The rotor position and speed information in permanent magnet synchronous motor (PMSM) drive systems plays a critical role in ensuring operational stability and safety. Position sensor failures can lead to inaccurate rotor position feedback, resulting in degraded speed regulation performance and abnormal motor operation. To address the offset fault and lock-in-place fault of position sensors, this paper proposes a full-speed-range position observer-based fault diagnosis method for PMSM position sensors. An improved sliding-mode-based mid-to-high-speed position signal identification method is developed, where an exponential composite function replaces the sign function, mitigating the chattering issue caused by the traditional sliding-mode approach. Additionally, an enhanced high-frequency pulsating signal injection-based zero/low-speed position signal identification method is proposed, employing a filterless high/low-frequency signal separation strategy to reduce system complexity. The full-speed-range position observer is utilized to provide real-time rotor position and speed information, which is then compared with the output of the position sensor to achieve effective diagnosis of offset and lock-in-place faults. The proposed method is validated through a MATLAB/Simulink simulation model.