This paper focuses on the specific growth rate estimation problem in a Polyhydroxybutyrate bioplastic production process by industrial fermentation. The kinetics of the process are unknown and there are uncertainties in the model parameters and inputs. During the first hours of the growth phase of the process, biomass concentration can be measured online by an optical density sensor, but as cell density increases this method becomes ineffective and biomass measurement is lost. An asymptotic observer is developed to estimate the growth rate for the case without biomass measurement based on corrections made by a pH control loop. Furthermore, an exponential observer based on the biomass measurement is developed to estimate the growth rate during the first hours, which gives the initial condition to the asymptotic observer. Error bounds and robustness to uncertainties in the models and in the inputs are found. The estimation is independent of the kinetic models of the microorganism. The characteristic features of the observer are illustrated by numerical simulations and validated by experimental results.