Robust Optimal Control of a Microbial Batch Culture Process

Abstract

This paper considers the microbial batch culture process for producing 1,3-propanediol (1,3-PD) via glycerol fermentation. Our goal was to design an optimal control scheme for this process, with the aim of balancing two (perhaps competing) objectives: (i) the process should yield a sufficiently high concentration of 1,3-PD at the terminal time and (ii) the process should be robust with respect to changes in various uncertain system parameters. Accordingly, we pose an optimal control problem, in which both process yield and process sensitivity are considered in the objective function. The control variables in this problem are the terminal time of the batch culture process and the initial concentrations of biomass and glycerol in the batch reactor. By performing a time-scaling transformation and introducing an auxiliary dynamic system to calculate process sensitivity, we obtain an equivalent optimal control problem in standard form. We then develop a particle swarm optimization algorithm for solving this equivalent problem. Finally, we explore the trade-off between process efficiency and process robustness via numerical simulations.

Appendix

The explicit formulas for the derivatives of $\tilde{\mu }$ in (20)–(21) are given below.

$$\begin{aligned} \frac{\partial \tilde{\mu }(\tau )}{\partial \sigma _1}&=-\mu _m\frac{\tilde{x}_2(\tau )}{(\tilde{x}_2(\tau )+k_2)^2}\prod _{i=2}^5\Big (1-\frac{\tilde{x}_i(\tau )}{x_i^*}\Big ),\quad \quad \frac{\partial \tilde{\mu }(\tau )}{\partial \sigma _k}=0,\quad k=2,3,\ldots ,9,\\ \frac{\partial \tilde{\mu }(\tau )}{\partial x_1}&=0, \quad \quad \frac{\partial \tilde{\mu }(\tau )}{\partial x_2}=\mu _m\left[ \displaystyle \frac{k_2}{(\tilde{x}_2(\tau )+k_2)^2}\left( 1-\displaystyle \frac{\tilde{x}_2(\tau )}{x_2^*}\right) -\displaystyle \frac{\tilde{x}_2(\tau )}{x_2^*(\tilde{x}_2(\tau )+k_2)}\right] \\&\qquad \qquad \qquad \qquad \qquad \quad \prod _{i=3}^5\Big (1-\displaystyle \frac{\tilde{x}_i(\tau )}{x_i^*}\Big ),\\ \frac{\partial \tilde{\mu }(\tau )}{\partial x_j}&=-\mu _m\displaystyle \frac{\tilde{x}_2(\tau )}{x_j^*(\tilde{x}_2(\tau )+k_2)}\prod ^5_{i=2,i\ne j} \Big (1-\displaystyle \frac{\tilde{x}_i(\tau )}{x_i^*}\Big ),\quad j=3,4,5. \end{aligned}$$

The explicit formulas for the derivatives of $\tilde{q}_2$ in (20)–(21) are given below.

$$\begin{aligned} \frac{\partial \tilde{q}_2(\tau )}{\partial \sigma _1}&=\frac{1}{Y_2}\frac{\partial \tilde{\mu }(\tau )}{\partial \sigma _1},\quad \quad \frac{\partial \tilde{q}_2(\tau )}{\partial \sigma _2}=1, \quad \quad \frac{\partial \tilde{q}_2(\tau )}{\partial \sigma _6}=-\frac{\tilde{\mu }(\tau )}{Y_2^2}, \quad \quad \frac{\partial \tilde{q}_2(\tau )}{\partial \sigma _k}=0,\\&k=3,4,5,7,8,9,\\ \frac{\partial \tilde{q}_2(\tau )}{\partial x_1}&=0, \quad \quad \frac{\partial \tilde{q}_2(\tau )}{\partial x_2}=\frac{1}{Y_2}\frac{\partial \tilde{\mu }(\tau )}{\partial x_2},\quad \quad \frac{\partial \tilde{q}_2(\tau )}{\partial x_j}=\frac{1}{Y_2}\frac{\partial \tilde{\mu }(\tau )}{\partial x_j},\quad j=3,4,5. \end{aligned}$$

The explicit formulas for the derivatives of $\tilde{q}_i$, $i=3,4,5$, in (20)–(21) are given below.

$$\begin{aligned} \displaystyle \frac{\partial \tilde{q}_i(\tau )}{\partial \sigma _k}= \left\{ \begin{array}{ll} Y_i\displaystyle \frac{\partial \tilde{\mu }(\tau )}{\partial \sigma _1},\quad \quad ~~~~ &{} \text{ if }\quad k=1,\\ 1, \quad \quad \quad \quad \quad \quad ~ &{} \text{ if } \quad \sigma _k=m_{i},\\ \tilde{\mu }(\tau ),\quad \quad \quad \quad ~~~ &{} \text{ if } \quad \sigma _k=Y_i,\\ 0,\quad \quad \quad \quad \quad \quad \ &{} \text{ otherwise, } \end{array} \right. \\ \end{aligned}$$

$$\begin{aligned} \frac{\partial \tilde{q}_i(\tau )}{\partial x_1}&=0, \quad \quad \frac{\partial \tilde{q}_i(\tau )}{\partial x_2}={Y_i}\frac{\partial \tilde{\mu }(\tau )}{\partial x_2},\quad \quad \frac{\partial \tilde{q}_i(\tau )}{\partial x_j}=Y_i\frac{\partial \tilde{\mu }(\tau )}{\partial x_j},\quad j=3,4,5. \end{aligned}$$