Dynamic optimization of 1,3-propanediol fermentation process:A switched dynamical system approach

Xiang Wu,Yuzhou Hou,Kanjian Zhang,Ming Cheng

1 School of Mathematical Sciences,Guizhou Normal University,Guiyang 550001,China

2 School of Life Sciences,Guizhou Normal University,Guiyang 550001,China

3 School of Electrical Engineering,Southeast University,Nanjing 210096,China

4 School of Automation,Southeast University,Nanjing 210096,China

5 Key Laboratory of Measurement and Control of CSE,Ministry of Education,Southeast University,Nanjing 210096,China

Keywords:Process control Optimization Mathematical modeling Switched system State-dependent switching Global optimization

ABSTRACT This paper considers a dynamic optimization problem (DOP) of 1,3-propanediol fermentation process(1,3-PFP).Our main contributions are as follows.Firstly,the DOP of 1,3-PFP is modeled as an optimal control problem of switched dynamical systems.Unlike the existing switched dynamical system optimal control problem,the state-dependent switching method is applied to design the switching rule.Then,in order to obtain the numerical solution,by introducing a discrete-valued function and using a relaxation technique,this problem is transformed into a nonlinear parameter optimization problem (NPOP).Although the gradient-based algorithm is very efficient for solving NPOPs,the existing algorithm is always trapped in a local minimum for such problems with multiple local minima.Next,in order to overcome this challenge,a gradient-based random search algorithm (GRSA) is proposed based on an improved gradient-based algorithm (IGA) and a novel random search algorithm (NRSA),which cannot usually be trapped in a local minimum.The convergence results are also established,and show that the GRSA is globally convergent.Finally,a DOP of 1,3-PFP is provided to illustrate the effectiveness of the GRSA proposed by this paper.

1.Introduction

As is well-known,1,3-propanediol is an important chemical material,which is widely used in polymers,medicines,lubricants,cosmetics,food and so on because of its unique symmetrical structure [1–3].The production approaches of 1,3-propanediol can be divided into the following two categories:one is microbial conversion;the other is chemical synthesis [4–6].Compared with the method of chemical synthesis,the approach of glycerol microbial conversion for 1,3-propanediol is more attractive in industry production,because it has a large number of cheap availability renewable feedstocks and is more environmentally safe [7–9].Glycerol can be converted to 1,3-propanediol by using fermentation process.In generally,the fermentation process can be divided into the following three types:continuous fermentation,batch fermentation,and fed-batch fermentation [10–12].During the reaction,the batch fermentation requires all substrate being added at the beginning of the reaction and nothing being added or removed,the continuous fermentation demands fresh medium being added continuously to replenish consumed substrate at a certain speed and old medium being drained at the same speed,and the fed-batch fermentation needs fresh medium being added to prevent nutrient depletion and nothing being removed.Among the three types,the fed-batch fermentation has been attracting researchers from various fields in engineering and science because of its high productivity [13–15].In order to raise the productivity of the fed-batch,the substrate concentration need to be controlled at an appropriate level.In the experiment,the addition of substrate will be determined by using a given feeding time sequence,which is usually obtained by experience [16–18].However,considering the high cost,it is impossible that many experiments are conducted to obtain the optimal feeding time sequence by using various glycerol feeding rates.Thus,mathematical modeling and dynamic optimization for the 1,3-PFP is very necessary [19–21].

Switched dynamical systems have presented a flexible modeling approach for various practical systems,such as flight obstacle avoidance system [22],train control system [23],hybrid electric vehicle [24],and chemical process system [25].Switched dynamical systems are usually formed by some continuous-time or discrete-time subsystems and a switching rule.Applying an appropriate switching rule can obtain stability and satisfactory performance[26–28].Generally speaking,there exist the following four common switching rules:time-dependent switching [29],average dwell time switching [30],state-dependent switching[31],and minimum dwell time switching [32].Recently,switched dynamical system optimal control problems are becoming increasingly attractive because of their significance in theory and industry production [33–35].Due to the discrete nature of switching rules,it is very challenging that switched dynamical system optimal control problems are solved by directly applying the classical optimal control methods such as the maximum principle [36] and the dynamic programming method[37].In additions,analytical methods also cannot be adopted to achieve a solution for switched dynamical system optimal control problems because of their nonlinear nature [38–40].Thus,in recent work,two well-known numerical computation methods are developed for switched dynamical system optimal control problems to obtain numerical solutions.One is the bi-level approach [41].The basic idea of this approach is that the switched dynamical system optimal control problem is divided into two optimization problems,which can be solved at different levels by using various available numerical algorithms.The other is the embedding approach [42–43].The basic idea of this approach is as follows.Firstly,a relaxed problem,which will be optimized over the convex closure of the original input set,is introduced.Then,the solution of the switched dynamical system optimal control problem can be obtained by projecting the solution of this relaxed problem back to the original input space.Besides above two numerical computation methods,many other available numerical algorithms are also proposed for obtaining the solution of switched dynamical system optimal control problems.Unfortunately,most of these algorithms depend on the following assumption:the time-dependent switching strategy is applied to design the switching rules,which indicates that the system dynamic must be continuously differentiable with respect to the state [44–47].Thus,the assumption is improper,because some small perturbations of the system state may lead to the dynamic equations being changed discontinuously.Thus,the solution obtained is usually not optimal.

In this paper,we consider a dynamic optimization problem(DOP) of 1,3-propanediol fermentation process improved gradient-based algorithm (1,3-PFP).Our main contributions are as follows.Firstly,the DOP of 1,3-PFP is modeled as an optimal control problem of switched dynamical systems.Unlike the existing switched dynamical system optimal control problem,the statedependent switching method is applied to design the switching rule.In generally,solving this problem analytically is very difficult due to its nonlinear nature.Then,in order to obtain the numerical solution,by introducing a discrete-valued function and using a relaxation technique,this problem is transformed into a NPOP,which can be solved by using any gradient-based numerical optimization algorithm.Over the past decades,many iterative methods have been developed to solving the nonlinear parameter optimization problem(NPOP)by using the information of the objective function.The idea of the iterative methods is usually that an iterative sequence is generated such that the corresponding objective function value sequence is monotonically decreasing.However,the existing algorithms have the following disadvantage:if an iteration is trapped to a narrow curved valley bottom of the objective function,then the monotone scheme will enforce iterations to creep along the bottom of the narrow curved valley.This will cause very short steps or even undesired zigzagging.Following that,the iterative methods will lose their efficiency.Then,in order to overcome this challenge,an improved gradient-based algorithm(IGA)is proposed based on a novel search method.In this novel search method,it is not required that the objective function value sequence is always monotonically decreasing.And a large number of numerical experiments shows that this novel search method can effectively improve the convergence speed of this algorithm,when an iteration is trapped to a curved narrow valley bottom of the objective function.Although the IGA is a very efficient numerical optimization method,this algorithm is usually trapped in a local minimum for optimization problems with multiple local minima.Next,in order to overcome this difficulty,a gradient-based random search algorithm(GRSA)is proposed based on the IGA and a NRSA.The convergence results of the GRSA are established,and show that this algorithm is globally convergent.Finally,a DOP of 1,3-PFP is provided to illustrate the effectiveness of this approach proposed by this paper.Numerical simulation results show that this approach proposed by this paper is low time-consuming,has faster convergence speed,and obtains a better result than the existing methods.

The rest of this paper is organized as follows.Section 2 presents the DOP of 1,3-PFP.In Section 3,by introducing a discrete-valued function and using a relaxation technique,this problem is transformed into a NPOP,which can be solved by using any gradientbased numerical optimization algorithm.A GRSA is proposed in Section 4.In Section 5,the convergence results of this algorithm are established.In Section 6,a DOP of 1,3-PFP is provided to illustrate the effectiveness of this approach proposed by this paper.

2.Problem Formulation

In this section,dynamic optimization for a fed-batch fermentation process of converting glycerol to 1,3-propanediol is modeled as a optimal control problem of switched systems under statedependent switching.

In generally,a fed-batch fermentation process of converting glycerol to 1,3-propanediol switches between the following modes:the batch mode and the feeding mode.No input feed will be added in the batch mode.However,there exist alkali and glycerol,which will be added continuously to the fermentor in the feeding mode.It’s important to note that mode switches will occur when the concentration of glycerol reaches a certain threshold.

wheref1:R4×R→R4andf2:R4×R×R→R4denote two given continuously differentiable functions;x0?R4denotes a given initial system state;u（t）denotes the input feeding ratesatisfies the following bound constraint:

bi,i=1,2,are two given constants.

Note that there exists no input feeding in the batch mode,which implies that the control inputu（t）does not occur,when the batch mode (Subsystem 1) is active.Furthermore,the bound constraint described by Eq.(3) is used,only when the feeding mode (Subsystem 2) is active.

Since there exists no input feed in the batch mode (Subsystem 1),Subsystem 1 denotes the natural fermentation process.However,there exist two processes(the natural fermentation process and the fermentation process due to the input feed) in the feeding mode (Subsystem 2).Thus,the function,u（t）,t）can be described by

whereg:R4×R×R→R4denotes a given continuously differentiable function.That is to say,dx（t）/dt=g（x（t）,u（t）,t）denotes the fermentation process due to the input feed.

Generally speaking,as the biomass grows,the consumption of glycerol increases.Since no new glycerol is added,the concentration of glycerol will eventually become very low during the batch mode (Subsystem 1).Then,the batch mode (Subsystem 1) will switch into the feeding mode(Subsystem 2),and the state switching condition is given by

where α2is a parameter to be optimized,and α2satisfies the following bound constraint:

andbi,i=3,4,are two given constants.

In additions,if the concentration of glycerol becomes very high during the feeding mode (Subsystem 2),the growth of cells will be inhibited.Thus,if too much glycerol is added,then it is necessary that the feeding mode (Subsystem 2) switches into the batch mode (Subsystem 1),and the state switching condition is given by

where α1is a parameter to be optimized,and α1satisfies the following bound constraint:

andbi,i=5,6,are two given constants.

Our main objective is that the concentration of 1,3-propanediol is maximized at the terminal timetf,where the terminal timetfis a given constant.Note thatandare equivalent.Then,the DOP of 1,3-PFP can be stated as follows:

Problem 1.Given the switched dynamical system described by Eq.(1) with the initial condition Eq.(2),choose a control input (input feeding rate)u（t）?Rand two parameters αi?R,i=1,2 such that the objective function

is minimized subject to the state switching conditions described by Eqs.(5)and(7);the bound constraints described by Eqs.(3),(6),and(8).

Remark 1.Note that the 1,3-PFP is complicated.For simplicity’s sake,we suppose that the parametersbi,i=1,2,???,6 and the terminal timetfare all given constants in the model,and there is no alkali input in batch mode.In the future,we will continue to study the DOP of 1,3-PFP,in which the bound constraints described by Eqs.(3),(6),and (8) become three stochastic constraints,the terminal timetfis also a decision variable,and there exists alkali input in batch mode.

3.Problem Transformation and Relaxation

In generally,solving Problem 1 analytically is very difficult due to its nonlinear nature.In this section,in order to obtain the numerical solution,by introducing a discrete-valued function and using a relaxation technique,this problem is transformed into a NPOP,which can be solved by using any gradient-based numerical optimization algorithm.

3.1.Problem transformation

In the switched dynamical system described by Eq.(1)with the initial condition Eq.(2),the system dynamics are usually nonsmooth because of the switching between Subsystems 1 and 2 dependents on the system state.Note that the existing approaches for optimal control problems of switched dynamical systems can be applied,only when the switching between subsystems dependents on the system time rather than the system state.Thus,in order to overcome this difficulty,a more tractable equivalent optimal control problem will be presented in this subsection.

one can transform the switched dynamical system described by Eq.(1) into an equivalent nonlinear dynamical system as follows:

Then,Problem 1 can be equivalently written as the following nonlinear dynamical system optimal control problem with a discrete-valued variable β（t）and three continuous variablesu（t）,α1,α2.

Problem 2.Given the nonlinear dynamical system described by Eq.(11) with the initial condition Eq.(2),find a control input(input feeding rate)u（t）?R,two parameters αi?R,i=1,2,and a discrete variable β（t）? ｛0,1｝ such that the objective function described by Eq.(9) is minimized subject to the state switching condition

and the bound constraints described by Eqs.(3),(6),and (8).

3.2.Problem relaxation

Note that Problem 2 is a nonlinear constrained optimization problem with three continuous decision variablesu（t）?R,α1?R,α2?R,and a discrete decision variable β（t）? 0,1｛ ｝.Thus,it is very difficult to obtain the optimal solution of Problem 2 by using standard numerical optimization approaches such as the interior-point method and the sequential quadratic programming approach,which are developed for solving nonlinear unconstrained optimization problems with only continuous decision variables.In order to overcome this difficulty,a relaxation optimization problem will be introduced in this subsection.

Now a relaxation optimization problem with only continuous decision variables can be introduced as follows.

Problem 3.Given the nonlinear dynamical system described by Eq.(11)with the initial condition Eq.(2),find a control input(input feeding rate)u（t）?R,two parameters αi?R,i=1,2,and a continuous variable β（t）?[0,1] such that the objective function

is minimized subject to the state switching condition described by Eq.(12)and the bound constraints described by Eqs.(3),(6),and(8),where γ>0 is a penalty parameter and ?:[0,1]→[0,+∞）is a strictly monotone decreasing differentiable function and satisfies the following condition:

Remark 2.In order to obtain the optimal solution of Problem 3,the penalty parameter γ>0 will be sufficiently large,which implies that

Then,by using the equality described by Eq.(15)together with β（t）?[0,1],one can obtain β（t）? ｛0,1｝.

In order to prove theoretically the equivalence relation between Problems 2 and 3,another relaxation optimization problem is provided as follows.

Problem 4.Given the nonlinear dynamical system described by Eq.(11)with the initial condition Eq.(2),find a control input(input feeding rate)u（t）?R,two parameters αi?R,i=1,2,and a continuous variable β（t）?[0,1] such that the objective function

3.3.A NPOp

Generally speaking,solving Problem 3 analytically is very difficult due to its nonlinear nature.Thus,by using the time-scaling transformation,an equivalent NPOP,which can be solved by using any gradient-based numerical optimization algorithm,will be achieved for Problem 3 in this subsection.

Suppose thatN≥1 is a given fixed integer and the set Γ is defined by

where τ=[τ1,???,τN-1]Tand τi,i=1,???,N-1,denote the switching times for the switched dynamical system described by Eq.(1).It should be noted that the switching times are not independent variables to be optimized and they can be obtained indirectly by using the system state trajectory.Then,Problem 3 can be written as the following equivalent problem:

Problem 5.Given the nonlinear dynamical system

with the initial condition (2),find a（σ,ξ,α1,α2）?RN×RN×R×Rsuch that the objective function

is minimized subject to the state switching condition

the bound constraints described by Eqs.(6),(8),and

Since the switching times are unknown,it is very difficult to obtain the gradient of the objective function described by Eq.(25).In order to overcome this difficulty,a time-scaling transformation is provided to transform variable switching times into fixed times as follows:

Lett（s）:[0,N]→Rbe a continuously differentiable function,which satisfies the following ordinary differential equation:

with the boundary condition

Next,by applying the time-scaling transform described by Eqs.(30) and (31) to the nonlinear dynamical system described by Eq.(24),the objective function described by Eq.(25),and the system state constraint described by Eq.(26),Problem 5 can be transformed into an equivalent NPOP with fixed switching times as follows.

Problem 6.Given the nonlinear dynamical system

4.A GRSa

Although the gradient-based algorithm is very efficient for solving NPOPs,this algorithm is always trapped in a local minimum for such problems with multiple local minima.In order to overcome this challenge,a GRSA is proposed based on an IGA and a NRSA,which cannot usually be trapped in a local minimum.Firstly,an IGA is briefly introduced.

4.1.An IGA

4.1.1.Gradient formulae

In order to obtain the numerical solution of Problem 6,the gradient formulae of the objective function described by Eq.(36) will be presented in this subsection.

For the simplicity of notation,the objective function described by Eq.(34) is be written as:

and λ（s）is referred to as a costate satisfying the following ordinary differential equation:

Now,the gradient formulae can be provided by the following theorem:

Theorem 2.For anys?[0,N],the gradient formulae of the objective function described by Eq.(36) with respect to the decision variables σ,ξ,?,α1,and α2are provided by

Proof.By using the Hamiltonian function Eq.(39),the nonlinear dynamical system Eq.(33),and the objective function Eq.(36),we have

Similarly,one can obtain the gradient formulae described by Eqs.(43)–(46).This completes the proof of Theorem 2.

4.1.2.Algorithm

Over the past decades,many iterative methods have been developed to solving the nonlinear parameter optimization problem by using the information of the objective function.In generally,the idea of the iterative methods is that an iterative sequence is generated such that the corresponding objective function value sequence is monotonically decreasing,i.e.,and the sequencecan converge to a local optimal solution of this problem for a given initial point ω0.Unfortunately,the existing methods have the following two disadvantages:If an iteration is trapped to a curved narrow valley bottom of the objective function,then the iterative methods will lose their efficiency due to the target with objective function value monotonically decreasing may leading to very short iterative steps;The Armijo-type search methods may break down for very short iterative steps due to rounding errors andwhere In order to overcome this challenge,an IGA is proposed for solving Problem 1 by using a novel search method in this subsection.It is not required that the objective function value sequence is always monotonically decreasing in this novel search method.A large number of numerical experiments show that this novel search method can effectively improve the convergence speed of this algorithm,when an iteration is trapped to a curved narrow valley bottom of the objective function.

For the sake of notation,define ω=andbe gradientat ω,which is a row vector.Then,this algorithm is presented as follows.

Step 3.Set

wheredkdenotes the search direction and μkdenotes the step satisfying either the following Wolfe-like condition:

or the following Armijo-like condition:

Step 4.Select θk?[θmin,θmax],and set

Proof.Define a functionEk:R→Ras follows:

The upper boundQkofBkin Theorem 2 (i.e.,Bk≤Qk) will be established by using induction.Clearly,by using the definition ofB0and the Eq.(55),we haveB0=Q0.Suppose thatBi≤Qifor any 0 ≤i

By using this equality Eqs.(60),we have

due toEkbeing a monotone increasing function.Note thatBi≤Qifor any 0 ≤i

Then,Eqs.(61) and (62) imply thatBk≤Qkfor anyk.

As is known to all,the conventional Armijo and Wolfe conditions will be satisfied,as long asis bounded below.Note thatfor anyk.Then,a step μkcan be choosed to satisfy the Armijo-like or Wolfe-like conditions described in Algorithm 1.This completes the proof of Theorem 3.

Remark 3.Theorem 3 shows that the inequalityalways holds for any θksatisfying 0 ≤θk≤1.This indicates that the novel search method described in Algorithm 1 is welldefined.

4.2.A NRSa

Although the IGA described in Algorithm 1 is a very efficient numerical optimization method,this algorithm is usually trapped in a local minimum for optimization problems with multiple local minima.In order to overcome this difficulty,this subsection will present a NRSA for solving Problem 1,which cannot usually be trapped in a local minimum.

4.3.A GRSa

In this subsection,a GRSA will be proposed for solving Problem 1 based on the IGA and the NRSA.This algorithm not only can easily and quickly obtain local minima,but also cannot be trapped in a local minimum.

Algorithm 3.A GRSAStep 1.Initial ω0,the maximum number of iterationK?N+,the tolerance ε>0,andk=0.

Step 2.Ifk

Step 3.By using the IGA(Algorithm 1 described by Section 4.1),solve Problem 6 with the initial point ωk.Suppose thatis the solution obtained,and go to Step 4.

5.Convergence Analysis

This section will establish the convergence results of the GRSA proposed by Section 4.

5.1.Convergence analysis for the IGA

for any ω on the line segment connecting ωkand ωk+μkadk,if the Armijo-like condition is applied.Then,a lower bound for the iteration step obtained by using the novel search method described in Algorithm 1 can be provided by the following lemma.

Lemma 1.If the Wolfe-like condition is satisfied,then we have

Proof.The results of this theorem will be proved in the following two cases:

Case 1.It is assumed that the iteration step μksatisfies the Wolfelike condition.Then,by using the inequality described by Eq.(51),one can obtain

which indicates that the inequality described by Eq.(66) is true.

Case 2.It is assumed that the iteration step μksatisfies the Armijolike condition.Ifaμk≥e,then we have μk≥which implies that the inequality described by Eq.(67)is true.Ifaμk

by using the inequality described by Eq.(59),sincepkdenotes the largest integer such that μk=satisfying the inequality described by Eq.(50).In additions,by using the inequality described by Eq.(65),one has

which together with Eq.(70) implies that the inequality described by Eq.(67) is also true.This completes the proof of Lemma 1.

Suppose that the following condition is satisfied:

Assumption 1.There are two real numbersq1>0 andq2>0 satisfy the following two inequalities

for any sufficiently largek.

Theorem 4.Suppose that the objective functiondescribed by Eq.(36)is bounded below and Assumption 1 is true.When the Wolfelike condition is applied,we assume thatis a Lipschitz continuous function with a real numberF>0 on the following level set:

Proof.Firstly,the following inequality will be established under three cases:

Case 1.When the Armijo-like condition is applied andaμk≥e,by using the inequalities described by Eq.(50) and Eq.(72),we have

which indicates that the inequality described by Eq.(77) is true.

Case 2.When the Armijo-like condition is applied andaμk≥e,by using the inequality described by Eq.(67),one obtain

which together with the inequality described by Eq.(50) implies

By using Assumption 1,from the inequality described by Eq.(79),we have

which indicates that the inequality described by Eq.(77) is true.

Case 3.When the Wolfe-like condition is applied,the process of proof is the same as the analysis of Case 2 apart from the inequality Eq.(78) being replaced by the inequality Eq.(66).

Next,by using Eqs.(53) and (54) and Eq.(77),one can obtain

Note that the objective functiondescribed by Eq.(36) is bounded below.Then,by using the inequality Eq.(59),we can derive thatBkis bounded below,which together with the inequality described by Eq.(80) implies

Suppose that the Euclidean norm ‖gk‖ is bounded away from 0.Then,this contradicts with the inequality Eq.(81) due toCk+1≤k+2,which can be obtained by using the inequality Eq.(60).Thus,this condition described by Eq.(75) is true.Furthermore,if θmax<1,by using the inequality Eq.(60),we have

which together with Eq.(81)implies this condition described by Eq.(76) is true.This completes the proof of Theorem 4.

5.2.Convergence analysis for the GRSA

Theorem 5.The GRSA (Algorithm 3 described by Section 4.3)converges to a global minimum of Problem 1.

Proof.Note that Problems 1,2,3,5,and 6 are equivalent.Then,by using the local convergence of the gradient-based algorithm(Algorithm 1) and the global convergence of the random search algorithm (Algorithm 2),it can be directly proved that the GRSA(Algorithm 3) converges to a global minimum of Problem 1.This completes the proof of Theorem 5.

6.Numerical Results

In this section,a DOP of 1,3-PFP is provided to illustrate the effectiveness of the GRSA (Algorithm 3 described by Section 4.3).

The functionsf1（x（t）,t）andg（x（t）,u（t）,t）in the switched dynamical system described by Eq.(1) can be defined by

where the functions φ,φ1,φ2,andu（t）denote the growth rate of cell,the formation rate of 1,3-propanediol,the consumption rate of substrate,the input feeding raterespectively;the constantslandmdenote the proportion of glycerol and the concentration of glycerol in the input feed,respectively;and the functions φ,φ1,and φ2are given by

The model parameters of the DOP for 1,3-PFP are presented byl=0?5709,m=10759?0000 mmol ?L-1,n1=

Then,the GRSA (Algorithm 3 described by Section 4.3) is adopted to solve the DOP of 1,3-PFP by using Matlab 2010a [48]on an Intel Pentium Dual-core PC with 2.60 GB of RAM.The optimal objective function value isJ*==-1179?4592 and the optimal values of the parameters α1and α2are 582?2597 and 244?9503,respectively.The optimal input feeding rateu*and the corresponding numerical simulation results are presented by Figs.1–5.

Fig.1.The optimal input feeding rate: u*.

Fig.2.The optimal concentrationof 1,3-propanediol: x1 （t ）.

Note that Problem 6 is an optimal control problem of nonlinear dynamical systems with state and input constraints.Thus,the interior penalty method (IPM) developed by Malisaniet al.[49] can also be applied for solving the DOP of 1,3-PFP.In order to compare with the GRSA proposed by us,the IPM developed by Malisaniet al.[49] is also used for solving the DOP of 1,3-PFP with the same model parameters under the same condition,and the numerical comparison results are presented by Fig.6 and Table 1.

Fig.6 shows that the GRSA proposed by us take only 42 iterations to obtain the satisfactory result=1179?4592,while the IPM developed by Malisaniet al.[49] takes 106 iterations to achieve the satisfactory result=1046?5302.That is,the iterations of the GRSA proposed by us is reduced by 60?3774% .In order to further illustrate the effectiveness of the method proposed by this paper,we solve the DOP of 1,3-PFP with the same model parameters and the experienced-based fed-batch operation under the same condition,and the corresponding concentrationx1（t）of 1,3-propanediol at terminal timetfis 893?1622.Compared with the result based on experienced-based operation,the value ofobtained by using the GRSA is increased by 32?0543 % .The above discussion and analysis together with Table 1 shows that the GRSA is an effective alternative method.

Table 1 The comparison results between the IPM developed by Malisani et al.[49] and the GRSA proposed by us

To sum up,the above numerical simulation results indicate that the GRSA proposed by us is low time-consuming,has faster convergence speed,and can obtain a better result than the IPM developed by Malisaniet al.[49].In additions,compared with the result based on experienced-based operation,the GRSA proposed by us can also obtain a better result.That is,an effective numerical computation method is presented for solving the DOP of 1,3-PFP.

Fig.3.The optimal volume of fluid （L ）: x2 （t ）.

Fig.4.The optimalconcentrationof biomass: x3 （t ）.

7.Conclusions

In this paper,the DOP of 1,3-PFP is modeled as an optimal control problem of switched dynamical systems under statedependent switching.Then,by introducing a discrete-valued function and using a relaxation technique,this problem is transformed into a NPOP.Next,based on an IGA and a NRSA,this NPOP is solved by using a GRSA,which cannot usually be trapped in a local minimum.The convergence results are also established,and show that this algorithm is globally convergent.Finally,numerical simulation results for a DOP of 1,3-PFP show that the approach proposed by this paper is low time-consuming,has faster convergence speed,and obtains a better objective function value than the existing methods.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Fig.5.The optimal concentration of glycerol: x4 （t ）.

Fig.6.Convergence rates for the IPM developed by Malisani et al.[49] and the GRSA developed by us.

Acknowledgements

This work was supposed by the National Natural Science Foundation of China(61963010 and 61563011),and the special project for cultivation of new academic talent and innovation exploration of Guizhou Normal University in 2019 (11904-0520077).