Abstract
Considering both extracellular and intracellular environments, we propose a nonlinear switched system composed of 16 modes based on the different metabolic mechanism to model the microbial continuous fermentation process of glycerol to 1,3-propanediol (1,3-PD). The existence and continuity of sensitivities of the system states with respect to the decision variables and unknown parameters are discussed. Taking the dilution rate and the feeding glycerol concentration as control variables, the sensitivity functions as constraints, an optimal control problem is presented with the concentration of 1,3-PD at the terminal time as performance index. According to sensitivity analysis, we obtain the gradient formulas of performance index and constraints with respect to the decision variables and the parameters to be identified. The sequential quadratic programming algorithm is adopted to seek a numerical solution. Numerical results indicate that under the obtained optimal dilution rate and feeding glycerol concentration, the productivity of 1,3-PD at the terminal time is increased significantly.
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References
Zeng, A.P., Ross, A., Biebl, H., Tag, C., Günzel, B., Deckwer, W.D.: Multiple product inhibition and growth modeling of Clostridium butyricum and Klebsiella pneumoniae in glycerol fermentation. Biotechnol. Bioeng. 44(8), 902–911 (1994)
Zeng, A.P., Deckwer, W.D.: A kinetic model for substrate and energy consumption of microbial growth under substrate-sufficient conditions. Biotechnol. Progress 11(1), 71–79 (1995)
Zeng, A.P.: A kinetic model for product formation of microbial and mammalian cells. Biotechnol. Bioeng. 46(4), 314–324 (2004)
Xiu, Z., Zeng, A.P., An, L.: Mathematical modeling of kinetics and research on multiplicity of glycerol bioconversion to 1,3-propanediol. J. Dalian Univ. Technol. 40(4), 428–433 (2000)
Gao, C., Feng, E., Wang, Z., Xiu, Z.: Parameters identification problem of the nonlinear dynamical system in microbial continuous cultures. Appl. Math. Comput. 169(1), 476–484 (2005)
Li, X., Feng, E., Xiu, Z.: Stability and optimal control of microorganisms in continuous culture. J. Appl. Math. Comput. 22(1), 425–434 (2006)
Gao, C., Feng, E., Wang, Z., Xiu, Z.: Nonlinear dynamical systems of bio-dissimilation of glycerol to 1,3-propanediol and their optimal controls. J. Ind. Manag. Optim. 1(3), 377–388 (2005)
Ma, Y.F., Xiu, Z.L., Sun, L.H., Feng, E.M.: Hopf bifurcation and chaos analysis of a microbial continuous culture model with time delay. Int. J. Nonlinear Sci. Numer. Simul. 7(3), 305–308 (2006)
Ye, J., Feng, E., Lian, H., Xiu, Z.: Existence of equilibrium points and stability of the nonlinear dynamical system in microbial continuous cultures. Appl. Math. Comput. 207(2), 307–318 (2009)
Lian, H., Feng, E., Li, X., Ye, J., Xiu, Z.: Oscillatory behavior in microbial continuous culture with discrete time delay. Nonlinear Anal. Real World Appl. 10(5), 2749–2757 (2009)
Sun, Y.Q., Qi, W.T., Teng, H., Xiu, Z.L., Zeng, A.P.: Mathematical modeling of glycerol fermentation by Klebsiella pneumoniae: concerning enzyme-catalytic reductive pathway and transport of glycerol and 1,3-propanediol across cell membrane. Biochem. Eng. J. 38(1), 22–32 (2008)
Zhai, J., Ye, J., Wang, L., Feng, E., Yin, H., Xiu, Z.: Pathway identification using parallel optimization for a complex metabolic system in microbial continuous culture. Nonlinear Anal. Real World Appl. 12, 2730–2741 (2011)
Wang, J., Ye, J., Wang, L., Liu, S., Feng, E.: Modelling of a nonlinear switching system in microbial continuous culture and its parameter optimization via sensitivity functions. Appl. Math. Modell. 39, 2036–2047 (2015)
Wang, J., Ye, J., Feng, E., Xiu, Z.: Modeling and parameter estimation of a nonlinear switching system in fed-batch culture with ph feedback. Appl. Math. Modell. 36, 4887–4897 (2012)
Zhai, J., Jiang, G., Ye, J.: Optimal dilution strategy for a microbial continuous culture based on the biological robustness. Numer. Algebra Control Optim. 5(1), 59–69 (2015)
Victor, J.L., Yogeshwar, S.: Computation of the gradient and sensitivity coeffients insum of squares minimization problems with differential equation models. Comput. Chem. Eng. 21, 1471–1479 (1997)
Galán, S., Feehery, W.F., Barton, P.I.: Parametric sensitivity functions for hybrid discrete/continuous systems. Appl. Numer. Math. 31, 17–47 (1999)
Barton, P.I., Lee, C.K.: Modelling, simulation, sensitivity analysis, and optimization of hybrid systems. ACM Trans. Model. Comput. Simul. 2(4), 256–289 (2002)
Lu, X.: Basics of Optimization Methods. Tongji University Publishing House, Shanghai (2003)
Zhu, X., Pu, D.: Sequential quadratic programming with a exible step acceptance strategy. Appl. Math. Modell. 36, 3992–4002 (2012)
Sun, W., Xu, C., Zhu, D.: Optimization Methods. Higher Education Press, Beijing (2004)
Acknowledgments
This work was supported by Natural Science Foundation of Shandong Province, China (Grant No. ZR2014FM029) and A Project of Shandong Province Higher Educational Science and Technology Program, China (Grant No. J14LI06).
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Hu, Q., Zhai, J. & Feng, E. Modelling and optimal control of a nonlinear switched system in microbial continuous fermentation. Japan J. Indust. Appl. Math. 33, 683–700 (2016). https://doi.org/10.1007/s13160-016-0227-7
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DOI: https://doi.org/10.1007/s13160-016-0227-7
Keywords
- Continuous fermentation
- 1, 3-propanediol
- Sensitivity functions
- Nonlinear switched system
- Switching optimal control