<p>In this work, a virtual fault derivative based iterative unknown input observer (UIO) design method is investigated to achieve the convergent fault estimation for a class of nonlinear systems with Lipschitz nonlinearities, faults and input disturbances. A virtual fault derivative differential equation is introduced to design an initial UIO with more relaxed tuning parameters. Furthermore, under the designed UIO, the unknown input disturbance term can be eliminated in the obtained error dynamics. Based on the initial UIO, an accumulated compensation factor is utilized to design a set of iterative UIOs for the system so as to acquire the state and fault estimates with high precision. In certain conditions, the obtained integral sequence can be proved to be uniformly convergent and results in the convergence of the state and fault estimation. Finally, two illustrative examples are provided to test the proposed convergent estimation scheme.</p>