Koopman modeling and optimal control for microbial fed-batch fermentation with switching operators

The modeling of microbial fed-batch fermentation with switching operators to product 1,3-propanediol (1,3-PD) still maintains a challenge because it is strongly of nonlinearity and uncertainty. Machine learning methods for learning such models have become a hot research topic, but the interpretability of existing techniques remains a challenging problem. Recently, the Koopman operator, which is a linear operator governing the eigenfunction evolution along trajectories of a nonlinear dynamical system with switching operators, has been studied for modeling complex dynamics. In this paper, we propose a Koopman modeling method based on an interpretable Koopman operator. The predominant merit of using the Koopman operator is to offer a linear infinite dimensional description of a nonlinear dynamical system with switching operators. In the proposed method, an enhanced learning-based extended dynamic mode decomposition (enhanced-EDMD) algorithm based on a novel eigenfunction construction method is proposed to obtain a finite-dimensional approximation of the Koopman operator. The convergence analysis of the enhanced-EDMD algorithm is also studied. Furthermore, to maximize the productivity of 1,3-PD and minimize the total variation in the optimal control within a time frame, an algorithm combining the model predictive control method with the enhanced learning-based EDMD (denoted by MPC-Enhanced-EDMD), based on gradient-based optimization and exact penalty function method, is proposed for devising optimal feeding rate of glycerol evolving with time. Numerical simulations are conducted by demonstrating the effectiveness of the enhanced-EDMD algorithm on the dynamics prediction and the MPC-Enhanced-EDMD method on the optimal feeding rates of glycerol.