This paper presents a nonlinear observer design methodology for a class of Lipschitz nonlinear systems via convex optimization. A sufficient condition for the existence of an observer gain matrix to stabilize the estimation error dynamics is given in term of a quadratic stability margin. In addition, the observer gain matrix is optimally designed by minimizing the magnitude of elements of the observer gain matrix to reduce the amplification of sensor measurement noise. Furthermore, when disturbances considered as unknown deterministic inputs are imposed on the error dynamics in an additive form, the observer gain matrix is redesigned to minimize an induced ℒ 2 gain between the disturbance to the estimation error as well as the effect of measurement noise. Finally a systematic design algorithm is applied to a flexible joint robot system.