2015, 5(4): 339-349. doi: 10.3934/naco.2015.5.339

Optimal control of microbial fed-batch culture involving multiple feeds

1. 

College of Information Science and Engineering, Shandong University of Science and Technology, Qindao, Shandong 266510, China, China, China

Received  December 2014 Revised  October 2015 Published  October 2015

In this paper, we consider optimal control problem in the fed-batch fermentation of glycerol by Klebsiella pneumoniae with open loop glycerol input and pH logic control, while the feeding volume of glycerol is regarded as the control variable. To maximize the concentration of 1,3-PD at the terminal time, an optimal control model is established, and a computational approach is constructed to solve the control model. Finally, the numerical simulations show that the terminal concentration of producing 1,3-PD has been increased obviously by employing the optimal feeding strategy.
Citation: Jinggui Gao, Xiaoyan Zhao, Jinggang Zhai. Optimal control of microbial fed-batch culture involving multiple feeds. Numerical Algebra, Control and Optimization, 2015, 5 (4) : 339-349. doi: 10.3934/naco.2015.5.339
References:
[1]

H. Biebl, K. Menzel, A. P. Zeng and W. D. Deckwer, Microbial production of 1, 3-propanediol, Applied Microbiology and Biotechnology, 52 (1999), 289-297.

[2]

G. M. Cheng, L. Wang and Q. Lin and R. C. Loxton, Robust optimal control of a microbial batch culture process, Journal of Optimization Theory and Applications, 167 (2015), 342-362. doi: 10.1007/s10957-014-0654-z.

[3]

X. Y. Dang, The calculation of ph of some kinds solution, Journal of Gansu Normal Colleges, 2 (2007), 45-48.

[4]

C. X. Gao, E. M. Feng, Z. T. Wang and Z. L. Xiu, Nonlinear dynamical systems of biodissimilation of glycerol to 1,3-propanediol and their optimal controls, Journal of Industrial and Management Optimization, 1 (2005), 377-388. doi: 10.3934/jimo.2005.1.377.

[5]

C. Gao, K. Li, E. Feng and Z. Xiu, Nonlinear impulsive system of fed-batch culture in fermentative production and its properties, Chaos, Solitons & Fractals, 28 (2006), 271-277. doi: 10.1016/j.chaos.2005.05.027.

[6]

C. Gao, E. Feng, Z. Wang and Z. Xiu, Nonlinear dynamical systems of bio-dissimilation of glycerol to 1, 3-propanediol and their optimal controls, Journal of Industrial and Management Optimization, 1 (2005), 377-384. doi: 10.3934/jimo.2005.1.377.

[7]

J. G. Gao, B. Y. Shen, E. M. Feng and Z. L. Xiu, Modelling and optimal control for an impulsive dynamical system in microbial fed-batch culture, Computational and Applied Mathematics, 32 (2013), 275-290. doi: 10.1007/s40314-013-0012-z.

[8]

J. G. Gao, X. Y. Zhao, E. M. Feng and Z. L. Xiu, Modelling and parameter identification for a hybrid dynamical system in microbial fed-batch culture, International Journal of Computer Mathematics, DOI:10.1080/00207160.2014.998656, 2015. doi: 10.1080/00207160.2014.998656.

[9]

J. G. Gao, E. M. Feng and Z. L. Xiu, Metabolic system identification and optimization in continuous culture, International Journal of Computer Mathematics, 89 (2012), 1426-1444. doi: 10.1080/00207160.2012.689289.

[10]

H. Huang, C. S. Gong and G. T. Tsao, Production of 1, 3-propanediol by klebsiella pneumoniae, Applied biochemistry and biotechnology, 98 (2002), 687-698.

[11]

J. Kennedy and R. Eberhart, Particle swarm optimization, In Neural Networks, 1995. Proceedings, IEEE International Conference on, 4 (1995), 1942-1948.

[12]

Q. H. Li, L. Li, T. K. Tai and S. L. Xie, An improved pso-based of harmony search for complicated optimization problems, International Journal of Hybrid Information Technology, 1 (2008), 91-98.

[13]

C. Y. Liu, Z. H. Gong, E. 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.

[14]

A. P. Nikolai, A. P. Viktor, M. S. Anatolii and V. S. Natalia, Differential Equations with Impulse Effects, De Gruyter, 2010. doi: 10.1515/9783110218176.

[15]

S. J. Pirt et al, Principles of Microbe and Cell Cultivation, Blackwell Scientific Publications, Springer Verlag, New York, 1975.

[16]

S. Sattayasamitsathit, P. Methacanon and P. Prasertsan, Enhance 1,3-propanediol production from crude glycerol in batch and fed-batch fermentation with two-phase ph-controlled strategy, Electronic Journal of Biotechnology, 6 (2011), 1-12.

[17]

A. I. Selvakumar and K. Thanushkodi, A new particle swarm optimization solution to nonconvex economic dispatch problems, Power Systems, IEEE Transactions on, 22 (2007), 42-51.

[18]

G. Wang, E. M. Feng and Z. L. Xiu, Modeling and parameter identification of microbial biconversion in fed-batch cultures, Process Control, 18 (2008), 458-464.

[19]

H. Wang, E. Feng and Z. Xiu, Optimality condition of the nonlinear impulsive system in fed-batch fermentation, Nonlinear Analysis: Theory, Methods & Applications, 68 (2008), 12-23. doi: 10.1016/j.na.2006.10.027.

[20]

L. Wang, Determining the transport mechanism of an enzyme-catalytic complex metabolic network based on biological robustness, Bioprocess Biosyst. Eng., 36 (2013), 433-441.

[21]

L. Wang, J. Ye, E. Feng and Z. 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.

[22]

L. Wang, G. M. Cheng, E. M. Feng, Tao 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.

[23]

Z. L. Xiu, A. P. Zeng and L. J. An, Mathematical modelling of kinetics and research on multiplicity of glycerol bioconversion to 1,3-propanediol, Journal of Dalian University of Technology, 40 (2000), 428-433.

[24]

X. H. Yang, Z. F. Yang and J. Q. Li, Solving nonlinear optimal problems by hybrid accelerating genetic algorithm, Journal of Beijing Normal University, 39 (2003), 551-556.

[25]

J. Ye, Y. Zhang, E. Feng, Z. Xiu and H. Yin, Nonlinear hybrid system and parameter identification of microbial fed-batch culture with open loop glycerol input and ph logic control, Applied Mathematical Modelling, 36 (2011), 357-369. doi: 10.1016/j.apm.2011.05.059.

[26]

J. Yu, L. Xi and S. Wang, An improved particle swarm optimization for evolving feedforward artificial neural networks, Neural Processing Letters, 26 (2007), 217-231.

[27]

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.

[28]

A. P. Zeng, A. Ross, H. Biebl, C. Tag, B. Guenzel and W. D. Deckwer, Multiple product inhibition and growth modeling of clostridium butyricum and klebsiella pneumoniae in glycerol fermentation, Biotechnology and bioengineering, 44 (1994), 902-911.

[29]

A. P. Zeng and H. Biebl, Bulk chemicals from biotechnology: the case of 1, 3-propanediol production and the new trends, Tools and Applications of Biochemical Engineering Science, 15 (2002), 239-259.

show all references

References:
[1]

H. Biebl, K. Menzel, A. P. Zeng and W. D. Deckwer, Microbial production of 1, 3-propanediol, Applied Microbiology and Biotechnology, 52 (1999), 289-297.

[2]

G. M. Cheng, L. Wang and Q. Lin and R. C. Loxton, Robust optimal control of a microbial batch culture process, Journal of Optimization Theory and Applications, 167 (2015), 342-362. doi: 10.1007/s10957-014-0654-z.

[3]

X. Y. Dang, The calculation of ph of some kinds solution, Journal of Gansu Normal Colleges, 2 (2007), 45-48.

[4]

C. X. Gao, E. M. Feng, Z. T. Wang and Z. L. Xiu, Nonlinear dynamical systems of biodissimilation of glycerol to 1,3-propanediol and their optimal controls, Journal of Industrial and Management Optimization, 1 (2005), 377-388. doi: 10.3934/jimo.2005.1.377.

[5]

C. Gao, K. Li, E. Feng and Z. Xiu, Nonlinear impulsive system of fed-batch culture in fermentative production and its properties, Chaos, Solitons & Fractals, 28 (2006), 271-277. doi: 10.1016/j.chaos.2005.05.027.

[6]

C. Gao, E. Feng, Z. Wang and Z. Xiu, Nonlinear dynamical systems of bio-dissimilation of glycerol to 1, 3-propanediol and their optimal controls, Journal of Industrial and Management Optimization, 1 (2005), 377-384. doi: 10.3934/jimo.2005.1.377.

[7]

J. G. Gao, B. Y. Shen, E. M. Feng and Z. L. Xiu, Modelling and optimal control for an impulsive dynamical system in microbial fed-batch culture, Computational and Applied Mathematics, 32 (2013), 275-290. doi: 10.1007/s40314-013-0012-z.

[8]

J. G. Gao, X. Y. Zhao, E. M. Feng and Z. L. Xiu, Modelling and parameter identification for a hybrid dynamical system in microbial fed-batch culture, International Journal of Computer Mathematics, DOI:10.1080/00207160.2014.998656, 2015. doi: 10.1080/00207160.2014.998656.

[9]

J. G. Gao, E. M. Feng and Z. L. Xiu, Metabolic system identification and optimization in continuous culture, International Journal of Computer Mathematics, 89 (2012), 1426-1444. doi: 10.1080/00207160.2012.689289.

[10]

H. Huang, C. S. Gong and G. T. Tsao, Production of 1, 3-propanediol by klebsiella pneumoniae, Applied biochemistry and biotechnology, 98 (2002), 687-698.

[11]

J. Kennedy and R. Eberhart, Particle swarm optimization, In Neural Networks, 1995. Proceedings, IEEE International Conference on, 4 (1995), 1942-1948.

[12]

Q. H. Li, L. Li, T. K. Tai and S. L. Xie, An improved pso-based of harmony search for complicated optimization problems, International Journal of Hybrid Information Technology, 1 (2008), 91-98.

[13]

C. Y. Liu, Z. H. Gong, E. 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.

[14]

A. P. Nikolai, A. P. Viktor, M. S. Anatolii and V. S. Natalia, Differential Equations with Impulse Effects, De Gruyter, 2010. doi: 10.1515/9783110218176.

[15]

S. J. Pirt et al, Principles of Microbe and Cell Cultivation, Blackwell Scientific Publications, Springer Verlag, New York, 1975.

[16]

S. Sattayasamitsathit, P. Methacanon and P. Prasertsan, Enhance 1,3-propanediol production from crude glycerol in batch and fed-batch fermentation with two-phase ph-controlled strategy, Electronic Journal of Biotechnology, 6 (2011), 1-12.

[17]

A. I. Selvakumar and K. Thanushkodi, A new particle swarm optimization solution to nonconvex economic dispatch problems, Power Systems, IEEE Transactions on, 22 (2007), 42-51.

[18]

G. Wang, E. M. Feng and Z. L. Xiu, Modeling and parameter identification of microbial biconversion in fed-batch cultures, Process Control, 18 (2008), 458-464.

[19]

H. Wang, E. Feng and Z. Xiu, Optimality condition of the nonlinear impulsive system in fed-batch fermentation, Nonlinear Analysis: Theory, Methods & Applications, 68 (2008), 12-23. doi: 10.1016/j.na.2006.10.027.

[20]

L. Wang, Determining the transport mechanism of an enzyme-catalytic complex metabolic network based on biological robustness, Bioprocess Biosyst. Eng., 36 (2013), 433-441.

[21]

L. Wang, J. Ye, E. Feng and Z. 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.

[22]

L. Wang, G. M. Cheng, E. M. Feng, Tao 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.

[23]

Z. L. Xiu, A. P. Zeng and L. J. An, Mathematical modelling of kinetics and research on multiplicity of glycerol bioconversion to 1,3-propanediol, Journal of Dalian University of Technology, 40 (2000), 428-433.

[24]

X. H. Yang, Z. F. Yang and J. Q. Li, Solving nonlinear optimal problems by hybrid accelerating genetic algorithm, Journal of Beijing Normal University, 39 (2003), 551-556.

[25]

J. Ye, Y. Zhang, E. Feng, Z. Xiu and H. Yin, Nonlinear hybrid system and parameter identification of microbial fed-batch culture with open loop glycerol input and ph logic control, Applied Mathematical Modelling, 36 (2011), 357-369. doi: 10.1016/j.apm.2011.05.059.

[26]

J. Yu, L. Xi and S. Wang, An improved particle swarm optimization for evolving feedforward artificial neural networks, Neural Processing Letters, 26 (2007), 217-231.

[27]

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.

[28]

A. P. Zeng, A. Ross, H. Biebl, C. Tag, B. Guenzel and W. D. Deckwer, Multiple product inhibition and growth modeling of clostridium butyricum and klebsiella pneumoniae in glycerol fermentation, Biotechnology and bioengineering, 44 (1994), 902-911.

[29]

A. P. Zeng and H. Biebl, Bulk chemicals from biotechnology: the case of 1, 3-propanediol production and the new trends, Tools and Applications of Biochemical Engineering Science, 15 (2002), 239-259.

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