Economically optimal batch diafiltration via analytical multi-objective optimal control

• Nonlinear multi-objective control problem solved via Pontryagin's minimum principle.
• Optimal state-feedback policy is derived analytically.
• The solution is generalized in terms of the weights of multiple objectives.
• Selected case studies show applicability and benefits of the developed approach.

This paper studies the problem of economically oriented optimal operation of batch membrane diafiltration processes that are designed to concentrate the valuable components of the solution and to purge the impurities from it. We consider a complex economical objective that accounts for the total operational costs comprising a cost of consumed diluant, costs related to duration of processing, and a cost of product loss. The optimization problem is formulated as a multi-objective optimal control problem in order to investigate the impact of operational cost factors on optimal operation policy. This is achieved thanks to the use of the analytical approach that exploits Pontryagin's minimum principle. We show that the economically optimal control strategy is to carry out an operation involving saturated (bang-bang or constraint-tracking) control modes and a singular arc. For the most common cases of diafiltration problems, it turns out that the switching of the consecutive control modes can be realized in the state feedback fashion, i.e. the entire optimal operation is defined analytically in the space of process states. We demonstrate the applicability of the presented approach and we illustrate achievable benefits, over traditional control methods for the batch diafiltration processes, on two case studies taken from the literature.