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Practical algorithm for stochastic optimal control problem about microbial fermentation in batch culture

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Abstract

How to add glycerol to maximize production of 1,3-propanediol (1,3-PD) is a critical problem in process control of microbial fermentation. Most of the existing works are focusing on modelling this process through deterministic-based differential equations. However, this process is not deterministic, but intrinsically stochastic considering nature of interference. Thus, it is of importance to consider stochastic microorganism. In this paper, we will modelling this process through stochastic differential equations and maximizing production of 1,3-PD is formulated as an optimal control problem subject to continuous state constraints and stochastic disturbances. A modified particle swarm algorithm through integrating the hybrid Monte Carlo sampling and path integral is proposed to solve this problem. The constraint transcription, local smoothing and time-scaling transformation are introduced to handle the continuous state constraints. Numerical results show that, by employing the obtained optimal control governed by stochastic dynamical system, 1,3-PD concentration at the terminal time can be increased compared with the previous results.

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References

  1. Yong, J.M., Zhou, X.Y.: Stochastic Controls: Hamiltonian Systems and HJB Equations. Springer, New York (1999)

    Book  MATH  Google Scholar 

  2. Gong, Z.H., Liu, C.Y., Wang, Y.J.: Optimal control of switched systems with multiple time-delays and a cost on changing control. J. Ind. Manag. Optim. (2017) https://doi.org/10.3934/jimo.2017042

  3. Kushner, H.J.: Necessary conditions for continuous parameter stochastic optimization problems. SIAM J. Control Optim. 10, 550–565 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bismut, J.M.: An introductory approach to duality in optimal stochastic control. SIAM Rev. 20(1), 62–78 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  5. Peng, S.G.: A general stochastic maximum principle for optimal control problem. SIAM J. Control. Optim. 28, 966–979 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  6. Wu, C., Teo, K.L., Wu, S.: MinCmax optimal control of linear systems with uncertainty and terminal state constraints. Automatica 49(6), 1809–1815 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chai, J., Wu, C., Zhao, C., Chi, H.L., Wang, X., Ling, B.W.K., Teo, K.L.: Reference tag supported RFID tracking using robust support vector regression and Kalman filter. Adv. Eng. Inform. 32, 1–10 (2017)

    Article  Google Scholar 

  8. Broek, B.V.D., Wiegerinck, W., Kappen, B.: Stochastic optimal control of state constrained systems. Int. J. Control 84(3), 597–615 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Li, A., Feng, E.M., Sun, X.L.: Stochastic optimal control and algorithm of the trajectory of horizontal wells. J. Comput. Appl. Math. 212(2), 419–430 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  10. Zeng, A.P., Biebl, H., Schlieker, H.: Pathway analysis of glycerol fermentation by K. pneumoniae: regulation of reducing equivalent balance and product formation. Enzyme Microbiol. Technol 15, 770–779 (1993)

    Article  Google Scholar 

  11. Xiu, Z.L.: Research progress on the production of 1,3-propanediol by fermentation. Microbiology 27, 300–302 (2000)

    Google Scholar 

  12. Gong, Z.H.: A multistage system of microbial fed-batch fermentation and its parameter identification. Math. Comput. Simul. 80, 1903–1910 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  13. Gao, C.X., Feng, E.M., Wang, Z.T., Xiu, Z.L.: Nonlinear dynamical systems of biodissimilation of glycerol to 1,3-propanediol and their optimal controls. J. Ind. Manag. Optim. 1, 377–388 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  14. Zhao, C., Wu, C., Chai, J., Wang, X., Yang, X., Lee, J.M., Kim, M.J.: Decomposition-based multi-objective firefly algorithm for RFID network planning with uncertainty. Appl. Soft Comput. 55, 549–564 (2017)

    Article  Google Scholar 

  15. Gtinzel, B.: Mikrobielle Herstellung von 1,3-Propandiol durch Clostridium butyricum und adsorptive Abtremutng von Diolen, Ph.D Dissertation, TU Braunschweig, Germany (1991)

  16. Wang, J., Ye, J.X., Yin, H.C., Feng, E.M., Wang, L.: Sensitivity analysis and identification of kinetic parameters in batch fermentation of glycerol. J. Comput. Appl. Math. 236, 2268–2276 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  17. Zhu, X., Feng, E.M.: Joint estimation in batch culture by using unscented Kalman filter. Biotechnol. Bioproc. Eng. 17, 1238–1243 (2012)

    Article  Google Scholar 

  18. Wang, J., Ye, J., Feng, E.M., Yin, H., Xiu, Z.: Modeling and identification of a nonlinear hybrid dynamical system in batch fermentation of glycerol. Math. Comput. Model. 54, 618–624 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  19. Xu, G., Liu, Y., Gao, Q.: Multi-objective optimization of a continuous bio-dissimilation process of glycerol to 1,3-propanediol. J. Biotechnol. 219, 59–71 (2016)

    Article  Google Scholar 

  20. Liu, C., Gong, Z., Teo, K.L., Sun, J., Caccetta, L.: Robust multi-objective optimal switching control arising in 1,3-propanediol microbial fed-batch process. Nonlinear Anal. Hybrid 25, 1–20 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  21. Zhang, J.X., Yuan, J.L., Feng, E.M., Yin, H.C., Xiu, Z.L.: Strong stability of a nonlinear multi-stage dynamic system in batch culture of glycerol bioconversion to 1,3-propanediol. Math. Model. Anal. 21(2), 159–173 (2016)

    Article  MathSciNet  Google Scholar 

  22. Wang, L., Xiu, Z.L., Feng, E.M.: a stochastic model of microbial bioconversion process in batch culture. Int. J. Chem. React. Eng. 2011(9), A82 (2011)

    Google Scholar 

  23. Wang, L., Feng, E.M., Xiu, Z.L.: Modeling nonlinear stochastic kinetic system and stochastic optimal control of microbial bioconversion process in batch culture. Nonlinear Anal.-Model. 18(1), 99–111 (2013)

    MathSciNet  MATH  Google Scholar 

  24. Wang, L., Ye, J.X., Feng, E.M., Xiu, Z.L.: An improved model for multistage simulation of glycerol fermentation in batch culture and its parameter identification. Nonlinear Anal.-Hybrid Syst. 3(4), 455–462 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  25. Lin, Q., Loxton, R., Teo, K.L.: The control parameterization method for nonlinear optimal control: a survey. J. Ind. Manag. Optim. 10, 275–309 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  26. Teo, K.L., Goh, C.J., Wong, K.H.: A Unified Computational Approach to Optimal Control Problems. Long Scientific Technical, Essex (1991)

    MATH  Google Scholar 

  27. Kennedy, J., Eberhart, R.C.: Particle swarm optimization. In: Proceedings of IEEE International Conference on Neural Networks, Perth, Australia, pp. 1942–1948 (1995)

  28. Du, K.L., Swamy, M.N.S.:: Particle swarm optimization. In: Search and Optimization by Metaheuristics, pp. 153–173. Springer, London (2016)

  29. Michalewicz, Z.: A survey of constraint handling techniques in evolutionary computation methods. In: Proceedings of the 4th Annual Conference on Evolutionary Programming (pp. 135-155), MIT Press, Cambridge (1995)

  30. Cheng, G.M., Wang, L., Loxton, R., Lin, Q.: Robust optimal control of a microbial batch culture process. J. Optim. Theory Appl. 161(1), 342–362 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  31. Wang, L., Feng, E.M., Ye, J.X., Xiu, Z.L.: Modeling and Properties of Nonlinear Stochastic Dynamical System of Continuous Culture, Complex Sciences, pp. 458–466. Springer, Berlin (2009)

    Google Scholar 

  32. Higham, D.J.: An algorithm introduction to numerical simulation of stochastic differential equations. SIAM Rev. 43(3), 525–546 (2001)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grants Nos. 11371164, 11771008 and 61473326), the National Natural Science Foundation for the Youth of China (Grants Nos. 11401073 and 11501574), the Fundamental Research Funds for Central Universities in China and the Natural Science Foundation of Shandong Province in China (Grant Nos. ZR2015FM014, ZR2015AL010 and ZR2017MA005).

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Authors and Affiliations

  1. School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, Liaoning, People’s Republic of China

    Lei Wang & Jinlong Yuan

  2. Department of Mathematics, School of Science, Dalian Maritime University, Dalian, 116026, Liaoning, People’s Republic of China

    Jinlong Yuan

  3. Australasian Joint Research Centre for Building Information Modelling, School of Built Environment, Curtin University, Perth, 6845, Australia

    Changzhi Wu & Xiangyu Wang

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  1. Lei Wang
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  2. Jinlong Yuan
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  3. Changzhi Wu
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  4. Xiangyu Wang
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Correspondence to Changzhi Wu.

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Wang, L., Yuan, J., Wu, C. et al. Practical algorithm for stochastic optimal control problem about microbial fermentation in batch culture. Optim Lett 13, 527–541 (2019). https://doi.org/10.1007/s11590-017-1220-z

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  • DOI: https://doi.org/10.1007/s11590-017-1220-z

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