This paper deals with nonlinear observer design for a class of nonlinear systems with nonlinear output measurements. The proposed methodology is based on the use of Linear Matrix Inequalities (LMIs) to handle a problem of W 1,2 -convergence criterion. Some new assumptions and convenient Young's formulation are used to get less conservative LMI conditions compared to the literature. Indeed, the obtained LMIs contain additional decision variables, which render the conditions more general. Furthermore, due to the presence of nonlinearities in both the process dynamics and the output measurements, the class of systems studied in this paper is more general than those available in the literature in the same LMI context. Two numerical examples are presented to show the effectiveness and superiority of the new design procedure compared to the H ∞ method.