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## Time-delay optimal control of a fed-batch production involving multiple feeds

 1 School of Mathematics and Information Science, Shandong Technology and Business University, Yantai 264005, China 2 School of Computer Science and Technology, Shandong Technology and Business University, Yantai 264005, China

* Corresponding author: Chongyang Liu

Received  March 2018 Revised  September 2018 Published  September 2019

In this paper, we consider time-delay optimal control of 1, 3-propan-ediol (1, 3-PD) fed-batch production involving multiple feeds. First, we propose a nonlinear time-delay system involving feeds of glycerol and alkali to formulate the production process. Then, taking the feeding rates of glycerol and alkali as well as the terminal time of process as the controls, we present a time-delay optimal control model subject to control and state constraints to maximize 1, 3-PD productivity. By a time-scaling transformation, we convert the optimal control problem into an equivalent problem with fixed terminal time. Furthermore, by applying control parameterization and constraint transcription techniques, we approximate the equivalent problem by a sequence of finite-dimensional optimization problems. An improved particle swarm optimization algorithm is developed to solve the resulting optimization problems. Finally, numerical results show that 1, 3-PD productivity increases considerably using the obtained optimal control strategy.

Citation: Chongyang Liu, Meijia Han. Time-delay optimal control of a fed-batch production involving multiple feeds. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020099
##### References:
 [1] B. Bao, H. Yin and E. Feng, Computation of impulsive optimal control for 1, 3-PD fed-batch culture, J. Process Contr., 34 (2015), 49-55. doi: 10.1016/j.jprocont.2015.07.005. Google Scholar [2] F. Barbirato, E. H. Himmi, T. Conte and A. Bories, 1, 3-Propanediol production by fermentation: An interesting way to valorize glycerin from the ester and ethanol industries, Ind. Crop Prod., 7 (1998), 281-289. doi: 10.1016/S0926-6690(97)00059-9. Google Scholar [3] A. Bryson and Y. Ho, Applied Optimal Control, Halsted Press, New York, 1975. Google Scholar [4] 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, J. Ind. Manag. Optim., 1 (2005), 377-388. doi: 10.3934/jimo.2005.1.377. Google Scholar [5] Z. Gong, C. Liu and Y. Wang, Optimal control of switched systems with multiple time-delays and a cost on changing control, J. Ind. Manag. Optim., 14 (2018), 183-198. doi: 10.3934/jimo.2017042. Google Scholar [6] V. K. Gorbunov, The parameterization method for optimal control problems, Comput. Math. Math. Phys., 19 (1979), 18-30. Google Scholar [7] J. He, W. Xu, Z. Feng and X. Yang, On the global optimal solution for linear quadratic problem of switched system, J. Ind. Manag. Optim., 15 (2019), 817-832. doi: 10.3934/jimo.2018072. Google Scholar [8] J. Kennedy and R. C. Eberhart, Particle swarm optimization, Proceedings of the 1995 IEEE International Conference on Neural Networks, Perth, Australia, (1995), 1942–1948. doi: 10.1109/ICNN.1995.488968. Google Scholar [9] J. V. Kurian, A new polymer platform for the future-sorona from corn derived 1, 3-propanediol, J. Polym. Environ., 13 (2005), 159-167. doi: 10.1007/s10924-005-2947-7. Google Scholar [10] B. Li, C. Xu, K. L. Teo and J. Chu, Time optimal Zermelo's navigation problem with moving and fixed obstacles, Appl. Math. Comput., 224 (2013), 866-875. doi: 10.1016/j.amc.2013.08.092. Google Scholar [11] B. Li, C. J. Yu, K. L. Teo and G. R. Duan, An exact penalty function method for continuous inequality constrained optimal control problem, J. Optimiz. Theory App., 151 (2011), 260-291. doi: 10.1007/s10957-011-9904-5. Google Scholar [12] H. Q. Li, L. Li, T. H. Kim and S. L. Xie, An improved PSO-based of harmony search for complicated optimization problems, Internat. J. Hybrid Inform. Technol., 1 (2008), 57-64. Google Scholar [13] Q. Lin, R. Loxton and K. L. Teo, The control parameterization method for nonlinear optimal control: A survey, J. Ind. Manag. Optim., 10 (2014), 275-309. doi: 10.3934/jimo.2014.10.275. Google Scholar [14] C. Liu, Sensitivity analysis and parameter identification for a nonlinear time-delay system in microbial fed-batch process, Appl. Math. Model., 38 (2014), 1448-1463. doi: 10.1016/j.apm.2013.07.039. Google Scholar [15] C. Liu, Optimal control of a switched autonomous system with time delay arising in fed-batch processes, IMA J. Appl. Math., 80 (2015), 569-584. doi: 10.1093/imamat/hxt053. Google Scholar [16] C. Liu and Z. Gong, Optimal Control of Switched Systems Arising in Fermentation Processes, Springer-Verlag, Berlin, 2014. doi: 10.1007/978-3-662-43793-3. Google Scholar [17] C. Liu, Z. Gong and E. Feng, Modelling and optimal control for nonlinear multistage dynamical system of microbial fed-batch culture, J. Ind. Manag. Optim., 5 (2009), 835-850. doi: 10.3934/jimo.2009.5.835. Google Scholar [18] C. Liu, Z. Gong, K. L. Teo, J. Sun and L. Caccetta, Robust multi-objective optimal switching control arising in 1, 3-propanediol microbial fed-batch process, Nonlinear Anal-Hybri., 25 (2017), 1-20. doi: 10.1016/j.nahs.2017.01.006. Google Scholar [19] C. Liu, Z. Gong, H. W. J. Lee and K. L. Teo, Robust bi-objective optimal control of 1, 3-propanediol microbial batch production process, J. Process Contr., 78 (2019), 170-182. doi: 10.1016/j.jprocont.2018.10.001. Google Scholar [20] C. Liu, R. Loxton and K. L. Teo, A computational method for solving time-delay optimal control problems with free terminal time, Syst. Contr. Lett., 72 (2014), 53-60. doi: 10.1016/j.sysconle.2014.07.001. Google Scholar [21] Y. Mu, D. J. Zhang, H. Teng, W. Wang and Z. L. Xiu, Microbial production of 1, 3-propanediol by Klebsiella pneumoniae using crude glycerol from biodiesel preparation, Biotechnol. Lett., 28 (2006), 1755-1759. doi: 10.1007/s10529-006-9154-z. Google Scholar [22] K. E. Parsopoulos and M. N. Vrahatis, Particle swarm optimization method in multiobjective problems, Proceedings of the 2002 ACM Symp. Appl. Comput., (2002), 603-607. doi: 10.1145/508791.508907. Google Scholar [23] R. W. H. Sargent and G. R. Sullivan, The development of an efficient optimal control package, Proceedings of the 8th IFIP Conference on Optimization Techniques, W$\ddot{{u}}$rzburg, Germany, 7 (2005), 158–168. doi: 10.1007/BFb0006520. Google Scholar [24] R. K. Saxena, P. Anand, S. Saran and J. Isar, Microbial production of 1, 3-propanediol: Recent developments and emerging opportunities, Biotechnol Adv., 27 (2009), 895-913. doi: 10.1016/j.biotechadv.2009.07.003. Google Scholar [25] J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, Springer-Verlag, New York, 1980. Google Scholar [26] K. L. Teo, G. J. Goh and K. H. Wong, A Unified Computational Approach to Optimal Control Problems, Longman Scientific & Technical, Exssex, 1991. Google Scholar [27] G. Wang, E. Feng and Z. Xiu, Vector measure as controls for explicit nonlinear impulsive system of fed-batch culture, J. Math. Anal. Appl., 351 (2009), 120-127. doi: 10.1016/j.jmaa.2008.09.054. Google Scholar [28] Z. Xiu, B. Song, L. Sun and A. Zeng, Theoretical analysis of effects of metabolic overflow and time delay on the performance and dynamic behavior of a twostage fermentation process, Biochem. Eng. J., 11 (2002), 101-109. Google Scholar [29] F. Yang, K. L. Teo, R. Loxton, V. Rehbock, B. Li, C. Yu and L. Jennings, VISUAL MISER: An efficient user-friendly visual program for solving optimal control problems, J. Ind. Manag. Optim., 12 (2016), 781-810. doi: 10.3934/jimo.2016.12.781. Google Scholar [30] J. Ye, H. Xu, E. Feng and Z. Xiu, Optimization of a fed-batch bioreactor for 1, 3-propanediol production using hybrid nonlinear optimal control, J. Process Contr., 24 (2014), 1556-1569. Google Scholar [31] C. Yu, Q. Lin, R. Loxton. K. L. Teo and G. Wang, A hybrid time-scaling transformation for time-delay optimal control problems, J. Optimiz. Theory App., 169 (2016), 876-901. doi: 10.1007/s10957-015-0783-z. Google Scholar [32] C. Yu, K. L. Teo, L. Zhang and Y. Bai, A new exact penalty function method for continuous inequality constrained optimization problems, J. Ind. Manag. Optim., 6 (2010), 895-910. doi: 10.3934/jimo.2010.6.895. Google Scholar [33] J. B. Yu, L. F. Xi and S. J. Wang, An improved particle swarm optimization for evolving feedforward artificial neural networks, Neural Process Lett., 26 (2007), 217-231. doi: 10.1007/s11063-007-9053-x. Google Scholar [34] A. P. Zeng and H. Biebl, Bulk-chemicals from biotechnology: The case of microbial production of 1, 3-propanediol and the new trends, Adv. Biochem. Eng. Biotechnol., 74 (2002), 239-259. doi: 10.1007/3-540-45736-4_11. Google Scholar

show all references

##### References:
 [1] B. Bao, H. Yin and E. Feng, Computation of impulsive optimal control for 1, 3-PD fed-batch culture, J. Process Contr., 34 (2015), 49-55. doi: 10.1016/j.jprocont.2015.07.005. Google Scholar [2] F. Barbirato, E. H. Himmi, T. Conte and A. Bories, 1, 3-Propanediol production by fermentation: An interesting way to valorize glycerin from the ester and ethanol industries, Ind. Crop Prod., 7 (1998), 281-289. doi: 10.1016/S0926-6690(97)00059-9. Google Scholar [3] A. Bryson and Y. Ho, Applied Optimal Control, Halsted Press, New York, 1975. Google Scholar [4] 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, J. Ind. Manag. Optim., 1 (2005), 377-388. doi: 10.3934/jimo.2005.1.377. Google Scholar [5] Z. Gong, C. Liu and Y. Wang, Optimal control of switched systems with multiple time-delays and a cost on changing control, J. Ind. Manag. Optim., 14 (2018), 183-198. doi: 10.3934/jimo.2017042. Google Scholar [6] V. K. Gorbunov, The parameterization method for optimal control problems, Comput. Math. Math. Phys., 19 (1979), 18-30. Google Scholar [7] J. He, W. Xu, Z. Feng and X. Yang, On the global optimal solution for linear quadratic problem of switched system, J. Ind. Manag. Optim., 15 (2019), 817-832. doi: 10.3934/jimo.2018072. Google Scholar [8] J. Kennedy and R. C. Eberhart, Particle swarm optimization, Proceedings of the 1995 IEEE International Conference on Neural Networks, Perth, Australia, (1995), 1942–1948. doi: 10.1109/ICNN.1995.488968. Google Scholar [9] J. V. Kurian, A new polymer platform for the future-sorona from corn derived 1, 3-propanediol, J. Polym. Environ., 13 (2005), 159-167. doi: 10.1007/s10924-005-2947-7. Google Scholar [10] B. Li, C. Xu, K. L. Teo and J. Chu, Time optimal Zermelo's navigation problem with moving and fixed obstacles, Appl. Math. Comput., 224 (2013), 866-875. doi: 10.1016/j.amc.2013.08.092. Google Scholar [11] B. Li, C. J. Yu, K. L. Teo and G. R. Duan, An exact penalty function method for continuous inequality constrained optimal control problem, J. Optimiz. Theory App., 151 (2011), 260-291. doi: 10.1007/s10957-011-9904-5. Google Scholar [12] H. Q. Li, L. Li, T. H. Kim and S. L. Xie, An improved PSO-based of harmony search for complicated optimization problems, Internat. J. Hybrid Inform. Technol., 1 (2008), 57-64. Google Scholar [13] Q. Lin, R. Loxton and K. L. Teo, The control parameterization method for nonlinear optimal control: A survey, J. Ind. Manag. Optim., 10 (2014), 275-309. doi: 10.3934/jimo.2014.10.275. Google Scholar [14] C. Liu, Sensitivity analysis and parameter identification for a nonlinear time-delay system in microbial fed-batch process, Appl. Math. Model., 38 (2014), 1448-1463. doi: 10.1016/j.apm.2013.07.039. Google Scholar [15] C. Liu, Optimal control of a switched autonomous system with time delay arising in fed-batch processes, IMA J. Appl. Math., 80 (2015), 569-584. doi: 10.1093/imamat/hxt053. Google Scholar [16] C. Liu and Z. Gong, Optimal Control of Switched Systems Arising in Fermentation Processes, Springer-Verlag, Berlin, 2014. doi: 10.1007/978-3-662-43793-3. Google Scholar [17] C. Liu, Z. Gong and E. Feng, Modelling and optimal control for nonlinear multistage dynamical system of microbial fed-batch culture, J. Ind. Manag. Optim., 5 (2009), 835-850. doi: 10.3934/jimo.2009.5.835. Google Scholar [18] C. Liu, Z. Gong, K. L. Teo, J. Sun and L. Caccetta, Robust multi-objective optimal switching control arising in 1, 3-propanediol microbial fed-batch process, Nonlinear Anal-Hybri., 25 (2017), 1-20. doi: 10.1016/j.nahs.2017.01.006. Google Scholar [19] C. Liu, Z. Gong, H. W. J. Lee and K. L. Teo, Robust bi-objective optimal control of 1, 3-propanediol microbial batch production process, J. Process Contr., 78 (2019), 170-182. doi: 10.1016/j.jprocont.2018.10.001. Google Scholar [20] C. Liu, R. Loxton and K. L. Teo, A computational method for solving time-delay optimal control problems with free terminal time, Syst. Contr. Lett., 72 (2014), 53-60. doi: 10.1016/j.sysconle.2014.07.001. Google Scholar [21] Y. Mu, D. J. Zhang, H. Teng, W. Wang and Z. L. Xiu, Microbial production of 1, 3-propanediol by Klebsiella pneumoniae using crude glycerol from biodiesel preparation, Biotechnol. Lett., 28 (2006), 1755-1759. doi: 10.1007/s10529-006-9154-z. Google Scholar [22] K. E. Parsopoulos and M. N. Vrahatis, Particle swarm optimization method in multiobjective problems, Proceedings of the 2002 ACM Symp. Appl. Comput., (2002), 603-607. doi: 10.1145/508791.508907. Google Scholar [23] R. W. H. Sargent and G. R. Sullivan, The development of an efficient optimal control package, Proceedings of the 8th IFIP Conference on Optimization Techniques, W$\ddot{{u}}$rzburg, Germany, 7 (2005), 158–168. doi: 10.1007/BFb0006520. Google Scholar [24] R. K. Saxena, P. Anand, S. Saran and J. Isar, Microbial production of 1, 3-propanediol: Recent developments and emerging opportunities, Biotechnol Adv., 27 (2009), 895-913. doi: 10.1016/j.biotechadv.2009.07.003. Google Scholar [25] J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, Springer-Verlag, New York, 1980. Google Scholar [26] K. L. Teo, G. J. Goh and K. H. Wong, A Unified Computational Approach to Optimal Control Problems, Longman Scientific & Technical, Exssex, 1991. Google Scholar [27] G. Wang, E. Feng and Z. Xiu, Vector measure as controls for explicit nonlinear impulsive system of fed-batch culture, J. Math. Anal. Appl., 351 (2009), 120-127. doi: 10.1016/j.jmaa.2008.09.054. Google Scholar [28] Z. Xiu, B. Song, L. Sun and A. Zeng, Theoretical analysis of effects of metabolic overflow and time delay on the performance and dynamic behavior of a twostage fermentation process, Biochem. Eng. J., 11 (2002), 101-109. Google Scholar [29] F. Yang, K. L. Teo, R. Loxton, V. Rehbock, B. Li, C. Yu and L. Jennings, VISUAL MISER: An efficient user-friendly visual program for solving optimal control problems, J. Ind. Manag. Optim., 12 (2016), 781-810. doi: 10.3934/jimo.2016.12.781. Google Scholar [30] J. Ye, H. Xu, E. Feng and Z. Xiu, Optimization of a fed-batch bioreactor for 1, 3-propanediol production using hybrid nonlinear optimal control, J. Process Contr., 24 (2014), 1556-1569. Google Scholar [31] C. Yu, Q. Lin, R. Loxton. K. L. Teo and G. Wang, A hybrid time-scaling transformation for time-delay optimal control problems, J. Optimiz. Theory App., 169 (2016), 876-901. doi: 10.1007/s10957-015-0783-z. Google Scholar [32] C. Yu, K. L. Teo, L. Zhang and Y. Bai, A new exact penalty function method for continuous inequality constrained optimization problems, J. Ind. Manag. Optim., 6 (2010), 895-910. doi: 10.3934/jimo.2010.6.895. Google Scholar [33] J. B. Yu, L. F. Xi and S. J. Wang, An improved particle swarm optimization for evolving feedforward artificial neural networks, Neural Process Lett., 26 (2007), 217-231. doi: 10.1007/s11063-007-9053-x. Google Scholar [34] A. P. Zeng and H. Biebl, Bulk-chemicals from biotechnology: The case of microbial production of 1, 3-propanediol and the new trends, Adv. Biochem. Eng. Biotechnol., 74 (2002), 239-259. doi: 10.1007/3-540-45736-4_11. Google Scholar
Optimal feeding rates of glycerol and alkali in Phs. Ⅱ-Ⅵ
Concentration changes of biomass, glycerol and 1, 3-PD with respect to fermentation time
1, 3-PD productivity changes with respect to fermentation time. Stars represent the 1, 3-PD productivity in experiment [21], and solid line denotes the 1, 3-PD productivity in this work
Phase characteristics in fed-batch process [18]
 Phase Start time (h) End time (h) Number of processes Process duration (s) Feeding Batch Feeding Batch Ⅰ 0 5.3300 0 1 0 19188 Ⅱ 5.3300 6.1078 28 28 5 95 Ⅲ 6.1078 7.1356 37 37 7 93 Ⅳ 7.1356 8.8300 61 61 8 92 Ⅴ 8.8300 12.1356 119 119 7 93 Ⅵ 12.1356 15.8300 133 133 6 94 Ⅶ 15.8300 18.0800 81 81 4 96 Ⅷ 18.0800 19.8300 63 63 3 97 Ⅸ 19.8300 23.8300 144 144 2 98 Ⅹ 23.8300 24.1633 12 12 1 99
 Phase Start time (h) End time (h) Number of processes Process duration (s) Feeding Batch Feeding Batch Ⅰ 0 5.3300 0 1 0 19188 Ⅱ 5.3300 6.1078 28 28 5 95 Ⅲ 6.1078 7.1356 37 37 7 93 Ⅳ 7.1356 8.8300 61 61 8 92 Ⅴ 8.8300 12.1356 119 119 7 93 Ⅵ 12.1356 15.8300 133 133 6 94 Ⅶ 15.8300 18.0800 81 81 4 96 Ⅷ 18.0800 19.8300 63 63 3 97 Ⅸ 19.8300 23.8300 144 144 2 98 Ⅹ 23.8300 24.1633 12 12 1 99
The kinetic parameters and critical concentrations in system (1) [14]
 $\Delta_1$ $k_1$ $m_2$ $Y_2$ $\Delta_2$ $k_2$ $m_3$ 0.8 0.28 1.927 0.0063 6.8489 17.7296 -3.2819 $Y_3$ $\Delta_3$ $k_3$ $m_4$ $Y_4$ $\Delta_4$ $k_4$ 80.6096 10.3687 15.50 -0.97 33.07 5.74 85.71 $c_1$ $c_2$ $c_3$ $c_4$ $x_{*1}$ $x_{*2}$ $x_{*3}$ 0.025 0.06 2.81 65.5226 0.01 0 0 $x_{*4}$ $x_{*5}$ $x^{*}_1$ $x^{*}_2$ $x^{*}_3$ $x^{*}_4$ $x^{*}_5$ 0 0 9 2039 1036 1026 360.9
 $\Delta_1$ $k_1$ $m_2$ $Y_2$ $\Delta_2$ $k_2$ $m_3$ 0.8 0.28 1.927 0.0063 6.8489 17.7296 -3.2819 $Y_3$ $\Delta_3$ $k_3$ $m_4$ $Y_4$ $\Delta_4$ $k_4$ 80.6096 10.3687 15.50 -0.97 33.07 5.74 85.71 $c_1$ $c_2$ $c_3$ $c_4$ $x_{*1}$ $x_{*2}$ $x_{*3}$ 0.025 0.06 2.81 65.5226 0.01 0 0 $x_{*4}$ $x_{*5}$ $x^{*}_1$ $x^{*}_2$ $x^{*}_3$ $x^{*}_4$ $x^{*}_5$ 0 0 9 2039 1036 1026 360.9
The bounds of feeding rates in Phs.Ⅱ-Ⅹ [21]
 Phases Upper bounds ($u_1$, $u_2$) Lower bounds ($u_1$, $u_2$) Ⅱ-Ⅲ 0.2524 0.1682 Ⅳ 0.2390 0.1594 Ⅴ-Ⅵ 0.2524 0.1682 Ⅶ 0.2657 0.1771 Ⅷ 0.2924 0.1949 Ⅸ-Ⅹ 0.3058 0.2038
 Phases Upper bounds ($u_1$, $u_2$) Lower bounds ($u_1$, $u_2$) Ⅱ-Ⅲ 0.2524 0.1682 Ⅳ 0.2390 0.1594 Ⅴ-Ⅵ 0.2524 0.1682 Ⅶ 0.2657 0.1771 Ⅷ 0.2924 0.1949 Ⅸ-Ⅹ 0.3058 0.2038
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