Abstract
The variant of solving the problem of optimal control of biomass selection from the apparatus using the Pontryagin maximum principle is presented in the article. The maximum productivity of the device as an optimality criterion was chosen. On a finite-dimensional space the Pontryagin function, the Hamiltonian function, and the end function were defined. To describe the dynamics of the development of the yeast population, the Ferhulst-Gause equation was used. It is proved that optimal control (in the chosen criterion) is possible with the found ratios of the coefficient of natural growth e and the coefficient β taking into account the intraspecies competition of microorganisms. To determine the optimal concentration of microorganisms and the biomass selection value, it is necessary to determine the numerical values of the coefficients of the model e and β during the cultivation process. The method for determining the model coefficients and the optimal biomass selection from the apparatus was presented. The algorithm and the system for optimal control of the selection of biomass from the apparatus were developed. They differ in that here the found theoretical relationships of the named coefficients are used. To study the optimal control system, a computer experiment was carried out using experimental data on the hourly accumulation of microorganisms taken at the Voronezh yeast plant. The data were obtained for a 13 hour culture process with selection of the culture medium after 6 hours. The selection is conducted in such a way that the accumulation (yeast concentration) of X during sampling remains unchanged with a constant culture medium selection. That is, the volume of biomass equal to the volume of the medium supplied with the feeding is taken. The graphs of the hourly change in yeast concentration and the dynamics of selection were obtained. The results analysis shows that the use of such a system in the process of controlling yeast production makes it possible to shorten the cultivation time by 1 hour.
Highlights
The variant of solving the problem of optimal control of biomass selection from the apparatus using the Pontryagin maximum principle is presented in the article
It is proved that optimal control
the coefficient β taking into account the intraspecies competition of microorganisms
Summary
Представлен вариант решения задачи оптимального управления отбором биомассы из аппарата с использованием принципа максимума Понтрягина. Для определения оптимального значения концентрации микроорганизмов и величины отбора биомассы необходимо по ходу процесса культивирования определять численные значения коэффициентов модели ε и β. Представлен метод определения коэффициентов модели и оптимального значения отбора биомассы из аппарата. Разработаны алгоритм и система оптимального управления отбором биомассы из аппарата, отличающаяся тем, что в ней используются найденные теоретические соотношения названных коэффициентов. Для исследования системы оптимального управления был проведен машинный эксперимент с использованием опытных данных о почасовом накоплении микроорганизмов, снятых на Воронежском дрожжевом заводе. Данные были получены для 13 часового процесса культивирования с отбором культуральной среды после 6 часа. Что накопление (концентрация дрожжей) Х во время отбора остается неизменной при постоянном отборе культуральной среды. Ключевые слова: оптимальное управление, принцип максимума Понтрягина, отбор биомассы, уравнение Ферхюльста– Гаузе, система оптимального управления
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