Subspace Decomposition-Based Reconstruction Modeling for Fault

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Subspace Decomposition based Reconstruction Modeling for Fault Diagnosis in Multiphase Batch Processes Chunhui Zhao, and Furong Gao Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/ie401019k • Publication Date (Web): 19 Sep 2013 Downloaded from http://pubs.acs.org on September 23, 2013

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Industrial & Engineering Chemistry Research

Subspace Decomposition based Reconstruction Modeling for Fault Diagnosis in Multiphase Batch Processes Chunhui Zhao1,2*, Furong Gao1 1

State Key Laboratory of Industrial Control Technology, Department of Control Science and Engineering, Zhejiang University, Hangzhou, 310027, China

2

Key Laboratory of System Control and Information Processing, Ministry of Education, Shanghai, 200240, China

Abstract In the present work, a fault diagnosis strategy is developed for multiphase batch processes based on comprehensive subspace decomposition and reconstruction modeling method. Phase nature of fault processes is analyzed and a principal component of fault deviations (PCFD) modeling algorithm is proposed in each phase to characterize the fault effects responsible to out-of-control monitoring statistics in two different monitoring subspaces, principal component subspace (PCS) and residual subspace (RS) respectively. Phase based reconstruction models are then developed in each monitoring subspace based on those significant fault deviations which can better explain fault characteristics. The action of fault diagnosis is then taken by finding the specific reconstruction model that can well eliminate alarming signals. Critical-to-diagnosis phases are identified to provide more reliable fault isolation results. The proposed in-line fault diagnosis method can give an enhanced understanding of different fault characteristics across different phases and it is easier to identify the fault root cause in each local phase. Its performance on fault diagnosis is illustrated by data from a multiphase batch process, injection molding. Keywords: Multiphase batch process; fault reconstruction; critical-to-diagnosis phase; fault diagnosis; subspace decomposition; multivariate statistical analysis.

*

Corresponding author: Tel: 86-571-87951879, Fax: 86-571-87951879, E-mail address: [email protected] 1

ACS Paragon Plus Environment


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1 Introduction To improve the product quality and maintain the process safety, proper process monitoring and fault diagnosis[1-9], which has become one of the most active research areas in process control over the last few decades, is of importance and necessary for industrial processes. Different methods have been developed and reported for fault detection and diagnosis to solve the concerned problems in various practical applications. Multivariate statistical analysis, such as principal component analysis (PCA)[10] and partial least squares (PLS)[11,12], have been widely used in the field of statistical process monitoring (SPM) for process data analysis and process improvements. As summarized by Qin[1], the tasks involved in SPM typically include fault detection, fault estimation, and fault identification and diagnosis, etc. Owing to the data-based nature of SPM, the multivariate statistical methods extract the underlying characteristics from measurement data in an empirical way, which actually can reveal the important information about process operation status. For fault detection, they define the normal operation regions by accommodating the acceptable process variations. When the process moves outside the desired operating region, it is concluded that an “unusual and faulty” change in process behaviors has occurred. After a fault is detected, it is hoped that people can quickly identify an assignable cause for the abnormality and take the necessary corrective actions to eliminate the abnormal conditions. However, the procedure can not be readily completed only by human operators, especially for those complex chemical plants. There has been tremendous interest in diagnosing the possible root causes of a fault situation. Automatic fault diagnosis strategies for chemical process operations are more desirable and drawing increasing attention. Different methods[13-19] have been reported for fault diagnosis based on historical measurement data, such as fisher discriminant analysis[16,19], steady-state based approach[18], support vector machine[16] and structured residual-based approaches[14,16,19] 2


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etc. In the multivariate statistical process monitoring field, the method of contribution plots[6,20,21], has been widely used to isolate the root faulty variables based on the assumption that the out-of-control monitoring statistics are contributed significantly by those faulty variables. Although this approach does not need prior fault knowledge except for a normal statistical monitoring model, it, however, may result in confusing results because of the correlations across process variables. That is, the contribution from one variable may be propagated to other variables, resulting in “smearing” effects[22] between the contributing and noncontributing variables. If the actual fault directions can be obtained, the fault can be diagnosed without ambiguity[22,23], including recovering fault-free data and estimating the fault magnitude. In the context of PCA models, the concept of fault reconstruction was defined in the work of Dunia and Qin[24] where faults are characterized by fault directions or subspaces to estimate the fault-free part of the measurement data. The procedure to restore normal conditions by applying a corrective change in the data is called data reconstruction, and the procedure for identifying a fault by reconstruction for a given type of faults is called fault diagnosis via reconstruction. In their work, the fault detectability, identifiability and reconstructability were addressed regarding the applicability of reconstruction method. Further, a combined index that mixes together the SPE and Hotelling T2 monitoring statistics was developed for both fault detection and fault reconstruction in the work of Yue and Qin[25]. Also the method for the extraction of fault feature subspace or directions was proposed by performing singular value decomposition on the averaged historical fault data. However, the above work did not consider the multiplicity of fault patterns after the disturbances happened. Considering the varying fault characteristics during process evolvement in continuous processes, a multiple-time-region based fault reconstruction modeling method[26] was developed for online fault diagnosis of continuous processes. They divided the whole fault continuous process into different time regions according to the 3



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changes of fault patterns so that the similar fault characteristics were classified into the same time region while those different ones were grouped into different regions. Multiple reconstruction models were then developed to capture the varying reconstruction relationships for fault diagnosis. For fault reconstruction technique, it is critical to get an accurate characterization of the fault feature directions and subspaces, along which, the fault-free part can be restored so that the out-of-control monitoring statistics can be brought back to normal. In previous work[24,25], the concerned information is in general obtained by performing statistical analysis, such as PCA, on the fault measurement data. However, the conventional PCA based reconstruction modeling methods focus on the general distribution information of fault data and may not well discriminate fault from normal statuses, which thus may not work efficiently to recover the fault-free data as well as the real fault cause. For reconstruction, since the basic idea is to remove the out-of-control monitoring statistics, the fault deviations that are responsible to the alarming signals should be focused on. The calculation of monitoring statistics in fault detection system thus should be considered for the development of fault reconstruction model in fault diagnosis. Moreover, the subject of reconstruction based fault diagnosis may arouse new issues and demand specific solutions when it refers to multiphase batch processes. For batch processes which are common in chemical, pharmaceutical, and food industries, they in general operate in multiple phases which show different underlying characteristics[22]. Multivariate statistical techniques, such as multiway principal component analysis (MPCA)[27] and multiway partial least squares (MPLS)[28] which were introduced by Nomikos and Macgregor for batch process monitoring, may not work efficiently for multiphase batch processes. Considering that the multiplicity of phases is an inherent nature of many batch processes and each phase exhibits different underlying behaviors, phase partition and phase-based monitoring strategies have been developed and 4


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achieved wide applications[7,8,29-32]. Zhao et al.[30] separated between-phase transitions from steady phases and developed sub-PCA monitoring models to capture the changes of process characteristics from one phase to another. Yu and Qin[7] partitioned all the sampling points into different clusters corresponding to different operation phases in the batch process by combining the Gaussian mixture model with hybrid unfolding of a multiway data matrix. Further, Yu developed a novel multiway kernel localized Fisher discriminant analysis (MKLFDA) method for batch process monitoring which can capture process nonlinearity, identify multiple phases and separate normal and faulty batches. Besides the multiplicity of phases for normal batch processes which has been commonly recognized, the phase nature under fault conditions is an interesting issue and should be analyzed and addressed in detail for fault diagnosis. It is natural to imagine that the fault characteristics in each local phase may be neglected if the whole fault batch is treated as a single analysis subject. Moreover, different phases may be of different significance for fault isolation where only in some phases we can well characterize and reconstruct the fault effects, called critical-to-diagnosis phases here. To avoid the bad influences of uncritical phases, it is thus natural to identify the critical phases which are expected to provide more accurate diagnosis results. The fact of multiple fault characteristics across different phases shows the necessity of performing phase based fault diagnosis. In the present work, phase based subspace decomposition and reconstruction modeling method is developed for fault diagnosis in multiphase batch processes. How the batch-wise fault deviations change relative to normal and evolve across different phases should be studied in detail for fault identification. A principal component of fault deviations (PCFD) modeling algorithm is proposed to characterize the alarm-responsible fault effects in principal component subspace (PCS) and residual subspace (RS) of monitoring system respectively. It is based on the basic principle of reconstruction method which is supposed to remove the out-of-control monitoring statistics 5



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of fault detection results. Based on the proposed PCRD algorithm, the phase-based fault diagnosis strategy is then developed. In each local phase, major fault deviations responsible to the out-of-control monitoring statistics in two different monitoring subspaces are extracted by phase based PCFD (PPCFD) algorithm respectively, which can well characterize the fault effects on the abnormal monitoring signals. Critical-to-diagnosis phases are identified which can provide more reliable fault isolation results. For comparison, the multiway version of PCFD algorithm (MPCFD) is also developed for fault diagnosis which treats the whole batch process as a single subject for reconstruction modeling. The rest of this paper is organized as follows. First, principal component analysis based fault detection is preliminarily revisited. Then Section 3 describes the proposed PCFD algorithm. The phase-based PCFD algorithm is then developed in Section 4 for reconstruction modeling and fault diagnosis in batch processes where phase nature of fault characteristics is discussed and a comprehensive subspace decomposition of fault deviations in each phase is conducted using PCFD algorithm. In Illustrations section, its applications to injection molding, a multiphase batch process, are presented, suggesting its feasibility and efficacy for fault diagnosis. Finally, conclusions are drawn in the last section. 2 Principal Component Analysis based Fault Detection In this subsection, the PCA based fault detection system is given a simple revisit. In general, it uses two subspaces (PCS and RS) to monitor different types of process variations. Two different monitoring statistics are used, T2 and SPE, reflecting the abnormal changes in each subspace. Let X be an N × J -dimensional normal data matrix in which the rows are observations (N) and the columns are process variables (J). It is assumed that X is normalized to have zero mean and unit variance. PCA is performed on it to decompose the systematic information and residuals from X : 6


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T = XP

(1)

X = TP T + E = XPP T + E

where, T ( N × R ) are the PCA scores derived from the measurement data X based on loadings P ( J × R ) . R denotes the number of retained principle components (PCs). In this way, the systematic information in X is characterized by TP T and separated from the residuals E , which are deemed to be noise. From the projection perspective, the PCA model can be formulated in another way: % =X % ˆ +X X = XPPT + X ( I − PPT ) = XΩ + XΩ

(2)

where I is a J × J unity matrix. Ω ( PPT ) is the projector with respect to the column % ( I − PPT ) is the corresponding anti-projector. By projecting X on space of P while Ω

different projectors, the original measurement space is partitioned into two different % ), respectively. ˆ ) and RS ( X subspaces, PCS ( X

When a fault occurs, the faulty sample vector ( x ) can be represented as:

x = x ∗ + Σf

(3)

where, x ∗ is the normal portion; Σ is an orthonormal matrix that spans the fault systematic subspace with a dimension of J × R f . R f denotes the major fault directions along which the fault systematic information varies. f represents the fault scores in the fault systematic subspace so that its Euclidean length f

represents the magnitude of the fault.

Consequently, the projections of x onto PCS and/or RS may have significant increases, which could be detected by two different monitoring statistics. Hoteling-T2 is used to detect the deviations in the PCS:

T 2 = x T PΛ −1P T x = Λ −1/2 P T ( x∗ + Σf ) = Λ −1/2 P T x∗ + Λ −1/2 P T Σf 2

2

(4)

where, Λ is the covariance of PCs from the normal training data, which is calculated by

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Λ = TT T ( N − 1) .

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denotes the Euclidean length of a vector.

SPE is a measure to detect the deviations in the RS:

% 2= Ω % ( x ∗ + Σf ) = Ωx % ∗ + ΩΣf % SPE = Ωx 2

2

(5)

From Eqs. (4) and (5), clearly, the inclusion of the fault scores f may result in changes of the above two monitoring statistics. When the increase is significant enough to make the monitoring statistics go out of the confidence regions, the fault is detected. The confidence limits for the two monitoring statistics T2 and SPE can be calculated by a F -distribution and a weighted Chi-squared distribution with significance factor α

( α =0.01 or 0.05)

respectively: 2

T ~

R ( N 2 − 1) N ( N − R)

FR , N − R ,α

SPE ~ g χ h ,α 2

(6)

(7)

where, g = ν / 2m and h = 2(m )2 / ν , in which, m is the average of all the SPE values calculated for normal training data, and ν is the corresponding variance. 3 The Proposed PCFD algorithm

Here, the proposed PCFD algorithm is presented for the development of fault reconstruction models regarding the influences of fault deviations on two different monitoring statistics. The algorithm is used to explain the major fault deviations responsible to out-of-control monitoring statistics in systematic subspace and residual subspace of PCA monitoring system respectively. First, two data sets are prepared, X ( N × J )

and

X f ( N f × J ) , each being composed of the same number of variables and maybe different

number of samples. X denotes the normal data and X f denotes one data set collected from one fault case where subscript f denotes fault data. The data in X ( N × J ) has been 8


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centered and scaled to be of zero mean and unit standard deviation. And then the mean and standard deviation from the normal data are used to preprocess fault data X f ( N f × J ) as what is done for fault detection so that the preprocessed X f can cover the fault deviation information departing from the normal center which is, in general, larger than that covered in normal data set X . To bring T 2 and SPE back to normal, the PCFD reconstruction modeling algorithm is implemented as follows: (1). Development of PCA monitoring models

Perform conventional PCA algorithm on X ( N × J ) to develop monitoring models,

T = XP ˆ T  X = XPP , where T ( N × R ) and P ( J × R ) are principal components (PCs) and the  T E = XPe Pe  ˆ +E X = X corresponding principal loadings; E ( N × J ) and Pe ( J × Re ) are PCA residuals and the corresponding residual loadings where the subscript e denote residual subspace. R is the number of retained PCs in PCS which is determined to keep most of normal variability (85% here). This reveals the first largest variation directions of normal process data which are also the monitoring directions of T2 statistic for fault detection. Re is the number of retained directions in the residual subspace Re = J − R . (2). Projection of fault data on monitoring models

Projecting the fault data X f ( N f × J ) onto the monitoring systematic subspace and

)  X f = X f PPT residual subspace  ( , where the fault deviations that are monitored by T 2 T X = X P P  f f e e statistic and SPE statistic are obtained separately. Also, it is noted that no fault information is 9



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) ( missing since the sum of two different projections ( X f + X f ) is in fact equal to the original

[ P Pe ][ P Pe ]

T

fault data space ( X f ) resulting from orthonormal matrix and thus [ P Pe ][ P Pe ]

T

= PPT + Pe Pe T where

[P Pe ]

is

is identity matrix. The alarming monitoring

statistics are generated by those significant out-of-control fault deviations in systematic and residual subspaces. Therefore, the fault directions that are responsible to the out-of-control monitoring statistics in two different monitoring subspaces are supposed to be extracted. (3). PCA decomposition of fault deviations in PCS

) PCA is performed on X f to extract the first principal directions that can explain the major

relative

fault

deviations

in

systematic

subspace

of

monitoring

system:

)ˆ ) ) ) T ) −1 ) T ) ) ) T X = X = X f Pf Pf ) ) f f Pf ( Pf Pf ) Pf  , where Pf ( J × R f ) are the principal loadings of ) ) )ˆ E f = X f − X f

) fault deviations in PCS, revealing the first largest directions of fault deviations. R f is the number of retained fault PCs that are used to bring the alarming T 2 statistic back to normal.

)ˆ X f denote the principal fault systematic deviations that have been explained by the PCA

) reconstruction model Pf in systematic subspace of monitoring system. The left fault ) ) variations E f after the correction of Pf should be kept within the normal region so that the reconstructed T2 monitoring statistics will stay well within the confidence limit that has been predefined based on normal data X . (4). PCA decomposition of fault deviations in RS

( PCA is performed on X f to extract the first principal directions that can explain the major

relative

fault

deviations

in

residual

subspace

10


of

monitoring

system:

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(ˆ ( ( ( T ( −1 ( T ( ( ( T X = X = X f Pf Pf ( ( f f Pf ( Pf Pf ) Pf  , where Pf ( J × R f ) are the principal loadings of ( ( (ˆ E f = X f − X f

( fault deviations in RS, revealing the first largest directions of fault deviation. R f is the least number of required fault PCs to bring the alarming SPE statistics back to normal, which

( (ˆ satisfies R f ≤ Re . X denote the principal fault systematic deviations that have been f ( explained by the PCA reconstruction model Pf in residual subspace of monitoring system. ( ( The left fault variations E f after the correction of Pf should be kept within the normal region so that the reconstructed SPE monitoring statistics will stay well within the confidence limit that has been predefined based on normal data.

) ) ( ( Based on the above algorithm, two reconstruction models, Pf ( J × R f ) and Pf ( J × R f ) , are developed for the reconstruction of T 2 and SPE monitoring statistics respectively. Those fault directions which have caused out-of-control T 2 /SPE statistic are separated from the others and used to correct abnormal data for fault reconstruction. For fault diagnosis visa reconstruction, it tries to remove the alarming signals of monitoring statistics by fault data correction. Therefore, the fault effects responsible to alarming signals should be analyzed from the perspective of fault detection. For the calculation of two monitoring statistics, the fault data are projected to monitoring models in PCS and RS respectively for fault detection. Therefore, for fault correction, the fault information in two different subspaces should be analyzed which is the projections of fault data onto two monitoring subspaces as calculated in Step (2). So PCA is performed on the two different projections as shown in Steps (3) and (4) of PCFD algorithm. In this way, the major fault effects that are responsible to alarming signals of two different monitoring statistics can be more efficiently extracted and used to remove alarms respectively. In

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comparison, direct PCA on fault data only extracts the general fault variations, which, however, can not clearly distinguish different fault variations that are responsible to alarming T2 and SPE respectively. Also, those major general fault deviations by PCA do not necessarily

have the largest influences on one specific monitoring statistic. The underlying principle of the proposed PCFD algorithm is simply illustrated in Figure 1 taking the example for the fault reconstruction modeling of T2. A two-dimensional normal PCA monitoring systematic subspace ( P ) is spanned by two orthogonal distribution directions, p1 and p 2 , as indicated by the grey ellipse. The normal region for the acceptable process variations is enclosed in this systematic plane. p e is the one-dimensional PCA residual model which is orthogonal to the plane. For the calculation of T2 statistic, the fault data which are not plotted here are projected onto the PCA systematic subspace forming a

) plane as indicated by the white ellipse X f . Out-of-control T2 statistic indicates some ) abnormality. To reveal the effects of fault deviations on T2, X f is then decomposed by PCA ) ) where two loadings p f ,1 and p f ,2 are available, which are responsible to the alarming T2 statistic and could be the possible fault reconstruction directions. By removing the abnormality along one or both of the two candidate directions, the out-of-control T2 can be brought below the confidence limit. The similar underlying principle is used for the reconstruction modeling of SPE in residual subspace of PCA monitoring system. 4 Phase-based PCFD Algorithm for Fault Diagnosis In this section, multiphase nature of fault characteristics is analyzed and phase based reconstruction models are developed based on the proposed PCFD algorithm for fault diagnosis in multiphase batch processes. 4.1 Phase nature of fault characteristics In general, many batches characterized by prescribed processing of raw materials for a 12


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finite duration to convert them to products are carried out in a sequence of discrete steps, which are called phases caused by operational or phenomenological (chemical reactions, microbial activities, etc.) events. For normal batch processes, it commonly knows the multiplicity of phases is an inherent nature of many batch processes and each phase exhibits different operation behaviors. As mentioned before, data collected from each processing unit and each phase may have different variable correlation structures requiring different monitoring actions. They also provide important information about the progress of the batch. If they are treated together, the effects of different phases may be hidden, provided less information for process understanding and analysis. Therefore, it is better to divide a normal batch process into several local phases and multiple phase models can thus be developed in different phases. In this way, the behavior of each phase can be seen and thus more comprehensive process understanding can be expected. A variety of techniques have been suggested for modeling and monitoring multiphase processes based on multivariate statistical analysis techniques[7,8,29-33]. For post-analysis purpose, most of them depend on blocking/grouping of the variables in each phase, such as multiblock statistical analysis method[33] and multiway modeling method[29,34] in each phase, to improve process monitoring insight. Several extensions of the conventional multiblock and multiway algorithms have also been reported[33], such as a hierarchical principal component analysis (HPCA) technique, pathway multiblock method, and consensus multiblock method, to name a few. They were developed to solve different monitoring problems in multiphase batch processes. Under fault status, each phase may be influenced differently by the disturbances, regarding the fault magnitude, fault correlations and fault evolvement speed, to name a few. That is, each phase presents different fault effects on the monitoring subspaces, revealing different reconstruction relationships. For fault diagnosis, the nature of fault phase should be 13



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considered so that a more specific fault location can be identified as well as the fault cause. Moreover, different phases may play different roles in fault diagnosis. For some phases, the relative fault deviations can be well modeled for easier reconstruction and more reliable fault diagnoses, called critical-to-diagnosis phases here. In contrary, for some phases, the fault deviations may be hard to be characterized which may not be useful but will result in confusing diagnosis results, called uncritical phases. The identification of critical-to-diagnosis phases is thus necessary to remove the bad influences of uncritical phases and improve fault diagnosis performance. In multiphase modeling, one important issue is how to get the phase marks (i.e., phase division) before designing the phase model. Various strategies[29,30] have been reported from different viewpoints and based on different principles, providing a rich database for phase division. In the present work, the phase mark is identified from normal batches using indicator variables (stroke and displacement as the indicator variables for the case study in this work), which in general agrees well with the actual operation phases. Then the consecutive fault patterns within the same phase can be collected as one modeling data set and the fault patterns from different phases are classified into different data sets. C phases, representing C kinds of fault characteristic or relative fault deviations in comparison with the normal case, can be modeled separately. The advantage of phase based fault diagnosis is that one can better capture the changing fault deviations and get more specific fault isolation in different phases. 4.2 Phase-based fault subspace decomposition Based on the separation of multiple phases from the whole fault batch processes, each fault phase will be analyzed and modeled for fault diagnosis. From the perspective of fault detection, the fault deviations responsible to the out-of-control monitoring statistics are in fact the concerned fault effects for reconstruction. To remove the fault influences and bring 14


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the alarming signals back to normal, the phase-based relationships between normal and fault patterns will be analyzed using the proposed PCFD algorithm. Using the proposed algorithm, the out-of-control fault deviations within each local phase will be approximated by one representative statistical model while those fault patterns across different phases will be characterized by multiple statistical models. Fault diagnosis is then performed in each local phase by finding the fault reconstruction directions that can well reconstruct the fault effects relative to normal. Critical-to-diagnosis phases will be identified which are deemed to yield more reliable fault isolation performance. Phase based fault modeling and diagnosis allows different diagnosis actions in different phases and can reflect the changes of inherent fault characteristics to improve the fault understanding and diagnosis efficiency. Figure 2 presents the proposed modeling procedure for reconstruction based fault diagnosis. In each batch run, assume that J process variables are measured online at k=1,2,…, K time instances throughout the batch. Then, process observations collected from similar I normal batches can be organized as a three-way array X( I × J × K ) . At each time, the time-slice can be separated as X k ( I × J ) where subscript k denotes process time. The means of each column are subtracted to approximately eliminate the main non-linearity. And each variable is scaled to unit variance to handle different measurement units, thus giving each equal weight. In the present work, the batches are of equal length without special declaration so that the specific process time can be used as an indicator for data normalization. Similarly, for each fault type, the process observations can be arranged into a three-way array X f ( I f × J × K ) . It is noted that the number of fault batches can be different from that in normal case but the variable dimension should be consistent. At each time, the time-slice fault data matrix X f ,k ( I f × J ) is preprocessed using normalization information obtained from

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normal data, covering the fault deviations relative to normal status. Then both normal data and fault data are arranged by batch-unfolding those normalized time slices in each local phase,

X c ( I × JK c )

and

X f ,c ( I f × JK c ) , where subscript c denotes phase and

c=1,2,3, …,C. K c denotes phase duration. As shown in Figure 3, the fault modeling data

X f ,c ( I f × JK c ) is arranged and the normal data are also arranged in the similar way. They are then used for development of monitoring models and reconstruction models as well as fault diagnosis via reconstruction in each phase. The specific procedure is described as follows. (1), In each phase, the phase-representative PCA monitoring models (systematic model

Pc ( JK c × Rc ) and residual model Pe ,c ( JK c × Re ,c ) ) are developed from normal phase data X c ( I × JK c ) for fault detection where Rc and Re ,c denote the model dimension in two different monitoring subspaces; the subscript e means the residual subspace. Based on the developed monitoring models, two monitoring statistics are calculated as below:

t c T = x c T Pc e c T = x c T ( I − Pc Pc T ) Tc 2 = ( t c − tc ) Σc −1 ( t c − tc ) T

(8)

SPEc = e c T e c

where, t c ( Rc × 1) is the PCA score vector extracted from the current phase data ( x c ( JK c × 1) ); tc denotes the mean score vectors of I batches in Phase c, which is in general zero vector resulting from the mean-centering during data preprocessing. Σc is the phase variance-covariance matrix of scores in PCA systematic subspace for normal training data; and e c T ( JK c × 1) is the final PCA residual in each phase space. The phase-representative confidence limits are also calculated for two monitoring statistics

Tc 2 and SPEc . In each phase, the batch-wise variation of original measurement samples is 16


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deemed to follow a multivariate Gaussian distribution. This premise provides an important basis for deriving the confidence limits of monitoring statistics. So the phase-representative control limit of T2 in the systematic subspace is defined by the F -distribution with α as the significance factor[35]. Similarly, in the residual subspace, the phase-representative confidence limit of SPE can be approximated by a weighted Chi-squared distribution[36]. (2), By performing the proposed PCFD method shown in Section 3 on the phase-based fault

) ) data and normal data, we can get the reconstruction models ( P f ,c JK c × Rc

(

)

and

( ( P f ,c JK c × Rc ) that capture the fault effects on monitoring statistics T2 and SPE respectively.

(

)

) ( Rc and Rc indicate the dimensions of the two fault reconstruction models. They are used to correct fault effects, x c T Pc Pc T and x c T Pe ,c Pe ,c T , which are the projected fault data onto PCS and RS of PCA monitoring system:

(

) ) ) −1 ) ) e c T = x c T Pc Pc T I − Pf ,c ( Pf ,c T Pf ,c ) Pf ,c T

(

)

( ( ( −1 ( ( e c T = x c T Pe ,c Pe ,c T I − Pf ,c ( Pf ,c T Pf ,c ) Pf ,c T

)

(9)

) ( ) ( where, ec and e c are the corrected data by Pf ,c and Pf ,c in two different subspaces respectively.

) ( (3) The corrected data e c and ec are then re-projected onto the PCA monitoring system developed from normal data in Step (1) as shown in Eq. (8) to update two monitoring statistics:

) t c *T = e c T Pc ( e c *T = e c T ( I − Pc Pc T ) Tc*2 = ( t c * − tc ) Σc −1 ( t c* − tc ) T

(10)

SPEc* = e c*T e c * In comparison with the original alarming monitoring statistics, both new monitoring

17



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statistics Tc*2 and SPEc * are expected to be well within the predefined confidence limits if the right fault reconstructed models are used. The proper dimension of reconstruction models in each phase is thus determined to be the least number of reconstruction directions that are required to bring the previous alarming monitoring statistics under the limits. Also they can be modified by cross-validation to get the best reconstruction results for validation data. 4.3 End-of-phase fault detection and diagnosis In previous section, the core algorithm has been formulated and the phase based reconstruction models are prepared for each fault. In post-analysis of abnormalities, the fault diagnosis is performed at the end of each phase when fault is detected in the current local phase. For fault detection, the two monitoring statistics Tc 2 and SPEc are calculated as shown in Eq. (8) and are compared with their respective predefined confidence limits based on normal data. In each phase space, if both statistics are within the predefined normal regions, the current operation in this phase space can be deemed to be operating following the normal rule. On the contrary, when out-of-control monitoring alarms are issued, some abnormal behaviors are happening in the current local phase. Based on the indicated out-of-control monitoring statistics, T2 and/or SPE, fault reconstruction should be performed. It tries to find out the specific fault reconstruction model in each local phase that can well correct the alarming monitoring statistics. The fault cause can thus be identified as indicated by the good reconstruction model. For each fault batch, phase-based fault reconstruction is performed at the end of each phase to check the possible fault cause. It is hoped that the fault causes can be accurately determined and agree well with the real case. Since the fault cause is directly judged based on the fault reconstruction results, it is hoped that only the right reconstruction model can correct the fault. Therefore, in the present work, two types of fault diagnosis analyses are performed 18


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to evaluate the performance of fault reconstruction: (a) Same-fault analysis where a fault reconstruction model is identified on one fault’s training dataset and the normal dataset and then applied to the testing data of the same fault case; (b) Cross-fault analysis where a fault reconstruction model is identified on one fault’s training dataset and the normal dataset and then applied to the testing data of a different fault case. Clearly, they are distinguished by whether the testing data are from the same fault data from which the reconstruction model is identified. For same-fault analysis, the right reconstruction model is used and good diagnosis performance means alarming statistics should be eliminated. In contrast, for cross-fault analysis, the wrong reconstruction model is used so that good diagnosis performance means none of alarming statistic signals can be eliminated. Correspondingly, the evaluation index for fault reconstruction can be defined by calculating two ratios, the missing reconstruction ratio (MRR) ( R* % ) and false reconstruction ratio (FRR) ( R • % ):

R* % = •

R %=

Nf* Nf Nf• Nf

× 100 (11)

× 100

where N f * , indicating the missing reconstruction, is the number of out-of-control T2/SPE statistics that fail to be brought back to normal region using the right reconstruction model.

N f • , revealing the false reconstruction, is the number of out-of-control T2/SPE statistics that are falsely brought back to normal region using the wrong reconstruction model. N f is the total number of fault batches. Clearly, MRR is for the evaluation of same-fault analysis results and FRR is for cross-fault 19



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analysis results. Smaller missing and false reconstruction ratios mean better fault diagnosis performance is achieved. Ideally, it is hoped that 0% missing and false reconstruction ratios (MRR and FRR) can be obtained, which means that all out-of-control alarming statistics can only be reconstructed using the right fault reconstruction model and thus the fault cause is diagnosed correctly. However, due to various factors, such as the similarity between different fault types, unexpected variations different from fault training samples, etc., it is impractical to get such perfect reconstruction performance. It is noted that using phase-based method, the two ratios can be calculated for each local phase, revealing their respective significance for fault diagnosis. Considering different phases may present different fault diagnosis performance, critical-to-diagnosis phases can be identified based on reconstruction performance for fault training data by checking the two reconstruction ratios. The critical-to-diagnosis phases are supposed to have the lowest sum values of MRR and FRR indices. For phases with the same results, both can be judged to be critical phases. In critical phases, the fault diagnosis results are more reliable whereas the results in those uncritical phases may not be trusted enough. Also, for T2 and SPE, their fault reconstruction ratios may be different, revealing different difficulties in reconstructing the fault effects in systematic and residual subspaces of monitoring system. So the identified critical phases may be different for the reconstruction of the two monitoring statistics. Figure 4 presents the flow diagram of the proposed post-analysis fault diagnosis procedure. (1). For the new fault phase sample where the process abnormality is detected by T2 and/or

SPE monitoring statistics, x new ( JK c × 1) , the specific phase is indicated by the current process time. Preprocess it with the corresponding normalization information from normal training data, x new ( JK c × 1) . (2). For each alternative fault type, if the current phase is critical which has been identified based on training data, based on the indicated out-of-control monitoring statistics, T2 and/or 20


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SPE, fault reconstruction is performed as shown in Eqs. (9) and (10) using each of possible phase-based reconstruction models developed from multiple candidate faults. It tries to find by which fault model and in which phase the specific faulty monitoring statistics can be brought back to normal. It is noted for each alternative fault, only the results in critical phases can be trusted. (3). If the concerned out-of-control statistics can be reduced to below the confidence limits in each critical phase, the concerned fault is initially diagnosed. Further, check whether the fault diagnosis results are the same as those in previous critical phases and whether the results are consistent with each other for two monitoring statistics if both of them can detect the process abnormalities. (4). If the concerned faulty statistics cannot be reduced to below the normal region by any fault model in the current critical phase, then the current batch process can be considered to be operating under a new fault status which does not belong to the current fault library. Maybe fault model updating and supplement is needed which is a different issue and not addressed here.

5 Illustrations and Discussions 5.1 Process description Injection molding, a key process in polymer processing, transforms polymer materials into various shapes and types of products. An injection molding process consists of three major operation phases, injection of molten plastic into the mold, packing-holding of the material under pressure, and cooling of the plastic in the mold until the part becomes sufficiently rigid for ejection. Besides, plastication takes place in the barrel in the early cooling phase, where polymer is melted and conveyed to the barrel front by screw rotation, preparing for next cycle. It is a multiphase batch process and has been widely used in our previous work for process monitoring and quality analysis [37]. An injection molding machine is well instrumented in a 21



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lab of Hong Kong University of Science and Technology and the authors have rich expertise knowledge of the injection molding. It can be readily implemented for experiments, in which, all key process conditions such as the temperatures, pressures, displacement and velocity can be online measured by their corresponding transducers, providing abundant process information. It provides an ideal candidate for application and verification of the proposed phase-based reconstruction modeling and fault diagnosis strategy. The material used in this work is high-density polyethylene (HDPE). Nine process variables are selected for modeling, which can be collected online with a set of sensors. They can reveal the process operation status and are shown in Table 1. Totally 47 normal batch runs are conducted under normal operation conditions which are used to developed the PCA monitoring system. Beside, three types of faults are considered, each composed of 48 batches, including: (1) Thermocouple fault simulation where the thermocouple measurement value of nozzle temperature is shown to be 90% of actual value. For this fault case, the real nozzle temperature will be higher than that in normal case since the nozzle will be heated until the shown temperature reaches the setting point which actually already goes beyond the normal value. (2) Heating fault simulation where the power to heat the nozzle is reduced to 80% of that under normal status. In this fault case, due to the sacrificed heating power, the real nozzle temperature is lower than the normal value. (3) Material fault simulation where blue polypropylene (PP) is added to the original material HDPE. In this fault case, due to different density and viscosity values between HDPE and PP, the process may show different characteristics under the same condition. All batches are unified to have even duration (526 samples in this experiment), which, thus, results in the three-way descriptor array X( I • × 9 × 526) where I • denotes the number of 22


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batches for both normal and fault cases. For each fault case, thirty-nine batches are used for modeling, while the other nine cycles are used for model validation. It is noted that for all fault cases the disturbances act throughout the whole batch so that the different effects on multiple phases can be studied. 5.2 Phase based fault reconstruction modeling In the present work, four main phases (injection, packing-holding, plastication and cooling) are considered for the test of the proposed reconstruction modeling and fault diagnosis method. Using process time, stroke and displacement as the indicator variables, the landmarks of different operation phases are identified. Four phases are separated from the whole batch process, which cover 68, 160, 87 and 211 measurement samples respectively. In each phase, batch-wise data unfolding is performed as shown in Figure 3. Four phase-representative normal data sets are obtained corresponding to each phase,

X1 ( I • × 612) , X 2 ( I • ×1440) , X3 ( I • × 783) and X 4 ( I • ×1899) where the subscript c is used to indicate the phase index (c=1,2,3,4) and I • = 47 . The number of columns in each normal phase data X c is calculated by the product of phase duration and variable number in each time-slice. Similarly, the phase-representative fault data are arranged as X f ,c ( I • × 9 K c ) for each fault case where K c indicates the time duration of each phase and I • = 48 . The normal phase-representative data sets are normalized to be of zero mean and unit variance which are then used for the development of PCA monitoring system. Here the dimension of PCA systematic model is chosen to keep up to 85% of the normal process variation for the calculation of T2. The left information is kept in the PCA residual subspace for the calculation of SPE monitoring statistic. For the fault data, 39 batches are used as training data while the left nine batches are for testing. First all variables in the fault space are preprocessed by the normalization information obtained from normal training data, covering

23



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the fault characteristics departing from normal status. Then the abnormality in the testing batches is detected by projecting them onto the lower-dimensional PCA systematic subspace and residual subspace defined based on good phases. Based on the monitoring results, all faults can be clearly detected by the phase-based monitoring system where multiple monitoring results can be calculated in different phases for each batch. Also the multiway PCA monitoring model is developed for fault detection where only single monitoring result can be obtained for each testing batch and all the abnormality can be detected by the out-of-control signals. The fault batches will be reconstructed for fault diagnosis using the proposed phase based PCFD (PPCFD) in comparison with multiway PCFD (MPCFD) reconstruction method. It is noted that for PPCFD method, multiple fault diagnosis results can be obtained in different phases for each batch while for MPCFD method which treats the whole batch process as a single subject and applies PCFD algorithm for reconstruction modeling, only one fault diagnosis result can be available for each batch. For the proposed PPCFD method, the critical-to-diagnosis phases are identified for the reconstruction of different faults which should give the lowest MRR and FRR values. As shown in Table 2, the critical-to-phase identification results are summarized for each fault type. For simplicity, only the results for critical phases are shown. For different faults, different phases are judged to be critical. Also, it is noted that not both monitoring statistics can be reconstructed. For example, only T2 can be reconstructed for Fault 1 while only SPE can be corrected for Fault 2 in their respective critical phases. Here we define the specific statistic that can be believed for fault diagnosis as critical reconstruction statistic. So for the diagnosis of different faults, we will focus on different critical statistics and different critical phases. Also for the concerned three faults, the dimensions of reconstruction models are shown in Table 2. Clearly, for different faults, the fault deviations in PCA systematic subspace and residual subspace are decomposed differently. Comparatively, the fault 24


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reconstruction model is also developed using multiway modeling method which treats the whole batch process as a single subject. It in general needs more directions for the reconstruction. For fair comparison, the best reconstruction results are shown for the two methods. Comparing the results for the two methods, PPCFD method can focus on the critical process periods/phases and extract the important information of fault deviations, while MPCFD has to work hard to extract the key information from the whole batch process. It is nature to guess that the information of critical phases may be hidden or badly influenced by the other unessential phases if different phases are not separated from each other. 5.3 Phase based fault diagnosis For fault diagnosis, despite impractically, it is ideally hoped that only the right reconstruction model can correct the faulty data, i.e., removing the out-of-control statistical signals; and those wrong reconstruction models can not restore the normal monitoring statistics to avoid false isolation of fault cause. They are checked by same-fault analysis and cross-fault analysis respectively. As shown in Figure 5, the monitoring and diagnosis results for three different faults using the models developed from Fault 1 are presented for testing batches in the critical phase of Fault 1. For Fault 1, it is same-fault analysis while for the other two faults, Faults 2 and 3, it is cross-fault analysis. In Phase IV which has been identified as the critical-to-diagnosis phase of Fault 1, all the out-of-control T2 statistic values are well brought back to normal for testing batches of Fault 1 and not corrected for testing batches of Fault 2 and Fault 3. Therefore, the fault cause is correctly identified as Fault 1. Also the results for multiway method are comparatively shown in Figure 6 where the real fault cause is also identified since only the reconstruction models of T2 (critical-to-diagnosis statistic as judged in Table 2) developed from Fault 1 can correct the alarming signals. So for the diagnosis of Fault 1, PPCFD and MPCFD methods give the similar results. The results for identification of Fault 2 using the two different methods are compared in 25



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Figures 7 and 8. For PPCFD method, in Phase I which identified as the critical phase for Fault 2, the same-fault analysis results shown in Figure 7 (b) denotes only one batch is missed for the reconstruction of critical statistic SPE. The cross-fault analysis results shown in Figures 7 (a) and (c) reveal that no alarming SPE values are falsely corrected. Therefore, the fault cause is correctly identified as Fault 2. The reconstruction results of T2 will not be used for fault diagnosis since it is not critical reconstruction statistic which has been indicated by the results shown in Table 2. In comparison, for MPCFD method as shown in Figure 8, the diagnosis results of T2 can not clearly distinguish Fault 2 from Fault 3 while the reconstruction of SPE in same-fault analysis reveals too high missing ratio. It means that the MPCFD models developed from Fault 2 do not work well for identification of real fault cause. The same-fault analysis results for all fault cases are summarized in Table 3 where the missing reconstruction ratio (MRR) is calculated in critical phases and compared between the concerned two methods, PPCFD and MPCFD. Besides, to illustrate the superiority of PCFD algorithm for reconstruction modeling, phase based PCA (PPCA) models are also developed for reconstruction based fault diagnosis. It is hoped that the right fault models can bring out-of-control monitoring statistics back to normal as many as possible, giving a MRR value as low as possible. Clearly, PPCFD method shows more critical reconstruction statistics and lower MRR values which means better reconstruction performance in same-fault analysis is achieved in comparison with that by MPCFD and PPCA methods. For PPCA method, only T2 is useful for fault reconstruction in critical phases for each fault type since this method does not distinguish the fault effects in monitoring PCS and RS. The cross-fault analysis results are summarized in Table 4 where PPCFD, PPCA and MPCFD methods are compared. In general, it is hoped that the wrong fault models can not recover the normal part so that the fault cause is not falsely identified. From the available 26


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numeric values, PPCFD method can give better false reconstruction ratio (FRR) than that of PPCA and MPCFD methods in cross-fault analysis. MPCFD method can not be trusted for the diagnosis of Fault 2 where neither of T2 and SPE statistics can be used for reconstruction based fault diagnosis. Combining the same-fault analysis and cross-fault analysis results, the fault cause is better identified by PPCFD method in comparison with PPCA and MPCFD methods. Based on the above results, clearly, by the consideration of phase nature, the changing fault characteristics are better distinguished and the developed local phase models are more accurate. Also, the underlying fault characteristics can be well captured and understood by PCFD algorithm in comparison with conventional PCA algorithm. The comparison results have demonstrated that the separation of different local phases and analysis of fault effects in different monitoring subspaces can make fault reconstruction easier and thus fault diagnosis better.

6 Conclusions In the present work, the phase nature of fault batch processes is analyzed using a principal component of fault deviations (PCFD) algorithm regarding the reconstruction of fault effects for fault diagnosis. First, fault effects in two different monitoring subspaces are analyzed by PCFD algorithm for reconstruction modeling in each phase. Each fault phase can be characterized by a representative statistical model and their different roles in fault diagnosis are also investigated. Critical phases are identified and can be focused on to provide more reliable fault diagnosis results. It is found that each fault may distort two monitoring subspaces and multiple phases differently so that the separation of different reconstruction subspaces and different phases can improve the fault understanding and diagnosis efficiency. The case study on a multiphase batch process, injection molding, has demonstrated the performance of phase-based fault diagnosis method. 27



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Acknowledgement This work is supported by Program for New Century Excellent Talents in University (NCET-12-0492), Zhejiang Provincial Natural Science Foundation of China (LR13F030001), the Fundamental Research Funds for the Central Universities (2012QNA5012) and the Foundation of Key Laboratory of System Control and Information Processing, Ministry of Education, P.R. China.

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List of Figure Captions Figure 1 Illustration of PCFD based fault reconstruction modeling in PCA systematic subspace ( p1 and p 2 are the two orthogonal principal directions in PCA model P ; p e is the one-dimensional PCA residual model; p1 and p 2 are the two orthogonal principal

) directions decomposed from the fault deviations X f which are obtained by projecting faulty data onto the two-dimensional PCA systematical model P .) Figure 2 Flow chart of the proposed phase based fault reconstruction modeling method Figure 3 Illustration of the phase-based data arrangement in cth phase using batch-wise unfolding and data normalization Figure 4 Flow chart of end-of-phase fault diagnosis procedure Figure 5 Fault reconstruction results for (a) Fault 1 (b) Fault 2 and (c) Fault 3 (left: the original fault detection results; right: the reconstruction results) using models developed from Fault 1 in critical-to-diagnosis phase (Phase IV) of Fault 1 and nine testing batches using phase based fault diagnosis method (Black dot line: monitoring/reconstructed statistics; red dashed line: the 99% control limit) Figure 6 Fault reconstruction results for (a) Fault 1 (b) Fault 2 and (c) Fault 3 (left: the original fault detection results; right: the reconstruction results) using models developed from Fault 1 and nine testing batches using multiway fault diagnosis method (Black dot line: monitoring/reconstructed statistics; red dashed line: the 99% control limit) Figure 7 Fault reconstruction results for (a) Fault 1 (b) Fault 2 and (c) Fault 3 (left: the original fault detection results; right: the reconstruction results) using models developed from Fault 2 in critical-to-diagnosis phase (Phase I) of Fault 2 and nine testing batches using phase based fault diagnosis method (Black dot line: monitoring/reconstructed statistics; red dashed line: the 99% control limit) Figure 8 Fault reconstruction results for (a) Fault 1 (b) Fault 2 and (c) Fault 3 (left: the 32


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original fault detection results; right: the reconstruction results) using models developed from Fault 2 and nine testing batches using multiway fault diagnosis method (Black dot line: monitoring/reconstructed statistics; red dashed line: the 99% control limit)

List of Table Captions Table 1 Nine process variables for injection molding process Table 2 Fault reconstruction modeling results (dimension of reconstruction model and critical-to-diagnosis phases) for phase based method and multiway method Table 3 Same-fault diagnosis results evaluated by missing reconstruction ratio (MRR) (%) for phase based method in critical phases and multiway method Table 4 Cross-fault diagnosis results evaluated by false reconstruction ratio (FRR) (%) for (a) phase based method in critical-to-diagnosis phases (b) multiway method

33



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

pe

) p f ,1

p1

P

) Xf

) p f ,2 p2

Figure 1 Illustration of PCFD based fault reconstruction modeling in PCA systematic subspace ( p1 and p 2 are the two orthogonal principal directions in PCA model P ; p e is the one-dimensional PCA residual model; p1 and p 2 are the two orthogonal principal

) directions decomposed from the fault deviations X f which are obtained by projecting faulty data onto the two-dimensional PCA systematical model P .)

34


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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


Measurement data X and X f for the normal case and one fault case

Data normalization for X ; preprocess X f using the normalization information from X

Phase division and phase data unfolding

Phase based monitoring model development based on X

Phase based fault reconstruction modeling using PCFD algorithm

Fault correction on X and X f

Update monitoring statistics using phase based monitoring models

Critical-to-diagnosis phase identification

Fault reconstruction model library archiving

Figure 2 Flow chart of the proposed phase based fault reconstruction modeling method

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Time Kc

X f ,c ( I f × J × K c ) Batches If

Variables J

If

X f ,1,c J

1

X c ( I × JK c )

I

X1,c

If

X 2,c

X f ,1,c 1

X f , k ,c Kc J

2J

J

1

X f ,c ( I f × JK c )

X f ,2,c

X k ,c Kc J

2J

X f ,2,c J

X f , k ,c

2J

Kc J

Figure 3 Illustration of the phase-based data arrangement in cth phase using batch-wise unfolding and data normalization

36


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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


New data x new

Process monitoring by PCA models

Yes

Monitoring statistics stay below confidence limits? No Some fault is detected

No

Is it in critical-to-diagnosis phase? Yes Fault reconstruction using candidate models

Corrected fault data are re-projected onto PCA monitoring models

Reconstructed monitoring statistics are below the confidence limits？

No

Unknown fault case; considering library updating

Yes Possible fault causes are preliminarily known

Satisfying the consistency principle?

No

Yes Fault cause is determined

Figure 4 Flow chart of end-of-phase fault diagnosis procedure

37



25 Reconstructed T

2

200

T

2

150 100 50 0 5 Batches

15 10

10

0

5 Batches

10

0

5 Batches

10

0

5 Batches

10

0

5 Batches

10

500 Reconstructed SPE

2000 1500 SPE

20

5 0

1000 500 0

400

300

200 0

5 Batches

10

(a)

300 Reconstructed T

2

600

T

2

400

200

0

200

100

0 0

5 Batches

10


5000 4000 SPE

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

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3000 2000 1000 0

800 600 400 200

0

5 Batches

10

(b)

38


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70 Reconstructed T

2

100

T

2

80 60 40 20

60 50 40 30 20

0

5 Batches

10

0

5 Batches

10

0

5 Batches

10


1000 800 SPE

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


600 400 200

400

300

200 0

5 Batches

10

(c) Figure 5 Fault reconstruction results for (a) Fault 1 (b) Fault 2 and (c) Fault 3 (left: the original fault detection results; right: the reconstruction results) using models developed from Fault 1 in critical-to-diagnosis phase (Phase IV) of Fault 1 and nine testing batches using phase based fault diagnosis method (Black dot line: monitoring/reconstructed statistics; red dashed line: the 99% control limit)

39



25

x 10 Reconstructed T

T

2

2

30

1

20

10

0

0 0

5 Batches

10

0

28

10

5 Batches

10

5 Batches

10

5 Batches

10

x 10 Reconstructed SPE

10

6 SPE

5 Batches 26

x 10 8

4 2

5

0

0 0

5 Batches

10

0

(a) 25

x 10 300 Reconstructed T

2

6

T

2

4

2

200

100

0

0 0

5 Batches

10

0

30

27

x 10

x 10 8 Reconstructed SPE

4 3 SPE

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

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2 1 0

6 4 2 0

0

5 Batches

10

0

(b) 40


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24


2

3

T

2

2

1

0

80 60 40 20

0

5 Batches

10

0

28

5 Batches

10

5 Batches

10

27

x 10


6 4 SPE

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


2

0

1.5 1 0.5 0

0

5 Batches

10

0

(c) Figure 6 Fault reconstruction results for (a) Fault 1 (b) Fault 2 and (c) Fault 3 (left: the original fault detection results; right: the reconstruction results) using models developed from Fault 1 and nine testing batches using multiway fault diagnosis method (Black dot line: monitoring/reconstructed statistics; red dashed line: the 99% control limit)

41



23


2

4

2

3 T

2 1

400

200

0

0 0

5 Batches

10

0

5 Batches

10

0

5 Batches

10

27


SPE

10

5

1000

500

0

0 0

5 Batches

10

(a) 24


2

4

T

2

3 2 1 0

30 20 10 0

0

5 Batches

10

0

5 Batches

10

0

5 Batches

10

28


4 3 SPE

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

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2 1 0

200 150 100 50

0

5 Batches

10

(b)

42 ACS Paragon Plus Environment

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23


2

15

T

2

10

5

0

30

20

10 0

5 Batches

10

0

5 Batches

10

0

5 Batches

10

27


8 6 SPE

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


4 2 0

300

200

100 0

5 Batches

10

(c) Figure 7 Fault reconstruction results for (a) Fault 1 (b) Fault 2 and (c) Fault 3 (left: the original fault detection results; right: the reconstruction results) using models developed from Fault 2 in critical-to-diagnosis phase (Phase I) of Fault 2 and nine testing batches using phase based fault diagnosis method (Black dot line: monitoring/reconstructed statistics; red dashed line: the 99% control limit)



25

x 10 Reconstructed T

T

2

2

200

1

150 100 50

0

0 0

5 Batches

10

0

5 Batches

10

5 Batches

10

0

5 Batches

10

0

5 Batches

10

28

25

x 10


8

SPE

6 4 2 0

4

2

0 0

5 Batches

10

0

(a) 25


2

6

T

2

4

2

0

20

10

0 0

5 Batches

10

30


4 3 SPE

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

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2 1 0

4000

2000

0 0

5 Batches

10

(b)


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24


2

3

T

2

2 1

40 20

0

0 0

5 Batches

10

0

5 Batches

10

0

5 Batches

10

28

x 10

30


6 4 SPE

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


2 0

20

10 10

10 0

10 0

5 Batches

10

(c) Figure 8 Fault reconstruction results for (a) Fault 1 (b) Fault 2 and (c) Fault 3 (left: the original fault detection results; right: the reconstruction results) using models developed from Fault 2 and nine testing batches using multiway fault diagnosis method (Black dot line: monitoring/reconstructed statistics; red dashed line: the 99% control limit)



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

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Table 1 Nine process variables for injection molding process No.

Variable’s descriptions

Unit

1

Pressure valve opening

%

2

Flow valve opening

%

3

Screw stroke

4

Screw velocity

5

Injection pressure

6

Nozzle temperature

7

Barrel temperature zone 1 °C

8


9


mm mm/sec Bar °C

Table 2 Fault reconstruction modeling results (dimension of reconstruction model and critical-to-diagnosis phases) for phase based method and multiway method Fault No.

Method PPCFD MPCFD

Fault 1

Fault 2

T2

T2

SPE

Fault 3 T2

SPE

Dim

CP No.

Dim

CP No.

Dim

CP No.

Dim

CP No.

3/1899*

IV

---

---

---

---

250/612

I

130/4734*

---

---

---

*

Dim

SPE CP No.

----4/1899 IV 69/4734

Dim

CP No.

260/612 160/1899 ---

I IV

The fraction denotes the ratio between the dimension of reconstruction model and the number of modeling variables --- It denotes that the reconstruction statistics of the concerned fault are not critical and believable. So they are not shown here.


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Table 3 Same-fault diagnosis results evaluated by missing reconstruction ratio (MRR) (%) for phase based method in critical phases and multiway method Model No.

Fault 1

T2

Fault 2

SPE T2

Fault 3

T2

SPE

SPE

Method PPCFD MPCFD PPCA

IV* 0 0 IV 66.67

IV ----IV ---

I I --- 11.11 ----I I 0 ---

I ---

IV 0 0

I 0

IV 33.33

*

I 0

IV 0

--I IV --- ---

It denotes the critical-to-diagnosis phases --- It denotes that the reconstruction statistics of the concerned fault are not critical and believable.



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

Table 4 Cross-fault diagnosis results evaluated by false reconstruction ratio (FRR) (%) for (a) phase based method in critical-to-diagnosis phases Testing Fault No. Fault 1 Fault 2 Fault 3 T2 SPE T2 SPE T2 SPE

Modeling Fault No. PPCFD 1 IV* ** ** 0 --0 ----method 2 I --0 ** ** 0 3 I --0 --0 ** ** IV 22.22 11.11 0 11.11 ** ** PPCA 1 IV ** ** 0 --44.44 --method 2 I 0 --** ** 44.44 --3 I 22.22 --22.22 --** ** IV 22.22 --0 --** ** * It denotes the critical-to-diagnosis phases --- It denotes that the reconstruction statistics of the concerned fault are not critical and believable. So they are not shown here. ** It means the same-fault analysis results which are not shown here. (b) multiway method Fault 2 Fault 3 Testing Fault No. Fault 1 2 2 T SPE T SPE T2 SPE Modeling Fault No. Fault 1 ** ** 0 --0 --Fault 2 --- --** ** --- --Fault 3 0 --- 11.11 --- ** ** --- It denotes that the reconstruction statistics of the concerned fault are not critical and believable. So they are not shown here. ** It means the same-fault analysis results which are not shown here.


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Subspace Decomposition-Based Reconstruction Modeling for Fault

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