The inspection of inhomogeneous transverse and longitudinal wall thicknesses, which determines the quality of reducing pipe during the production of seamless steel reducing pipe, is lags and difficult to establish its mechanism model. Aiming at the problems, we proposed the quality prediction model of reducing pipe based on EOSELMRPLS algorithm, which taking into account the production characteristics of its timevarying, nonlinearity, rapid intermission, and data echelon distribution. Key contents such as analysis of data time interval, solving of mean value, establishment of regression model, and model online prediction were introduced and the established prediction model was used in the quality prediction and iteration control of reducing pipe. It is shown through experiment and simulation that the prediction and iteration control method based on EOSELMRPLS model can effectively improve the quality of steel reducing pipe, and, moreover, its maintenance cost was low and it has good characteristics of real time, reliability, and high accuracy.
As seamless tubes are widely used in various fields such as automobile, aviation, petroleum, chemical industry, architecture, boiler, and military industry and are playing a very important role in national economy, the seamless tubes are called industrial blood vessels. With the rapid development of economy, the service fields of seamless tube expand ceaselessly and the requirements for product quality also become more and more urgent and higher. The working procedure of seamless steel pipe consists of piercing, tube rolling, and tube reducing. Due to the restriction of mandril rigidity during piercing and tube rolling, it is difficult to obtain the seamless steel pipe whose diameter is below 70 mm on tube rolling train. Also, even a smallsized seamless steel pipe hot rolled whose diameter is greater than 70 mm is not expected to be produced by rolling a small tubular billet because this will decrease the train productivity by leaps and bounds. Therefore, it is reasonable to produce steel pipe with a small diameter using reducing mode. Owing to the use of reducing process, people can use tubular billet with a large diameter for piercing and rolling, larger reducing can be achieved in reducing mill train, and hence long pipe with a small diameter is obtained, which is an effective technical measure in increasing the output, expanding product variety, and reducing consumption. Various countries throughout the world tend to adopt continuous rolling process with high efficiency in steel pipe extending working procedure to produce shell with single specification and alter technical process in reducing working procedure to obtain finished pipes in different specifications.
Since tension reducing mill is the last forming equipment in steel pipe hot rolling production and has big influence on the steel pipe quality, the deviation of wall thickness is an important index in deciding the steel pipe quality. However, as the mechanism model for pipe reducing process is limited and the quality monitoring of the pipe is accomplished by the periodical spotcheck of technical personnel, the inspection results lag severely. Thus, it is significant in theory and economy to establish the wall thickness prediction model of steel pipe reducing with a sufficient accuracy. Macrea and Cepisca [
It takes dozen of seconds to produce a piece of steel pipe using reduction. Thus it can be seen that the seamless steel pipe reducing production process is a typical rapid intermittent one. Plentiful production data on site provide convenience for our use of soft measurement method during intermittent production process. Data from reducing production process take on echelon distribution. In the meantime, as the product specifications often change, it is very difficult to guarantee the same adjustment of stand this time as that of last time for products with the same specification. This will lead to the occurrence of a certain timevarying in model. These problems result in low precision of modeling established by traditional intermittent process modeling method such as multiway PLS. The modeling is difficult to be used in prediction and control of field products. Aiming at the production process characteristics such as timevarying, nonlinearity, rapid intermission, and data echelon distribution, we proposed to use EOSELMRPLS (ensemble of online sequentialextreme learning machinerecursive partial least square) algorithm to establish the quality prediction model for reducing production. Under the circumstance of ensuring high precision, the model has high flexibility and adaptability and can be well used in the prediction and control of product quality on production site.
In order to establish accurately the quality prediction model of reducing pipe, one should firstly analyze the factors influencing the quality of reducing pipe to prevent incomplete information and existed redundancy during modeling from decreasing the precision of model. It is known through the analysis of process characteristic that the effect of various independent variables on the quality of reducing pipe during different time intervals is different. Part of variables only exist in some time intervals and can be treated as factors observing quality of reducing pipe, whereas another part of variables run through entire production process and has large influence on the quality of reducing pipe.
Steel pipe reducing process is substantially a hollow body sinking continuous rolling process. As shown in Figure
Bite stage: bite stage means the stage from pipe head to enter the first stand till the pipe head to enter the last stand. As shown in Figure
Stable rolling stage: from pipe head to enter the last stand till the pipe tail still not to be away from the first stand, when whole mill train rolls the same piece of steel pipe, and rolling load and speed are all stable, called stable rolling stage. As shown in Figure
Steel leaving stage: from pipe tail away from the 1st stand till away from the last stand. As shown in Figure
The relation between time and displacement of reducing tube.
The heating temperature of shell will bring about the variation of metallic resistance to deformation and result in the variation in rolling force and average tension coefficient. The higher the heating temperature of shell is, the lower the metallic resistance to deformation is, the smaller the rolling force is, and the smaller the external diameter undulation of steel pipe is. In addition, the heating temperature of shell is a function of steel pipe diameter and obvious positive correlation exists between the heating temperature and the diameter. The higher the heating temperature of shell is, the bigger the thermal external diameter of steel pipe is after metal is subjected to thermal expansion.
It is generally recognized that friction factor plays a role concerning the effect of rolling speed on metallic transverse inhomogeneous deformation during reduction of shell. The lower the rolling speed is, the larger the friction force is, and hence the more favorable it reduces the inhomogeneity of steel pipe wall thickness.
When the shell reduces, it is difficult to control its wall thickness because its inner surface is not supported by mandril and the pipe wall of the shell will be in a free varying state with changing rolling process condition. Moreover, the inhomogeneity wall thickness of the shell will be inherited to the finished steel pipe after reduction. Therefore, it is an important condition to improve the wall thickness homogeneity of shell in order to guarantee the wall thickness precision of finished steel pipe.
When tension reducing mill is used for the reducing of shell, due to the existence of tension, the diameter of shell decreases while the wall thickness thins. Under stable tension condition, metallic transverse deformation is small, which is favorable for the improvement of the precision of steel pipe wall thickness. However, tension cannot be established or tension undulation occurs among stands when the head of pierced pipe enters in turn various reducing roll stands and the tail end of shell leaves in turn various reducing roll stands. Thus the inhomogeneity of longitudinal wall thickness in the steel pipe certainly will occur.
The output of quality prediction model is the quality of reducing pipe. The quantification index of judging the quality of reducing pipe is the transverse and longitudinal wall thickness. Transverse wall thickness inhomogeneity is the ratio of maximum wall thickness deviation to nominal wall thickness. Its calculation formula is shown as follows:
The size of inhomogeneity of longitudinal wall thickness in steel pipe is determined by the difference between the mean value of rough pipe frontend wall thickness and the mean value of rearend wall thickness. Its calculation formula is shown as follows:
Since linear PLS model cannot describe correctly the nonlinear relation between independent variable
External relation model:
Internal relation model:
Since neural network has the capability of fitting nonlinearity, during the modeling of batch process, nonlinear multiway PLS method that internal model adopts neural network gains extensive application. As traditional feedforward neural network adopts gradient learning algorithm during training, parameters in network needs iteration and update. Not only does the training time last long but also it easily results in the issues of local minimum and excessive training [
In supervised batch learning, the learning algorithms use a finite number of inputoutput samples for training [
In real applications, the training data may arrive chunkbychunk or onebyone; hence, the batch ELM algorithm has to be modified for this case so as to make it online sequential [
The output weight matrix
If
Equation (
Given a chunk of initial training set
Suppose that we have another chunk of data
Considering both
For sequential learning, we have to express
Combining (
When
Let
Equation (
EOSELM consists of many OSELM networks with the same number of hidden nodes and the same activation function for each hidden node [
Assume the output of each OSELM network is
The difference of nonlinear RPLS modeling method based on OSELM from linear PLS method is that it uses ELM to establish internal nonlinear model and in the meantime and achieve the update of internal and external models. This method reserves linear external model, extracts through PLS the attributive information of process, eliminates the colinearity of data, reduces the dimension of input variable, then adopts ELM to establish nonlinear internal model between input score vector matrix and output score vector, and raises the nonlinear processing capability of internal model. Thus, EOSELMRPLS method has the advantages of PLS and ELM, that is, the characteristics of robustness and feature extraction of PLS method and quick nonlinear processing capability of ELM as well as the precision accuracy through model realtime update.
The modeling and testing steps of nonlinear RPLS method based on EOSELM are as follows.
Assign two standardized data matrices,
Deploy the batch data of batch process, use crossvalidation method to determine the number of latent variable, adopt linear PLS method to calculate the score vector matrices
Assign the node number of ELM hidden layer and activation function (e.g., sigmoid function), use ELM to establish nonlinear model between internal model
When new batch data
According to formula (
Use testing data to check model precision. Conduct PLS decomposition on the testing data
Introduce
As there are many specifications of reducing pipe products, one cannot predict accurately the quality of steel pipe using single model. One must classify data in accordance with the specifications of products. As shown in Table
Size table of reducing tube.
Ser. number  External diameter of shell  Wall thickness of shell  External diameter of finished pipe  Wall thickness of finished pipe  Number of variable stands  Number of total stands 

1  152.5  4.25  42.2  3.56  18  24 
2  152.5  10.25  70  10.25  12  18 
3  152.5  7.75  139.7  7.72  12  16 
4  152.5  6  73.03  5.51  12  12 
5  152.5  9.75  114.3  10  10  16 
6  152.5  25  121  25  10  12 
7  152.5  6  139.7  6.2  10  10 
Variable table for modeling of reducing tube quality.
Ser. number  Variable name  Variable meaning 

1 

Motor speed, current, and torque of Number 1 reducing mill 
2 

Motor speed, current, and torque of Number 2 reducing mill 
3 

Motor speed, current, and torque of Number 3 reducing mill 
4 

Motor speed, current, and torque of Number 4 reducing mill 
5 

Motor speed, current, and torque of Number 5 reducing mill 
6 

Motor speed, current, and torque of Number 6 reducing mill 
7 

Motor speed, current, and torque of Number 7 reducing mill 
8 

Motor speed, current, and torque of Number 8 reducing mill 
9 

Motor speed, current, and torque of Number 9 reducing mill 
10 

Motor speed, current, and torque of Number 10 reducing mill 
11 

Motor speed, current, and torque of Number 11 reducing mill 
12 

Motor speed, current, and torque of Number 12 reducing mill 
13 

The temperature of shell 
14 

The transverse wall thickness of shell 
15 

The longitudinal wall thickness of shell 
Prior to establishing quality prediction model of reducing pipe, one must preprocess modeling data and conduct batch treatment, time interval division, mean value treatment, and twodimensional spread on the modeling data. On the basis of obtaining threedimensional data, one must process the process data into subsections according to different time intervals of production operation. In this work, according to the variation of roll current, one firstly classified the production process of reducing pipe into bite subtime interval, stable rolling subtime interval, and steel leaving subtime interval. Then according to the sequence of roll addition, bite and steel leaving stages were divided in detail. Bite stage was divided into eleven subtime intervals and in the same way, steel leaving stage was divided into eleven subtime intervals. After process variables in various time intervals required by modeling were determined, one took the mean value of each process variable in this time interval. Practical data treatment is shown in Figure
The relation between time and variables of reducing tube.
Adding shell heating temperature
Mean values of aforementioned three stage data were arranged from left to right and a data vector
Frame chart of reducing tube quality prediction model.
Similarly, one reorganized the production data of 25 pieces of reducing pipes to form test data matrix
Prediction result of transverse wall thickness.
Prediction result of longitudinal wall thickness.
It is seen in Figures
After quality model was obtained by calculation, iteration learning control method based on mathematical model was applied to improve the product quality of reducing pipe. Due to the modelplant mismatch and unknown disturbances from batch to batch, the final quality does not always meet the desired product quality in real industry. Batchtobatch iterative learning control can be used to solve this problem by using the information of previous batch and currant batch to revise the next batch input trajectory. The following model is used to express the inputoutput relationship:
For the
The prediction of the
Let
The same as above, the errors for the
Assume that the model prediction errors for the
The objective of the ILC is to control the input trajectory in order to make the final product quality achieve the desired target. By solving the following optimal quadratic objective function (
It is to be noted that the change of control trajectory
The batchtobatch iterative learning control scheme of the EOSELMRPLS based method is illustrated in Figure
Scheme of ILC based on OSELMRPLS.
Table
Iterative learning calculating result of wall thickness of reducing tube.
Ser. number  Variable name  Initial value  First learning value  Second learning value  Third learning value  Fourth learning value 

1  Rotary speed of 
1137.52  1136.51  1134.25  1133.15  1132.53 
2  Rotary speed of 
1133.82  1132.65  1131.58  1130.52  1130.22 
3  Rotary speed of 
1288.24  1288.93  1289.36  1290.05  1290.22 
4  Rotary speed of 
1409.32  1408.31  1407.15  1406.53  1406.17 
5  Rotary speed of 
1485.26  1483.84  1482.37  1480.98  1480.68 
6  Rotary speed of 
1458.81  1459.84  1460.37  1460.64  1460.87 
7  Rotary speed of 
1179.95  1178.68  1177.55  1177.09  1176.95 
8  Rotary speed of 
1110.95  1109.72  1108.66  1107.85  1107.62 
9  Rotary speed of 
1290.43  1291.42  1292.37  1293.21  1293.38 
10  Rotary speed of 
1220.67  1219.55  1218.76  1217.84  1217.52 
11  Rotary speed of 
1143.38  1141.75  1140.53  1139.61  1138.92 
12  Rotary speed of 
1103.21  1102.33  1101.36  1100.52  1100.22 
13  Current of Number 1 
35.46  34.75  34.24  33.97  33.91 
14  Current of Number 2 
52.48  51.62  51.28  51.07  50.98 
15  Current of Number 3 
41.54  42.37  42.97  43.31  43.42 
16  Current of Number 4 
134.52  133.75  133.24  133.94  133.87 
17  Current of Number 5 
218.21  217.47  216.84  216.37  216.28 
18  Current of Number 6 
332.46  333.23  333.86  334.32  334.44 
19  Current of Number 7 
324.46  323.71  323.14  322.71  322.62 
20  Current of Number 8 
302.35  301.43  300.85  300.18  300.02 
21  Current of Number 9 
326.87  327.95  328.56  328.83  328.92 
22  Current of Number 10 
324.37  323.65  323.15  322.97  322.88 
23  Current of Number 11 
325.84  324.77  324.24  323.95  323.86 
24  Current of Number 12 
274.94  274.04  273.65  273.13  273.04 
25  Torque of Number 1 
251.64  252.31  252.88  253.23  253.32 
26  Torque of Number 2 
320.48  321.25  321.93  322.35  322.52 
27  Torque of Number 3 
62.44  63.12  63.34  63.52  63.57 
28  Torque of Number 4 
445.85  446.87  447.66  448.02  448.21 
29  Torque of Number 5 
652.48  650.36  648.82  647.74  647.48 
30  Torque of Number 6 
1042.68  1040.24  1039.07  1038.21  1037.98 
31  Torque of Number 7 
1204.15  1203.26  1202.77  1202.35  1202.12 
32  Torque of Number 8 
1251.42  1250.15  1249.14  1248.48  1248.22 
33  Torque of Number 9 
1175.25  1176.37  1177.06  1177.84  1178.12 
34  Torque of Number 10 
1214.85  1213.45  1212.84  1212.25  1212.07 
35  Torque of Number 11 
1316.47  1317.57  1318.16  1318.54  1318.75 
36  Torque of Number 12 
1224.46  1223.24  1222.63  1222.22  1221.97 
37  The temperature 
886  892  896  898  901 
Iterative learning control effect of transverse wall thickness.
Iterative learning control effect of longitudinal wall thickness.
The production process of reducing pipe continuous rolling has the characteristics of typical multitimeintervals and dynamic multivariables. In the meantime, there are many specifications of products, and minor variation of model will occur due to the adjustment of stand for same specification of product; in the meantime, the production of wall reduction and diameter reduction possesses the characteristics of typical multitimeintervals and dynamic multivariables. The production process was classified into three big time intervals such as bite stage, stable rolling stage, and steel leaving stage influenced by different variables, and according to the sequence that roll touches steel, the production process was further classified into 23 subtimeintervals; we proposed the reducing production process model based on EOSELMRPLS algorithm. This algorithm overcame the shortcomings that quality modeling algorithm has complex structure, big calculation load, and bad capability in processing nonlinear problem during conventional intermittent process. Based on the quality prediction model of reducing pipe, we applied iteration learning control technique to the control system of wall thickness deviation and improved the production quality of reducing pipe. Field data simulation and the test results of reducing production in Baogang Iron and Steel Group Steel Pipe Subcompany revealed the validity of this method.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This research is supported by National Natural Science Foundation of China (Grant no. 61203214), Provincial Science and Technology Department of Education Projects, the General Project (L2013101), National Natural Science Foundation of China (Grant nos. 61203103, 61374146, and 61374147). The study is sponsored by National Natural Science Foundation of China (61203214).