Multi-objective optimization of a nonlinear switched time-delay system in microbial fed-batch process
Introduction
The microbial conversion of glycerol induced by Klebisella pneumoniae (K.pneumoniae) to 1,3-Propanediol (1,3-PD) has raised the extensive concerns due to its pollution-free production process, easy availability of renewable feedstock and higher productivity [1], [2], [3], [4]. Among the various fermentation modes which include continuous, batch and fed-batch mode [5], [6], [7], the fed-batch mode is the most efficient in reduction of substrate inhibition and improvement of productivity of product than other fermentation modes.
The fed-batch fermentation is typically implemented by switching between batch mode (in which the feeding of substrates are closed) and feeding mode (in which the feeding of substrates are opened). Liu et al. [8] modelled the fed-batch process as a nonlinear switched system. According to whether glycerol and alkali are added at the same time, fed-batch fermentation can be divided into two types: coupling and non-coupling. For coupled fed-batch fermentation, Wang et al. [9] proposed a nonlinear switched system and discussed the non-Zeno behaviors of the system and some basic properties of the solutions, including the existence, uniqueness, boundedness and regularity. Then the sensitivity of parameters was analyzed in [10]. For uncoupled fed-batch fermentation, Ye et al. [11] established a nonlinear hybrid system model according to the rules of alkali feeding determined by observation equation of pH. Based on this system model, the parameter identification and optimal control problem were discussed in [12], [13]. However, the above literatures only considered the concentrations of extracellular substances. In 2008, Sun et al. [14] firstly proposed a mathematical model involving concentration changes of intracellular substances and the effects of two key enzymes. Considering the changes in the concentration of substances in the intracellular and extracellular environment, Niu et al. [15] studied the optimal control problem by taking the maximization of product concentration at the terminal time as the performance index. Based on the principle of gene regulation, the regulatory network model including gene-mRNA-enzyme-products was further considered in [16].
Time-delay systems have an active area of research, for example, see [17], [18], [19], [20]. It is worth mentioning that the time-delay is not considered in the above fermentation processes. In fact, fermentation process is also influenced by time-delays [21], [22]. Some reasons may be responsible for the occurrence of time-delays in the fed-batch process: the glycerol and 1,3-PD have to be transported across the cell membrane requiring a certain amount of time; the inhibitory effects of 3-hydroxypropionaldehy (3-HPA) on the substrate conversion and the product generation requires an accumulation process of 3-HPA. Yu et al. [23] studied the stability analysis of genetic regulatory networks with switching parameters and time-delays. Considering the time-delays of microbial growth process and the changes in concentration of extracellular substances, Liu et al. [24] established a nonlinear switched time-delay system for coupled fed-batch fermentation, in which the sensitivity analysis of system parameters was also discussed. However, only one objective is considered in the above study. In order to maximize the productivity of target product and minimize the consumption rate of substrate, a multi-objective optimization problem for coupled fed-batch fermentation was studied in [25], where only the extracellular environment and a given time-delay were considered. Based on the above study, a robust multi-objective optimal switching control model was proposed in [26]. In order to further improve the accuracy of the model, it is significant to study the multi-objective optimization problem with time-delays for uncoupled fed-batch fermentation, in which the intracellular environment (including the regulation of dha regulator) and the extracellular environment should be comprehensive considered.
In this paper, by comprehensive considering the regulation of dha regulator, the intracellular and extracellular environment, we establish a nonlinear switched time-delay system model for uncoupled fed-batch fermentation. According to the above two reasons that may cause the occurrence of time-delays in the fed-batch process, the unknown time-delays of extracellular 1,3-PD, intracellular glycerol, 3-HPA and 1,3-PD are considered in system. In order to determine the optimal values of time-delays, a biological robustness is defined by a weighted sum of the expectation and variance of the relative deviation between states before and after the time-delays are perturbed. In actual fermentation, for improving the economic benefit of fermentation, we hope to obtain the maximum yield of 1,3-PD and the minimum consumption of glycerol in the shortest fermentation time, so it is necessary to optimize the value of the terminal fermentation time. Thus, taking the minimization of biological robustness, the maximization of 1,3-PD productivity and the minimization of consumption rate of glycerol as the objectives, we present a multi-objective optimization problem involving the time-delay system and continuous state inequality constraints. The feeding rates of glycerol and alkali, switching instants between batch and feeding modes, terminal time and time-delays are regarded as the decision variables. For solving the multi-objective optimization problem, we convert it into a sequence of single-objective problems by using NC method [27]. Considering that the gradient formulas of objective functions and constraints with respect to time-delays and terminal time are not obvious and the processes of solving are complicated, especially the process of solving the gradient formulas of biological robustness that contains expectation, variance and multiple time-delays, we give the gradient formulas of targets and constraints in Theorems 1–3 by using variation principle and time-scaling transformation technology. Then, an optimization algorithm based on SQP algorithm is constructed to solve these single-objective problems. By a large scale calculation, a set of Pareto solutions which can provide reference value for actual experiment are obtained.
The remainder of the paper is organized as follows. Section 2 describes the nonlinear switched time-delay system for the fed-batch fermentation. Section 3 presents a biologic robustness and a multi-objective problem. The multi-objective optimization strategy is formulated in Section 4. The numerical solutions of the multi-objective problem are given in Section 5. Finally, Section 6 gives the concluding remarks.
Section snippets
Fed-batch fermentation model
Fed-batch fermentation begins with batch mode, then glycerol and alkali (usually NaOH) are discontinuously added to the reactor for keeping the concentration of glycerol and the value of pH in required levels. There are four modes in fed-batch fermentation process.
batch process (neither glycerol feeding nor alkali feeding);
semibatch process with alkali feeding only;
semibatch process with glycerol feeding only;
semibatch process with both glycerol and alkali feeding.
In
Biological robustness
Considering that the values of time-delays are unknown, we intend to employ the idea of biological robustness to identify time-delays. In [35], [36], Kitano argued that robustness is a property allowing a system to preserve functionality when faced with external and internal perturbations. The definition of biological robustness should meet the case that the smaller the changes in state variables provoked by small shift in unknown parameters for the overall fermentation process, the more robust
Normal constraint method
Consider the following single-objective problems (SPi), i ∈ I3, where Ji is the ith component of J. The optimal solutions and optimal values for the problems (SPi), i ∈ I3 are regarded as i ∈ I3, respectively. To apply the NC method, we require three anchor points, and the ith anchor point is denoted by . The hyper plane containing all anchor points is
Numerical results
Under anaerobic conditions at 37∘C, a fed-batch fermentation starts from the batch process with initial state vector . The pH is controlled in 6.48–6.52, i.e., and . Glycerol is fed once every 100 s. The lower and upper bounds of tf are and respectively. The upper bounds of time-delays are . In problem the number m is taken as 6.
In Algorithm (MPA-SQP), the initial values of ε, γ are
Conclusions
In this paper, we propose a nonlinear time-delay switched system model involving unknown time-delays and free terminal times to describe the fed-batch fermentation process in which both the regulation of dha regulator, extracellular and intracellular environment are considered. For determining optimal values of time-delays, maximizing productivity of 1,3-PD and minimizing consumption of glycerol in actual fermentation, we present a multi-objective optimization problem involving the time-delay
Declaration of Competing Interest
None.
Acknowledgments
This work was supported by Natural Science Foundation of Shandong Province, China (Grant No. ZR2014FM029) and the Fundamental Research Funds for the Central Universities, China (Grant No. DUT19LK37).
References (38)
- et al.
Microbial production of 1,3-propanediol: recent developments and emerging opportunities
Biotechnol. Adv.
(2009) - et al.
High concentration and productivity of 1, 3-propanediol from continuous fermentation of glycerol by klebsiella pneumoniae
Enzyme Microb. Technol.
(1997) - et al.
Existence of equilibrium points and stability of the nonlinear dynamical system in microbial continuous cultures
Appl. Math. Comput.
(2009) - et al.
Optimal switching control for microbial fed-batch culture
Nonlinear Anal.: Hybrid Syst.
(2008) - et al.
Modelling and properties of a nonlinear autonomous switching system in fed-batch culture of glycerol
Commun. Nonlinear Sci. Numer. Simul.
(2012) - et al.
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
Nonlinear Anal.: Real World Appl.
(2011) - et al.
Nonlinear hybrid system and parameter identification of microbial fed-batch culture with open loop glycerol input and ph logic control
Appl. Math. Model.
(2012) - et al.
Optimization of a fed-batch bioreactor for 1,3-propanediol production using hybrid nonlinear optimal control
J. Process Control
(2014) - et al.
Mathematical modeling of glycerol fermentation by pneumoniae: Concerning enzyme-catalytic reductive pathway and transport of glycerol and 1,3-propanediol across cell membrane
Biochem. Eng. J.
(2008) - et al.
Optimal control of nonlinear switched system in an uncoupled microbial fed-batch fermentation process
J. Franklin Inst.
(2018)