Robust optimal control of a nonlinear impulsive time-delay system for 1,3-PD fed-batch culture

In this paper, taking the feeding process as a form of impulsive and considering the time-delay in fermentation process. A robust model with the time-delay system as the control variable and the time-delay system as the constraint is established. In order to solve this optimal control problem, we have propose an particle swarm optimization method to solve problem. Numerical results show that 1,3-PD yield at the terminal time increases compared with the experimental result.


Introduction
The main raw of a new polyester material is 1,3-Propanediol (1,3-PD) is the production, antifreeze and protective agent. Traditional chemical synthesis of 1,3-PD requires expensive catalysts and complex environment. In recent years, microbial fermentation technology has received attention from experts and scholars at home and abroad.
There are three types of microbial fermentation for the production of 1,3-PD. Fed-batch fermentation includes both batch and continuous fermentation modes. Compared to other fermentation methods, fed-batch fermentation has the ability to overcome the inhibition of catabolic metabolites produced during the chemical production process. Therefore, the industrial production of 1,3-PD has been widely used in the replenishment fermentation of wholesale fermentation [1]. During fed-batch fermentation materials are added to the fermenter in batches to maintain a suitable fermentation environment and improve the yield of 1,3-PD. Therefore, the optimal control of glycerol and alkali addition in fed-batch fermentation has received much attention. The literature [2] propose the optimal switching control problem for the fermentation processes of the replenishment batch is studied. However, the above optimal control studies ignore the time-delay phenomenon in the reaction process and the robustness of the intracellular material.
In this article, a impulsive time-delay differential equation with control is proposed to describe the time-delay replenishment batch, using the addition of glycerol and alkali as control vectors. In order to maximize production, we use the end-moment 1,3-PD production as a performance indicator, we promose a optimal control model with time-delay.

Nonlinear impulsive time-delay system
In the production of replenishment sub-wholesale fermentation, alkali and glycerol are added to the fermenter in batches to provide sufficient nutrients. Based on the fermentation process, this paper assumes that : Under assumptions (H1)-(H3), biomass, substance and reaction in an intermittent process can expressed bythe differential equation the above parameters values are given in the literature [3].
Then, the whole fed-batch process can be written as

Optimal control model
During batch fermentation process, the producer expects to maximize the yield at end moment. Therefore, with the addition of glycerol and alkali as the control function and the yield of 1,3-PD at the terminal moment as the performance index, the following optimal control model is proposed in this paper: ]. , [

Numerical solution method
In this article, in order to overcome the localization of the particle swarm algorithm, a new velocity and position strategy is proposed. In addition, a new strategy of dealing with position outside parameter bounds is also introduced.
(1) (Updating strategy) In the previous iterations, balance development and exploitation by modifying the update speed and location as follows.
w and 2 w are the inertia weights.
(2) (Processing parameter bounds) Assuming that the position component of the particle violates the parameter bound constraint, the particle update position component reflected back by parameters.

Numerical results
In the numerical experiments, we use the same settings initial condition for the experimental to optimize the feeding amount. The initial concentration, initial volume of fermenting material, and end , respectively. Based on experiment, the fermentation process is divided into eight stages. Material are added at 100s in each stage. Table 1 and Table 2 given the upper and lower bounds of the amounts of glycerol and alkali added at each stage. Applying the PSO algorithm to solve OCP, the obtained optimal feeding strategy of material are listed in Table 3. From Figure 1, we can obtain that product is better than experiment data.

Conclusions
In this article, an impulsive time-delay system with multiple control variables to descr-ible the 1, 3-PD fed-batch process. Taking the 1, 3-PD at terminal time as performance index, the optimal control model of impulsive time-delay is established. The improved PSO is applied to sol-ve the optimal control problem. Numerical shows that the production of 1,3-PD is significantly hi-gher than experimental data at terminal time.