(568s) Run-to-Run Model Predictive Control of Batch Crystallization

The production of crystals with desired size and shape distributions from batch crystallization processes have become of great interest to the pharmaceutical industry, because those characteristics significantly influence the bioavailability of drugs. Recently, model predictive control (MPC) has received a growing attention within the crystallization process community, owing to its unique ability to handle constraints on the inputs, outputs, and the rate of changes of inputs. Another key benefit of using MPC is in its applicability to dynamic and/or multi-input multi-output (MIMO) systems. However, one of the key challenges in the use of MPC to the batch crystallization process is the plant/model mismatch, which persists from run to run and it may deteriorate further due to the lack of in situ measurements for the product qualities [1]. Motivated by this, the main ideas of run-to-run (R2R) control are integrated with conventional MPC formulation in an effort to reduce the model mismatch and suppress the uncertainties in the system using the post-batch measurements. The major benefit in the utilization of R2R control is in its unique nature to compensate for drifting processes through correlations developed to model output variations as a function of changes in the process environment [2]-[4]. As a result, small changes have to be made to the optimal jacket temperature computed from the MPC using nominal system parameters in order to produce crystals with desired size and shape distributions from the batch crystallizer.

The present work focuses on the modeling of a batch crystallization process to produce tetragonal HEW lysozyme crystals through kinetic Monte Carlo (kMC) simulation methods in the way described in [5] using the rate equations originally developed by [6]. The kMC simulation is regarded as a representation of a real crystallization process, and the potential disturbances considered in this work include uncertainties in the solubility as well as the nucleation and crystal growth rates. Then, a population balance equation (PBE) is presented to describe the evolution of the crystal volume distribution, and the method of moments (MOM) is applied to derive a reduced-order moment model, which is used for the design of a MPC scheme. Initially, the optimal jacket temperature trajectory is computed by using nominal values for the kinetic parameters and applied to the crystallizer. After the first batch run, a multivariable nonlinear optimization (MNO)  problem is solved for the identification of the kinetic parameters while minimizing the difference between the size and shape distributions estimated through model prediction and the post-batch measurements (e.g., measured final crystal size and shape distributions) obtained from the kMC simulation. Furthermore, the constraints on the nucleation and crystal growth rates are imposed in the MNO  problem so that the kinetic parameters computed from the MNO result in the generation of physically meaningful rates. Then, the MPC is re-solved to compute a set of new optimal jacket temperatures using the reduced-order model with the new estimated kinetic parameters. This iterative optimization-based two-step control strategy continues until it provides negligible improvement from the previous step. We observe that the result is particularly more sensitive to the solubility than the kinetic parameters for the nucleation and crystal growth rates.

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[2] Wang Y, Gao F, Doyle FJ. Survey on iterative learning control, repetitive control, and run-to-run control. J. of Process Control. 2009;19:1589-1600.

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[4] Sachs E, Guo RS, Ha A. On-line process optimization and control using the sequential design of experiments. Symposium on VLSI Technology, Honolulu, HI, 1990.

[5] Kwon JS, Nayhouse M, Christofides PD, Orkoulas G. Modeling and control of crystal shape in continuous protein crystallization. Chem. Eng. Sci. 2014;107:47-57.

[6] Durbin SD, Feher G. Simulation of lysozyme crystal growth by the Monte Carlo method. J. Cryst. Growth. 1991;110:41-51.