非线性切换动力系统的最优控制及应用
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摘要
本文以微生物批式流加发酵甘油生产1,3-丙二醇为背景,研究了该发酵过程的非线性动力系统及其最优控制.该项研究,不仅可推动非线性动力系统、最优控制理论以及优化算法的研究,还可提高目标产物产量,为1,3-丙二醇的大规模产业化生产提供参考.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”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.将甘油和碱的流加看作一个连续过程并且以甘油的流加速度作为控制函数,本文建立了非线性多阶段动力系统描述微生物批式流加发酵甘油生产1,3-丙二醇过程.对于该动力系统,研究了其解的存在唯一性、关于控制函数的Lipschitz连续性和一致有界性等性质.以终端时刻1,3-丙二醇的浓度最大化为性能指标,建立了多阶段最优控制模型.利用有界变差理论,证明了多阶段最优控制问题最优控制的存在性.进一步,基于控制参数化方法和改进的粒子群算法提出了一种求解多阶段最优控制问题的全局优化算法.数值结果表明：应用所得的最优甘油流加策略,终端时刻1,3-丙二醇的浓度比实验数据提高了16.04%.
2.在微生物批式流加发酵甘油生产1,3-丙二醇过程中,恰当的甘油流加开始和结束时刻对于提高1,3-丙二醇的产量具有重要的意义.因此,以甘油的流加开始和结束时刻作为控制函数,本文提出了自治切换系统来描述该过程.以终端时刻1,3-丙二醇的浓度最大化为性能指标,建立了受连续状态约束的最优控制模型.利用约束转化方法和光滑近似技术处理最优控制问题中的连续状态不等式约束.基于罚函数法和性能指标的一阶、二阶梯度信息,构造了一种求解该最优控制模型的计算方法.数值结果表明：终端时刻1,3-丙二醇的浓度比以往所得结果有显著提高.
3.在微生物发酵甘油生产1,3-丙二醇过程中,甘油对细胞生长既提供营养又产生底物抑制作用.因此,将发酵罐内甘油浓度维持在一定的范围内对提高1,3-丙二醇的产量非常重要.以甘油浓度作为切换规则并且以甘油流加速度作为控制函数,本文建立了状态依赖的切换动力系统描述微生物批式流加发酵过程.研究了该系统解的存在唯一性、一致有界性、一致Lipschitz连续性和正则性等性质.以终端时刻1,3-丙二醇的浓度最大化为性能指标,建立了最优切换控制模型.讨论了该最优切换控制问题最优控制的存在性.由于在最优切换控制问题中切换次数事先未知,将最优切换控制问题转化为一个等价的双层优化问题.最后,基于启发式算法和控制参数化方法构造了一种求解双层优化问题的计算方法.数值结果表明：终端时刻1,3-丙二醇的浓度比实验数据有显著提高并且大大地减少了切换次数.这为实际批式流加发酵提供了一定的指导作用.
This dissertation investigates the nonlinear dynamical systems and their optimal control problems in fed-batch fermentation of glycerol bioconversion to 1,3-propanediol (1,3-PD) by Klebsiella pneumoniae. The research can not only develop nonlinear dy-namical system, optimal control theory and optimization algorithm, but also increase the productivity of product and provide certain reference for commercial process of 1,3-PD by fermentation. Therefore, it is very interesting both in theory and in practice. In addition, the research is supported by the National Natural Science Foundation of China, the Major State Basic Research Development Program of China and the National High Technology Research and Development Program of China. The main contributions obtained in this dissertation are summarized as follows.
1. Taking the feeding of glycerol and alkali as a time-continuous process, we propose a nonlinear multistage dynamical system, in which the feeding rate of glycerol is the control function, to describe the microbial fed-batch fermentation of glycerol to 1,3-PD. Some properties, i.e., the existence and uniqueness, Lipschitzian conti-nuity in control function and uniform boundedness of solutions to the system, are proved. To maximize the concentration of 1,3-PD at the terminal time, a multistage optimal control model is presented. The existence of optimal control is established using theory of bounded variation. Based on the discretization method and an im-proved Particle Swarm Optimization algorithm, a global optimization algorithm is constructed to seek the optimal solution. Numerical results show that, by employing the optimal control policy, the concentration of 1,3-PD at the terminal time can be increased by 16.04% compared with the experimental data.
2. It is decisive for improving the productivity of 1,3-PD to optimize the switching instants between the feeding and batch processes in microbial fed-batch fermen- tation. Thus, an autonomous switched system with variable switching instants is proposed to formulate the fermentation process. To maximize the concentration of 1,3-PD at the terminal time, we present an optimal control model involving the autonomous switched system and subject to continuous state constraints. A con-straint transcription approach and a local smoothing technique are introduced to deal with the continuous state inequality constraints. On the basis of a penalty function and the first and second gradients of the cost function, a gradient-based algorithm is developed to solve the optimal control model. Numerical results show that the concentration of 1,3-PD at the terminal time can be increased considerably compared with previous results.
3. The feeding of glycerol should be kept in a given range such that it can not only provide sufficient nutrition for cells growth but also effectively avoid the inhibitory effect of excessive glycerol on the cells growth. A switched system with state depen-dent switching, in which the feeding rate of glycerol is taken as the control function, to describe the microbial fed-batch fermentation process. Some properties such as the existence and uniqueness, uniform boundedness, uniformly Lipschitzian conti-nuity and regularity of solutions, are discussed. To maximize the concentration of 1,3-PD at the terminal time, an optimal switching control model is presented. The existence of optimal control is also ascertained. Since the number of switchings is not known a prior. The optimal switching control model is equivalently transcribed into a two-level optimization problem. Finally, a solution method is developed on the basis of a heuristic approach and the control parametrization method. Numerical results show that, by employing the obtained optimal strategy,1,3-PD concentration at the terminal time can be increased considerably and the number of switchings is also decreased greatly. This provides guidance to the actual industrial production of 1,3-PD which is beneficial to optimize the biochemical engineer.
1.将甘油和碱的流加看作一个连续过程并且以甘油的流加速度作为控制函数,本文建立了非线性多阶段动力系统描述微生物批式流加发酵甘油生产1,3-丙二醇过程.对于该动力系统,研究了其解的存在唯一性、关于控制函数的Lipschitz连续性和一致有界性等性质.以终端时刻1,3-丙二醇的浓度最大化为性能指标,建立了多阶段最优控制模型.利用有界变差理论,证明了多阶段最优控制问题最优控制的存在性.进一步,基于控制参数化方法和改进的粒子群算法提出了一种求解多阶段最优控制问题的全局优化算法.数值结果表明：应用所得的最优甘油流加策略,终端时刻1,3-丙二醇的浓度比实验数据提高了16.04%.
2.在微生物批式流加发酵甘油生产1,3-丙二醇过程中,恰当的甘油流加开始和结束时刻对于提高1,3-丙二醇的产量具有重要的意义.因此,以甘油的流加开始和结束时刻作为控制函数,本文提出了自治切换系统来描述该过程.以终端时刻1,3-丙二醇的浓度最大化为性能指标,建立了受连续状态约束的最优控制模型.利用约束转化方法和光滑近似技术处理最优控制问题中的连续状态不等式约束.基于罚函数法和性能指标的一阶、二阶梯度信息,构造了一种求解该最优控制模型的计算方法.数值结果表明：终端时刻1,3-丙二醇的浓度比以往所得结果有显著提高.
3.在微生物发酵甘油生产1,3-丙二醇过程中,甘油对细胞生长既提供营养又产生底物抑制作用.因此,将发酵罐内甘油浓度维持在一定的范围内对提高1,3-丙二醇的产量非常重要.以甘油浓度作为切换规则并且以甘油流加速度作为控制函数,本文建立了状态依赖的切换动力系统描述微生物批式流加发酵过程.研究了该系统解的存在唯一性、一致有界性、一致Lipschitz连续性和正则性等性质.以终端时刻1,3-丙二醇的浓度最大化为性能指标,建立了最优切换控制模型.讨论了该最优切换控制问题最优控制的存在性.由于在最优切换控制问题中切换次数事先未知,将最优切换控制问题转化为一个等价的双层优化问题.最后,基于启发式算法和控制参数化方法构造了一种求解双层优化问题的计算方法.数值结果表明：终端时刻1,3-丙二醇的浓度比实验数据有显著提高并且大大地减少了切换次数.这为实际批式流加发酵提供了一定的指导作用.
This dissertation investigates the nonlinear dynamical systems and their optimal control problems in fed-batch fermentation of glycerol bioconversion to 1,3-propanediol (1,3-PD) by Klebsiella pneumoniae. The research can not only develop nonlinear dy-namical system, optimal control theory and optimization algorithm, but also increase the productivity of product and provide certain reference for commercial process of 1,3-PD by fermentation. Therefore, it is very interesting both in theory and in practice. In addition, the research is supported by the National Natural Science Foundation of China, the Major State Basic Research Development Program of China and the National High Technology Research and Development Program of China. The main contributions obtained in this dissertation are summarized as follows.
1. Taking the feeding of glycerol and alkali as a time-continuous process, we propose a nonlinear multistage dynamical system, in which the feeding rate of glycerol is the control function, to describe the microbial fed-batch fermentation of glycerol to 1,3-PD. Some properties, i.e., the existence and uniqueness, Lipschitzian conti-nuity in control function and uniform boundedness of solutions to the system, are proved. To maximize the concentration of 1,3-PD at the terminal time, a multistage optimal control model is presented. The existence of optimal control is established using theory of bounded variation. Based on the discretization method and an im-proved Particle Swarm Optimization algorithm, a global optimization algorithm is constructed to seek the optimal solution. Numerical results show that, by employing the optimal control policy, the concentration of 1,3-PD at the terminal time can be increased by 16.04% compared with the experimental data.
2. It is decisive for improving the productivity of 1,3-PD to optimize the switching instants between the feeding and batch processes in microbial fed-batch fermen- tation. Thus, an autonomous switched system with variable switching instants is proposed to formulate the fermentation process. To maximize the concentration of 1,3-PD at the terminal time, we present an optimal control model involving the autonomous switched system and subject to continuous state constraints. A con-straint transcription approach and a local smoothing technique are introduced to deal with the continuous state inequality constraints. On the basis of a penalty function and the first and second gradients of the cost function, a gradient-based algorithm is developed to solve the optimal control model. Numerical results show that the concentration of 1,3-PD at the terminal time can be increased considerably compared with previous results.
3. The feeding of glycerol should be kept in a given range such that it can not only provide sufficient nutrition for cells growth but also effectively avoid the inhibitory effect of excessive glycerol on the cells growth. A switched system with state depen-dent switching, in which the feeding rate of glycerol is taken as the control function, to describe the microbial fed-batch fermentation process. Some properties such as the existence and uniqueness, uniform boundedness, uniformly Lipschitzian conti-nuity and regularity of solutions, are discussed. To maximize the concentration of 1,3-PD at the terminal time, an optimal switching control model is presented. The existence of optimal control is also ascertained. Since the number of switchings is not known a prior. The optimal switching control model is equivalently transcribed into a two-level optimization problem. Finally, a solution method is developed on the basis of a heuristic approach and the control parametrization method. Numerical results show that, by employing the obtained optimal strategy,1,3-PD concentration at the terminal time can be increased considerably and the number of switchings is also decreased greatly. This provides guidance to the actual industrial production of 1,3-PD which is beneficial to optimize the biochemical engineer.
引文
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