微生物发酵中几类问题的建模、优化和最优控制
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摘要
1,3—丙二醇是一种重要的化工原料。近几年,微生物发酵法生产1,3—丙二醇受到国内外学者的广泛关注。本文以克雷伯氏杆菌歧化甘油生产1,3—丙二醇为研究背景,研究了该发酵过程中几类问题的建模、优化和最优控制。该项研究,一方面可以丰富非线性动力系统、最优控制、双层规划以及优化理论与算法的研究:另一方面可以为1,3—丙二醇的大规模产业化生产提供理论参考。因此,该项研究具有一定的理论意义与应用价值。另外,该项研究得到国家自然科学基金项目、“973计划”及“863计划”项目的资助。本论文研究的内容与取得的主要结果可概括如下:
1.针对间歇发酵中微生物的生长特点,以底物和多产物对微生物生长的综合抑制力作为控制函数,建立了含有控制和参数的非线性动力系统描述该过程,并证明了该系统的主要性质。以实验数据和计算值之间误差平方和最小为性能指标,以上述含控制和参数的非线性动力系统为约束条件,建立了系统辨识模型。该系统辨识模型为一个最优控制问题,它的直接求解很困难。根据微生物生长特点,我们采用离散化技术将其转化为一个参数辨识问题,并构造了改进的粒子群算法求得系统辨识模型的一个较满意的解。数值结果表明:该算法具有较好的收敛性,并且实验观测值和计算值之间的误差比已有文献降低了6到16个百分点。
2.根据批式流加实际发酵过程,将甘油和碱的注入看作一个连续过程而不是脉冲形式,首次提出了一个非线性多阶段动力系统描述该过程。此外,为了确定系统中的参数值,建立了参数辨识模型,证明了参数的可辨识性,并构造改进的单纯形法求得最优参数。数值结果表明实验观测值和计算值之间的误差比已有文献降低了1.16到40.97个百分点,因而,辨识后的多阶段动力系统能更好地描述批式流加发酵过程。在批式流加发酵中,甘油和碱的合理注入速度是提高1,3—丙二醇生产强度的关键。因而,将流加速度看成一个随时间变化的控制函数,建立了含控制的非线性多阶段动力系统,并证明了该系统的一些主要性质。以终端时刻1,3—丙二醇的生产强度最大为性能指标,以上述含控制的多阶段动力系统和连续状态不等式约束为约束条件,建立了多阶段最优控制模型。最后利用不可微优化理论得到了最优控制问题的最优性条件,并证明了最优性条件和最优性函数零点的等价性。
3.通量平衡分析是代谢网络分析中一个非常有效的工具,它能够预测细胞内的通量分布,然而准确的预测依赖于合理的目标函数。基于代谢网络,首次提出了一个非线性双层规划模型推断厌氧条件下克雷伯氏杆菌歧化甘油生产1,3—丙二醇的代谢目标函数,利用K-K-T最优性条件将上述双层规划模型转化成一个带有互补松弛条件的非线性规划,证明了该双层规划最优解的存在性,构造了有效的算法来求解并分析了算法的收敛性。由于观测实验数据会存在一定的误差,所以研究上述所得目标函数关于允许实验误差的鲁棒性是非常必要的。因此,提出一组非线性双层规划模型来判断目标函数的鲁棒性。数值结果表明所得目标函数具有较好的鲁棒性。
1,3-propanediol(1,3-PD) is an important chemical raw material.In recent years,the production of 1,3-PD by microbial fermentation has been widely investigated.The dissertation investigates the modeling,optimization and optimal control of several problems in glycerol bioconversion to 1,3-PD by Klebsiella pneumoniae(K.pneumoniae).The research not only can enrich the theory and the application of nonlinear dynamical systems, optimal control,bilevel programming and optimization theory and algorithm,but also can provide theoretical guide for the commercial production of 1,3-PD.Hence,this research is very interesting in both theory and practice.In addition,the research is supported by national natural science foundation,"973 program" and "863 program".The main contributions are summarized as follows:
1.According to the characteristic of microbial growth in batch culture,we establish a controlled dynamical system with parameters to formulate this process by taking the total inhibitions of substrate and multiple products to the cell growth as the control function.Subsequently,some properties of the proposed system are proved. Taking the minimal error between the experimental data and calculated values as the performance index,and the above controlled dynamical system as the constraint, we present a system identification model,which is an optimal control problem.It is difficult to solve the system identification model directly.So,we transform it into a parameter identification problem by discretizing technique combining the characteristic of microbial growth.Finally,an improved particle swarm algorithm is constructed to obtain a satisfactory solution of the system identification problem. Numerical results show that the above algorithm is of good convergence and the errors are cut down by 6%-16%.
2.In view of the actual fed-batch fermentation process,we firstly propose a nonlinear multistage dynamical system,taking the feeding of glycerol and alkali as a time-continuous process instead of an impulsive form,to formulate the process.To determine the parameters in the multistage system,we establish a parameter identification model and prove the identifiability of optimal parameters.Finally,an improved simplex method is constructed to solve the identification model.Numerical results show that the error is cut down greatly and the proposed system can formulate the fed-batch culture better.Since a proper feeding rate is required during the fed-batch process,we propose a controlled multistage dynamical system by taking the feeding rate as the control function.Some important properties of the system are also proved.To maximize the productivity of 1,3-PD at the terminal time,an optimal control model subject to our proposed controlled multistage system and continuous state inequality constraints is presented.Finally,we obtain an optimality condition for the optimal control problem by non-differentiable optimization theory,and prove the equivalence between the optimality condition and zeros of the optimality function.
3.Flux balance analysis(FBA) is an effective tool in the analysis of metabolic network. It can predict the flux distribution of engineered cells,whereas the accurate prediction depends on the reasonable objective function.Basing on the metabolic network,we firstly propose a nonlinear bilevel programming model to infer the metabolic objective function of anaerobic glycerol metabolism by K.pneumoniae for 1,3-PD production.Making use of the K-K-T optimality condition of the lower level problem,the bilevel programming model is equivalently transcribed into a nonlinear programming with complementary and slackness conditions.We give the existence theorem of the optimal solution to the above model.An efficient algorithm is proposed to solve the model and its convergence is also simply analyzed.It is necessary to investigate the robustness of the obtained objective function because of the experimental errors.Hence,a series of nonlinear bilevel programming models is established to analyze the robustness.Numerical results show that the obtained objective function is of good robustness.
1.针对间歇发酵中微生物的生长特点,以底物和多产物对微生物生长的综合抑制力作为控制函数,建立了含有控制和参数的非线性动力系统描述该过程,并证明了该系统的主要性质。以实验数据和计算值之间误差平方和最小为性能指标,以上述含控制和参数的非线性动力系统为约束条件,建立了系统辨识模型。该系统辨识模型为一个最优控制问题,它的直接求解很困难。根据微生物生长特点,我们采用离散化技术将其转化为一个参数辨识问题,并构造了改进的粒子群算法求得系统辨识模型的一个较满意的解。数值结果表明:该算法具有较好的收敛性,并且实验观测值和计算值之间的误差比已有文献降低了6到16个百分点。
2.根据批式流加实际发酵过程,将甘油和碱的注入看作一个连续过程而不是脉冲形式,首次提出了一个非线性多阶段动力系统描述该过程。此外,为了确定系统中的参数值,建立了参数辨识模型,证明了参数的可辨识性,并构造改进的单纯形法求得最优参数。数值结果表明实验观测值和计算值之间的误差比已有文献降低了1.16到40.97个百分点,因而,辨识后的多阶段动力系统能更好地描述批式流加发酵过程。在批式流加发酵中,甘油和碱的合理注入速度是提高1,3—丙二醇生产强度的关键。因而,将流加速度看成一个随时间变化的控制函数,建立了含控制的非线性多阶段动力系统,并证明了该系统的一些主要性质。以终端时刻1,3—丙二醇的生产强度最大为性能指标,以上述含控制的多阶段动力系统和连续状态不等式约束为约束条件,建立了多阶段最优控制模型。最后利用不可微优化理论得到了最优控制问题的最优性条件,并证明了最优性条件和最优性函数零点的等价性。
3.通量平衡分析是代谢网络分析中一个非常有效的工具,它能够预测细胞内的通量分布,然而准确的预测依赖于合理的目标函数。基于代谢网络,首次提出了一个非线性双层规划模型推断厌氧条件下克雷伯氏杆菌歧化甘油生产1,3—丙二醇的代谢目标函数,利用K-K-T最优性条件将上述双层规划模型转化成一个带有互补松弛条件的非线性规划,证明了该双层规划最优解的存在性,构造了有效的算法来求解并分析了算法的收敛性。由于观测实验数据会存在一定的误差,所以研究上述所得目标函数关于允许实验误差的鲁棒性是非常必要的。因此,提出一组非线性双层规划模型来判断目标函数的鲁棒性。数值结果表明所得目标函数具有较好的鲁棒性。
1,3-propanediol(1,3-PD) is an important chemical raw material.In recent years,the production of 1,3-PD by microbial fermentation has been widely investigated.The dissertation investigates the modeling,optimization and optimal control of several problems in glycerol bioconversion to 1,3-PD by Klebsiella pneumoniae(K.pneumoniae).The research not only can enrich the theory and the application of nonlinear dynamical systems, optimal control,bilevel programming and optimization theory and algorithm,but also can provide theoretical guide for the commercial production of 1,3-PD.Hence,this research is very interesting in both theory and practice.In addition,the research is supported by national natural science foundation,"973 program" and "863 program".The main contributions are summarized as follows:
1.According to the characteristic of microbial growth in batch culture,we establish a controlled dynamical system with parameters to formulate this process by taking the total inhibitions of substrate and multiple products to the cell growth as the control function.Subsequently,some properties of the proposed system are proved. Taking the minimal error between the experimental data and calculated values as the performance index,and the above controlled dynamical system as the constraint, we present a system identification model,which is an optimal control problem.It is difficult to solve the system identification model directly.So,we transform it into a parameter identification problem by discretizing technique combining the characteristic of microbial growth.Finally,an improved particle swarm algorithm is constructed to obtain a satisfactory solution of the system identification problem. Numerical results show that the above algorithm is of good convergence and the errors are cut down by 6%-16%.
2.In view of the actual fed-batch fermentation process,we firstly propose a nonlinear multistage dynamical system,taking the feeding of glycerol and alkali as a time-continuous process instead of an impulsive form,to formulate the process.To determine the parameters in the multistage system,we establish a parameter identification model and prove the identifiability of optimal parameters.Finally,an improved simplex method is constructed to solve the identification model.Numerical results show that the error is cut down greatly and the proposed system can formulate the fed-batch culture better.Since a proper feeding rate is required during the fed-batch process,we propose a controlled multistage dynamical system by taking the feeding rate as the control function.Some important properties of the system are also proved.To maximize the productivity of 1,3-PD at the terminal time,an optimal control model subject to our proposed controlled multistage system and continuous state inequality constraints is presented.Finally,we obtain an optimality condition for the optimal control problem by non-differentiable optimization theory,and prove the equivalence between the optimality condition and zeros of the optimality function.
3.Flux balance analysis(FBA) is an effective tool in the analysis of metabolic network. It can predict the flux distribution of engineered cells,whereas the accurate prediction depends on the reasonable objective function.Basing on the metabolic network,we firstly propose a nonlinear bilevel programming model to infer the metabolic objective function of anaerobic glycerol metabolism by K.pneumoniae for 1,3-PD production.Making use of the K-K-T optimality condition of the lower level problem,the bilevel programming model is equivalently transcribed into a nonlinear programming with complementary and slackness conditions.We give the existence theorem of the optimal solution to the above model.An efficient algorithm is proposed to solve the model and its convergence is also simply analyzed.It is necessary to investigate the robustness of the obtained objective function because of the experimental errors.Hence,a series of nonlinear bilevel programming models is established to analyze the robustness.Numerical results show that the obtained objective function is of good robustness.
引文
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