Novel Monitoring Strategy Combining the Advantages of the Multiple

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A novel monitoring strategy combining the advantages of multiple modeling strategy and GMM for multi-mode processes Shumei Zhang, Fuli Wang, Shuai Tan, Shu Wang, and Yuqing Chang Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.5b00373 • Publication Date (Web): 04 Nov 2015 Downloaded from http://pubs.acs.org on November 10, 2015

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Industrial & Engineering Chemistry Research

A novel monitoring strategy combining the advantages of multiple modeling strategy and GMM for multi-mode processes Shumei Zhang1*, Fuli Wang1,2, Shuai Tan3, Shu Wang1,2 ,Yuqing Chang1,2 1. College of Information Science&Engineering, Northeastern University 2. Northeastern University Stat Key Laboratory of Integrated Automation of Process Industry Technology and Research Center of National Metallurgial Automation 3. Key Laboratory of Advanced Control and Optimization for Chemical Processes of Ministry of Education，East China University of Science and Technology

Abstract: Multiple modeling strategy and GMM have been widely used to monitor the multimode processes. On the basis of a deterministic view, multiple modeling strategy builds the specific model for each mode, which can extract more accurate information for monitoring. However, multiple modeling strategy is unable to deal with the situation in which the online mode information cannot be determined, and it is easy to lead to a severe error when an inappropriate model is used for monitoring. GMM builds a mixture model for the whole process from a probabilistic view. It unites all the models probabilistically for monitoring without having to identify the mode information. However, it may be badly performed for some specific modes

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because some irrelevant models of other modes are introduced by GMM. Besides, it may not efficiently capture the local features especially for the complex processes with transitional modes. In this paper, a novel monitoring strategy, which combines the advantages of multiple modeling strategies and GMM, is proposed for multimode processes. All possible models are probabilistically united for monitoring when the mode cannot be identified for sure. If the mode can be determined completely, the corresponding model is deterministically used for monitoring. To evaluate the feasibility and efficiency of the proposed method, the Tennessee Eastman challenge is demonstrated to compare the proposed method with multi-PCA and traditional GMM.

Key words: process monitoring; multimode process; Gaussian mixture model (GMM); transitional mode

1. INTRODUCTION During the last few decades, multivariate statistical process control (MSPC)[1-2]schemes, such as principal component analysis (PCA) component analysis (ICA)

[5]

[3]

, partial least-squares (PLS)

[4]

and independent

, have been widely used to satisfy growing requirements of safety

and high product quality in modern industrial processes. To overcome specific limitations of the conventional methods, several extensions or combined approaches of traditional MSPC methods, like kernel PCA, dynamic PCA/ICA, multi-way PCA/PLS, ICA-PCA, etc. have been applied to diverse processes [6-10]. However, most conventional MSPC methods and their extensions have poor performances for multimode processes owing to their assumptions that the process should be operated under a


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stable mode. In industrial practice, operating conditions of modern industrial processes often change due to various factors, such as changes of market demands, variations in product specifications, alternations of feedstock, different manufacturing strategies and fluctuations in the external environment [11,12]. To overcome the deficiency of multimode process monitoring, some efforts have been reported to approach the multimode process monitoring issue and they can be categorized by the following categories: (a) global modeling;(b) adaptive models; (c) robust models; (d)multiple models;(e) mixture models. The characteristics of variables such as mean and covariance in different operating conditions are significantly different. The global modeling methods [11-14] have been committed to construct a model that can describe different structures of all the modes. However, most of global modeling methods can only eliminate influence caused by diversity of mean in different modes, but cannot deal with the differences of the covariance structures of each mode. Therefore, owing to the fact that a global PCA model is actually the statistical average of all operation modes, a global PCA model may lose important local information and may lead to low resolution for some modes when the modes have different covariance structures. A global PCA model is the simplest solution but it would have good performance if and only if the variables in different modes have the similar characteristics, or it would fail to represent each operating mode precisely due to statistical averaging. The key of adaptive modeling strategy is how to make a new normal sample adapt to the mode changes. On this basis, adaptive PCA and PLS methods were developed

[15-20]

. However,

adaptive approach are effective to address slow and normal process changes and ineffective in responding to the mode alterations. Besides, in most cases, they lack the ability of distinguishing


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the deliberate changes in the process operating conditions from process faults, especially when the feed is changed manually and the process varies quite significantly. On the other hand, these approaches are carried out blindly with a high computational load on the model construction which is undesirable in practice, despite whether a process change has been identified or not. The robust modeling strategy focuses on the part information of the measurement data which are not influenced by normal changes in operating condition. Kano et al. [20-22] employed external analysis to remove the influence of external variables from operation data and then used PCA or ICA to monitor the process. These approaches are robust to operating mode changes, but the selection of the external variables mainly depends on experience and process knowledge. However, it’s difficult to obtain process knowledge from modern complex processes and this problem greatly restricts their application in many cases when the external variables are unknown or cannot be determined. Moreover, the process variations related to external variables are not modeled and monitored. Through the above analysis, we can find that the above three methods are more applicable to some specific processes because their applicable conditions greatly restricts their use in many cases. In contrast, multiple modeling strategy and mixture modeling strategy are used more often due to their few restrictions. The idea of multiple modeling strategy is very simple and direct: Since multimode processes consist of different modes, multiple local models should be built respectively to match each operation mode. The specific model is used in online monitoring. Employing the multiple modeling strategies, a series of studies[23-30] based on the multiple PCA/PLS model methods have been reported to handle the intrinsic multimodality of process data. They are successfully applied to multistage batch processes and multimode continuous processes.


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Multiple modeling strategy is an obvious choice due to its simple idea. It can extract more accurate information and perform well in monitoring the determined modes. However, several issues should be considered in its application. First, in the preliminary step, a priori process knowledge is required to segment the historical data into multiple groups that correspond to different modes. Second, the most suitable model should be determined for every new sample in the online monitoring phase. Multiple modeling strategy cannot be used when the online mode cannot be determined and it may lead to a severe error with high false alarm rate when an unsuitable model is used for monitoring. Mixture modeling technique hopes to build a mixture model that is well suited to representing the data sources driven by different operating modes. Under the guide of this theoretical framework, Yu and Qin[31] proposed a finite Gaussian mixture model (FGMM) and Bayesian inference-based probabilistic approach for fault detection under various modes. Then some developed methods [32-36] based on GMM are further proposed for multimode process monitoring. GMM uses a probabilistic view to solve the problem. It unites all the local models in the whole process without identifying the mode. Once the mixture model has been built, it is very easy to perform in online monitoring. However, it may poorly perform for some specific modes because some irrelevant models may be introduced. Besides, it may not efficiently capture the local features for the complex processes. Though the multimode processes have caused wide public concern, most researches largely focused on stable modes without considering transitional mode between two stable modes. In fact, the process turns into transitional mode when process changes from one stable mode to another stable mode. The transitional mode cannot be ignored because it may lead to loss of production time, off-grade materials, and lack of reproducibility of product grades [30].


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The transitional mode monitoring has attracted increasing attention in recent years, but it has not been well-solved up till now. Yew Seng Ng[37] proposed a new adjoined modeling technique using overlapping PCA models to ensure smooth evolution of the monitoring, but it cannot identify the transitional mode or the stable modes. A stage-based sub-PCA modeling method is developed by Lu[25] to model and monitor multistage batch processes, but the hard stage partition algorithm neglects the stage-to-stage transiting characteristics. To overcome the issue, Zhao et al.[26] proposed a soft transitional model based on user-specified parameters and Yao et al.[38] improved the approach to identify the weighted parameters automatically by solving the optimization problems. However, the approach is not valid for the complex transitions which have their specific characteristic and cannot be completely described by the weighted sum of characteristics of two neighboring stable modes. The papers

[29-30, 39-40]

by Shuai Tan et al. also

address this issue, but their approaches are either multiple model-based or too complex to be applied in real chemical processes. Due to the dynamic characteristics of transitional modes, a transitional mode may be divided into several sub-modes and corresponding sub-models should be constructed. Thus in the online monitoring phase, the most suitable model should be determined for every new sample. “Minimum SPE criterion”is the most popular technology to choose the specific local model[23]. For the complex processes with many modes, the number of the models is large and frequent model switching will cause high false alarm rate. In short, the transitional mode is the process with dynamic information which needs a series of sub-models to grasp more accurate dynamic details. Besides, some transitional modes are very similar at the very beginning and it is hard to distinguish them from each other. In this case, it can easily lead to false mode identification results.


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As mentioned above, GMM-based methods have aroused the attention of many scholars due to their advantages in multimode process monitoring. However, most GMM-based methods are focused on the stable modes without considering the transitional modes. On account of the importance of transitional modes, GMM-based method should also describe the characteristics of transitional modes. Unlike a stable mode corresponding to a Gaussian component, a transitional mode may be described by several Gaussian components. That is, transitional modes will greatly increase the number of the Gaussian components in GMM. Thus the large amounts of estimated parameters will lead to huge computational cost and high complexity. Meanwhile, owing to the fact that the duration of stable modes is much longer than transitional modes, the information contained in transitional modes may be overwhelmed when all historical data are selected to train a GMM. GMM may not efficiently capture the local features of transitional modes when stable modes and transitional modes are modeled together. In this case, the transitional mode between two stable modes may be falsely alarmed. So the transitional modes should be modeled separately. Through the above analysis, there are advantages and disadvantages in both multiple modeling strategy and GMM. Combining their advantages, a novel monitoring strategy is proposed in this paper. First, different GMM are built for stable modes and transitional modes respectively. This can solve the problem that some transitional information may be lost when a GMM is built for the whole process. Meanwhile, a GMM is constructed for a transitional mode instead of several PCA sub-models, which can greatly reduce the number of sub-models. Second, the deterministic idea of multiple modeling strategy and the probabilistic view of GMM have been absorbed in the proposed paper. When the mode can be decided for sure, the specific model is utilized for monitoring, which can avoid the bad effects of those irrelevant models. If the mode cannot be


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identified for certain, all possible models should be combined for monitoring with certain probabilities. This can avoid the severe error which results from wrong identification of multiple mode strategy. The paper is organized as follows. Section 2 introduces process monitoring based on GMM. In section 3, multiple GMM are trained for stable modes and each transitional mode. In Section 4, the proposed monitoring strategy is introduced. The new proposed approach is demonstrated on the Tennessee Eastman challenge process in Section 5. Finally Section 6 outlines the concluding remarks of this paper.

2. PROCESS MONITORING BASED ON GMM GMM has been successfully applied to process monitoring in the past two decades. In Gaussian mixture model, for an arbitrary data sample x ∈ R J which may come from M possible Gaussian distributions, the probability density function of the J-dimensional sample point x can be expressed as follows: M

p ( x | θ ) = ∑ ω m p ( x | θm )

(1)

m =1

where M is the number of Gaussian components included in GMM; ωm denotes the weight of the m-th component, which is the prior probability of the m-th Gaussian component satisfying

0 ≤ ωm ≤ 1 and

M

∑ω

m

=1 . θm = { µm , Σ m } consisting of the elements of the mean µm and the

m =1

covariance matrix Σ m , represents the density function parameter set. p( x | θm ) denotes the multivariate Gaussian probability density function as given as follows: p ( x | θm ) =

1 (2π ) Σ m J

 1  exp  − ( x − µm )T Σ m −1 ( x − µm )   2 

(2)


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The maximum likelihood distribution parameters of GMM can be estimated using the EM algorithm and the F–J algorithm[41] is employed to adjust the number of Gaussian components M. Given

the

{{

training

Θ( ) = ω1( ) , µ1( ) , Σ1( 0

0

0

0)

X = [ x1 , x2 ,..., xn ] ∈ R J ×n

data

} ,...{ω ( ) , µ( ) , Σ ( ) }} 0 M

0 M

0 M

and

an

initial

estimate

, the modified expectation and maximization

procedure is iterated as follows:

(1) E-step The posterior probability can be obtained using Bayesian theorem, p (θ |xi )= (s)

(s) m

ωm(s) p (xi |θ m(s) ) M

∑ω

(s) m

(3)

p (xi |θ ) (s) m

m =1

where p (s) (θ m(s) |xi ) denotes the posterior probability of the i-th training sample within the m-th Gaussian component at the s-th iteration.

(2) M-step n  p (s) (θ m(s) |xi )xi ∑   µm(s+1) = i=1n  p (s) (θ m(s) |xi ) ∑  i=1  n T  p (s) (θ m(s) |xi ) ( xi -µm(s+1) )( xi -µm(s+1) ) ∑  (s+1) i=1  Σm = n  p (s) (θ m(s) |xi ) ∑  i=1  n  p (s) (θ m(s) |xi ) ∑  ωm(s+1) = i=1  n  

(4)

where µm(s+1) , Σ m(s+1) , ωm(s+1) are the mean, covariance, and prior probability of the m-th Gaussian component at the (s+1)-th iteration, respectively.


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In order to monitor the industrial processes, a probability monitoring index BIP is proposed by Yu and Qin as follows Mc

BIP = ∑ p ( Cm | xt ) pL( m ) ( xt )

(5)

m =1

where p ( Cm | xt ) denotes the posterior probability of the t-th training sample xt to the m-th Gaussian component. pL(

m)

( xt )

is the local Mahalanobis distance-based probability index

relative to each Gaussian component and is defined as

pL(

m)

( xt ) = p {D ( ( x, Cm ) | x ∈ Cm ) ≤ D ( ( xt , Cm ) | xt ∈ Cm )}

(6)

where D ( xt , Cm ) represents the Mahalanobis distance of the training sample xt to the m-th Gaussian component. Since 0 ≤ pL(

m)

( xt ) ≤ 1 , it can be obtained that Mc

0 ≤ BIP ≤ ∑ p ( Cm | xt ) = 1

(7)

m =1

Under a prespecified confidence level (1 − α )100% , the process can be considered abnormal when BIP > 1 − α . Noticing that the faulty symptom with BIP is not as evident as distance based indices (e.g., T2 or SPE), Xie etc.

[33]

directly utilize the Mahalanobis distance as local index and thus the

integrated global monitoring statistics is defined as Mc

BID = ∑ p ( Cm | xt ) D ( xt , Cm )

(8)

m =1

The upper control limit for BID , denoted as BIDlmt, is calculated from the F-distribution, which is

BIDlmt =

J (n 2 − 1) FJ ,n − J ;γ n( n − J )

(9)


10

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where n and J represent the number of samples and variables respectively. FJ ,n − J ;γ is a F distribution with J and n-J degrees of freedom under given significance level γ . In this paper, the statistics BID is used for monitoring.

3. OFFLINE MODE IDENTIFICATION AND MODELING For multi-mode processes, offline mode identification plays a key role in selecting training data for Gaussian mixture model. However, the prior process knowledge is difficult to obtain in most complex industrial processes. Some automatic clustering techniques, such as k-means clustering[42-43], fuzzy c-means clustering method[44], subtractive clustering algorithm[45], etc. can be employed for offline mode identification. For the multi-mode processes with between-mode transitions, the clustering techniques can only be used for preliminary identification because they cannot determine that the cluster belongs to stable mode or transitional mode. Different from the traditional multiple modeling methods, we build a single model for a transitional mode rather than multiple sub-models. That is, all the sub-modes which belong to a transitional mode are put together to build a GMM for the transitional mode. So the clustering result of automatic clustering techniques should be further analyzed to obtain deep mode information. In industrial practice, stable mode plays the primary role in the process and it occupies the most production time to yield high productivity. The process variables in stable mode are running in one steady state, in other words, the characteristics of variables such as mean and covariance are constant. Thus, one stable mode will be clustered together. In contrast to stable mode, the transitional mode is the dynamic process between two stable modes and the duration time of transitional mode is much shorter than stable modes. The variables of the transitional mode keep changing in time direction, so the transitional mode will be separated into one or


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more sub-modes by the clustering techniques. Therefore, there will be a series of short submodes between two long neighboring stable modes. The clustering results can be further analyzed according to this conclusion. Considering a multimode process with J process variables measured over sampling points t ( t = 1, 2,..., K ), historical data collected in normal working condition are composed of a twodimension array X ( K × J ) . The steps of offline mode identification and modeling are as follows:

Step1: Cluster the data by k-means algorithm. A variant k-means algorithm is adopted to classify data samples xt ( t = 1, 2,..., K ) into C number of clusters. The clustering result are directly associated with the operation time because the historical data are extracted along the sampling time of a multimode process. The detailed steps of improved k-means clustering algorithm can be found in [42, 43].

Step2: Sort the data by time and assign them to different segments according to the clustering result. The successive data that belong to the same cluster can be assigned to a segment. Thus the offline training data are divided into S data segments which belong to C clusters. The segments are sequenced in time direction. The s-th segment is defined as XI s ( s = 1, 2,..., S ) with ns S

successive data samples contained in XI s , where ns satisfies

∑n

s

= K . The membership

s =1

function can be defined as u ( XI s ) = c , where c = 1, 2,..., C .

Step3: Delete the singular points. The singular points last much shorter than sub-mode, so XI s is assumed to be the segment of the singular points if ns ≤ h . Parameter h is the minimum duration of the sub-modes. All the singular points are deleted directly and the remaining segments are rearranged in time direction.


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Step4: Determine the mode information. Owing to the fact that stable mode lasts much longer than the sub-stages of the transitions,

XI s is assumed to belong to stable mode if ns ≥ H . Parameter H is the minimum stable mode duration. Then all the segments between two neighboring stable mode (i.e. stable mode A and stable mode B) are assumed to belong to a transitional mode (i.e. the transitional mode AB).

Step 5: Build the offline models for each mode separately. All the data samples which belong to the same stable mode are collected together to build a single Gaussian model. The mean and variance can be directly estimated by sample mean and sample variance. The historical data that belongs to the same transitional mode are collected together to train its transitional GMM. The details of offline modeling are shown in Scheme 1.

Scheme 1. Flowchart of offline mode identification and modeling.


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4. ONLINE MODE IDENTIFICATION AND MONITORING ALGORITHM The modes in most multimode processes do not occur in a specified order. Therefore, when a new data sample xnew has been obtained, there is no priori information to identify the mode of

xnew . In this case, the most suitable model needs to be determined for every new sample in the online monitoring phase. However, it may lead to severe errors when an inappropriate model is used for monitoring, because false alarms may be caused if the data sample does not belong to the corresponding operation mode. In industrial process, some transitional modes are very similar and it is hard to distinguish them from each other at the very beginning. To overcome this issue, all possible transitional models are probabilistically united for monitoring when the mode cannot be identified for sure. After running for a period of time, it is enough to distinguish the mode from each other, and the mode of the current sample can be identified for sure. The corresponding model is used for monitoring according to the online mode identification result. In conclusion, when the most suitable model for a new data sample can be decided for sure, the corresponding model is deterministically used for monitoring. On the contrary, if the mode cannot be fully determined in some ambiguous stages, all possible models should be probabilistically utilized to monitor the new data sample.

4.1 Joint monitoring model All possible models will be probabilistically united for monitoring when the mode cannot be identified for sure. We simply take the transitions which begin from mode A as an example. There are L transitional modes that begin from stable mode A. Define the l-th transitional model Ml

as TRlA and the GMM model of TRlA is pl ( x | θ ) = ∑ ωil p ( x | θil ) . The detailed steps are shown as i =1

follows:


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Step1: For the l-th transitional mode TRlA , the probability density of the samples xk is calculated. A l

p

Ml

( xk ) = ∑ ωil p( xk | θil )

(10)

i =1

Step 2: For the data sample xk , the condition is judged: A pmax ( xk ) ≥ α A ptr ( xk )

(11)

A where pmax ( xk ) is the maximum probability and ptrA ( xt ) is the sum of all the transitional

L

A probability density, pmax ( xk ) = max ( plA ( xk ) ) and ptrA ( xt ) = ∑ plA ( xt ) . l =1,2,..., L

l =1

Step 3: If the condition (11) is satisfied, it is considered that the process definitely enters the A transitional model corresponding to the maximum probability pmax ( xk ) . Otherwise, it illustrates

that some transitional modes are very similar at the beginning and it’s hard to distinguish them. Therefore, the joint monitoring for the new data sample xk is carried out by all possible transitional models.

Step 4: The weight of l-th ( l = 1, 2,..., L ) transitional model TRlA in joint monitoring model is defined as follow:

plA ( xt ) wl = A ptr ( xt )

(12)

L

where the weight wl satisfies 0 ≤ wl ≤ 1 and ∑ wl = 1 . l =1

Step 5: The joint statistic and its corresponding confidence limit can be defined as L

BID = ∑ wl BID(l ) t

(13)

l =1


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L

BIDlmt = ∑ wl BID(l )lmt

(14)

l =1

where BID( l ) t is the statistic of the data sample xt in the l-th ( l = 1, 2,..., L ) transitional model and BID( l )lmt is the confidence limit of corresponding statistic in the l-th ( l = 1, 2,..., L ) transitional

model TRlA .

4.2 The proposed monitoring strategy We assume that the multimode process begins with the stable mode. The purpose of online mode identification is to select the right model to monitor the current sample. For the first data sample, the probability density of the data under different stable modes is calculated. The specific model corresponding to the biggest probability density is used for monitoring. Assume that the current mode type at time (t-1) has been known and it belongs to stable mode, where t is the current sampling time. There are three possible cases for the current sample: (1) The process is normal and the mode type remains the same; (2) The process turns into the transitional mode; (3) The process is abnormal (fault or un-modeled mode). The current stable model is used to monitor the current sample. If the statistics are under the control limits, the process is deemed to be operating normally and the mode type does not change. Otherwise, if the current sample goes beyond the control limits, the process may turn into the transitional mode (case 2) or become abnormal (case 3). It needs further analysis to identify the current state. The condition (11) is calculated to judge whether a joint monitoring model should be used. If the condition (11) is satisfied, the specific model corresponding to the maximum probability is used to monitor the current sample. Otherwise, all possible models are united together for monitoring the current sample. If the statistics are under the new control limits, the process is


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normal and it turns into the transitional mode. Otherwise, if the statistics exceed the control limits, the process is assumed to be abnormal. When the process enters the transitional mode, there are two cases: (1) the process is monitored by a specific model; (2) the process is monitored by a joint monitoring model. For case 1, the current model is used to monitor the current sample. If the statistics are under the control limits, the process is normal and the mode type remains the same. The current model is used to monitor the next data sample. Otherwise, the model of the next stable mode is used to monitor the current sample. If the statistics are under the new control limits, the process is assumed to turn into the next stable mode. Otherwise, if the statistics go beyond the control limits, the process is deemed to be abnormal. For case 2, the condition (11) is calculated to judge whether a specific model or a joint monitoring model should be used. If the condition (11) is satisfied, the specific model is used to monitor the current sample. Otherwise, a joint monitoring model is used to monitor the current sample. If the statistics are under the control limits, the process is normal and the process is still in the transitional mode. Otherwise, if the statistics go beyond the control limits, the joint model of all possible stable modes is used to monitor the sample. If the statistics are under the new control limits, the process enters the stable mode. Otherwise, the process is assumed to be abnormal. When the process enters the stable modes from the transitional modes, there are two cases: (1) the process is monitored by the model of the next stable mode; (2) the process is monitored by a joint monitoring model of all possible stable modes. For case 1, the current specific model is used to monitor the data sample. For case 2, the probability density of the data under all possible stable modes is calculated and the specific model corresponding to the biggest probability density is utilized for monitoring.


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t=1; the sample xt

Choose the stable model corresponding to the biggest probability density Use the specific stable model for process monitoring

t=t+1; Update the sample N

The current sampling goes beyond the control limit

Normal

Y The transitional mode is determined for sure

N

Y Use the specific transitional model for process monitoring

Y

All possible transitional models are probabilistically united for monitoring

The current sampling goes beyond the control limit N

Fault or Un-modeled mode

Enter the transitional mode

The transitional mode is determined for sure

Y

N All possible transitional models are probabilistically united for monitoring

t=t+1; Update the sample

N Normal


Use the specific transitional model for process monitoring t=t+1; Update the sample N

Normal


Y

Y

All possible stable models are probabilistically united for monitoring

N


Use the next stable model for process monitoring

Y

Y Enter the stable modes


The current sampling goes beyond the control limit N


Enter the next stable mode

Scheme 2. Flow chart of online monitoring


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The detailed procedures of the proposed approach are shown in Scheme 2. It is worth noting that in practical application, the process is assumed to be abnormal if and only if consecutive d (d equals to 10 in this paper) samples go beyond the control limits rather than a single sample, and this can effectively avoid false alarms. Besides, in most cases, it is enough to distinguish two different transitional modes from each other when the transitional mode is coming to an end. That is, the specific model is more likely to be used for monitoring before the process enters the stable mode from the transitional mode.

5. APPLICATION AND RESULTS The Tennessee Eastman process presented by Downs and Vogel[46]has been widely applied to evaluate and test the performance of various monitoring techniques. As shown in Figure1, the process consists of five major unit operations: a reactor, a product condenser, a recycle compressor, a product stripper, and a vapor–liquid separator. There are 41 measured variables (including 22 continuous process measurements and 19 composition measurements) and 12 manipulated variables in TE process. A set of 21 programmed faults is introduced to the process. Detailed descriptions of the TE process are well explained in a book of Chiang et al[47]. In the present paper, 15 continuous process variables among the 41 output variables are selected for monitoring purposes, which are tabulated in Table 1. Studies have shown that the reactor pressure of TE process have great influences on the cost of production, while G/H mass ratio decide the final components of the product. Therefore, different operation points were designed to simulate a typical multimode process by changing reactor pressure and G/H mass ratio according to the production requirements. Four different stable modes are considered here for the simulation of multimode processes, which are listed in Table 2.


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Figure 1. Tennessee Eastman system Table 1. Monitoring variables in the TE process No.

Variable description

No.

Variable description

1

A feed

9

Product separator temperature

2

D feed

10

Product separator pressure

3

E feed

11

Product separator underflow

4

A and C feed

12

Stripper pressure

5

Recycle flow

13

Stripper temperature

6

Reactor feed rate

14

Reactor cooling water outlet temperature

7

Reactor temperature

15

Separator cooling temperature

8

Purge rate

water

outlet

The simulation data were separated into two parts: the training data sets and the testing data sets. The training data are obtained by running the simulation 400 h under normal operation conditions with a sampling interval of 0.01 h. The set points of reactor pressure and desired G/H


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mass ratio are changed to simulate a continuous multimode process, which are shown in Table 3. The initial set points of reactor pressure and the desired G/H mass ratio are 2800kpa and 90/10, respectively. After 50h, the reactor pressure is reduced to 2600 kPa while the desired G/H mass ratio remains unchanged. Thus, the process enters the transitional mode and continues for some time until the process reaches a new stable mode. The operation points are changed for seven times, simulating four stable modes and six transitional modes. The drift in production status causes changes in most variables, and the typical trends of 15 variables are presented in Figure 2.

Figure 2. Trends of 15 variables of the training data Table 2. Operating modes of Tennessee Eastman process Mode type

Set point of Reactor Pressure

desired G/H mass ratio

Stable mode A

2800kpa

90/10

Stable mode B

2600kpa

90/10

Stable mode C

2400kpa

90/10

Stable mode D

2800kpa

80/20


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Table 3. Operation Changes of Training Data Operational change

Description

Operation time

1

The set point of reactor pressure is 2800kpa and the initial desired G/H mass ratio is set as 90/10

2

The reactor pressure is reduced to 2600kpa

51st hour

3


101st hour

4

The reactor pressure is increased to 2800kpa

161st hour

5

The desired G/H mass ratio is changed from 90/10 to 221st hour 80/20

6

The desired G/H mass ratio is set as 90/10

291st hour

7


351st hour

To examine the effectiveness of the proposed monitoring approach for complex multimode processes, three test scenarios are designed, as shown in Table 4.

Table 4. Three Test Cases of Tennessee Eastman process Test Case 1

Stable mode A, normal Transitional mode AB, normal Stable mode B, normal

Operation time 1-500 501-1310 1311-1800

2

Stable mode D, normal Stable mode D, fault: Random variation in C feed temp. (stream 4)

1-800 801-1800

3

Stable mode B, normal Transition BC, normal Transition BC, fault: random variation in reactor cooling water inlet temp

1-500 501-1000 1001-1800

State Description

Figure 3 shows the offline mode identification of the training data. Then the data which belong to a same mode should be trained to construct its corresponding model. In this simulation, four single-Gaussian models have been trained for four stable modes with six GMM for transitional modes. To compare the monitoring performance of the proposed method with multi-PCA and conventional GMM, a GMM for all the historical data has been trained and thirty-two sub-PCA


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models (including four PCA models for 4 stable modes and 28 sub-models for six transitions) have been built.

Figure 3. Offline mode identification Figure 4 shows the monitoring results of Multi-PCA in test case 1. Both T2 and SPE statistics are under their own control limits of stable mode A from 1st to the 500th samples. When the two statistics go beyond the control limits from the 501st sample under the model of stable mode A,

“Minimum SPE criterion”is adopted to find the most suitable model. The SPE statistics of three transitions from 501st to the 515th samples are shown in Figure 5. Therefore, the model of AC transitional mode corresponding to the minimum SPE is chosen for the process from the 501st samples. Both T2 and SPE statistics are under the control limit until the 636th sample. The process is considered to be abnormal from the 636th sample to the 1312th sample because the statistics exceed the control limit, and this gives a wrong conclusion. When the data re-enters the stable mode, “Minimum SPE criterion” is used again to determine the mode type. The specific model of stable mode B is utilized for monitoring after the 1313th sample.


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Figure 4. Monitoring charts of Multi-PCA in test case 1.

Figure 5. SPE statistics of three transitional modes in case1 As shown in Figure 6, AB transitional mode and AC transitional mode are very similar at the beginning and it is hard to distinguish them from each other, so it is easy to choose the wrong model based on“Minimum SPE criterion”. Owing to the deterministic characteristic of Multi-


24

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PCA, the specific model is utilized to monitor if the model is determined, which will lead to the severe error once the wrong model is chosen. As shown in Table 5, the severe error, which is due to the wrong mode identification from the 501st samples, leads to high false alarm rates for T2 and SPE as 34.30% and 34.25%, respectively.

Figure 6. Variables’ trends of three transitional modes in normal operation The monitoring results of GMM are presented in Figure 7. Most of the statistics are under the control limit, while the samples from the 501st sample to 567th sample go beyond the control limit. It is concluded that the GMM which is trained for the entire process cannot capture the information of the first sub-stage of AB transition. From Figure 3, we can find that the duration of AB-1 sub-stage is very short. So the information contained in this sub-stage may be overwhelmed when all historical data are selected to train the GMM, and in this case, the false alarm rates will be increased. The Table 5 gives the false alarm rates for BID as 3.45%.


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Figure 7. Monitoring charts of GMM in test case1 Figure 8 gives the online mode identification result of the proposed method in the first test case. As shown in Figure 8, Figure 8 (a) gives the corresponding state of the data. Figure 8 (b) shows the logarithm of the probability density under corresponding stable modes when the process is running at the stable modes. The logarithms of the probability density under three transitional modes are shown from Figure 8(c) to Figure 8 (e), respectively. Figure 8 (f) gives the weight of three transitional models in the joint monitoring model at the beginning of the transition when the mode cannot be identified for sure. The monitoring results of the proposed method are presented in Figure 9. The statistics are under the corresponding control limits for the first 500 samples. When the BID statistics exceed the control limit under the model of stable mode A, the probability density under three transitions which start from mode A are calculated, which is shown from Figure 8 (c) to Figure 8 (e). We can find that the probability density under AB mode and AC mode are both high at the beginning, while the probability density under AD mode is very small. So the AB model and AC model are probabilistically united for monitoring from the 501st sample to 619th samples. The specific AB


26

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model is chosen to monitor from 620th samples to 1310th samples until that the statistics go beyond the control limit. After the 1310th sample, the model of mode B is utilized for monitoring. From the Figure 8 (a), we can find that probability density under next stable mode B is big after the 1310th samples. Combining the advantages of multiple-models strategy and the mixture strategy, the proposed method avoids the severe error of multi-PCA. It simultaneously improves the performance of capturing the information of transitional modes with the lowest false alarm rates as 0.25%.

Table 5. The simulation results of three test cases

statistics

False positive ratio

False negative ratio

Detecting delays for disturbances

T2 SPE

0.3430 0.3425

~ ~

~ ~

GMM

BID

0.0345

~

~

the proposed method

BID

0.0025

~

~

T2 SPE

0.0138 0.0100

0.2093 0.0341

25 16

GMM

BID

0. 0113

0.0412

15

the proposed method

BID

0.0125

0.0294

15

T2

0.0380

0.1800

14

SPE

0.1110

0.0500

12

GMM

BID

0.1370

0.0950

0

the proposed method

BID

0.0090

0.0275

0

methods Case1 Multi-PCA

Case2 Multi-PCA

Case3 Multi-PCA

The second test case starts with the normal operation at stable mode D for the first 800 samples, along with a random variation in C feed temp from the 801st sample to the 1800th sample. The fault detection results of three methods are shown in Figures 10, 11, and 12, respectively. All the


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three monitoring methods detect this fault successfully, and their faulty symptoms are quite obvious. The SPE of multi-PCA approach detects this fault at 0.17 h after the fault is introduced, and it is 0.09 h earlier than T2 statistics. The GMM also presents clear features of random variation with its statistics, and the time delay is 0.17 h. The proposed method detects the disturbance with 0.16 h time delay, which is the minimum value for all the three methods. For the normal data of stable modes, all the three methods perform well with the low false alarm rates no more than 1.38%. From Table 5, it can be seen that T2 statistics of multi-PCA perform poorly while SPE statistics performs better on avoiding false negative rates for abnormal samples and their false negative rates are 20.93% and 3.41%, respectively. In contrast, GMM performs better than multi-PCA for abnormal data and the false negative rates of its statistics are 4.12%. The proposed method has the best performance with false negative rates of BID as 2.94%. Compared to the proposed method, the false alarm rate of GMM is lower while the false negative rates are higher for the stable modes. That is because GMM, which describes all the modes in a model, has a looser control limit. In the last test case, the process is first running at stable mode B and enters BC transitional mode after 500 samples. Then a fault of random variation in reactor cooling water inlet temp occurs at the sample 1001. Figures 13−15 show the monitoring performances using multi-PCA, GMM and the proposed methods. From Figure 13, we can find that the T2 statistics of multi-PCA do not show clear faulty indications until the 1015th sample, which is 0.15 h delay after the disturbance is introduced while the SPE of multi-PCA approach detects this fault at 0.13 h after the fault is introduced. The plots of Figure 14 and Figure 15 show that GMM and the proposed method detect this disturbance timely with no time delay. It is easily observed from Figure 13 and Figure 14 that both the multi-PCA and GMM indices perform poorly on both detecting


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faulty measurements precisely and avoiding false alarms for normal samples. In the SPE plot of multi-PCA, many normal samples jump above the corresponding control limit with false alarms triggered. The overall false alarm rate of 11.10% is unsatisfactory. Meanwhile, the false negative rate of T2 statistics is as high as 18.0% with large numbers of faulty samples undetected. In Figure 14, the fault is warned timely, but there are too many false alarms for the normal data, especially from the 846th sample to 1000th sample. That is because the information of some substages of BC transitional model is lost when GMM is offline trained and thus BC mode cannot be completely described by GMM, resulting in the high false alarm rates of 13.7%,. Apparently, the statistics of GMM also detect the fault with lots of false negatives, and the performance of fault detection is unsatisfactory as the false negative rates of BID reach up to 9.5%. In Figure 15, the fault is warned timely, and the probability density can also reflect the state of the process. The probability density under stable mode is very large at the first 500 samples, while the probability density under BC transitional mode is large when the process enters the transitions from the 501st sample to 1000th sample. When the fault occurs, both the probability density under BC mode and stable mode are small. Compared with the former two monitoring algorithms, the proposed method obtained the best performance, with significantly reduced missed detections and false alarms. Only 0.9% of normal samples trigger false alarms while the false negative rate is as low as 2.38%. Therefore, it is confirmed that the proposed method has significant superiority for monitoring the multimode processes with complex transitions.


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Figure 8. Online mode identification of the proposed GMM in test case 1:(a) the mode identification of the data; (b) the logarithm of probability density under stable mode; (c) the logarithm of probability density under transitional mode AB; (d) the logarithm of probability density under transitional mode AC;(e) the logarithm of probability density under transitional mode AB; (f) the weight of each transitional mode at the beginning of the transition

Figure 9. Monitoring charts of the proposed GMM in case1


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Figure 10. Monitoring charts of Multi-PCA in test case 2

Figure 11. Monitoring charts of GMM in test case 2.


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Figure 12. Monitoring charts of the proposed method in test case 2.

Figure 13. Monitoring charts of Multi-PCA in test case 3.


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Figure 14. Monitoring charts of GMM in test case 3

Figure 15. Monitoring results of the proposed method in test case 3: (a) the mode identification of the data; (b) the logarithm of probability density under stable mode; (c) the logarithm of probability density under transitional mode BC; (d) the monitoring charts of the proposed method


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6. CONCLUSIONS In the present paper, a novel monitoring method has been proposed for the multimode processes with between-mode transitions. The specific model is chosen when the mode is identified for sure, while all possible models are combined together when the mode cannot be decided definitely. Compared to the existing multiple methods and mixture modeling strategy, the proposed method has both deterministic and probabilistic characteristics. Hence, it can represent the process information more precisely than the traditional mixture methods and avoid severe errors on account of wrong identification of the multiple methods. The feasibility and efficiency of the proposed method have been evaluated through three case studies in Tennessee Eastman Chemical process. The monitoring results are compared to those of multi-PCA and GMM and it is shown that the process behavior interpretations and the capability of detecting process faults have been improved by the proposed method.

AUTHOR INFORMATION Corresponding Author *Email: [email protected]. Address: P.O. Box 131, Northeastern University, NO. 3-11, Wenhua Road, Heping District, Shenyang, P. R. China.

Notes The authors declare no competing financial interest.

ACKNOWLEDGMENTS The authors gratefully acknowledge the support from the following foundations: National Natural Science Foundation of China (61533007, 61374146 and 61403072), Stat Key Laboratory of Synthetical Automation for Process Industries Fundamental Research Funds


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(2013ZCX02-04), and the Fundamental Research Funds for East China University of Science and Technology (22A201514050).

REFERCENCES (1) AlGhazzawi, A.; Lennox, B. Monitoring a complex refining process using multivariate statistics. Control Eng. Pract. 2008, 16, 294–307. (2) Qin, S. J.. Statistical process monitoring: Basics and beyond. J. Chemometr. 2003, 17 (8-9), 480-502. (3) Jackson, J. E. A user’s guide to principal components. New York: Wiley, 1991. (4) Geladi, P.; Kowalski, B. R. Partial least-squares regression. Analytica Chimica Acta. 1986, 185, 1–17. (5) Hyvarinen, A.; Oja, E. Independent component analysis: algorithms and applications. Neural Networks. 2000, 13, 411-430. (6) Choi, J. H.; Lee, J. M.; Choi, S. W.; Lee, D.; Lee, I. B.. Fault identification for process monitoring using kernel principal component analysis. Chem. Eng. Sci. 2005, 60 (1), 279–288. (7) Chen, J.; Liu, K. C.. On-line batch process monitoring using dynamic PCA and dynamic PLS models. Chem. Eng. Sci. 2002, 57 (1), 63–75. (8) Wold, S.; Geladi, P.; Esbensen, K.; Öhman, J.. Multi-way principal components-and PLSanalysis. J. Chemometr. 1987, 1, 41–56. (9) Lee, J. M.; Yoo, C. K.; Lee, I. B. Statistical monitoring of dynamic processes based on dynamic independent component analysis. Chem. Eng. Sci. 2004, 59, 2995-3006.


35


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 36 of 41

(10) Ge, Z. Q.; Song, Z. H.. Process Monitoring Based on Independent Component AnalysisPrincipal Component Analysis (ICA-PCA) and Similarity Factors. Ind. Eng. Chem. Res. 2007, 46, 2054-2063. (11) Ma, H. H.; Hu, Y.; Shi, H. B.. A novel local neighborhood standardization strategy and its application in fault detection of multimode process. Chemom. Intell. Lab. Syst. 2012, 118, 287– 300. (12) Tong, C.; Palazoglu, A.; Yan, X. F.. An adaptive multimode process monitoring strategy based on mode clustering and mode unfolding. J. Process Control. 2013, 23, 1497–1507. (13) Hwang, D. H.; Han, C. H.. Real-time monitoring for a process with multiple operating modes. Control Eng. Pract. 1999, 7, 891–902. (14) Lane, S.; Martin, E. B.; Kooijmans, R.; Morris, A. J.. Performance monitoring of a multiproduct semi-batch process. J. Process Control. 2001, 11, 1–11. (15) Li, W.; Yue, H. H.; Valle-Cervantes, S.; Qin, S. J.. Recursive PCA for adaptive process monitoring. J. Process Control. 2000, 10, 471–486 (16) Qin, S. J.. Recursive PLS algorithms for adaptive data monitoring. Comput. Chem. Eng.

1998, 22, 503–514. (17) Wang, X.; Kruger, U.; Irwin, G. W. Process monitoring approach using fast moving window PCA. Ind. Eng. Chem. Res. 2005, 44, 5691–5702 (18) Wang, X.; Kruger, U.; Lennox, B. Recursive partial least squares algorithms for monitoring complex industrial processes. Control Eng. Prac. 2003, 11(6), 613–632.


36

Page 37 of 41

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(19) Liu, X. Q.; Kruger, U.; Littler, T.; Xie, L.; Wang, S. Q.. Moving window kernel PCA for adaptive monitoring of nonlinear processes. Chemom. Intell. Lab. Syst. 2009, 96, 132–143. (20) Kano, M.; Hasebe, S.; Hashimoto, I.; Ohno, H.. Evolution of multivariate statistical process control: application of independent component analysis and external analysis. Comput. Chem. Eng. 2004, 28, 1157–1166. (21) Yamamoto, T.; Shimameguri, A.; Ogawa, M.; Kano, M.; Hashimoto, I.. Application of statistical process monitoring with external analysis to an industrial monomer plant. In: ADCHEM. 2004, 405–410. (22) Ge, Z. Q.; Yang, C. J.; Song, Z. H.; Wang, H. Q.. Robust online monitoring for multimode processes based on nonlinear external analysis. Ind. Eng. Chem. Res. 2008, 47, 4775–4783. (23) Zhao, S. J.; Zhang, J.; Xu, Y. M.. Monitoring of processes with multiple operation modes through multiple principle component analysis models. Ind. Eng. Chem. Res. 2004, 43, 7025– 7035. (24) Natarajan, S.; Srinivasan, R.. Multi-model based process condition monitoring of offshore oil and gas production process. Chem. Eng. Res. Des. 2010, 88, 572–591. (25) Lu, N. Y.; Gao, F. R.; Wang, F. L. A sub-PCA modeling and online monitoring strategy for batch processes. AIChE J. 2004, 50 (1), 255–259. (26) Zhao, C. H.; Wang, F. L.; Lu, N. Y.; Jia, M. X.. Stage-based soft-transition multiple PCA modeling and on-line monitoring strategy for batch processes. J. Process Control. 2007, 17 (9), 728–741.


37


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 38 of 41

(27) Zhao, C. H.; Wang, F. L.; Mao, Z. Z.; Lu, N. Y.; Jia, M. X.. Improved Batch Process Monitoring and Quality Prediction Based on Multi-phase Statistical Analysis. Ind. Eng. Chem. Res. 2008, 47, 835-849. (28) Zhao, C. H.; Gao, F. R.; Wang, F. L.. Phase-based Joint Modeling and Spectroscopy Analysis for Batch Processes Monitoring. Ind. Eng. Chem. Res. 2010, 49 (2), 669–681. (29) Wang, F. L.; Tan, S.; Peng, J.; Chang, Y. Q.. Process monitoring based on mode identification for multi-mode process with transitions. Chemom. Intell. Lab. Syst. 2012, 110(1), 144-155. (30) Tan, S.; Wang, F. L.; Peng, J.; Chang, Y. Q.; Wang, S.. Multimode Process Monitoring Based on Mode Identification. Ind. Eng. Chem. Res. 2012, 51(1), 374-388. (31) Yu, J.; Qin, S. J.. Multimode process monitoring with Bayesian inference-based finite Gaussian mixture models. AIChE J. 2008, 54, 1811–1829. (32) Yu, J.; Qin, S. J.. Multiway Gaussian mixture model based multi-phase batch process monitoring. Ind. Eng. Chem. Res. 2009, 48 (18), 8585-8594. (33) Xie, X.; Shi, H. B.. Dynamic multimode process modeling and monitoring using adaptive Gaussian mixture models. Ind. Eng. Chem. Res. 2012, 51, 5497–5505. (34) Yu, J.. A nonlinear kernel Gaussian mixture model based inferential monitoring approach for fault detection and diagnosis of chemical processes. Chem. Eng. Sci. 2012, 68, 506–519. (35) Yu, J.. A particle filter driven dynamic Gaussian mixture model approach for complex process monitoring and fault diagnosis. J. Process Control. 2012, 22, 778–788.


38

Page 39 of 41

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(36) Yoo, C. K; Villez, K.; Lee, I. B.; Rose´n, C.; Vanrolleghem, P. A.. Multi-model statistical process monitoring and diagnosis of a sequencing batch reactor. Biotechnol Bioeng. 2007, 96, 687–701. (37) Nga, Y. S.; Srinivasan, R.. An adjoined multi-model approach for monitoring batch and transient operations. Comput. Chem. Eng. 2009, 33 (4), 887–902. (38) Yao, Y.; Gao, F.. Phase and transition based batch process modeling and online monitoring. J. Process Control. 2009, 19, 816–826. (39) Zhu, Z. B.; Song, Z. H.; Palazoglu, A.. Transition Process Modeling and Monitoring Based on Dynamic Ensemble Clustering and Multiclass Support Vector Data Description. Ind. Eng. Chem. Res. 2011, 50, 13969–13983. (40) Wang, J.; Wei, H. T.; Cao, L. L.; Jin, Q. B.. Soft-Transition Sub-PCA Fault Monitoring of Batch Processes. Ind. Eng. Chem. Res. 2013, 52, 9879−9888. (41) Figueiredo, M.; Jain, A. K.. Unsupervised learning of ﬁnite mixture models. IEEE Trans Pattern Anal Mach Intell. 2002, 24, 381–396. (42) Jain, A. K.; Murty, M. N.; Flynn, P. J.. Data Clustering: A Review. ACM Comput. Surv.

1999, 31, 264. (43) Abdulrahman Alguwaizani. Degeneracy on K-means clustering. Electronic Notes in Discrete Mathematics. 2012, 39, 13–20. (44) Marsili-Libelli, S.; Muller, A.. Adaptive fuzzy pattern recognition in the anaerobic digestion process. Pattern Recogn. Lett.. 1996, 17, 651–659.


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Page 40 of 41

(45) Chiu, S. L.. Fuzzy model identification based on cluster estimation. J. Intell Fuzzy Syst.

1994, 2, 267-278. (46) Downs, J. J.; Vogel, E. F.. Plant-wide industrial process control problem. Comput Chem Eng. 1993, 17, 245–255. (47) Chiang, L. H.; Russell, E. L.; Braatz, R. D.. Fault Detection and Diagnosis in Industrial Systems. Springle-Verlag: London, 2001.


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For table of Contents Only 39x30mm (300 x 300 DPI)


Novel Monitoring Strategy Combining the Advantages of the Multiple

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