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This work proposes a proportional-integral fault estimation observer for nonlinear systems by means of convex structures. Unlike the traditional approaches that consider quadratic Lyapunov functions, this work employs nonquadratic ones. As a result, less conservative conditions in the form of linear matrix inequalities are obtained. Furthermore, an performance is employed in order to ensure robustness against sensor noise and disturbances; thus, the designed observer estimates both the state and the actuator faults even in the presence of disturbances. The performance of the proposed methodology is illustrated through a numerical example and a three-tank hydraulic system.