This paper is devoted to the observer-based control design for linear systems with quantized output. To this end, the output dynamic controller issued by the Luenberger observer is considered. Thus, through appropriate sector conditions which embed the quantization error, the global ultimate boundedness stability problem is tackled for the closed-loop system. Namely, to deal with the discontinuity induced by the quantizer, the whole treatise is carried out considering Krasovskii solutions to the closed-loop system. Furthermore, the controller is designed via a convex optimization setup, providing a constructive method that can be efficiently implemented.