Abstract

Estimates are given of the stability of stationary states of a process of microbiological synthesis with the use of mathematical models of kinetics based on an unstructured approach. The object of research is a continuous biotechnological process, in which, in addition to biomass, the target product is obtained. The stability was assessed with the Hurwitz matrix. The process uses a component that forms an additional amount of the main substrate. Five stages of the solution of the problem are given: the description of a mathematical model that takes into account biological and technological limitations; the determination of stationary states, which requires assessment of the stability conditions and the calculation of process indices for the adopted stationary states; the formation of a system of equations of the first approximation (a system of linear differential equations of the first order with constant coefficients) and estimation of the coefficients corresponding to the accepted stationary states; the formation of the Hurwitz matrix and the calculation of its elements; and the calculation of indices of necessary and sufficient stability conditions based on the Hurwitz matrix. Three options are given for the calculation of elements of the Hurwitz matrix: the obtainment of the characteristic equation from a system of four differential equations of the first approximation. The calculated relations for the matrix coefficients are determined and the coefficients are calculated as a result of the solution of the equation according to the system of the first approximation with the introduction of eigenvalues. The stability of three stationary states is estimated for the flow rate D = 0.1, 0.2 and 0.3 h^–1.

中文翻译：

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用于生产乳酸的生物技术过程的非平稳状态的稳定性

摘要

使用基于非结构化方法的动力学数学模型，对微生物合成过程的稳态稳定性进行了估计。研究的对象是一个连续的生物技术过程，在这个过程中，除了生物质之外，还获得了目标产品。使用 Hurwitz 矩阵评估稳定性。该工艺使用了一种成分，该成分形成了额外数量的主基板。给出了解决问题的五个阶段： 描述考虑到生物和技术限制的数学模型；确定静止状态，这需要评估稳定性条件并计算所采用静止状态的过程指数；形成一阶近似方程组（具有常数系数的一阶线性微分方程组）并估计对应于已接受的静止状态的系数；Hurwitz 矩阵的形成及其元素的计算；以及基于Hurwitz矩阵的充分必要稳定条件指标的计算。为计算 Hurwitz 矩阵的元素提供了三个选项：从一阶近似的四个微分方程的系统中获得特征方程。矩阵系数的计算关系被确定，并且系数被计算为根据引入特征值的一阶近似系统解方程的结果。D = 0.1、0.2 和 0.3 h ^–1。