Spatial-Statistical Local Approach for Improved Manifold-Based

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Spatial-statistical local approach for improved manifold-based process monitoring Nan Li, Yan Weiwu, and Yupu Yang Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.5b00257 • Publication Date (Web): 17 Aug 2015 Downloaded from http://pubs.acs.org on August 23, 2015

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Industrial & Engineering Chemistry Research

Spatial-statistical local approach for improved manifold-based process monitoring Nan Li, Weiwu Yan and Yupu Yang

Department of Automation, Shanghai Jiao Tong University, and Key Laboratory of System Control and Information Processing, Ministry of Education of China, Shanghai 200240, China

Abstract: In this article, a new manifold-based process monitoring scheme which incorporates statistical local approach into neighborhood preserving embedding (NPE) is proposed to monitor changes in the local structure of process data. This method not only inherits the ability of NPE to discover the local structure of data, but also implements online fault detection by monitoring the local information changes of new observations. Moreover, the incorporation of statistical local approach allows that no assumptions have to be made on data distribution, since the constructed new monitoring vectors approximately follow multivariate Gaussian distributions. Thus the confidence limits of two statistics constructed for process monitoring can be easily determined by  2 or F distributions. Furthermore, the new developed method can also improve the detection sensitivity significantly. To evaluate the performance of this new monitoring scheme, it



Corresponding author: [email protected].

ACS Paragon Plus Environment


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was tested in the Tennessee Eastman (TE) benchmark process and the experimental results have demonstrated its superiority.

Keywords: process monitoring, manifold learning, statistical local approach, neighborhood preserving embedding, local information

1 Introduction

Process monitoring is very essential for ensuring the plant and staff safety and the yield and quality of products.1-3 With the advent and development of distributed control system (DCS) and supervisory control and data acquisition (SCADA) system, multivariate statistical process monitoring (MSPM) has been widely used for monitoring of large-scale industrial processes.4-11 Among various MSPM methods, principal component analysis (PCA) and partial least squares (PLS) are the most representative and have been introduced into process monitoring for decades.4, 6, 12-18 Both PCA and PLS project the original data onto a low dimensional subspace that retains the most variations of the original data space. In other words, the global structure is preserved.19-21 However, due to the complexity of industrial processes, the direct implementation of these traditional methods has encountered many problems. Therefore, for getting more effective monitoring strategies, researchers have done a lot of work to extend these PCA/PLS based methods. Dynamic PCA (DPCA) and dynamic PLS (DPLS) are the dynamic extensions of PCA and PLS, taking account of not only the correlations but also the autocorrelations.22-24


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Multi-way PCA (MPCA) and Multi-way PLS (MPLS) are two solutions for batch process, and have been applied to biochemical processes, semiconductor processes, etc.25-28 Adaptive PCA and adaptive PLS have been used to monitor processes with time-variant behaviors.29-33 In order to deal with the nonlinear correlations between process variables, kernel PCA (KPCA) and kernel PLS (KPLS) have been proposed by combining PCA and PLS with the kernel trick.34-42 Moreover, Multi-block PCA and Multi-block PLS have been developed for monitoring the plant-wide process which always has many different parts and operation units.43-47 Although PCA and PLS can capture the global structure of data19-21, the detailed local structure information between data points is ignored. However, the inner data structure represents the detailed relationship between different data samples, which is also an important aspect of data set, and the loss of this crucial information may have great impact on dimension reduction performance and thus the process monitoring efficiency will also be influenced.19, 20 As opposed to the globality-based data projection techniques like PCA and PLS19-21, recently developed manifold learning techniques seek to discover the local structure of data. Some representative manifold learning methods are Isomap48, locally linear embedding (LLE)49, Laplacian eigenmaps (LE)50, locality preserving projections (LPP)51 and neighborhood preserving embedding (NPE).52 Among these methods, LPP and NPE are respectively the linear forms of LE and LLE. Since Isomap, LLE and LE yield mappings defined only on the training data set, it is very difficult to naturally evaluate the mappings on the testing data set. However, for LPP and NPE, it is easy to do so due to the mapping matrices obtained from the training data set. Consequently, LPP and



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NPE have been applied to process monitoring recently to overcome some limitations existing in PCA/PLS-based monitoring methods. Hu and Yuan respectively applied LPP-based and NPE-based MSPM to batch process monitoring.53-55 A nonlinear dimensionality reduction method called generalized orthogonal LPP (GOLPP) was proposed for nonlinear fault detection and diagnosis in manufacturing processes.56 Moreover, some researchers implemented process monitoring by integrating the local information with the global/nonlocal information and claimed that their monitoring methods outperform those based on PCA and LPP/NPE.19-21, 57-59 Similar to traditional PCA-based methods, two statistics, T2 and SPE, were also constructed for process monitoring in these manifold-based methods. However, the constructed statistics take no account of the local information of the testing data. Moreover, due to the complexity of industrial processes, the data distribution is typically non-Gaussian. Thus the Gaussian assumption in some manifold-based methods is typically unreasonable.19,

20, 53-55, 57, 58

Although kernel density

estimation (KDE)60 has been commonly used to estimate the confidence limit of non-Gaussian distributed data and was also applied in some manifold-based monitoring methods21, the implementation of KDE is very time-consuming and its efficiency is heavily affected by the parameter selection. Poor parameter selection also tends to produce disappointing results. Statistical local approach is a novel method for abrupt change detection, proposed by Basseville, etc.61 This method first constructs a monitoring function describing the system, and then achieves the purpose of fault detection by monitoring the changes in the model parameters of the function rather than performing a direct analysis of the recorded variables. Based on the


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central limit theorem, a parameter deviation can be presented as a change in the mean of a Gaussian probability density function, no matter how the data is distributed. Thus statistical local approach is very appropriate for non-Gaussian fault detection and Gaussian monitoring statistics can be established to simplify the monitoring task. An additional advantage of incorporating statistical local approach is that the monitoring statistics based on improved residuals (improved residuals will be introduced in the subsequent sections) are more sensitive to incipient fault conditions.62-66 This is because statistical local approach can play a role similar to cumulative sum.65 Interested readers can refer to this literature5 to get detailed discussion and proof for this property. To address the problems mentioned above, in this paper, a novel method called spatial-statistical local approach (SSLA) is developed by introducing the statistical local approach into one of the aforementioned manifold learning methods, NPE. With the incorporation of the statistical local approach, the constructed improved residuals not only take account of the local information of the testing data but also approximately follow Gaussian distributions. For the purpose of fault detection, two new statistics are constructed of which the confidence limits can be easily estimated by  2 or F distributions. The rest of the paper is organized as follows. First, some preliminaries including NPE and the statistical local approach are given in Section 2. Next, Section 3 presents the details of the development of the proposed SSLA. Then in Section 4, the new monitoring strategy based on SSLA is described in detail. In order to evaluate the monitoring performance of the proposed monitoring strategy, the



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SSLA-based method is applied to the Tennessee Eastman (TE) process in Section 5. Finally, this paper is concluded in Section 6.

2 Preliminaries

In this section, NPE and the statistical local approach are first briefly reviewed before SSLA is developed in the subsequent sections.

2.1 Neighborhood preserving embedding (NPE)

Different from PCA which aims at preserving the global Euclidean structure, NPE aims at preserving the local neighborhood structure of the data set. Let X  [x1, x2 ,

, xn ] 

mn

denote

the data set with m variables and n data points. Then the algorithmic procedure of NPE on the data set, X , can be shown as follows including three steps 52: 1. Constructing an adjacency graph: Let G denote a graph with n nodes. The i-th node corresponds to the data point x i . There are two ways to construct the adjacency graph: k nearest neighbors (KNN) and ε neighborhood. With KNN, put a directed edge from node i to j if x j is among the k nearest neighbors of x i . With ε neighborhood, put an edge between nodes i and j if x j  xi   2. Computing the weights: Let W denote the weight matrix with Wij having the weight of the edge from node i to j, or 0 if there is no such edge. The weighs on the edges can be computed by solving the following minimization problem:


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n

2

n

min  xi   Wij x j W

i 1

j 1

(1)

n

s.t. Wij  1 j 1

3. Computing the projections: The objective of NPE is to find the projection matrix by solving the following minimization problem: n

n

i 1

j 1

2

min  y i   Wij y j P

(2)

s.t.PT XXT P  I d where P 

d m

is the projection matrix, y i  PT xi and I d denotes a unity matrix with size d.

This problem can be transformed into the following generalized eigenvector problem:

XMXT p   XXT p

(3)

where M  (I n  W)T (I n  W) . Let the column vectors p1 , p 2 , ordered according to their eigenvalues, 1  2 

, p d be the solutions of (3),

 d . Thus, the embedding can be expressed

as xi  yi  PT xi，P = [p1 , p2 ,

, pd ]

(4)

For process monitoring, two statistics, T2 and SPE, are constructed as54:

T 2  yTnewy1y new SPE  x new  x new

2

 x new  By new

(5) 2

(6)

where Σy  YT Y (n  1) is the sample covariance matrix of Y under normal conditions; x new is the reconstruction of x new and B is the reconstruction matrix.



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2.2 Statistical local approach 66

The statistical local approach is detailed in ref

. Denote θ 0 and θ as parameters

representing normal and abnormal behaviors of the process. The statistical local approach assumes the abnormal parameter θ is close or local to θ 0 and thus writes θ as follows: θ  θ0 

θ K

(7)

where θ is an unknown but fixed vector. The hypothesis test involved in the statistical local approach is given as follows: H 0 : θ  θ 0 , H1 : θ  θ 0 

θ K

(8)

A vector-valued function,  (θ0 , x k ) , can be defined as a primary function if  (θ0 , x k ) satisfies the following conditions: 1. Eθ ( (θ0 , x k ))  0 , if θ  θ 0 ; 2. Eθ ( (θ0 , x k ))  0 , if θ  θ 0 but θ is in the neighborhood of θ 0 ; 3.

 (θ, xk ) is differentiable with respect to θ ;

4.

 (θ0 , x k ) exists in the vicinity of θ 0 ;

where Eθ () is the expectation when the actual system parameter value is θ . Based on the primary residual function defined above, an improved residual can be further defined as follows:

 θ (θ0 , K ) 

1 K

K

 (θ , x ) k 1

0

k

(9)

where  θ (θ0 , K ) is the improved residual when the actual system parameter is θ .The central


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limit theorem shows that  θ0 (θ0 , K ) asymptotically follows a multi-normal distribution of zero mean and covariance K

K

(θ0 )  lim  E ( (θ0 , xi ) T (θ0 , x j )) K 

(10)

i 1 j 1

if no fault condition is present. In addition, the central limit theorem also shows that if the actual system parameter values is θ  θ0 

θ , the covariance of  θ (θ0 , K ) is equal to that of K

 θ (θ0 , K ) , but the mean of  θ (θ0 , K ) departs from zero. Using this fact, a Hotelling’s T 2 can 0

be defined as TK2   θ (θ0 , K)T (θ0 )1 θ (θ0 , K)

(11)

Then the null and alternative hypothesis in (8) can be reformulated as follows: H0 : TK2  T02 , H1 : TK2  T02

(12)

where T02 is the confidence limit for a  2 or F distribution.

3 Spatial-statistical local approach (SSLA)

This section incorporates the statistical local approach into NPE, developing the required monitoring functions (primary residuals) and proving their sufficiency for process monitoring.

3.1 Deriving the primary residuals

To derive the primary residuals, we first modify the optimization problem, (2), for NPE slightly as



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min PT E (zzT )P P

s.t .P CP  I d T

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(13) n

T where E () is the statistical expectation operator, C  XX and z  x   w j x j with x j j 1

being one neighbor of x and w j being the weight from x to x j . Then the ith projection vector p i can be formulated as:

pi

arg max{pT E (zzT )p  i (pT Cp 1)}

(14)

p

where i is the Lagrangian multiplier (eigenvalue). The solution of (14) is given by

 {pT E (zzT )p  i (pT Cp  1)}  pi  arg   0 p    arg 2 E (zz )p  2i Cp = 0

(15)

T

By defining gi  2zzT pi  2iCpi  2ti z  2iCpi

(16)

pi  arg E (gi )  0

(17)

Epi (gi )  0

(18)

(15) can be simplified to

Consequently,

which satisfies the condition 1 given in Section 2 for the primary residuals. Moreover, it can be easily seen that the other three conditions for the primary residuals are also satisfied for g i . Therefore, G = [g1T , gT2 ,

, gTm ]T can be used as the monitoring function for monitoring the

changes in the eigenvectors, and the primary residual can be calculated from this monitoring function.


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Similarly, the monitoring function for detecting the changes in the eignevalues can also be developed. By premultiplying (16) by pTi , we have

pTi gi  2pTi zzT p  2ipTi Cpi  2ti2  2i

(19)

hi  2ti2  2i

(20)

Define hi as

It is very easy to see that h = [h1 , h2 ,

, hm ]T also satisfies the four conditions for the primary

residuals. Hence, it can also be used as a monitoring function (a primary residual) for monitoring the change in the eigenvalues. For monitoring the main and minor local structures separately, G and h can be divided into two parts respectively: Gma  [g1T , gT2 ,

Gmi  [gTd 1 , gTd 2 , hma = [h1 , h2 , hmi = [hd 1 , hd 2 ,

, gTd ]T

, gTm ]T , hd ]T , hm ]T

(21) (22) (23) (24)

3.2 Sensitivity of the primary residuals

In this subsection, the sensitivity of the monitoring functions (primary residuals) is investigated by examining whether: 1. The primary residuals G ma and h ma are sensitive to changes in the eigenvectors and eigenvalues associated with the main local structure;



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2. The primary residuals G mi and h mi are sensitive to changes in the eigenvectors and eigenvalues associated with the minor local structure. The investigation results are given in the following two theorems. Theorem 1: For detecting changes related to the main local structure, G ma and h ma are both sufficient in detecting changes in the eigenvectors or eigenvalues. Theorem 2: For detecting changes related to the minor local structure, G mi and h mi are both sufficient in detecting changes in the eigenvectors or eigenvalues. The proof for these two theorems takes the proving way of Ref 62 as reference and is provided in the appendix section. The dimensions of the primary residuals h ma and h mi are d and m-d, respectively. Nevertheless, the dimensions of the primary residuals G ma and G mi are d∙m and (m-d)∙m, respectively. Obviously, the computation burdens of h ma and h mi are smaller than those of

G ma and G mi . Furthermore, since both sets of the primary residuals are sensitive in detecting changes in the eigenvectors or eigenvalues, it is preferable to employ h ma and h mi .


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4 SSLA-based monitoring

4.1 Construction of improved residuals and monitoring statistics

Based on the primary residuals and the analysis detailed in the previous section, two improved residuals are constructed with h ma and h mi in this section. The two improved residuals are given as follows:

 ma 

1 K

 mi 

1 K

K

h k 1

ma

(25)

mi

(26)

K

h k 1

Since the sensitivity of improved residuals reduces as K increases, a moving window approach can be used to increase the sensitivity of the improved residuals and reduce the computation burdens

62-65

. By applying the moving window approach to the kth sample, these two improved

residuals become:

 ma ,k 

k 1  hma w i k  w1

(27)

 mi ,k 

k 1  hmi w i k  w1

(28)

where k denotes the current sampling instance, and w is the length of the moving window. As Eq. (11), two Hotelling’s T2 statistics can be defined for the above two improved residuals: T 1 Tma2   ma , k  ma ma , k

(29)

1 Tmi2   miT ,k mi  mi ,k

(30)



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2 The confidence limits of Tma and Tmi2 can be easily estimated by two  2 or F distributions,

respectively. In our research, the confidence limits of these two statistics are estimated by the approximate 2 distributions, Tlimit, 

g  h2 . The control limits are obtained by fitting weighted

2

distributions to the reference distributions generated from normal operating data. In 2 Tlimit, 

2 g  h2 , g is a weighting parameter included to account for the magnitude of Tma or Tmi2 ,

and h accounts for the degrees of freedom. If a and b are the estimated mean and variance of the 2 or Tmi2 , then g and h can be estimated by g  b / 2a and h  2a 2 / b . Tma

4.2 Discussion on the two statistics 2 Although Tma and Tmi2 are both derived from the “improved residuals”, they cannot be

interrupted as SPE statistics. The full name of SPE is squared prediction error which is used as a measure for the reconstruction error. For example, PCA reconstructs the original data space using the principal component space, and SPE is computed with the reconstruction error, and the SPE statistic in NPE is computed in the similar way. Different from SPE, the derived primary residuals and improved residuals are computed from the monitoring function to measure the deviation of the model parameters. In this paper, the monitored model parameters are the generalized eigenvalues and eigenvectors, and they are not for measuring the reconstruction error 2 of the process data. Therefore, Tma and Tmi2 , and SPE are two kinds of different statistics.


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The globality-based methods like PCA can detect the changes of the global structure of the process data, however, our proposed the SSLA-based process monitoring is aimed at detection the changes of the local structure of the process data. As we know, T2 in PCA summarizes the information in the principal component subspace, and SPE in PCA summarizes the information in the residual subspace, and the two subspaces are orthogonal. This means T2 and SPE can cover all the global structure information of the process data. Similar to the PCA-based methods, our proposed SSLA-based method can cover all the local structure information of the process 2 2 data. It can be seen from the definitions of Tma and Tmi2 that Tma summarizes the main local

structure information of the process data, and Tmi2 summarizes all the left minor local structure information. Although the projection vectors are not pair-wise orthogonal, the low dimensional 2 data related to Tma are note correlated with the low dimensional data related to Tmi2 . Therefore,

2 can cover all the local structure information for process monitoring. Tmi2 that Tma

4.3 Outline of monitoring strategy

The scheme of the SSLA-based process monitoring is outlined as follows: Offline modeling: (1) Acquire a sufficiently large training data set X = [x1 , x2 ,

, xn ] 

mn

under the normal

operating condition and normalize it. (2) Determine the K nearest neighbors for each sample, x i , and compute the weights, W , by solving the optimization problem (1).



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n

(3) Compute the reconstruction error z i for x i by z i  xi   Wij x j and center it. j1

(4) Solve the optimization problem (13) by solving the eigenvalue problem, Sp  Cp , where

S  E (zzT ) which can be estimated by S 

1 n T z i zTi and C  XX .  n i 1

(5) Based on the information quantified by the eigenvalues, determine the numbers of the dimensions of h ma and h mi . (6) Calculate the primary residuals, h ma and h mi , for normal operating data via Eqs. (19) and (20). Then the improved residuals  ma,k and  mi ,k can be calculated by Eqs. (27) and (28). 2 (7) Calculate the monitoring statistics, Tma and Tmi2 for the normal operating data via Eqs.

(29) and (30). 2 (8) Estimate the confidence limits of Tma and Tmi2 .

Online monitoring (1) Obtain a new observation data, x , and normalize it. (2) Determine the K nearest neighbors of x from the training data set, X , and compute the n

2

n

weights, w , by solving min x   w j x j s.t. w j  1 . w

j 1

j 1

n

(3) Compute the reconstruction error, z , by z  x   w j x j and center it. j 1


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(4) Calculate the primary residuals, h ma and h mi , for the observation data via Eqs. (19) and (20). Then the improved residuals  ma,k and  mi ,k can be calculated by Eqs. (27) and (28) after the width of the moving window, w, is given. 2 (5) Calculate the monitoring statistics, Tma and Tmi2 for the new observation data via Eqs.

(29) and (30). 2 (6) Monitor whether Tma or Tmi2 exceeds its corresponding confidence limit.

To show the monitoring flow more intuitively, the steps of SSLA for process monitoring are illustrated in Figure 1.

Figure 1. Flowchart of SSLA-based process monitoring



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5 Case study

In this section, the developed SSLA-based monitoring scheme is applied to the TE benchmark process to evaluate its monitoring performance. To highlight the superiority of the proposed method, the monitoring results of NPE-based method are used for comparison. The Tennessee Eastman process developed by Downs and Vogel

67

has been widely used to

test the performance of various monitoring methods. This process consists of five main unit operations: a reactor, a condenser, a separator, a stripper, and a recycle compressor. There are four gaseous reactants (A, C, D, and E), two liquid products (G and H), a byproduct (F), and an insert (B). The gaseous reactants are fed to reactor where the liquid products are produced. The plant-wide control structure recommended in

68

is chosen in this study. The control structure of

the TE process is shown schematically in Figure 2. This process has 22 continuous process variables, 19 non-continuous process variables and 12 manipulated variables. In this study, all 22 continuous variables and 11 manipulated variables except for the agitation speed of the reactor’s stirrer are selected for process monitoring. The details of these 33 variables are listed in Table 1. Moreover, 21 known simulated faults are listed in Table 2.


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Figure 2. Control structure of the TE process

No. Description

No. Description

1

A feed (stream 1)

18

Stripper temperature

2

D feed (stream 2)

19

Stripper stream flow

3

E feed (stream 3)

20

Compressor work

4

Total feed (stream 4)

21

Reactor cooling water outlet temp

5

Recycle flow (stream 8)

22

Condenser cooling water outlet temp

6

Reactor feed rate (stream 6)

23

D feed flow (stream 2)

7

Reactor pressure

24

E feed flow (stream 3)

8

Reactor level

25

A feed flow (stream 1)

9

Reactor temperature

26

Total feed flow (stream 4)



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10

Purge rate

27

Compressor recycle valve

11

Separator temperature

28

purge valve

12

Separator level

29

Separator pot liquid flow (stream 10)

13

Separator pressure

30

Stripper liquid product flow

14

Separator under flow (stream 10)

31

Stripper steam valve

15

Stripper level

32

Reactor cooling water flow

16

Stripper pressure

33

Condenser cooling water flow

17

Stripper under flow (stream 11)

Table 1. Monitored variables in the TE process

Fault index

Description

IDV(1)

A/C feed ratio, B composition constant (stream Step 4)

IDV(2)

B composition, A/C ratio constant (stream 4)

Step

IDV(3)

D feed temperature (stream 2)

Step

IDV(4)

Reactor cooling water inlet temperature

Step

IDV(5)

Condenser cooling water inlet temperature

Step

IDV(6)

A feed loss (stream 1)

Step

IDV(7)

C header pressure loss-reduced availability Step (steam 4)

IDV(8)

A, B, C feed composition (stream 4)

Random variation

IDV(9)

D feed temperature (stream 2)

Random variation


Type

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IDV(10)

C feed temperature (stream 4)

Random variation

IDV(11)

Reactor cooling water inlet temperature

Random variation

IDV(12)

Condenser cooing water inlet temperature

Random variation

IDV(13)

Reaction kinetics

Slow drift

IDV(14)

Reactor cooling water valve

Sticking

IDV(15)

Condenser cooling water valve

Sticking

IDV(16)

Unknown

IDV(17)

Unknown

IDV(18)

Unknown

IDV(19)

Unknown

IDV(20)

Unknown

IDV(21)

The valve of stream 4 was fixed at the steady state position

Table 2. Process faults of the TE process Both the training data set collected under the normal conditions and each testing data set collected under an abnormal fault condition contain 960 samples, and their sampling interval is 3 min. Each programmed fault begins to be introduced into the process at the 161th sample. The number of the nearest neighbors is selected as K=10 for both NPE and SSLA. For monitoring the TE process, 15 eigenvectors corresponding to the minimal 15 eigenvalues are selected when NPE is applied. For fair comparison, when SSLA is applied, the minimal 15 eigenvalues and their corresponding eigenvectors are used to compute h ma and  ma,k , and the rest is used to



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compute h mi and  mi ,k . The width of the moving window is selected as w  70 . Moreover, the significance level is selected as 0.01 to estimate the control limits for the two methods. The monitoring results (fault detection rates) of these 21 faults are tabulated in Table 3. There, the monitoring results of some faults are set to be boldface. It can be seen that the fault detection rates of these faults are significantly improved by SSLA. As for the rest of the faults, the monitoring results of SSLA are comparable with those of NPE since the fault magnitudes are so large that they can be detected easily by both SSLA and NPE. To further demonstrate the superiority of SSLA-based monitoring method, the detailed monitoring results of four particular faults are given and analyzed as follows.

Fault detection rates (%) NPE Fault index

SSLA

T2

SPE

Tma2

Tmi2

1

99.13

99.88

99

99.63

2

98.38

96.13

98.13

97.5

3

1.13

2.38

62.63

67.25

4

56.63

99.88

100

100

5

99.75

100

99.88

99.88

6

100

100

100

100

7

100

100

100

99.88

8

97.25

89.13

97.13

100

9

2.13

2.13

46.25

51.38


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10

36

40.13

97.5

97

11

54.63

61

99.25

98.88

12

98.75

97.25

100

100

13

94

95.25

94

94.88

14

99.5

99.75

99.88

99.75

15

1.25

2.25

31.38

71.38

16

22.38

42.13

100

100

17

77.63

95.88

96.5

100

18

89.25

90.38

89.38

93

19

11.63

18

100

99.5

20

33.75

56.88

89.75

91.25

21

45

46.88

72

87.88

Table 3. Monitoring results of the TE process Fault 4 is a step change in the reactor cooling water inlet temperature. After this fault occurs, the reactor temperature increases suddenly in a very short period of time, however the other process variables are left almost unaltered. This situation is caused by the closed control loop. The reactor temperatures under the normal conditions and the faulty conditions are given in Figure 3. Compared to the normal operating conditions, the mean and variance of each process variable both have changes less than 2% 4. This makes the detection for fault 4 more challenging. Figure 4 presents the monitoring results of NPE and SSLA for fault 4. Comparing the results shown in Figure 4(a) and (b), it can be seen that the monitoring performance is greatly improved, especially by the Tma2 statistic. This step fault is successfully detected after it is introduced in the



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process, since all the values of the Tma2 statistics keep above the control limit. However, many values of the T 2 statistic stay below the corresponding control limit which means that this fault cannot be detected continuously.

Figure 3. The reactor temperature: (a) Normal conditions (b) Faulty conditions


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Figure 4. Monitoring charts of fault 4: (a) NPE (control limits, T 2 : 31.35, SPE: 11.01), and (b) SSLA (control limits, Tma2 : 51.30, Tmi2 : 56.91) Fault 10 is a random change in the temperature of stream 4 (C feed), which leads to the random variation of the stripper temperature and other variables with larger variances. The stripper temperature under the normal conditions and the faulty conditions are shown in Figure 5. The monitoring results of NPE and SSLA for fault 10 are shown in Figure 6. From Figure 6(a), we can see clearly that not only the T2 statistic of NPE fails to detect many faulty samples but also the SPE statistic does. However, the monitoring results of SSLA given in Figure 6(b) present significant improvements. Almost all the faulty samples are detected by the two statistics of SSLA, Tma2 and Tmi2



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Figure 5. The stripper temperature: (a) Normal conditions (b) Faulty conditions

Figure 6. Monitoring charts of fault 10: (a) NPE (control limits, T 2 : 31.35, SPE: 11.01), and (b) SSLA (control limits, Tma2 : 51.30, Tmi2 : 56.91) To further test the sensitivity of the developed method, the monitoring results of a small fault and an unknown fault are examined. Fault 15 is a small fault which causes the sticking problem


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of the condenser cooling water valve. According to Chiang et al. 4, traditional statistics can hardly detect this fault. Figure 7 shows the monitoring results of NPE and SSLA for fault 15. Obviously, almost all the faulty samples fail to be detected by both T2 and SPE of NPE, which can be seen from Figure 7(a). Nevertheless, from Figure 7(b) one can see that the monitoring performance for this fault is significantly improved by SSLA. This fault can be continuously detected by both Tma2 and Tmi2 of SSLA in the late period of the process. Moreover, Tmi2 can also detect the appearance of this fault successfully in the early period of introducing this fault. Fault 19 is one of the faults of which the causes are unknown. The monitoring results of NPE and SSLA for fault 19 are plotted in Figure 8. It can be clearly seen from Figure 8(a) that neither T2 nor SPE of NPE can continuously detect this fault with a lot of faulty samples below the control limits. However, both Tma2 and Tmi2 of SSLA can always continuously detect this fault almost from the time when the fault is introduced, which can be seen from Figure 8(b).



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Figure 7. Monitoring charts of fault 15: (a) NPE (control limits, T 2 : 31.35, SPE: 11.01), and (b) SSLA (control limits, Tma2 : 51.30, Tmi2 : 56.91)

Figure 8. Monitoring charts of fault 19: (a) NPE (control limits, T 2 : 31.35, SPE: 11.01), and (b) SSLA (control limits, Tma2 : 51.30, Tmi2 : 56.91)


Page 29 of 42

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6 Conclusions

As opposed to PCA/PLS-based monitoring methods which implement process monitoring by mining the global structure information of data, manifold-based methods (LPP and NPE are two commonly used methods) conduct process monitoring by extracting the local structure information of data. However, the existing manifold-based methods usually analyze the new observations directly without taking account of their corresponding local information. Thus they cannot monitor the changes in the local structure of data and may lose sensitivity for some small-magnitude faulty samples. Additionally, the diversity of data distribution will also affect heavily the performance of these existing manifold-based methods. The developed SSLA-based method in this paper has addressed these issues. The monitoring vectors (primary residuals and improved residuals) and statistics of the proposed method are constructed with the reconstruction error vector of each sample instead of each sample itself. Thus the local information of new observations is taken fully into account. By analyzing the sensitivity of primary residuals, it has been proved that the constructed monitoring vectors and statistics can be used for monitoring the changes in the local structure of data. Moreover, due to the incorporation of statistical local approach, the constructed improved residuals will always approximately follow multivariate Gaussian distributions irrespective of the actual data distribution. Hence, the control limit of each monitoring statistic can be easily determined by a  2 or F distribution and the performance of this method is less affected by the diversity of data distribution. As to the detection sensitivity,



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the experimental results in the TE benchmark process have demonstrated that the proposed method can significantly improved the detection sensitivity for some faults which cannot be detected continuously by the NPE-based method. In this study, statistical local approach is just used to improve NPE-based monitoring method, and in essence NPE is still a linear dimension reduction technique. Hence, in future work, further efforts may be devoted to explore incorporating statistical local approach into other linear or nonlinear manifold-based methods.

Acknowledgements

This work was supported in part by the National Natural Science Foundation of China (61273161) and the National High Technology Research and Development Program of China (“863” Program) (2011AA040605).

Appendix

Proof of Theorems 1 and 2: Rewrite Eq. (14) as follows:

pi

arg max{pT Sp  i (pT Cp 1)} p

(31)

where S  E (zzT ) . One can notice that if vector z is centered, S becomes the covariance matrix of z . To investigate the impacts of changes in the eigenvector p i and the eigenvalue

i , denote the departures of p i and i as p i and i , respectively. Thus p*i and i*


Page 31 of 42

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representing the changed eigenvectors and eigenvalues can be expressed as p*i  pi  p

(32)

i*  i  

(33)

1. Directional changes in the ith eigenvector Assume that the direction of p i changes but i remains unchanged. According to Eq. (31), rewrite Eq. (15) as follows:

2Spi  2i Cpi = 0

(34)

A change in the local structure caused by the directional changes of the eigenvectors produces a different S , S*p , thus Eq. (34) becomes:

2S*ppi  2i Cpi = ei  0

(35)

which implies that Ep* (g i ) departures from zero, i.e. Ep* (g i )  2(S  i I m )pi  0 . Hence, i

i

changes in the directions of the eigenvectors can be detected. Next, we examine whether i can also reflect such directional changes. Let p1* ,p*2 , denote the eigenvectors obtained from S*p , then:

2S*pp*i  2i Cp*i = 0

(36)

By subtracting Eq. (35) from Eq. (36), we have:

2S*p p  2i Cp = ei

(37)

Finally, by pre-multiplying Eq. (37) by pTi , we further get:

2(pTi S*p  2i pTi C)p  pTi ei   i  0


(38)

,p*m


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Page 32 of 42

which implies that Ep* ( hi ) departures from zero, i.e. Ep* ( hi )  0 when the direction of p i i

i

changes. From the above, the primary residuals, G and h , are both sufficient to detect the directional changes of the eigenvectors. 2. Changes in the ith eigenvalue Under the assumption that the eigenvalues change but the directions of the eigenvectors remain unchanged, Eq. (34) becomes: 2S* pi  2iCpi  fi  0

(39)

where S* denotes the changed matrix S produced by the changes of the eigenvalues. This implies that E * (g i ) departures from zero, i.e. E * (g i )  fi  0 . Hence, changes of the i

i

eigenvalues can be detected by G . Let 1* , 2* ,

, m* denote the eigenvalues obtained from S* , then

2S* pi  2i*Cpi  0

(40)

Subtracting Eq. (39) from Eq. (40) gives rise to:

2Cpi  fi  0

(41)

Finally, pre-multiplying Eq. (41) by pTi produces: 2  pTi fi   i  0

(42)

which implies that E * ( hi ) departures from zero, i.e. E * (hi )  0 . Hence, changes of i

i

eigenvalues can be detected by h .


Page 33 of 42

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From the above, the primary residuals, G and h , are both sufficient to detect the changes of the eigenvalues.

□

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Spatial-Statistical Local Approach for Improved Manifold-Based

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