Sequential Geometric Programming Method for Parameter Estimation of a Nonlinear System in Microbial Continuous Fermentation

Xu, Gongxian; Liu, Zijia

doi:https://doi.org/10.1155/2023/8072920

International Journal of Chemical Engineering

On this page

Abstract Introduction Discussion Conclusions Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2023 | Article ID 8072920 | https://doi.org/10.1155/2023/8072920

Sequential Geometric Programming Method for Parameter Estimation of a Nonlinear System in Microbial Continuous Fermentation

Gongxian Xu¹and Zijia Liu¹

Academic Editor: Jitendra Kumar

Received11 Mar 2023

Revised08 Sept 2023

Accepted15 Sept 2023

Published10 Oct 2023

Abstract

This paper addresses the problem of parameter estimation for the microbial continuous fermentation of glycerol to 1,3-propanediol. A nonlinear dynamical system is first presented to describe the microbial continuous fermentation. Some mathematical properties of the dynamical system in the microbial continuous fermentation are also presented. A parameter estimation model is proposed to estimate the parameters of the dynamical system. The proposed estimation model is a large-scale, nonlinear, and nonconvex optimization problem if the number of experimental groups is large. A sequential geometric programming (SGP) method is proposed to efficiently solve the parameter estimation problem. The results indicated that our proposed SGP method can yield smaller errors between the experimental and calculated steady-state concentrations than the existing seven methods. For the five error indices considered, that is, the concentration errors of biomass, glycerol, 1,3-propanediol, acetic acid, and ethanol, the results obtained using the proposed SGP method are better than those obtained using the methods in the literature (Xiu et al., Gao et al., Sun et al., Sun et al., Li and Qu, Wang et al., and Zhang and Xu), with improvements of approximately 71.86–95.03%, 52.08–94.87%, 99.70–99.98%, 5.39–90.29%, and 12.67–80.83%, respectively. This concludes that the established dynamical system can better describe the microbial continuous fermentation. We also present that our established dynamical system has multiple positive steady states in some fermentation conditions. We observe that there are two regions of multiple positive steady states at relatively high values of substrate glycerol concentration in feed medium.

1. Introduction

1,3-Propanediol (1,3-PDO) plays a key role in many industry fields, as it has extensive applications on a large commercial scale [1, 2]. In the production of 1,3-PDO, the microbial fermentation of glycerol to 1,3-PDO is attracting extensive attention because of its green production process [1]. In recent years, much research has been directed toward the development of the microbial fermentation process of glycerol, including the metabolic engineering and synthetic biology strategies in the biomanufacturing of 1,3-PDO and the mathematical modeling, optimization, and control of such processes [1, 3–35]. For example, Zhu et al. [1] reviewed the advances in metabolic engineering and synthetic biology techniques in the microbial production of 1,3-PDO. Fokum et al. [3] reviewed the recent developments in the biomanufacturing strategies of 1,3-PDO from glycerol. Wang et al. [4] reprogrammed the metabolism of Klebsiella pneumoniae to efficiently produce 1,3-PDO. The conducted metabolic engineering manipulations can dramatically reduce the accumulation of acetate. Lee et al. [5] reviewed the advances in biological and chemical techniques for the 1,3-PDO production from glycerol. Asopa et al. [6] used Saccharomyces cerevisiae to produce 1,3-PDO and butyric acid through microbial fermentation of glycerol. Gupta et al. [7] used the new producer, Clostridium butyricum L4, to develop a fed-batch fermentation process of crude glycerol into 1,3-PDO. The developed fermentation process can obtain a high yield of 1,3-PDO. Liu et al. [8], Wang et al. [9], Gao et al. [10], and Xu et al. [11] addressed the optimization models and methods to optimize the fermentation processes of glycerol. Pan et al. [12] addressed the theoretical study of feedback control for a two-stage fermentation process of 1,3-PDO. To deal with the challenges of the online measurement of the microbial fermentation process, Zhang et al. [13] presented a robust soft sensor to efficiently predict the concentrations of 1,3-PDO and glycerol. Xu and Li [14] presented the mathematical optimization approach to optimize the metabolic objective for glycerol metabolism into 1,3-PDO production. Xu et al. [15] proposed a two-stage approach to efficiently solve the parameter identification problem of the microbial batch process of glycerol. Pröschle et al. [16] designed the advanced controller to control the fed-batch fermenter of glycerol to 1,3-PDO. Emel’yanenko and Verevkin [17] addressed the thermodynamic properties of 1,3-PDO. Rodriguez et al. [18] proposed the kinetic model to describe the fermentation process of the raw glycerol into 1,3-PDO. Silva et al. [19] addressed the multiplicity study of steady states in a microbial fermentation process of 1,3-PDO. Liu and Zhao [20] presented an optimal switching technique to control the 1,3-PDO fed-batch production. Yuan et al. [21] proposed a robust feedback method to control the nonlinear switched system of 1,3-PDO fed-batch production. Liberato et al. [22] used both crude glycerol and corn steep liquor in 1,3-PDO production using a Clostridium butyricum strain.

Xiu et al. [23], Gao et al. [24], Sun et al. [25], Sun et al. [26], Li and Qu [27], Wang et al. [28], and Zhang and Xu [29] used the excess kinetic models, S-system, and fractional-order model to mathematically describe the microbial continuous fermentation of glycerol to 1,3-PDO. However, a comparison study suggested that the steady-state concentrations calculated by these works significantly violate the experimental data (refer to Section 5). For example, the errors of glycerol concentrations reached more than 43% in [24–27]. This concludes that the mathematical models established by these researchers cannot satisfactorily describe the real bioprocess. To better describe the microbial continuous fermentation of glycerol, it is necessary to present new mathematical modeling or parameter estimation methods.

For this purpose, in the present study, we address the problem of parameter estimation for the microbial continuous fermentation of glycerol to 1,3-PDO. First, a nonlinear dynamical system is presented to describe the microbial continuous fermentation. Then, some mathematical properties of the dynamical system in the microbial continuous fermentation are also presented in terms of the estimated parameters, reactant concentrations, and fermentation conditions. Section 3 proposes a parameter estimation model to estimate the value of the parameter vector in the dynamical system of the microbial continuous fermentation. Section 4 proposes a sequential geometric programming (SGP) method to efficiently solve the nonlinear, nonconvex parameter estimation problem. Section 5 presents the computation results obtained using the proposed SGP algorithm and also presents a comparative study to demonstrate that the proposed SGP algorithm can yield smaller errors between the experimental and calculated steady-state concentrations than the other seven methods. Additionally, we investigate the multiple positive steady states of our proposed dynamical system in Section 5. Finally, we provide the conclusions of the present work in Section 6.

2. Nonlinear Dynamical System of Microbial Continuous Fermentation

2.1. Nonlinear Dynamical System

In the microbial continuous fermentation of glycerol to 1,3-PDO by Klebsiella pneumonia, the substrate glycerol is continuously added to the fermenter, and equal volumes of substrate glycerol, reaction products, and cells are extracted from the fermenter. The concentration of various substances in the fermenter is in a constant state. The main products of the microbial continuous fermentation include 1,3-PDO, acetic acid, and ethanol [23]. Figure 1 presents the schematic of the microbial continuous fermentation in the fermenter. In this figure, is the volume flow of feed medium into the fermenter, L/h; denotes the concentration of substrate glycerol in feed medium, mmol/L; is the volume of fermentation broth, L; represents the biomass, g/L; and , , , and represent the concentrations of glycerol, 1,3-PDO, acetic acid, and ethanol, respectively, mmol/L.

A process can be modeled by some modeling methods, such as the neural network modeling techniques [36, 37] and the ODE (ordinary differential equation) methods [38]. Based on the basic conservation law and the previous literature [23], in this study, the material balance equations of the microbial continuous fermentation are written as the following five-dimensional nonlinear ODEs:where is the fermentation time, h; represents the terminal time of the microbial continuous fermentation; represents the dilution rate, h⁻¹; is the nonlinear function that denotes the specific growth rate of cells, h⁻¹; is the nonlinear function representing the specific consumption rate of glycerol, mmol/(g·h); and , , and are the nonlinear functions denoting the specific formation rates of 1,3-PDO, acetic acid, and ethanol, respectively, mmol/(g·h). Considering the nature of the microbial continuous fermentation, we set the rates () to be .

The rates () in (1)–(5) are expressed as the following equations:

In these equations, is the model parameter vector to be estimated later with () and (), where is defined aswhere and when and and when .

Under certain experimental conditions, the maximum value of is h⁻¹, and the Monod saturation constant is mmol/L. The critical values of , , , , and are g/L, mmol/L, mmol/L, mmol/L, and mmol/L, respectively. Therefore, microbial fermentation system (1)–(6) will work in the subset of , expressed as follows:

In addition, the dilution rate and glycerol concentration in the feed will stay within certain limits, i.e., , where is expressed as follows:where and .

By introducing the expressions of functions () into (1)–(6), we obtain the following reformulations of the microbial continuous fermentation:

Now, we perform some transformations, as follows:

Then, we obtain the following dynamical system with a power function structure:

The above dynamical system can be further represented as

In this representation, , , and the functions and are defined as

Dynamical system (14)–(16) is a differential-algebraic system.

Remark 1. Compared to the model (1)–(6), the advantages of the transformed dynamical system (14)–(16) are as follows: (1) it is still a nonlinear model that can describe the nonlinear fermentation process and (2) it involves a special power function structure that can be used to propose a novel SGP method for the parameter estimation problem of the microbial continuous fermentation.

2.2. Mathematical Properties of the Dynamical System

In this subsection, we consider the properties of dynamical system (14)–(16).

Property 2. For , the functions () and () provided in dynamical system (14)–(16) are continuously differentiable on , i.e., , , and the functions and are continuous in on .

Proof. By the definitions of the functions () and (), it can be easily verified that the conclusion is valid.

Property 3. For , the functions () and () provided in dynamical system (14)–(16) are locally Lipschitz continuous on with respect to .

Proof. Let . By the mean value theorem, there exist () and () such thatAs , , and are bounded sets, the derivatives of () and () are bounded on from Property 2. Thus, and are bounded. For any , letThen, we haveThese conclude that () and () are locally Lipschitz continuous on with respect to .

Property 4. For , dynamical system (14)–(16) has a unique solution, expressed by , and is continuous on with respect to .

Proof. For and , we know and , and the functions and are continuous in on from Property 2. Therefore, dynamical system (14)–(16) has a unique solution, expressed by . According to the continuous dependence of solutions on parameters in nonlinear differential equations, is continuous on with respect to .

Property 5. Let the solution set of dynamical system (14)–(16) be is the solution of dynamical system (14)–(16) for θ ∈ Θ}.
Then, is a compact set in .

Proof. From the definition of set , is a bounded closed set in . Therefore, is a compact set in . By Properties 2 and 4, we obtain that the mapping from to is continuous. This concludes that is a compact set in .

Property 6. For and , the vector functionsatisfies the following linear growth condition with respect to :where and .

Proof. By the derivation of dynamical system (14)–(16) in Section 2.1, we obtainThen, for , , and , we haveAs , , , and , we obtainThen, by (), we haveLet .
Then, .
Thus, Let and , and then we obtain . and because and . This completes the proof of Property 6.

Remark 7. It can be proven that and in Property 6 are dependent on the operating parameters and . Thus, there will be different and values for different operating conditions of the microbial continuous fermentation.

Property 8. The functions () and () in dynamical system (14)–(16) are signomial functions.

Remark 9. By Property 8, we observe that () and () in dynamical system (14)–(16) involve a special structure in the form of signomial functions. This type of mathematical function is often found in geometric programming (GP) problems [39, 40]. In Section 4, we will propose a novel GP method for the parameter estimation problem of the microbial continuous fermentation.

3. Parameter Estimation Model of the Dynamical System

To estimate the value of parameter vector in dynamical system (14)–(16) of microbial continuous fermentation, we will first propose a parameter estimation model in this section.

Given certain fermentation condition , we can measure the steady-state concentrations of all reactants in the microbial continuous fermentation. Now, we have different sets of experimental steady-state data that correspond to different fermentation conditions (). Let (, , ) be the experimental steady-state concentrations of biomass (), glycerol (), 1,3-PDO (), acetic acid (), and ethanol () under fermentation conditions , and let (, ) be the corresponding steady-state concentrations of variables () calculated by the steady-state conditions. To keep the sum of the squared steady-state concentration deviations from the experimental data minimized, we propose the following optimization model to estimate parameter in dynamical system (14)–(16) of the microbial continuous fermentation:subject to satisfying the steady-state constraints:and the bound constraints to the variables:where (), , the equality constraints are the steady-state conditions, and the last three constraints control the corresponding variables to stay within certain limits.

Remark 10. In parameter estimation problem (28)-(30), the number of optimization variables is , the number of equality constraints is , the number of lower bound constraints is , and the number of upper bound constraints is . Therefore, if the number of experimental groups is large, then problem (28)-(30) will be a large-scale, nonlinear, and nonconvex optimization problem.

4. SGP Method for the Parameter Estimation Model

As stated previously, proposed parameter estimation model (28)-(30) of the dynamical system is a nonlinear, nonconvex optimization problem. To efficiently solve it, we propose an SGP method in this work.

As there is an implicit requirement that the optimization variables are positive in the framework of GP, we first denote with () and (). Additionally, we replace the expression with both and . The inequality can be further written as

Then, model (28)-(30) can be represented as the following equivalent formulations:subject to satisfyingwhere . It can be observed that each of equality constraints (35)–(42) of this problem includes a ratio of certain two posynomials. This type of constraint form is often found in complementary geometric programming. Problem (32)–(47) is an intrinsically nonconvex NP-hard problem.

Replacing each of equality constraints (35)–(42) with two inequality constraints, we can rewrite problem (32)–(47) assubject to satisfyingwhere ; denotes the weighting coefficient with a sufficiently large value. We can easily observe that if (), then problem (48)-(49) is equivalent to problem (32)–(47). The reason why is used here instead of is that the optimization variables of GP must be positive. The introduction of the penalty term can guarantee that at the optimal solution of problem (48)-(49).

By using some derivations to those inequalities involving variables (), we obtain the equivalent problem, as follows:subject to satisfying

It is well known that the standard GP involves a posynomial objective and monomial equality and/or posynomial inequality constraints. This type of optimization problem can be solved very efficiently because it is convex with the logarithmic transformation. Problem (50)-(51) is not a standard GP because many of its inequality constraints are not legal posynomial ones. To deal with this issue, an efficient condensation method is used to transform these inequality constraints into valid posynomial ones. This approach is to approximate every posynomial function in the denominator of inequality constraints by using a monomial function.

Let be a posynomial where are the monomials. Then, using the arithmetic-geometric mean inequality, we obtainwhere are calculated through

Here, is a given point. We have . Inequality (52) implies that can be replaced with , where is a posynomial.

Applying the approach above to problem (50)-(51), we have the following problem:subject to satisfyingwhere and (, ) are the monomial functions approximated through (52). Problem (54)-(55) is a standard GP.

We now summarize the proposed SGP method for the parameter estimation model of the dynamical system in the microbial continuous fermentation and provide the following SGP algorithm, denoted as Algorithm 1.

	Step 0. Choose the initial values , , , , and () of optimization variables , , , , and . Given the solution accuracy and initial weighting coefficient , set .
	Step 1. For the given , , , , and (), construct the monomials and (, ) using (52).
	Step 2. Solve problem (54)-(55) to attain , , , , and () with the weighting coefficient .
	Step 3. If , then stop.
	Step 4. Update with . Set and continue from Step 1.

Remark 11. In the implementation of the proposed Algorithm 1, numerical computation problems may occur when is very small. To deal with this issue, we can transform () into by and obtain the following problem similar to standard GP (54)-(55).subject to satisfyingwhere ; and (, ) are the monomial functions approximated through (52). In problem (56)-(57), the number of upper bound constraints is , which is fewer than that of (54)-(55).
Based on Algorithm 1 and the above analysis, we present Algorithm 2.

	Step 0. Choose the initial values , , , , and () of optimization variables , , , , and . Given the solution accuracy and initial weighting coefficient , select (). Set .
	Step 1. For the given , , , , and (), construct the monomials and (, ) using (52).
	Step 2. Solve problem (54)-(55) to attain , , , , and () with the weighting coefficient .
	Step 3. If , then go to Step 4. Else, go to Step 5.
	Step 4. Choose with . Set and continue from Step 1.
	Step 5. Set , and go to Step 6.
	Step 6. For the given , , , , and (), construct the monomials and (, ) using (52).
	Step 7. Solve problem (56)-(57) to attain , , , , and () with the weighting coefficient .
	Step 8. If , then stop. Else, go to Step 9.
	Step 9. Choose with . Set and continue from Step 6.

5. Computational Results and Discussion

In this section, we apply the proposed SGP method (Algorithm 2) to solve nonconvex parameter estimation model (28)–(30) of dynamical system (14)–(16). Experimental steady-state data (, , ) under 21 different fermentation conditions () were drawn from the literature [23]. Here, the number of experimental groups is 21, which means that both problems (54)-(55) and (56)-(57) are large-scale, nonlinear optimization problems. Table 1 presents the number of optimization variables, the number of equality constraints, the number of inequality constraints, the number of lower bound constraints, and the number of upper bound constraints for problems (54)-(55) and (56)-(57). In the implementation of Algorithm 2, the following parameters were set: h⁻¹, mmol/L, g/L, mmol/L, mmol/L, mmol/L, mmol/L, h⁻¹, h⁻¹, (), (), (), (), , and . Additionally, we set () if . We set if , where .

After 53 iterations, Algorithm 2 stops with and (). Table 2 presents the optimal values of the estimated parameters () obtained using Algorithm 2. Figure 2 provides the comparison between the experimental data and steady-state concentrations calculated using the proposed SGP approach. Table 3 presents the comparison between the proposed SGP method and the approaches used in the literature [23–29]. In this table, , , , , and denote the error functions of biomass, glycerol, 1,3-PDO, acetic acid, and ethanol, respectively, and are defined aswhere (, , ) are the experimental steady-state concentrations and (, , ) are the corresponding positive steady-state concentrations of variables () calculated by the steady-state equations with optimal model parameters. As can be observed in Table 3, our proposed SGP approach can produce smaller errors between the experimental and calculated steady-state concentrations than the other seven methods. As observed for the seven error indices considered, , , , , , , and in Table 3, the results obtained using the proposed SGP method are better than those by the methods used in [23–29], with improvements of approximately 71.86–95.03%, 52.08–94.87%, 99.70–99.98%, 5.39–90.29%, 12.67–80.83%, 50.12–89.05%, and 70.01–88.37%, respectively. We can also observe that the methods used in the literature [24–27] yield very large errors of glycerol concentrations reaching more than 43%. These conclude that our established dynamical system can better describe the microbial continuous fermentation.

The experimental studies indicated that in the microbial continuous fermentation, the metabolic overflow of products and their inhibition on cell growth can give rise to multiple steady-state phenomena. To investigate whether our established dynamical system of the microbial continuous fermentation has multiple steady states, we can compute the steady-state equations (i.e., , ) to find the positive steady states under different fermentation conditions . After some computations, we can observe that our proposed dynamical system has multiple positive steady states in some fermentation conditions . As an illustration, Figure 3 presents the experimental and computational results of the microbial continuous fermentation under 1000 different fermentation conditions, where h⁻¹, mmol/L, and the red squares denote the experimental data. From Figure 3, we can observe that the dynamical system has three types of positive steady states: (1) one positive steady state at mmol/L and mmol/L, (2) three positive steady states at mmol/L, and (3) two positive steady states at mmol/L. This concludes that the proposed dynamical system has multiple positive steady states under some fermentation conditions .

6. Conclusions

This work has studied the problem of parameter estimation for the microbial continuous fermentation. A nonlinear dynamical system has been first presented to describe the microbial continuous fermentation. To estimate the value of the parameter vector in the dynamical system, a parameter estimation model as presented in (28)–(30) has been proposed. Model (28)–(30) can minimize the sum of the squared steady-state concentration deviations from the experimental data and has many optimization variables and constraints. Therefore, if the number of experimental groups is large, then problem (28)–(30) will be a large-scale, nonlinear, and nonconvex optimization problem. To efficiently solve problem (28)–(30), an SGP method has been proposed. The results indicated that our proposed algorithm can yield smaller errors between the experimental and calculated steady-state concentrations than the existing methods in the literature [23–29]. This concludes that the established dynamical system can better describe the microbial continuous fermentation and that the proposed SGP method is valid. The proposed framework in this work can also be applied to the parameter estimation of other continuous (bio)chemical processes. We also observe that there are two regions of multiple positive steady-states at relatively high values of substrate glycerol concentration in feed medium.

Data Availability

The data used to support the findings of this study are included within the article and are also available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (nos. 11101051, 61976027, and 62273056), Natural Science Foundation of Liaoning Province (no. 2022-MS-371), and Liaoning Revitalization Talents Program (no. XLYC2008002).

References

F. Zhu, D. Liu, and Z. Chen, “Recent advances in biological production of 1,3-propanediol: new routes and engineering strategies,” Green Chemistry, vol. 24, no. 4, pp. 1390–1403, 2022.
View at: Publisher Site | Google Scholar
M. Laura, T. Monica, and V. Dan-Cristian, “The effect of crude glycerol impurities on 1,3-propanediol biosynthesis by Klebsiella pneumoniae DSMZ 2026,” Renewable Energy, vol. 153, pp. 1418–1427, 2020.
View at: Publisher Site | Google Scholar
E. Fokum, H. M. Zabed, J. Yun, G. Zhang, and X. Qi, “Recent technological and strategical developments in the biomanufacturing of 1,3-propanediol from glycerol,” International journal of Environmental Science and Technology, vol. 18, no. 8, pp. 2467–2490, 2021.
View at: Publisher Site | Google Scholar
W. Wang, X. Yu, Y. Wei, R. Ledesma-Amaro, and X. Ji, “Reprogramming the metabolism of Klebsiella pneumoniae for efficient 1,3-propanediol production,” Chemical Engineering Science, vol. 236, Article ID 116539, 2021.
View at: Publisher Site | Google Scholar
C. S. Lee, M. K. Aroua, W. M. A. W. Daud et al., “A review: conversion of bioglycerol into 1,3-propanediol via biological and chemical method,” Renewable and Sustainable Energy Reviews, vol. 42, pp. 963–972, 2015.
View at: Publisher Site | Google Scholar
R. P. Asopa, M. M. Ikram, and V. K. Saharan, “Valorization of glycerol into 1,3-propanediol and organic acids using biocatalyst Saccharomyces cerevisiae,” Bioresource Technology Reports, vol. 18, Article ID 101084, 2022.
View at: Publisher Site | Google Scholar
P. Gupta, M. Kumar, R. P. Gupta, S. K. Puri, and S. S. V. Ramakumar, “Fermentative reforming of crude glycerol to 1,3-propanediol using Clostridium butyricum strain L4,” Chemosphere, vol. 292, Article ID 133426, 2022.
View at: Publisher Site | Google Scholar
C. Liu, Z. Gong, K. L. Teo, R. Loxton, and E. Feng, “Bi-objective dynamic optimization of a nonlinear time-delay system in microbial batch process,” Optimization Letters, vol. 12, no. 6, pp. 1249–1264, 2018.
View at: Publisher Site | Google Scholar
J. Wang, X. Zhang, J. Ye, J. Wang, and E. Feng, “Optimizing design for continuous conversion of glycerol to 1,3-propanediol using discrete-valued optimal control,” Journal of Process Control, vol. 104, pp. 126–134, 2021.
View at: Publisher Site | Google Scholar
X. Gao, J. Zhai, and E. Feng, “Multi-objective optimization of a nonlinear switched time-delay system in microbial fed-batch process,” Journal of the Franklin Institute, vol. 357, no. 17, pp. 12609–12639, 2020.
View at: Publisher Site | Google Scholar
G. Xu, Y. Liu, and Q. Gao, “Multi-objective optimization of a continuous bio-dissimilation process of glycerol to 1,3-propanediol,” Journal of Biotechnology, vol. 219, pp. 59–71, 2016.
View at: Publisher Site | Google Scholar
D. Pan, X. Wang, J. Wang, H. Shi, G. Wang, and Z. Xiu, “Optimization and feedback control system of dilution rate for 1,3-propanediol in two-stage fermentation: a theoretical study,” Biotechnology Progress, vol. 38, no. 1, Article ID e3225, 2022.
View at: Publisher Site | Google Scholar
A. Zhang, K. Zhu, X. Zhuang et al., “A robust soft sensor to monitor 1,3-propanediol fermentation process by Clostridium butyricum based on artificial neural network,” Biotechnology and Bioengineering, vol. 117, no. 11, pp. 3345–3355, 2020.
View at: Publisher Site | Google Scholar
G. Xu and C. Li, “Identifying the shared metabolic objectives of glycerol bioconversion in Klebsiella pneumoniae under different culture conditions,” Journal of Biotechnology, vol. 248, pp. 59–68, 2017.
View at: Publisher Site | Google Scholar
G. Xu, D. Lv, and W. Tan, “A two-stage method for parameter identification of a nonlinear system in a microbial batch process,” Applied Sciences, vol. 9, no. 2, Article ID 337, 2019.
View at: Publisher Site | Google Scholar
T. Pröschle, D. Ávalos, A. Sepúlveda, F. Llull, F. Ibáñez, and J. R. Pérez-Correa, “Fermentation of glycerol using Clostridium butyricum for the production of 1,3-Propanediol in a Fed-batch bioreactor using advanced controllers,” in Proceedings of the 20th International Conference on Modeling & Applied Simulation (MAS 2021), pp. 170–175, Cracow, Poland, September 2021.
View at: Google Scholar
V. N. Emel’yanenko and S. P. Verevkin, “Benchmark thermodynamic properties of 1,3-propanediol: comprehensive experimental and theoretical study,” The Journal of Chemical Thermodynamics, vol. 85, pp. 111–119, 2015.
View at: Publisher Site | Google Scholar
A. Rodriguez, M. Wojtusik, F. Masca, V. E. Santos, and F. Garcia-Ochoa, “Kinetic modeling of 1,3-propanediol production from raw glycerol by Shimwellia blattae: influence of the initial substrate concentration,” Biochemical Engineering Journal, vol. 117, pp. 57–65, 2017.
View at: Publisher Site | Google Scholar
J. P. Silva, Y. B. Almeida, I. O. Pinheiro, A. Knoelchelmann, and J. M. F. Silva, “Multiplicity of steady states in a bioreactor during the production of 1,3-propanediol by Clostridium butyricum,” Bioprocess and Biosystems Engineering, vol. 38, no. 2, pp. 229–235, 2015.
View at: Publisher Site | Google Scholar
C. Liu and Q. Zhao, “Optimal switching control of 1,3-propanediol fed-batch production with a cost on smooth feeding rate variation,” Nonlinear Analysis: Hybrid Systems, vol. 49, Article ID 101372, 2023.
View at: Publisher Site | Google Scholar
J. Yuan, C. Wu, C. Liu, K. L. Teo, and J. Xie, “Robust suboptimal feedback control for a fed-batch nonlinear time-delayed switched system,” Journal of the Franklin Institute, vol. 360, no. 3, pp. 1835–1869, 2023.
View at: Publisher Site | Google Scholar
V. S. Liberato, F. F. Martins, C. Maria S Ribeiro, M. Alice Z Coelho, and T. F. Ferreira, “Two-waste culture medium to produce 1,3-propanediol through a wild Clostridium butyricum strain,” Fuel, vol. 322, Article ID 124202, 2022.
View at: Publisher Site | Google Scholar
Z. Xiu, B. Song, Z. Wang, L. Sun, E. Feng, and A.-P. Zeng, “Optimization of dissimilation of glycerol to 1,3-propanediol by Klebsiella pneumoniae in one- and two-stage anaerobic cultures,” Biochemical Engineering Journal, vol. 19, no. 3, pp. 189–197, 2004.
View at: Publisher Site | Google Scholar
C. Gao, E. Feng, Z. Wang, and Z. Xiu, “Parameters identification problem of the nonlinear dynamical system in microbial continuous cultures,” Applied Mathematics and Computation, vol. 169, no. 1, pp. 476–484, 2005.
View at: Publisher Site | Google Scholar
Y. Sun, W. Qi, H. Teng, Z. Xiu, and A. P. Zeng, “Mathematical modeling of glycerol fermentation by Klebsiella pneumoniae: concerning enzyme-catalytic reductive pathway and transport of glycerol and 1,3-propanediol across cell membrane,” Biochemical Engineering Journal, vol. 38, no. 1, pp. 22–32, 2008.
View at: Publisher Site | Google Scholar
Y. Sun, J. Ye, X. Mu et al., “Nonlinear mathematical simulation and analysis of dha regulon for glycerol metabolism in Klebsiella pneumoniae,” Chinese Journal of Chemical Engineering, vol. 20, no. 5, pp. 958–970, 2012.
View at: Publisher Site | Google Scholar
X. Li and R. Qu, “Parameter identification and terminal steady-state optimization for s system in microbial continuous fermentation,” International Journal of Biomathematics, vol. 5, no. 2, Article ID 1250020, 2012.
View at: Publisher Site | Google Scholar
D. Wang, J. Zhai, and E. Feng, “Fractional order modeling and parameter identification for a class of continuous fermentation,” Journal of Systems Science and Mathematical Sciences, vol. 40, pp. 1517–1530, 2020.
View at: Google Scholar
J. Zhang and G. Xu, “Parameter identification and nonlinear analysis of the continuous bio-dissimilation process of glycerol,” in Proceedings of the 2022 15th International Congress on Image and Signal Processing, BioMedical Engineering and Informatics (CISP-BMEI), pp. 1–6, IEEE, Beijing, China, November 2022.
View at: Google Scholar
D. S. Kong, E. J. Park, S. Mutyala et al., “Bioconversion of crude glycerol into 1,3-propanediol(1,3-PDO) with bioelectrochemical system and zero-valent iron using Klebsiella pneumoniae L17,” Energies, vol. 14, no. 20, Article ID 6806, 2021.
View at: Publisher Site | Google Scholar
A. G. D. O. Paranhos and E. L. Silva, “Statistical optimization of H₂, 1,3-propanediol and propionic acid production from crude glycerol using an anaerobic fluidized bed reactor: interaction effects of substrate concentration and hydraulic retention time,” Biomass and Bioenergy, vol. 138, Article ID 105575, 2020.
View at: Publisher Site | Google Scholar
X. Wu, Y. Hou, K. Zhang, and M. Cheng, “Dynamic optimization of 1,3-propanediol fermentation process: a switched dynamical system approach,” Chinese Journal of Chemical Engineering, vol. 44, pp. 192–204, 2022.
View at: Publisher Site | Google Scholar
L. Wang, J. Yuan, C. Wu, and X. Wang, “Practical algorithm for stochastic optimal control problem about microbial fermentation in batch culture,” Optimization Letters, vol. 13, no. 3, pp. 527–541, 2019.
View at: Publisher Site | Google Scholar
P. Mu, L. Wang, and C. Liu, “A control parameterization method to solve the fractional-order optimal control problem,” Journal of Optimization Theory and Applications, vol. 187, no. 1, pp. 234–247, 2020.
View at: Publisher Site | Google Scholar
H. Bei, L. Wang, J. Sun, and L. Zhang, “A multistage feedback control strategy for producing 1,3-propanediol in microbial continuous fermentation,” Complexity, vol. 2019, Article ID 6252607, 9 pages, 2019.
View at: Publisher Site | Google Scholar
S. Wang, P. Takyi-Aninakwa, S. Jin, C. Yu, C. Fernandez, and D.-I. Stroe, “An improved feedforward-long short-term memory modeling method for the whole-life-cycle state of charge prediction of lithium-ion batteries considering current-voltage-temperature variation,” Energy, vol. 254, Article ID 124224, 2022.
View at: Publisher Site | Google Scholar
S. Wang, Y. Fan, S. Jin, P. Takyi-Aninakwa, and C. Fernandez, “Improved anti-noise adaptive long short-term memory neural network modeling for the robust remaining useful life prediction of lithium-ion batteries,” Reliability Engineering & System Safety, vol. 230, Article ID 108920, 2023.
View at: Publisher Site | Google Scholar
L. T. Biegler, Nonlinear Programming: Concepts, Algorithm, and Applications to Chemical Processes, SIAM, Philadelphia, PA, USA, 2010.
S. Boyd, S.-J. Kim, L. Vandenberghe, and A. Hassibi, “A tutorial on geometric programming,” Optimization and Engineering, vol. 8, no. 1, pp. 67–127, 2007.
View at: Publisher Site | Google Scholar
S. Islam and W. A. Mandal, Fuzzy Geometric Programming Techniques and Applications, Springer, Singapore, 2019.

Copyright

Copyright © 2023 Gongxian Xu and Zijia Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation

Download other formats

Order printed copies

Views

218

Downloads

205

Citations