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The robust static-output-feedback controller design is studied for a class of uncertain nonlinear systems with multiobjective constraints. The parameter uncertainties are allowed to be unstructured but norm-bounded. The nonlinear dynamics is described by vector-valued function. The aim is the design of static-output-feedback controller such that the closed-loop system is of robust stability, the magnitude of the control input is within the given bound, the prescribed constraint on the disturbance rejection is guaranteed, and the upper bound of quadratic cost function is optimized, simultaneously. The controller design is based on the Lyapunov stability theory and multiobjective optimization strategy, by using linear matrix inequality (LMI) approach. Matrix variable transformation technique is adopted in the design to convert the non-convex LMI constraints into convex ones. A numerical example is provided to illustrate the effectiveness of the proposed design algorithm.