Research Article | Open Access
Optimal Control for Multistage Nonlinear Dynamic System of Microbial Bioconversion in Batch Culture
In batch culture of glycerol biodissimilation to 1,3-propanediol (1,3-PD), the aim of adding glycerol is to obtain as much 1,3-PD as possible. Taking the yield intensity of 1,3-PD as the performance index and the initial concentration of biomass, glycerol, and terminal time as the control vector, we propose an optimal control model subject to a multistage nonlinear dynamical system and constraints of continuous state. A computational approach is constructed to seek the solution of the above model. Firstly, we transform the optimal control problem into the one with fixed terminal time. Secondly, we transcribe the optimal control model into an unconstrained one based on the penalty functions and an extension of the state space. Finally, by approximating the control function with simple functions, we transform the unconstrained optimal control problem into a sequence of nonlinear programming problems, which can be solved using gradient-based optimization techniques. The convergence analysis and optimality function of the algorithm are also investigated. Numerical results show that, by employing the optimal control, the concentration of 1,3-PD at the terminal time can be increased, compared with the previous results.
The bioconversion of glycerol to 1,3-propanediol (1,3-PD) has recently received more and more attention throughout the world due to its environmental safety, high region specificity, cheaply available feedstock, and relatively high theoretical molar yield . Many researchs have been carried out including the quantitative description of the cell growth kinetics of multiple inhibitions, the metabolic overflow kinetics of substrate consumption and product formation [2–4], open-loop substrate input and pH logic control , enzyme-catalytic reductive pathway and transport of glycerol and 1,3-propanediol across cell membrane , parameter identification of biochemical systems  and feedback control and pulse feeding  for the models of the continuous cultures, feeding strategy of glycerol , and optimal control  and optimality condition  in fed-batch culture.
Compared with continuous and feed-batch cultures, glycerol fermentation in batch culture can obtain the highest production concentration and molar yield 1,3-PD to glycerol . So nonlinear dynamical systems in this culture have been extensively considered in recent years [13–15]. In batch culture of glycerol biodissimilation to 1,3-propanediol (1,3-PD), the aim of adding glycerol is to obtain as much 1,3-PD as possible. In this paper, based on the previous model in , taking the yield intensity of 1,3-PD as the performance index and the initial concentration of biomass, glycerol and terminal time as the control vector, we propose an optimal control model subject to a multistage nonlinear dynamical system and constraints of continuous state. A computational approach is constructed to seek the solution of the above model in two aspects. On the one hand transform the optimal control problem into the one with fixed terminal time and transcribe it into an unconstrained one based on the penalty functions and an extension of the state space; on the other hand, by approximating the control function with simple functions, we transform the unconstrained optimal control problem into a sequence of nonlinear programming problems, which can be solved using gradient-based optimization techniques. The convergence analysis and optimality function of the algorithm are also investigated. Numerical results show that, by employing the optimal control, the concentration of 1,3-PD at the terminal time can be increased, compared with the previous results.
This paper is organized as follows. In Section 2, a nonlinear dynamical system of batch culture is proposed. In Section 3, we propose an optimal control model, develop a computational approach to solve the optimal control model, and prove the convergence of algorithm. Section 4 illustrates the numerical results. Finally, conclusions are provided in Section 5.
2. Nonlinear Dynamical System
On the basis of our previous literature(see ), mass balances of biomass, substrate, and products in batch culture can be formulated as the following nonlinear dynamical system: where , , , , and are biomass, glycerol, 1,3-PD, acetate, and ethanol concentrations at time in the reactor, respectively. denotes the initial state, and is the terminal time of the fermentation process. is as state vector. The specific growth rate of cells , specific consumption rate of substrate, and specific formation rate of products , , are expressed by the following equations on the basis of [13, 16]:
In batch culture, the initial concentrations of biomass, glycerol, and the terminal time can be chosen as control variables. Let be the control vector. The solution of system (2.1) with respect to control vector is defined by .
Based on the factual fermentation, there exist critical concentrations, outside which cells cease to grow, of biomass, glycerol, 1,3-PD, acetate and ethanol. Hence, it is biologically meaningful to restrict the concentrations of biomass, glycerol, products, and the volume of culture fluid in a set and the control vector in a admissible control set defined respectively, as follows:
Let denote the space of continuous bounded functions on with values in , equipped with the sup-norm topology, that is, for , , , where is the Euclidean norm.
3. Optimal Control Problem
The optimal control problem using the yield intensity of 1,3-PD at the terminal time as cost functional, based on the controlled multistage nonlinear dynamical system (2.1), can be formulated as follows:
From the theory on continuous dependence of solutions on parameters and our previous literature (see ), we know that is continuous relative to , so is continuous on . Moreover, is a closed bounded convex subset of . Hence we know the optimal control must exist, namely, such that , for all .
3.1. Differentiability with respect to the Control Vector
In this subsection, our aim is to show the differentiability and the gradient information of solutions of the system (2.1) with respect to the control vector. To begin with, we discuss some properties of the function .
Proposition 3.1. For the system (2.1), and are continuous in on an open set in .
Proof. It follows by inspection that the function by definition and (2.3).
Using Theorems I-1-4 and II-1-2 in , we can show that the system (2.1) has a unique solution , and the solution satisfies the integral equation and are the initial value of the system (2.1), due to the differentiability of with respect to the initial vector, and we have where is the vector in with entries 0 except for 1 at the th entry, and is the matrix whose th column is , . From this speculation, we obtain the following result.
Proposition 3.2. Partial derivatives and exist and are continuous in . Furthermore, is the unique solution of the initial-value problem
By virtue of the result of Proposition 3.2, we can obtain the value of and . The following Proposition gives a formula to compute the value of .
Proposition 3.3. Partial derivatives exist and are continuous in . Furthermore,
Proof. The existence and continuity of can be directly obtained by the function and implicit function theorem. Next, we derive the formula (3.4), for , and we have: then, by the integral mean value theorem, there exists a constant , such that Let , then We obtain the desired result.
3.2. Model Transformation
The optimal control problem (3.5) is not a standard case because the terminal time is free. Using the method in Section 6.8.1 of , the (3.5) can be transformed into the one with fixed terminal time. Treating as an unknown parameter and using the transformation , the (3.5) is converted to (3.8) as follows:
3.3. Semi-Infinite Optimization with Inequality Constrained
For the optimal control problem , it is difficult to cope with the continuous state inequality constraints, that is, this is a semi-infinite optimization problem. To overcome the difficulty, let The condition, , for all , is equivalently transcribed into where .
We now define an approximate problem denoted by (3.13), where the smoothed state constraint is treated as the penalty function:
By similar arguments as those given in , (3.13) is coincident with (3.5) as . On this basis, (3.5) can be explored by solving a sequence of approximate (3.13). Each of these (3.13) is viewed as a smooth nonlinear mathematical programming problem.
3.4. Optimization Algorithm
In this subsection, similar with the approach based on gradient in [22, 23], we proposed an algorithm based on gradient of to solve the (3.13). can be derived by solving the ordinary differential equation (3.3) and computing (3.4). The admissible control set can be called a “box” because of its rectangular shape, we use the classic gradient projection method to cope with “box.”
The projection of an arbitrary u onto the feasible set is defined as follows. The th component is given by
Thus, the control variable obtained by projecting the steepest descent direction at onto the feasible set is given by where is optional step size.
On the basis of the above analysis, we can obtain an optimal control for (3.13) as shown in the following algorithm.
Algorithm 3.4. Step 1. Set constants , and is positive constant. Set , compute by equations (3.3) and (3.4), if , stop. Else, then go to Step 2.Step 2. Compute the step-size with Armijo line search rules, where is any integer such that
go to Step 3.Step 3. If , stop. Otherwise, compute using the equation defined by (3.14), replace by , and go to Step 1.
Note that, due to the boundedness of the function , it is very easy to find a satisfying (3.15) and (3.16), using the following subprocedure, which uses the last used step length , as the starting point for the computation of the next one.
Subprocedure of Algorithm 3.4
Step 1. If , set . Else, set .Step 2. If satisfies (3.15) and (3.16). Else, set , stop.Step 3. If satisfies (3.15) and not (3.16), replace by , and go to Step 2. If satisfies (3.16) and not (3.15), replace by , and go to Step 2.
For Algorithm 3.4, we see that is continuous, that for all , and that if and only if , that is, that is an optimality function for the problem (3.13). So, we have the following theorem to guarantee the convergence of the algorithm.
Theorem 3.5. If is such that , then is computed by Algorithm 3.4 using a finite number of function evaluations and any accumulation point of this sequence satisfies .
Proof. We apply Theorem in  with , then the desired result can be obtained immediately.
Remark 3.6. Although the approach we are using here to deal with the constraints of continuous state is similar with the one mentioned in , There are still three main difference between them. First of all, our approach is applied for the batch culture, while the one in  is used for the fed-batch culture. Secondly, the control variables in the two approaches are different. Our variable controls the initial and terminal points, and their variable controls the switching time. Finally, we use the gradient-based algorithm to numerically solve the problem. Their algorithm is an improved Particle Swarm Optimization (PSO) algorithm, not gradient based.
4. Numerical Results
According to the model and algorithm mentioned above, we have programmed the software and applied it to the optimal control problem of microbial fermentation in batch culture. The system parameters are listed in Table 1 (see [9, 16]).
The basic data are listed, respectively, as follows.
Boundary Value of Control Vector
mmol/L, mmol/L, mmol/L, mmol/L, , and h.
Boundary Value of State Vector
mmol/L, mmol/L, mmol/L, mmol/L, mmol/L, mmol/L. mmol/L, mmol/L, and , mmol/L.
We adopt , and in the procedure. Then, by Algorithm 3.4, the optimal control vector and objective function are and 54.5911, respectively. Numerical results show that, by employing the optimal control, the concentration of 1,3-PD at the terminal time can be increased, compared with the previous results.
In this paper, based on the previous model in , taking the yield intensity of 1,3-PD as the performance index and the initial concentration of biomass, glycerol, and terminal time as the control vector, we propose an optimal control model subject to a multistage nonlinear dynamical system and constraints of continuous state. A computational approach is constructed to seek the solution of the above model in two aspects. The convergence analysis and optimality function of the algorithm are also investigated. Numerical results show that, by employing the optimal control, yield intensity of 1,3-PD at the terminal time can be increased, compared with the previous results.
Our current tasks accommodate the modeling and simulation of the fermentation process. Moreover, the stability and reachability of the improved model need to be discussed.
This work was supported by the National Natural Science Foundation of China (no. 10871033), the Natural Science Foundation for the Youth of China (no. 11001153) the Shandong Province Natural Science Foundation of China (no. ZR2010AQ016) and the Fundamental Research Funds for the Central Universities.
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