well as simulation results are provided under the conditions of motor parameter and load torque variations. Finally, it is clearly confirmed that the proposed control method can accurately control the speed of a PMSM.