基于时间尺度变换的一类微生物批式流加发酵混杂切换系统的最优控制

doi:10.15960/j.cnki.issn.1007-6093.2025.04.017

摘要/Abstract

摘要：

本文研究了一类微生物批式流加发酵生产1,3-丙二醇(1,3-PD) 的过程优化问题。该生物过程本质上可刻画为一个同时具有自治切换和受控切换的混杂切换系统。本文通过时间尺度变换, 将受控切换时刻映射到固定时刻, 得到一系列具有自治切换的子系统；然后, 以甘油流加时刻和发酵终端时刻为控制变量, 以终端时刻1,3-PD生产强度最大化为目标函数, 建立最优控制模型, 利用约束转换技术处理连续状态不等式约束；最后, 构造一个竞争粒子群算法求解所提出的最优控制问题, 并比较了不同控制子周期长度下的最优控制结果。数值结果表明, 在减少甘油切换次数的情况下, 通过选取适当的甘油流加时长, 1,3-PD生产强度也可以达到较高的水平。

关键词: 最优控制, 混杂切换系统, 时间尺度变换, 竞争粒子群算法

Abstract:

This paper considers the process control of the microbial production of 1,3-propanediol in fed-culture. This microbial process is in essential described as a hybrid switching system in which both autonomous switchings and controlled switchings are involved. By using time-scaling transformation, the controlled switching times are transformed to a sequence of integer time points in the new time horizon, resulting in a system composed of subsystems with autonomous switchings. Then, taking the feeding instants and the terminal time as control variables, we formulate an optimal control model with the productivity of 1,3-propanediol as performance index. The continuous state inequality constraints in the model are handled by the constraint transformation technique. A competitive particle swarm algorithm is constructed to solve the proposed optimal control problem and the optimal control strategies under different length of controlled subperiods are discussed. Numerical results show that, even if the number of switchings is reduced, it is still possible to achieve a relatively high productivity of 1,3-propanediol by sophisticated setting of the length of glycerol feeding time.

Key words: optimal control, hybrid switching system, time-scaling transformation, competitive particle swarm algorithm

中图分类号:

O232

王佳, 黄誉之, 叶剑雄. 基于时间尺度变换的一类微生物批式流加发酵混杂切换系统的最优控制[J]. 运筹学学报（中英文）, 2025, 29(4): 219-230.

Jia WANG, Yuzhi HUANG, Jianxiong YE. Optimal control of a hybrid switching system in a class of microbial fed-batch culture via time-scaling transformation[J]. Operations Research Transactions, 2025, 29(4): 219-230.

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参考文献 21

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参数	临界值	参数	临界值	参数	临界值	参数	临界值
$ d_s $	0.02	$ Y_2 $	0.013 133 4	$ {\rm pH}^* $	6.6	$ x_{2*} $	0
$ m_2 $	0	$ Y_3 $	97.785 9	$ x_1^* $	10	$ x_{3*} $	0
$ m_3 $	$ -2.670 \;3 $	$ Y_4 $	15.383 4	$ x_2^* $	2 039	$ x_{4*} $	0
$ m_4 $	0.002 921 45	$ Y_5 $	10.733 1	$ x_3^* $	1 370	$ x_{5*} $	0
$ m_5 $	0.040 800 5	$ \Delta_2 $	12.013 9	$ x_4^* $	1 026	$ x_{6*} $	0
$ K_2^* $	105.592	$ \Delta_3 $	7.559 7	$ x_5^* $	360.96	$ x_{7*} $	2
$ K_3^* $	99.793 1	$ \Delta_4 $	0.010 352	$ x_6^* $	500
$ K_4^* $	104.997	$ \Delta_5 $	0.102 197	$ x_7^* $	5
$ K_5^* $	90.622 6	$ {\rm pH}_* $	6.4	$ x_{1*} $	0

粒子数	$ x_3/t $的平均最优值	标准差
105	43.919 1	0.057 8

$ \delta/s $	粒子数	$ x_3/t $的最优值
100	105	43.965 0
200	105	44.180 5
300	105	44.271 8
400	105	44.272 5
1 800	105	44.341 6
3 600	105	44.327 4

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