Robust Optimal Control for a Microbial Batch Culture Processes: Incorporating Free Terminal Time and Sensitivity Analysis

This paper focuses on the robust optimal control problem of an enzymatic-catalytic dynamic system in batch culture with uncertain parameters. To enhance the productivity of 1,3-propanediol (1,3-PD) in batch culture, we define the initial concentrations of biomass and glycerol as well as the terminal time of the fermentation process as control variables. By considering the nonlinear characteristics of the enzymatic-catalytic system as a primary constraint, we formulate a robust optimal control model that simultaneously evaluates the production efficiency of 1,3-PD and its sensitivity to uncertain parameters. The main objective of this model is to identify an optimal control strategy that not only strengthens the robustness of the system but also maximizes the yield of 1,3-PD. Given that the terminal time of this robust optimal control problem is unfixed and non-standard terms exist in the optimal control formulation, we adopt the time-scale transformation method and introduce an auxiliary dynamic system for calculating system sensitivity. Subsequently, we develop an improved particle swarm optimization algorithm to solve this equivalent problem. Finally, the trade-off between production efficiency and process robustness is systematically assessed through numerical simulations. The results demonstrate that sacrificing a marginal 3.9% of 1,3-PD production efficiency can lead to a 25.3% enhancement in system robustness. The primary contribution of this study lies in proposing a novel control strategy that effectively addresses the uncertainty of system parameters, while enhancing the production efficiency of 1,3-PD and the stability of the production process. The innovation of this control strategy resides in its comprehensive consideration of the impacts of multiple variables on system robustness, providing valuable theoretical support for practical applications.