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doi: 10.3934/jimo.2021040
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## Multistage optimal control for microbial fed-batch fermentation process

 1 School of Science, University of Science and Technology Liaoning, Anshan, Liaoning, China 2 School of Mathematics and Information Science, Shandong Technology and Business University, Yantai, Shangdong, China 3 School of Mathematical Sciences, Dalian University of Technology, Dalian, Liaoning, China

* Corresponding author: Email address: zhaohuagong@163.com

Received  September 2020 Revised  January 2021 Early access March 2021

Fund Project: This work was supported by the Natural Science Foundation of China(No. 11771008), the China Scholarship Council (No. 201902575002), and the Natural Science Foundation of Shandong Province, China (No. ZR2019MA031)

In this paper, we consider multistage optimal control of bioconversion glycerol to 1, 3-propanediol(1, 3-PD) in fed-batch fermentation process. To maximize the productivity of 1, 3-PD, the whole fermentation process is divided into three stages according to the characteristics of microbial growth. Stages 2 and 3 are discussed mainly. The main aim of stage 2 is to restrict accumulation of 3-hydroxypropionaldehyde and maximize the biomass in the shortest time, and the purpose of stage 3 is to get high productivity of 1, 3-PD. With these different objectives, multi-objective optimal control problems are proposed in stages 2 and 3. In order to solve the above optimal control problems, the multi-objective problems are transformed to the corresponding single-objective problems using the mass balance equation of biomass and normalization of the objective. Furthermore, the single-objective optimal control problems are transformed to two-level optimization problems by the control parametrization technique. Finally, numerical solution methods combined an improved Particle Swarm Optimization with penalty function method are developed to solve the resulting optimization problems. Numerical results show that the productivity of 1, 3-PD is higher than the reported results.

Citation: Xiaohong Li, Mingxin Sun, Zhaohua Gong, Enmin Feng. Multistage optimal control for microbial fed-batch fermentation process. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021040
##### References:

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##### References:
The concentrations of biomass, 1, 3-PD and 3-HPA under the obtained optimal control
The concentration of glycerol under the obtained optimal control
The indices of (O-PL) and (O-PS)
The critical concentrations and kinetic parameters in system (1) [5]
 $x_1^*$ $x_2^*$ $x_3^*$ $x_4^*$ $x_5^*$ $x_6^*$ $x_7^*$ $x_8^*$ 10 2039 2000 1026 360 2039 275 2000 $m_2$ $m_3$ $m_4$ $m_5$ $Y_2$ $Y_3$ $Y_4$ $Y_5$ 2.2 -2.69 -0.97 5.26 0.0082 67.69 33.07 11.66 $k_2^*$ $k_3^*$ $k_4^*$ $\bigtriangleup_2$ $\bigtriangleup_3$ $\bigtriangleup_4$ $n_2$ $n_3$ 11.43 15.50 85.71 28.58 26.59 5.74 1 3 $n_4$ $n_5$ $k_s$ $V_0$ $r$ $p_1$ $p_2$ $p_3$ 3 3 0.28 5 0.75 30.0688 3.8179 679.913 $p_4$ $p_5$ $p_6$ $p_7$ $p_8$ $p_9$ $p_{10}$ $p_{11}$ 58.5244 3.9251 8.3591 59.266 2.2919 2478.52 19.6651 136.563 $p_{12}$ $p_{13}$ $p_{14}$ $p_{15}$ $p_{16}$ $p_{17}$ $k_1$ $k_2$ 22.4736 0.7205 5.6354 5.5599 1.7492 1.4570 0.53 0.14
 $x_1^*$ $x_2^*$ $x_3^*$ $x_4^*$ $x_5^*$ $x_6^*$ $x_7^*$ $x_8^*$ 10 2039 2000 1026 360 2039 275 2000 $m_2$ $m_3$ $m_4$ $m_5$ $Y_2$ $Y_3$ $Y_4$ $Y_5$ 2.2 -2.69 -0.97 5.26 0.0082 67.69 33.07 11.66 $k_2^*$ $k_3^*$ $k_4^*$ $\bigtriangleup_2$ $\bigtriangleup_3$ $\bigtriangleup_4$ $n_2$ $n_3$ 11.43 15.50 85.71 28.58 26.59 5.74 1 3 $n_4$ $n_5$ $k_s$ $V_0$ $r$ $p_1$ $p_2$ $p_3$ 3 3 0.28 5 0.75 30.0688 3.8179 679.913 $p_4$ $p_5$ $p_6$ $p_7$ $p_8$ $p_9$ $p_{10}$ $p_{11}$ 58.5244 3.9251 8.3591 59.266 2.2919 2478.52 19.6651 136.563 $p_{12}$ $p_{13}$ $p_{14}$ $p_{15}$ $p_{16}$ $p_{17}$ $k_1$ $k_2$ 22.4736 0.7205 5.6354 5.5599 1.7492 1.4570 0.53 0.14
 Algorithm 1 Algorithm AL to solve (PL-TL) Step 1:Set integers $N^0>0$ and $d>0$, parameter $C>0$, and error $\varepsilon>0$, let $i=0$.Step 2: Suppose current switching number is $N^i$, enter following sub-loop to solve (I-PL).  Step 2.1: Set penalty factor $M_0>0, k=0$.  Step 2.2: Solve (I-PL-M) using improved PSO algorithm. Its optimal solution is denoted by $(\tau^k,v^k_l)$, optimal index by $JL(N^i)$.  Step 2.3: If $M_i[(\mu -D(t,\tau^k,v^k_l))\mid_{t=t_2}]^2<\varepsilon$, then $(\tau^k,v^k_l)$ is the optimal solution of (I-PL), and go to Step 3. Otherwise, go to Step 2.4.  Step 2.4: Set $M_{k+1}=CM_k, k:=k+1$, and go to Step 2.2.Step 3: Set $N^{i+1}=N^i+d, i:=i+1$, and go to Step 2. Otherwise, go to Step 4.Step 4: If $|JL(N^{i-1})-JL(N^i)|<\varepsilon$, then $N^{i-1}$ is the optimal switching number, and $(\tau^k,v^k_l)$ is the optimal solution of (PL-TL), stop. Otherwise, set $N^{i+1}=N^i+d, i:=i+1$, and go to Step 2.
 Algorithm 1 Algorithm AL to solve (PL-TL) Step 1:Set integers $N^0>0$ and $d>0$, parameter $C>0$, and error $\varepsilon>0$, let $i=0$.Step 2: Suppose current switching number is $N^i$, enter following sub-loop to solve (I-PL).  Step 2.1: Set penalty factor $M_0>0, k=0$.  Step 2.2: Solve (I-PL-M) using improved PSO algorithm. Its optimal solution is denoted by $(\tau^k,v^k_l)$, optimal index by $JL(N^i)$.  Step 2.3: If $M_i[(\mu -D(t,\tau^k,v^k_l))\mid_{t=t_2}]^2<\varepsilon$, then $(\tau^k,v^k_l)$ is the optimal solution of (I-PL), and go to Step 3. Otherwise, go to Step 2.4.  Step 2.4: Set $M_{k+1}=CM_k, k:=k+1$, and go to Step 2.2.Step 3: Set $N^{i+1}=N^i+d, i:=i+1$, and go to Step 2. Otherwise, go to Step 4.Step 4: If $|JL(N^{i-1})-JL(N^i)|<\varepsilon$, then $N^{i-1}$ is the optimal switching number, and $(\tau^k,v^k_l)$ is the optimal solution of (PL-TL), stop. Otherwise, set $N^{i+1}=N^i+d, i:=i+1$, and go to Step 2.
The initial state and final state in three stages
 $x(0)=(2.7,217.4,0,0,0,0,0,0)^\top$ $x(t_1)=(3.125,26.557,121.263,40.906,20.783,15.887,13.143,92.883)^\top$ $x(t_2)=(8.007, 82.786,492.433,120.359, 88.621, 53.972,110.348,456.382)^\top$ $x(t_3)=(7.763,223.007, 1321.761,180.664,102.715,202.643,126.896, 1135.526)^\top$
 $x(0)=(2.7,217.4,0,0,0,0,0,0)^\top$ $x(t_1)=(3.125,26.557,121.263,40.906,20.783,15.887,13.143,92.883)^\top$ $x(t_2)=(8.007, 82.786,492.433,120.359, 88.621, 53.972,110.348,456.382)^\top$ $x(t_3)=(7.763,223.007, 1321.761,180.664,102.715,202.643,126.896, 1135.526)^\top$
The obtained optimal control
 terminal time feeding and batch time feeding rate switch times stage 2 $t_2=15.171$ $\tau=(0.006,0.042)$ $v_l=0.538$ $N=205$ stage 3 $t_3=24.292$ $\tau=(0.018,0.058)$ $v_s=1.264$ $N=120$
 terminal time feeding and batch time feeding rate switch times stage 2 $t_2=15.171$ $\tau=(0.006,0.042)$ $v_l=0.538$ $N=205$ stage 3 $t_3=24.292$ $\tau=(0.018,0.058)$ $v_s=1.264$ $N=120$
The comparison between our results and previous results
 result in [13] result in [21] result in [32] our result 1, 3-PD concentration 975.319 1416.7 679.22 1321.761 fermentation time 24.16 39 14.9167 24.292 1, 3-PD productivity 40.3691 36.3256 45.5342 54.4114
 result in [13] result in [21] result in [32] our result 1, 3-PD concentration 975.319 1416.7 679.22 1321.761 fermentation time 24.16 39 14.9167 24.292 1, 3-PD productivity 40.3691 36.3256 45.5342 54.4114
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