This paper presents theory for stability analysis and design for a class of observer-based feedback control systems. Relaxation of the controllability and observability conditions imposed in the Yakubovich-Kalman-Popov (YKP) lemma can be made for a class of nonlinear systems described by a linear time-invariant system (LTI) with a feedback-connected cone-bounded nonlinear element. It is shown how a circle-criterion approach can be used to design an observer-based state feedback control which yields a closed-loop system with specified robustness characteristics. The approach is relevant for design with preservation of stability when a cone-bounded nonlinearity is introduced in the feedback loop. Important applications are to be found in nonlinear control with high robustness requirements.