Knowl Inf Syst DOI 10.1007/s10115-013-0716-9 REGULAR PAPER
Data mining-based flatness pattern prediction for cold rolling process with varying operating condition Ningyun Lu · Bin Jiang · Jianhua Lu
Received: 3 May 2013 / Revised: 3 September 2013 / Accepted: 29 November 2013 © Springer-Verlag London 2014
Abstract Data-rich environments in modern rolling processes provide a great opportunity for more effective process control and more total quality improvement. Flatness is a key geometrical feature of strip products in a cold rolling process. In order to achieve good flatness, it is necessary to reveal the factors that often influence the flatness quality, to develop a general flatness pattern prediction model that can handle the varying operating condition during the rolling of products with different specifications and to realize an effective flatness feedback control strategy. This paper develops a practical data mining-based flatness pattern prediction method for cold rolling process with varying operating condition. Firstly, the high-dimensional process measurements are projected onto a low-dimensional space (i.e., the latent variable space) using locality preserving projection method; at the same time, the Legendre orthogonal polynomials are used to extract the basic flatness patterns by projecting the high-dimensional flatness measurements into several flatness characteristic coefficients. Secondly, a mixture probabilistic linear regression model is adopted to describe the relationships between the latent variables and the flatness characteristic coefficients. Case study is conducted on a real steel rolling process. Results show that the developed method has not only the satisfactory prediction performance, but good potentials to improve process understanding and strip flatness quality. Keywords Data mining · Flatness pattern prediction · Locality preserving projection (LPP) · Mixture probabilistic linear regression model (MPLR) · Varying operating condition
N. Lu · B. Jiang (B) College of Automation Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, Jiangsu, China e-mail:
[email protected] J. Lu School of Computer Science and Engineering, Southeast University, Nanjing, Jiangsu, China
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1 Introduction In modern manufacturing industries, the term “big data” is being used to capture and characterize many challenges associated with collecting, managing, analyzing, and visualizing the massive amounts of industrial production data [24,29]. For instance, in a steel rolling process, the recorded data covers customer order and static data, various process data and end-product quality-related data, and the others. The data can be recorded in real-time at a sampling rate of up to several measurements per millisecond [6]. Therefore, there are tens of thousands of measurements for various measurable variables per strip. While these data contain rich information about the process and the product, it provides a great opportunity for more effective process control and more total quality improvement. However, due to the 3Vs (Volume, Velocity and Variety) of industrial big data, there remain many challenging tasks unsolved. For example, most of the current data-based statistical process monitoring and quality control strategies cannot effectively handle varying operating condition. This paper takes the 3Vs of the cold rolling process data into consideration and attempts to present an effective online quality prediction method for the process with varying operating condition. For cold rolling of strip products, flatness is a key geometrical feature, which greatly influences the final quality of a rolled product. Flatness is usually defined as the degree to which the surface of a flat product approaches to a plane [4]. Flatness defects are often encountered in strip rolling. Products with poor flatness are more likely to break during later manufacturing phases and move imprecisely along the industrial facilities, harming their own surfaces [20]. Thus, to achieve the required flatness of the rolled product, it is necessary to reveal the factors that might influence the flatness, to develop a general flatness pattern prediction model that can handle the varying operating conditions during the rolling of products with different specifications and to realize an effective flatness feedback control strategy. The performance of flatness feedback control highly depends on timely flatness measurements and accurate flatness pattern recognition. With respect to flatness measurements, both mechanical-and optical-based systems are currently in use for online flatness inspection of rolled products. The main drawback of mechanical systems is their high maintenance requirements, while the high cost of the machine vision subsystem is the main drawback of optical systems [20]. Furthermore, these flatness inspection systems are usually installed behind the rolling stand, which inevitably introduces time delay in the flatness control system [13,14]. In the Ref. [13], the flatness inspection system is placed just after the last stand of a five-stand tandem cold mill, about five meters apart, but it still causes observable delay in the flatness measurement signal. The time delay problem in flatness measurements has aroused more and more concerns in the rolling process. Considerable research efforts have been made into flatness prediction, most of which focus on mechanical modeling based on 3D semi-analytical models (e.g., enhanced Slab Method), Finite Difference (FDM) or Finite Element (FEM) methods [21]. For example, a coupled approach for flatness prediction in cold rolling of thin strip has been presented by analyzing the influences of in-bite plastic deformation and post-bite buckling and by introducing a buckling criterion into a FEM model of strip and roll deformation [1]. This approach can be used for detecting several types of flatness defects; however, developing such relationships between strip flatness and process parameters based on mechanical analysis is rather complicated and time-consuming [15]. In the recent decade, models based on data mining techniques are being increasingly used in mechanical property prediction and shape defect analysis in the steel industry. The reason may be threefold [10]. First of all, it is very hard to develop accurate mechanical
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models with affordable cost. Current knowledge about how the process conditions and the steel grade affect the final quality is still fragmented. Secondly, a huge number of data have been recorded in the rolling mills, as mentioned before. And thirdly, many data mining and modeling methods have been intensively researched. For example, an artificial neural network (ANN)-based model was developed to predict the mechanical properties (yield strength, ultimate tensile strength, and elongation of the hot rolled steel strips/coils) [22]. Similarly, artificial intelligence technique has been used to improve the prediction of rolling force in a hot-rolling mill, where an online learning neural network for both long-term learning and short-term learning was developed [30]. Multiple linear regression and artificial neural networks were evaluated to predict the post-rolling flatness, where the author claimed that both the techniques are suitable for the prediction purpose, but multiple linear regression is preferable [34]. Moreover, data mining techniques, such as fuzzy neural model, least squares support vector regression, clustering algorithm, and so on, have been reported for flatness pattern prediction and recognition in various rolling processes [12,15,40]. The scope of this paper is to investigate a data mining-based flatness pattern prediction method for the rolling processes with varying operating condition. Varying operating condition is a common phenomenon in the steel industry, particularly in tandem cold strip rolling, to improve the productivity and to cope with the increasing diversity of cold rolling products. The mill operates over a wide range of conditions. The setup values of the width, thickness and steel grade of the preceding coil are changed to different setup values of the next coil without stopping the rolling mill [37]. Different from the traditional cold rolling processes which can be considered to have quasi-steady-state character, a tandem cold strip rolling process is usually characterized by its high complexity, nonlinearity, time-variation and varying operating condition. Because of this, the rolling process data can be considered as “big data,” as it indeed has high volume, velocity and variety. The desire for an effective flatness prediction technique for such rolling processes with varying operating condition is apparent in steel industry. However, the aforementioned flatness prediction or flatness pattern recognition methods are based on data mining in a single operating region, which would meet problems when they are applied to the process with varying operating condition. This is because, the relationship between process parameters and flatness patterns may change significantly when the process is running under different operating conditions (i.e., the mill is processing the coils with different widths, thicknesses or steel grades). As a result, a multi-mode flatness pattern prediction strategy will be desirable. In the areas of process modeling, control, soft sensor design, process monitoring or quality prediction, considerable research efforts have been reported to approach the multiple operating condition issue [7,10,23,39]. The general idea behind the use of a multiple model approach is the representation of a multi-mode process with a set of relatively simple, accurate, and easy-to-interpret local models that are integrated together in some manner. Mixture probabilistic model is an attractive technique with successful applications in process monitoring and soft censoring. For example, a Bayesian inference-based finite Gaussian mixture model was used for multi-mode process monitoring of complex chemical industrial processes [39]. A mixture probabilistic principle component regression model (MPPCR) was developed for soft sensing of difficult-to-measure quality variables in multi-mode chemical processes [10]. When applying a mixture model method to rolling processes, we must consider the issues caused by the “big data” (i.e., the high-dimensional high-speed data flow generated from hundreds of process and quality variables with sampling period as short as 20 ms [34]). The flatness pattern prediction method should be accurate both locally and globally, with as low computational burden as possible.
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This paper aims to develop such a practical data mining-based flatness pattern prediction method. Considering the high volume of process data, the high-dimensional process measurements are projected onto a low-dimensional space (i.e., the latent variable space) using locality preserving projection (LPP) [11]; at the same time, the Legendre orthogonal polynomials are used to extract the basic flatness patterns by projecting the high-dimensional flatness measurements into several flatness characteristic coefficients. Considering the high variety of process data collected from different operating conditions, a mixture probabilistic regression model is adopted to describe the relationships between the latent variables and the flatness characteristic coefficients. During online implementation, the expectation maximization (EM) algorithm is used to assign the new data to different operating modes with corresponding posterior probabilities. The goal is to develop a global flatness prediction model over different operating conditions. The main contributions of this paper can be summarized as below. 1. LPP is adopted for dimensionality reduction, because the locality preserving ability of LPP is particularly suitable for the multi-mode processes. In addition, LPP is a linear algorithm. It is easy to implement so that it is suitable for practical application [11]. 2. To handle the changing process features under varying operating conditions, a mixture probabilistic regression model is investigated for flatness pattern prediction in the cold rolling process. 3. Relationships between the strip flatness patterns and the process parameters are revealed from process data, which lay solid foundation for the subsequent flatness feedback control. 4. The application results show that, the proposed approach can deal with the high-samplingfrequency rolling process data with satisfying prediction performance. The remainder of this paper is structured as follows. In Sect. 2, the basic algorithms including LPP, mixture probabilistic regression model and the basic flatness patterns extracted by Legendre polynomials are introduced, followed by a global flatness pattern prediction method. Subsequently, a case study is given in Sect. 3 by applying the proposed method to a real steel cold rolling process. Finally, conclusions are made in Sect. 4. 2 A global flatness pattern prediction method 2.1 Locality preserving projection (LPP) In a general dimensionality reduction problem [11], given a set of points x1 , x2 , . . . , xn in R p which actually lie on a low-dimensional manifold, the objective is to find a transformation matrix that maps these n points to a set of points z1 , z2 , . . . , zn in Rm (m p), such that zi can “represent” xi , where zi = T xi . To achieve this, the adjacency graph G = (X, W ), where X represents the original data set and W is a weighting matrix, should be first constructed with edges connecting nearby points to each other. That is, if nodes xi and x j in G are “close” according to some criterion (e.g., ε-neighborhoods, or k nearest neighbors), nodes xi and x j are connected by an edge. The next step is to choose the weights in W . W is a sparse symmetric n×n matrix. Wi j is the weight of the edge joining vertices i and j. If there is no such edge, Wi j = 0. In the originally proposed LPP method, two variations were given for weighting the edges: Heat-kernel and Simple-minded. By Heat-Kernel, if nodes i and j are connected, Wi j = e−xi −x j /t where t ∈ R is a parameter [2]. By Simple-minded, Wi j = 1 if and only if vertices i and j are connected by an edge.
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Data mining-based flatness pattern prediction Table 1 Major steps of LPP algorithm
Input Output Objective
n High-dimensional original data: X = {xi }i=1 ∈ Rp n Low-dimensional manifold: Z = {zi }i=1 ∈ R m (m p) Find a transformation matrix = (ϕ1 , ϕ2 , . . . , ϕm ) such that zi = T xi
Step 1
Construct adjacency graph: G = (X, W ) where n samples in X correspond to n nodes in G. Options: (a). ε-neighborhoods; (b). k nearest neighbors Determine the weighting matrix W ∈ R n×n .
Step 2
Options: (a). Wi j = 1or 0; Step 3
Step 4
(b). Wi j = e
−xi −x j t
Compute the transformation matrix = (ϕ1 , ϕ2 , . . . , ϕm ) by solving the eigenvectors ϕ and eigenvalues λ of X L X T ϕ = λX D X T ϕ where Dii = j W ji ; L = D − W. Compute the output zi = T xi
The final step is to compute the mapping matrix . Considering the problem of mapping the graph G to a line so that connected points stay as close together as possible. Let z = (z1 , z2 , . . . , zn )T be such a map. We can obtain a “good” map by minimizing the objective function: (zi − z j )2 Wi j (1) i, j
under proper constraints to ensure that if xi and x j are close then zi and z j are close as well. Suppose ϕ is a transformation vector (a column vector in ), zT = ϕ T X . By simple algebra formulation, we can obtain ϕ by computing the eigenvectors and eigenvalues for the generalized eigenvector problem: X L X T ϕ = λX D X T ϕ (2) where D is a diagonal matrix, Dii = j W ji ; L = D − W , is the Laplacian matri x. The ith column vector of X is xi . Let the column vectors ϕ1 , . . . , ϕm be the solution of the above equation, ordered according to their eigenvalues, λ1 < · · · < λm . Thus, the embedding is presented as follows: xi → zi = T xi , = (ϕ1 , ϕ2 , . . . , ϕm )
(3)
where zi is a m-dimensional vector, and is a p × m matrix. The above steps can be summarized by Table 1. From the above derivations, it is clear that LPP is a linear dimension reduction method which can optimally preserve the local neighborhood information in a certain sense. Compared with those nonlinear dimensionality reduction techniques like locally linear embedding (LLE) [26], ISOMAP [31], or Laplacian eigenmaps [2], LPP can be simply applied to evaluate the map for new test points. It has become a natural alternative to principal component analysis (PCA)-based techniques in exploratory data analysis. 2.2 Mixture probabilistic model The multiple model approach is an appealing mathematical framework for the modeling and analysis of complex industrial processes with varying operating condition. The basic principle
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of this modeling framework is to approximate a complex process with a finite collection of local models. Finite mixture probabilistic models are typical multiple model approaches [8,18]. A variety of finite mixture models have been developed recently, such as Gaussian mixture model (GMM) [19,39], mixture of probabilistic linear regression (MPLR) [9,25], and mixture probabilistic principal component analysis (MPPCA) or regression (MPPCR) model [5,10,33]. In this paper, a MPLR model is adopted because it is effective as a general mixture of probabilistic model. Assume that X ∈ R n× p and Y ∈ R n×q are matrices consisting of n samples collected from p process variables x = [x1 , x2 , . . . , x p ] and q quality variables y = [y1 , y2 , . . . , yq ]. In a general linear regression (LR), the objective is to estimate a mapping function, y = ˜x + ε, or
(4a)
yˆ = ˜x
(4b)
where x˜ is an argument vector, x˜ = [xT , 1]T , X˜ = [X, 1]; ∈ R p×q is the regression ˜X ˜ T )−1 , ε ˜ T (X matrix which can be calculated by unbiased least square estimation, = Y X is considered as measurement noise which is assumed to be Gaussian, i.e., p(ε) = N (0, σ 2 I); yˆ is the estimation of y. In a mixture regression model method for multi-mode processes, assume that the number K of operating mode, K , is known and fixed. Thus, there are K local spaces (or modes) {Sk }k=1 in the overall operating space S. The local linear regression model for kth subspace can be represented as, y = k x˜ + ε, or
(5a)
yˆ = k x˜
(5b)
where the regression matrix k ∈ R p×q will be estimated below. The global regression model is a weighted combination of all local regression models, where the weights are given by the posterior probabilities p(k|x)(k = 1, . . . , K ). That is to say, p(k|x) represents a conditional probability of the local operating space Sk given x. The formulation of MPLR can be written as, yˆ MPLR =
K
p(k|x)k x˜
(6)
k=1
In Eq. (6), yˆ MPLR is the final output by the MPLR model, which is the weighted sum of the estimated outputs by each local regression model (Eq. 5b). Given the density p(x|k), then the posterior probability p(k|x) can be calculated by using the Bayes’ theorem, wk p(x|k) p(k|x) = K j=1 w j p(x| j)
(7)
where wk denotes a prior probability p(k) whichis the mixing proportion of each sub regresK K sion model and subject to k=1 wk = 1 (i.e., k=1 p(k) = 1); given that an individual x belongs to mode k, there is a density p(x|k). Generally, the calculation of p(x|k) and p(k) is a density estimation problem. Expectation maximization (EM) algorithm can be an effective tool for such estimation [9]. The remaining problem is how to estimate the regression matrix, k . According to [25], define the diagonal matrix Rk as diag(Rk ) = [r1,k , . . . , ri,k , . . . , rn,k ], where pi,k = p(xi |k)
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and ri,k = p(k|xi ) = k by,
wk pi,k K , j=1 w j p j,k
then we can get an approximately optimal estimation of
k = Y Rk X˜ T ( X˜ Rk X˜ T )−1
(8)
2.3 Basic flatness patterns Flatness defects are frequently encountered in strip rolling, which are mainly caused during rolling and cooling. Their origins may be the irregularly shaped sections of the incoming material and the non-uniform pressure applied to the entire transversal section of the strip during rolling, or the uneven temperature across the width of the strip during cooling. Bad flatness is predominantly characterized by bad leveling, edge waviness and center buckle. Other flatness defects include quarter buckle and herringbone. But in practice, unsatisfying shape quality may show different patterns comprised of different combinations of the above common defects. Flatness pattern recognition is an important component in flatness control system, which converts the practical flatness signals into several characteristic parameters, and determines the types and magnitudes of flatness defects to provide useful information for online flatness feedback control. Traditionally, there are six basic patterns: left waves, right waves, center waves, doubleedge waves, quarter waves and edge-center waves. With the development of modern shape control technology (such as the dynamic shape roll (DSR) developed by VAI Clecim), more complex flatness patterns are introduced such as asymmetric defects (e.g., the right-one-third waves and left-one-third waves [27]) and higher-order defects [36]. Polynomial fit is the most popular method to extract flatness patterns. In this paper, the shape of the flatness will be approximated by orthogonal polynomials because the coefficients of the polynomials can be used to effectively describe the most important flatness defects and therefore be very useful in a flatness feedback control system [35]. Legendre polynomials are used in this case due to their simplicity and accuracy. A Legendre polynomial of grade g + 1(g ≥ 1) can be calculated using the following equation: L g+1 (v) =
2g + 1 g L g (v) − L g−1 (v) g+1 g+1
(9)
where v is the width variable along the transversal section of the strip; the Legendre polynomial of grade 0 is L 0 (v) = 1. Using this equation, the following polynomials can be calculated, L 1 (v) = v, 1 L 2 (v) = (3v 2 − 1), 2 1 L 3 (v) = (5v 3 − 3v), 2 1 L 4 (v) = (35v 4 − 30v 2 + 3), 8 ...
(10)
Graphical representation of the normalized polynomials can be seen in Fig. 1. From Fig. 1, it is easy to observe that the first-order polynomials correspond to the linear flatness
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Linear flatness
(a)
1
0.5 L1(v)
−L1(v)
0
−0.5
(b)
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0
0.5 −L2(v)
−0.5 L2(v)
−1 −1
−0.5
0
0.5
−1 −1
1
−0.5
0.5
0
0.5
1
0.5
1
(d)
1
Quartic flatness
Cubic flatness
(c)
0
v
v
L3(v)
−L3(v)
−0.5
0.5
L4(v)
0
−0.5 −L4(v)
−1 −1
−0.5
0
0.5
1
−1
−0.5
0
v
v
Fig. 1 Basic flatness patterns based on the Legendre polynomials
defects (i.e., left or right waves mainly caused by bad leveling); the second-order polynomials correspond to the quadratic flatness defects (i.e., double-edge waves and center waves); the third-order polynomials correspond to the cubic flatness defects (i.e., left or right-one-third waves); and the fourth-order polynomials correspond to the quartic flatness defects (i.e., quarter waves and edge-center waves). If DSR technology is used in the shape control system, higher-order polynomials are needed for featuring more complex flatness defects. In general, the Legendre polynomial fit of the flatness profile can be given by, y = f (v) =
g
ci L i (v)
(11)
i=0
where y is flatness measurement along the strip width (a function of width variable v), ci (i = 0, . . . , g) are flatness characteristic coefficients that can be approximated using the least-sqaures method. When a strip is perfectly flat, the coefficients have no statistically significance. On the contrary, significant coefficients indicate the occurrence of the corresponding flatness defects. In general, odd coefficients indicate the presence of asymmetric flatness defects and even coefficients indicate the presence of symmetric flatness defects. According to this, the feedback control system can thereby adopt proper actions to remove the flatness defects as quick as possible. 2.4 Knowledge on operating condition In authors’ previous work, the relationships between process variables and flatness defects were investigated through several knowledge mining techniques (principal component analy-
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Online prediction
×
×
×
×
×
×
=
=
Σ
Θ
∼
=
Θ =
Σ =
Fig. 2 Scheme of the proposed flatness prediction method
sis, clustering analysis and correlation analysis) by analyzing the data from a DSR installed tandem cold rolling mill (the same process will be studied in this paper) [41]. We have drawn a conclusion that, strip width is the major factor that causes the difference among data-based process models for different operating conditions; while incoming and exit thickness make no difference. Furthermore, discussions about the impact of strip width on various process setups (such as bending force influence ratio, rolling force influence ration) have also been reported [28]. According to the above, strip width is chosen as an indicator variable for distinguishing between various operating conditions. The number of operating modes in the mixture model is relevant to how the strip width range is partitioned, which is actually prior information for the proposed flatness pattern prediction method. Besides, process also shows different characters for different flatness defects. In practise, we can learn the mode number K from data through the data mining technique such as various clustering algorithms. 2.5 Major steps of the proposed method The scheme of the proposed flatness pattern prediction method can be depicted by Fig. 2. Detailed explanations are given below. Offline Modeling Phase: (1) Training data set As a data mining-based method, it is important to collect proper training data to ensure that the training dataset is the best representative of the real world scenario. For the cold rolling process considered in this paper, strip width is chosen as the indicator variable for distinguishing different operating conditions. The training dataset should cover the entire operating conditions (i.e., all strip width grades).
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After obtaining a proper training dataset from historical database, two sub datasets are generated: X n× p and Yn×q , where X contains process measurements (such as rolling speed, rolling force, exit tension, bending force, and exit thickness), Y contains flatness measurements, n is the total number of samples, p is the number of process variables, and q is the number of flatness measuring points. All the variables in the datasets should be denoised and normalized to have zero means and unity variances. Then, different processing steps are applied for process measurements and flatness measurements, respectively. (2) Dimension reduction for process measurements by LPP Considering the high dimensionality and complex correlation of process variables, LPP is adopted to reduce the dimensionality on one hand and to preserve the local correlation structure of process variables on the other hand. When applying LPP, one should determine options such as Distance Metric (i.e., Euclidean, Mahalanobis or Hamming distance), Neighbor Mode (i.e., k-nearest neighbors (k-nn) or ε-neighborhoods) and Weight Mode (i.e., Heat-Kernel or 0–1 weighting). If k − nn and Heat-Kernel are used, one should also determine the major parameters such as the k in k-nn and the t in HeatKernel. In addition, the intrinsic dimension should also be estimated in LPP. The existing approaches for estimating the intrinsic dimension can be roughly divided into two groups: eigenvalue or projection methods, and geometric methods [16]. In eigenvalue methods, intrinsic dimension is determined by the number of eigenvalues greater than a given threshold. Eigenvalue methods are widely used because it is easy to implement, e.g., by plotting the eigenvalues and looking for a clear-cut boundary. By applying LPP, one gets, zi = T xi (i = 1, . . . , n) where zi is a vector with a lower dimension m (intrinsic dimension), Z n×m = [z1 , . . . , zn ]T ; ∈ R n×n is the mapping matrix obtained from an LPP algorithm. (3) Flatness feature extraction by the Legendre polynomials The number of basic patterns (i.e., the number of Legendre polynomials used for fitting the flatness measurements) depends on the application situations. For example, for the mill stand with CVC (continuously variable crown), it allows control of globally parabolic shape defects but cannot compensate asymmetric shape defects; therefore, six basic patterns (including left waves, right waves, center waves, double-edge waves, quarter waves and edge-center waves) are enough, where the Legrande polynomials with grade 1, 2 and 4 are used for polynomial fitting. But for the mill stand with DSR technology, since it can correct higher-order defects, more Legrande polynomials are required for better feature extraction. By applying feature extraction based on the Legrande polynomials, one can get a flatness feature matrix, Cn×r , from the flatness measurement matrix, Yn×q , where r is the number of characteristic coefficients. (4) Develop mixture probabilistic linear regression model between Z n×m and Cn×r K , and estimate the The first step is to divide the dataset Z n×m into K sub sets, {Z n×m }k=1 prior probability functions p(k) and p(zi |k)(k = 1, . . . , K ) using EM algorithm. Then, the posterior p(k|zi ) is obtained by Eq. (7).
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In order to get the regression matrix of local probabilistic linear regression model k using Eq. (8), one needs Rk = diag([r1,k , . . . , ri,k , . . . , rn,k ]), where ri,k = p(k|zi ) = wk pi,k , wk denotes p(k) and pi,k = p(zi |k). j w j p j,k The local models’ parameter k will be used for online flatness pattern prediction. Online Prediction Phase: Based on the above-developed mixture probabilistic linear regression model, the flatness pattern can be predicted using the online process measurements with negligible time delay. Given a new process data sample, xnew , its projection on the low-dimensional space is, znew = T xnew Its posterior probability in each operating mode can be calculated using Eq. (7), wk p(znew |k) p(k|znew ) = K j=1 w j p(znew | j) Then the predicted flatness pattern can be obtained by cˆ MPLR =
K
p(k|znew )k z˜ new
k=1
In order to visualize the predicted pattern, a reconstruction step can be applied by y=
g
ci L i (v)
i=0
3 Case study 3.1 Process description The application process is a real tandem cold rolling mill with DSR installed on the fifth stand. DSR is an advanced technology used for flatness control on cold rolling mills. This technology uses a sleeve rotating around a stationary beam equipped with seven individually controlled pads. Each pad is actuated by a hydraulic roll load cylinder which allows the pressure to be individually adjusted at each pad location. This ensures the control of sleeve deformation and of load distribution across the product width in the roll gap [36]. Since the rolling pressure is distributed on all pads, a multivariable flatness control strategy is developed, where the pressures on seven pads and the bending pressure are control variables. It allows optimal flatness control because it is possible to determine the impact of each actuator (bending, DSR pads...) on each coefficient of the shape error and to compensate for the higher- order defects. The application process has multiple operating conditions because flying gauge control strategy is used. In addition, in the multivariable flatness feedback control strategy, strip width is divided into 10 grades, as shown in Table 2. For different strip width grades, the process shows quite different statistical behaviors. It indeed brings difficulties for data-based modeling, controlling, process monitoring and quality prediction of such a rolling process.
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W10
W09
W08
W07
W06
Width (mm)
850–949
950–1049
1050–1149
1150–1249
1250–1349
Grades
W05
W04
W03
W02
W01
Width (mm)
1350–1449
1450–1549
1550–1649
1650–1749
>1750
Table 3 Selected process variables for flatness prediction
No.
Measured variables
1
G5 rolling speed
mpm
2
G5 rolling force
T
3
G5 exit tension
KN
4
Errors between the actual bending measurements and the bending reference Errors between the actual tilting measurements and the tilting reference Pressures of 7 DSR pads
um
Position difference of balance cylinder (between the drive and operation sides) Position difference of screw-down cylinder (between the drive and operation sides) Temperatures of 7 DSR pads
um
5 6–12 13 14 15–21
Unit
um T
um ◦C
3.2 Data acquisition and preprocessing Due to the restriction of the production condition, the training dataset only contains 21 rolling batches, covering 7 strip width grades. Fourteen of them are coils with flatness defects, and the rest 7 batches are normal coils with different strip widths. The original training dataset is DATA(2041 × 57). Although the process consists of hundreds of measured process variables, only 21 process variables are finally selected for predicting the flatness quality in this case study, as shown in Table 3. The rest 36 variables in DATA(2041 × 57) are flatness measurements, because there are 36 flatness measuring points along the strip width direction. Thus, we can obtain process measurements X (2041 × 21) and flatness measurements Y (2041 × 36). The measurements are normalized to have zero means and unity variances. Two additional rolling batches, TEST1(110 × 57) and TEST2(255 × 57) with different types of flatness defects, are also collected for verification. These test datasets are normalized using the means and variances of the training dataset. 3.3 Dimension reduction by LPP The major options in the LPP algorithm are listed in Table 4. The intrinsic dimension is determined by principal component analysis (PCA). When the accumulation of the first m biggest principal components is larger than the pre-specified PCA Ratio, the intrinsic dimension is m. In this case, PCA Ratio is set as 0.90, the intrinsic dimension is 11. Thus, after applying LPP, process measurements X (2041×21) are transformed to be Z (2041×11). This means, the 11 latent variables can cover over 90 % information in the original 21
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Data mining-based flatness pattern prediction Table 4 Options in the LPP algorithm
Options
Value/method
Metric Neighbor mode
Euclidean k-nn
k
5
Weight mode
Heat Kernel
t
5
PCA Ratio
0.90
3rd latent variable
2 1 0 −1 −2 −3 4 4
2 0
2
2nd latent variable −2
0 −4
−2
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Fig. 3 Projection of the original process measurements into LPP subspace
process variables, and at the same time, the local correlation structures are well preserved, as illustrated in Fig. 3. 3.4 Flatness feature extraction Since the application mill is installed with DSR, six Legrande polynomials (grades 1, 2, 3, 4, 6 and 8) are used for fitting more complex flatness defects, where 1 (231v 6 − 315v 4 + 105v 2 − 5) 16 1 L 8 (v) = (6435v 8 − 12012v 6 + 6930v 4 − 1260v 2 + 35) 128 L 6 (v) =
Thus, the flatness characteristic coefficient matrix for the training data is C(2041 × 6). Figure 4 illustrates several actually measured flatness profiles in the rolling mill and the corresponding calculated flatness characteristic coefficients. Those figures show the complexity of the flatness features and the necessity of using high-order Legrande polynomials. 3.5 Results during offline modeling The mode number K is determined by a modified k-means clustering method, which can find K automatically without any prior knowledge. K is 20 in this case. The prior probability functions p(k) and p(zi |k)(k = 1, . . . , K ) are estimated using a recursive EM
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Fig. 7 Graphical representation for a part of measured flatness in TEST1
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algorithm [3]. Figure 5 shows the predictions of the flatness characteristic coefficients, ci (i = 1, 2, 3, 4, 6, 8), for all training data, where solid curves are characteristic coefficients estimated from the actual flatness measurements, and dotted curves are predictions by the developed method. Figure 6 shows the predictions of several actually measured flatness profiles in details. In both Figs. 5 and 6, the prediction curves are well overlapped with
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the actual characteristic coefficients’ curves, indicating that the proposed method has good performance for global flatness prediction during offline modeling phase. 3.6 Online flatness prediction results Testing dataset 1 (TEST1) contains typical wavy edges and quarter pockets, as shown in Fig. 7. The right wave, left wave and quarter pockets appear alternatively. Figure 8 shows the posterior probability for the samples in TEST1, where the flatness pattern of this testing coil well matches Mode 10 of the training dataset. Figure 9 presents the predictions of flatness characteristic coefficients, from which, we can observe that, c4 is the most significant and stable coefficient, but c1 is fluctuating clearly. As explained in Sect. 2.3, the coefficients ci (i = 1, 2, 3, 4, 6, 8) correspond to the ith order flatness defect. Thus, the significant c4 means a possible quartic flatness defect, and the fluctuating c1 implies alternative edge waves, which are consistent with the observations in Fig. 7. For details, Fig. 10 shows the 3-dimensional graphical representation of flatness predictions, which are reconstructed from the predictions of the flatness characteristic coefficients in Fig. 9. The flatness profile predictions for samples Nos. 30, 40, 60 and 80 in TEST1 are presented in Fig. 11, and the corresponding flatness characteristic coefficients are shown in Fig. 12. Those profiles and coefficients look slightly different at different points, but the pattern is fixed, as all samples in TEST1 belong to Mode 10.
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Testing dataset 2 (TEST2) contains a flatness defect caused by unsatisfied crown control, as shown in Fig. 13, where the variations of thickness across the width direction are not even. Figure 14 shows the posterior probability for TEST2, where the first 16 samples do not match with any existing modes, and the remaining samples match with Mode 3. This mismatch problem may be originated from the incompleteness of the training dataset. This suggests that, for a data mining-based method, the quality of training dataset is vitally important. Figure 15 presents the predictions of flatness characteristic coefficients. Figure 16 is the 3-dimensional graphical representation of flatness predictions. The flatness profile prediction and the corresponding flatness characteristic coefficients for sample No. 80 are presented in Fig. 17. It is clear that, c2 and c4 are significant, indicating that the flatness of this testing coil is a combination of quadratic and quartic patterns. 3.7 Knowledge mining for flatness control Since a linear dimension reduction method, LPP, and a linear regression model, MPLR, are used in the developed global flatness prediction method, it is possible to trace back and develop the relationship between the original process variables and the flatness coefficients,
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By the contribution plot [38], it is easy to reveal the key process variables that have significant impacts with the flatness defect. Take the second case (TEST2) in Sect. 3.6 as an example, Fig. 18 shows the contribution plots for c2 and c4 , respectively, where the process variables including Rolling force (No. 2), Bending control performance (No. 4), Pressures of DSR pads (Nos. 6, 8, 9 and 10), and Temperatures of DSR pads (Nos. 16 and 19) are key variables for flatness control to compensate the flatness defect. This result is in accord with the knowledge obtained in our previous work [41].
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4 Conclusions Considering the 3Vs of the cold rolling process data, a practical data mining-based flatness pattern prediction method for cold rolling process with varying operating conditions has been presented in this paper, where LPP is used for dimensionality reduction and MPLR is used to develop a global prediction model. LPP is better than PCA for its locality preserving ability. LPP outperforms the other manifold learning methods such as LLE and ISOMAP for its linearity, which is easy to implement online. As for multimode modeling, GMM and MPPCR can also be used in this study, but the computational cost is much higher than MPLR. The results of case study have shown that the developed method has not only the satisfac-
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tory prediction performance, but also good potentials to improve process understanding and strip flatness quality for the operation engineer. In addition, the proposed flatness pattern prediction method is a pure data-based method. It is easy to implement with low hardware cost. And more importantly, it has almost no time delay problem compared with the existing flatness inspection systems. With the accurate real-time flatness pattern information, we can reasonably expect the flatness control system to achieve better strip quality. Acknowledgments We gratefully acknowledge the financial support of National Natural Science Foundation of China (Nos. 61073059, 61374141 and 61034005), Jiangsu Provincial Natural Science Foundation of China (BK2010409), and the Fundamental Research Funds for the Central Universities (NS2012039).
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Ningyun Lu received her Ph.D. degree from Northeastern University, Shenyang, China, in 2004. From 2004 to 2005, she worked as a Research Associate in Hong Kong University of Science and Technology. Currently, she is an Associate Professor in College of Automation Engineering at Nanjing University of Aeronautics and Astronautics, Nanjing, China. Her research interest includes data-driven fault prognosis and diagnosis and its applications to various industrial processes.
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Bin Jiang obtained the Ph.D. degree in Automatic Control from Northeastern University, Shenyang, China, in 1995. He had ever been postdoctoral fellow, research fellow and visiting professor in Singapore, France, USA and Canada, respectively. Now he is a Chair Professor of Cheung Kong Scholar Program in Ministry of Education and Dean of College of Automation Engineering in Nanjing University of Aeronautics and Astronautics, China. He currently serves as Associate Editor or Editorial Board Member for a number of journals such as IEEE Trans. On Control Systems Technology; Int. J. of Control, Automation and Systems; Nonlinear Analysis: Hybrid Systems; Int. J. of Applied Mathematics and Computer Science; Acta Automatica Sinica; Journal of Astronautics. He is a senior member of IEEE, Chair of Control Systems Chapter in IEEE Nanjing Section, a member of IFAC Technical Committee on Fault Detection, Supervision, and Safety of Technical Processes. His research interests include fault diagnosis and fault tolerant control and their applications.
Jianhua Lu received his Ph.D. degree from Northeastern University at Shenyang, China, in 2003. From 2003 to 2004, he worked as a Research Assistant in the Chinese University of Hong Kong. From 2004 to 2005, he is a visiting scholar in Hong Kong University of Science and Technology. He is currently an Associate Professor in School of Computer Science and Engineering at Southeast University, Nanjing, China. His research interests include data mining, database and e-commerce.
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