shown that the monitoring performance of the proposed method is considerably better than that of the .... cases of similar data sets (top) and different data sets ...
Proceedings of IFAC Symposium on Advanced Control of Chemical Processes (ADCHEM), Vol.I, pp.231-236, Pisa, Italy, June 14-16
DISSIMILARITY OF PROCESS DATA FOR STATISTICAL PROCESS MONITORING Manabu Kano ∗ Koji Nagao ∗ Hiromu Ohno ∗∗ Shinji Hasebe ∗ Iori Hashimoto ∗ ∗
∗∗
Dept. of Chem. Eng., Kyoto University, Kyoto, Japan Dept. of Chem. Sci. and Eng., Kobe University, Kobe, Japan
Abstract: For monitoring chemical processes, multivariate statistical process control (MSPC) has been widely used. In the present work, a new process monitoring method is proposed. The proposed method utilizes a change in distribution of process data, since the distribution reflects the corresponding operating condition. In order to quantitatively evaluate the difference between two data sets, the dissimilarity index is defined. The proposed method and the conventional SPC methods are applied to monitoring problems of the Tennessee Eastman process. The results have clearly shown that the monitoring performance of the proposed method is considerably better c 2000 IFAC than that of the conventional methods. Copyright Keywords: fault detection, monitoring, statistical process control, pattern recognition
1. INTRODUCTION
multivariate statistical process control (MSPC) has been developed (Kresta, et al., 1991).
For the successful operation of any processes, it is important to detect process upsets, equipment malfunctions, or other special events as early as possible and then to find and remove the factors causing those events. In chemical processes, data-based approaches, rather than model-based approaches, have been widely used for process monitoring because it is often difficult to develop detailed physical models. The data-based approaches are referred to as statistical process control (SPC), and conventional SPC charts such as Shewhart charts, CUSUM charts, and EWMA charts have been widely used for monitoring univariate processes. Such univariate SPC charts, however, do not function well for multivariable processes with correlated variables. In addition, chemical processes are becoming more heavily instrumented, and process data are more frequently recorded. This causes a data overload, and a great deal of data is wasted. Therefore, in order to extract useful information from process data and utilize it for process monitoring,
MSPC is based on the chemometric techniques such as principal component analysis (PCA) and partial least squares (PLS). Wise and Gallagher (1996) reviewed some of the chemometric techniques and their application to chemical process monitoring and dynamic process modeling. They defined chemometrics as the science of relating measurements made on a chemical system to the state of the system via application of mathematical or statistical methods. Extension of MSPC to monitor time-varying batch processes by using multiway PCA and PLS (Wold, et al., 1987) was made by Nomikos and MacGregor (1994). Another extension to handle very large processes via multiblock PCA and PLS was made by MacGregor, et al. (1994). PCA is a tool for data compression and information extraction. PCA finds linear combinations of variables that describe major trends in a data set. For monitoring a process by
using PCA, control limits are set for two kinds of statistics, T 2 and Q, after a PCA model is developed. Q is the sum of squared errors, and it is a measure of the amount of variation not captured by the first few principal components. A measure of the variation within the PCA model is given by Hotelling’s T 2 statistic. T 2 is the sum of normalized squared scores, and it is a measure of the distance from the multivariate mean to the projection of the operating point on the subspace formed by the PCA model. Many successful applications have shown the practicability of MSPC. The conventional MSPC method described above, however, does not always function well, because it cannot detect the change of correlation among process variables if scores (T 2 ) and errors (Q) are inside the control limits. In the present work, in order to improve the performance of process monitoring, a new statistical process monitoring method is proposed. The proposed method is based on the idea that a change of operating condition can be detected by monitoring a distribution of time-series data, which reflects the corresponding operating condition. In order to quantitatively evaluate the difference between two data sets, a new index representing dissimilarity is defined. The performance of the proposed monitoring method and the conventional methods is compared by using simulated data obtained from the Tennessee Eastman process.
2. MONITORING BASED ON DISSIMILARITY OF DATA SETS A change of operating condition can be detected by monitoring distribution of process data, since the distribution reflects the corresponding operating condition. In this section, the dissimilarity index is introduced for quantitatively evaluating the difference between distributions of process data. Then, a monitoring method based on the dissimilarity index is described.
expansion is a well known technique used for feature extraction or dimension reduction in the pattern recognition area, and it is mathematically equivalent to PCA. Consider the following two data sets:
(i)
(i)
x11 x12 (i) (i) x21 x22 Xi = .. .. . . (i) (i) xNi 1 xNi 2
The concept of similarity or dissimilarity is often used for classifying a set of data. In cluster analysis, for example, the degree of dissimilarity between two classes is measured by the distance between barycenters of the data, and two classes with the smallest degree of dissimilarity are combined for generating a new class. In order to evaluate the difference between two data sets, a classification method based on the Karhunen-Loeve (KL) expansion (Fukunaga and Koontz, 1970) is utilized in this work. The KL
,
i = 1, 2 (1)
where Ni is the number of samples of i-th data set X i and P is the number of variables. Here, each column of X i is assumed to be mean-centered. The covariance matrices are given by Ri =
1 T X Xi Ni i
(2)
and the covariance matrix of the mixture of both data sets is given by T 1 X1 X1 X2 N X2 N2 N1 R1 + R2 (3) = N N where N = N1 + N2 . Applying eigenvalue decomposition to R, an orthogonal matrix P 0 which satisfies R=
RP 0 = P 0 Λ
(4)
is derived. Λ is a diagonal matrix whose diagonal elements are eigenvalues of R. By defining a transformation matrix P as 1
P = P 0 Λ− 2
(5)
the following equation can be derived. P T RP = I
(6)
When the data matrices X i are transformed into r Yi=
2.1 Dissimilarity
(i)
· · · x1P (i) · · · x2P . .. . .. (i) · · · xNi P
1 Ni X i P 0 Λ− 2 N
(7)
the covariance matrices of the transformed data matrices 1 T Y Yi Ni i Ni T P Ri P = N satisfy the following equation: Si =
(8)
S1 + S2 = I Applying eigenvalue decomposition covariance matrices S i ,
(9) to
the
4
Y 1 Y2
2 y2
x2
2
near zero. On the contrary, D should be near one when data sets are quite different from each other.
4 X 1 X2
0
D = 0.04
0
2.2 Monitoring Method −2 −4 −4
−2
−2
0 x
2
4
−4 −4
−2
1
X 1 X
Y 1 Y
2
2
y2
x2
4
4
0
−2 −4 −4
2
1
4 2
0 y
D = 0.43
2
0
−2
−2
0 x
2
4
−4 −4
−2
1
0 y
2
4
1
Fig. 1. Transformation of data sets Xi (left) into Yi (right) for evaluating the dissimilarity. The cases of similar data sets (top) and different data sets (bottom). (i)
(i)
(i)
S i wj = λj w j
(10)
is derived. Here, wj and λj are the eigenvectors and the corresponding eigenvalues, respectively. From Eqs. (9) and (10), (1) (1) (1) wj S 2 w j = 1 − λj
(11)
can be derived. This equation means that the eigevectors of S 2 are the same as those of S 1 and that the following relationship is satisfied. (1)
1 − λj
(2)
= λj
(12)
As a result, both transformed data sets have the same set of principal components, and the corresponding eigenvalues of the covariance matrices are reversely ordered. Thus, after the above transformation, the most important correlation for data set 1 becomes equivalent to the least important correlation for data set 2, and vice versa. When data sets are quite similar to each other, the eigenvalues λj must be near 0.5. On the other hand, when data sets are quite different from each other, the largest and the smallest eigenvalues should be near 1 and 0, respectively. The above mathematical explanation can be physically understood by using Fig. 1. Finally, the dissimilarity index, D, is defined for evaluating the dissimilarity of data sets. D=
P 4 X (λj − 0.5)2 P
(13)
j=1
The index D changes between zero and one. When two data sets are similar to each other, D must be
In order to detect a change of operating condition, the reference data set representing a normal operating condition should be defined, and the dissimilarity between the reference data set and the data set representing a current operating condition should be used as an index for monitoring. For applying the proposed monitoring method based on the dissimilarity, a reference data set and a control limit are determined by the following procedure. (1) Acquire time-series data when a process is operated under a normal condition. Then, normalize each column of the data matrix, i.e. adjust it to zero mean and unit variance. (2) Determine the size of time-window, w. Generate many data sets with w samples from the data by moving the time-window. Then, select a reference data set. (3) Calculate the index D. (4) Determine the control limit of D. The data used for selecting the reference data set and determining the control limit should not be the same. In this work, the control limit is determined so that 99% of calculated values of D are below the limit value, and the remaining 1% are above the limit value. Such a control limit can be easily determined, and it is practicable. For on-line monitoring, the data matrix representing a current operating condition is scaled by using the mean and the variance obtained at step (1), and then the index D is calculated. If the index is outside the control limit, the operating condition is judged to be abnormal.
3. APPLICATION In this section, the proposed monitoring method as well as the conventional SPC methods are applied to the monitoring problems of the Tennessee Eastman process.
3.1 Tennessee Eastman process The simulator of the Tennessee Eastman process was developed by Downs and Vogel (1993). The process consists of a reactor/separator/recycle arrangement involving two simultaneous gas-liquid exothermic reactions and two
FIC
FIC
FIC
Purge Compressor
FIC
Condenser
JIC
PI
TIC
Separator
LIC
Feed D LIC FIC
FIC
PI
XE TIC
FIC
FC
FI
Steam
Reactor FI
FIC
LIC
Stripper
Analyzer
Feed E
TIC
XB
TI
PIC Reactor Cooling
TIC
Analyzer
Condenser Cooling
Feed C
FC
Analyzer
Feed A
G H
Product
Fig. 2. Decentralized control system of the Tennessee Eastman process.
Case 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16-20
Table 1 Process disturbances. Disturbance Type A/C feed ratio Step B comp. Step D feed temp. step Reactor cooling water Step (RCW) inlet temp. Condenser cooling water Step (CCW) inlet temp. A feed loss Step C header pressure loss Step A, B, C feed comp. Random D feed temp. Random C feed temp. Random RCW inlet temp. Random CCW inlet temp. Random Reaction kinetics Slow drift RCW valve Sticking CCW valve Sticking Unknown Unknown
additional byproduct reactions, and it has 12 manipulated variables and 41 measurements. The simulator includes 20 programmed disturbances listed in Table 1. The control system utilized for dynamic simulations is the decentralized PID control system proposed by McAvoy and Ye (1994), which is shown in Fig. 2. The sampling interval of process variables is set to be three minutes.
3.2 Monitoring Results The proposed monitoring method (referred to as DISSIM hereafter) is compared with the univariate SPC method (referred to as USPC) and the conventional MSPC method (referred to as cMSPC). In USPC, each variable is independently
Table 2 Process variables utilized for monitoring. A feed D feed E feed A and C feed Recycle flow Reactor feed rate Reactor temperature Purge rate Product separator temperature Product separator pressure Product separator underflow Stripper pressure Stripper temperature Stripper steam flow Reactor cooling water outlet temperature Separator cooling water outlet temperature monitored. The monitored indexes of cMSPC are Ti2 and Qi . The subscript i denotes the number of adopted principal components, which is determined so that the best performance is given. In addition, D is the monitored index in DISSIM. The control limits of all indexes are determined by the following steps. (1) Each monitoring method is applied to the data representing a normal operating condition, and each index is calculated. (2) The control limit of each index is determined so that the number of samples outside the control limit are 1% of the entire samples. Then, each monitoring method is evaluated by the following steps. (1) Each monitoring method is applied to the data in Cases 1-20, and each index is calculated.
(2) For the data obtained after the occurrence of an event, the percentage of the samples outside the control limit is calculated in each simulation. Then, the mean of those percentages of 10 different simulations is calculated in each case. Since the control limits are determined so that they represent 99% confidence limits, a monitoring method is regarded to be successful in detecting the event if the mean calculated in step (2) is considerably higher than 1%. On the contrary, when the mean is less than or close to 1%, the monitoring method is regarded to be not functioning well. This measure for evaluating the performance of process monitoring methods is termed reliability in this work. A total of 16 variables, selected by Chen and McAvoy (1998) for the monitoring purpose, are utilized. These 16 variables are listed in Table 2. Static Monitoring; The data matrix for static monitoring with DISSIM at step k becomes Y (k) =
y(k) y(k −1) .. .
to that used for building statistical models in inferential control systems (Kano, et al., 2000). The data matrix for dynamic monitoring with DISSIM at step k becomes the following matrix: Y (k) =
y(k −s) y(k −1−s) .. .
y(k) y(k −1) .. .
(15)
y(k −w+1−s) y(k −w+1) where s denotes the interval. The performance of dynamic monitoring depends strongly on s. Although more complicated matrix for dynamic monitoring can be adopted, the above simple case is taken into account for simplicity. In this work, s = 5 is selected by trial and error. The results of dynamic monitoring are summarized in Table 4. The reliability of DISSIM is considerably better than that of cMSPC in almost all cases. In addition, the performance of dynamic monitoring is much better than that of static monitoring. These results show the superiority of the proposed method, DISSIM, over conventional SPC methods and the usefulness of dynamic monitoring.
(14)
y(k −w+1) where y is the row vector of monitored process variables and w denotes the time-window size. The results of static monitoring are summarized in Table 3. Each simulated data includes 100 samples after an event occurs. As expected, cMSPC functions better than USPC in almost all cases. Since the performance of cMSPC depends on the number of principal components, its selection is crucial. In this application, 90% of variation can be explained by first 6 principal components. However, 6 principal components are not sufficient for the monitoring purpose. The best performance is given by using 11 principal components. The reliability of DISSIM is considerably better than that of cMSPC in Cases 3, 4, 5, 9, 11, 12, 14, 15, 16, and 19. In most of these cases, the reliability of cMSPC is less than 30%. That is, DISSIM can detect the small events which are difficult to detect by using cMSPC. In other cases, the reliability of both methods is comparable to each other. It should be noted that the reliability of DISSIM depends on the time-window size. Dynamic Monitoring; Using past measurements as monitored variables may be useful for capturing the correlation among process variables, because the process dynamics can be taken into account (Ku, et al., 1995). This approach is analogous
4. CONCLUSION In order to improve the performance of process monitoring, a new monitoring method is proposed. The proposed method utilizes the change in distribution of process data, since the distribution reflects the corresponding operating condition. In order to quantitatively evaluate the difference between two data sets, a new index representing dissimilarity is defined. The proposed monitoring method and the conventional methods are applied to the monitoring problems of the Tennessee Eastman process. The results have shown that the reliability of the proposed method, DISSIM, is considerably better than that of the conventional methods. The dynamic monitoring with DISSIM is quite successful. It should be noted, however, that it is important to determine the adequate size of time-window for successful functioning of the proposed monitoring method, since its performance depends on the time-window size. Furthermore, the dissimilarity index proposed in this work can be used for fault identification. If process data obtained from several types of faulty operation are available, such faults can be identified by using the similarity index, defined as S = 1 − D, between predefined faulty data and the data when a fault is detected.
Table 3 Comparison of monitoring methods by reliability (%). (static monitoring) Case Index xi T62 Q6 2 T11 Q11 DISSIM D Method USPC cMSPC
Case Index xi T62 Q6 2 T11 Q11 DISSIM D Method USPC cMSPC
1
2
3
4
5
6
7
8
9
10
99.2 98.2 99.9 99.3 100.0 99.0 99.0
96.4 92.4 96.1 93.5 97.6 93.3 93.2
5.1 4.3 5.2 3.3 22.8 58.3 48.2
3.5 2.6 2.1 0.7 4.1 15.2 0.9
4.0 2.7 1.2 0.5 3.2 11.2 2.7
98.7 100.0 100.0 100.0 100.0 100.0 100.0
59.1 100.0 100.0 100.0 100.0 100.0 100.0
44.0 83.1 87.0 84.3 88.2 87.0 85.4
11.9 4.2 6.5 4.3 17.5 60.2 40.9
43.4 73.3 80.0 76.9 81.8 81.9 78.9
11
12
13
14
15
16
17
18
19
20
8.3 3.1 6.9 1.2 24.0 58.6 32.5
7.8 4.3 3.7 2.1 6.2 37.1 2.2
42.4 74.9 75.0 74.9 76.2 75.9 71.7
35.7 11.5 49.4 14.4 71.0 96.0 93.8
4.2 2.4 1.9 0.9 3.2 12.6 2.0
13.6 4.6 15.8 11.5 19.8 75.9 52.3
53.8 67.8 67.4 66.9 68.7 68.4 64.8
50.7 47.1 53.9 48.4 55.2 60.2 50.7
8.0 3.8 13.7 12.2 10.6 66.3 50.9
43.5 67.2 67.0 67.7 68.5 70.1 67.1
w
100 200 w
100 200
Table 4 Comparison of monitoring methods by reliability (%). (dynamic monitoring) Case Index 2 T19 Q19 DISSIM D Method cMSPC
Case Index 2 T19 Q19 DISSIM D Method cMSPC
1
2
3
4
5
6
7
8
9
10
99.0 99.9 98.9
92.1 97.2 94.5
5.0 14.6 71.9
0.9 2.0 23.8
0.8 1.0 23.1
100.0 100.0 100.0
100.0 100.0 100.0
83.6 87.8 87.9
7.1 17.3 76.2
75.6 81.1 84.6
11
12
13
14
15
16
17
18
19
20
0.3 24.4 77.5
2.7 3.9 55.0
74.0 75.1 78.2
22.2 85.3 96.9
0.8 0.8 20.6
9.5 34.4 82.9
66.3 67.7 73.2
49.1 54.4 62.4
7.5 28.9 80.8
66.7 68.4 72.2
w
100 w
100
ACKNOWLEDGMENT The authors gratefully acknowledge funding from the Japan Society for the Promotion of Science (JSPS-RFTF96R14301).
REFERENCES Chen, G. and T. J. McAvoy (1998). Predictive on-line monitoring of continuous processes. J. Proc. Cont., 8, 409-420. Downs, J. J. and E. F. Vogel (1993). A plant-wide industrial process control problem. Computers chem. Engng, 17, 245-255. Fukunaga, K. and W. L. G. Koontz (1970). Application of the Karhunen-Loeve wxpansion to feature selection and ordering. IEEE Trans. Comput., C-19, 311-318. Kano, M., K. Miyazaki, S. Hasebe, and I. Hashimoto (2000). Inferential control system of distillation compositions using dynamic Partial Least Squares regression. J. Proc. Cont., 10, 157-166. Kresta, J. V., J. F. MacGregor, and T. E. Marlin (1991). Multivariate statistical monitoring
of process operating performance. Can. J. Chem. Eng., 69, 35-47. Ku, W., R. H. Storer, and C. Georgakis (1995). Disturbance detection and isolation by dynamic principal component analysis. Chemometrics and intelligent laboratory systems, 30, 179-196. MacGregor, J. F., C. Jaeckle, C. Kiparissides, and M. Koutoudi (1994). Process monitoring and diagnosis by multiblock methods. AIChE J., 40, 826-838. McAvoy, T. J. and N. Ye (1994). Base control for the Tennessee Eastman problem. Computers chem. Engng, 18, 383-413. Nomikos, P. and J. F. MacGregor (1994). Monitoring batch processes using multiway principal component analysis. AIChE J., 40, 1361-1375. Wise, B. M. and N. B. Gallagher (1996). The process chemometrics approach to process monitoring and fault detection. J. Proc. Cont., 6, 329-348. Wold, S., P. Geladi, K. Esbesen, and J. Ohman (1987). Multi-way principal components- and PLS-analysis. J. Chemometrics, 1, 41-56.
