微生物发酵中的非线性动力系统及最优控制
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摘要
本文以生物化工领域中的一个实际课题——甘油微生物歧化方法生产1,3-丙二醇的间歇和批式流加发酵——为背景,研究了两类非线性动力系统的参数辨识与最优控制问题。首先根据发酵过程的特性和动态行为,分别建立符合各自特性的非线性动力系统,论述了系统的主要性质并对其中的参数进行了辨识。此外为了寻找最优策略,建立了最优控制模型,研究了最优控制问题的最优性条件。该项研究,一方面可以丰富非线性动力系统、最优控制理论与算法的理论和应用,另一方面可以降低消耗、节约成本为1,3-丙二醇的大规模产业化生产提供理论指导,因此该项研究具有一定的理论意义与应用价值。本论文研究的内容与取得的主要结果可概括如下:
1.在甘油微生物歧化生产1,3-丙二醇的间歇发酵过程中,依微生物的生长特点将整个发酵过程分为发育期、生长期和稳定期三个阶段,利用在不同阶段上发酵过程的动态行为不相同这一特点建立了描述间歇发酵过程的非线性多阶段动力系统,论述了系统解的存在唯一性。为了确定系统中参数的值,建立了参数辨识模型,并证明了参数的可辨识性。然后构造求解参数辨识问题的优化算法,数值结果显示辨识后的模型较已有的模型计算值与实验值之间的误差要小的多,说明该非线性多阶段动力系统更适合描述间歇发酵过程。
2.根据甘油微生物歧化生产1,3-丙二醇批式流加发酵的特点,分别用连续函数和离散函数描述发酵过程中的连续动态行为和离散事件,建立了描述批式流加过程的混杂动力系统,讨论了系统解的存在唯一性,解对参数的连续依赖性,并根据实验数据建立了参数辨识模型,构造了求解参数辨识问题的优化算法,数值结果显示经参数辨识后的混杂动力系统不仅有效地克服了已有模型中甘油浓度的计算值会出现负值的缺陷,而且计算值与实验值之间的平均相对误差由已有模型的18.36%降低到1.9%,说明了混杂动力系统在描述批式流加发酵过程上的有效性。
3.对于间歇发酵,建立以初始状态为控制变量,以描述发酵过程的非线性多阶段动力系统为约束的终端最优控制模型,论证了最优控制问题的可控性,并推导出该最优控制问题的一阶最优性条件,为求解最优控制问题的优化算法的停机准则设计和收敛性分析提供理论依据。然后,以批式流加发酵过程为背景,建立以离散时刻添加甘油的量为控制变量,以高彩霞建立的描述批式流加发酵过程的非线性脉冲动力系统为约束的终端最优控制模型,并研究了此类最优控制问题的一阶最优性条件。此外,还研究了以混杂动力系统为约束的终端最优控制问题,并构造了求解该最优控制问题的优化算法。数值结果显示本文算法将生产强度从38.448mmol/(L·h)提高到42.13mmol/(L·h),该结论为1,3-丙二醇的实际生产提供了理论指导。
This dissertation investigates the nonlinear dynamical systems and their optimal control problems with the bio-dissimilation of glycerol to 1,3-propanediol by Klebsiella pneumoniae in the background. Two different kinds of nonlinear dynamical systems are presented to describe the batch and the fed-batch fermentation. Properties of these systems and their parameters identification problem are discussed as well. Moreover, in order to find the optimal control strategy we propose optimal control models and obtain optimality conditions. These results can not only develop the theory and the application of nonlinear dynamical systems and optimal control, but also reduce experimental cost and provide certain guidance for industrialization of 1,3-propanediol production. Therefore, this research is very interesting in both theory and practice. The main results, obtained in this dissertation, are summarized as bellow:
1. The batch fermentation of glycerol to 1,3-Propanediol could be divided into three stages according to the growth habit of Klebsiella pneumoniae, which are development stage, growth stage and steady stage, and on different stage the dynamical system is different as well. Using this property, we employ a nonlinear multistage dynamical system to describe batch fermentation and discuss the existence and uniqueness of solution to system. In order to identify values of parameters of this system such that the model can simulate the fermentation as exactly as possible, a parameters identification model is proposed and the existence of optimal solution is proved. Then an optimization algorithm is constructed to solve this problem. Numerical results show that errors between experimental and computational results are less than before, which implies that the nonlinear multistage dynamical system is fit for simulating such fermentation.
2. The hybrid dynamical system of fed-batch fermentation and its parameters identification problem are investigated. Since there are continuously dynamic behavior and discrete events in the process of fed-batch fermentation of glycerol to 1,3-Propanediol, we propose a hybrid dynamical system to describe this kind of fermentation. The existence of solution to system together with the dependence of solution to parameters is discussed. Moreover, a parameters identification model is developed and an optimization algorithm is constructed to solve this parameters identification problem. Numerical results show that the average relative error of computational and experimental results of this system is only 1.9%, but that of the other system is 18.36%, which demonstrates that hybrid system has much advantage in describing fed-batch fermentation.
3. Optimal control problems of the nonlinear multistage dynamical system and the nonlinear impulsive dynamical system are investigated. For batch fermentation, we present an optimal control model in which control variable is initial state and the nonlinear multistage dynamical system is constraint condition. The existence of solution to control model is discussed and the optimality condition is obtained. Then we propose an optimal control model for fed-batch fermentation, where volume of glycerol adding at impulsive moments are control variables and nonlinear impulsive dynamical system is constraint condition. The optimality condition of this problem is obtained as well. Furthermore, an optimal control problem with hybrid dynamical system as constraint condition is investigated and an optimization algorithm is constructed. Numerical result shows that the intensity of production of glycerol at terminal moment is improved from 38.448mmol/(L·h) to 42.13mmol/(L·h), which shows that the optimal control is necessary. The result provides guidance for the industrialization of 1,3-propanediol production by fermentation.
1.在甘油微生物歧化生产1,3-丙二醇的间歇发酵过程中,依微生物的生长特点将整个发酵过程分为发育期、生长期和稳定期三个阶段,利用在不同阶段上发酵过程的动态行为不相同这一特点建立了描述间歇发酵过程的非线性多阶段动力系统,论述了系统解的存在唯一性。为了确定系统中参数的值,建立了参数辨识模型,并证明了参数的可辨识性。然后构造求解参数辨识问题的优化算法,数值结果显示辨识后的模型较已有的模型计算值与实验值之间的误差要小的多,说明该非线性多阶段动力系统更适合描述间歇发酵过程。
2.根据甘油微生物歧化生产1,3-丙二醇批式流加发酵的特点,分别用连续函数和离散函数描述发酵过程中的连续动态行为和离散事件,建立了描述批式流加过程的混杂动力系统,讨论了系统解的存在唯一性,解对参数的连续依赖性,并根据实验数据建立了参数辨识模型,构造了求解参数辨识问题的优化算法,数值结果显示经参数辨识后的混杂动力系统不仅有效地克服了已有模型中甘油浓度的计算值会出现负值的缺陷,而且计算值与实验值之间的平均相对误差由已有模型的18.36%降低到1.9%,说明了混杂动力系统在描述批式流加发酵过程上的有效性。
3.对于间歇发酵,建立以初始状态为控制变量,以描述发酵过程的非线性多阶段动力系统为约束的终端最优控制模型,论证了最优控制问题的可控性,并推导出该最优控制问题的一阶最优性条件,为求解最优控制问题的优化算法的停机准则设计和收敛性分析提供理论依据。然后,以批式流加发酵过程为背景,建立以离散时刻添加甘油的量为控制变量,以高彩霞建立的描述批式流加发酵过程的非线性脉冲动力系统为约束的终端最优控制模型,并研究了此类最优控制问题的一阶最优性条件。此外,还研究了以混杂动力系统为约束的终端最优控制问题,并构造了求解该最优控制问题的优化算法。数值结果显示本文算法将生产强度从38.448mmol/(L·h)提高到42.13mmol/(L·h),该结论为1,3-丙二醇的实际生产提供了理论指导。
This dissertation investigates the nonlinear dynamical systems and their optimal control problems with the bio-dissimilation of glycerol to 1,3-propanediol by Klebsiella pneumoniae in the background. Two different kinds of nonlinear dynamical systems are presented to describe the batch and the fed-batch fermentation. Properties of these systems and their parameters identification problem are discussed as well. Moreover, in order to find the optimal control strategy we propose optimal control models and obtain optimality conditions. These results can not only develop the theory and the application of nonlinear dynamical systems and optimal control, but also reduce experimental cost and provide certain guidance for industrialization of 1,3-propanediol production. Therefore, this research is very interesting in both theory and practice. The main results, obtained in this dissertation, are summarized as bellow:
1. The batch fermentation of glycerol to 1,3-Propanediol could be divided into three stages according to the growth habit of Klebsiella pneumoniae, which are development stage, growth stage and steady stage, and on different stage the dynamical system is different as well. Using this property, we employ a nonlinear multistage dynamical system to describe batch fermentation and discuss the existence and uniqueness of solution to system. In order to identify values of parameters of this system such that the model can simulate the fermentation as exactly as possible, a parameters identification model is proposed and the existence of optimal solution is proved. Then an optimization algorithm is constructed to solve this problem. Numerical results show that errors between experimental and computational results are less than before, which implies that the nonlinear multistage dynamical system is fit for simulating such fermentation.
2. The hybrid dynamical system of fed-batch fermentation and its parameters identification problem are investigated. Since there are continuously dynamic behavior and discrete events in the process of fed-batch fermentation of glycerol to 1,3-Propanediol, we propose a hybrid dynamical system to describe this kind of fermentation. The existence of solution to system together with the dependence of solution to parameters is discussed. Moreover, a parameters identification model is developed and an optimization algorithm is constructed to solve this parameters identification problem. Numerical results show that the average relative error of computational and experimental results of this system is only 1.9%, but that of the other system is 18.36%, which demonstrates that hybrid system has much advantage in describing fed-batch fermentation.
3. Optimal control problems of the nonlinear multistage dynamical system and the nonlinear impulsive dynamical system are investigated. For batch fermentation, we present an optimal control model in which control variable is initial state and the nonlinear multistage dynamical system is constraint condition. The existence of solution to control model is discussed and the optimality condition is obtained. Then we propose an optimal control model for fed-batch fermentation, where volume of glycerol adding at impulsive moments are control variables and nonlinear impulsive dynamical system is constraint condition. The optimality condition of this problem is obtained as well. Furthermore, an optimal control problem with hybrid dynamical system as constraint condition is investigated and an optimization algorithm is constructed. Numerical result shows that the intensity of production of glycerol at terminal moment is improved from 38.448mmol/(L·h) to 42.13mmol/(L·h), which shows that the optimal control is necessary. The result provides guidance for the industrialization of 1,3-propanediol production by fermentation.
引文
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