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Preface: Advances in theory and real world applications of control and dynamic optimization
June  2020, 13(6): 1623-1635. doi: 10.3934/dcdss.2020095

## A linear optimal feedback control for producing 1, 3-propanediol via microbial fermentation

 1 School of Maritime Economics and Management, Dalian Maritime University, Dalian 116026, China 2 Collaborative Innovation Center for Transport Studies@Dalian Maritime University, Dalian 116026, China 3 School of Mathematical Science, Dalian University of Technology, Dalian, Liaoning 116024, China 4 Department of Mathematics, Loyola Marymount University, Los Angeles CA 90045, USA

* Corresponding author: Lei Wang

Received  January 2018 Revised  April 2018 Published  September 2019

Fund Project: 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(Grants No. DUT19LK37), and the Natural Science Foundation of Shandong Province in China(Grants Nos. ZR2015FM014, ZR2015AL010 and ZR2017MA005)

In this paper, we consider a multistage feedback control strategy for the production of 1, 3-propanediol(1, 3-PD) in microbial fermentation. The feedback control strategy is widely used in industry, and to the best of our knowledge, this is the first time it is applied to 1, 3-PD. The feedback control law is assumed to be linear of the concentrations of biomass and glycerol, and the coefficients in the controller are continuous. A multistage feedback control law is obtained by using the control parameterization method on the coefficient functions. Then, the optimal control problem can be transformed into an optimal parameter selection problem. The time horizon is partitioned adaptively. The corresponding gradients are derived, and finally, our numerical results indicate that the strategy is flexible and efficient.

Citation: Honghan Bei, Lei Wang, Yanping Ma, Jing Sun, Liwei Zhang. A linear optimal feedback control for producing 1, 3-propanediol via microbial fermentation. Discrete and Continuous Dynamical Systems - S, 2020, 13 (6) : 1623-1635. doi: 10.3934/dcdss.2020095
##### References:
 [1] T. Atkins and M. Escudier, Linear Feedback Control, A Dictionary of Mechanical Engineering, 2013. [2] M. E. Fisher, W. J. Grantham and K. L. Teo, Neighbouring extremals for nonlinear systems with control constraints, Dyn. Control, 5 (1995), 225-240.  doi: 10.1007/BF01968675. [3] C. X. Gao, Z. T. Wang, E. M. Feng and Z. L. Xiu, Parameter identification and optimization of process for bio-dissimilation of glycerol to 1, 3-propanediol in batch culture, J. Dalian Univ. Technol, 46 (2006), 771-774. [4] K. K. Gao, X. Zhang, E. M. Feng and Z. L. Xiu, Sensitivity analysis and parameter identification of nonlinear hybrid systems for glycerol transport mechanisms in continuous culture, Journal of Theoretical Biology, 347 (2014), 137-143.  doi: 10.1016/j.jtbi.2013.12.025. [5] C. X. Gao, Z. T. Wang, E. M. Feng and Z. L. Xiu, Nonlinear dynamical systems of bio-dissimilation of glycerol to 1, 3-propanediol and their optimal controls, J. Ind. Manag. Optim, 1 (2005), 377-388.  doi: 10.3934/jimo.2005.1.377. [6] C. Jiang, K. L. Teo, R. Loxton and G. R. Duan, A neighbouring extremal solution for an optimal switched impulsive control problem, Journal of Industrial and Management Optimization, 8 (2012), 591-609.  doi: 10.3934/jimo.2012.8.591. [7] X. H. Li, E. M. Feng and Z. L. Xiu, Optimal control and property of nonlinear dynamic system for microorganism in batch culture, Gongcheng Shuxue Xuebao, 23 (2006), 7-12. [8] Q. Lin, R. Loxton, K. L. Teo and Y. H. Wu, Optimal feedback control for dynamic systems with state constraints: An exact penalty approach, Optimization Letters, 8 (2014), 1535-1551.  doi: 10.1007/s11590-013-0657-y. [9] Q. Lin, R. Loxton and K. L. Teo, The control parameterization method for nonlinear optimal control: A survey, Journal of Industrial and Management Optimization, 10 (2014), 275-309.  doi: 10.3934/jimo.2014.10.275. [10] Q. Lin, R. Loxton, K. L. Teo, Y. H. Wu and C. J. Yu, A new exact penalty method for semi-infinite programming problems, Journal of Computational and Applied Mathematics, 261 (2014), 271-286.  doi: 10.1016/j.cam.2013.11.010. [11] C. Y. Liu, Z. H. Gong and K. L. Teo, Robust parameter estimation for nonlinear multistage time-delay systems with noisy measurement data, Applied Mathematical Modelling, 53 (2018), 353-368.  doi: 10.1016/j.apm.2017.09.007. [12] C. Y. Liu, Z. H. Gong and H. C. Yin, Modelling and optimal control for nonlinear multistage dynamical system of microbial fed-batch culture, Rocky Mountain Journal of Mathematics, 5 (2009), 835-850.  doi: 10.3934/jimo.2009.5.835. [13] R. Loxton, K. L. Teo and V. Rehbock, Optimal control problems with multiple characteristic time points in the objective and constraints, Automatica, 44 (2008), 2923-2929.  doi: 10.1016/j.automatica.2008.04.011. [14] R. C. Loxton, Optimal control problems involving constrained, switched, and delay system, Ph. D Thesis, Curtin University of Technology in Perth, 2010. [15] R. Loxton, Q. Lin, V. Rehbock and K. L. Teo, Control parameterization for optimal control problems with continuous inequality constraints: New convergence results, Numer. Algebra Control Optim., 2 (2012), 571-599.  doi: 10.3934/naco.2012.2.571. [16] R. Loxton, K. L. Teo, V. Rehbock and K. F. C. Yiu, Optimal control problems with a continuous inequality constraint on the state and the control, Automatica, 45 (2009), 2250-2257.  doi: 10.1016/j.automatica.2009.05.029. [17] R. Loxton, K. L. Teo and V. Rehbock, Robust suboptimal control of nonlinear systems, Appl. Math. Comput, 217 (2011), 6566-6576.  doi: 10.1016/j.amc.2011.01.039. [18] D. G. Luenberger and Y. Ye, Linear and Nonlinear Programming, Springer, 2008. [19] J. J. Lv, L. P. Pang and E. M. Feng, Asymptotical stability of a nonlinear non-differentiable dynamic system in microbial continuous cultures, Advances in Difference Equations, 2017 (2017), 1-14.  doi: 10.1186/s13662-017-1288-x. [20] J. Nocedal and S. J. Wright, Numerical Optimization, , Springer series in operations research, 2006. [21] R. Steuer, Computational approaches to the topology, stability and dynamics of metabolic networks, Phytochemistry, 68 (2007), 2139-2151.  doi: 10.1016/j.phytochem.2007.04.041. [22] Y. Q. Sun, W. T. Qi and H. Teng, Mathematical modeling of glycerol fermentation by Klebssiella pneumoniae: concerning enzyme-catalytic reductive pathway and transport of glycerol and 1, 3-propanediol across cell membrane, Biochem. Eng. J., 38 (2008), 22-32. [23] K. L. Teo and C. J. Goh, A simple computational procedure for optimization problems with functinal inequality constraints, IEEE Transactions on Automatic Control, 32 (1987), 940-941. [24] K. L. Teo, K. H. Wong and C. J. Goh, A Unified Computational Approach to Optimal Control Problems, Long Scientific Technical, Essex, 1991. [25] L. Wang, J. L. Yuan, C. Z. Wu and X. Y. Wang, Practical algorithm for stochastic optimal control problem about microbial fermentation in batch culture, Optim Lett, 13 (2019), 527-541.  doi: 10.1007/s11590-017-1220-z. [26] L. Wang, E. M. Feng and Z. L. Xiu, Modeling nonlinear stochastic kinetic system and stochastic optimal control of microbial bioconversion process in batch culture, Nonliear Anal. Model. Control, 18 (2013), 99-111. [27] L. W. Wang, X. G. Liu and Z. Y. Zhang, A new sensitivity-based adaptive control vector parameterization approach for dynamic optimization of bioprocesses, Bioprocess and Biosystems Engineering, 40 (2017), 181-189.  doi: 10.1007/s00449-016-1685-7. [28] Z. L. Xiu, A. P. Zeng and L. J. An, Mathematical simulation and multi steady state study of glycerol biological disproportionation process, Journal of Dalian University of Technology, 40 (2000), 428-433. [29] A. P. Zeng, A. Rose and H. Biebl, Multiple product inhibition and growth modeling of Clostridium butyricum and Klebsiella pneumonia in fermentation, Biotechnology and Bioengineering, 44 (1994), 902-911. [30] Y. D. Zhang, E. M. Feng and Z. L. Xiu, Robust analysis of hybrid dynamical systems for 1, 3-propanediol transport mechanisms in microbial continuous fermentation, Mathematical and Computer Modelling, 54) (2011), 3164-3171.  doi: 10.1016/j.mcm.2011.08.010. [31] P. P. Zhang, X. G. Liu and L. Ma, Optimal Control Vector Parameterization Approach with a Hybrid Intelligent Algorithm for Nonlinear Chemical Dynamic Optimization Problems, Chemical Engineering & Technology, 38 (2015), 2067-2078.  doi: 10.1002/ceat.201400796.

show all references

##### References:
 [1] T. Atkins and M. Escudier, Linear Feedback Control, A Dictionary of Mechanical Engineering, 2013. [2] M. E. Fisher, W. J. Grantham and K. L. Teo, Neighbouring extremals for nonlinear systems with control constraints, Dyn. Control, 5 (1995), 225-240.  doi: 10.1007/BF01968675. [3] C. X. Gao, Z. T. Wang, E. M. Feng and Z. L. Xiu, Parameter identification and optimization of process for bio-dissimilation of glycerol to 1, 3-propanediol in batch culture, J. Dalian Univ. Technol, 46 (2006), 771-774. [4] K. K. Gao, X. Zhang, E. M. Feng and Z. L. Xiu, Sensitivity analysis and parameter identification of nonlinear hybrid systems for glycerol transport mechanisms in continuous culture, Journal of Theoretical Biology, 347 (2014), 137-143.  doi: 10.1016/j.jtbi.2013.12.025. [5] C. X. Gao, Z. T. Wang, E. M. Feng and Z. L. Xiu, Nonlinear dynamical systems of bio-dissimilation of glycerol to 1, 3-propanediol and their optimal controls, J. Ind. Manag. Optim, 1 (2005), 377-388.  doi: 10.3934/jimo.2005.1.377. [6] C. Jiang, K. L. Teo, R. Loxton and G. R. Duan, A neighbouring extremal solution for an optimal switched impulsive control problem, Journal of Industrial and Management Optimization, 8 (2012), 591-609.  doi: 10.3934/jimo.2012.8.591. [7] X. H. Li, E. M. Feng and Z. L. Xiu, Optimal control and property of nonlinear dynamic system for microorganism in batch culture, Gongcheng Shuxue Xuebao, 23 (2006), 7-12. [8] Q. Lin, R. Loxton, K. L. Teo and Y. H. Wu, Optimal feedback control for dynamic systems with state constraints: An exact penalty approach, Optimization Letters, 8 (2014), 1535-1551.  doi: 10.1007/s11590-013-0657-y. [9] Q. Lin, R. Loxton and K. L. Teo, The control parameterization method for nonlinear optimal control: A survey, Journal of Industrial and Management Optimization, 10 (2014), 275-309.  doi: 10.3934/jimo.2014.10.275. [10] Q. Lin, R. Loxton, K. L. Teo, Y. H. Wu and C. J. Yu, A new exact penalty method for semi-infinite programming problems, Journal of Computational and Applied Mathematics, 261 (2014), 271-286.  doi: 10.1016/j.cam.2013.11.010. [11] C. Y. Liu, Z. H. Gong and K. L. Teo, Robust parameter estimation for nonlinear multistage time-delay systems with noisy measurement data, Applied Mathematical Modelling, 53 (2018), 353-368.  doi: 10.1016/j.apm.2017.09.007. [12] C. Y. Liu, Z. H. Gong and H. C. Yin, Modelling and optimal control for nonlinear multistage dynamical system of microbial fed-batch culture, Rocky Mountain Journal of Mathematics, 5 (2009), 835-850.  doi: 10.3934/jimo.2009.5.835. [13] R. Loxton, K. L. Teo and V. Rehbock, Optimal control problems with multiple characteristic time points in the objective and constraints, Automatica, 44 (2008), 2923-2929.  doi: 10.1016/j.automatica.2008.04.011. [14] R. C. Loxton, Optimal control problems involving constrained, switched, and delay system, Ph. D Thesis, Curtin University of Technology in Perth, 2010. [15] R. Loxton, Q. Lin, V. Rehbock and K. L. Teo, Control parameterization for optimal control problems with continuous inequality constraints: New convergence results, Numer. Algebra Control Optim., 2 (2012), 571-599.  doi: 10.3934/naco.2012.2.571. [16] R. Loxton, K. L. Teo, V. Rehbock and K. F. C. Yiu, Optimal control problems with a continuous inequality constraint on the state and the control, Automatica, 45 (2009), 2250-2257.  doi: 10.1016/j.automatica.2009.05.029. [17] R. Loxton, K. L. Teo and V. Rehbock, Robust suboptimal control of nonlinear systems, Appl. Math. Comput, 217 (2011), 6566-6576.  doi: 10.1016/j.amc.2011.01.039. [18] D. G. Luenberger and Y. Ye, Linear and Nonlinear Programming, Springer, 2008. [19] J. J. Lv, L. P. Pang and E. M. Feng, Asymptotical stability of a nonlinear non-differentiable dynamic system in microbial continuous cultures, Advances in Difference Equations, 2017 (2017), 1-14.  doi: 10.1186/s13662-017-1288-x. [20] J. Nocedal and S. J. Wright, Numerical Optimization, , Springer series in operations research, 2006. [21] R. Steuer, Computational approaches to the topology, stability and dynamics of metabolic networks, Phytochemistry, 68 (2007), 2139-2151.  doi: 10.1016/j.phytochem.2007.04.041. [22] Y. Q. Sun, W. T. Qi and H. Teng, Mathematical modeling of glycerol fermentation by Klebssiella pneumoniae: concerning enzyme-catalytic reductive pathway and transport of glycerol and 1, 3-propanediol across cell membrane, Biochem. Eng. J., 38 (2008), 22-32. [23] K. L. Teo and C. J. Goh, A simple computational procedure for optimization problems with functinal inequality constraints, IEEE Transactions on Automatic Control, 32 (1987), 940-941. [24] K. L. Teo, K. H. Wong and C. J. Goh, A Unified Computational Approach to Optimal Control Problems, Long Scientific Technical, Essex, 1991. [25] L. Wang, J. L. Yuan, C. Z. Wu and X. Y. Wang, Practical algorithm for stochastic optimal control problem about microbial fermentation in batch culture, Optim Lett, 13 (2019), 527-541.  doi: 10.1007/s11590-017-1220-z. [26] L. Wang, E. M. Feng and Z. L. Xiu, Modeling nonlinear stochastic kinetic system and stochastic optimal control of microbial bioconversion process in batch culture, Nonliear Anal. Model. Control, 18 (2013), 99-111. [27] L. W. Wang, X. G. Liu and Z. Y. Zhang, A new sensitivity-based adaptive control vector parameterization approach for dynamic optimization of bioprocesses, Bioprocess and Biosystems Engineering, 40 (2017), 181-189.  doi: 10.1007/s00449-016-1685-7. [28] Z. L. Xiu, A. P. Zeng and L. J. An, Mathematical simulation and multi steady state study of glycerol biological disproportionation process, Journal of Dalian University of Technology, 40 (2000), 428-433. [29] A. P. Zeng, A. Rose and H. Biebl, Multiple product inhibition and growth modeling of Clostridium butyricum and Klebsiella pneumonia in fermentation, Biotechnology and Bioengineering, 44 (1994), 902-911. [30] Y. D. Zhang, E. M. Feng and Z. L. Xiu, Robust analysis of hybrid dynamical systems for 1, 3-propanediol transport mechanisms in microbial continuous fermentation, Mathematical and Computer Modelling, 54) (2011), 3164-3171.  doi: 10.1016/j.mcm.2011.08.010. [31] P. P. Zhang, X. G. Liu and L. Ma, Optimal Control Vector Parameterization Approach with a Hybrid Intelligent Algorithm for Nonlinear Chemical Dynamic Optimization Problems, Chemical Engineering & Technology, 38 (2015), 2067-2078.  doi: 10.1002/ceat.201400796.
The evolution of the time grids in 100 hours
The changes of control parameters, $\xi_1(t)$ and $\xi_2(t)$, over time
The change of concentration of biomass, glycerol, 1, 3-PD, acetate and ethanol in continuous culture process in 100 hours
The values of some parameters used in Eqs. (1) - (4)
 $i$ $m_i$ $Y_i$ $\Delta{q_i}$ $k_i$ $b_i$ $c_i$ 1 - - - - 0.025 0.06 2 2.20 0.0082 28.58 11.43 5.18 50.45 3 -2.69 67.69 26.59 15.50 - - 4 -0.97 33.07 5.74 85.71 - -
 $i$ $m_i$ $Y_i$ $\Delta{q_i}$ $k_i$ $b_i$ $c_i$ 1 - - - - 0.025 0.06 2 2.20 0.0082 28.58 11.43 5.18 50.45 3 -2.69 67.69 26.59 15.50 - - 4 -0.97 33.07 5.74 85.71 - -
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