This study evaluates the robust fault estimation problem of systems with actuator and sensor faults though the simultaneous use of unknown input disturbances and measurement noise. Specifically, an augmented descriptor system is preliminarily developed by creating an augmented state consisting of system states and sensor faults. Next, a novel fast adaptive unknown input observer (FAUIO) is proposed for the system to enhance its fault estimation performance. The existence condition of the novel FAUIO is then introduced for linear time-invariant systems with unknown input disturbances. Furthermore, the proposed FAUIO is extended to a class of Lipschitz nonlinear systems with unknown input disturbances and measurement noise to investigate the robust fault estimation problem. Accordingly, an H∞ performance index is employed to attenuate the influence of disturbances on fault estimation. Moreover, the linear matrix inequality (LMI) technique is applied to solve the designed FAUIO. Finally, the effectiveness of the developed FAUIO is validated via the simulation of two examples.