Nonlinear impulsive system of microbial production in fed-batch culture and its optimal control

In this study the optimal control of fed-batch glycerol fermentation is investigated based on an impulsive dynamical system. Considering the sudden increase of the glycerol and alkali in fed-batch culture of biodissimilation of glycerol to 1,3-propanediol, this paper proposes a nonlinear impulsive system of fed-batch culture. The existence, uniqueness and regularity properties of piecewise solution for the system are proved. In view of the controllability of volumes of glycerol added to the reactor instantaneously, the paper constructs an optimal control model based on the nonlinear impulsive system and the existence of the optimal control is obtained. The control variables here are the moments and the sizes of jumps in the states at the discrete instants and the objective is to maximize the productivity of 1,3-propanediol over one cycle.     

Optimal switching control for microbial fed-batch culture

In fed-batch culture of glycerol bio-dissimilation to 1,3-propanediol (1,3-PD), the aim of adding glycerol is to obtain as much 1,3-PD as possible. Hence, a proper feed strategy is required during the process. In this paper, we present an optimal switching control model based on our proposed controlled switching system. Some properties of the controlled switching system are obtained. Subsequently, we prove the existence of optimal control. In order to deduce the optimality conditions, we transcribe the optimal switching control model into an equivalent one with fixed switching instants and parameters. Finally, the optimality conditions of the equivalent problem are investigated by calculus of variations.     

微生物批式流加发酵多阶段最优控制的最优性条件

在微生物批式流加发酵生产1,3一丙二醇(1,3-PD)过程中,关键是如何控制甘油和碱的流加速度.本文将流加速度看成一个随时间变化的控制函数,提出一个带控制的多阶段动力系统描述批式发酵过程,并证明了系统的一些性质.以终端时刻1,3-PD的生产强度最大为性能指标,以上述动力系统和连续状态不等式为约束条件建立了最优控制模型,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.     

Optimal switching control of a fed-batch fermentation process

Considering the hybrid nature in fed-batch culture of glycerol biconversion to 1,3-propanediol (1,3-PD) by Klebsiella pneumoniae, we propose a state-based switching dynamical system to describe the fermentation process. To maximize the concentration of 1,3-PD at the terminal time, an optimal switching control model subject to our proposed switching system and constraints of continuous state inequality and control function is presented. Because the number of the switchings is not known a priori, we reformulate the above optimal control problem as a two-level optimization problem. An optimization algorithm is developed to seek the optimal solution on the basis of a heuristic approach and control parametrization technique. Numerical results show that, by employing the obtained optimal control strategy, 1,3-PD concentration at the terminal time can be increased considerably.     

Optimal control and optimization algorithm of nonlinear impulsive delay system producing 1,3-Propanediol

According to the controllability of pulse times and the amount of jumps in the states at these times in the process of fed-batch culture producing 1,3-propanediol, this paper proposes a terminal optimal control model, whose constraint condition is the nonlinear impulsive delay system. The existence of optimal control is discussed and an optimization algorithm which is applied to each subinternal over one cycle for this optimal control problem is constructed. Finally, the numerical simulations show that the terminal intensity of producing 1,3-propanediol has been increased obviously.     

Optimality condition of the nonlinear impulsive system in fed-batch fermentation

Based on the nonlinear impulsive system of fed-batch fermentation, which is used in the process of bio-dissimilation of glycerol to 1,3-propanediol by Klebsiella pneumoniae, this paper is concerned with the optimal control of the volumes of infused glycerol at impulsive moments. The optimal control model is developed. The authors study the properties of both the nonlinear impulsive system and the solution of it, and ascertain the existence of the optimal control. Then the optimality condition is obtained, which can be used as a necessary condition that the optimal control should satisfy.     

Discrete optimal control model and bound error for microbial continuous fermentation

A discrete optimal control model is given according to the discrete nonlinear dynamic system of continuous fermentations of glycerol to 1,3-propanediol (1,3-PD). The property of some major functions and their bounds on approximation errors are studied in this paper. Then, the conclusion that the optimality function of discrete optimal control model is the consistent approximation to one of the continuous optimal control model is proved. The results presented in this work can be used as guidelines for the optimal algorithm and its convergence.     

Optimal control for nonlinear dynamical system of microbial fed-batch culture

In fed-batch culture of glycerol bio-dissimilation to 1, 3-propanediol (1, 3-PD), the aim of adding glycerol is to obtain as much 1, 3-PD as possible. So a proper feeding rate is required during the process. Taking the concentration of 1, 3-PD at the terminal time as the performance index and the feeding rate of glycerol as the control function, we propose an optimal control model subject to a nonlinear dynamical system and constraints of continuous state and non-stationary control. A computational approach is constructed to seek the solution of the above model in two aspects. On the one hand we transcribe the optimal control model into an unconstrained one based on the penalty functions and an extension of the state space; on the other hand, by approximating the control function with simple functions, we transform the unconstrained optimal control problem into a sequence of nonlinear programming problems, which can be solved using gradient-based optimization techniques. The convergence analysis of this approximation is also investigated. Numerical results show that, by employing the optimal control policy, the concentration of 1, 3-PD at the terminal time can be increased considerably.     

Modelling and well-posedness of a nonlinear hybrid system in fed-batch production of 1,3-propanediol with open loop glycerol input and pH logic control

A fed-batch fermentation of glycerol by Klebsiella pneumoniae with open loop substrate input and pH logic control is considered in this paper. We propose a nonlinear hybrid system to describe this process, which consists of discrete variables to represent the flow rate of glycerol and alkali and continuous variables for substance concentrations. The hybrid system includes both time- and state-based switchings, which are respectively determined by a pre-assigned times sequence and an output equation of the pH. It is proved that the hybrid system is non-Zeno. Some basic properties of solutions to the hybrid system are also explored, including existence, uniqueness, boundedness and continuous dependence on initial state and parameters. Additionally, a numerical simulation is carried out to show that the proposed hybrid system can describe the fed-batch culture properly.     

Bilevel parameters identification for the multi-stage nonlinear impulsive system in microorganisms fed-batch cultures

Considering the abrupt jump of the substrate and different characters of the bacterial in the lag phase, the exponential phase and the stationary phase this paper proposes the multi-stage nonlinear impulsive system for the fed-batch fermentation from glycerol to 1,3-propanediol (1,3-PD) and establishes the bilevel identification system for its sensitive parameters. The properties of the solutions for the nonlinear multi-stage dynamical system are investigated and identifiability of the parameters is proved. Finally an optimal algorithm is constructed to obtain the optimal solution of the identification model and the numerical example is then discussed to illustrate the algorithm.     

Sensitivity analysis and parameter identification for a nonlinear time-delay system in microbial fed-batch process

Developing suitable dynamic models of bioprocess is a difficult issue in bioscience. In this paper, considering the microbial metabolism mechanism, i.e., the production of new biomass is delayed by the amount of time it takes to metabolize the nutrients, in glycerol bioconversion to 1,3-propanediol, we propose a nonlinear time-delay system to formulate the fed-batch fermentation process. Some important properties are also discussed. Then, in view of the effect of time-delay and the high number of kinetic parameters in the system, the parametric sensitivity analysis is used to determine the key parameters. Finally, a parameter identification model is presented and a global optimization method is developed to seek the optimal key parameters. Numerical results show that the nonlinear time-delay system can describe the fed-batch fermentation process reasonably.     

Nonlinear hybrid system and parameter identification of microbial fed-batch culture with open loop glycerol input and pH logic control

In this paper, we study the fed-batch fermentation of glycerol by Klebsiella pneumoniae with open loop glycerol input and pH logic control using a flow of alkali as manipulated variable. A nonlinear hybrid system is developed to describe this process. We prove the finiteness of switching numbers of the system in terms of bounded variation and explore the existence and uniqueness of the solutions. Additionally, a parameter identification problem is proposed and an asynchronous parallel particle swarm optimization (PSO) algorithm is constructed to solve it. Numerical results show the effectiveness of the algorithm and reveal that the proposed model could describe the fed-batch process properly.     

An improved model for multistage simulation of glycerol fermentation in batch culture and its parameter identification

The bioconversion of glycerol to 1,3-propanediol is a complex bioprocess. In this study, we aim to explore a novel model for describing the multistage cell growth in batch culture. This is achieved by a modification at the specific rate of cell growth in consideration of its time-dependent changes. The existence, uniqueness and boundedness of solutions to the system and the continuity of solutions with respect to the parameters are discussed. In addition, a parameter identification problem of the system is developed and a feasible optimization algorithm is constructed to solve it. Numerical result shows that the improved model could describe the batch culture well.     

Pathway identification using parallel optimization for a complex metabolic system in microbial continuous culture

The bio-dissimilation of glycerol to 1,3-propanediol (1,3-PD) by Klebsiella pneumoniae (K. pneumoniae) is a complex bioprocess due to the multiple inhibitions of substrate and products onto the cell growth. In consideration of the fact that both the inhibition mechanisms of 3-hydroxypropionaldehyde (3-HPA) onto the cell growth and the transport systems of glycerol and 1,3-PD across the cell membrane are still unclear, we consider 72 possible metabolic pathways, and establish a novel mathematical model which is represented by an eight-dimensional nonlinear dynamical system. The existence, uniqueness, continuous dependence of solutions to the system and the compactness of the solution set are explored. On the basis of biological robustness, we give a quantitative definition of robustness index of the intracellular substances. Taking the robustness index of the intracellular substances together with the relative error between the experimental data and the computational values of the extracellular substances as a performance index, a parameter identification model is proposed for the nonlinear dynamical system, in which 43848 continuous variables and 1152 discrete variables are involved. A parallel particle swarm optimization — pathways identification algorithm (PPSO-PIA) is constructed to find the optimal pathway and parameters under various experiments conditions. Numerical results show that the optimal pathway and the corresponding dynamical system can describe the continuous fermentation reasonably.