The State Feedback Control Of The Microbial Culture Process

With the development of science and technology, biomathematics has been used in many domains such as biological technology, medicine dynamics, economy, population dynamics and epidemiology. Mathematical models of differential equations play an im-portant role in describing biological dynamics. Mathematically, these models explain all kinds of biological behaviors, which allows people to understand biological complexity scientifically so that some interactions of population can be intend to control. The re-search in mathematical biology which models by normal differential equations are mainly concentrated on two branches:continuous dynamical systems and impulsive dynamical systems. Since many changes in the law of nature shows impulsive effect, impulsive differ-ential equations are suitable for the mathematical simulation of the evolutionary process in which the parameters undergo a short-term rapid change in their values. Especially, the impulsive dynamical systems are suitable for the mathematical modeling of evolutionary processes which experience a change of state abruptly owing to instantaneous perturba-tions. The presence of impulses gives the system a mixed nature, both continuous and discrete. Therefore the theory of impulsive dynamical systems is much richer than the corresponding theory of dynamical systems without impulsive effects. In this thesis, we mainly study the modelling、simulation and optimization of the microorganism culture process with feedback control. The thesis includes four chapters and the main results of this dissertation may be summarized as follows:Chapter 1, Introduction. In section 1.1, the background and the significance of the research is introduced. In section 1.2, the research status of Impulsive differential equations in the biological dynamics is reviewed briefly. In section 1.3, the preliminaries is introduced including some of the basic theory of Impulsive dynamical system.Chapter 2, The state feedback control of microbial batch culture model. In sec-tion 2.1, the feedback control of a batch culture model with constant biomass yield and Monod’s kinetics is presented. The condition for the existence and stability of a positive period-1 solution is evidenced by an analytical method. It also presents the complete expression of the period of period-1 solution. In addition, it is shown that a positive period-2 solution does not exist. In section 2.2, the feedback control of the batch culture model with variable biomass yield is presented. In section 2.2.1, Monod’s kinetics and linearly biomass yield is studied. In section 2.2.2, extended Monod’s kinetics and Sigmoid biomass yield is studied. In section 2.3, the feedback control of the batch culture model with growth-associated product inhibition is presented. For each dynamic model, the condition for the existence and stability of a positive period-1 solution are evidenced by an analytical method. It also presents the complete expression of the period of period-1 solution. In addition, it is shown that a positive period-2 solution does not exist. Fol-lowing this, simulations are given to verify the theoretical results. At last, followed by a presentation of the optimization of the bioprocess.Chapter 3, The state feedback control of microbial chemostat model. In section 3.1, the feedback control of single species microbial chemostat model is studied. In section 3.1.1, a universal chemostat model with constant biomass yield and impulsive effect is presented. In section 3.1.2, a chemostat model with sigmoid biomass yield and impulsive effect is presented. Under the two models, the condition for the existence of a positive period-1 solution are evidenced by Bendixson theorem. It also discusses the position of the period-1 solution and the existence of positive period-2 solution, which also indicates the nonexistence of the period-κ(κ≥3) solution, thus the analyzed systems are not chaos. Following this, simulations are given to verify the theoretical results. At last, followed by a presentation of the optimization of the bioprocess. In section 3.2, the feedback control of two species microbial chemostat model is studied. The existence of boundary period-1 solution is shown, and the stability of this solution is discussed by the constructing of the poincare mapping.Chapter 4, The feedback control of microbial turbidostat model. The dynamic prop-erties of the model with different specific growth rate is discussed. Followed by a presen-tation of the optimization.