For systems with unknown input disturbances, a robust state-space observer is designed to simultaneously estimate the system state, the finitely times derivatives of the fault, and the fault signal at the same time. The considered fault signal can be allowed to be unbounded provided its finitely times derivative is bounded. This design approach can then be applied to the vehicle lateral dynamic systems. The data recorded from the industry are used to make simulation. Simulated results demonstrate the efficiency of the proposed design approach.