Predictive control of crystal size distribution in protein crystallization
This work focuses on the modelling, simulation and control of a batch protein crystallization process that is used to produce the crystals of tetragonal hen egg-white (HEW) lysozyme. First, a model is presented that describes the formation of protein crystals via nucleation and growth. Existing experimental data are used to develop empirical models of the nucleation and growth mechanisms of the tetragonal HEW lysozyme crystal. The developed growth and nucleation rate expressions are used within a population balance model to simulate the batch crystallization process. Then, model reduction techniques are used to derive a reduced-order moments model for the purpose of controller design. Online measurements of the solute concentration and reactor temperature are assumed to be available, and a Luenberger-type observer is used to estimate the moments of the crystal size distribution based on the available measurements. A predictive controller, which uses the available state estimates, is designed to achieve the objective of maximizing the volume-averaged crystal size while respecting constraints on the manipulated input variables (which reflect physical limitations of control actuators) and on the process state variables (which reflect performance considerations). Simulation results demonstrate that the proposed predictive controller is able to increase the volume-averaged crystal size by 30% and 8.5% compared to constant temperature control (CTC) and constant supersaturation control (CSC) strategies, respectively, while reducing the number of fine crystals produced. Furthermore, a comparison of the crystal size distributions (CSDs) indicates that the product achieved by the proposed predictive control strategy has larger total volume and lower polydispersity compared to the CTC and CSC strategies. Finally, the robustness of the proposed method (achieved due to the presence of feedback) with respect to plant-model mismatch is demonstrated. The proposed method is demonstrated to successfully achieve the task of maximizing the volume-averaged crystal size in the presence of plant-model mismatch, and is found to be robust in comparison to open-loop optimal control strategies.