An observer-based robust control strategy is proposed for controlling overflow metabolism cultures operated in fed-batch mode. In order to maximize the biomass productivity, the controller is designed to regulate the inhibitory by-product concentration at small levels keeping the substrate concentration close to its critical level. To this end, a reduced order nonlinear model of the bioprocess dynamics is obtained and a partial feedback linearizing strategy is applied. The resulting free linear dynamics is designed by means of a convex optimization problem aiming at mitigating the effects of non canceled nonlinearities and model uncertainties. An adaptive extended Luenberger observer is also designed for estimating the byproduct concentration from the measurements of biomass and substrate concentrations. Realistic numerical simulations demonstrate that the proposed observer based robust control strategy is able to maximize the biomass concentration despite large disturbances on the measurements (30 % for substrate and 15 % for biomass concentrations).