Comprehensive Subspace Decomposition with Analysis of Between

Article pubs.acs.org/IECR

Comprehensive Subspace Decomposition with Analysis of BetweenMode Relative Changes for Multimode Process Monitoring Chunhui Zhao,*,†,‡ Wei Wang,† Yan Qin,† and Furong Gao†,§ †

State Key Laboratory of Industrial Control Technology, Department of Control Science and Engineering, Zhejiang University, Hangzhou, 310027, China ‡ Key Laboratory of Advanced Control and Optimization for Chemical Processes, Shanghai, 200237, China § Department of Chemical and Biomolecular Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong SAR ABSTRACT: Multimode processes may be operating under different statuses, revealing different process characteristics and variations. It can be an interesting issue to check how the process variations change from one mode to another and thus how they can be used for multimode analysis and monitoring. In the present work, a between-mode relative analysis algorithm is proposed to explore the between-mode relationship. Comprehensive subspace decomposition is performed in each mode regarding their relative changes from the reference mode and influences on process monitoring. Each mode is separated into three different systematic subspaces and one residual subspace based on the between-mode changes of process variations, including the increased part, the decreased part, and the unchanged part. These different variations are then modeled respectively and different monitoring statistics are defined for online monitoring to identify mode affiliation and detect the fault status. By separating and monitoring different types of between-mode relative variations, the proposed method can not only efficiently distinguish different modes and detect faults but also offer enhanced process understanding. Theoretical support is framed and the related statistical characteristics are analyzed. Its feasibility and performance are illustrated with multimode data generated from the Tennessee Eastman (TE) benchmark process. modeling methods18−20 develop different local models to match the specific characteristics of each operation mode. The mode-specific separate modeling methods may be effective in modeling the general variations within each mode where they decompose the general distribution information in each process mode based on the size of distribution variance. That is, they focus on separately describing the variations within each isolated mode for process monitoring. However, they rarely analyze the between-mode relationship in which process variations may have interesting changes from one mode to another, and correspondingly the monitoring statistics will respond to these changes. The statistical analysis, modeling, and real-time monitoring of multimode processes is a challenging and yet interesting issue, which has drawn increasing attention. For continuous processes, the multimode problem has been widely addressed and reported regarding fault detection and diagnosis. Yu et al.21 have used Bayesian inference-based method to determine different operations modes for multimode process monitoring. Also, Yu22 proposed a novel Bayesian inference-based Gaussian mixture contribution method to isolate the faulty variables in chemical processes with multiple operating modes. In his work, multiple modes are identified corresponding to different operating conditions via iterative estimation before model development. Instead of separation of multiple modes before model development, some

1. INTRODUCTION Driven by urgent demands of maintaining process safety and improving product quality, process monitoring has become an important issue that has drawn increasing attention in modern industrial processes, including fault detection and diagnosis. Data-driven methods1−11 have been widely developed with a vast amount of measurement data collected from various sensors. Multivariate statistical analyses, such as principal component analysis (PCA)12,13 and partial least-squares (PLS),14,15 have been widely used for process analysis, monitoring, and fault diagnosis. These techniques analyze the underlying correlations of measurement variables and define the normal operation regions by accommodating the acceptable variations. When the process moves outside the desired operating region, it is concluded that an “unusual and faulty” change has occurred. Therefore, the quality of monitoring models and confidence regions are important to the accuracy of online monitoring. It is thus important to obtain monitoring models that can well describe the process status and close confidence regions that can only enclose the normal process behaviors. In practice, industrials processes frequently go through multiple modes due to various factors, such as alterations of feedstocks and compositions, different manufacturing strategies, fluctuations in the external environment, and various product specifications. Different operation modes may have different process characteristics which cannot be well accommodated by a single statistical model. Instead of a global modeling method16,17 in which a uniform model is developed to accommodate all operation modes, mode-specific separate © 2015 American Chemical Society

Received: Revised: Accepted: Published: 3154

November 5, 2014 March 3, 2015 March 11, 2015 March 11, 2015 DOI: 10.1021/ie504380c Ind. Eng. Chem. Res. 2015, 54, 3154−3166

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Industrial & Engineering Chemistry Research

2. METHODOLOGY In the present work, the focus is not to distinguish multiple modes before model development. Instead, it assumes that multiple modes have been predefined with different data sets from different modes for model development. To highlight the difference of the proposed algorithm to describe each mode, it will be compared with the mode-specific separate modeling method which develops monitoring models and supervises the changes of process variations in each mode by describing each specific mode separately. As mentioned before, the betweenmode relationship can be useful to understand multiple modes and model development, which will be explored by the proposed relative analysis algorithm. 2.1. Analysis of Between-Mode Relative Changes. For multimode processes, between-mode analysis can reveal more interesting information. The general idea is to analyze the relative changes from one mode to another and thus decompose these different kinds of variations for monitoring. However, one important question is what kinds of relative changes should be focused on in between-mode analysis for the purpose of online monitoring? To solve this problem, a comprehensive subspace decomposition and feature extraction algorithm is proposed to analyze the differences between two modes from the perspective of process monitoring. It is used to decompose the relative changes and their effects on monitoring results on the basis of PCA monitoring system. One reference mode is chosen at random and the others are alternative modes which will be related to the reference one to reveal the relative changes regarding their different influences on monitoring results. In the context of a conventional PCA-based monitoring system, the relative changes from the reference mode to the alternative mode can be quantitatively evaluated along monitoring directions. Here, three types are considered, including the increased part, the decreased part, and the unchanged part. For each alternative mode, four subspaces are decomposed based on how the variations change in comparison with the reference mode. It is noted that although the proposed algorithm is presented based on the PCA monitoring method, it can be readily generalized to other statistical analysis methods. It is noted that a relative analysis algorithm2 has been proposed for reconstruction-based fault diagnosis in which only the increased part was extracted to develop the reconstruction model for disturbance removal since only the increased variations have caused out-of-control monitoring statistics. In contrast, the objective here is to distinguish two modes and thus to monitor each mode more comprehensively. Therefore, the betweenmode relative analysis algorithm in the present work is different from our previous work.2 First, two mode-representative data sets are prepared for the reference mode and the alternative mode, Xr (Nr × J) and Xa (Na × J) (where subscript r means the reference mode and a denotes the alternative mode, Nr and Na are the number of samples in the reference mode and the alternative mode, respectively), each being composed of the same number of variables and maybe different number of samples. Independently, the data in Xr and Xa have been centered and scaled to be of zero mean and unit standard deviation. Then for each mode, they have different normalization information. For simplicity, the normalized data in each mode are still described by Xr and Xa, respectively. The specific modeling procedure is described as follows.

research work focused on the mode understanding and changes of process variations from one mode to another. Adaptive modeling methods23,24 have been developed to adjust monitoring models following the changes of multiple modes. However, they have to clearly distinguish the normal process changes and the abnormality in order to avoid false model updating. Besides, they only consider the difference among different modes. Instead, Zhao et al. proposed that there is a certain similarity among multiple modes despite of the changes from one mode to another. They proposed a two-step multiset analysis algorithm,25 termed MsPCA, to relate the inherent variable correlations over multiple data spaces. Some common bases which are supposed to represent the similarity of variable correlations among different modes are identified and extracted as a linear combination of the original observations. These common bases which enclose a common subspace are then used as monitoring directions for multimode monitoring.26 Zhang et al.27 then extended the idea of common subspace to a nonlinear version for process monitoring. However, the above methods pay more attention to the similarity of variable correlations without considering the variations. Sometimes, even the variable correlations are quite similar to each other, the associated variations may be different, resulting in different values of monitoring statistics. For batch processes, the multimode problem also has been reported,28 which considers multiphase and multimode statuses at the same time. It is noted that the concept of “mode” is different from “phase” in a batch cycle in the present work. For multimode processes, some interesting questions may arise: how the process variations change from one mode to another and how they influence the development of monitoring models? From the monitoring perspective, the between-mode changes of variations along different monitoring directions have not been well explored yet. These questions are very meaningful to be discussed and addressed, which may provide useful information for multimode modeling and process monitoring. In the present work, inspired by the above recognition, multimode processes are analyzed and modeled for online monitoring. The between-mode relationship is explored by analyzing the relative changes of process variations between two different modes for the development of monitoring models. One mode is chosen as the reference mode and the others are the alternative modes. The relative changes from one mode to another are given a comprehensive analysis and decomposition, which are reflected by three different major parts, the increased variations, decreased variations, and the similar variations. It is based on the fact that different relative variations in each alternative mode in comparison with those in the reference mode have different influences on the development of monitoring models. Therefore, for each alternative mode, the measurement space is decomposed into four subspaces on the basis of how the variations of each alternative mode change in comparison with those of the reference mode. The proposed method can efficiently distinguish different types of process variations between different modes and effectively model them for online process monitoring. It is noted that the relative changes are analyzed based on two steady modes and the problem of between-mode transition is not analyzed which needs further efforts and investigation since some important issues should be in particular addressed regarding the transition characteristics, its duration, and so on. The performance of the proposed algorithm is illustrated by a typical multimode continuous process. 3155

DOI: 10.1021/ie504380c Ind. Eng. Chem. Res. 2015, 54, 3154−3166

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Industrial & Engineering Chemistry Research Step (1). PCA Modeling on the Reference Mode. Perform PCA algorithm on the reference data set Xr to get the monitoring models,

Correspondingly, the variations associated with the concerned directions are larger in the alternative mode in comparison with those in the reference mode. The number of the retained principal directions is Rr,l and Rer,l, respectively. Subscript l denotes larger. (c) Find the monitoring directions corresponding to Ratioi smaller than TCtrL and Δi smaller than ECtrL, composing e e (J × Rr,s ), respectively. Pr,s (J × Rr,s ) and Pr,s Correspondingly, the variations associated with the related directions are smaller in the alternative mode in comparison with those in the reference mode. The number of the retained principal directions is Rr,s and Rer,s, respectively. Subscript s denotes smaller. The between-mode variations are separated into three parts by considering the values of the two indices, which are explained as follows: (i) If the Ratioi value is equal to 1 or Δi is equal to zero, it means that the corresponding variation in the alternative mode is similar to that in the reference mode along the concerned monitoring directions. That is, the two modes present similar variations and can share the same monitoring models. (ii) If the Ratioi value is significantly larger than 1 or Δi is significantly larger than zero, it means that the corresponding variation in the alternative mode is larger than that in the reference mode along the concerned monitoring directions. (iii) If the Ratioi value is significantly smaller than 1 or Δi is smaller than zero, it means that the corresponding variation in the alternative mode is smaller than that in the reference mode along the concerned monitoring directions. Step (4). Reconstruction of Between-Mode Changes. Pr,l, Pr,n and Pr,s represent three different subspaces separated from PCS, and Per,l, Per,n and Per,s represent three different subspaces separated from RS. Also it is noted that the orthogonality of directions in each model is retained. Then the directions extracted from PCS and RS can be integrated into a combined model which reveals different between-mode relative changes in the alternative mode:

Tr = X rPr Er = X r(I − Pr Pr T) = X rPer PeT r

(1)

where Tr (Nr × Rr) and Pr (J × Rr) are principal components (PCs) and the corresponding principal loadings in PCS. Rr is the number of retained PCs determined by cumulative explained variance rate(CEVR).12 Pr in fact reveal the first directions with the largest variation of reference data set which are also the monitoring directions. Er (Nr × J) are PCA residuals and Per (J × Rer ) are the corresponding monitoring directions in RS which are directly associated with the remaining J − Rr PCs. Rer is the number of retained monitoring directions in RS, Rer = J − Rr. Step (2). Using the Reference Model to Check the Alternative Mode. Project Xa onto Pr (J × Rr) to get PCs and onto Per (J × Rer ) to get residuals: Ta = X aPr Ea =

X aPer PeT r

R re

=

∑ X aper,iper,i T

(2)

i=1

Compare the variation between the alternative data set and the reference data set along different monitoring directions as Ratioi =

var(Ta, i)

(i = 1, 2, ..., R r)

var(Tr, i) 2

2

Δi = || X aper, iper, i T || − || X rpr,ei per, i T ||

(3)

(i = 1, 2, ..., R re) (4)

where var(•) denotes the PC variance around the center obtained from the reference normal case (zero here) and subscript i denotes the ith monitoring direction or PC. So Ratio is a Rr-dimensional vector composed of Ratioi. The double lines, ∥ ∥, denote the Euclidean length. So Δ is a Rer -dimensional vector composed of Δi. The calculation paths of the two indices (Ratioi and Δi) are different which may result from the fact that the two monitoring statistics (T2 and SPE) are calculated in a different way. Step (3). Separation of the between-Mode Variations. Sort the values of Ratio index and Δ index, respectively. Confine a confidence region for Ratio index by defining an upper limit TCtrU and a lower limit TCtrL. Also, confine a confidence region for Δ index by defining an upper limit ECtrU and a lower limit ECtrL. A simple discussion of defining the concerned limits can be seen in the illustration section of this paper. (a) Keep the directions with Ratioi and Δi values well within the predefined confidence regions. The concerned monitoring directions are Pr,n (J × Rr,n) in the PCS and Per,n (J × Rer,n) in the RS. Correspondingly, the variations associated with the concerned directions are similar between the alternative mode and the reference mode. The number of the retained principal directions is Rr,n and Rer,n in PCS and RS, respectively. Subscript n denotes normal. (b) Find the monitoring directions corresponding to Ratioi larger than TCtrU and Δi larger than ECtrU, composing P r,l (J × R r,l ) and Pr,le (J × Rr,le ), respectively.

Π r,l(J × Jr,l ) = [Pr,l , Per,l ] Π r,s(J × Jr,s ) = [Pr,s , Per,s] Π r,n(J × Jr,n ) = [Pr,n , Per,n]

(5)

where Πr,l, Πr,s and Πr,n are orthonormal matrices where the directions are orthogonal with each other. In this way, the directions with increased variations, decreased variations, and unchanged variations in the alternative mode are integrated into a combined model, respectively. Jr,l = Rr,l + Rer,l, Jr,s = Rr,s + Rer,s and Jr,n = Rr,n + Rer,n. It is clear that Rr = Rr,n + Rr,l + Rr,s and Rer = Rer,n + Rer,l + Rer,s. Therefore, J = Jr,n + Jr,l + Jr,s, resulting from J = Rr + Rer . Πr,l reveals the monitoring directions chosen from the PCA model of the reference mode along which the associated variations in the alternative mode increase in comparison with those in the alternative mode. Πr,s reveals the monitoring directions chosen from the PCA model of the reference mode along which the associated variations in the alternative mode decrease in comparison with those in the reference mode. Πr,n reveals the monitoring directions chosen from the PCA model of the reference mode along which the 3156


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monitoring system is obtained, and for the alternative mode, a four-model monitoring system is available. The models associated with different between-mode changes are summarized in Table 1 regarding their different statistical meanings.

associated variations in the alternative mode stay similar to those in the reference mode. The process variations along Πr,l, Πr,s, and Πr,n in the alternative mode are then calculated respectively by ⎧ X̂ = X Π Π T a r,l r,l ⎪ a,l ⎪ ⎨ X̂ a,s = X aΠ r,sΠ r,s T ⎪ ⎪ X̂ = X Π Π T a r,n r,n ⎩ a,n

Table 1. Subspace Decomposition Results for Each Alternative Mode by the Proposed Method in Comparison with the Mode-Specific Separate Modeling Method for Multimode Processes (6)

method

In this way, the systematic variations that are possibly responsible to the between-mode changes are separated from those unchanged ones where the increased and decreased parts are also distinguished from each other. Step (5). PCA Modeling on Different between-Mode Variations. Although the directions in Πr,l, Πr,s and Πr,n, respectively, are orthogonal with each other, they are not ordered based on decreased variance in the alternative mode. PCA is then performed on X̂ a,l, X̂ a,s and X̂ a,n to explain their major systematic variations in order for each kind of betweenmode relative changes, respectively: ⎧ X̂ = T P T + E a,l a,l a,l ⎪ a,l ⎪ T ⎨ X̂ a,s = Ta,sPa,s + Ea,s ⎪ ⎪ X̂ = T P T + E a,n a,n a,n ⎩ a,n

proposed

subspace

statistical meanings

Pa,l

increasing systematic subspace that covers variations that are larger than those in the reference mode invariable systematic subspace that covers variations that stay similar to those in the reference mode decreasing systematic subspace that covers variations that are smaller than those in the reference mode residual subspace that covers residuals after the explanation of systematic models systematic subspace for calculation of monitoring statistic T2 residual subspace for calculation of monitoring statistic SPE

Pa,n Pa,s Ea mode-specific separate modeling

P E

Also, the subspace separation for different modes is illustrated in Figure 1. The process variations in each mode are (7)

where Ra,l, Ra,s, and Ra,n denote the number of PCs kept in Pa,l, Pa,s and Pa,n. They are determined by CEVR to keep most of the data variability. Ea,l, Ea,s, and Ea,n are the residuals in X̂ a,l, X̂ a,s and X̂ a,n after the explanation of Pa,l, Pa,s and Pa,n, respectively. In general, 100% CEVR can be used to calculate Pa,l where Ea,l thus converges to zeros since X̂ a,l cover increased variations which may not be residuals. A smaller CEVR value can be used to calculate Pa,n since X̂ a,n may also cover some residuals if they are similar between the two modes. A much smaller CEVR can be used for Pa,s since the decreased variations X̂ a,s may cover more residuals. In this way, different variations in the alternative mode relative to the reference mode are separated from each other and integrated orderly by PCA. Three systematic PCA models (Pa,l, Pa,n, and Pa,s) are developed to reveal the betweenmode relative changes. Then different monitoring statistics will be calculated to check the specific process variations. By the above five steps, the between-mode relationship is explored and the relative changes are separated into three systematic parts regarding their influences on development of monitoring models. It is noted that on the basis of different between-mode relationships, the four-model monitoring system can converge to different results, which can be regarded as extreme cases of the four-model monitoring system. For example, if the alternative mode only has a little related properties with the reference mode, the subspace covered by Pa,n may disappear and the four subspaces may converge to three subspaces (Pa,l,Pa,s, and the final residual subspace). 2.2. Between-Mode Model Development. For the mode-specific separate modeling method, without considering the between-mode relationship, each mode can be described by a general two-model monitoring system by the conventional PCA algorithm. Instead, based on the analysis of between-mode relative changes and comprehensive subspace decomposition, different process variations can be extracted and distinguished from each other. For the reference mode, a two-model

Figure 1. A schematic of between-mode relative analysis-based subspace decomposition.

decomposed into different parts and modeled by different models for process monitoring. In this way, the between-mode relationship is understood and made good use of for process monitoring to check how process variations change from one mode to another. For the reference mode, the two-model monitoring system is developed as follows: Tr = X rPr Er = X rPr,ePr,e T 3157

(8) DOI: 10.1021/ie504380c Ind. Eng. Chem. Res. 2015, 54, 3154−3166

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Figure 2. Flowchart of online mode identification and fault detection using the proposed method.

For the alternative mode, the monitoring statistics can be calculated as

where Tr are the systematic scores and Er are the residuals corresponding to two different monitoring subspaces spanned by Pr and Pe,r, respectively. For each alternative mode, the systematic scores are calculated as

T −1 Ta,l, k 2 = (ta,l, k − ta,l ̅ ) Σa,l (ta,l, k − ta,l ̅ ) T −1 Ta,s, k 2 = (ta,s, k − ta,s ̅ ) Σa,s (ta,s, k − ta,s ̅ )

Ta,l = X aPa,l

T −1 Ta,n, k 2 = (ta,n, k − ta,n ̅ ) Σa,n (ta,n, k − ta,n ̅ )

Ta,s = X aPa,s Ta,n = X aPa,n

SPEa, k , i = ea, k Tea, k

(9)

where, Ta,l, Ta,s and Ta,n are systematic scores in each alternative mode with increased, decreased, and unchanged variations in PCS in comparison with the reference mode. After the explanation of these systematic models, the final residuals in each alternative mode are thus calculated as Ea = X a − X aPa,l Pa,l T − X aPa,sPa,s T − X aPa,nPa,n T

where ta̅ ,l, ta̅ ,s, and ta̅ ,n denote the mean vectors calculated from Ta,l, Ta,s, and Ta,n, respectively, which are all zero vectors due to the data preprocessing. Σa,l, Σa,s, and Σa,n are diagonal matrices with elements being the variance of each PC in scores Ta,l, Ta,s, and Ta,n, respectively. The meanings of different subspaces are summarized in Table 1 regarding each alternative mode a. Assuming the process variables follow a multivariate normal distribution, the control limit in the systematic subspace for each mode is defined by the F-distribution with α as the significance factor12,29 and in the residual subspace, the representative confidence limit of SPE for each mode can be approximated by a weighted chi-squared distribution.30,31 2.3. Online Monitoring Strategy. Whenever a new observation at the kth time interval, xnew(J × 1), is available, it is first preprocessed using the data normalization information from the reference mode. The normalized new observation is then projected onto the two-model monitoring system developed from training data of the reference mode, and the monitoring statistics are calculated using eqs 8 and 11:

(10)

where Ea is the final residuals after the explanation of three systematic models that are responsible to the between-mode changes, which in fact include the left residuals in Πr,l after the explanation of Pa,l, the left residuals in Πr,s after the explanation of Pa,s and the left residuals in Πr,n after the explanation of Pa,n. For process monitoring, it is interesting to check how variations evolve along time direction in different subspaces. The monitoring statistics are calculated in different monitoring subspaces for the reference mode and the alternative mode at each time k. For the reference mode Tr, k 2 = (t r, k − t r̅ )T Σr −1(t r, k − t r̅ ) SPEr, k = er, k Ter, k

(12)

(11)

where subscript k denotes the kth sampling time. tr̅ denotes the mean vector calculated from Tr which is zero due to the data preprocessing. Σr is a diagonal matrix with elements being the variance of each PC in scores Tr.

t new,r T = x new TPr enew,r T = x new TPr,ePr,e T 3158

(13) DOI: 10.1021/ie504380c Ind. Eng. Chem. Res. 2015, 54, 3154−3166

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of between-mode relative changes. Comprehensive subspaces are decomposed based on between-mode changes. (3) In different subspaces, analyze the underlying characteristics and extract the monitoring models from each subspace. Derive the confidence limits of monitoring statistics in different subspaces for each mode. B. Online Application. (1) For new sampling data, xnew(J × 1), process it with the data normalization information from the reference mode. (2) Adopt the two-model monitoring system and calculate two monitoring statistics, T2 and SPE statistics. Compare them with the confidence limits of the reference mode. (3) If both statistics are observed to be below the confidence limits of the reference mode in both subspaces, the current observation is considered to be behaving normally and belong to this operation mode. Moreover, the mode membership should be consistent for a certain time. (4) Otherwise, the current observation may be a fault or in another normal mode. Repreprocess it using data normalization information from different alternative modes and calculate the corresponding monitoring statistics by adopting different models. (5) If all statistics stay well within the normal regions defined for one alternative mode, then the mode affiliation of the current observation is judged. Otherwise it is some fault, and fault diagnosis can be performed to identify the possible fault cause.

Tnew,r 2 = (t new,r − t r̅ )T Σr −1(t new,r − t r̅ ) SPEnew,r = enew,r Tenew,r

(14)

(T2r,new

Compare the values of two monitoring statistics and SPEr,new) with the predefined control limits for the reference mode in PCS and RS, respectively. If both monitoring statistics stay well within the predefined normal regions, the current sample can be deemed to be operating according to the normal operation rule of the reference mode. Also, considering that the operation mode will not change abruptly and the mode membership should be consistent with its previous one, the mode affiliation can be readily determined by combining the current monitoring result and previous ones. On the contrary, if consistent alarming signals are issued, it reveals that the current process is operating in some fault or the alternative mode status. To distinguish this, further, the four-model monitoring system defined for different alternative modes will be tested in turn using eqs 9, 10, and 12 to see which alternative mode can best match the current operation pattern: t new,a,l T = x new TPa,l t new,a,n T = x new TPa,n t new,a,s T = x new TPa,s T

T

T

T

T

enew,a = x new − x new Pa,l Pa,l − x new Pa,nPa,n − x new TPa,sPa,s T

3. ILLUSTRATIONS AND DISCUSSIONS In this section, the proposed between-mode analysis and monitoring method is applied to the well-known Tennessee Eastman (TE) benchmark chemical process with multiple modes generated by setting different operating conditions. The proposed method is also compared with the mode-specific separate modeling method regarding the process understanding and fault detection performance. 3.1. Process Description. The TE process32 has been widely used for testing various process monitoring and fault diagnosis methods. It contains two blocks of process variables: 41 measured variables and 11 manipulated variables. The details on the process description can be found in ref 32. The 41 measured variables (as shown in Table 2) are used in the present work for modeling and include twenty-two continuous measurement variables which are sampled with an interval of three min and 19 composition variables which are sampled with time delays that vary from six min to 15 min. As mentioned in ref 32, all process measurements include Gaussian noise with standard deviation typical of the measurement type. As a complex chemical process, the TE process provides a good simulation platform to validate the online monitoring performance of the proposed method. The base case from Downs and Vogel32 is used as the reference mode, and three different modes as specified by Ricker33 are considered as the alternative modes, which are summarized in Table 3. Among those operating modes, the product composition can shift between 9:1 and 1:1, while the production rate is either specified at a particular target or maximized. It is noted that the Downs and Vogel base case has a different operation status from alternative mode no. 1 although they have the same desired G/H mass ratio and production (kg/h), which was explained in ref 33. The decentralized regulatory control which was designed by Ricker33 is deployed on the TE process with the continuous measurement variables operated under close-loop conditions. 3.2. Model Development Based on Between-Mode Relative Analysis. Six hundred normal observations from each mode are used for between-mode analysis and monitoring

T

(15)

T −1 Tnew,a,l 2 = (t new,a,l − ta,l ̅ ) Σa,l (t new,l − ta,l ̅ ) T −1 Tnew,a,n 2 = (t new,a,n − ta,n ̅ ) Σa,n (t new,a,n − ta,n ̅ ) T −1 Tnew,a,s 2 = (t new,a,s − ta,s ̅ ) Σa,s (t new,a,s − ta,s ̅ )

SPEnew,a = enew,a Tenew,a

(16)

Compare the values of four monitoring statistics calculated at each time with the predefined control limits in PCS and RS for each alternative mode, respectively. If all monitoring statistics stay well within the predefined normal regions, the current sample can be deemed to be operating according to the normal operation rule of one alternative mode. On the contrary, if consistent alarming signals are issued for all modes, it reveals that the current process is having some fault. Then some fault diagnosis action can be taken in time to check the fault cause. The flowchart for online application to judge between the normal alternative mode and fault status is shown in Figure 2. It is noted that the reference mode is described by the conventional two-model monitoring method while the alternative modes are described by the proposed four-model monitoring system. Therefore, in the present work, the analysis and monitoring of alternative modes will be focused on in the illustration section. 2.4. Outline of the Proposed Strategy. The betweenmode analysis-based multimode modeling and monitoring method can be summarized as follows: A. Model Development. (1) Collect the measurement data sets for some predefined modes to get a multimode library where the mode affiliation has been well-known. Normalize each mode separately to make the normalized data have zero mean and unit variance, respectively. (2) Choose one reference mode and associate it with each of the other modes for analysis 3159


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process variability. Then each alternative mode is associated with the reference mode to decompose the between-mode relative changes. Here, 100% CEVR is used to calculate Pa,l from X̂ a,l and 95% CEVR is used to calculate Pa,n from X̂ a,n. 90% CEVR is used to calculate Pa,s from X̂ a,s since this part covers smaller variations than those in the reference mode. The modeling results are summarized in Table 4 using the proposed method in comparison with the mode-specific separate modeling method. The dimensions of different monitoring subspaces decomposed using the two methods are presented with the percent of data variations covered in each subspace. For the conventional mode-specific separate modeling method, only two subspaces are decomposed in which 36 directions can account for more than 97% of the original process variations. The proposed method can decompose three systematic subspaces and one residual subspace for process monitoring to check the changes of process status in each alternative mode. Here, 1.3 and 0.3 are set as the upper limits of Ratioi and Δi indices, respectively, while 0.7 and −0.3 are set as the lower limits of the two indices, respectively. Of course, the limits can be adjusted for the two indices. In general, to separate more directions of Pa,l, a smaller upper limit can be set for Ratioi and Δi indices, respectively. To separate more directions of Pa,s, a larger lower limit can be set for two indices, respectively. The parameters of upper/lower limits reflect the compromise between mode similarity and dissimilarity. In general, their values can be set by trial and error so that each alternative model can be better separated from the reference mode. Unfortunately, there is no definite criterion or uniform standard to strictly quantify them. Therefore, their determination is inevitably affected more or less by artificial subjectivity factors. With the same modeling parameters, it is found that Pa,l can cover 30% of the original process variations in each alternative mode, Pa,n can cover about 60%, and Pa,n covers 10%. This means that each of the alternative modes share a similar ratio of variations with the reference mode as indicated by Dim and Var % indices in each subspace. However, it does not mean that these three alternative modes are operating with similar process characteristics. This can be seen from the comparison of model coefficients shown in Figure 3 where monitoring models have been developed on the basis of an analysis of between-mode relative changes. The model coefficients along the first direction in each systematic subspace are quite different from each other for the three alternative modes as shown in Figure 3 for the proposed method. Also the model coefficients along the first direction of PCA systematic subspace are plotted in Figure 4 for three alternative modes using the mode-specific separate modeling method. Clearly, based on relative analysis, the significant variables that are responsible for the between-mode relative changes are revealed, which can provide more interesting information in comparison with the mode-specific

Table 2. Process Continuous Measurement Variables in the TE Process Used for Modeling variable no. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

continuous measurement variables A feed (stream 1) D feed (stream 2) E feed (stream 3) A and C feed (stream 4) recycle flow (stream 4) reactor feed rate (stream 6) reactor pressure reactor level Reactor temperature purge rate (stream 9) product separator temperature product separator level product separator pressure product separator underflow (stream 10) stripper level stripper pressure stripper underflow (stream 11) stripper temperature stripper steam flow compressor work reactor cooling water outlet temperature separator cooling water outlet temperature

variable no.

composition variables

1 2 3 4 5 6 7 8 9 10 11 12 13 14

A (stream 6) B (stream 6) C (stream 6) D (stream 6) E (stream 6) F (stream 6) A (stream 9) B (stream 9) C (stream 9) D (stream 9) E (stream 9) F (stream 9) G (stream 9) H (stream 9)

15 16 17 18 19

D (stream 11) E (stream 11) F (stream 11) G (stream 11) H (stream 11)

Table 3. Reference Mode and Three Different Alternative Modes of TE Process Used in the Present Work mode no.

desired G/H mass ratio

desired production (kg/h)

reference mode alternative mode no. 1 alternative mode no. 2 alternative mode no. 3

50/50 50/50 90/10 50/50

14076 14076 11111 maximum

model development. All variables in the normal data space are mean-centered and scaled to unit variance for each mode. Starting from the reference mode, the between-mode relative changes are analyzed to decompose the variations in the process space of each alternative mode. Different monitoring models are then developed for each mode. A two-model monitoring system is developed for the reference mode while a four-model monitoring system is obtained for each alternative mode. First, PCA monitoring models are developed for the reference mode, including a 28-dimensional systematic subspace and a 21-dimensional residual subspace. The number of PCs is determined by CEVR to keep 85% of the original

Table 4. Comparison of Fault Subspace Decomposition Results (Dimension of Each Subspace (Dim) and the Percent of Data Variation (Var%) Explained in Each Subspace) mode-specific separate modeling method subspace systematic, P

residual, E

proposed method subspace systematic, Pa,l

systematic, Pa,n

systematic, Pa,s

residual, Ea

alternative mode no.

Dim

Var%

Dim

Var%

Dim

Var%

Dim

Var%

Dim

Var%

Dim

Var%

1 2 3

36 36 34

97.25 97.17 97.05

6 6 8

2.75 2.83 2.95

16 14 15

30.40 27.42 26.12

21 22 20

56.71 60.88 60.96

3 3 4

11.28 8.59 10.06

1 2 3

1.61 3.11 2.85

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Figure 4. Model coefficients along the first direction of one systematic subspace (P) using the mode-specific separate modeling method for three different alternative modes.

the reference mode to alternative mode no. 1. In contrast, the conventional mode-specific separate modeling method can only reveal that three modeling variables (variable nos. 7, 13, and 16) are significant to cover the largest general variations along the first distribution direction in this alternative mode. The similar scenarios can also be observed for the other alternative modes regarding the two methods where the significant variables, however, are different. This reveals that different variables are responsible for different types of relative changes from the reference mode to each alternative mode. 3.3. Online Process Monitoring. On the basis of the developed monitoring models, online process monitoring is performed to check the changes of different types of process variations. For normal data in one alternative mode, it is hoped that only the monitoring models developed from the same mode can well accommodate them with no alarm signals in order to correctly identify the mode affiliation. First, for the normal data in alternative mode no. 1, the online monitoring results using models developed from the same mode are presented in Figure 5 by comparing the proposed method and the mode-specific separate modeling method. In general, both methods show that all the monitoring statistics stay within the normal region although the conventional method gives more false alarms in the SPE monitoring chart. In contrast, using the monitoring models developed from the other alternative modes, the monitoring statistics cannot stay below the confidence limit as shown in Figures 6 and 7 using two methods, respectively. In this way, the mode affiliation of the current process is correctly identified for both methods since

Figure 3. Model coefficients along the first directions of three different systematic subspaces (Pa,l, Pa,n, and Pa,s) using the proposed method for (a) alternative mode no. 1, (b) alternative mode no. 2, and (c) alternative mode no. 3.

conventional modeling method in which each mode was described separately. For example, the 22 modeling variable is significantly responsible for the increased relative changes from 3161


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Figure 5. Online monitoring results for normal process data in alternative mode no. 1 using models developed from alternative mode no. 1 by (a) the proposed method and (b) the mode-specific separate modeling method (dotted line, monitoring statistics; red dashed line, 99% monitoring confidence limit).

developed from alternative mode no. 1, T2a,l and T2a,n monitoring statistics can quickly issue alarms as a response to the disturbance. Using the models developed from the other two alternative modes, significant alarms are presented even during the normal process region (the first 100 samples) since the adopted monitoring models do not match the current process mode. Clearly, none of these models can accommodate the current process fault which is thus correctly detected. For fault no. 3 in different alternative modes, the models developed from the same alternative mode are used to check the process status, respectively. The results are shown in Figure 9 for the proposed

only the models developed from alternative mode no. 1 can correctly indicate the normal status. Four different faults are considered and imposed in each alternative mode for fault detection, which are described in Table 5. All these disturbances are imposed from the 100th sample in each alternative mode. Here only random variations and slow-drift problems are considered, which represent slower and smaller disturbances in comparison with step changes. For Fault no. 1 that happens in alternative mode no. 1, the online monitoring results using models developed from different alternative modes are presented in Figure 8. If models are 3162


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Figure 7. Online monitoring results for normal process data in alternative mode no. 1 using models developed from (a) alternative mode no. 2 and (b) alternative mode no. 3 by the mode-specific separate modeling method (dotted line, monitoring statistics; red dashed line, 99% monitoring confidence limit).

Figure 6. Online monitoring results for normal process data in alternative mode no. 1 using models developed from (a) alternative mode no. 2 and (b) alternative mode no. 3 by the proposed method (dotted line, monitoring statistics; red dashed line, 99% monitoring confidence limit).

Table 5. Three Faults Happening in Different Alternative Modes of TE Process

method where all abnormalities are detected in each alternative mode. Also it is noted that T2a,l monitoring statistic in the Pa,l subspace (i.e., the increased part) and T2a,n monitoring statistic in the Pa,n subspace (i.e., the invariable part) are more sensitive to detect this fault. For all faults, the fault detection performance is evaluated by time delay which is the time difference between the detection time when the fault is first observed and the real time when the fault happens in each alternative mode. From the results shown in Table 6, the comparison between the two methods reveals similar detection time delay for each fault in each alternative mode. For the mode-specific separate modeling method, the SPE monitoring statistic is more sensitive to fault detection which agrees well with the common recognition. Instead, since the proposed method analyzes the between-mode relative changes and performs a comprehensive subspace decomposition, the sensitivity of T2 has been clearly improved. Also, it is

fault no.

process variable

type

1 2 3

A, B, C feed composition (stream 4) D feed temperature (stream 2) reactor cooling water inlet temperature

random variation random variation random variation

noted that the Pa,l monitoring model is more useful for the timely detection of a fault. It is noted that the results shown here used the base case from Downs and Vogel32 as the reference mode. To reveal the effects of the choice of reference mode, the other modes can be chosen as a new reference mode and then between-mode analysis is performed again. With a different mode as the reference mode, similar conclusions are drawn although the results show the differences to a certain extent. In general, from the results it is hard to say which mode is better to serve as a reference mode. 3163


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Figure 8. Online monitoring results for fault no. 1 in alternative mode no. 1 using models developed from (a) alternative mode no. 1 (b) alternative mode no. 2 and (c) alternative mode no. 3 by the proposed method (dotted line, monitoring statistics; red dashed line, 99% monitoring confidence limit).

Figure 9. Online monitoring results for fault no. 3 in (a) alternative mode no. 1, (b) alternative mode no. 2, and (c) alternative mode no. 3, respectively, using models developed from the same alternative mode by the proposed method (dotted line, monitoring statistics; red dashed line, 99% monitoring confidence limit). 3164


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Table 6. Comparison of Online Fault Detection Performance (Time Delay (Samples)) for Three Faults Using the Proposed Method and the Mode-Specific Separate Modeling Method alternative mode no. 1

a

fault no. 1 2 3 4

proposed method

mode-specific separate modeling

31 (Pa,l and Pa,s)a 61 (Pa,l) 5 (Pa,l and Pa,n) 45 (Pa,s)

31 (E) 68 (E) 5 (P and E) 46 (E)

alternative mode no. 2

alternative mode no. 3

proposed method


proposed method


31 (Pa,l and Pa,s) 61 (Pa,l) 5 (Pa,l and Pa,n) 50 (Pa,s)

31 (E) 67 (E) 5 (P and E) 53 (P)

11 (Pa,l) 34 (Pa,l) 1 (Pa,l and Pa,n) 5 (Pa,l)

12 (E) 34 (E) 2 (P and E) 5 (E)

The value in the bracket indicates the monitoring subspace where the fault can be detected with the smallest time delay. (3) Subrahmanya, N.; Shin, Y. C. A Data-Based Framework for Fault Detection and Diagnostics of Non-linear Systems with Partial State Measurement. Eng. Appl. Artif. Intell. 2013, 26, 446−455. (4) He, B.; Zhang, J.; Chen, T.; Yang, X. H. Penalized Reconstruction-Based Multivariate Contribution Analysis for Fault Isolation. Ind. Eng. Chem. Res. 2013, 52, 7784−7794. (5) Van. Den. Kerkhof, P.; Vanlaer, J.; Gins, G.; Impe, J. F. M. V. Analysis of Smearing-out in Contribution Plot Based Fault Isolation for Statistical Process Control. Chem. Eng. Sci. 2013, 104, 285−293. (6) Bin, Z.; Sconyers, C.; Byington, C.; Patrick, R.; Orchard, M. E.; Vachtsevanos, G. A Probabilistic Fault Detection Approach: Application to Bearing Fault Detection. IEEE Trans. Ind. Electron. 2011, 58, 2011−2018. (7) Zhao, C. H.; Lu, N. Y. Statistical Monitoring and Quality Analysis for Batch Processes; Science Press: Beijing, 2014. (8) Kruger, U.; Kumar, S.; Littler, T. Improved Principal Component Monitoring Using the Local Approach. Automatica 2007, 43, 1532− 1542. (9) Muradore, R.; Fiorini, P. A PLS-Based Statistical Approach for Fault Detection and Isolation of Robotic Manipulators. IEEE Trans. Ind. Electron. 2012, 59, 3167−3175. (10) Liu, J. L.; Chen, D.-S. Fault Isolation Using Modified Contribution Plots. Comput. Chem. Eng. 2014, 61, 9−19. (11) Raich, A.; Cinar, A. Statistical Process Monitoring and Disturbance Diagnosis in Multivariable Continuous Processes. AIChE J. 1996, 42, 995−1009. (12) Jackson, J. E. A User’s Guide for Principal Components; Wiley: New York, 2005. (13) Wold, S.; Esbensen, K.; Geladi, P. Principal Component Analysis. Chemom. Intell. Lab. Syst. 1987, 2, 37−52. (14) Burnham, A. J.; Viveros, R.; MacGregor, J. F. Frameworks for Latent Variable Multivariate Regression. J. Chem. 1996, 10, 31−45. (15) Dayal, B. S.; MacGregor, J. F. Improved PLS Algorithms. J. Chem. 1997, 11, 73−85. (16) Hwang, D. H.; Han, C. H. Real-Time Monitoring for a Process with Multiple Operation Modes. Control Eng. Pract. 1999, 7, 891−902. (17) Lane, S.; Martin, E. B.; Kooijmans, R.; Morris, A. J. Performance Monitoring of a Multi-product Semi-batch Process. J. Process Control 2011, 11, 1−11. (18) Yoo, C. K.; Villez, K.; Lee, I.; Rosen, C.; Vanrolleghem, P. A. Multi-model Statistical Process Monitoring and Diagnosis of a Sequencing Batch Reactor. Biotechnol. Bioeng. 2007, 96, 687−701. (19) Zhao, S. J.; Zhang, J.; Xu, Y. M. Monitoring of Processes with Multiple Operating Modes through Multiple Principle Component Analysis Models. Ind. Eng. Chem. Res. 2004, 43, 7025−7035. (20) Zhao, S. J.; Zhang, J.; Xu, Y. M. Performance Monitoring of Processes with Multiple Operating Modes through Multiple PLS Models. J. Process Control 2006, 16, 763−772. (21) Yu, J.; Qin, S. J. Multimode Process Monitoring with Bayesian Inference-Based Finite Gaussian Mixture Models. AIChE J. 2008, 54, 1811−1829. (22) Yu, J. A New Fault Diagnosis Method of Multimode Processes Using Bayesian Inference Based Gaussian Mixture Contribution Decomposition. Eng. Appl. Artif. Intel. 2013, 26, 456−466.

4. CONCLUSIONS The between-mode relationship is decomposed from the perspective of process monitoring by analysis of relative changes. Instead of a separate model development for each mode to describe the general data distribution information, the process variations are related with the reference mode and decomposed into different parts, reflecting different types of between-mode changes. This allows different monitoring models built in different subspaces to supervise different types of process variations. Also, the between-mode relative changes can be better understood. The case study on TE process has demonstrated the performance of the proposed multimode analysis and monitoring method. By decomposing different monitoring subspaces based on between-mode analysis, each mode can be clearly looked into. Moreover, the enhanced process understanding can be obtained since more sufficient process information is obtained. Also meaningful future work is deserved, which can focus on how to better utilize the between-mode relative changes for fault diagnosis and prognosis. It is noted that there are also other alternative methods to analyze between-mode relative changes, such as the Fisher discriminant analysis algorithm (FDA).34,35 The FDA algorithm can analyze the between-mode changes but from a different perspective. Its potential application to analyze multimode processes can be an interesting issue to be addressed in the future.

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AUTHOR INFORMATION

Corresponding Author

*Tel: 86-571-87951879. Fax: 86-571-87951879. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work is supported by the Program for the National Natural Science Foundation of China (No. 61422306, No. 61433005, and No. 61273166), the Zhejiang Provincial Natural Science Foundation of China (LR13F030001), the Fundamental Research Funds for the Central Universities, and New Century Excellent Talents in University (NCET-12-0492).

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REFERENCES

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