Collaboration Strategy Based on Conflict Resolution for Flatness Actuator Group

Jiangsu Provincial Engineering Laboratory of Intelligent Manufacturing Equipment, Industrial Technology Research Institute of Intelligent Equipment, Nanjing Institute of Technology, Nanjing 211167, China School of Mechanical Engineering, Jiangsu University of Science and Technology, Zhenjiang 212003, China State Key Laboratory of Rolling and Automation, Northeastern University, 3-11 Wenhua Road, Shenyang, China Transportation Engineering College, Dalian Maritime University, No. 1 Linghai Road, Dalian, China


Introduction
With the steel industry promotion and development, the strip flatness quality receives more and more attention [1]. Some researchers have tried to improve the flatness control effect by establishing a high-precision flatness closed-loop control algorithm. Zhang et al. adopted GA to optimize PIDNN and proposed flatness intelligent control method based on GA-PIDNN for 900 HC reversible cold rolling mill in this paper [2]. Wang et al. proposed a new multivariable optimization algorithm with global convergence for a cold rolling mill flatness control [3]. Prinza et al. developed a new feedforward control approach for the thickness profile of the strip in a tandem hot rolling mill [4]. However, they relied too much on the computing power of the controller. In addition, some researchers have analyzed the effect of a single type of actuator on the strip flatness. Wang et al. presented an investigation on the shape prediction and control of strip [5]. Voronin et al. showed the distribution of the roll gap along the length of the roll body according to the horizontal displacement of the work rolls [6]. However, they did not consider the uncoordinated regulating behaviors between the flatness actuator group.
In the actual application process, the main incongruous actuator group behaviors are as follows: When a symmetrical flatness defect is detected by shapemeter roll, the work roll tilting may participate in the flatness regulating process [7][8][9]. However, the additional flatness change is caused by work roll tilting, which will consume the regulating margin of other actuators in the flatness closed-loop control system [10][11][12]. When the direction of work roll bending is opposite to the direction of intermediate roll bending, since the regulating efficiency curves of these two actuators are both concave, the offset between the effects of the two actuators on the strip flatness cannot be avoided [13][14][15]. When the intermediate roll shifting is alternately decreased and increased, the massive thermal deformation is generated in this contact area between work roll and intermediate roll, which can lead to serious roll wear [16][17][18].
In the existing flatness control system, the optimal regulating amount of all the actuators is merely calculated. Nevertheless, there are a large amount of incongruous actuator group behaviors which have a bad influence on flatness quality in the actual application process [19][20][21]. erefore, in response to the above questions, a collaboration strategy based on conflict resolution for the flatness actuator group has been proposed in this paper. On the basis of the original flatness control system, the coordination between the actuator group is made achievable according to the matching degree between regulating characteristics of flatness actuator and flatness defect.

Flatness Actuator Regulating Characteristics
Different types of flatness actuator have complex differences in the effect of strip flatness [22]. In Figure 1, work roll bending, intermediate roll bending, and intermediate roll shifting have the ability to eliminate symmetrical flatness defect. Work roll tilting has the ability to eliminate asymmetric flatness defect. Work roll bending, work roll tilting, and intermediate roll shifting are in high sensitivity. Intermediate roll bending is in low sensitivity. e characteristic of flatness actuator in high sensitivity is that it can cause huge flatness changes with very little adjustment. e effectiveness of the flatness control system has an important influence on the quality of the strip flatness. In the process of eliminating flatness defects, high effectiveness can be maintained through flatness actuator in high sensitivity. Simultaneously, it works with flatness actuator in low sensitivity to improve profile control accuracy.

External Evaluation Function.
rough the external evaluation function, we can determine whether the effect of eliminating flatness defect meets the requirement. e expression of external evaluation function is as follows: where n 1 is the number of measuring sections. g i is the weight factor. mes i is the measuring flatness.

External Constraint Condition.
e external constraint condition is determined according to the upper and lower limit of the flatness actuator. e expression of external constraint condition is as follows: where Δu WB (n 2 ) is the adjustment of work roll bending in the n 2 cycle. v WB (n 2 − 1) is the actual value of work roll bending in the n 2 − 1 cycle. v WB (n 2 + 1) is the actual value of work roll bending in the n 2 + 1 cycle. Δu WB (n 2 ) is the adjustment of intermediate roll bending in the n 2 cycle. v IB (n 2 − 1) is the actual value of intermediate roll bending in the n 2 − 1 cycle. v IB (n 2 + 1) is the actual value of intermediate roll bending in the n 2 + 1 cycle. Δu IS (n 2 ) is the adjustment of intermediate roll shifting in the n 2 cycle. v IS (n 2 − 1) is the actual value of intermediate roll shifting in the n 2 − 1 cycle. v IS (n 2 + 1) is the actual value of intermediate roll shifting in the n 2 + 1 cycle. Δu WT (n 2 ) is the adjustment of work roll tilting in the n 2 cycle. v WT (n 2 − 1) is the actual value of work roll tilting in the n 2 − 1 cycle. v WT (n 2 + 1) is the actual value of work roll tilting in the n 2 + 1 cycle. u WB is the upper limit of work roll bending. It represents the maximum value that the work roll bending can output. l WB is the lower limit of work roll bending. It represents the minimum value that the work roll bending can output. u IB is the upper limit of intermediate roll bending. It represents the maximum value that the intermediate roll bending can output. l IB is the lower limit of intermediate roll bending. It represents the minimum value that the intermediate roll bending can output. u IS is the upper limit of intermediate roll shifting. It represents the maximum value that the intermediate roll shifting can output. l IS is the lower limit of intermediate roll shifting. It represents the minimum value that the intermediate roll shifting can output. u WT is the upper limit of work roll tilting. It represents the maximum value that the work roll tilting can output. l WT is the lower limit of work roll tilting. It represents the minimum value that the work roll tilting can output.

Actual Flatness Condition Discriminating Factor.
ere are three coefficients: linear coefficient, quadratic coefficient, and edge coefficient. And the role of virtual flatness curve is that the local condition of actual flatness can be described quantitatively by these coefficients. e expression of virtual flatness curve T(j) is as follows: where X 1 is the linear coefficient of virtual flatness curve. X 2 is the quadratic coefficient of virtual flatness curve. X 3 is the edge coefficient of virtual flatness curve. m 3 is the number of measuring sections occupied by strip. Its range is from 1 to 38. T(j) is the virtual flatness in j section. e actual flatness condition discriminating factor includes the single-wave distinguishing factor X 1 e, the symmetrical distinguishing factor X 2 e, and edge distinguishing factor X 3 e. e actual flatness condition discriminating factor is plugged into expression of virtual flatness curve and T(j, X 1 e, X 2 e, X 3 e) is achieved. e mean square error is calculated between T(j, X 1 e, X 2 e, X 3 e) and the actual flatness M(j). When the mean square error reaches the minimum value within the constraints ll 1 ≤ X 1 e ≤ ul 1 , ll 2 ≤ X 2 e ≤ ul 2 , and ll 3 ≤ X 3 e ≤ ul 3 , T(j, X 1 e, X 2 e, X 3 e) is equivalent to the actual flatness M(j). e expression of calculating actual flatness condition discriminating factor is as follows:   (M(j) − T(j, X 1 e, X 2 e, X 3 e)) 2 where M(j) is the actual flatness in j section. X 1 e is the single-wave distinguishing factor. X 2 e is the symmetrical distinguishing factor. X 3 e is the edge distinguishing factor. u 1 is the upper limit of single-wave distinguishing factor. It represents the maximum value of single-wave distinguishing factor. l 1 is the lower limit of single-wave distinguishing factor. It represents the minimum value of single-wave distinguishing factor. u 2 is the upper limit of symmetrical distinguishing factor. It represents the maximum value of symmetrical distinguishing factor. l 2 is lower limit of symmetrical distinguishing factor. It represents the minimum value of symmetrical distinguishing factor. u 3 is the upper limit of edge distinguishing factor. It represents the maximum value of edge distinguishing factor. l 3 is the lower limit of edge distinguishing factor. It represents the minimum value of edge distinguishing factor.

Flatness Actuator Group Collaboration Strategy.
Not only can the analysis of actual flatness condition be conducted in real time but also the reasonable adjustment strategy is intelligently selected in the intelligent flatness control system [23][24][25]. As a consequence, the overall regulation capacity of flatness adjustment actuator after the combination is made to match with the flatness defect. When the single-wave distinguishing factor is greater than the upper limit of linear reasonable range, the local flatness status is single wave in the drive side. When the single-wave distinguishing factor is less than the lower limit of linear reasonable range, the local flatness status is single wave in the operating side. When the single-wave distinguishing factor is within the linear reasonable range, the local flatness status is symmetrical between the drive side and the operating side. If the actual flatness status is unsymmetrical, the following are the regulation strategies: e adjustment of work roll tilting is relatively big. If the actual flatness status is symmetrical, the following are the regulation strategies: e adjustment of work roll tilting is relatively small.
When the expression X 1 e > u l or X 1 e < l l is satisfied, the following are the regulation strategy A: where u l is the upper limit of linear reasonable range. It represents critical value of single wave in the drive side. l l is the lower limit of linear reasonable range. It represents the critical value of single wave in the operating side.
When the expression l l < X 1 e < u l is satisfied, the following are the regulation strategy B: When the symmetrical distinguishing factor is greater than the upper limit of quadratic reasonable range, the local flatness status is severe central wave. When the symmetrical distinguishing factor is less than the lower limit of quadratic reasonable range, the local flatness status is severe bilateral wave. When the symmetrical distinguishing factor is within the quadratic reasonable range, the local flatness status is slight central wave or bilateral wave. If the severe flatness defect appears in rolling, the top priority is the speed of eliminating flatness deviation.
erefore, the following are the regulation strategies: e adjustment of work roll bending is relatively big, while the adjustment of intermediate roll bending is relatively small. If the slight flatness defect appears in rolling, the top priority is the accuracy of eliminating flatness deviation. erefore, the following are the regulation strategies: e adjustment of work roll bending is relatively small, while the adjustment of intermediate roll bending is relatively big.
When the expression X 2 e > u q or X 2 e < l q is satisfied, the following are the regulation strategy C: where u q is the upper limit of quadratic reasonable range. It represents critical value of severe central wave. l q is the lower limit of quadratic reasonable range. It represents critical value of severe bilateral wave. When the expression l q < X 2 e < u q is satisfied, the following are the regulation strategy D: When the edge distinguishing factor is greater than the upper limit of edge reasonable range, the local flatness status is severe edge drop. When the edge distinguishing factor is less than the lower limit of edge reasonable range, the local flatness status is tight flatness in the outermost section. When the edge distinguishing factor is within the edge reasonable range, the local flatness status is slight edge drop. If the severe edge drop appears in rolling, the following are the regulation strategies: e adjustment of intermediate roll shifting is relatively big. If the slight edge drop appears in rolling, the following are the regulation strategies: e adjustment of intermediate roll shifting is relatively small.
When the expression X 3 e > u e or X 3 e < l e is satisfied, the following are the regulation strategy E: where u e is the upper limit of edge reasonable range. l e is the lower limit of edge reasonable range.
When the expression l e < X 3 e < u e is satisfied, the following are the regulation strategy F: rough collaboration strategy, the flatness control system can intelligently select the optimal adjusting mode according to the actual flatness status. e flowchart of collaboration strategy for flatness actuator group is shown in Figure 2. e collaboration strategy for the flatness actuator group includes flatness analysis module, strategy matching module, and coordinated adjustment computing module. First of all, the method of calculating the equivalent flatness curve is used to extract the flatness defect characteristics for the measured flatness value. Secondly, the adjustment strategy that matches the actual flatness is selected by solving the flatness distinguishing factor. Finally, the Topkis-Veinott algorithm and genetic algorithm are jointly optimized to obtain the coordinated adjustment of the actuator group. e specific requirements of the strip steel flatness in the downstream process are different. Different specifications of strip steel flatness control accuracy are also different. erefore, the determination of these coefficients requires comprehensive consideration of strip steel specifications and target flatness coefficients.

Flatness Actuator Group Coordinated Adjustment.
e collaboration strategy strategies A ∼ F were originally formulated for different flatness conditions. When the flatness condition matches the adjustment strategy, the collaboration strategy can effectively avoid the uncoordinated regulating behaviors in the flatness actuator group. Every collaboration strategy can be called in a loop. When the flatness condition does not match the adjustment strategy, the current strategy is abandoned, and other strategies are selected based on the judgment conditions. e external evaluation function is considered as the objective function of calculating coordinated adjustment. e external constraint condition and the flatness actuator group collaboration strategy are seen together as constraint condition of calculating coordinated adjustment. e single-wave situation is taken as an example. e expression of calculating coordinated adjustment is as follows:

Coordinated Algorithm Based on Topkis-Veinott and Genetic Algorithm
In order to achieve actual flatness condition discriminating factor and flatness actuator group coordinated adjustment, the coordinated algorithm is proposed based on Topkis-Veinott and genetic algorithm. Its main advantage is as follows.
In the coordinated algorithm, both the searching definiteness and randomness are taken into account. e probabilistic search is adopted in the transfer direction of search point. And the deterministic search is adopted in transfer relation of search point. is algorithm design can provide high search speed and flexibility, and the situation of missing optimal point can be avoided. Moreover, in the coordinated algorithm, multipoint searching and singlepoint searching are simultaneously carried through. is algorithm design can provide a more extensive search scope and more abundant search information.
Expression (4) and expression (11) are equivalent to the following function optimization problem: where f TV (x TV ) is the objective function of function optimization problem. g TVi (x TV ) ≥ 0 is nonlinear and linear inequality constraints of function optimization problem. m TV is the number of nonlinear and linear inequality constraints. x TV � (x TV1 , x TV2 , . . . , x TVN TV ) T is the variable vector. N TV is the number of variables. e flowchart of coordinated algorithm is shown in Figure 3. Its step is as follows: TV is selected as the initial point of coordinated algorithm. e expression ε TV > 0 and the expression k TV � 0 are satisfied. x (0) TV is the initial point of coordinated algorithm. ε TV is the iteration accuracy of coordinated algorithm. k TV is the iteration number of coordinated algorithms.
(2) e programming problem A is established as follows: e optimal solution of programming problem A is (P TV ) T , and it is the result after k TV iterations. x TV is a point in the iterative process. P TV � (P TV1 , P TV2 , . . . , P TVn TV ) T is the descent direction vector of point x TV . n TV is the dimension of vector P TV . y TV � max ∇f TV (x TV ) T P TV , −∇g TVi (x TV ) P TV , i ∈ I TV } is the decision parameter of terminating iteration. ∇f TV (x TV ) is the partial derivative of objective function. ∇g TVi (x TV ) is the partial derivative of inequality constraints.
(3) e programming problem A is transformed into an equivalent programming problem B: M is the number of inequality constraints. a i is the lower limit of constrained domain. b i is the upper limit of constrained domain.  hj (xi) ≥ 0 ⑰ output optimal value of coordinated algorithm . a i is set to −1. b i is set to 1. An initial population is generated. (5) e maximum generation T, population number M population , cross probability P cross , and mutation probability P mutation are assigned to a starting value. (6) e fitness F fitness of each individual in the population is calculated. (7) When the condition t < T is satisfied, turn to (8).
When the condition t < T is not satisfied, turn to (13). (8) e selective probability P select and accumulative probability P accumulate of each individual in the population are calculated. A random number in interval [0, 1] is generated. If the random number is less than P accumulate (1), the first individual is selected. If the random number is more than P accumulate (k − 1) and less than P accumulate (k), the k individual is selected. e best individuals get multiple copies. Medium individual keeps steady. e worst individual is dead. M population individuals are randomly selected on the basis of selective probability P select . e copies of the best individual are related P accumulate . And it is a calculated value. e medium individual is an individual who has a higher fitness than the eliminated individual and has not reached the optimal fitness. It is the medium one after sorting all the individuals as their fitness values. (9) A random number in interval [0, 1] is generated. If the random is less than cross probability P cross , the individual is crossed. e individuals are selected from the population for mating. e offspring goes into the new population. e unmated individuals are directly copied into the new population. (10) e mutation opportunity of each individual is equipotent. A random number in interval [0, 1] is generated. When the random is less than P mutation , the individual is mutated. e individuals are selected for mutating in the new population. e original individual is replaced by the individual after mutating. (11) Set t � t + 1. Turn to (6). (12) e individual of the maximum F fitness is decoded.
x(t) after decoding is the optimal value. x(t) is the optimal solution (P (k TV ) When the terminal condition |Z(k)| < ε Z is satisfied, turn to (18). When the terminal condition |Z(k)| < ε Z is not satisfied, turn to (14). (14) λ U TVk TV is the upper bound of the search step size factor in the k TV iteration. λ TVk TV � max . . . , m TV } is the search step size factor in the k TV iteration. λ TVk TV can be deachieved by linear search technology.
(15) e following one-dimensional search problem is solved: is calculated. en, we can turn to (2).

Field Test Experiment.
e collaboration strategy for flatness actuator group is adopted to a flatness control system of a 1450 mm five-stand cold rolling mill. e C language program is written according to the collaboration strategy; the custom function block for the coordinated algorithm that can be called directly in Step 7 environment is generated through the Function Block generator tool. e conventional method is to use the least square method to solve the optimal adjustment amount of each flatness actuator. However, the strategy of flatness actuator is not matched to the actual flatness of the strip. e collaboration strategy is encapsulated into the coordinated regulating module and it is embedded into the original flatness control system. e main hardware of SIMATIC TDC is shown in Table 1. e initial value of coordinated algorithm parameter is shown in Table 2. e algorithm comparison chart is shown in Figure 4. e flatness control system equipment distribution is shown in Figure 5. e operation interface of the flatness control system is shown in Figure 6. e 1450 mm five-stand cold rolling mill production line is shown in Figure 7.
GA is done in a probabilistic way, but this randomness may cause nonconvergence. Topkis-Veinott algorithm uses a deterministic search method. e transfer from one search point to another has a certain transfer direction and transfer relationship. e coordinated algorithm takes into account the determinism and randomness of search. e probabilistic search technology is used for the transfer direction of the search point. e deterministic search technology is used for the transfer relationship of search point. is ensures high search speed and flexibility. And it avoids the situation where the best point cannot be searched all the time. In Figure 4, the objective function value of GA maintains a decreasing trend in the initial iteration stage. However, as the number of iterations increases, the objective function value of GA fluctuates greatly. Topkis-Veinott algorithm can ensure the trend of continuous reduction of the objective function value. But its number of iterations is relatively large. e value of the objective function of the coordinated algorithm maintains a decreasing trend. And the convergence is reached in a small number of iterations. e flatness detecting device is ABB shapemeter roll. e flatness regulating device of the six-roll UCM rolling mill includes work roll tilting, work roll bending, intermediate roll bending, intermediate roll shifting, and selective work roll cooling. SIMATIC TDC controller communicates with HMI, PDA, and L2 server via Industrial Ethernet. e independent computer is used for monitoring and diagnostics of SIMATIC TDC controllers (Table 3).

Flatness Control Effect of Different Rolling Speed.
When the rolling speed is different, the control effect with using the flatness actuator group collaboration strategy is compared with the control effect with using the conventional method. e experimental parameter of the flatness control effect test of different rolling speed is shown in Table 4. e flatness control effect of different rolling speed is shown in Figure 8. e compensation efficiency of strip rolling speed is as follows: where c s is compensation efficiency of strip rolling speed. It represents the compensation efficiency for the flatness deviation caused by the speed change. d 1 is the average flatness deviation of 910 m/min rolling speed with the conventional method. d 2 is the average flatness deviation of 1100 m/min rolling speed with the conventional method. In Figure 8, when the rolling speed is 910 m/min and the control method is changed from conventional model to collaboration strategy model, the average flatness deviation is decreased in every measuring section. e maximum decreasing magnitude is 3.81 I. It indicates that if the wide      Mathematical Problems in Engineering strip is rolled, the control effect of using collaboration strategy model is better than the control effect of using conventional model. When the conventional model is used and the rolling speed is changed from 1100 m/min to 910 m/ min, the average flatness deviation is increased in every measuring section. e maximum increasing magnitude is 3.17 I. It indicates that if the conventional model is used, the control effect of low rolling speed is worse than control effect of high rolling speed. When the rolling speed is changed from 910 m/min to 1100 m/min and the control method is changed from collaboration strategy model to conventional model, the change of average flatness deviation is small. e maximum changing magnitude is 1.33 I. It indicates that the rolling speed can be compensated by using collaboration strategy model. e maximal compensation efficiency of strip rolling speed is 51.89%.

Flatness Control Effect of Different Rolling Force.
When the rolling force is different, the control effect with using the flatness actuator group collaboration strategy is compared with the control effect with using conventional method. e experimental parameter of the flatness control effect test of different rolling force is shown in Tables 5 and 6. e flatness control effect of different rolling force is shown in Figure 9. e compensation efficiency of rolling force is as follows: where c f is the compensation efficiency of rolling force. It represents the compensation efficiency for the flatness deviation caused by the change of rolling force. e 1 is the average flatness deviation of 8300 kN ∼ 8900 kN with conventional method. e 2 is the average flatness deviation of 7700 kN ∼ 8300 kN with conventional method. In Figure 9, when the rolling force is 8300 kN ∼ 8900 kN and the control method is changed from conventional model to collaboration strategy model, the average flatness deviation is decreased in every measuring section. e maximum decreasing magnitude is 2.52 I. It indicates that if the strip is rolled in large rolling force, the control effect of using collaboration strategy model is better than the control effect of using conventional model. When the conventional model is used and the rolling force is changed from  7700kN-8300 kN to 8300 kN ∼ 8900 kN, the average flatness deviation is increased in every measuring section. e maximum increasing magnitude is 2.71 I. It indicates that if the conventional model is used, the control effect of large rolling force is worse than control effect of little rolling force. When the rolling force is changed from 7700 kN-8300 kN to 8300 kN ∼ 8900 kN and the control method is changed from conventional model to collaboration strategy model, the change of average flatness deviation is small. e maximum changing magnitude is 1.15 I. It indicates that the rolling force can be compensated by using collaboration strategy model. e maximal compensation efficiency of rolling force is 42.88%.

Flatness Control Effect of Different Rolling Reduction.
When the rolling reduction is different, the control effect with using the flatness actuator group collaboration strategy is compared with the control effect with using conventional method. e experimental parameter of the flatness control effect test of different rolling reduction is shown in Table 7. e flatness control effect of different rolling reduction is shown in Figure 10. e compensation efficiency of rolling reduction is as follows: where c r is the compensation efficiency of rolling reduction. It represents the compensation efficiency for the flatness deviation caused by the change of the rolling reduction. f 1 is the average flatness deviation of 32.96% rolling reduction with conventional method. f 2 is the average flatness deviation of 15.89% rolling reduction with conventional method (Table 8).
In Figure 10, when the rolling reduction is 32.96 and the control method is changed from conventional model to collaboration strategy model, the average flatness deviation is decreased in every measuring section. e maximum decreasing magnitude is 5.89 I. It indicates that if the strip is rolled in high rolling reduction, the control effect of using collaboration strategy model is better than the control effect of using conventional model. When the conventional model is used and the rolling reduction is changed from 15.89% to 32.96%, the average flatness deviation is increased in every measuring section. e maximum increasing magnitude is 2.93 I. It indicates that if the conventional model is used, the control effect of high rolling reduction is worse than control effect of low rolling reduction. When the rolling reduction is         changed from 15.89% to 32.96% and the control method is changed from conventional model to collaboration strategy model, the change of average flatness deviation is small. e maximum changing magnitude is 1.31 I. It indicates that the rolling reduction can be compensated by using collaboration strategy model. e maximal compensation efficiency of rolling reduction is 36.77%.

Conclusion
(1) e flatness actuator group collaboration strategy is created on account of the actual flatness condition discrimination factor. In the newly raised collaboration strategy model, the actual flatness situation can be calculated and identified. What is more, the overall regulation capacity of flatness adjustment actuator after the combination is made to match with the flatness defect so that the flatness control system can give full play to its potential. (2) In online test experiment using collaboration strategy model, a preliminary finding is achieved: When the strip rolling speed is increased from 910 m/min to 1100 m/min, the maximal compensation efficiency of rolling speed is 51.89%. When the rolling force is increased from 7700 kN-8300 kN to 8300 kN ∼ 8900 kN, the maximal compensation efficiency of rolling force is 42.88%. When the rolling reduction is increased from 15.89% to 32.96%, the maximal compensation efficiency of rolling reduction is 36.77%. us, it can be seen that the flatness control effect is improved under rolling conditions of low rolling speed, large rolling force, and high rolling reduction by using collaboration strategy.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.