Stochastic Optimal Control of Seeded Batch Crystallizer Applying the Ito Process

Minimization of operation costs and the enhancement in product quality have been major concerns for all industrial processes. Predetermined operating conditions can help to achieve the goals for efficient production. These conditions can be determined using an optimal control analysis of batch crystallization process characterized by determination of the time-varying profiles for process parameters. Batch crystallization is associated with parameters such as temperature, supersaturation, and agitation. Some process parameters, such as solubility and crystal lattice, are functions of fundamental properties of the system. Thus, the process parameters can have various associated physical and engineering sources of uncertainties, which, in turn, would prevent the real process operation to be optimal. These static uncertainties result in dynamic uncertainties, because of the unsteady state nature of the process. This paper presents a novel approach to solve optimal control problems in batch crystallization involving uncertainties. These uncertainties are modeled as a special class of stochastic processes called Ito processes. The resulting stochastic optimal control problem is solved using Ito's Lemma, stochastic calculus, and stochastic maximum principle. The comparison between the results for the deterministic and the stochastic optimal temperature profile show that, when uncertainties are present, the stochastic optimal control approach presented in this paper gives better results, in terms of maximum crystal growth, represented in terms of crystallization moments. Percentage improvements of similar to 3% and similar to 6% are observed for the stochastic optimal profile, in comparison to the deterministic and linear cooling cases, in the presence of parametric uncertainty.