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doi: 10.3934/jimo.2021040

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 Published  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:
[1]

V. S. Bisaria and A. Kondo, Bioprocessing of Renewable Resources to Commodity Bioproducts, John Wiley & Sons Inc., New Jersey, 2014. doi: 10.1002/9781118845394.  Google Scholar

[2]

C. X. GaoE. M. FengZ. T. Wang and Z. L. Xiu, Parameters identification problem of the nonlinear dynamical system in microbial continuous cultures, Applied Mathematics and Computation, 169 (2005), 476-484.  doi: 10.1016/j.amc.2004.10.048.  Google Scholar

[3]

Z. H. Gong, C. Y. Liu and Y. S. Yu, Modeling and parameter identification involving 3-hydroxypropionaldehyde inhibitory effects in glycerol continuous fermentation, Mathematical Problems in Engineering, 2012 (2012), Art. ID 690587, 18 pp. doi: 10.1155/2012/690587.  Google Scholar

[4]

Z. H. Gong, A multistage system of microbial fed batch fermentation and its parameter identification, Mathematics and Computers in Simulation, 80 (2010), 1903-1910.  doi: 10.1016/j.matcom.2009.12.011.  Google Scholar

[5]

G. D. JinX. H. LiY. Wang and E. M. Feng, Fed batch fermentation dynamic system and parameters identification, Journal of Liaoning University of Science and Technology, 38 (2015), 315-320.   Google Scholar

[6]

C. Y. Liu, Modelling and parameter identification for a nonlinear time-delay system in microbial batch fermentation, Applied Mathematical Modelling, 37 (2013), 6899-6908.  doi: 10.1016/j.apm.2013.02.021.  Google Scholar

[7]

C. Y. LiuZ. H. GongK. L. TeoR. Loxton and E. M. Feng, Bi-objective dynamic optimization of a nonlinear time-delay system in microbial batch process, Optimization Letters, 12 (2018), 1249-1264.  doi: 10.1007/s11590-016-1105-6.  Google Scholar

[8]

C. Y. LiuZ. H. GongH. W. J. Lee and K. L. Teo, Robust bi-objective optimal control of 1, 3-propanediol microbial batch production process, Journal of Process Control, 78 (2019), 170-182.  doi: 10.1016/j.jprocont.2018.10.001.  Google Scholar

[9]

Z. H. GongC. Y. LiuK. L. Teo and and J. Sun, Distributionally robust parameter identification of a time-delay dynamical system with stochastic measurements, Applied Mathematical Modelling, 69 (2019), 685-695.  doi: 10.1016/j.apm.2018.09.040.  Google Scholar

[10]

C. Y. LiuZ. H. GongK. L. Teo and S. Wang, Modelling and optimal state-delay control in microbial batch process, Applied Mathematical Modelling, 89 (2021), 792-801.  doi: 10.1016/j.apm.2020.07.051.  Google Scholar

[11]

C. Y. LiuZ. H. GongE. M. Feng and H. C. Yin, Modelling and optimal control for nonlinear multistage dynamical system of microbial fed batch culture, Journal of Industrial and Management Optimization, 5 (2009), 835-850.  doi: 10.3934/jimo.2009.5.835.  Google Scholar

[12]

C. Y. Liu, Optimal control for nonlinear dynamical system of microbial fed batch culture, Journal of Computational and Applied Mathematics, 232 (2009), 252-261.  doi: 10.1016/j.cam.2009.06.006.  Google Scholar

[13]

C. Y. LiuZ. H. GongE. M. Feng and H. C. Yin, Optimal switching control of a fed-batch fermentation process, Journal of Global Optimization, 52 (2012), 265-280.  doi: 10.1007/s10898-011-9663-8.  Google Scholar

[14]

C. Y. LiuR. LoxtonQ. Lin and K. L. Teo, Dynamic optimization for switched time-delay systems with state-dependent switching conditions, SIAM Journal on Control and Optimization, 56 (2018), 3499-3523.  doi: 10.1137/16M1070530.  Google Scholar

[15]

C. Y. Liu and Z. H. Gong, Optimal Control of Switched Systems Arising in Fermentation Processes, Springer-Verlag, New York, 2014. doi: 10.1007/978-3-662-43793-3.  Google Scholar

[16]

C. Y. LiuZ. H. GongK. L. Teo and E. M. Feng, Multi-objective optimization of nonlinear switched time delay systems in fed batch process, Applied Mathematical Modelling, 40 (2016), 10533-10548.  doi: 10.1016/j.apm.2016.07.010.  Google Scholar

[17]

C. Y. LiuZ. H. GongK. L. TeoJ. Sun and L. Caccettal, Robust multi-objective optimal switching control arising in 1, 3-propanediol microbial fed-batch process, Nonlinear Analysis: Hybrid Systems, 25 (2017), 1-20.  doi: 10.1016/j.nahs.2017.01.006.  Google Scholar

[18]

C. Y. Liu and M. J. Han, Time-delay optimal control of a fed-batch production involving multiple feeds, Discrete and Continuous Dynamical Systems-Series S, 13 (2020), 1697-1709.  doi: 10.3934/dcdss.2020099.  Google Scholar

[19]

H. LiuJ. WangD. Zhang and Z. L. Xiu, Fermentative production of 1, 3-propanediol by Klebsiella pneumoniae in fed-batch culture, Food and Fermentation Industries, 27 (2001), 4-7.   Google Scholar

[20]

T. NiuJ. G. ZhaiH. C. YinE. M. FengC. Y. Liu and Z. L. Xiu, Multi-objective optimisation of nonlinear switched systems in uncoupled fed-batch fermentation, International Journal of Systems Science, 51 (2020), 1798-1813.  doi: 10.1080/00207721.2020.1780338.  Google Scholar

[21]

T. NiuJ. G. ZhaiH. C. Yin and E. M. Feng, Optimal control of nonlinear switched system in an uncoupled microbial fed-batch fermentation process, Journal of the Franklin Institute, 355 (2018), 6169-6190.  doi: 10.1016/j.jfranklin.2018.05.012.  Google Scholar

[22]

Y. Q. SunJ. T. ShenL. YanJ. J. ZhouL. L. JiangY. ChenJ. L. YuanE. M. Feng and Z. L. Xiu, Advances in bioconversion of glycerol to 1, 3-propanediol: Prospects and challenges, Process Biochemistry, 71 (2018), 134-146.   Google Scholar

[23]

Y. Q. SunW. T. QiH. TengZ. L. Xiu and A. P. Zeng, Mathematical modeling of glycerol fermentation by Klebsiella pneumoniae: Concerning enzyme-catalytic reductive pathway and transport of glycerol and 1, 3-propanediol across cell membrane, Biochemical Engineering Journal, 38 (2008), 22-32.   Google Scholar

[24]

Y. Q. Sun, Nonlinear Mathematical Simulation and Analysis of Enzyme-catalytic Kinetics and Genetic Regulation for Glycerol Dissimilation by Klebsiella pneumoniae, Ph.D thesis, Dalian University of Technology in Dalian, 2010. Google Scholar

[25]

L. WangJ. X. YeE. M. Feng and Z. L. Xiu, An improved model for multistage simulation of glycerol fermentation in batch culture and its parameter identification, Nonlinear Analysis: Hybrid Systems, 3 (2009), 455-462.  doi: 10.1016/j.nahs.2009.03.003.  Google Scholar

[26]

L. WangG. M. ChengE. M. FengT. Su and Z. L. Xiu, Analysis and application of biological robustness as performance index in microbial fermentation, Applied Mathematical Modelling, 39 (2015), 2048-2055.  doi: 10.1016/j.apm.2014.10.022.  Google Scholar

[27]

L. WangJ. L. YuanC. Z. Wu and X. Y. Wang, Practical algorithm for stochastic optimal control problem about microbial fermentation in batch culture, Optimization Letters, 13 (2019), 527-541.  doi: 10.1007/s11590-017-1220-z.  Google Scholar

[28]

J. WangH. XuK. L. TeoJ. Sun and J. X. Ye, Mixed-integer minimax dynamic optimization for structure identification of glycerol metabolic network, Applied Mathematical Modelling, 82 (2020), 503-520.  doi: 10.1016/j.apm.2020.01.042.  Google Scholar

[29]

G. X. Wang, Ordinary differential equation, Math in Economics, (2015), 187–212. doi: 10.1142/9789814663823_0005.  Google Scholar

[30]

Z. L. XiuB. H. SongL. H. Sun and A. P. Zeng, Theoretical analysis of effects of metabolic overflow and time delay on the performance and dynamic behavior of a two-stage fermentation process, Biochemical Engineering Journal, 11 (2002), 101-109.  doi: 10.1016/S1369-703X(02)00033-5.  Google Scholar

[31]

J. X. YeA. Li and J. G. Zhai, A measure of concentration robustness in a biochemical reaction network and its application on system identification, Applied Mathematical Modelling, 58 (2018), 270-280.  doi: 10.1016/j.apm.2017.07.026.  Google Scholar

[32]

J. X. YeH. XuE. M. Feng and Z. L. Xiu, Optimization of a fed-batch bioreactor for 1, 3-propanediol production using hybrid nonlinear optimal control, Journal of Process Control, 24 (2014), 1556-1269.   Google Scholar

[33]

J. X. YeE. M. FengH. C. Yin and Z. L. Xiu, Modelling and well-posedness of a nonlinear hybrid system in fed-batch production of 1, 3-propanediol with open loop glycerol input and pH logic control, Nonlinear Analysis: Real World Applications, 12 (2011), 364-376.  doi: 10.1016/j.nonrwa.2010.06.022.  Google Scholar

[34]

J. L. YuanJ. XieM. HuangH. FanE. M. Feng and Z. L. Xiu, Robust optimal control problem with multiple characteristic time points in the objective for a batch nonlinear time-varying process using parallel global optimization, Optimization and Engineering, 21 (2020), 905-937.  doi: 10.1007/s11081-019-09472-z.  Google Scholar

[35]

A. P. Zeng and W. D. Deckwer, A kinetic model for substrate and energy consumption of microbial growth under substrate-sufficient conditions, Biotechnology Progress, 11 (1995), 71-79.  doi: 10.1021/bp00031a010.  Google Scholar

show all references

References:
[1]

V. S. Bisaria and A. Kondo, Bioprocessing of Renewable Resources to Commodity Bioproducts, John Wiley & Sons Inc., New Jersey, 2014. doi: 10.1002/9781118845394.  Google Scholar

[2]

C. X. GaoE. M. FengZ. T. Wang and Z. L. Xiu, Parameters identification problem of the nonlinear dynamical system in microbial continuous cultures, Applied Mathematics and Computation, 169 (2005), 476-484.  doi: 10.1016/j.amc.2004.10.048.  Google Scholar

[3]

Z. H. Gong, C. Y. Liu and Y. S. Yu, Modeling and parameter identification involving 3-hydroxypropionaldehyde inhibitory effects in glycerol continuous fermentation, Mathematical Problems in Engineering, 2012 (2012), Art. ID 690587, 18 pp. doi: 10.1155/2012/690587.  Google Scholar

[4]

Z. H. Gong, A multistage system of microbial fed batch fermentation and its parameter identification, Mathematics and Computers in Simulation, 80 (2010), 1903-1910.  doi: 10.1016/j.matcom.2009.12.011.  Google Scholar

[5]

G. D. JinX. H. LiY. Wang and E. M. Feng, Fed batch fermentation dynamic system and parameters identification, Journal of Liaoning University of Science and Technology, 38 (2015), 315-320.   Google Scholar

[6]

C. Y. Liu, Modelling and parameter identification for a nonlinear time-delay system in microbial batch fermentation, Applied Mathematical Modelling, 37 (2013), 6899-6908.  doi: 10.1016/j.apm.2013.02.021.  Google Scholar

[7]

C. Y. LiuZ. H. GongK. L. TeoR. Loxton and E. M. Feng, Bi-objective dynamic optimization of a nonlinear time-delay system in microbial batch process, Optimization Letters, 12 (2018), 1249-1264.  doi: 10.1007/s11590-016-1105-6.  Google Scholar

[8]

C. Y. LiuZ. H. GongH. W. J. Lee and K. L. Teo, Robust bi-objective optimal control of 1, 3-propanediol microbial batch production process, Journal of Process Control, 78 (2019), 170-182.  doi: 10.1016/j.jprocont.2018.10.001.  Google Scholar

[9]

Z. H. GongC. Y. LiuK. L. Teo and and J. Sun, Distributionally robust parameter identification of a time-delay dynamical system with stochastic measurements, Applied Mathematical Modelling, 69 (2019), 685-695.  doi: 10.1016/j.apm.2018.09.040.  Google Scholar

[10]

C. Y. LiuZ. H. GongK. L. Teo and S. Wang, Modelling and optimal state-delay control in microbial batch process, Applied Mathematical Modelling, 89 (2021), 792-801.  doi: 10.1016/j.apm.2020.07.051.  Google Scholar

[11]

C. Y. LiuZ. H. GongE. M. Feng and H. C. Yin, Modelling and optimal control for nonlinear multistage dynamical system of microbial fed batch culture, Journal of Industrial and Management Optimization, 5 (2009), 835-850.  doi: 10.3934/jimo.2009.5.835.  Google Scholar

[12]

C. Y. Liu, Optimal control for nonlinear dynamical system of microbial fed batch culture, Journal of Computational and Applied Mathematics, 232 (2009), 252-261.  doi: 10.1016/j.cam.2009.06.006.  Google Scholar

[13]

C. Y. LiuZ. H. GongE. M. Feng and H. C. Yin, Optimal switching control of a fed-batch fermentation process, Journal of Global Optimization, 52 (2012), 265-280.  doi: 10.1007/s10898-011-9663-8.  Google Scholar

[14]

C. Y. LiuR. LoxtonQ. Lin and K. L. Teo, Dynamic optimization for switched time-delay systems with state-dependent switching conditions, SIAM Journal on Control and Optimization, 56 (2018), 3499-3523.  doi: 10.1137/16M1070530.  Google Scholar

[15]

C. Y. Liu and Z. H. Gong, Optimal Control of Switched Systems Arising in Fermentation Processes, Springer-Verlag, New York, 2014. doi: 10.1007/978-3-662-43793-3.  Google Scholar

[16]

C. Y. LiuZ. H. GongK. L. Teo and E. M. Feng, Multi-objective optimization of nonlinear switched time delay systems in fed batch process, Applied Mathematical Modelling, 40 (2016), 10533-10548.  doi: 10.1016/j.apm.2016.07.010.  Google Scholar

[17]

C. Y. LiuZ. H. GongK. L. TeoJ. Sun and L. Caccettal, Robust multi-objective optimal switching control arising in 1, 3-propanediol microbial fed-batch process, Nonlinear Analysis: Hybrid Systems, 25 (2017), 1-20.  doi: 10.1016/j.nahs.2017.01.006.  Google Scholar

[18]

C. Y. Liu and M. J. Han, Time-delay optimal control of a fed-batch production involving multiple feeds, Discrete and Continuous Dynamical Systems-Series S, 13 (2020), 1697-1709.  doi: 10.3934/dcdss.2020099.  Google Scholar

[19]

H. LiuJ. WangD. Zhang and Z. L. Xiu, Fermentative production of 1, 3-propanediol by Klebsiella pneumoniae in fed-batch culture, Food and Fermentation Industries, 27 (2001), 4-7.   Google Scholar

[20]

T. NiuJ. G. ZhaiH. C. YinE. M. FengC. Y. Liu and Z. L. Xiu, Multi-objective optimisation of nonlinear switched systems in uncoupled fed-batch fermentation, International Journal of Systems Science, 51 (2020), 1798-1813.  doi: 10.1080/00207721.2020.1780338.  Google Scholar

[21]

T. NiuJ. G. ZhaiH. C. Yin and E. M. Feng, Optimal control of nonlinear switched system in an uncoupled microbial fed-batch fermentation process, Journal of the Franklin Institute, 355 (2018), 6169-6190.  doi: 10.1016/j.jfranklin.2018.05.012.  Google Scholar

[22]

Y. Q. SunJ. T. ShenL. YanJ. J. ZhouL. L. JiangY. ChenJ. L. YuanE. M. Feng and Z. L. Xiu, Advances in bioconversion of glycerol to 1, 3-propanediol: Prospects and challenges, Process Biochemistry, 71 (2018), 134-146.   Google Scholar

[23]

Y. Q. SunW. T. QiH. TengZ. L. Xiu and A. P. Zeng, Mathematical modeling of glycerol fermentation by Klebsiella pneumoniae: Concerning enzyme-catalytic reductive pathway and transport of glycerol and 1, 3-propanediol across cell membrane, Biochemical Engineering Journal, 38 (2008), 22-32.   Google Scholar

[24]

Y. Q. Sun, Nonlinear Mathematical Simulation and Analysis of Enzyme-catalytic Kinetics and Genetic Regulation for Glycerol Dissimilation by Klebsiella pneumoniae, Ph.D thesis, Dalian University of Technology in Dalian, 2010. Google Scholar

[25]

L. WangJ. X. YeE. M. Feng and Z. L. Xiu, An improved model for multistage simulation of glycerol fermentation in batch culture and its parameter identification, Nonlinear Analysis: Hybrid Systems, 3 (2009), 455-462.  doi: 10.1016/j.nahs.2009.03.003.  Google Scholar

[26]

L. WangG. M. ChengE. M. FengT. Su and Z. L. Xiu, Analysis and application of biological robustness as performance index in microbial fermentation, Applied Mathematical Modelling, 39 (2015), 2048-2055.  doi: 10.1016/j.apm.2014.10.022.  Google Scholar

[27]

L. WangJ. L. YuanC. Z. Wu and X. Y. Wang, Practical algorithm for stochastic optimal control problem about microbial fermentation in batch culture, Optimization Letters, 13 (2019), 527-541.  doi: 10.1007/s11590-017-1220-z.  Google Scholar

[28]

J. WangH. XuK. L. TeoJ. Sun and J. X. Ye, Mixed-integer minimax dynamic optimization for structure identification of glycerol metabolic network, Applied Mathematical Modelling, 82 (2020), 503-520.  doi: 10.1016/j.apm.2020.01.042.  Google Scholar

[29]

G. X. Wang, Ordinary differential equation, Math in Economics, (2015), 187–212. doi: 10.1142/9789814663823_0005.  Google Scholar

[30]

Z. L. XiuB. H. SongL. H. Sun and A. P. Zeng, Theoretical analysis of effects of metabolic overflow and time delay on the performance and dynamic behavior of a two-stage fermentation process, Biochemical Engineering Journal, 11 (2002), 101-109.  doi: 10.1016/S1369-703X(02)00033-5.  Google Scholar

[31]

J. X. YeA. Li and J. G. Zhai, A measure of concentration robustness in a biochemical reaction network and its application on system identification, Applied Mathematical Modelling, 58 (2018), 270-280.  doi: 10.1016/j.apm.2017.07.026.  Google Scholar

[32]

J. X. YeH. XuE. M. Feng and Z. L. Xiu, Optimization of a fed-batch bioreactor for 1, 3-propanediol production using hybrid nonlinear optimal control, Journal of Process Control, 24 (2014), 1556-1269.   Google Scholar

[33]

J. X. YeE. M. FengH. C. Yin and Z. L. Xiu, Modelling and well-posedness of a nonlinear hybrid system in fed-batch production of 1, 3-propanediol with open loop glycerol input and pH logic control, Nonlinear Analysis: Real World Applications, 12 (2011), 364-376.  doi: 10.1016/j.nonrwa.2010.06.022.  Google Scholar

[34]

J. L. YuanJ. XieM. HuangH. FanE. M. Feng and Z. L. Xiu, Robust optimal control problem with multiple characteristic time points in the objective for a batch nonlinear time-varying process using parallel global optimization, Optimization and Engineering, 21 (2020), 905-937.  doi: 10.1007/s11081-019-09472-z.  Google Scholar

[35]

A. P. Zeng and W. D. Deckwer, A kinetic model for substrate and energy consumption of microbial growth under substrate-sufficient conditions, Biotechnology Progress, 11 (1995), 71-79.  doi: 10.1021/bp00031a010.  Google Scholar

Figure 1.  The concentrations of biomass, 1, 3-PD and 3-HPA under the obtained optimal control
Figure 2.  The concentration of glycerol under the obtained optimal control
Figure 3.  The indices of (O-PL) and (O-PS)
Table 1.  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.
Table 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 $
Table 3.  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 $
Table 4.  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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