This paper presents an Unknown Input Observer (UIO) for a class of Lipschitz nonlinear systems affected by disturbances/faults and having slow internal dynamics after the decoupling of the Unknown Input (UI). Thanks to the Asymptotic Decoupling notion, the constraint related to the internal dynamics is bypassed. The nonlinear UIO presents time varying dynamics that allows to obtain the Asymptotic Decoupling notion. It will be shown that the proposed approach allows to enhance the convergence rate of the UIO. The convergence enhancement of the state estimation error is demonstrated via a Lyapunov analysis and the established conditions are expressed in terms of Linear Matrix Inequalities. Finally, simulations will be given in order to illustrate the proposed approach with some comparisons to existing techniques.