This paper presented a PID control scheme for a synchronous motor with permanent magnets. PID controllers are the most used controllers in industrial applications and it always presents difficulty to realise the differential part. Moreover, in the last decades a visible effort has been made in the field of sensorless control in which observer and approximation of state variables are used to avoid sensors in the control loop. A polynomial Kalman filter is used to approximate its derivative actions to avoid disturbances and spikes due to the differentiation. The Kalman filter estimates two coefficients at each sampling time, which are necessary to calculate the polynomial of the first order with which the tracking error is intended to be modelled. Simulation results are shown to demonstrate the effectiveness of the approximation.