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A feedback controller design which guarantees both finite-time boundedness and H∞ attenuation for a class of nonlinear systems with conic type nonlinearities and additive disturbances is presented. Conditions which guarantee the existence of a robust state-feedback controller for maintaining a bound on the transient response and satisfying an H∞ bound in the steady state for this class of systems are derived. A solution for the controller gain is obtained through the application of linear matrix inequality techniques. The controller developed is robust for all nonlinearities satisfying the conic inequality and all admissible disturbances. We conclude the paper with a numerical example illustrating the applicability of the controller design.