The problem of unknown input observer design for non-linear Lipschitz systems is considered. A new dynamic framework which is a generalization of previously used linear unknown input observers is introduced. The additional degrees of freedom offered by this dynamic framework are used to deal with the Lipschitz non-linearity. The necessary and sufficient condition that ensures asymptotic convergence of the new observer is presented, and the equivalence between this condition and an H-infinity optimal control problem which satisfies the standard regularity assumptions in the H-infinity optimization theory is shown. Based on these results, a design procedure that is solvable using commercially available software is presented. A simulation example is given to illustrate the proposed design.