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Comparison Between Statistical and Observer-Based Approaches for Fault Detection and Isolation in a Chemical Process
10th International Symposium on Process Systems Engineering - PSE2009 Rita Maria de Brito Alves, Claudio Augusto Oller do Nascimento and Evaristo Chalbaud Biscaia Jr. (Editors) © 2009 Elsevier B.V. All rights reserved.
1257
Comparison Between Statistical and ObserverBased Approaches for Fault Detection and Isolation in a Chemical Process Thiago de Sá Feitala,b, Uwe Krugerb, José Carlos Pintoa, Enrique L. Limaa a
Programa de Engenharia Química/COPPE, Universidade Federal do Rio de Janeiro, 68502, Rio de Janeiro, Brazil b Department of Electrical Engineering, The Petroleum Institute, P.O. Box 2533, Abu Dhabi, U.A.E.
Abstract This paper summarizes the results of a comparison including three different fault detection and isolation (FDI) methods for dynamic systems. The techniques studied have recently been proposed in the literature and are Dynamic Principal Component Analysis (DPCA), Canonical Variate Analysis (CVA) and Subspace Model Identification (SMI). The aim of this study is to contrast the performance of each method in detecting and isolating incipient fault conditions. Utilizing real data from a debutaniser distillation tower, this study yields that the observer approach based on an identified SMI model is most sensitive for fault detection but performs poorly in isolating the fault condition. This method failed to correctly diagnose the fault condition using fault isolation approach. In contrast, DPCA offered a correct picture of this event using variable reconstruction and contribution charts, whilst CVA only yielded satisfactory results using variable reconstruction. For this study, it is therefore concluded that both approaches have complementary strengths and weaknesses. Keywords: Subspace Methods, Fault Detection and Isolation, Chemical Process
1. Introduction Based on the ever growing requirement for safe and reliably operating processes in the chemical industry, fault detection and isolation has received considerable attention over the past decades. The research literature has shown two main approaches: (i) data driven multivariate statistical techniques (Chiang et al., 2001; MacGregor et al., 2005) and the use of model-based state-space systems (Isermann, 2005). These approaches yielded similar fault detection and isolation methods. Given the large number of variables that are typically recorded in complex chemical, the use of mechanistic first principal models is difficult in practice, which has led to significant research activities on data-driven methods. The aim of this article is to investigate the potential of utilizing model-based methods based on the observer design for fault detection and isolation. The required model is provided by subspace model identification, a data driven methodology that has been extensively studied over the past decade (Verhaegen, 1994; Van Overschee and De Moor, 1996). This data-driven and model-based methodology is contrasted with competitive multivariate methods for dynamic systems. Chiang et al. (2001) summarized that DPCA and CVA are such methods that identify a dynamic monitoring model in a reduced dimensional latent variable space.
1258
T. Feital et al.
For DPCA and CVA, process monitoring is based on the use of univariate statistics and fault isolation relies on variable reconstruction (Qin and Li, 2001) and contribution charts (Miller et al., 1998). In a similar fashion to the univariate fault detection indices of DPCA and CVA, the paper proposes the use of the T2 and SPE statistics on the basis of the estimated states and the output estimation error, respectively, for the model-based approach. To examine the utility of the model- and observer-based approach with DPCA and CVA, the article summarizes the results of an industrial process. The objective of this study is to contrast the performance of each method in detecting fault conditions and in diagnosing the recorded event.
2. Fault Detection and Isolation Methods This section briefly reviews DPCA, CVA and the proposed model- and observerbased approaches for FDI. 2.1. Dynamic Principal Component Analysis (DPCA) DPCA (Chiang et al., 2001) relies on a time series arrangement of a set of recorded process variables. An eigendecomposition of the covariance matrix of this augmented variable set forms then the basis for constructing a model plane and a complementary residual subspace. The retained components of this eigendecomposition relate to the most dominant eigenvalues of this decomposition and span the model plane. The residual subspace is consequently spanned by the discarded eigenvectors. Geometrically, the orthogonal projection of the augmented data vector, z, onto the model plane and the residual subspace is z PP z and z I PP z, respectively, such that the following equality holds z z z. 2.2. Canonical Variate Analysis (CVA) Similar to DPCA, CVA performs FDI in a model plane and a residual subspace defining state sequences, x, (Chiang et al., 2001). CVA relies on augmented vectors that include time lagged values of the process input and output variables. More precisely, the lagged terms are time-series arrangements of the output variables that relate to future, f, and past measurements and an arrangement of the input variables for past measurements. The state sequences are the first d dominant canonical variates, / x J p U Σ p, where Ud is a matrix storing the first d columns of U, /
/
Σ Σ Σ UΣV . Moreover, p is a vector storing the past time-series arrangements of the input and output variables and the different Σ are covariance and cross-covariance matrices. The residual vector of the state-space model in term of the past is given by the r remaining state variables, x J p. 2.3. Subspace Model Identification (SMI) The state space model that the observer-based FDI scheme requires is provided by an SMI technique. Based on the numerically stable and efficient QR and SVD decompositions, the MOESP algorithm (Verhaegen, 1994) has gained attention. In addition, Chen (1999) reported that the subspace identification offers various practical advantages, mainly for industrial process systems, over the classical realization from input/output identified models (Chen, 1999). This algorithm relies on future and past arrangements of the input and output variables to form a total of four block Hankel matrices, where the samples are stored row wise. These matrices are then stacked prior
Comparison Between Statistical and Observer-Based Approaches for Fault Detection and Isolation in a Chemical Process
1259
to the application of a QR decomposition. Together with a subsequent SVD decomposition, the quadruple of state space matrices, A, B, C and D can be extracted from the R-matrix of the QR decomposition and the left singular vectors of the SVD decomposition. 2.4. Observer Designs The observer-based approach for FDI traditionally uses process variables to estimate output variables through a state-space model in a closed-loop design. The provided output estimation error can then be monitored as residuals (Patton and Chen, 1997). As the design of such observers may offer degrees of freedom, many approaches have been proposed to tackle uncertainties and disturbances issues. This paper uses the traditional Luenberger state observer designed by pole placement. 2.5. Univariate Statistical Fault Detection Indices For fault detection, two traditional univariate statistics are employed for multivariate statistical process monitoring, the Hotelling’s T2 statistic:
T t Λ t
(1)
which follows an F-distribution and a squared prediction error:
SPE r r
(2)
for which approximate distributions have been proposed in Box (1954). In Equation 1, t denotes an uncorrelated mean-centered variable set and Λ, its covariance matrix. Table 1 shows the definition of these statistics for each method. Table 1 – Fault Detection Indices
Technique
Principal Subspace
Residual Subspace
Based on DPCA
t Λ t
z z
Based on CVA
x Λ x
x x
Based on observer
x Λ x
e e
The work in reference Nomikos and MacGregor (1995) allows the determination of confidence limits for each of these statistics with a significance of 0.01 for example. 2.6. Fault Isolation Indices This task relates to the determination of which process variables are mostly affected by a detected fault condition. The research literature has proposed two different concepts: (i) the use of contribution chart (Kourti and MacGregor, 1996) and (ii) variable reconstruction (Dunia and Qin, 1998). These methods are readily applicable for DPCA (Qin and Li, 2001) and CVA (Lee et al., 2006). The observer-based method relies on the use of generalized observers, as discussed in Isermann (2005). More precisely, each output variable is reconstructed in turn by eliminating the associated column from the observer setting and using the remaining output variables. Table 2 summarizes the isolation indices for each method in which a implies “affected”, i
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denotes the ith process variable and * represents a vector reconstructed by excluding the ith process variable. Table 2 Isolation Indices
Variable Contribution
Variable Reconstruction
Technique
Principal Subspace
Residual Subspace
Residual Subspace
Based on DPCA
t " λ" p",% z%
z% z%
(' &() ' & ) & &
x" j",% z%
x " j",% z%
(' +() ' + +) +
-
e% e%
( ,() ' ,'
Based on CVA Based on observer
,) ,
3. Industrial Application Example This section presents a debutaniser process used to contrast the performance of each method in detecting and isolating a fault condition. The process is designed to purify a mixture of hydrocarbons into butane from a fresh feed which is product of a depropaniser distillation. Therefore, the fresh feed is not steady and can interfere in the top and bottom temperatures by an incipient change. The recorded fault data set contains a small drop in the fresh feed followed by a larger one, which some minutes later affects all the temperatures of the process. In such undesired conditions, the concentration of the head-product is altered, increasing the concentration of pentane. In order to manage the quality of this process, it is required a dynamic model and a fast FDI method. The industrial data were recorded at a sampling interval of 30 seconds, including 8000 samples describing normal operation conditions for the training data set and 4744 samples containing the fault condition for the testing data set. The input variables are: the reflux flow (u1), fresh feed flow (u2) and fresh feed temperature (u3); and the output variables are: mid-tray temperature (y1), top-tray temperature (y2), butane product flow (y3), bottom-tray temperature (y4) and inlet temperature (y5).
4. FDI Results using DPCA, CVA and Observer-Design This section summarizes the application of the presented fault detection and isolation methods to the recorded data sets. The subspace methods, CVA and MOESP, relied on 2 block rows, whilst for the DPCA, Parallel Analysis showed that the number of principal components does not change with the number of lag variables from 5 lag variables on. Table 3 gives the number of selected components for each method, where for CVA and MOESP it was selected according to the Scree test (Chiang et al., 2001). The observer design was based on the selection of poles inside the unit circle. For fault detection, three parameters were used to assess the performance of each method: type I error (false alarm rate), type II error (misdetection rate) and the detection delay. Table 3 also summarizes the fault detection results and indicates that the observer-based approach outperformed the statistical-based ones in the last two parameters. Ideally, the methods should have values around 1% for the type I error due to the confidence limit chosen. Such result can be explained due to the fact that the observer-based approach has a well-defined dynamic structure, which guarantee quick and accurate responds compared to CVA-based approach. Regarding to DPCA-based
Comparison Between Statistical and Observer-Based Approaches for Fault Detection and Isolation in a Chemical Process
1261
approach, which dynamic structure is based on an ARX model, the detection delay was significant once the influence of the lag variables (fault-free) in the moment of the fault can be high.
Method DPCA CVA OBS
l 5 1 1
n 7 5 5
Table 3 - Fault Detection Results Type I Error (%) Type II Error (%) Detection Delay (min) T2 T2 T2 SPE SPE SPE 0.7 0.4 0
1.3 2.3 0.1
20.7 27.4 5.4
24.8 22.4 14.4
57 84.5 27
51.5 50.5 34
Next, the performance of each method for fault isolation was contrasted. Table 4 summarizes the results of each method using contribution charts and Table 5, using variable reconstruction approach. Letters a and b in Tables 4 and 5 stands for the first and the second fault detection indices, respectively. The fault isolation step via contribution charts showed divergent results among the methods. DPCA-based approach presented the best description of the fault since the residuals were able to detect the minimal increasing in the fresh feed, and the T2 contributions could reveal the most affected variable. From this result, the root cause of the fault can be clearly identified and the fault diagnosis can be correctly performed as a trouble in the feed stream. CVA-based approach had similar result compared to DPCA only for the model subspace. However, the “faulty” variable (u2) was missed in such way that impaired the correct fault diagnosis. Finally, the observer-based approach was unable to make a reasonable description of the fault since the proposed fault detection index for the state variables (OBSa) does not allow a kind of contribution and the residuals are restrictive only to the output variables.
Table 4 - Fault Isolation Results Via Variable Contribution Variable Contribution u2 u3 y1 y2 y3 y4 y5 u1 Methods DPCAa 40.3 3.8 46.2 734.2 64.4 34.2 6.1 43.8 DPCAb 16.3 66.6 1.5 20.5 1.1 17.5 0.3 2.3 CVAa 0.9 1 4.9 49.8 0.5 2.3 21.8 8.4 CVAb 3.7 2.8 4.3 54.7 157.1 2.7 37.4 27.9 OBSa OBSb 1576 576 263 359 4626 The variable reconstruction approach also had good results compared to the contribution charts, concerning the statistical methods. On the other hand, the bank of generalized observers did not show any improvement. Although this approach can make an analysis of the input variables as well as the output variables, the results of these industrial data sets showed that it does not work well. The observers designed for the input variables monitoring were able to predict correctly the output variables so that the influence from the input variables was not significant and the fault isolation turned out to be impractical.
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Table 5 - Fault Isolation Results Via Variable Reconstruction Variable Reconstruction u u u3 y1 y2 y3 y4 y5 Methods 1 2 DPCAb 0.89 0.52 0.96 0.87 0.98 0.84 1.00 0.94 CVAb 44.80 32.98 45.05 36.13 42.71 44.75 34.31 42.48 OBSb 0.000 0.000 0.000 0.019 0.005 0.009 0.009 0.010
5. Conclusions This paper studied the performance of the most traditional fault detection and isolation methods in a chemical process. In order to make a reasonable comparison, univariate statistics for the state variables and the output estimation errors of the observer-based approach were proposed. Results from a real process data showed a very accurate detectability in the case of the observer-based method. The dynamic structure of such method can be handled by pole placement allowing different types of respond. For the fault isolation step, only statistical-based methods were able to make a good description of the fault. Therefore, none of the methods showed a significant advantage in both fault detection and isolation steps simultaneously. A hybrid approach unifying the strengths of such methods will be the focus of future works.
References C. Lee, S. W. Choi, I.-B. Lee, 2006, Variable reconstruction and sensor fault identification using canonical variate analysis, Journal of Process Control, 16, 747-761. C. T. Chen, 1999, Linear Systems Theory and Design. Oxford University Press, 3rd Edition. G. E. P. Box, 1954, Some Theorems on Quadratic Forms Applied in the Study of Analysis of Variance in One-Way Classification, The Annals of Mathematical Statistics 25, 290-302. J. Chen, R. J. Patton, 1999, Robust Model-Based Fault Diagnosis for Dynamic Systems, Kluwer Academic Publishers. J. F. MacGregor, H. Yu, S. G. Munoz 1, J. Flores-Cerrillo, 2005, Data-based latent variable methods for process analysis, monitoring and control, Computers and Chemical Engineering 29, 1217–1223. L. H. Chiang, E. L. Russell, R. D. Braatz, 2001, Fault Detection and Diagnosis in Industrial Systems, Springer. M. Verhaegen, 1994, Identification of the deterministic part of MIMO state space models given in innovations form from input-output data, Automatica 30, 61-74. P. Nomikos, J. F. MacGregor, 1995, Multivariate SPC Charts for Monitoring Batch Processes, Technometrics 37, 1, 41-58. P. Van Overschee, B. De Moor, 1996, Subspace Identification for Linear Systems – Theory, Implementation and Application, Kluwer Academic Publishers R. Dunia, S. J. Qin, 1998, A Unified Geometric Approach to Process and Sensor Fault Identification and Reconstruction: The Unidimensional Fault Case, Computers and Chemical Engineering 22, 7-8, 927-943. R. Isermann, 2005, Model-based fault-detection and diagnosis – status and applications, Annual Reviews in Control 29, 71–85. R. J. Patton, J. Chen, 1997, Observer-based fault detection and isolation: robustness and applications, Control Engineering Practice 5, 5, 671-682. S. J. Qin, W. Li, 2001, Detection and Identification of Faulty Sensors in Dynamic Processes, AIChE Journal 47, 7, 1581-1593. T. Kourti, J. F. MacGregor, 1996, Multivariate SPC Methods for Process and Product Monitoring, Journal Of Quality Technology 28, 4, 409-428.
1257
Comparison Between Statistical and ObserverBased Approaches for Fault Detection and Isolation in a Chemical Process Thiago de Sá Feitala,b, Uwe Krugerb, José Carlos Pintoa, Enrique L. Limaa a
Programa de Engenharia Química/COPPE, Universidade Federal do Rio de Janeiro, 68502, Rio de Janeiro, Brazil b Department of Electrical Engineering, The Petroleum Institute, P.O. Box 2533, Abu Dhabi, U.A.E.
Abstract This paper summarizes the results of a comparison including three different fault detection and isolation (FDI) methods for dynamic systems. The techniques studied have recently been proposed in the literature and are Dynamic Principal Component Analysis (DPCA), Canonical Variate Analysis (CVA) and Subspace Model Identification (SMI). The aim of this study is to contrast the performance of each method in detecting and isolating incipient fault conditions. Utilizing real data from a debutaniser distillation tower, this study yields that the observer approach based on an identified SMI model is most sensitive for fault detection but performs poorly in isolating the fault condition. This method failed to correctly diagnose the fault condition using fault isolation approach. In contrast, DPCA offered a correct picture of this event using variable reconstruction and contribution charts, whilst CVA only yielded satisfactory results using variable reconstruction. For this study, it is therefore concluded that both approaches have complementary strengths and weaknesses. Keywords: Subspace Methods, Fault Detection and Isolation, Chemical Process
1. Introduction Based on the ever growing requirement for safe and reliably operating processes in the chemical industry, fault detection and isolation has received considerable attention over the past decades. The research literature has shown two main approaches: (i) data driven multivariate statistical techniques (Chiang et al., 2001; MacGregor et al., 2005) and the use of model-based state-space systems (Isermann, 2005). These approaches yielded similar fault detection and isolation methods. Given the large number of variables that are typically recorded in complex chemical, the use of mechanistic first principal models is difficult in practice, which has led to significant research activities on data-driven methods. The aim of this article is to investigate the potential of utilizing model-based methods based on the observer design for fault detection and isolation. The required model is provided by subspace model identification, a data driven methodology that has been extensively studied over the past decade (Verhaegen, 1994; Van Overschee and De Moor, 1996). This data-driven and model-based methodology is contrasted with competitive multivariate methods for dynamic systems. Chiang et al. (2001) summarized that DPCA and CVA are such methods that identify a dynamic monitoring model in a reduced dimensional latent variable space.
1258
T. Feital et al.
For DPCA and CVA, process monitoring is based on the use of univariate statistics and fault isolation relies on variable reconstruction (Qin and Li, 2001) and contribution charts (Miller et al., 1998). In a similar fashion to the univariate fault detection indices of DPCA and CVA, the paper proposes the use of the T2 and SPE statistics on the basis of the estimated states and the output estimation error, respectively, for the model-based approach. To examine the utility of the model- and observer-based approach with DPCA and CVA, the article summarizes the results of an industrial process. The objective of this study is to contrast the performance of each method in detecting fault conditions and in diagnosing the recorded event.
2. Fault Detection and Isolation Methods This section briefly reviews DPCA, CVA and the proposed model- and observerbased approaches for FDI. 2.1. Dynamic Principal Component Analysis (DPCA) DPCA (Chiang et al., 2001) relies on a time series arrangement of a set of recorded process variables. An eigendecomposition of the covariance matrix of this augmented variable set forms then the basis for constructing a model plane and a complementary residual subspace. The retained components of this eigendecomposition relate to the most dominant eigenvalues of this decomposition and span the model plane. The residual subspace is consequently spanned by the discarded eigenvectors. Geometrically, the orthogonal projection of the augmented data vector, z, onto the model plane and the residual subspace is z PP z and z I PP z, respectively, such that the following equality holds z z z. 2.2. Canonical Variate Analysis (CVA) Similar to DPCA, CVA performs FDI in a model plane and a residual subspace defining state sequences, x, (Chiang et al., 2001). CVA relies on augmented vectors that include time lagged values of the process input and output variables. More precisely, the lagged terms are time-series arrangements of the output variables that relate to future, f, and past measurements and an arrangement of the input variables for past measurements. The state sequences are the first d dominant canonical variates, / x J p U Σ p, where Ud is a matrix storing the first d columns of U, /
/
Σ Σ Σ UΣV . Moreover, p is a vector storing the past time-series arrangements of the input and output variables and the different Σ are covariance and cross-covariance matrices. The residual vector of the state-space model in term of the past is given by the r remaining state variables, x J p. 2.3. Subspace Model Identification (SMI) The state space model that the observer-based FDI scheme requires is provided by an SMI technique. Based on the numerically stable and efficient QR and SVD decompositions, the MOESP algorithm (Verhaegen, 1994) has gained attention. In addition, Chen (1999) reported that the subspace identification offers various practical advantages, mainly for industrial process systems, over the classical realization from input/output identified models (Chen, 1999). This algorithm relies on future and past arrangements of the input and output variables to form a total of four block Hankel matrices, where the samples are stored row wise. These matrices are then stacked prior
Comparison Between Statistical and Observer-Based Approaches for Fault Detection and Isolation in a Chemical Process
1259
to the application of a QR decomposition. Together with a subsequent SVD decomposition, the quadruple of state space matrices, A, B, C and D can be extracted from the R-matrix of the QR decomposition and the left singular vectors of the SVD decomposition. 2.4. Observer Designs The observer-based approach for FDI traditionally uses process variables to estimate output variables through a state-space model in a closed-loop design. The provided output estimation error can then be monitored as residuals (Patton and Chen, 1997). As the design of such observers may offer degrees of freedom, many approaches have been proposed to tackle uncertainties and disturbances issues. This paper uses the traditional Luenberger state observer designed by pole placement. 2.5. Univariate Statistical Fault Detection Indices For fault detection, two traditional univariate statistics are employed for multivariate statistical process monitoring, the Hotelling’s T2 statistic:
T t Λ t
(1)
which follows an F-distribution and a squared prediction error:
SPE r r
(2)
for which approximate distributions have been proposed in Box (1954). In Equation 1, t denotes an uncorrelated mean-centered variable set and Λ, its covariance matrix. Table 1 shows the definition of these statistics for each method. Table 1 – Fault Detection Indices
Technique
Principal Subspace
Residual Subspace
Based on DPCA
t Λ t
z z
Based on CVA
x Λ x
x x
Based on observer
x Λ x
e e
The work in reference Nomikos and MacGregor (1995) allows the determination of confidence limits for each of these statistics with a significance of 0.01 for example. 2.6. Fault Isolation Indices This task relates to the determination of which process variables are mostly affected by a detected fault condition. The research literature has proposed two different concepts: (i) the use of contribution chart (Kourti and MacGregor, 1996) and (ii) variable reconstruction (Dunia and Qin, 1998). These methods are readily applicable for DPCA (Qin and Li, 2001) and CVA (Lee et al., 2006). The observer-based method relies on the use of generalized observers, as discussed in Isermann (2005). More precisely, each output variable is reconstructed in turn by eliminating the associated column from the observer setting and using the remaining output variables. Table 2 summarizes the isolation indices for each method in which a implies “affected”, i
1260
T. Feital et al.
denotes the ith process variable and * represents a vector reconstructed by excluding the ith process variable. Table 2 Isolation Indices
Variable Contribution
Variable Reconstruction
Technique
Principal Subspace
Residual Subspace
Residual Subspace
Based on DPCA
t " λ" p",% z%
z% z%
(' &() ' & ) & &
x" j",% z%
x " j",% z%
(' +() ' + +) +
-
e% e%
( ,() ' ,'
Based on CVA Based on observer
,) ,
3. Industrial Application Example This section presents a debutaniser process used to contrast the performance of each method in detecting and isolating a fault condition. The process is designed to purify a mixture of hydrocarbons into butane from a fresh feed which is product of a depropaniser distillation. Therefore, the fresh feed is not steady and can interfere in the top and bottom temperatures by an incipient change. The recorded fault data set contains a small drop in the fresh feed followed by a larger one, which some minutes later affects all the temperatures of the process. In such undesired conditions, the concentration of the head-product is altered, increasing the concentration of pentane. In order to manage the quality of this process, it is required a dynamic model and a fast FDI method. The industrial data were recorded at a sampling interval of 30 seconds, including 8000 samples describing normal operation conditions for the training data set and 4744 samples containing the fault condition for the testing data set. The input variables are: the reflux flow (u1), fresh feed flow (u2) and fresh feed temperature (u3); and the output variables are: mid-tray temperature (y1), top-tray temperature (y2), butane product flow (y3), bottom-tray temperature (y4) and inlet temperature (y5).
4. FDI Results using DPCA, CVA and Observer-Design This section summarizes the application of the presented fault detection and isolation methods to the recorded data sets. The subspace methods, CVA and MOESP, relied on 2 block rows, whilst for the DPCA, Parallel Analysis showed that the number of principal components does not change with the number of lag variables from 5 lag variables on. Table 3 gives the number of selected components for each method, where for CVA and MOESP it was selected according to the Scree test (Chiang et al., 2001). The observer design was based on the selection of poles inside the unit circle. For fault detection, three parameters were used to assess the performance of each method: type I error (false alarm rate), type II error (misdetection rate) and the detection delay. Table 3 also summarizes the fault detection results and indicates that the observer-based approach outperformed the statistical-based ones in the last two parameters. Ideally, the methods should have values around 1% for the type I error due to the confidence limit chosen. Such result can be explained due to the fact that the observer-based approach has a well-defined dynamic structure, which guarantee quick and accurate responds compared to CVA-based approach. Regarding to DPCA-based
Comparison Between Statistical and Observer-Based Approaches for Fault Detection and Isolation in a Chemical Process
1261
approach, which dynamic structure is based on an ARX model, the detection delay was significant once the influence of the lag variables (fault-free) in the moment of the fault can be high.
Method DPCA CVA OBS
l 5 1 1
n 7 5 5
Table 3 - Fault Detection Results Type I Error (%) Type II Error (%) Detection Delay (min) T2 T2 T2 SPE SPE SPE 0.7 0.4 0
1.3 2.3 0.1
20.7 27.4 5.4
24.8 22.4 14.4
57 84.5 27
51.5 50.5 34
Next, the performance of each method for fault isolation was contrasted. Table 4 summarizes the results of each method using contribution charts and Table 5, using variable reconstruction approach. Letters a and b in Tables 4 and 5 stands for the first and the second fault detection indices, respectively. The fault isolation step via contribution charts showed divergent results among the methods. DPCA-based approach presented the best description of the fault since the residuals were able to detect the minimal increasing in the fresh feed, and the T2 contributions could reveal the most affected variable. From this result, the root cause of the fault can be clearly identified and the fault diagnosis can be correctly performed as a trouble in the feed stream. CVA-based approach had similar result compared to DPCA only for the model subspace. However, the “faulty” variable (u2) was missed in such way that impaired the correct fault diagnosis. Finally, the observer-based approach was unable to make a reasonable description of the fault since the proposed fault detection index for the state variables (OBSa) does not allow a kind of contribution and the residuals are restrictive only to the output variables.
Table 4 - Fault Isolation Results Via Variable Contribution Variable Contribution u2 u3 y1 y2 y3 y4 y5 u1 Methods DPCAa 40.3 3.8 46.2 734.2 64.4 34.2 6.1 43.8 DPCAb 16.3 66.6 1.5 20.5 1.1 17.5 0.3 2.3 CVAa 0.9 1 4.9 49.8 0.5 2.3 21.8 8.4 CVAb 3.7 2.8 4.3 54.7 157.1 2.7 37.4 27.9 OBSa OBSb 1576 576 263 359 4626 The variable reconstruction approach also had good results compared to the contribution charts, concerning the statistical methods. On the other hand, the bank of generalized observers did not show any improvement. Although this approach can make an analysis of the input variables as well as the output variables, the results of these industrial data sets showed that it does not work well. The observers designed for the input variables monitoring were able to predict correctly the output variables so that the influence from the input variables was not significant and the fault isolation turned out to be impractical.
1262
T. Feital et al.
Table 5 - Fault Isolation Results Via Variable Reconstruction Variable Reconstruction u u u3 y1 y2 y3 y4 y5 Methods 1 2 DPCAb 0.89 0.52 0.96 0.87 0.98 0.84 1.00 0.94 CVAb 44.80 32.98 45.05 36.13 42.71 44.75 34.31 42.48 OBSb 0.000 0.000 0.000 0.019 0.005 0.009 0.009 0.010
5. Conclusions This paper studied the performance of the most traditional fault detection and isolation methods in a chemical process. In order to make a reasonable comparison, univariate statistics for the state variables and the output estimation errors of the observer-based approach were proposed. Results from a real process data showed a very accurate detectability in the case of the observer-based method. The dynamic structure of such method can be handled by pole placement allowing different types of respond. For the fault isolation step, only statistical-based methods were able to make a good description of the fault. Therefore, none of the methods showed a significant advantage in both fault detection and isolation steps simultaneously. A hybrid approach unifying the strengths of such methods will be the focus of future works.
References C. Lee, S. W. Choi, I.-B. Lee, 2006, Variable reconstruction and sensor fault identification using canonical variate analysis, Journal of Process Control, 16, 747-761. C. T. Chen, 1999, Linear Systems Theory and Design. Oxford University Press, 3rd Edition. G. E. P. Box, 1954, Some Theorems on Quadratic Forms Applied in the Study of Analysis of Variance in One-Way Classification, The Annals of Mathematical Statistics 25, 290-302. J. Chen, R. J. Patton, 1999, Robust Model-Based Fault Diagnosis for Dynamic Systems, Kluwer Academic Publishers. J. F. MacGregor, H. Yu, S. G. Munoz 1, J. Flores-Cerrillo, 2005, Data-based latent variable methods for process analysis, monitoring and control, Computers and Chemical Engineering 29, 1217–1223. L. H. Chiang, E. L. Russell, R. D. Braatz, 2001, Fault Detection and Diagnosis in Industrial Systems, Springer. M. Verhaegen, 1994, Identification of the deterministic part of MIMO state space models given in innovations form from input-output data, Automatica 30, 61-74. P. Nomikos, J. F. MacGregor, 1995, Multivariate SPC Charts for Monitoring Batch Processes, Technometrics 37, 1, 41-58. P. Van Overschee, B. De Moor, 1996, Subspace Identification for Linear Systems – Theory, Implementation and Application, Kluwer Academic Publishers R. Dunia, S. J. Qin, 1998, A Unified Geometric Approach to Process and Sensor Fault Identification and Reconstruction: The Unidimensional Fault Case, Computers and Chemical Engineering 22, 7-8, 927-943. R. Isermann, 2005, Model-based fault-detection and diagnosis – status and applications, Annual Reviews in Control 29, 71–85. R. J. Patton, J. Chen, 1997, Observer-based fault detection and isolation: robustness and applications, Control Engineering Practice 5, 5, 671-682. S. J. Qin, W. Li, 2001, Detection and Identification of Faulty Sensors in Dynamic Processes, AIChE Journal 47, 7, 1581-1593. T. Kourti, J. F. MacGregor, 1996, Multivariate SPC Methods for Process and Product Monitoring, Journal Of Quality Technology 28, 4, 409-428.