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In the present paper, sufficient conditions for the synthesis of robust Unknown Input Observers (UIOs) are proposed for a class of nonlinear systems, both in continuous and discrete time. The considered class is general enough to contain bilinear systems as well as Linear Parameter-Varying (LPV) systems with no parameter variation on the output matrix. The proposed conditions are numerically tractable, and are expressed in terms of Linear Matrix Inequalities (LMIs) or Linear Matrix Equalities (LMEs). Furthermore, the gain synthesis problem is shown to be formulated as a convex optimisation one, directly enabling the minimization of the influence of noisy measurements and model uncertainty. Simulations on energy systems are provided to illustrate the proposed methodologies.