Abstract
Our work is devoted to a class of optimal control problems of parabolic partial differential equations. Because of the partial differential equations constraints, it is rather difficult to solve the optimization problem. The gradient of the cost function can be found by the adjoint problem approach. Based on the adjoint problem approach, the gradient of cost function is proved to be Lipschitz continuous. An improved conjugate method is applied to solve this optimization problem and this algorithm is proved to be convergent. This method is applied to set-point values in continuous cast secondary cooling zone. Based on the real data in a plant, the simulation experiments show that the method can ensure the steel billet quality. From these experiment results, it is concluded that the improved conjugate gradient algorithm is convergent and the method is effective in optimal control problem of partial differential equations.
1. Introduction
Continuous caster is a machine by which the molten steel is solidified to slabs by spraying on it the cooling water. Because the production situation of secondary cooling zone is very bad, the set point of water volume in secondary cooling plays an important role in the continuous casting [1]. Undercooling of the strand in continuous cast secondary cooling can result in a liquid pool that is too long. However, overcooling can lead to the formation of cracks. The quality of steel billet depends on the behaviour of the surface temperature [2]. Optimal control problem of partial differential equations is encountered [3–5] in many applications ranging from engineering to science. The mathematical model arises in many engineering and physical processes such as heat conduction [2], fluid mechanics [6], and material sciences [7]. Many researchers considered a similar optimal control problem of partial differential equations [8, 9]: where is temperature (K). is the surface temperature (K). is the objective temperature. is a coefficient. is a controlled variable. is a guess for the controlled variable by a priori knowledge. , .
Li [10] applied conjugate gradient algorithm to estimate boundary condition. Lee et al. [11] proposed a repulsive particle swarm optimization algorithm to solve this problem. Farag et al. [12] used a modified partial quadratic interpolation method to solve parabolic optimal control problem. Based on the adjoint problem approach, Kaya and Erdem [13] studied an inverse parabolic problem. Based on the gradient of cost function that is Lipschitz continuous, Hasanov [14] proved existence of a quasisolution of the inverse problem and proposed a monotone iteration scheme. Hasanov [15] used the gradient method and proved the convergence of this method. However, the step size in their gradient algorithm is a constant. Step size largely infects the convergent speed of this gradient algorithm. Hasanov and Pektaş [16] considered an inverse source problem. They used a conjugate gradient algorithm to solve this problem, and the good results were obtained.
In this paper, a class of optimal control problems of partial differential equations is abstracted from optimization problem for set-point values in secondary cooling zone process. In our optimal control of partial differential equations problem, the gradient of cost function is proved Lipschitz continuous. We present an improved conjugate gradient algorithm to solve this problem and prove convergence of this algorithm. The simulation experiment shows that this algorithm can effectively solve the optimization problem for set-point values in secondary cooling zone, and the quality of billet is ensured.
2. Continuous Casting Operation Model of Secondary Cooling Zone
Continuous casting makes a liquid metal by a special set of cooling devices into a certain section in the shape of a casting process. As shown in Figure 1, , , and are the thickness direction of steel billet, the width direction of steel billet, and the length direction of steel billet.
2.1. Continuous Casting Heat Transfer Model
2.1.1. Assumptions
Based on the following key assumptions [17], continuous casting heat transfer model can be obtained.(1)One-dimensional heat transfer is considered, so heat transfer can be ignored in the length and the width direction of billet.(2)Steel billet at the same cooling zone cools uniformity.(3)The meniscus surface is assumed to be flat.(4)In mushy zone, we use the equivalent specific heat instead of specific heat.(5)The convection heat transfer process in the liquid phase and two-phase zone is equal to conduction heat transfer process.
2.1.2. The Differential Equation
Based on the above assumptions, we can get continuous casting heat transfer model for one-dimensional model. where is temperature (K). is times (s). is slab thickness direction and is a coefficient depending on physical parameters of the steel, which is calculated by where is steel density , is specific heat , and is thermal conductivity .
2.1.3. Boundary Conditions
Steel billet goes through the crystallizer, secondary cooling zone, and air cooling zone. The different cooling sections have different cooling characteristics, so the boundary conditions are shown as where is heat flux in continuous casting process which is calculated by where is the equivalent heat transfer coefficient in secondary cooling zone . is the steel billet surface temperature (K) and is cooling water temperature (K): where and are constants. is the width of steel billet. is the length of steel billet. is the spray cooling water volume. is the spray cooling air volume. The boundary condition at can be written as
2.2. Optimal Control Model of Continuous Cast Secondary Cooling Zone
Based on continuous casting heat transfer model, we have optimal control model of continuous cast secondary cooling zone: where is steel slab temperature (K). is the surface temperature of steel slabs . is the objective temperature of steel slabs , which depends on continuous casting technology. is times (s). is slab thickness direction . is a positive number. is initial casting temperature, which is usually a constant; , . Other parameters have already been defined in continuous casting transfer model.
Many researchers [18, 19] use , the surface temperature of steel billet to evaluate the quality of billet. We hope that the surface temperature is close to the objective temperature by controlling the boundary condition . In the actual production of continuous casting, the controlled variable cannot be too large, so we also give a penalty for the controlled variable.
According to (3) and (5), and in (1) are assigned, so optimal control problem in continuous casting secondary cooling zone transfers to a general optimal control problem in (1). Because of this, we only consider a general optimal control problem in later sections.
3. Adjoint Problem and Lipschitz Continuity of the Gradient of the Cost Function
3.1. Adjoint Problem
Based on the first optimization and then discrete method, we can get the adjoint equation firstly and then derive the optimality conditions, using numerical methods to solve this problem. The gradient of the cost function with respect to the control variable can be efficiently calculated by the adjoint problem. Now we give the strong version of the Lagrange function, and then we transfer all constraints into cost function by separate multipliers , , , and and perform partial integrations with respect to space and time, so the Lagrange function can be obtained:
The vibration of the function can be expressed as
According to (9) and (10), the following equation can be obtained:
According to (11), the following dual-equation can be obtained:
Now we obtained adjoint problem in our optimal control problem.
3.2. Lipschitz Continuity of the Gradient of the Cost Function
We can consider the variation of the cost function:
In order to prove that the gradient of the cost function is Lipschitz continuous, we give two other partial differential equations. According to (8), if we let , we obtain the following partial differential equation:
According to (12), the following partial differential equation can be obtained in a similar way:
Lemma 1. For all , the following identity holds:
Proof. It is easy to know that the following identity holds:
Thus boundary conditions of (12) show
The following equations can be obtained:
Lemma 1 can be proved.
Lemma 2. There exists a constant , so that the following equation can be obtained:
Proof. If we multiply both sides of (14) by and integrate on time and space, the following equation can be obtained:
It is easy to know that the following identity holds:
According to the boundary conditions of (14), the following equation can be derived:
Taking (22) and (23) into (21), we have the following equations:
We use the -inequality
On the right-hand side of (24), then we have
According to (24) and (26), the following equation can be derived:
From literature [14], the following equation can be obtained:
So the following equation can be obtained:
We can know that there is a constant which makes (20) established. Lemma 2 can be proved.
Lemma 3. Consider
Proof. According to (15), the weak solution of this partial differential equation is the following equation:
According to the boundary conditions of (14) and (15), the following equation can be obtained:
Lemma 3 can be proved.
Theorem 4. If the gradient of cost function is Lipschitz continuous, there is a constant to make the following equation hold:
Proof. Taking into account Lemmas 1, 2, and 3, we can obtain the following equation:
4. Conjugate Gradient Algorithm Solves Optimal Control of Parabolic Partial Differential Equations and Its Convergent Analysis
4.1. Conjugate Gradient Algorithm
After we prove that the gradient of cost function is Lipschitz continuous, we can use the conjugate gradient algorithm to solve this PDE optimization problem. If we choose the reasonable parameters in conjugate gradient algorithm, we can prove the convergence of conjugate gradient algorithm.
As shown in Figure 2, is iteration number. is iteration direction. is gradient of cost function. is step size. is controlled quantity. is cost function. is a parameter in conjugate gradient algorithm, and is a small constant.
Because we use FR conjugate method to solve this optimal problem, and can be calculated by the following equations:
Step size largely infects the convergent speed of conjugate gradient algorithm. If we choose Wolfe line search rules, we have the following formula [20]:
4.2. Convergence Analysis of Conjugate Gradient Algorithm
Theorem 5. If , , and are calculated by (35) and (36), the following equation holds:
Proof. According to (36), the following equations can be obtained:
According to (38) and (39), the following equation can be obtained:
According to the recursion formula of , the following equation can be represented:
If this algorithm is not convergent, (37) does not hold and there exists a constant , so that the following identity holds:
According to (39), we can obtain the following equations:
Since is the gradient of cost function, we have proven that is Lipschitz continuous. There exists a constant, so that the following inequality holds:
Then we can obtain the following equations:
The sequence is monotonic decline and has lower bound, so is convergent. Since , is convergent and we can conclude that is convergent and not convergent at the same time. Thus the assumption is invalid and we can prove that (37) holds, so this conjugate gradient algorithm is convergent.
4.3. An Optimal Choice of the Stopping in the Conjugate Gradient Algorithm
It is important to understand the behavior of the cost function value depending on the iteration number. For given smooth objective temperature , an optimal control problem is solved by conjugate gradient algorithm:
As shown in Figure 3, the behaviour of cost function value as a function of the iteration number consists of three phases: the initial phase of rapid decrease but very short duration, the second phase of slow decrease, and the third phase of almost constant behaviour.
When dealing with optimal control problems, the stopping parameter plays an important role in conjugate gradient algorithm. However, it is difficult to choose a good stopping parameter . If stopping parameter is too large, the result cannot converge to optimal solution. The iteration number is too large because stopping parameter is too small. As shown in Table 1, the best value of stopping parameter is achieved at .
5. Simulation Experiment
To test the effectiveness of the method in this paper, the simulation experiments are done in actual plant data. The experiment result proves the effectiveness of the method in this paper. Physical parameters of steel grade are given in Table 2.
Example 1. Experimental parameters are shown as
In this example, the objective temperature changes from 1173.16 (K) to 1273.16 (K), when time is equal to 10 seconds. Figure 4 shows that the actual temperature can follow the objective temperature very well, so the quality of billet can be ensured. From Figure 5, we can know that the maximum error is about 50 (K). The temperature error is very small in most of the time. Based on technological requirements in continuous casting process, the behaviour of the surface temperature ensures that the billet has a good quality. Figure 6 shows the controlled variable. Because of penalty of controlled variable, the controlled variable cannot be very large. When the controlled variable is less than zero, this process is a heating process. When the controlled variable is larger than zero, this process is a cooling process.
Example 2. Experimental parameters are shown as
In this example, the objective temperature is a sinusoidal curve which is continuous function. Figure 7 shows that the actual temperature can follow the objective temperature very well, so the quality of billet can be ensured. From Figure 8, we can know that the maximum error is about 8 (K). Because the objective temperature is a sinusoidal curve, the maximum error becomes smaller than Example 1. The temperature error is very small in most of the time. Based on technological requirements in continuous casting process, the behaviour of the surface temperature ensures that the billet has a good quality. Figure 9 shows the controlled variable. Because of a penalty of controlled variable, the controlled variable cannot be very large.
Example 3. Experimental parameters are shown as
In this example, the objective temperature is 1673.16 (K), but the initial temperature of billet is 1844.16 (K). Figure 10 shows that the actual temperature can follow the objective temperature very well. While the billet quality can be ensured, the surface temperature of billet can approach the objective temperature rapidly. From Figure 11, we can know that the maximum error is about 160 (K). Due to the penalties of control variable, the actual temperature cannot step to the objective temperature. Figure 12 shows the controlled variable. Because of penalty of controlled variable, the controlled variable cannot be very large.
6. Conclusions
In order to solve optimization problem for set-point values in secondary cooling process, this paper applies first optimization and then discrete method to this optimal problem. We prove that the gradient of cost function is Lipschitz continuous. Base on this, we propose an improved conjugate algorithm to solve this problem and prove that this algorithm is convergent.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors wish to thank the reviewers for their careful reading and very helpful feedback, which helped improve the paper as well as the important guiding significance to their researches. This work was supported by the National Natural Science Foundation of China (61333006, 61473074) and the Fundamental Research for Central Universities (N120708001).