For a class of nonlinear systems where actuator faults, sensor faults and disturbances exist simultaneously, a fault detection and estimation method is proposed based on finite-time unknown input observers in this paper. Firstly, the original system is decoupled into two reduced-order subsystems by linear nonsingular transformation, and one subsystem contains only disturbances, and the other subsystem contains both disturbances and faults. Then, new states are obtained by the first-order low-pass filters and the augmented system is formed with the subsystem, which can transform the sensor faults of the original system into the actuator faults of the augmented system. The unknown input observers are designed to detect the faults of the augmented system, and the disturbances and faults of the system can be estimated in finite time. The finite time convergences of the designed unknown input observers are proved by theoretical analysis. Finally, the simulation study is conducted on the permanent magnet synchronous motor (PMSM) speed system, and the simulation results verify the effectiveness of the proposed control method.