Robust multi-objective optimal switching control arising in 1,3-propanediol microbial fed-batch process

This paper considers optimal control of glycerol producing 1,3-propanediol (1,3-PD) via microbial fed-batch fermentation. The fed-batch process is formulated as a nonlinear switched time-delay system. In general, the time-delay in the fed-batch process cannot be exactly estimated. Our goal is to design an optimal switching control scheme to simultaneously maximize 1,3-PD productivity and 1,3-PD yield under time-delay uncertainty. Accordingly, we propose a robust multi-objective optimal switching control model, in which two objectives, i.e., 1,3-PD productivity and 1,3-PD yield, and their sensitivities with respect to uncertain time-delay are considered in the objective vector. The control variables in this problem are the feeding rate of glycerol, the switching instants and the terminal time of the process. By introducing an auxiliary dynamic system to calculate the objective sensitivities and performing a time-scaling transformation, we obtain an equivalent multi-objective optimal switching control problem in standard form. We then convert the equivalent multi-objective optimal control problem into a sequence of single-objective optimal switching control problems by using a modified normal boundary intersection method. A novel gradient-based single-objective solver combining control parameterization with constraint transcription technique is developed to solve these resulting single-objective optimal control problems. Finally, numerical results are provided to verify the effectiveness of the proposed solution approach.