Robust Dynamic Optimization for Batch Fermentation Nonlinear Fractional Switched Time-Delay Dynamical System Using Inexact Output Data

In this paper, in view of the dynamic behavior characterized by memory and heredity, we study a nonlinear fractional switched time-delay dynamical system (NFSTDDS) with unknown fractional orders, kinetic parameters, and switching time for the bioconversion of glycerol to 1,3-propanediol in batch fermentation. Some important properties of the NFSTDDS are discussed. Our main objective is to determine the unknown quantities of the NFSTDDS using inexact output data. To achieve this, we propose a dynamic optimization problem governed by the NFSTDDS to minimize the cost function defined by the expectation and variance of the relative error between the solution of the NFSTDDS and the inexact output data. The time-scaling transformation technique is utilized to convert the free switching time into a fixed one. Subsequently, we develop a numerical method to discretize the dynamic optimization problem, transforming it into a finite-dimensional optimization problem with a non-smooth term. A smoothing method is utilized to approximate the non-smooth term in the finite-dimensional optimization problem. Then, an analysis of approximation errors for the smoothing method is conducted. Moreover, we design an optimization algorithm that leverages the gradients of the cost function to address the resulting optimization problem. Numerical simulations indicate that the NFSTDDS offers a reasonably precise depiction of the batch fermentation process and the proposed optimization algorithm is both effective and practical in application.