For state-space systems with unknown process 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. The considered fault signal can be allowed to be unbounded provided its finitely times derivative is bounded. A numerical example is given to illustrate the design procedure. Vehicle lateral dynamic system plays an important role in the vehicle manoeuverability, stability and driving safety. An application example with the real data for the vehicle lateral dynamic system is finally provided to demonstrate the efficiency of the present estimation technique.