Optimal control of uncertain batch processes via Koopman expectation-assisted gradient methods

Batch processes are indispensable in the chemical industry for producing low-volume, high-value products, leveraging their inherent flexibility to adapt to evolving market demands. While process control optimization promises significant potential for improving efficiency and profitability, discrepancies between plant behavior and model predictions, stemming from imperfect models and measurements, can lead to suboptimal or even infeasible outcomes. To address these challenges, our work focuses on developing efficient gradient-based methods for optimizing batch processes under parametric uncertainty, with the goal of satisfying process chance constraints. Central to our approach is the use of the Koopman expectation method, a novel and computationally efficient alternative to Monte Carlo simulations for propagating uncertainty in nonlinear dynamic systems. We explore two optimization procedures: Direct iterative optimization, which fully integrates the Koopman expectation into the gradient calculation loop, and Backoff-assisted iterative optimization, which combines deterministic optimization with probabilistic constraint corrections informed by the Koopman expectation. The Direct iterative optimization demonstrates robustness and is straightforward to implement, albeit with significant computational demands. In contrast, the Backoff-assisted method significantly reduces computational burden while maintaining satisfactory optimization results. Both methods are successfully applied to a case study involving the minimum time problem in batch distillation, demonstrating their effectiveness in achieving constraint satisfaction under uncertain initial conditions and model parameters. The proposed methodology is further applied to a batch crystallization case study, illustrating its broader applicability and effectiveness across different batch processing scenarios.