微生物培养的状态反馈控制
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摘要
随着科学和技术的发展,生物数学已经被广泛的应用于各个领域,人们用它能够从数学的角度解释生物学行为和现象,从而达到对某些生物的相互作用进行有目的地控制.生物数学中以微分方程为模型的研究工作主要集中在连续动力系统和脉冲动力系统上.由于自然界的许多变化规律呈现出脉冲效应,因此用脉冲动力系统描述某些运动状态的快速变化或跳跃更为切合实际,脉冲动力系统理论比相应的连续动力系统的理论更加丰富和复杂.本文研究了微生物培养的状态反馈控制问题,给出了一系列带有状态反馈的微生物培养模型,并利用连续动力系统及脉冲动力系统的相关理论对模型的动力学性质进行了讨论分析,包括平衡点的存在性和稳定性、周期解的存在性和稳定性等,对具体模型进行优化问题研究.文章的结论为微生物培养的反馈控制提供了理论依据.
第一章是绪论.1.1节介绍了本文的研究背景及研究意义.1.2节简要叙述了脉冲微分方程在生物动力学中应用的国内外研究现状.1.3节是预备知识,给出脉冲动力系统的一些基本理论,包括解的存在性、连续性及稳定性,以及周期解存在和稳定的判别方法等.
第二章研究微生物分批培养模型的状态反馈控制.2.1节研究了脉冲添加营养基恒定得率系数Monod型分批培养模型.应用微分方程的分析方法,给出了模型阶1周期解存在的条件,并证明了阶2周期解的不存在性.进一步,对周期解的周期进行了刻画,给出了周期的表达式,并通过类似庞加莱准则证明了阶1周期解是稳定的.2.2节研究脉冲添加营养基变生长量分批培养模型.2.2.1节对线性得率系数Monod型增长率函数模型进行了理论分析;2.2.2节对Sigmoid得率系数扩展Monod型增长量函数模型进行了理论分析.2.3节研究了脉冲添加营养基产物抑制生长关联型分批培养模型.针对于不同的模型分别给出了阶1周期解存在的条件,并证明了阶2周期解的不存在性.通过类似庞加莱准则对阶1周期解的稳定性进行了分析,分别给出了各个模型阶1周期解周期的表达式.同时利用计算机对各模型进行了数值模拟,并给出了相应的生物结论.最后,以微生物产率为目标函数对该控制过程进行了优化处理.
第三章研究微生物培养恒化器模型的状态反馈控制.3.1节研究单一微生物恒化器模型的反馈控制.3.1.1节研究脉冲添加清质恒定得率系数恒化器统一模型,3.1.2节研究脉冲混合添加营养基与清质Sigmoid型得率系数恒化器模型.利用Bendixson定理给出了两个模型阶1周期解存在的条件,并对周期解的位置进行了判定.同时利用类似庞加莱准则对周期解的稳定性进行了分析.对阶2周期解的存在性进行了讨论,得到了阶κ(κ≥3)周期解的不存在性,进而证明了所研究的系统不存在混沌现象.最后,对Monod型增长率函数模型进行了数值模拟,同时以微生物产率为目标函数对该控制过程进行了优化分析.3.2节研究两个微生物共存于一种营养基中的恒化器模型反馈控制.分别研究了脉冲添加营养基和清质的两个微生物竞争与捕食恒化器模型,构造了庞加莱映射,并利用该映射对模型边界周期解的稳定性进行了讨论.
第四章给出了微生物培养的恒浊器模型,并对微生物连续培养模式下变流速的状态反馈控制进行研究.分析了具有不同增长率函数模型的动力学性质,包括平衡态的存在性和稳定性.并以微生物稳态产出为目标进行了优化分析.
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.
第一章是绪论.1.1节介绍了本文的研究背景及研究意义.1.2节简要叙述了脉冲微分方程在生物动力学中应用的国内外研究现状.1.3节是预备知识,给出脉冲动力系统的一些基本理论,包括解的存在性、连续性及稳定性,以及周期解存在和稳定的判别方法等.
第二章研究微生物分批培养模型的状态反馈控制.2.1节研究了脉冲添加营养基恒定得率系数Monod型分批培养模型.应用微分方程的分析方法,给出了模型阶1周期解存在的条件,并证明了阶2周期解的不存在性.进一步,对周期解的周期进行了刻画,给出了周期的表达式,并通过类似庞加莱准则证明了阶1周期解是稳定的.2.2节研究脉冲添加营养基变生长量分批培养模型.2.2.1节对线性得率系数Monod型增长率函数模型进行了理论分析;2.2.2节对Sigmoid得率系数扩展Monod型增长量函数模型进行了理论分析.2.3节研究了脉冲添加营养基产物抑制生长关联型分批培养模型.针对于不同的模型分别给出了阶1周期解存在的条件,并证明了阶2周期解的不存在性.通过类似庞加莱准则对阶1周期解的稳定性进行了分析,分别给出了各个模型阶1周期解周期的表达式.同时利用计算机对各模型进行了数值模拟,并给出了相应的生物结论.最后,以微生物产率为目标函数对该控制过程进行了优化处理.
第三章研究微生物培养恒化器模型的状态反馈控制.3.1节研究单一微生物恒化器模型的反馈控制.3.1.1节研究脉冲添加清质恒定得率系数恒化器统一模型,3.1.2节研究脉冲混合添加营养基与清质Sigmoid型得率系数恒化器模型.利用Bendixson定理给出了两个模型阶1周期解存在的条件,并对周期解的位置进行了判定.同时利用类似庞加莱准则对周期解的稳定性进行了分析.对阶2周期解的存在性进行了讨论,得到了阶κ(κ≥3)周期解的不存在性,进而证明了所研究的系统不存在混沌现象.最后,对Monod型增长率函数模型进行了数值模拟,同时以微生物产率为目标函数对该控制过程进行了优化分析.3.2节研究两个微生物共存于一种营养基中的恒化器模型反馈控制.分别研究了脉冲添加营养基和清质的两个微生物竞争与捕食恒化器模型,构造了庞加莱映射,并利用该映射对模型边界周期解的稳定性进行了讨论.
第四章给出了微生物培养的恒浊器模型,并对微生物连续培养模式下变流速的状态反馈控制进行研究.分析了具有不同增长率函数模型的动力学性质,包括平衡态的存在性和稳定性.并以微生物稳态产出为目标进行了优化分析.
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.
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