Quality-Related Statistical Process Monitoring ... - ACS Publications

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Quality-Related Statistical Process Monitoring Method Based on Global and Local Partial Least-Squares Projection Bin Zhong, Jing Wang,* Jinglin Zhou, Haiyan Wu, and Qibing Jin College of Information Science and Technology, Beijing University of Chemical Technology, Beijing 100029, China ABSTRACT: A novel quality-related statistical process monitoring method based on global and local partial least-squares projection (QGLPLS) is proposed in this paper. The main idea of the QGLPLS method is to integrate the advantages of localitypreserving projections (LPP) and partial least squares (PLS) and extract meaningful low-dimensional representations of high-dimensional process and quality data. QGLPLS can exploit the underlying geometrical structure that contains both global and local information pertaining to the sampled data, including the process variable and quality variable measurements. It is well-known that the PLS method can find only the global structure information and ignores the local features of data sets and that the LPP method can preserve local features of data sets well without considering the product quality variables. The capacity for the preservation of global and local projections of the proposed method is compared to that of the PLS and LPP methods; the comparison results demonstrate that the QGLPLS method can effectively capture meaningful information hidden in the process and quality data. Next, a unified optimization framework, i.e., global covariance maximum and local graph minimum in the process measurement and quality data space, is constructed, and QGLPLS-based T2 and squared prediction error statistic control charts are developed for online process monitoring. Finally, two typical chemical processes, the Tennessee Eastman process and the penicillin fermentation process, are used to test the validity and effectiveness of the QGLPLS-based monitoring method. The experimental results show that the obtained process monitoring performances are better than those when using traditional monitoring methods, such as PLS, principal component analysis, LPP, and global−local structure analysis.

1. INTRODUCTION Fault detection and diagnosis are extremely necessary in complex industrial systems and have been topics of concern for a growing number of engineers and researchers.1−6 A range of data-driven monitoring technologies have been widely used to extract useful information from a large number of highly correlated process variables and historical data sets; these monitoring technologies have been successfully applied to complex chemical process monitoring and fault detection.7−9 Two widely used statistical methods for extracting hidden variables are principal component analysis (PCA) and partial least-squares (PLS). Some extended methods based on these two fundamental techniques are addressed to effectively solve a range of monitoring issues for process industries with complex nonlinear, time series, and data unequal characteristics.10−14 Two types of statistics, Hoteling’s T2 and squared prediction error (SPE or Q), are used to detect the changing information on industry process in these PCA- and PLS-based approaches. However, these traditional techniques consider only the second-order statistics of covariance and not the higherorder statistics. These methods cannot effectively extract the characteristics of the higher-order statistics for the nonGaussian process, which is a common process in actual conditions. As a result, some abnormal conditions of chemical processes cannot be successfully identified using the traditional methods. Independent component analysis (ICA) is proposed to solve the unusual problem of a non-Gaussian process by mapping the multivariate statistical process data into the independent component (IC) hidden subspace.15−17 The ICA method extracts independent components that are assumed to be non-Gaussian and independent of each other via a nonGaussian maximize indicator. © 2016 American Chemical Society

Only process variables but not the quality variables of the products are taken into account when PCA- and ICA-based monitoring methods are used to create a statistical model. PLSbased methods obtain a projection space by using quality variables to guide the decomposition of the process variable sampling space that is able to reflect the changes between process measurements X and quality data Y. Thus, a PLS-based method has greater explanatory power than PCA- or ICA-based methods. The T2 statistic of the score space includes the variables that not only are associated with the quality output Y but also are perpendicular to Y, such that there is redundancy within the extracted feature information. A total projection to latent structures (T-PLS) was proposed by Qin and coworkers,18 in which the input space is divided into four different subspaces either associated with the output data or not. Moreover, four statistical parameters are constructed for process monitoring and fault detection in the T-PLS method. However, T-PLS has a disadvantage: the output-related monitoring indicator is used to detect only those quality variables that can be predicted from the process measurement variables. However, in many cases, the prediction ability of T-PLS is limited because of the large number of unmeasured confounders of the process measurement and the quality variables. As a result, most of the output variables cannot be obtained via prediction from the input variables, and the unpredictable output variables directly affect the monitoring Received: Revised: Accepted: Published: 1609

July 13, 2015 January 17, 2016 January 17, 2016 January 18, 2016 DOI: 10.1021/acs.iecr.5b02559 Ind. Eng. Chem. Res. 2016, 55, 1609−1622

Article

Industrial & Engineering Chemistry Research

their validity and effectiveness. The experimental results show better monitoring performances compared to traditional monitoring methods, such as PLS, PCA, LPP, and GLSA.

performance of T-PLS. A few years later, Qin and Zheng proposed an improved method called concurrent projection to latent structures (CPLS). In this method, the input space is categorized into five subspaces, and both the predictable output and unpredictable output are monitored simultaneously.19 Subsequently, Zhang et al. proposed an improved KCPLS algorithm and applied the method to a nonlinear process.20 However, the traditional PCA-, ICA-, and PLS-based methods use only the mean and variance information on an industrial process, which all belong to global structural information. The local characteristics of the adjacent structures cannot be extracted well. Furthermore, a series of algorithms based on the manifold learning have arisen, such as locally linear embedding (LLE) and isometric feature mapping (ISOMAP). These methods have been successfully applied to the nonlinear dimensionality reduction of high-dimensional data. In addition, these methods are used to maintain the local structure of the data in the study of data visualization and image processing.21−23 More recently, a novel algorithm, called locality-preserving projections (LPP) was proposed. LPP has the ability to utilize the linear approximation technique to achieve nonlinear dimensionality reduction while maintaining the local characteristics of the adjacent structures.24 Multiway locality-preserving projection (MLPP) was applied to the chemical process monitoring field in 2007.25 Because the LPP method utilizes the local structure to describe the global features and converts the nonlinear transformation via an approximate linear form, it shows excellent performance in the process monitoring field. Therefore, many scholars have focused their efforts on LPP and the extended methods in the field of chemical process monitoring.26,27 Recently, a novel process monitoring model combining the advantages of PCA and LPP has been proposed by many researchers.28−30 Zhang et al.28 proposed a method for fault detection and identification, named the global−local structure analysis (GLSA) model. GLSA takes advantage of PCA (extracting global features) and LPP (preserving the local structure), and its monitoring capability was found to be superior to that of either PCA or LPP alone. This method considered only the process measurement variables and did not take advantage of the product quality variables, i.e., the output variables. In general, it is extremely important to take into account the influence of the process variables on the quality data sets. To obtain the relationship between the measurement and the quality data of a chemical engineering process and simultaneously preserve the local characteristics, a novel quality-related statistical process monitoring method based on global and local partial least-squares projection (QGLPLS) is proposed in this paper. The QGLPLS method integrates the advantages of locality-preserving projections and partial least squares and extracts the meaningful low-dimensional representations from the high-dimensional process and quality data. QGLPLS can exploit the underlying geometrical structure, which contains both global and local information on sampled data, including the process variable and quality variable measurements. In this way, the QGLPLS method is able to not only identify the process measurement and product quality variables’ characteristics in a potential direction but also preserve (as much as possible) the local structural characteristics between the two hidden subspaces. The proposed QGLPLS-based monitoring method is applied to the Tennessee Eastman process and the stabilization stage of the penicillin fermentation process to test

2. QGLPLS-BASED PROCESS MONITORING 2.1. Review of the PLS and LPP Methods. The PLSbased method addresses the high-dimensional data by maximizing the covariance objective function and seeks the multidimensional potential direction between the process measurement and quality variable spaces. The PLS establishes the input−output relationship so that the quality data, which is difficult to obtain, can be easily inferred by the measurement data. In this manner, the goal of quality data prediction can be achieved. PCA-based methods extract only the latent variables from measurement variable space. However, PLS-based methods obtain the latent variables from both the input and output variables. Therefore, PLS not only is the best summary of the input variables information but also considers the extraction of the components that provide the best explanation of the output variables. As a result, PLS provides a deeper analysis and understanding of the industrial object. Given the input matrix X = [x1, x2,...,xn] ∈ Rk×n (process variables) and the output matrix Y = [y1, y2,...,ym] ∈ Rk×m (quality variables), where k is the sample time, n the number of process measurement variables, and m the number of quality variables, these data are decomposed into low-dimensional subspace as follows: X = TPT + E

(1)

Y = UQ T + F

(2)

Next, PLS is established in the following objective function: JPLS (wi , ci) = max cov(ti , ui) s.t. || wi || = 1, || ci || = 1

(3)

where T = [t1,...,td] ∈ R and P = [p1,...,pd] ∈ R are the score matrix and the load matrix for input data X, respectively. U = [u1,...,ud] ∈ Rk×d and Q = [q1,...,qd] ∈ Rm×d are the score matrix and load matrix for output data Y, respectively. ti = Xwi and ui = Yci, where wi and ci are the weight vectors. However, wi cannot connect ti to input data X directly.18 Let decomposition matrix R = [r1,...,rd] ∈ Rn×d, r1 = w1, and ri = Πji =−11(In − wjpTj )wi, i ≥ 2. Thus, the score matrix, T, can be calculated from X as T = XR. The maximization problem (eq 3) also can be translated into the following optimization problem: k×d

JPLS (wi , ci) = max w TiXTYci

n×d

(4)

The traditional PLS, PCA, or ICA methods consider only the global Euclidian structure of the process data, and the local structural characteristics are ignored. Currently, the amount of research regarding manifold learning is rapidly increasing, aiming at efficiently extracting the low-dimensional manifold structure from the high-dimensional data sets and further preserving the local structural features. One of the manifold learning algorithms is LPP, which explores the inherent geometric features and manifold structure from the input data sets by constructing a neighborhood graph. LPP can address the shortcomings of the traditional statistical method in terms of ignoring the local manifold structure of process data sets, and LPP uses the local structure to describe the global features and converts the nonlinear transformation into an approximately linear form. 1610

DOI: 10.1021/acs.iecr.5b02559 Ind. Eng. Chem. Res. 2016, 55, 1609−1622

Article

Industrial & Engineering Chemistry Research To project the sample data X = [xT1 xT2 ··· xTk ]T ∈ Rk×n into a new data space Φ = [ϕT1 ϕT2 ··· ϕTk ]T ∈ Rk×d (d ≪ n) via linear mapping, d projection vectors W = [w1, ..., wd] ∈ Rn×d should be found, i.e. ϕi = xiW

(i = 1, 2, ..., k)

PLS method to perform the relevant quality statistical analysis. As a result, the QGLPLS method is able not only to identify the latent characteristics direction for both the measurement and the quality data space but also to preserve (to the greatest extent possible) the local structural characteristics in the two hidden subspaces. We can consider both the manifold structure of process variables X and the product output variables Y by introducing parameters λ1 and λ2 to control the trade-off between the extraction of the global and local features. Therefore, the objective function of QGLPLS-based method is defined as

(5)

The optimal mapping matrix can be obtained from the following minimization problem: JLPP (w) = min

= min

1 2 1 2

k

∑ || ϕi − ϕj ||2 Sij i,j=1

JQGLPLS (w , c) = max{(w TXTYc) + λ1(w TXTS1Xw) + λ 2(c TY TS2Yc)}

k

s.t. w Tw = 1, c Tc = 1

i,j=1

k

= min

= max{(w TXTYc) + λ1(w TΘ1w) + λ 2(c TΘ2c}

∑ (xiw − xjw)T (xiw − xjw)Sij

(10)

k

∑ w Txi TSijxiw − ∑ w Txi TSijxjw i,j=1

where Θ1 = XT S1X ∈ Rn×n and Θ2 = YT S2Y ∈ Rm×m. The essence of the QGLPLS method is to find the target weight vectors w and c under the given performance function (eq 10). It is worth noting that the LPP constraints for input and output variable wTXTD1Xw = 1, cTYTD2Yc = 1 are removed in eq 10. Generally, the constraints of PLS wTw = 1, cTc = 1 and the constraints of LPP wTXTD1Xw = 1, cTYTD2Yc = 1 cannot be satisfied at the same time. Because we focus on the correlation between the input variable and the output variable first, so the data screening and embedding characteristics of LPP are given up partly, i.e., the LPP problem without constraints wTXTD1Xw = 1, cTYTD2Yc = 1 are used to improve the performance of PLS instead of PCA. The optimal vector w and c obtained from eq 10 will ensure the maximum correlation relationship between X and Y (PLS) and a relatively or local optimal data screening and embedding ability, respectively, for input X and output Y (local LPP, but not global LPP). The Lagrange function of eq 10 is

i,j=1

= min(w TXTDXw − w TXTSXw)

(6)

where Sij ∈ R is the neighborhood relationship matrix element between xi and xj, and its value increases as the distance between xi and xj decreases. If xi and xj are neighbors, then Sij = 2 e−∥xi − xj∥ /t, where t is a parameter that is used to adjust the size of matrix Sij; otherwise, Sij = 0. Here, we adopt K nearest neighbors to define whether xi and xj are the neighbors.24 D is a diagonal matrix with Dii = ∑j k= 1Sij. Defining L = D − S as a Laplacian matrix, eq 6 becomes k×k

JLPP (w) = min w TXTLXw

(7)

To minimize the objective function (eq 7), we can ensure that the relationship between projection points (ϕi, ϕj) in a lowdimensional space is still similar to that of (xi, xj) in the original space. In other words, low-dimensional data sets still preserve the local geometrical characteristics. The minimization problem of function 7 with linear mapping (eq 5) can also be written as min ϕTLϕ. Under the assumption ϕTDϕ = 1, the minimization problem (eq 7) is reduced to finding the argminϕT Dϕ = 1ϕTLϕ. The constraint ϕTDϕ = 1 aims to remove an arbitrary scaling factor from the embedding. Matrix D is a diagonal matrix that provides a natural measure on the data points. The larger the value of Dii (corresponding to ϕi), the more “important” ϕi.24,31 Under the constraint ϕTDϕ = 1, we can obtain wTXT DXw = 1 and the minimization problem (eq 7) becomes T T

JLPP (w) = 1 − min w X SXw

Ψ(w , c) = w TXTYc + λ1(w TΘ1w) + λ 2(c TΘ2c) − η1(w Tw − 1) − η2(c Tc − 1)

where η1 and η2 are Lagrange multipliers. Implementing the partial derivative of the above Lagrange objective function (eq 11) for w and c and setting them equal to zero gives

(8)

The minimization problem (eq 8) can be changed to a maximization problem that is equivalent to solving the following maximization problem: JLPP (w) = max w TXTSXw s.t. w TXTDXw = 1

(11)

∂Ψ = XTYc + 2λ1Θ1w − 2η1w = 0 ∂w

(12)

∂Ψ = Y TXw + 2λ 2 Θ2c − 2η2c = 0 ∂c

(13)

∂Ψ = w Tw − 1 = 0 ∂η1

(14)

∂Ψ = c Tc − 1 = 0 ∂η2

(15)

We can deduce the following equations from eqs 12 and 13: (9)

XTYc + 2λ1Θ1w = 2η1w

2.2. What is QGLPLS? A novel quality-related method based on the global and local partial least-squares projection method (QGLPLS) is proposed in this paper to obtain the relationship between the quality and measurement variables while maintaining the local characteristics as much as possible. The main idea of the QGLPLS method is to integrate the LPP method to preserve the local structural characteristics and the

(16)

Y TXw + 2λ 2 Θ2c = 2η2c

(17) T

T

Multiplying eqs 16 and 17 on the left by w and c , respectively, we obtain w TXTYc + 2λ1w TΘ1w = 2η1 1611

(18) DOI: 10.1021/acs.iecr.5b02559 Ind. Eng. Chem. Res. 2016, 55, 1609−1622

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Figure 1. QGLPLS-based method for process monitoring.

c TY TXw + 2λ 2c TΘ2c = 2η2

(19)

T T

It is known that w X Yc is a scalar quantity; thus, we have c TY TXw = (w TXTYc)T = w TXTYc

(20)

We add eq 18 to eq 19 and obtain 2w TXTYc + 2λ1w TΘ1w + 2λ 2c TΘ2c = 2η1 + 2η2

(21)

Equation 21 is simplified as follows: w TXTYc + λ1w TΘ1w + λ 2c TΘ2c = η1 + η2

(22)

Equation 22 shows that (η1 + η2) should be the maximum value if we want to obtain the maximum objective function (eq 10) of the QGLPLS. Next, let us consider how to find the appropriate pending vector w and c under the performance function (eq 10). We can deduce the relationship between the pending vector w and c from eqs 16 and 17. c = 0.5(η2Im − λ 2 Θ2)−1Y TXw

(23)

w = 0.5(η1In − λ1Θ1)−1XTYc

(24)

Figure 2. Original scurve and sample data set.

Similarly, we can obtain the optimal pending vector c. Substituting eq 24 into eq 17 results in the following:

T

−1 T

(0.5X Y (η2Im − λ 2 Θ2) Y X )w = (2η1In‐2λ1Θ1)w

(27)

(0.5Y TX (η1In − λ1Θ1)−1XTY )c = (2η2Im − 2λ 2 Θ2)c

(28)

If the trade-off parameters λ1 and λ2 are selected as λ1 = η1 and λ2 = η2, then eqs 26 and 28 will be simplified as follows:

Substituting eq 23 into eq 16 results in the following: XTY (2η2Im − 2λ 2 Θ2)−1Y TXw + 2λ1Θ1w = 2η1w

Y TX(2η1In − 2λ1Θ1)−1XTYc + 2λ 2 Θ2c = 2η2c

(0.5XTY (η2Im − η2 Θ2)−1Y TX )w = (2η1In − 2η1Θ1)w

(25)

(29) T

−1 T

(0.5Y X (η1In − η1Θ1) X Y )c = (2η2Im − 2η2 Θ2)c

(26) 1612

(30)


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Figure 3. Scurve data projection in the reduced dimensional space.

Figure 4. Flowchart of the Tennessee Eastman process.

Table 1. Description of the Input and Output Variables no.

contents

type

no.

contents

type

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Y1 Y2

A feed flow (stream 1) D feed flow (stream 2) E feed flow (stream 3) A and C feed flow (stream 4) recycle flow (stream 8) reactor feed rate (stream 6) reactor pressure reactor level reactor temperature purge rate (stream 9) separator temperature separator level separator pressure separator underflow (stream 10) stripper level stripper pressure composition of G (stream 9) composition of E (stream 11)

measured measured measured measured measured measured measured measured measured measured measured measured measured measured measured measured quality quality

17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33

stripper underflow (stream 11) stripper temperature stripper steam flow compressor work reactor cooling water outlet temp condenser cooling water outlet temperature D feed flow (stream 2) E feed flow (stream 3) A feed flow (stream 1) A and C feed flow (stream 4) compressor recycle value purge value (stream 9) separator pot liquid flow (stream 10) stripper steam product flow stripper steam valve reactor cooling water flow condenser cooling water flow

measured measured measured measured measured measured manipulated manipulated manipulated manipulated manipulated manipulated manipulated manipulated manipulated manipulated manipulated

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It is known that w is the feature vector of matrix ((In − Θ1)−1 X Y(Im − Θ2)−1YTX), and the corresponding eigenvalue is 4η1η2. Therefore, the optimal w is the maximum eigenvector corresponding to the maximum eigenvalue 4η1η2 of matrix ((In − Θ1)−1 XTY(Im − Θ2)−1YTX), according to the above analysis indicating that η1 and η2 should obtain the maximum value to maximize the objective function (eq 10). Similarly, the optimal vector c is the maximum eigenvector of matrix ((Im − Θ2)−1 YTX(In − Θ1)−1XTY) at the maximum eigenvalue 4η1η2. At this point, we have calculated the optimal weight vectors w and c for the establishment of the monitoring model of the QGLPLS method. 2.3. QGLPLS-Based Process Monitoring. We have obtained the optimal vectors w and c from eqs 31 and 32. Next, the process monitoring based on the QGLPLS method is considered. Calculating the score vectors t and u according to the optimal vector w and c

Finally, eqs 29 and 30 are equivalent to

T

((In‐Θ1)−1XTY (Im − Θ2)−1Y TX )w = 4η1η2w

(31)

((Im − Θ2)−1Y TX(In − Θ1)−1XTY )c = 4η1η2c

(32)

Table 2. Process Faults no.

contents

IDV (0) IDV (1)

normal operation A/C feed ratio, B composition constant (stream 4) B composition, A/C ratio constant (stream 4) reactor cooling water temperature A feed loss (stream 1) A, B, C feed composition (stream 4) C feed composition temperature reactor cooling water temperature

IDV (2) IDV IDV IDV IDV IDV

(4) (6) (8) (10) (11)

type − step step step step random variation random variation random variation

t = Xw

(33)

Figure 5. Y1 prediction based on PLS. 1614


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Figure 6. Y1 prediction based on the QGLPLS.

u = Yc

(34)

The statistics of T2 and SPE based on the QGLPLS method for the normal operating conditions are given as

Next, the load vectors p and q and the regression coefficient β are obtained as

p = XTt /(t Tt )

(35)

q = Y Tu/(u Tu)

(36)

β = u Tt /(t Tt )

(37)

T 2 = t Λ−1t T = t

(38)

̃ Ynormal = βTQ T

(39)

−1

}

tT

SPE = ||e||2 = ee T = (X − X̃ normal )(X − X̃ normal )T

(40) (41)

These statistics are the measurements of the potential variables and the residual subspaces for process monitoring. The confidence limits of the statistics of T2 and SPE can be estimated from the F and χ2 distributions, respectively18

Here, p is the load vector of t to the process variables X, q the load vector of u to the quality variables Y, and β the regression coefficient of u to t. Next, the input and output data sets X and Y can be reconstructed as X̃ normal = TPT = XRPT

{

1 TTT k−1

T2 ∼

A(k 2 − 1) FA , k − A k(k − A)

SPE ∼ gχh2 1615

(42) (43) DOI: 10.1021/acs.iecr.5b02559 Ind. Eng. Chem. Res. 2016, 55, 1609−1622

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Figure 7. Monitoring statistics for the LPP-, PLS-, GLSA-, and QGLPLS-based methods with the composition of E as the output variable.

Step 2: Set i ← 1. Step 3: Xi ← X, Yi ← Y. Step 4: Solve the following constrained optimization problem

If new test samples Xnew and Ynew are obtained during online process monitoring, then the corresponding score vector, prediction value, and residual matrix are calculated as follows:

Tnew = X new R

(44)

X̃ new = TnewPT = X new RPT

(45)

+ λ 2(ci TYi TS2Yc i i)}

̃ = βTnewQ T Ynew

(46)

s.t. wi Twi = 1, ci Tci = 1

enew = X new − X̃ new = X new (In − RPT)

(47)

T T JQGLPLS (wi , ci) = max{(wi TXi TYc i i) + λ1(wi Xi S1Xiwi)

This problem can be converted to determine the solutions of the eigenvalue problems of eqs 31 and 32 to find the optimal vector wi and ci. Step 5: Calculate the score vectors ti and ui using eqs 33 and 34, respectively, and compute load vectors pi and qi and the regression coefficient β using eqs 35, 36 and 37. Step 6: Iteratively calculate Xi+1 and Yi+1 as

Next, the statistics of T2 and SPE of new data are calculated as follows:

Tnew 2 = tnew Λ−1tnew T

(48)

SPEnew = (X new ‐X̃ new )(X new ‐X̃ new )T

(49)

Xi + 1 ← Xi − tipiT

The fault will be detected by comparing the new statistics with the corresponding control limits of eqs 42 and 43. The process monitoring flowchart based on QGLPLS is shown in Figure 1, and a detailed description is given in the following. Step 1: Collect the historical normal condition data sets, including process measurement X and quality data Y, and normalize these data sets as follows:

Yi + 1 ← Yi − βtiqiT Step 7: Compute the number of principal components using the cross-validation method. If i is the number of principal components, then take d = i and go to step 8. Otherwise, set X = Xi+1, Y = Yi+1, and i = i + 1 and return to step 3 until i = n. Step 8: Calculate the reconstruction values X̃ normal and Ỹnormal of X and Y by using eqs 38 and 39. Step 9: The T 2 and SPE statistics and their control limits are calculated from eqs 40−43. Step 10: Collect new test samples Xnew and Ynew, and then normalize them.

X = (X − X̅ ). /std_X Y = (Y − Y ̅ ). /std_Y

where X̅ and std_X are the mean and variance of original sampling data X, respectively. Y̅ and std_Y are the mean and variance of output data Y, respectively. 1616


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Figure 8. Scatter plots of the first two latent variables for the LPP, PLS, GLSA, and QGLPLS methods with the composition of E as output variable.

2.4. Global and Local Preservation Capacity Analysis. Swiss roll, twin peaks, and three-dimensional scurve data are the most common data for the manifold learning algorithm. Here, we use the scurve data sets as input space variables to test the capacity of global and local preservation of the LPP-, PLS-, and QGLPLS-based methods. Because the PLS and QGLPLS methods are related to the output variable Y, we define Y as the

Step 11: Calculate the score matrix, prediction value, and residual matrix using eqs 44−47. Step 12: Calculate the statistics T2 and SPE of the new data using eqs 48 and 49. Step 13: Compare T2new and SPEnew of the test data with the corresponding control limits to judge whether the process is abnormal. 1617


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Industrial & Engineering Chemistry Research Table 3. Fault Detection Rate (%) with Output Y1 (95% Control Limit) PCA IDV IDV IDV IDV IDV IDV IDV

(1) (2) (4) (6) (8) (10) (11)

LPP

GLSA

PLS

QGLPLS

T2

SPE

T2

SPE

T2

SPE

T2

SPE

T2

SPE

99.31 97.94 25.16 99.38 96.75 52.62 44.44

99.94 97.72 97.21 100 97.19 51.75 84.55

99 96.38 12.5 100 95.25 53.69 30.56

99.38 98.06 86.56 100 97.38 53.69 60.98

99.62 98.38 43.25 99.38 92.81 29.81 45.25

100 98.54 95.38 100 97.94 80 85.81

99.21 94.12 98.06 99.69 93.88 55.5 76.88

99.75 98.31 98.19 100 97.44 62.38 91.75

99.25 97.31 97.44 99.5 96.88 71.27 78.94

99.62 98.12 98.12 99 96 58.62 93.44

sum of three input space variables from the scurve data. The original scurve set and its sample data are shown in Figure 2. The different colors in Figure 2 indicate the different data distributions in three-dimensional space. The three different PLS-, LPP-, and QGLPLS-based methods can map the original scurve data in three-dimensional space into two-dimensional space; the different projection results are shown in Figure 3. In the present paper, K = 12 and t = 1 are selected for LPP, GLSA, and QGLPLS. It is obvious that the data in the two-dimensional space still remained relatively independent after dimensionality reduction of the LPP algorithm (Figure 3b). It can be concluded that LPP is a local preserving method that somewhat maintains the neighboring relations between sample points. The effect of the PLS method on dimensionality reduction is not as good as that of the LPP method, and the data overlap, especially at the boundary of different distribution ranges (Figure 3a). QGLPLS provides relatively clearer global and local preservation features among these three methods (Figure 3c). Meanwhile, the QGLPLS method can preserve the features in a lowdimensional manifold that are similar to the features of the original data. The target essence of fault detection in the offline modeling phase is to obtain the data fluctuation border with the normal operating conditions, which is used to determine whether the online data are incorrect. Thus, QGLPLS is an ideal dimension reduction method in the fault detection sense.

Table 4. Fault Detection Rate (%) with Output Y2 (95% Control Limit) PLS IDV IDV IDV IDV IDV IDV IDV

(1) (2) (4) (6) (8) (10) (11)

QGLPLS

T2

SPE

T2

SPE

99.5 98.25 56.88 99.75 93.69 62.38 61.66

100 98.69 99.44 100 97.81 55.22 91.94

99.06 97.81 67.19 99.44 97.06 67.94 81.38

100 95.94 98.75 100 97.81 63.12 91.75

good performance in regard to these faults, which can affect many observed variables. IDV (1), IDV (2), IDV (6), IDV (7), IDV (8), IDV (14), and IDV (18) belong to this type of fault. Other faults cause only a limited number of observed variables to deviate from their normal operation state. Thus, this type of fault detection is relatively more challenging; IDV (4), IDV (10), and IDV (11) belong to this category. In this paper, we select faults IDV (1), IDV (2), IDV (4), IDV (6), IDV (8), IDV (10), and IDV (11) as the test faults to verify the proposed QGLPLS method and to compare it with PLS, LPP, and other traditional statistical monitoring methods. If the QGLPLS method can easily detect faults of IDV (1), IDV (2), IDV (6), and IDV (8), it demonstrates that the QGLPLS method is feasible for process monitoring. Further, we anticipate that QGLPLS will be able to improve the fault detection capability of IDV (4), IDV (10), and IDV (11) to reflect the overall performance of QGLPLS. The details of the test faults are given in Table 2. Here, IDV (0) is the normal operation. All of the data sets include 960 samples, and the fault is added from 160 sample points to the end, i.e., faults occur in the eighth simulation hour. First, the process data sampled in the normal operating condition IDV (0) is used to construct the PLS model for output variable Y1. The output variable model is established by the PLS method, and the error meets the precision shown in Figure 5a,b. However, the prediction results in the fault operation condition IDV (1) and IDV (2) cannot be satisfactory. Next, IDV (0) is used to construct the QGLPLS model for output variable Y1, the composition of G; the modeling results are given in Figure 6a,b. The blue line is the actual sampled data, and the green line is the model output. The output in the fault operation condition IDV (1) and IDV (2) is predicted based on the QGLPLS model, as shown in Figures 6c,d. The comparative results between the actual fault data and the prediction data show that the relative modeling and prediction errors are normally less than ±2%. Here, the relative error is defined as err = (Y − Ỹ )/Y. When Figures 5 and 6 are

3. CASE STUDY 3.1. Monitoring Results Analysis for the Tennessee Eastman Process. The Tennessee Eastman (TE) process model has been widely used as a benchmark for the verification of monitoring methods.6 The flowchart of the TE process is shown in Figure 4; the process contains a total of five major units (reactor, condenser, compressor, separator, and stripper). The entire control strategy consists of 19 PI controllers, 11 set points, and 53 variables composed of 12 manipulated variables and 41 measured variables. Note that there are 22 measured variables with a sampling period of 3 min, and the remaining 19 variables are sampled with a 6−15 min time delay. The time delay has a critical influence on the product quality of the TE process. As a result, the QGLPLS method is used to predict the output before it is measured and to detect whether an outputrelated failure occurs. In this paper, 12 manipulated and 22 measurement variables are considered as the process input data X, and the output quality variables Y include the composition of G (stream 9) and the composition of E (stream 11). The related input and output variables are explained in Table 1. The TE simulation platform includes 21 preset faults, of which 16 are known faults and 5 are unknown faults. Most of the experiments demonstrated that most of the statistics show a 1618


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Industrial & Engineering Chemistry Research compared, the prediction results of QGLPLS are found to be better than those of the PLS method. The other prediction results, such as in the other fault conditions IDV (4) to IDV (11) or the prediction for output variable Y2, are similar to the above results. The relative prediction error is normally in the range of ±3%. We did not list all of them here for clarity. It can be concluded that QGLPLS model has a good prediction accuracy, which is acceptable for the following fault diagnosis.

The monitoring statistics based on the PLS, LPP, GLSA, and QGLPLS methods are shown in Figure. 7. Here, the composition of E is the output variable. We show only the process monitoring results of fault conditions IDV (4), IDV (10), and IDV (11) with T2 and Q (SPE) statistics. In addition, the scatter plots of the first two latent variables are shown in Figure. 8. Compared with the LPP-, PLS-, and GLSA-based methods, the QGLPLS-based method successfully separates the fault samples and the normal samples on the first two latent variables. From the scatter plots of the QGLPLS method, the difference between normal data set and fault data set is found to be more obvious. Thus, it is proven that the QGLPLS method has an advantage in process monitoring compared to the PLS, LPP, and GLSA methods. Table 3 shows the fault detection rate obtained from the different methods of the QGLPLS, PCA, PLS, LPP, and GLSA methods with output variable Y1. Table 4 gives the corresponding results with output variable Y2. Under the same operating conditions, we collect data sets over 10 monitoring results. The fault detection rate in Tables 3 and 4 is an average of 10 process monitoring results. Because the output variable is independent of the PCA, LPP, and GLSA methods, the monitoring results with composition of G are same as that of E, which is omitted in Table 4. The faults of IDV (1), IDV (2), IDV (6), and IDV (8) can influence a number of observed variables, so that most of the statistics are able to show very good performance for these faults. We find that several methods exhibit excellent performance for these relatively easily detected faults. However, there are only a limited number of process variables deviated from the normal operating points when some types of faults occur, such as IDV (4), IDV (10), and IDV (11). The detection for these faults is challenging. Because PLS- and QGLPLS-based methods consider the product quality data during the fault detection, their monitoring performance is far better for the methods that do not consider the output variables, such as PCA, LPP, and GLSA. Moreover, for the faults that are not easily detected, the proposed QGLPLS method demonstrates a clear superiority over the PLS method due to its global and local preservation ability. Next, let us compare the monitoring with different output variables: the composition of G and E. When the composition of G is selected as the quality variable, the fault detection rates obtained using the PLS and QGLPLS methods, especially for IDV (4), are better than those with the composition of E as the quality variable. It is also demonstrated that the appropriate

Figure 9. Flowchart of the penicillin fermentation process.

Table 5. Simulation Platform Process and Quality Variables of Pensim2.0 no. 1 2 3 4 5 6 7 8

variable description aeration rate

variable type

process variable agitator power process variable substrate feed rate process variable culture volume process variable carbon dioxide process concentration variable pH process variable fermenter process temperature variable acid flow rate process variable

variable description

variable type

9

alkali flow rate

10

cooling water flow rate substrate concentration dissolved oxygen concentration biomass concentration penicillin concentration generated heat

process variable process variable quality variable quality variable quality variable quality variable quality variable

no.

11 12 13 14 15

Figure 10. Monitoring statistics of the LPP, PLS, and QGLPLS methods with the aeration rate fault between 80 and 120 sampling time. 1619


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Figure 11. Scatter plots of first two latent variables of LPP-, PLS-, and QGLPLS-based methods with the aeration rate fault.

Figure 12. Regression coefficient histogram of the PLS and QGLPLS methods.

higher nonlinear feature and complexity than the TE process, which can be applied to test the nonlinear dimensionality reduction ability of the proposed QGLPLS method. Although penicillin is a typical batch process, its stationary process data can be seen as a continuous process. Thus, the preprocessing for unfolding three-dimensional data is not considered here.

selection of the quality variable will help to improve the process monitoring performance. 3.2. Monitoring Results Analysis for the Penicillin Fermentation Process. In this paper, we also collect the stationary process data from the penicillin fermentation process, Pensim2.0 platform. The penicillin fermentation process shows 1620


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Industrial & Engineering Chemistry Research The penicillin fermentation process is given in Figure 9,32 and we select 10 process variables as input X and 5 quality variables as output Y, as shown in Table 5. A slope fault, 10% decreasing, was added to the first process variable at 80−120 sampling time. Next, the PLS, LPP, and QGLPLS models were established to monitor the process variable and the product quality outputs. Figure 10 shows the monitoring statistics T2 and SPE obtained from the LPP, PLS, and QGLPLS methods. Obviously, the QGLPLS-based method has a better monitoring performance than the other two methods. The scatter plots of the first two latent variables from the LPP, PLS, and QGLPLS methods are shown in Figure 11. The QGLPLS-based method successfully separates the fault samples and the normal samples on the first two latent variables compared to the LPP and PLS-based methods. For all the quality variables, the QGLPLS-based method has a better explanation capability compared to the PLS-based method, as shown in Figure 12. The color bar from 1 to 10 on the right of the histogram corresponds to the numbers of the process variables.

(3) Ding, S. X. Data-driven design of monitoring and diagnosis systems for dynamic processes: A review of subspace technique based schemes and some recent results. J. Process Control 2014, 24, 431−449. (4) Tong, C.; El-Farra, N. H.; Palazoglu, A.; Yan, X. Fault Detection and Isolation in Hybrid Process Systems Using a Combined DataDriven and Observer-Design Methodology. AIChE J. 2014, 60, 2805− 2814. (5) Aumi, S.; Corbett, B.; Clarke-Pringle, T; Mhaskar, P. Data-Driven Model Predictive Quality Control of Batch Processes. AIChE J. 2013, 59, 2852−2861. (6) Ge, W.; Wang, J.; Zhou, J.; Wu, H.; Jin, Q. Incipient Fault Detection Based on Fault Extraction and Residual Evaluation. Ind. Eng. Chem. Res. 2015, 54, 3664−3677. (7) Yin, S.; Ding, S. X.; Xie, X.; Luo, H. A Review on Basic DataDriven Approaches for Industrial Process Monitoring. IEEE T. Ind. Electron 2014, 61, 6418−6428. (8) Rato, T. J.; Reis, M. S. Advantage of Using Decorrelated Residuals in Dynamic Principal Component Analysis for Monitoring Large-Scale Systems. Ind. Eng. Chem. Res. 2013, 52, 13685−13698. (9) Feital, T.; Kruger, U.; Dutra, J.; Pinto, J. C.; Lima, E. L. Modeling and Performance Monitoring of Multivariate Multimodal Processes. AIChE J. 2013, 59, 1557−1569. (10) Lu, N.; Gao, F.; Yang, Y.; Wang, F. PCA-Based Modeling and On-line Monitoring Strategy for Uneven-Length Batch Processes. Ind. Eng. Chem. Res. 2004, 43, 3343−3352. (11) Rato, T. J.; Reis, M. S. Defining the structure of DPCA models and its impact on process monitoring and prediction activities. Chemom. Intell. Lab. Syst. 2013, 125, 74−86. (12) Ge, Z.; Song, Z.; Gao, F. Nonlinear Quality Prediction for Multiphase Batch Processes. AIChE J. 2012, 58, 1778−1787. (13) Li, G.; Liu, B.; Qin, S. J.; Zhou, D. Quality Relevant Data-Driven Modeling and Monitoring of Multivariate Dynamic Processes: The Dynamic T-PLS Approach. IEEE Trans. Neural network 2011, 22, 2262−2271. (14) Zhao, L.; Zhao, C.; Gao, F. Between-Mode Quality Analysis Based Multimode Batch Process Quality Prediction. Ind. Eng. Chem. Res. 2014, 53, 15629−15638. (15) Yoo, C. K.; Lee, J.-M.; Vanrolleghem, P. A.; Lee, I.-B. On-line monitoring of batch processes using multiway independent component analysis. Chemom. Intell. Lab. Syst. 2004, 71, 151−163. (16) Lee, J. M.; Yoo, C. K.; Lee, I. B. Statistical process monitoring with independent component analysis. J. Process Control 2004, 14, 467−485. (17) Mori, J.; Yu, J.. Maximized Mutual Information Based NonGaussian Subspace Projection Method for Quality Relevant Process Monitoring and Fault Detection. 52nd IEEE Conference on Decision and Control, 2013; pp 10−13. (18) Zhou, D.; Li, G.; Qin, S. J. Total Projection to Latent Structures for Process Monitoring. AIChE J. 2010, 56, 168−178. (19) Qin, S. J.; Zheng, Y. Quality-Relevant and Process-Relevant Fault Monitoring with Concurrent Projection to Latent Structures. AIChE J. 2013, 59, 496−504. (20) Zhang, Y.; Sun, R.; Fan, Y. Fault diagnosis of nonlinear process based on KCPLS reconstruction [J]. Chemom. Intell. Lab. Syst. 2015, 140, 49−60. (21) Roweis, S. T.; Saul, L. K. Nonlinear Dimensionality Reduction by Locally Linear Embedding. Science 2000, 290, 2323−2326. (22) Tenenbaum, J. B.; de Silva, V.; Langford, J. C. A Global Geometric Framework for Nonlinear Dimensionality Reduction. Science 2000, 290, 2319−2323. (23) He, X.; Yan, S.; Hu, Y.; Niyogi, P.; Zhang, H.-J. Face recognition using laplacianfaces. IEEE Trans on Pattern Analysis and Machine Intell 2005, 27, 328−340. (24) He, X.; Niyogi, P. Locality Preserving Projections. Adv. Neural Inf. Process. Syst. 2003, 16, 153−160. (25) Hu, K.; Yuan, J. Multivariate statistical process control based on multiway locality preserving projections. J. Process Control 2008, 18, 797−807.

4. CONCLUSIONS In this paper, a new method QGLPLS was developed and successfully used for process monitoring. The QGLPLS method can effectively extract quality-related variation and maintain the global−local structural feature of process measurement variables during dimensionality reduction. Compared to the LPP and PLS methods, the QGLPLS method combines the inherent model structure of the process and quality variables and simultaneously relies on the global−local structural integration. As a result, the QGLPLS method has better global and local preservation ability, and it can maintain a clearer boundary in the projection process. Two chemical processes (the TE and penicillin fermentation processes) were used to test the monitoring ability of the proposed QGLPLS method. The modeling and prediction accuracy of the QGLPLS method was found to be higher than those of traditional statistical methods, such as PCA, PLS, LPP, and GLSA. Furthermore, the QGLPLS method showed better statistical monitoring performance for the highly nonlinear and complex industrial processes studied.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work is supported by the National Natural Science Foundation of China (61174128, 61473025, 61403017, and 61573050), the Beijing Natural Science Foundation (4132044), and the Fundamental Research Funds for the Central Universities of China (YS1404).

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REFERENCES

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