This paper proposes a theoretical analysis of a closed loop adaptive speed control system for control the inverter driven permanent magnet synchronous motor(PMSM). This control system utilizes a mechanically sensorless state observer for the generation of all controller feedback information. The observer processes measurements of stator frame voltage and current to produce estimates of rotor position and speed and rotor frame currents. It is shown that the identity observer, when properly formulated, has the same linearized error dynamics as the extended kalman filter(EKF). Consequently, it is shown that the gains within the identity observer can be designed in a manner identical to that of the EKF. In this way, the designability of the nonlinear observer is assured, as is the optimality of its performance for small errors. A sequence of simulation are performed and they demonstrate the successful performance.