×

Time-delay optimal control of a fed-batch production involving multiple feeds. (English) Zbl 1439.49009

Summary: 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.

MSC:

49J21 Existence theories for optimal control problems involving relations other than differential equations
49M37 Numerical methods based on nonlinear programming
34K34 Hybrid systems of functional-differential equations

Software:

Visual MISER
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] B. Bao; H. Yin; E. Feng, Computation of impulsive optimal control for 1, 3-PD fed-batch culture, J. Process Contr., 34, 49-55 (2015) · doi:10.1016/j.jprocont.2015.07.005
[2] F. Barbirato; E. H. Himmi; T. Conte; A. Bories, 1, 3-Propanediol production by fermentation: An interesting way to valorize glycerin from the ester and ethanol industries, Ind. Crop Prod., 7, 281-289 (1998) · doi:10.1016/S0926-6690(97)00059-9
[3] C. Gao; E. Feng; Z. Wang; Z. Xiu, Nonlinear dynamical systems of bio-dissimilation of glycerol to 1, 3-propanediol and their optimal controls, J. Ind. Manag. Optim., 1, 377-388 (2005) · Zbl 1076.92023 · doi:10.3934/jimo.2005.1.377
[4] C. Gao; E. Feng; Z. Wang; Z. Xiu, Nonlinear dynamical systems of bio-dissimilation of glycerol to 1, 3-propanediol and their optimal controls, J. Ind. Manag. Optim., 1, 377-388 (2005) · Zbl 1412.49011 · doi:10.3934/jimo.2005.1.377
[5] Z. Gong; C. Liu; Y. Wang, Optimal control of switched systems with multiple time-delays and a cost on changing control, J. Ind. Manag. Optim., 14, 183-198 (2018) · Zbl 0437.49021 · doi:10.3934/jimo.2017042
[6] V. K. Gorbunov, The parameterization method for optimal control problems, Comput. Math. Math. Phys., 19, 18-30 (1979) · Zbl 1438.49047 · doi:10.3934/jimo.2018072
[7] J. He; W. Xu; Z. Feng; X. Yang, On the global optimal solution for linear quadratic problem of switched system, J. Ind. Manag. Optim., 15, 817-832 (2019) · Zbl 1438.49047 · doi:10.3934/jimo.2018072
[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:10.1007/s10924-005-2947-7
[9] J. V. Kurian, A new polymer platform for the future-sorona from corn derived 1, 3-propanediol, J. Polym. Environ., 13, 159-167 (2005) · Zbl 1337.49071 · doi:10.1007/s10924-005-2947-7
[10] B. Li; C. Xu; K. L. Teo; J. Chu, Time optimal Zermelo’s navigation problem with moving and fixed obstacles, Appl. Math. Comput., 224, 866-875 (2013) · Zbl 1251.49026 · doi:10.1016/j.amc.2013.08.092
[11] B. Li; C. J. Yu; K. L. Teo; G. R. Duan, An exact penalty function method for continuous inequality constrained optimal control problem, J. Optimiz. Theory App., 151, 260-291 (2011) · Zbl 1251.49026 · doi:10.1007/s10957-011-9904-5
[12] H. Q. Li; L. Li; T. H. Kim; S. L. Xie, An improved PSO-based of harmony search for complicated optimization problems, Internat. J. Hybrid Inform. Technol., 1, 57-64 (2008) · Zbl 1276.49025 · doi:10.3934/jimo.2014.10.275
[13] Q. Lin; R. Loxton; K. L. Teo, The control parameterization method for nonlinear optimal control: A survey, J. Ind. Manag. Optim., 10, 275-309 (2014) · Zbl 1427.92042 · doi:10.3934/jimo.2014.10.275
[14] C. Liu, Sensitivity analysis and parameter identification for a nonlinear time-delay system in microbial fed-batch process, Appl. Math. Model., 38, 1448-1463 (2014) · Zbl 1315.93039 · doi:10.1016/j.apm.2013.07.039
[15] C. Liu, Optimal control of a switched autonomous system with time delay arising in fed-batch processes, IMA J. Appl. Math., 80, 569-584 (2015) · Zbl 1305.49002 · doi:10.1093/imamat/hxt053
[16] C. Liu and Z. Gong, Optimal Control of Switched Systems Arising in Fermentation Processes, Springer-Verlag, Berlin, 2014. · Zbl 1186.49024 · doi:10.3934/jimo.2009.5.835
[17] C. Liu; Z. Gong; E. Feng, Modelling and optimal control for nonlinear multistage dynamical system of microbial fed-batch culture, J. Ind. Manag. Optim., 5, 835-850 (2009) · Zbl 1378.49043 · doi:10.3934/jimo.2009.5.835
[18] C. Liu; Z. Gong; K. L. Teo; J. Sun; L. Caccetta, Robust multi-objective optimal switching control arising in 1, 3-propanediol microbial fed-batch process, Nonlinear Anal-Hybri., 25, 1-20 (2017) · Zbl 1378.49043 · doi:10.1016/j.nahs.2017.01.006
[19] C. Liu; Z. Gong; H. W. J. Lee; K. L. Teo, Robust bi-objective optimal control of 1, 3-propanediol microbial batch production process, J. Process Contr., 78, 170-182 (2019) · Zbl 1302.49042 · doi:10.1016/j.jprocont.2018.10.001
[20] C. Liu; R. Loxton; K. L. Teo, A computational method for solving time-delay optimal control problems with free terminal time, Syst. Contr. Lett., 72, 53-60 (2014) · Zbl 1302.49042 · doi:10.1016/j.sysconle.2014.07.001
[21] Y. Mu; D. J. Zhang; H. Teng; W. Wang; Z. L. Xiu, Microbial production of 1, 3-propanediol by Klebsiella pneumoniae using crude glycerol from biodiesel preparation, Biotechnol. Lett., 28, 1755-1759 (2006) · doi:10.1007/s10529-006-9154-z
[22] K. E. Parsopoulos; M. N. Vrahatis, Particle swarm optimization method in multiobjective problems, Proceedings of the 2002 ACM Symp. Appl. Comput., 603-607 (2002) · doi:10.1145/508791.508907
[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. · Zbl 0385.49018 · doi:10.1016/j.biotechadv.2009.07.003
[24] R. K. Saxena; P. Anand; S. Saran; J. Isar, Microbial production of 1, 3-propanediol: Recent developments and emerging opportunities, Biotechnol Adv., 27, 895-913 (2009) · Zbl 0423.65002 · doi:10.1016/j.biotechadv.2009.07.003
[25] J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, Springer-Verlag, New York, 1980. · Zbl 0747.49005
[26] K. L. Teo, G. J. Goh and K. H. Wong, A Unified Computational Approach to Optimal Control Problems, Longman Scientific & Technical, Exssex, 1991. · Zbl 1152.92030 · doi:10.1016/j.jmaa.2008.09.054
[27] G. Wang; E. Feng; Z. Xiu, Vector measure as controls for explicit nonlinear impulsive system of fed-batch culture, J. Math. Anal. Appl., 351, 120-127 (2009) · Zbl 1152.92030 · doi:10.1016/j.jmaa.2008.09.054
[28] Z. Xiu; B. Song; L. Sun; 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, 101-109 (2002) · Zbl 1325.49038 · doi:10.3934/jimo.2016.12.781
[29] F. Yang; K. L. Teo; R. Loxton; V. Rehbock; B. Li; C. Yu; L. Jennings, VISUAL MISER: An efficient user-friendly visual program for solving optimal control problems, J. Ind. Manag. Optim., 12, 781-810 (2016) · Zbl 1325.49038 · doi:10.3934/jimo.2016.12.781
[30] J. Ye; H. Xu; E. Feng; Z. Xiu, Optimization of a fed-batch bioreactor for 1, 3-propanediol production using hybrid nonlinear optimal control, J. Process Contr., 24, 1556-1569 (2014) · Zbl 1342.49003 · doi:10.1007/s10957-015-0783-z
[31] C. Yu; Q. Lin; R. Loxton. K. L. Teo; G. Wang, A hybrid time-scaling transformation for time-delay optimal control problems, J. Optimiz. Theory App., 169, 876-901 (2016) · Zbl 1203.90010 · doi:10.1007/s10957-015-0783-z
[32] C. Yu; K. L. Teo; L. Zhang; Y. Bai, A new exact penalty function method for continuous inequality constrained optimization problems, J. Ind. Manag. Optim., 6, 895-910 (2010) · Zbl 1203.90010 · doi:10.3934/jimo.2010.6.895
[33] J. B. Yu; L. F. Xi; S. J. Wang, An improved particle swarm optimization for evolving feedforward artificial neural networks, Neural Process Lett., 26, 217-231 (2007) · doi:10.1007/s11063-007-9053-x
[34] A. P. Zeng; H. Biebl, Bulk-chemicals from biotechnology: The case of microbial production of 1, 3-propanediol and the new trends, Adv. Biochem. Eng. Biotechnol., 74, 239-259 (2002) · doi:10.1007/3-540-45736-4_11
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.