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

Chongyang Liu; Meijia Han

doi:10.3934/dcdss.2020099

Article Contents

2020, Volume 13, Issue 6: 1697-1709. Doi: 10.3934/dcdss.2020099

This issue Previous Article Time-scaling transformation for optimal control problem with time-varying delay Next Article Optimal control of hybrid manufacturing systems by log-exponential smoothing aggregation

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

Chongyang Liu^1, , and
Meijia Han^2,

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
^* Corresponding author: Chongyang Liu

Received: February 28, 2018

Revised: August 31, 2018

Published: April 2020

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Abstract

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.

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Mathematics Subject Classification: Primary: 49J21, 49M37; Secondary: 34K34.

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Figure 1. Optimal feeding rates of glycerol and alkali in Phs. Ⅱ-Ⅵ

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Figure 2. Concentration changes of biomass, glycerol and 1, 3-PD with respect to fermentation time

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Figure 3. 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

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Table 1. Phase characteristics in fed-batch process [18]

Phase	Start time (h)	End time (h)	Number of processes		Process duration (s)
Phase	Start time (h)	End time (h)	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

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Table 2. 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

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Table 3. 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

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