Abstract
This paper is devoted to the study of numerical computation for a kind of time optimal control problem for the tubular reactor system. This kind of time optimal control problem is aimed at delaying the initiation time of the active control as late as possible, such that the state governed by this controlled system can reach the target set at a given ending time . To compute the time optimal control problem, we firstly approximate the original problem by finite element method and get a new approximation time optimal control problem governed by ordinary differential equations. Then, through the control parameterization method and time-scaling transformation, the approximation problem becomes an optimal parameter selection problem. Finally, we use Sequential Quadratic Program algorithm to solve the optimal parameter selection problem. A numerical simulation is given for illustration.
1. Introduction
Tubular reactors have extensive applications in the industrial production. A lot of reaction processes in chemical and biochemical engineering can be described by tubular reactor models, such as chlorine dioxide bleaching model [1]. The dynamic tubular reactors are typically described by nonlinear partial differential equations (PDEs), which include convection, reaction, and diffusion phenomena. For studies of such PDEs system, we would mention some works [2, 3]. Sometimes we regulate the reaction process artificially to improve the reactor rate or reduce the reaction time, by seeking some control strategies such as adding catalyst or placing a heating or cooling jacket. For this purpose, optimal control problems governed by such PDEs system arouse increasing attentions [4–7]. As a kind of important optimal control problem, time optimal control problem plays an important role in many fields of applications.
Generally speaking, time optimal control problems can be divided into two types. The first one is to find a minimal time and a control belonging to some constraint set which is acted upon from beginning time , such that the state governed by the controlled system can arrive at a given target set in the shortest time interval . This kind of problem is called the minimal time optimal control problem. The second one is to delay the initiation time of the active control as late as possible, such that the state governed by this controlled system can reach the target set at the given ending time This is a maximal time optimal control problem. There have been extensive researches on the first kind of time optimal control problem [8–11], but only a few works related to the second kind of time optimal control problem have been studied. For the second kind of time optimal control problem, we would mention the works [12, 13].
In this paper, we shall consider the second kind of time optimal control problem with boundary controls governed by a coupled tubular reactor system, which is described by nonlinear partial differential equations and corresponds to the following chemical reaction process [14, 15]: where , are two reactants and is the product of the chemical reaction. We assume that the kinetics depend on the concentrations of two reactants and , which has the form , where , are the kinetic constants and , are the concentrations of the reactants and , respectively. Moreover, we assume that the length of the reactor is A control is acted upon on the one side of the tubular reactor to control the reactor process, as depicted in Figure 1. More precisely, the above controlled process can be described as follows:where are the given initial data, is a given positive number, , are the dispersion coefficients, is the superficial fluid velocity, and is the boundary control. We denote by the solution to (2) corresponding to the control and the initial data
Let for Define Let for Define The second kind of time optimal control problem governed by (2) can be stated as follows: where is the solution to (2). We assume that, for given , there exists , such that satisfying the above assumption is called an admissible control for the problem . Obviously, the above assumption implies that the admissible control set is not empty. In the problem , represents the initiation time of the active control. We call the optimal time for the problem , and a control satisfying such that is called an optimal control for the problem . The goal of this problem is to find a maximal and a control such that , and the state of (2) driven by this control can reach the given target set at with the shortest actively controlled interval .
The time optimal control problems governed by PDEs are infinite dimensional control problems and are difficult in finding out analytical solutions. Thus, the numerical approximations of time optimal control problems attract a lot of attentions. Here, we would like to mention some related works [16–18] on the computations of time optimal control problems. However, the works mentioned above focused on the first kind of time optimal control problem. To the best of our knowledge, it seems that no attempts have been made to develop the computation for the second kind of time optimal control problem governed by nonlinear PDEs systems. In this paper, we shall consider the computation for the second kind of time optimal control problem . Firstly, we project the time optimal control problem by finite element method into an approximation problem . Although approximation problem is governed by ordinary differential equations, it is difficult to solve directly due to the unknown time variables. Then, we translate problem to an optimal parameter selection problem through the control parameterization method and time-scaling transformation. Finally, we use Sequential Quadratic Program (SQP) algorithm to solve the optimal parameter selection problem.
The rest of this paper is organized as follows. In Section 2, we give the finite element approximation for the time optimal control problem . In Section 3, we provide the control parameter method and time-scaling transformation to reduce the approximation problem to an optimal parameter selection problem. In Section 4, we give the procedure of solving the optimal parameter selection problem. In Section 5, we carry out some numerical experiments to illustrate the effectiveness of our approximation method.
2. The Finite Element Approximation of the Problem
In this section, we describe finite element approximation of the problem . Firstly, we introduce a standard triangulation of We decompose into finite number of subintervals , where is a positive integer and , are grid points which satisfy is called the mesh size of the triangulation Thus, we can write as Corresponding to each triangulation , we can define a finite dimensional space as follows: where stands for the space of all linear polynomials defined on the subinterval . Obviously, the basis of can be taken as , where , are continuous and piecewise linear polynomials that satisfy Thus, it is clear that is a space of dimensions and
Next we construct the approximation of (2). Let be the -projection from onto , defined byHere and in what follows, denotes the usual inner product of Moreover, if there is no risk to make any confusion, we will use to denote a function for a.e. Then, the finite element approximation of (2) can be defined as follows: seek functions and satisfying that and for a.e. , such thatNoting that , we can writeThe initial conditions can be written asSubstituting (13) into (12) and taking , we can obtain thatwhere denotes the derivative of with respect to the spacial variable and and denote the derivatives of and with respect to the time variable , respectively. Moreover, it follows from (11) and (14) thatTo simplify the notation, let where
Let Then, (15) and (16) can be rewritten as Define Thus, (19) is equivalent to the following equation:where , and
Now, we get a new time optimal control problem governed by ordinary differential equations as follows: where and is the solution to (21).
3. Control Parameterization Method and Time-Scaling Transformation
To solve the problem numerically, which is a time optimal control problem in the finite dimensional space, we apply the classical control parametrization method [19], where the control function can be approximated by the piecewise constant functions. We subdivide the time horizon into subintervals , where is a given integer and , satisfyHere each time point , is called a switching time. We allow the approximate control to switch at each switching time. To find the optimal switching time points, we assume that these switching points are not prefixed and are the parameters, which need to be chosen optimally in the following set: DefineWe approximate the control function by a constant vector on each subinterval: Then the control function can be approximated by the following piecewise constant control of the form It is clear that can switch at times Substituting (28) into (21), we can obtain that
Since the switching times are unknown, (29) is not easy to solve numerically. It is also difficult to integrate (29) accurately if some of the subintervals , are very short. To overcome these difficulties brought about by the unknown switching times, we make use of the time-scaling transformation [20–22]. Let be a new time variable, where is the number of the time subintervals. The time-scaling transformation between and can be established through the following differential equation: where is a piecewise constant function, which is defined by where Integrating (30) can yield thatwhere denotes the largest integer which is less than or equal to It follows from (32) that Noting that in (30), we have ThusEquality (34) shows that the time-scaling transformation maps to the th switching time Moreover, it also follows from (34) that By taking the approximation and making variable substitution , from (29) and (30), we can derive that where Let We denote by the solution to (36) corresponding to , where
Through the control parameterization method and time-scaling transformation, we can translate the approximation problem into an optimal parameter selection problem as follows: where is the solution to (36).
4. Solving Problem
In the problem , both , and , are decision variables, where , are the time parameters and , are the control parameters. The problem is essentially nonlinear mathematical programming problem and hence can be solved numerically with standard mathematical programming algorithms such as Sequential Quadratic Program algorithm. Now we present framework of solving the problem in Algorithm 1.
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We make the following finite difference scheme to solve , which is necessary in the third step of Algorithm 1. To simplify the notation, we write Firstly, we divide the time horizon into subintervals. Let and We obtain that where and at are known from the initial condition. Therefore, the computation of (39) is straightforward.
Following the procedure of Algorithm 1, we can solve the parameter selection problem . Let be the optimal parameter for the problem . By making use of (32), the approximation of and for the problem can be obtained.
5. Numerical Simulation and Discussion
In this section, a numerical simulation for the problem is presented. We take in (2). The control bounds are taken as and the target set bounds are
In this simulation, we divide the interval into equal subintervals. Thus, we have We choose the basis functions , as follows: We carry out the numerical simulation experiments within the MATLAB environment (version R2016a). We use personal computer with the following configuration: Intel Core i5-7200 2.50 GHz CPU, 8.00 GB RAM, 64-bit Windows 10 Operating System.
The following simulations are conducted by Algorithm 1. Using the piecewise constant control parameterization method with , we solve problem for The optimal time is given in Table 1. From Table 1, we can see that as increases from 10, 12, 15 to 20, the approximation value of varies from to , which means finer finite element triangulation can lead to shorter actively controlled interval . But when increases from 20, 30 to 40, the optimal time tends to be stable. Moreover, we see that the change of the value of has little effect on the approximation results. The optimal controls for when and are presented in Figure 2. It can be found in Figure 2 that the optimal control for the problem is bang-bang control, and the optimal control only switches one time. The corresponding states without control and with the optimal control acting on the interval for and are also presented in Figures 3 and 4, respectively.
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6. Conclusions
In this paper, we propose a computation method for the second kind of time optimal control problem for the tubular reactors systems, which are widely used in chemical and biochemical engineering. The aim of the second kind of time optimal control problem is to find a control such that the state governed by the tubular reactors system can arrive at a given target set at the ending time with the shortest duration of the control action. Currently, the computation of the second kind of time optimal control problem did not attract enough attentions of the researchers. Our approach can be summarized in three steps: A finite dimensional approximation problem can be obtained with finite element method; through control parameterization method and time-scaling transformation, we translate the approximation problem into an optimal parameter selection problem; we solve this optimal parameter selection problem by Sequential Quadratic Program algorithm. Moreover, we give an example for illustration to show the effectiveness of our approximation method.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
The authors appreciate the support of the National Natural Science Foundation of China (no. 51676182, no. 61374096), the Natural Science Foundation of Zhejiang Province (no. LY17C190008), and the Natural Science Foundation of Ningbo (no. 2014A610185).