Static Model Identification for Sendzimir Rolling Mill Using Noise Corrupted Operation Data

A Sendzimir rolling mill (ZRM), one of the rolling mill systems, is a machine used to obtain a steel strip with a desired shape in cold rolling. Model based controllers are mainly used for the shape control, but it is difficult to obtain the mathematical model of the ZRM, so model identification should be used. This study proposes a method identifying static model of the ZRM. To identify the static model of the ZRM, a mill matrix (<inline-formula> <tex-math notation="LaTeX">$G_{m}$ </tex-math></inline-formula>) is obtained that expresses the linear relation between the actuators, and the shape of the strip and the data obtained through the ZRM’s operation are used to obtain <inline-formula> <tex-math notation="LaTeX">$G_{m}$ </tex-math></inline-formula>. However, as the operation data are affected by large measurement noise, and the patterns of the multiple control inputs are not diverse, this results in an inaccurate estimation. Therefore, a data processing method using multiple valid sets of operation data is proposed to estimate <inline-formula> <tex-math notation="LaTeX">$G_{m}$ </tex-math></inline-formula>. Additionally, a <inline-formula> <tex-math notation="LaTeX">$G_{m}$ </tex-math></inline-formula> update method is suggested to estimate the <inline-formula> <tex-math notation="LaTeX">$G_{m}$ </tex-math></inline-formula> whenever a single pass operation is finished, according to the static model change of the plant. The proposed method is verified by comparing the results with the real operation data. This research will be helpful in all industries that use rolling mill machines such as 4-high mill, 6-high mill, and clustering mill in hot and cold rolling.


I. INTRODUCTION
Rolling mill system consists of several mills and the strip to be rolled passes between the mills to control the thickness and shape in hot and cold rolling [1]- [16], which affects the quality of the strip in the steel process. ZRM is one of the rolling mill systems used to roll hard steel strip and control its shape during cold rolling process [1]- [3], [17]- [25]. When a strip is passed through the ZRM, small work rolls above and below the strip alter its shape, flattening it. This is possible even for particularly thin or hard materials, as a cluster of large rolls transmit a strong load to the small work roll, which presses hard on the strip, shaping it to the desired flatness. Owing to this capability, the ZRM is utilized in cold steel rolling processing to produce a hard and thin steel strip product with a uniform shape. The outcome is a better quality final strip, with reduced strip tension imbalance, which prevents strip breakage.
The ZRM [1]- [3], [17]- [25] is more complex with many rolls than the other rolling mill systems used in hot The associate editor coordinating the review of this manuscript and approving it for publication was Yue Zhang . and cold rolling such as 4-high, 6-high mill, and cluster mill [5], [26]- [28]. The ZRM rolls cluster is composed of 8 AS-U-Rolls, 4 first and 6 second intermediate rolls (IMR), and 2 work rolls. The work rolls press vertically on the strip from the top and the bottom, with the other rolls pressing on the work roll to help the process. AS-U-Rolls, which are located on the outside of the cluster, are pushed up and pulled down by their actuators. The pressure is transferred to the work rolls through the second IMR and the first IMR rolls, in succession. Consequently, the work roll pressure determines the strip thickness. Additionally, one side of the first IMR is being tapered off. The pressure on the edge of the strip is adjusted by shifting the IMR in and out of the cluster [15], [23], [25], [29], and the leveling used in the steering control also affects the strip shape [17], [30]. To preclude the strip from being off-centered owing to rolling asymmetries such as tilted work rolls, the balance of the roll's left and right positions is adjusted by the leveling [17], [30]. The leveling is used not for flatness control, but for steering control, which can result in shape changing. The work rolls' pressure change through the AS-U-Roll actuators, IMR shifts, and leveling ultimately results in the strip shape change. VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ To control the strip's shape in the rolling mill systems including the ZRM, both non-model based and model based controls have been studied. Neural-fuzzy control [21], [22] and approach based on a wavelet radial basis function [20] have been proposed as non-model based control. Since non-model based control does not require the physical model, there is no cost of building a model. However, these controls do not provide optimal control inputs, as they are based on experience, and cannot guarantee the stability of the system. In contrast, model predictive control [5], [31], approaches based on singular value decomposition (SVD) [19], [32] and extended SVD [33] have been proposed as model based control. These controls give optimal control inputs for the desired performance, as they use the dynamic system and guarantee the stability. The model based controllers require a model of the system to control and various modeling approaches have been studied [5], [8]- [10], [18], [21], [23], [25]- [27], [34]- [38]. To obtain a mathematical model of the ZRM, accurate geometric and material information is needed for each roll, and then the complex physical relationship between the rolls needs to be determined. This entails additional cost, and there is no guarantee that the obtained model is accurate owing to incorrect parameter information and model changes such as roll deformation [26], [27], [31].
Therefore, to obtain the ZRM model in a straightforward manner, identification approaches using operation data of actuator inputs and the corresponding strip shape outputs have been proposed [33], [39]. The ZRM model has two components: a dynamic model, and a static model. The dynamic model contains information such as the delay from the work roll to the sensors, and the actuators. This information is simple and relatively easy to identify, as it can be approximated with a first-order system with delay [1]- [3]. The static model presents the steady-state change of the strip when input is applied to the actuators. This represents the complex relationship between the actuators and strip shape, and the static model is expressed as a linear model using a G m [1]- [3], [19] that represents the shape change resulting from the input variation at a certain operating point. Therefore, it is important to identify the accurate static model, as it is related to the steady-state shape control performance.
A method identifying the G m has been studied in [33], where the G m is identified by measuring the shape change when exciting the control input to each actuator. However, this method requires that an operation is initiated, and then suspended, only for the purpose of data collection, and the coil is squandered for the sole purpose of obtaining the data. Additionally, the operation causes work roll deformation and wear. This changes the parameters of the mill, and ultimately produces G m variations. As a result, the uncertainty causes large model errors and degraded control performance [13], [16], [40]. Therefore, to identify each new G m variant, the operation needs to be stopped periodically, causing additional costs. In addition, recursive least square (RLS) based identification method has been proposed in [39]. This identification method does not squander the coil. Using the input and output data, the G m is identified and updated iteratively. However, there is no analysis and verification of the identification method and only the control method was verified by a simulation. The operation data are not used and their issues are not solved, so there are drawbacks that the convergence rate is slow and the identification performance is low.
In this paper, we propose a least square based G m identification method by using data obtained during the operation, without suspending operation or consuming coils. In operation data, there are two main challenges with operation data: 1) in practice, the variance of inputs applied to each actuator is small, to prevent plant fault or strip breakage caused by a large shape change, which causes the output to be small. As a result, the output is significantly affected by the measurement noise; 2) the patterns of the multiple control input are not diverse, because the input pattern is determined by the experience of the operator, which results in a rank deficient of the least-square estimation matrix. Considering these data issues, the proposed G m identification method uses multiple operation data to identify a G m . The identification method based on massive data sets obtained for a year diminishes the effects of large measurement noise, and increases the variety of input patterns without stopping the operation. Therefore, the proposed method involves an operation data processing to collect as many data sets needed for G m identification as possible. The proposed data processing distinguishes valid data sets to be used for identification, from sparsely excited operation data generally not used for identification. Then, valid data sets composed of sparsely applied input and its steady-state shape changes are collected. In addition, we propose an G m update method to solve the model uncertainty problem caused by roll deformation and wear.
The three main contributions of the proposed method are as follows: • If enough operation data are given, an accurate static model can be obtained by the proposed G m identification method more simply than the physical model suggested in most existing papers. In addition, the proposed model identification method is available during operation whereas the existing identification method [33] causes the operation suspension and the additional coil consumption.
• To the best of our knowledge, there is no research that a static model of the ZRM is identified and analyzed by using the massive data obtained from the actual operation. The proposed identification method solves the operation data issues such as large measurement noise variance and insufficient applied input pattern. In addition, the proposed data processing can extract multiple data sets from the operation data where the control inputs are sparsely excited.
• The model uncertainty problem is solved as more accurate G m is updated by the proposed update method than updated by the RLS based method [39]. In addition, the proposed G m update method does not require additional cost to update such as operation suspension and coil consumption. In the next section (Section II), the ZRM model to be identified is presented. In Section III, the proposed data processing steps before identifying the G m , and the proposed update method for the estimated G m , are both described. In Section IV, to verify the proposed method, the output is then compared with that of the operation data and the results are analyzed.

II. THE ZRM MODEL
A steel strip is rolled through a ZRM to reduce the flatness error by changing the shape of strip (Fig. 1). As a result, it flattens a thin and hard strip, and it reduces the shape's imbalance to prevent strip breakage [1]- [7], [15], [41]. To reduce the flatness error, several work rolls press directly along the width of the strip. The movement of the work rolls is determined by adjusting As-U-Rolls, the intermediate rolls position, and the level difference by As-U-Roll actuators, IMR shift, and leveling. The change in the strip shape is measured by sensors placed in several zones along the width of the strip at shapemeter rolls, then rolled by a coiler. The ZRM's input vector is composed of u asu ∈ R N asu AS-U-Roll actuators, u IMR ∈ R 2 lower and upper IMR shifts, and u level ∈ R 1 a leveling, and expressed as u = [u asu , u IMR , u level ] T ∈ R N u where T denotes transpose, and N u = N asu + 3. In addition, the ZRM's output vector is composed of strip shape measured at shapemeter roll, and expressed as y = [y 1 , y 2 , . . . , y N y ] T ∈ R N y where N y is the number of strip-shape zones.

A. THE STATIC MODEL OF ZRM
The nonlinear static model of the ZRM is described as y = f (u) where y is the final shape after sufficient time since u is applied. It can be linearized at a certain operating point u 0 and y 0 as where u and y are the change of input and output from the operating point. Let the mill matrix G m := ∂f ∂t u=u 0 , the linearized static model of the ZRM is The G m ∈ R N y ×N u represents a linear combination of each input's effects on the shape of the strip. In other words, y is spanned by the G m columns. G m can be identified by using shape change data based on each input change. Sufficient variance of the shape change data is obtained by the sufficient large changes in the actuator inputs [33]. However, this data collection engenders a high cost owing to having to stop the operation for the input excitation, and owing to high coil wastage. Therefore, G m identification, using the data obtained during the operation, without incurring these additional costs is necessary. In the next section, an operation data processing and a G m identification are proposed. In addition, a G m update is proposed.

III. PROPOSED METHOD FOR G m IDENTIFICATION
The issues with the operation data for G m identification are that the patterns of the multiple control inputs are not diverse, the input variance is very small, and the influence of the measurement noise is large. Therefore, an identification method is proposed to address these issues and enable G m identification based on these data. First, an operation data processing step is performed to extract a valid inputs and outputs set. Then, a G m is identified using these data. Additionally, the G m update method is used to minimize model uncertainty due to model changes.
A. DATA PROCESSING data is defined as a single data set of ( u, y). It is important to use sufficient data to identify the G m because the more data are used, the more variety of input patterns are used. In addition, sufficient data can resolve the measurement noise problem of a small output variance being vulnerable to the noise. When the strip is passed through the ZRM, several u are sparsely applied at different times and the y are the shape change for the given control input deviation after sufficient time since the u are applied. Therefore, we propose a data processing to extract several data for G m identification from the operation data during a single pass.
The data extraction is divided into three steps: 1) a low pass filtering, 2) a steady-state condition check, and 3) an extraction of a data . In the first step, as the measured strip shape commonly includes high-frequency measurement noise, a low-pass filter with a 0.1 Hz cut-off bandwidth is used to reduce the measurement noise.
In the second step, when collecting the data , it is necessary to extract the u applied in steady state to exclude the influence of previously applied u. A searching window is defined to check the steady-state conditions. The window moves along the samples of operation data, and recognizes valid input changes in a single strip process. The window searches for the entire range of the operation data by setting the length of the window sufficiently long to identify the output change triggered by each input change. In this paper, VOLUME 8, 2020 the length of the searching window is set as 1250 samples (50 seconds), where the sampling time is set as 0.04 seconds. Whenever the searching window moves by 75 samples (3 seconds), the measured strip velocity signal, input, and output are examined with several conditions to find valid data sets: • A pre-steady-state section is defined as a steady-state section before an input is applied. The length of the section is set as 500 samples (20 seconds) from the starting point of the window. Likewise, post-steady-state section is defined as the steady-state section after an input is applied. It is also set as 500 samples. Between the two sections, 250 samples were set as the transient section. In the steady-state sections, the changes generated by the input sample are small, and extracted data should not be affected by other changes.
where u i,a:b is a vector of ith input signal from sample a to b. The difference represents the change generated by the input sample in the pre-steady state and post-steady state. ε is a small value, and is set as 0.001 in this paper.
• The differences between average pre-steady-state value and average post-steady-state value of each input is u i, avg = avg(u i,1:500 ) − avg(u i,751:1250 ) where avg(·) is average. The two norms of the difference vector u avg , whose components are u i, avg of i inputs, are larger than a threshold δ to verify whether the change in input is sufficient or not. The δ is set as 1 in this paper. In the third step, a data is obtained if the steady-state conditions are satisfied in the searching window, and u and y are calculated as the differences between average presteady-state values and average post-steady-state values of u and y. Then, the searching window moves for next excited input, and the steps above are repeated to extract more data.

B. G m ESTIMATION
Let U ∈ R N u ×N data input matrix expressed as U = u 1 , u 2 , . . . , u N data where N u is the number of actuators as input, N data is the number of the extracted data , and u k is kth input changes from the extracted data . Likewise, let Y ∈ R N y ×N data output matrix expressed as Y = y 1 , y 2 , . . . , y N data where y k is k th strip shape change by u k . Then, estimated mill matrix G m can be obtained based on least-squares estimation as where U T U U T −1 is the pseudo inverse U. To solve the problem with the operation data and to minimize the estimation error, N data should be large enough. The operation data do not include various patterns of multiple control input in U with small N data . In addition, a small input variance is applied and this results in a small strip change affected by the large measurement noise. Therefore, an estimation based on massive N data is necessary to prevent the lack of data diversity, and to reduce the effect of the measurement noise.
The elements of the obtained G m column may include noise components. Thus, the shape by G m are not smooth although they represents the shape change for an applied unit input. Therefore, each G m column is fitted with a polynomial curve as follows where g p i is a column vector, which is the ith column vector of G m polynomially curve fitted by 8th order, and it represents the effect of each actuator on the shape.

C. G m UPDATE
In practice, the ZRM continuously changes during operations owing to the wear or change of the rolls after multiple operations. As a result, the difference between G m and G m becomes large, leading to large uncertainty, which results in performance degradation of the flatness control. Therefore, to minimize the uncertainty, G m is updated by using only the latest data. Whenever a new set of u and y is obtained, the U and Y for G m are updated. The oldest data set is discarded, and the latest data set is appended as a column into U and Y to maintain a constant N data which is the number of data , for example, U = u 1 , u 2 , . . . , u N data and Y = y 1 , y 2 , . . . , y N data are given, and when new u N data +1 , y N data +1 are extracted from an operation data thorough the data processing, they are updated as U = u 2 , u 3 , . . . , u N data +1 and Y = y 2 , y 3 , . . . , y N data +1 . In addition, the N data should be set to an appropriate number. For example, a value that is neither too large, nor too small. If N data is too large, G m has a large uncertainty as U and Y contain outdated data, and if N data is too short, G m cannot be estimated due to rank deficient of the least-square estimation matrix as U contains a limited input pattern. In this paper, N data is experimentally determined as 1024. The steps of the modeling approach is summarized as follows: 1) Data processing -When the operation for the single pass is finished, input and output operation data are given. At this time, the output is passed through the low-pass filter to remove the high-frequency measurement noise of the output. 2) Data processing -The searching window moves along the input and the steady-state conditions are checked. 3) Data processing -If the steady-state conditions are satisfied in the searching window, the uand yare extracted as a data 4) G m update -Whenever a new data is given, the U and Y are updated. The U and Y are composed of latest N data data . 5) G m update -Using the estimation(4), new G m is obtained. Then, the updated G m is used for model based controller. Including the proposed static model identification method, the block diagram of the strip shape control in ZRM is shown in Fig. 2. In the next section, we analyze the results obtained by the proposed data processing and the proposed G m estimation method using the operational data, and verify the update method.

IV. RESULTS AND DISCUSSION
In this section, the result, data , obtained by the data processing is introduced. Also, estimated G p m for the obtained data is analyzed and the proposed method is verified by comparing with the operation data. The proposed method is compared with the RLS based identification method [39]. hlIn addition, the control results applying the initial G p m and the updated G p m to a model based controller [33] hlare shown.
For the results, 4405 operation data were used. In the ZRM, the number of As-U-Roll actuators, IMR shifts, and leveling are 7, 2, and 1. The number of strip-shape zones was 26. Then U, U ∈ R 10 , and Y, Y ∈ R 36 , so G m ∈ R 26×10 . Also, the width of strip was 1,055mm.

A. EXTRACTION OF u AND y
To show how to extract valid data from the input and output of a given operational data, the three steps of the proposed data processing were applied and the results are described, and the extracted input and output are analyzed. The case where As-U-Roll actuator 2 changed in a searching window, and the shape of the strip changed, is presented in Fig. 3. Only the left half shape of strip-shape zones is plotted, for efficient visual understanding, because AS-U-Roll 2 has a greater effect on the left section of the strip.
In Fig. 3 (a) and (d), the first step of the proposed data processing was applied. The searching window moved along the samples and checked for valid input and output changes. As a result, 3 sets of data were obtained at samples 5560, 7285, and 9835 respectively. At sample 7285, only As-U-Roll 2 changed, and this case was analyzed.
In Fig. 3 (b) and (e), the second step of the proposed data processing was applied. The input signal within the searching window was checked against steady-state conditions, to verify whether it is valid. In the window, the pre-steady-state and the post-steady-state sections, defined as 500 samples, are indicated by a blue dotted line. The two norm of the difference between the input signal and the input signal after  1 sample within both pre/post section ||u i,1:500 − u i,2:501 ||, ||u i,750:1249 − u i,751:1250 || for all i were obtained as 0 smaller than ε = 0.001. Furthermore, the two norms of the difference vector ||u 2, avg || were 1.05, larger than δ = 1. Therefore, it was verified that the u and y changes are valid, and subsequently the data needed to be extracted. Fig. 3 (c) and (f) show the u and y which were obtained by the third step of the proposed data processing. Only the step input of actuator 2 was applied to adjust the left shape; subsequently, the strip shape shows larger changes on the left side of the strip. However, the shape change was not smooth. This was caused by the output's measured noise, which cannot be eliminated by the low frequency bandpass filters. This problem could be solved by the proposed identification method, based on the use of massive extracted data.

B. G m ESTIMATION
Through the data processing steps, a total of 5393 data were extracted and used to verify the proposed identification method. Of those, 4854 data were used to estimate G m , and 539 data were used as test data, for verification purposes. Using the 4854 data , U and Y were obtained and applied to the estimation (4) for G m estimation.  Fig. 4. In other words, each G m column (g 1 , g 2 , . . . , g 10 ), and G p m (g as an effect of each actuator on the strip shape, are shown. The As-U-Roll actuators 1-3 ( Fig.4 (a)-(c)), and actuators 5-7 ( Fig.4 (e)-(g)) had a symmetrical effect on the left and the right strip shapes based on the center of actuator 4 ( Fig.4 (d)). Furthermore, the IMR shift 1 and 2 also had a symmetrical effect on the left and right strip shapes ( Fig.4 (h), (i)). However, they were not shown perfectly symmetric because the proposed method reflected minor asymmetric differences due to the deformation and wear of the rolls. Leveling had the effect of a linear function with a certain slope banded on the center of the strip shape. (Fig.4 (j)). Compared to the G m columns, the G p m columns looked physically reasonable, because radical change in G m is difficult to occur in practice between strip-shape zones. Therefore, the strip shape should be presented by the linear combination of weighted columns of G p m , rather than G m .   The NMSE can indicate how the estimated G m matches the actual ZRM static model G m . The NMSEs obtained from 539 test data were plotted (Fig. 5). To show the NMSE trend, the NMSE moving average was illustrated as a red line. The NMSE mean was 0.7324 (−1.3524 dB), the NMSE minimum was 0.0301 (−15.22 dB), and the NMSE maximum was 8.35 (9.22 dB). The large difference in performance was due to the abnormal operation data output caused by the measurement noise, even though it was considered as a reference. To identify the problem, the best result (Fig. 6) and worst result (Fig. 7) were shown and analyzed respectively. When the input u ( Fig.6 (a) and Fig. 7 (a)) is given, the y by the G p m was obtained as a linear combination of the u and it was compared with the y of the operation data by the actual model shown in the black line ( Fig.6 (b) and Fig. 7 (b)).
In the best case ( Fig. 6 (b)), the y by G p m matched with y of the operation data, within a small margin of error. However, even though the operation data was considered for reference, it included large measurement noise at the center of the shape, which was considered measurement noise. As a result, although the G p m is physically reasonable, it made NMSE increase (−10.95 dB). Notwithstanding, this result shows that the proposed identification method using massive data reduced the measurement noise present in most strip shapes.
In of the worst case ( Fig.7 (b)), the difference between the y of the G p m and operation data was large, so the identification can be seen as wrong. However, the operation data is considered abnormal due to a too large measurement noise, because the lower IMR shift (actuator 9) actually contributes to the right shape increase, but it did not show that acceptable physically change as the right shape decreased. Therefore, a large NMSE (3.91 dB) is obtained, even though the G p m results seem reasonable. Since we cannot find an VOLUME 8, 2020  ZRM to be controlled is the same as the updated G p m [4369]. As the model based controller, ESVD controller was used, which is the most recently suggested for the shape control [33]. The initial G p m and the updated G p m [4369] were used as the static model of the ESVD controller. Considering the ZRM environment, the parameters needed for the ESVD controller were experimentally determined. The control was to flatten the given strip shape which was determined as the initial shape of the data obtained by actual operation and is shown as blue line in Fig. 11.
In Fig. 11 [4369] showed better control results with lower NMSE than the ESVD controller using the initial G p m . Therefore, the control performance could be improved by using the proposed update method of reflecting model changes. As the above result, the proposed update method can be used simply without stopping operation, coil consumption, and slow convergence rate, unlike the existing methods.

V. CONCLUSION
This paper proposed an operation data processing and identification method for valid data sets for the static model of the ZRM. The operation data have three issues: diversity of multiple control input' patterns, small input and output size, and output distortion due to measurement noise. To address these issues, sufficient data are needed. The proposed data processing highlights how to extract valid input and output data sets out of one operation data. Subsequently, the proposed G m estimation method, using massive data, solves the operation data issues. Additionally, the estimated G m is fitted by polynomial to adapt physically as G p m . We verified the proposed method by comparing the strip shape by G p m for the given input of the operation data to the output of operation data. However, some of the test data showed poor results in the form of physically unreasonable data due to low SNR or sensor problems. These results can be considered to have low reliability. As many operations cause the change of the ZRM static model due to the wear or deformation of the roll, a fixed G p m degraded the performance. Therefore, we also solve the issue by using the proposed update method. Whenever the input is given, the G p m is updated to reduce the model uncertainty by discarding the old data set and using the new data set. By comparing several G p m based on the update method, we confirmed the matrix change caused by the ZRM change. In addition, we also compared the NMSE by the G p m with and without the update method, and we found that the update method is necessary to apply the system changes to the model simultaneously. Therefore, the proposed method could identify the static model in real-time, and could minimize the model uncertainty without calculating physical model or stopping the operation or wasting strip. This proposed method will be helpful to systems with cross directional control, and to rolling-mill systems that incur high costs due to operation stops. Furthermore, an accurate control model can be simply obtained by using the proposed method. Therefore, a future research will focus on designing a model predictive control for strip flatness control and robust controller using the identified model to solve the performance degradation due to the model uncertainty caused by the roll change.