Multi-objective optimal edge-drop control in tandem cold rolling of silicon steel strip

In cold rolling of silicon steel strip, edge drop directly affects the side cutting and yield of silicon steel. To improve the edge drop of silicon steel in the cold rolling process, a multi-stand coordinated control strategy and model based on multi-objective optimization are proposed. The first step is to analyze the influence of each control stand work roll shifting (WRS) on edge drop through the finite element model (FEM), then determine the edge drop control effectiveness of each work roll shifting actuator. Meanwhile, a fitting model is proposed to solve the limitation caused by the discrete change of efficiency factors in solving the optimal adjustments of each WRS. On this basis, the target model for silicon steel edge drop control is formulated, and the optimal adjustment model of WRS of each stand is established. To realize the optimal coordinated control among the rolling mill stands, a multi-objective optimal model of edge drop control has been established by the overall modeling method. The improved penalty function algorithm is developed to calculate the optimal adjustment of WRS for each stand. The experiments and application show that the proposed multi-objective optimal control strategy and model can effectively improve the accuracy of edge drop control in silicon steel continuous cold rolling.


Introduction
Silicon steel, known as electrical steel, has silicon content of about 0.3% or lower. The silicon element can reduce eddy currents, increase resistivity of matrix, and weaken magnetic aging phenomenon. However, silicon can make silicon steel more brittle. Due to the prevalence of edge drop defects, silicon steel needs to be edge cut, and the edging range is generally 20-30 mm, which causes a great waste of resources, and also has a large impact on the production efficiency of enterprises. To control the generation of edge drop phenomenon, many scholars have done a lot of research around the analysis of strip rolling mechanics, numerical modeling of WRS, and regulation of efficacy fit strategy. For example, Jiang et al. [1] simulated work roll flexural deformation in the cold rolling process by establishing a work roll mechanics model. Jiang analyzed the tighter work roll bite, the smaller deviation of plate thickness from the target setting after strip rolling. Cao et al. [2] performed a single stand FEM of a 4-roll EDC mill to study the deflection deformation due to work roll pressure in contact area. The conclusion shows that reducing the contact overhang area of work rolls would improve the edge drop phenomenon. Wang et al. [3] established a numerical model of WRS and designed a three-point edge drop deviation fit regulation strategy, which is based on the theory of metal shifting flow in the edge drop region. The edge drop 1 3 phenomenon is influenced by many factors such as strip thickness, width, rolling reduction, and rolling force. The complex working conditions in actual production cannot be fully simulated by simple rolling experiments and offline calculation methods [4,5]. Therefore, the traditional way of simulating idealized conditions has the matter of a single control target, which not only cannot reflect the real-time changes of the whole edge drop area but also cannot improve the local high point of the edge area. To improve the quality of silicon steel edge drop in the cold rolling process, a 1500-mm five-stand UCMW cold rolling mill is taken as the research object based on the single-point and three-point edge drop control models. [6] FEM is used to analyze the relationship between WRS and edge drop. In order to realize the coordinated multi-stand control of edge drop, a multi-objective optimization-based model is established to calculate the optimal WRS actuator amount for each stand. By using the improved penalty function method, the multi-objective optimization solves the optimal adjustment amount for each adjustment mechanism. This paper realizes the establishment, correction, and multi-objective control of the edge drop control efficiency coefficient model with real-time dynamic rolling, which provides a theoretical basis for industrial practice.

Silicon steel cold rolling edge drop closed-loop control
Edge drop closed-loop control is to calculate the required adjusting quantity to eliminate the deviation between actual and target edge drop by feedback model with the measured signal of thickness gauge under stable rolling conditions [7,8]. Then, continuously send out adjusting instructions to each edge drop-adjusting mechanism, and finally realize the continuous, dynamic, and real-time adjustment of strip edge drop area. Figure 1 shows an edge drop control system of a 1500-mm five-stand cold rolling mill. The first three stands of the mill are equipped with a WRS mechanism, and the mill exit is equipped with an edge drop sensor. The edge drop of silicon steel is measured online by thickness gauge during the rolling process, which is controlled in real time by the WRS mechanism of the first three stands. Therefore, in this paper, the high-precision edge drop closed-loop control system proposed includes the following three key models: 1. Model of regulating efficacy coefficient of WRS mechanism on edge drop Under the condition that work roll end taper is determined, online WRS is the regulating mechanism of the edge drop closed-loop control system. Obtaining the edge drop influence law of WRS and modeling its characterization is the basis for realizing high-precision edge drop closed-loop control.

Edge drop index evaluation model and target setting model
The edge drop index evaluation model is used to evaluate the degree of silicon steel edge drop and judge the quality of edge geometry control. The edge drop target setting model is used to provide target values for closed-loop control, which reflects the requirements of the downstream process for the dimensional accuracy of silicon steel products. 3. Multi-objective optimization model for optimal WRS in each stand To establish the real-time requirements of the edge drop closed-loop control process, a multi-objective optimization model for solving optimal WRS in each stand is proposed. At the same time, a global optimization algorithm with high computational efficiency, high computational accuracy, and strong convergence is also designed.

Rolling working condition feature space
The entire cold rolling process is divided into a threedimensional feature space characterized by several rolling condition points connected according to the various intervals of production process parameters, as shown in Fig. 2. Each node in Fig. 2 represents a rolling condition point, and the position of each rolling condition point in threedimensional feature space is determined by values of three process parameters: width, thickness, and rolling reduction. Each node in three-dimensional feature space corresponds to a set of edge drop control efficiency coefficient matrices for that rolling condition. Also, the corresponding edge drop control efficiency coefficient matrix can be obtained by establishing a FEM for the rolling process. The change of parameters at each working point in the silicon steel rolling process is continuous, and the effect of WRS on edge drop follows working parameters [9,10]. Thus, the value of edge drop control efficiency coefficients at non-nodes in threedimensional feature space can be solved by fitting the simulation results at neighboring nodes around them. In this way, simulation results of a finite number of working points can be used to obtain the edge drop control coefficient matrix for any working condition. For example, when an actual rolling condition is located at O in Fig. 2, the value of the edge drop control efficiency coefficient for this condition can be fitted from simulation results at eight nodes adjacent to it, such as 1, 2, 3, 4, 5, 6, 7, and 8.

The FEM effect law of WRS on edge drop
The rolling process parameters are selected, and the distribution strategy of the roll system and strip mesh is developed. ANSYS/LS-DYNA software was used to establish a three-dimensional finite element model of the silicon steel cold rolling process [11]. Considering the elastic flexural deformation and elastic squash of the roll, and the three-dimensional elastic-plastic deformation of the strip, the roll is set as an elastomer and the strip is set as an elastoplastic body. The restraint stress is applied at the roll neck, the bending roll force is applied at the unit rigid body connected to the roll neck, and the rotational speed is applied at the working roll neck. Also, ignore roll wear. According to the established three-dimensional FEM of the strip cold rolling process, different WRS amounts are given separately [12,13]. Based on calculating the influence law of pressure distribution between rolls and shifting thickness distribution of strip, the edge drop WRS control efficiency coefficient of each node is calculated respectively.
A finite element simulation model of the UCMW silicon steel mill is shown in Fig. 3. Table 1 shows the roll system parameters in a simulation model, which are derived from the production site of a 1500-mm silicon steel cold rolling mill. The thickness and reduction rate of each stand will change during the rolling process. Table 2 shows the rolling schedule for each stand. Figure 4 shows a set of relationship curves between WRS and corresponding silicon steel edge drop obtained by finite element simulation. The horizontal coordinate is WRS, and the vertical coordinate represents the corresponding edge drop variation. The location of edge drop measurement is based on the measurement position of the thickness gauge, which is 0-110 mm from the edge of the strip respectively. From Fig. 4, it can be obtained that the strip edge drop phenomenon is improved with WRS increasing. However, if WRS is too deep, it will lead to local thickening of strip and edge rise phenomenon. At the same time, the internal stress of the edge should be considered. The internal stress will become larger when the edge is locally thickened, and the edge cracking phenomenon will occur.
To verify the accuracy of the finite element analysis calculation, the simulation results of measurement points at 10, 40, 70, and 100 mm from the strip edge are compared with the measured values in actual production, and the results are shown in Fig. 5.
For example, in Fig. 5a, it can be seen that the overall deviation between finite element simulation and measured values under the same process conditions is small, and the overall law remains consistent. The reason is that strip thickness decreases while tension between the stands increases as rolling progress, resulting in changes in a range of efficacy of edge drop control at each stand [14][15][16]. So, a reasonable efficacy factor and strategy for each stand should be determined in industrial production.

Fitting model of WRS control efficacy
According to finite element simulation results, the edge drop control efficiency coefficient matrix is established: where eff i N is the amount of edge drop effect at the position N from the strip edge when the WRS is i. When the actual rolling working point falls at point O in Fig. 2, its eight boundary points are 1, 2, 3, 4, 5, 6, 7, and 8, and then fitted model of plate shape regulation efficacy coefficient at point O can be expressed as: where γ i is the weight of each boundary point, which needs to be determined by the similarity degree.
The similarity between each boundary point and the actual working point is determined by using a standard normal distribution function, as 10 10 eff 10 20 ⋯ eff 10 N eff 20 10 eff 20 where ω(d) is the similarity corresponding to the actual working point at distance d from the boundary point, and σ is the standard deviation of the normal distribution.
The distribution of similarity of each boundary point needs to determine the maximum distance d max between each boundary point, and 0.5 d max is brought into Eq. (3) to determine the appropriate value of σ. Then, the distance between each boundary point and the actual working point is brought into Eq. (3) respectively, and then the similarity degree can be obtained. According to the proportion of similarity of each boundary point in the sum of similarity of all boundary points, the weight factor of each boundary point can be obtained as follows:

Multi-objective optimal control of edge drop
To realize the high-precision control of edge drop, this paper proposes a control model optimization strategy based on a global optimization algorithm. By establishing an overall optimization model of each stand, the multi-objective coordinated control of multiple stands is realized. Then, each stand calculation of the optimal WRS actuator amount is completed by using the improved penalty function method.

Model of target set value of silicon steel edge drop after rolling
To solve the problem that each discrete point in the edge drop regulation efficacy matrix cannot be analyzed, curve fitting is performed for each discrete point in Fig. 4, and the curve equation is set as where f i,j (x i ) is the amount of edge drop effect of the ith adjustment mechanism at distance j from strip edge for different shifting x i , β is the intercept term of the polynomial fit, α 1 is the primary coefficient term in polynomial fit, and α 2 is the quadratic coefficient term in polynomial fit. The quadratic curve fitting was performed on the simulation results of each stand obtained from finite element analysis. The coefficients of each stand curve fitting are shown in Fig. 6.
For the quadratic function of the curve formula, the main consideration is the quadratic term coefficient α2. From Fig. 6, when the WRS of each stand is 0-20 mm, the first stand regulation efficiency is the highest, and the second and third stands can be ignored. With the WRS amount increasing, the second stand has the highest regulation efficiency when it reaches the middle edge drop area [17,18]. When WRS is to the edge of the strip, the first and second stand regulation efficiency decreases, the third stand regulation efficiency rises to the highest, and the regulation range of edge drop area of each stand gradually decreases. As a result, the edge drop characteristics of each stand are as follows: the first stand controls internal deviation of the strip, the second stand controls central deviation, and the third stand controls edge deviation.
According to the demand for a multi-objective control strategy, multiple edge drop target values need to be extended. Considering that strip has a certain deflection, the area method is used to describe the edge drop area, as shown in Fig. 7. The strip section in the middle region is roughly characterized by a quadratic curve, so a quadratic curve fitting is performed to obtain f(L) for the area of the strip where no edge drop occurs. The sharp reduction of strip Fitting F(L) and f(L), the control target at point X 0 from the edge position can be obtained as where λ is the control target coefficient, determined by production requirements.

Optimization model of each stand WRS actuator
In the establishment of edge drop regulation, the efficacy matrix has been the amount of depression as a characteristic parameter. At this time, set the optimization objective function of each stand without considering the amount of depression. The objective function of the first stand is established as shown in Eq. (7): The objective function of the second stand is as shown in Eq. (8): The objective function of the third stand is as shown in Eq. (9): where Min sn is the objective function of the n stand.Δ n is the control target corresponding to the nth point (mm). ΔS n is the increment of insertion adjustment corresponding to the nth stand. The total objective function is

Determination of starting point and boundary conditions of solution
In the control process, each feedback control is solved based on the best point of the previous solution, so starting points of the solution need to be added. The current equipment control quantities X 10 , X 20 , and X 30 , which in that order represent the WRS actuator of the first, the second, and the third stand in the current control cycle. If an original curve is y = f(x), and starting point of a solution is (x 0 , f(x 0 )), then the curve after adding starting point of a solution is The fitted curve after the first stand change is shown in Eq. (12): The fitted curve after the second stand change is shown in Eq. (13): The fitted curve after the third stand change is shown in Eq. (14): In addition to the starting point of the solution, there are also constraints of the boundary conditions. The boundary (8) f 3,n x i + f 3,n X 30 = + 1 x i + X 30 + 2 x i + X 30 2 conditions in the edge drop control system are the maximum shifting of a control device, and these boundary conditions need to be transformed into a penalty function to correct the objective function.
Initial boundaries are shown in Eq. (15): where lim x1 , lim x2 , and lim x3 are the first, second, and third stands corresponding to the maximum control amount, which are constraint boundaries. The initial boundaries are the upper and lower limits of different stands shifting. Because of the difference in each stand's thickness, reduction rate, and tensile force variation, which lead to distinct widths of shifting flow area. Thus, the initial boundaries are modified as shown in Eq. (16): Transforming the boundary condition Eq. (16) into a penalty function as shown in Eq. (17):

Solution of the objective function
The following modification of the objective function requires adding starting points and boundary conditions mentioned above. The objective function for the first stand after the coordinate change of the fitted curve is The objective function of the second stand after the change of coordinates of the fitted curve is The objective function of the third stand after the change of coordinates of the fitted curve is The total impact function of the first three stands is The final optimization objective function is where r (k) is the penalty function for the kth time, the reduction factor c is taken as 0.7, and r (0) is taken as 3. Set the initial feasible domain point as (× 1, × 2, × 3) = (1,1,1).
Under solving unconstrained optimization problems, the most rapid descent method reduces travel distance at each iteration, so the convergence speed is slow. The conjugate gradient method converges only linearly for general nonlinear functions. For the objective function seeking optimization problem with dimension n < 20, an improved penalty function method is used in this paper to solve the optimal adjustment amount, and the specific calculation steps are: Step1: Set initial point and related parameters. Select initial point x 0 , and n linearly independent search directions d 0 , d 1 , d 2 , …, d n−1 given the allowable error Err, so that.
Step2: Perform the first search. Let y 0 = x 1 , in turn perform a one-dimensional search along d 0 , d 1 , d 2 , ⋯ , d n−1 . For everything j = 1, 2, …, n are written as Step3: Select acceleration direction to search. Take the acceleration direction d(n) = y(n) − y(0), if ||d(n)||< Err meets, iterative computation terminates and obtained y(n) are the approximate optimal solution of this problem. Otherwise, the point y(n) starts a one-dimensional search along d(n) to find λ(n) such that: Step4: Adjust optimal search direction. In the original n-direction d(0), d (1), …, d(n − 1), replace d(0) with d(n) to form a new search direction, and return to Step2 to continue the solution. (20)

Analysis of experimental and application results
The developed edge drop control system has been applied to a 1500-mm 5-stand tandem UCMW cold rolling mill. The cold rolling mill is equipped with mechanical flatness actuators such as the work roll bending (WRB), the intermediate roll bending (IRB), the roll tilting (RT), and the intermediate roll shifting (IRS). The strip edge drop on the drive side and operation side is controlled by relative upper and lower WRS respectively [19][20][21]. The edge drop industrial controller is a SIMATIC S7-400 controller with integrated application module FM458, developed by SIEMENS. It is shown in Fig. 8.
To compare the advantages and disadvantages of four control strategies: single-point control method, empirical distribution method, three-point control method, and multipoint control method [22]. The corresponding parameters recorded by PDA (process data acquisition, by the iba Analyzer software provided by iba AG) have been analyzed. The test results were obtained as shown in Table 3.
(a) Edge drop control system human-machine interface (b) 1500mm five-stand cold rolling mill As can be seen from Table 3, the global optimization method stopped after 16 iterations when the allowable error is set to Err = 10 −3 , with the least number of iterations and the least total optimization time. In addition, the objective function minimum value of the global optimization method is r (k) = 0.005785, which is more accurate than the other three control strategies. By calculating the thickness control deviations for five points at 20, 40, 70, 90, and 120 mm positions for different control strategies, the comparison of adjustment deviations is obtained as shown in Fig. 9.
As can be seen from Fig. 9, the single-point control method will have control deviations in the central region. Mainly because the single control target can only the improve control accuracy of this target point, while the control effect on other location points is poor. The three-point control method cannot fully adapt to the changes of incoming materials and ignore the influence of the rear frame, resulting in the solved control quantities deviating more from the control target. Comparing the four control strategies, the multi-objective optimal control strategy and model for edge drop can effectively improve control quality in the cold rolling process.

Conclusion
To solve large deviation of control fit and discrete variation of regulation efficacy, an edge drop control method is proposed. The main conclusions are as follows: 1. To improve the silicon steel edge drop phenomenon and provide more accurate control objectives for the edge drop control system, a multi-stand coordinated control strategy and model based on multi-objective optimization are proposed. By dividing the characteristic space of rolling conditions, a FEM of the effect of WRS on edge drop is established. Then, a fitting model of the efficacy of WRS regulation is obtained. 2. To complete the establishment of a multi-objective optimal control model, this paper establishes a target set value model of edge drop control efficiency coefficient and solve the optimal adjustment amount of each mechanism with an improved penalty function method. 3. The theoretical analysis and application show that a closed-loop control strategy based on multi-objective optimal control reduces the adjustment deviation. Not only avoid the problem of a single control objective but also realize the online high-precision control demand of silicon steel edge drop.  Data availability All the authors certify that all data were obtained from field experiments.
Code availability All the authors certify that all the code runs successfully.

Declarations
Ethics approval Ethical approval was not involved in this study.
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Employment Pengfei Wang is an associate professor of Yanshan University.
Huagui Huang is a professor of Yanshan University. Xu Li is a professor of Northeastern University. Dewei Wang is an engineer of SWKD Thin Plate Technology Co., Ltd. Shuwei Duan is an engineer of MCC Captial Engineering & Research Incorporation.