Algorithm of using Hurwitz conditions to study stability of steady states of continuous fermentation processes with nonlinear kinetics

The authors of the article produce the algorithm to evaluate the stability of the steady states for microbiological synthesis using mathematical models for kinetics based on the unstructured approach. The object under study is a continuous biotechnological process aiming to obtain a target product, in addition to biomass. The estimation of stability is performed using Hurwitz matrix. In this process the use of raw materials containing components for additional producing of the primary substrate are possible. The algorithm includes five steps: formulation of the mathematical model that takes into account biological and technological limits; definition of steady states (one or several) for which it is necessary to evaluate conditions of stability and to calculate process parameters for the adopted steady states resulted from solving a system of nonlinear algebraic equations of the mathematical model; formulation of the equations of the first approximation (the system of first-order linear differential equations with constant coefficients) and evaluation of the coefficients corresponding to the adopted steady states; formation of Hurwitz matrix and calculation of its items; calculation of necessary and sufficient conditions for stability using Hurwitz matrix. Two methods of calculating Hurwitz matrix have been shown. The first method is implemented by transforming the system of four differential equations of the first approximation into one fourth-order ordinary differential equation. In the course of transformation there were determined calculated ratios for coefficients of the matrix. In the second method the coefficients are calculated by solving vector equation for the system of first approximation with the introduction of eigen values. The algorithm has been used for evaluating steady states of microbiological synthesis for lactic acid production from raw materials containing the component reproducing the main substrate. The estimation has been carried out for two steady states, one of which is optimal. For numerical values of the constants (from the literature) both steady states were stable.