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In this paper a new LMI (linear matrix inequality) condition is provided for the observer-based $\mathcal{H}_{\infty}$ stabilization of a class of nonlinear discrete-time systems. With the proposed design methodology, the observer and controller gains are computed simultaneously by solving only one inequality. Based on the Lyapunov theory and the use of mathematical artifacts such as matrix decomposition and the Young relation, the novel sufficient synthesis condition is expressed in terms of LMI, which can be easily solved by numerical tools. An application to a flexible link robot manipulator is provided to show the consistency of the proposed approach. A second numerical example is devoted to demonstrating the superiority and the lower conservatism of the proposed LMI compared to those available in the literature.

International audience; In this paper we address the problem of robust unknown input observer design for continuous nonlinear time-delay systems. To investigate the robustness of the studied system, we compare two approaches, the first based on a standard $\mathcal{H}_{\infty}$ method and the second with Sobolev norms. New sufficient synthesis conditions are given in linear matrix inequality form.