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

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Introduction
1,3-Propanediol (1,3-PDO) plays a key role in many industry felds, 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,.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 efciently produce 1,3-PDO.Te 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.Te 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 efciently 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 efciently solve the parameter identifcation 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 fractionalorder 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 signifcantly violate the experimental data (refer to Section 5).For example, the errors of glycerol concentrations reached more than 43% in [24][25][26][27].Tis 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.Ten, 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 efciently 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.

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.Te concentration of various substances in the fermenter is in a constant state.Te 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 fgure, F is the volume fow of feed medium into the fermenter, L/h; y sf denotes the concentration of substrate glycerol in feed medium, mmol/L; V is the volume of fermentation broth, L; y 1 represents the biomass, g/L; and y 2 , y 3 , y 4 , and y 5 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 diferential 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 fve-dimensional nonlinear ODEs: where t is the fermentation time, h; T represents the terminal time of the microbial continuous fermentation; d � F/V represents the dilution rate, h −1 ; r 1 is the nonlinear function that denotes the specifc growth rate of cells, h −1 ; r 2 is the nonlinear function representing the specifc consumption rate of glycerol, mmol/(g•h); and r 3 , r 4 , and r 5 are the nonlinear functions denoting the specifc formation rates of 1,3-PDO, acetic acid, and ethanol, respectively, mmol/(g•h).
Under certain experimental conditions, the maximum value of r 1 is a 1 h −1 , and the Monod saturation constant is a 2 mmol/L.Te critical values of y 1 , y 2 , y 3 , y 4 , and y 5 are y * 1 g/L, y * 2 mmol/L, y * 3 mmol/L, y * 4 mmol/L, and y * 5 mmol/L, respectively.Terefore, microbial fermentation system (1)- (6) will work in the subset of R 5 , expressed as follows: In addition, the dilution rate d and glycerol concentration in the feed y sf will stay within certain limits, i.e., (d, y sf ) T ∈ Y 2 , where Y 2 is expressed as follows: where d L > 0 and d U > 0.
By introducing the expressions of functions r i (i � 1, 2, • • • , 5) into (1)- (6), we obtain the following reformulations of the microbial continuous fermentation: Now, we perform some transformations, as follows: x 10 � θ 16 + dy 2 . ( Ten, we obtain the following dynamical system with a power function structure: International Journal of Chemical Engineering Te above dynamical system can be further represented as and the functions g i (y, x, d, y sf , θ) and h k (y, d, θ) are defned as Dynamical system ( 14)-( 16) is a diferential-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.

Mathematical Properties of the Dynamical System.
In this subsection, we consider the properties of dynamical system ( 14)-( 16).

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Proof.By the defnitions of the functions g i (y, x, d, y sf , θ) 14)-( 16) are locally Lipschitz continuous on Y 1 with respect to y.
By the mean value theorem, there exist As Y 1 , Y 2 , and Θ are bounded sets, the derivatives of Ten, we have Tese conclude that ) are locally Lipschitz continuous on Y 1 with respect to y. □ Property 4. For θ ∈ Θ, dynamical system ( 14)-( 16) has a unique solution, expressed by y(t; y 0 , d, y sf , θ), and y(t; y 0 , d, y sf , θ) is continuous on Θ with respect to θ.
Proof.From the defnition of set Θ, Θ is a bounded closed set in R 16 .Terefore, Θ is a compact set in R 16 .By Properties 2 and 4, we obtain that the mapping from θ ∈ Θ to □ Property 6.For ∀(d, y sf ) T ∈ Y 2 and ∀θ ∈ Θ, the vector function satisfes the following linear growth condition with respect to y: where 0 < b 1 < +∞ and 0 < b 2 < +∞.
Remark 9.By Property 8, we observe that g i (y, x, d, y sf , θ) 14)-( 16) involve a special structure in the form of signomial functions.Tis 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.

Parameter Estimation Model of the Dynamical System
To estimate the value of parameter vector θ in dynamical system ( 14)-( 16) of microbial continuous fermentation, we will frst propose a parameter estimation model in this section.
Given certain fermentation condition (d, y sf ) T ∈ Y 2 , we can measure the steady-state concentrations of all reactants in the microbial continuous fermentation.Now, we have m diferent sets of experimental steady-state data that correspond to diferent fermentation conditions ) be the experimental steady-state concentrations of biomass (y 1 ), glycerol (y 2 ), 1,3-PDO (y 3 ), acetic acid (y 4 ), and ethanol (y 5 ) under fermentation conditions (d n , y n sf ) T ∈ Y 2 , and let ) calculated by the steady-state conditions.To keep the sum of the squared steady-state concentration deviations from the experimental data y n i minimized, we propose the following optimization model to estimate parameter θ in dynamical system ( 14)-( 16) of the microbial continuous fermentation: min subject to satisfying the 15m steady-state constraints: International Journal of Chemical Engineering and the bound constraints to the 15m + 16 variables: where 10 , 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 15m + 16, the number of equality constraints is 15m, the number of lower bound constraints is 15m + 16, and the number of upper bound constraints is 5m + 16.Terefore, if the number m of experimental groups is large, then problem ( 28)-( 30) will be a large-scale, nonlinear, and nonconvex optimization problem.

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 efciently 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 frst denote θ with ω j � θ j (j ≠ 5, 9) and ω j � −θ j (j � 5, 9).Additionally, we replace the expression min y n ,x n ,θ f �  m n�1  5 i�1 (y n i − y n i ) 2 with both min y n ,x n ,ω,p p and  m n�1  5 i�1 (y n i − y n i ) 2 ≤ p. Te inequality  m n�1  5 i�1 (y n i − y n i ) 2 ≤ p can be further written as Ten, model ( 28)-( 30) can be represented as the following equivalent formulations: min subject to satisfying where ω � (ω where u � (u 1 , u 2 , . . ., u 14m ) T ∈ R 14m ; ρ > 0 denotes the weighting coefcient with a sufciently large value.We can easily observe that if u l � 0 (l � 1, 2, • • • , 14m), then problem (48)-( 49) is equivalent to problem (32)-(47).Te reason why u l > 0 is used here instead of u l � 0 is that the optimization variables of GP must be positive.Te introduction of the penalty term ρ 14m l�1 u l can guarantee that u l ≈ 0 at the optimal solution of problem ( 48)-(49).By using some derivations to those inequalities involving variables u l (l � 1, 2, • • • , 14m), we obtain the equivalent problem, as follows: min subject to satisfying 10 International Journal of Chemical Engineering It is well known that the standard GP involves a posynomial objective and monomial equality and/or posynomial inequality constraints.Tis type of optimization problem can be solved very efciently 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 efcient condensation method is used to transform these inequality constraints into valid posynomial ones.Tis approach is to approximate every posynomial function in the denominator of inequality constraints by using a monomial function.

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Let z(c) �  e q e (c) be a posynomial where q e (c) are the monomials.Ten, using the arithmetic-geometric mean inequality, we obtain where β e are calculated through Here, c > 0 is a given point.We have  z(c) � z(c).Inequality (52) implies that  z(c)/z(c) ≤ 1 can be replaced with  z(c)/ z(c) ≤ 1, where  z(c) is a posynomial.Applying the approach above to problem (50)-(51), we have the following problem: min subject to satisfying International Journal of Chemical Engineering ) are the monomial functions approximated through (52).In problem (56)-(57), the number of upper bound constraints is 5m + 16, which is fewer than that of (54)-(55).
Based on Algorithm 1 and the above analysis, we present Algorithm 2.
Te experimental studies indicated that in the microbial continuous fermentation, the metabolic overfow 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., dy i /dt � 0, i � 1,2, • • • , 5) to fnd the positive steady states under diferent fermentation conditions (d, y sf ) T .After some computations, we can observe that our proposed dynamical system has multiple positive steady states in some fermentation conditions (d, y sf ) T ∈ Y 2 .As an illustration, Figure 3   International Journal of Chemical Engineering   16 International Journal of Chemical Engineering

Conclusions
Tis work has studied the problem of parameter estimation for the microbial continuous fermentation.A nonlinear dynamical system has been frst 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.Terefore, if the number of experimental groups is large, then problem ( 28)-( 30) will be a large-scale, nonlinear, and nonconvex optimization problem.To efciently solve problem ( 28)-( 30), an SGP method has been proposed.Te results indicated that our proposed algorithm can yield smaller errors between the experimental and calculated steady-state concentrations than the existing International Journal of Chemical Engineering methods in the literature [23][24][25][26][27][28][29].Tis concludes that the established dynamical system can better describe the microbial continuous fermentation and that the proposed SGP method is valid.Te 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.

Figure 2 :
Figure2: Comparison between the experimental data and steady-state concentrations calculated using the proposed approach.