Modelling and parameter identification for a hybrid dynamical system in microbial fed-batch culture

In this paper, we consider the modelling and identification in the fed-batch fermentation of glycerol by Klebsiella pneumoniae with open-loop glycerol input and pH logic control. Taking into account the hybrid characteristic of the fed-batch operation, we present a nonlinear hybrid dynamical system, which is developed by embedding the discrete process of adding glycerol and alkali into the dynamical system of batch culture, to formulate this fermentation process. Some important properties of the solution to the proposed system are then discussed, including the existence, uniqueness, boundedness and regularity. To estimate the unknown parameters in the system, a parameter identification problem subject to continuous state inequality constraints is proposed, and its identifiability is also proved. Subsequently, the parametric sensitivity functions of the system are given and utilized to obtain the requisite gradient information for further numerical computation. Finally, to solve the identification problem, a gradient-based algorithm is constructed in conjunction with constraint transcription and the smoothing approximation technique. Numerical simulations show that the validated hybrid system is fit for describing the fed-batch processes as observed from the experimental results.