Combination Method of Principal Component and Wavelet Analysis for Multivariate Process Monitoring and Fault Diagnosis

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4198 Ind. Eng. Chem. Res. 2003, 42, 4198-4207

Ningyun Lu,†,‡ Fuli Wang,‡ and Furong Gao*,† Department of Chemical Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, and School of Information Science and Engineering, Northeastern University, Shenyang, Liaoning, China

Product quality and operation safety are important aspects of industrial processes, particularly those with large numbers of correlated process variables. Principal component analysis (PCA) has been widely used in multivariate process monitoring for its ability to reduce process dimensions. PCA and other statistical techniques, however, have difficulties in differentiating faults with similar time-domain process characteristics. A wavelet-based time-frequency approach is developed in this paper to improve PCA-based methods by extending the time-domain process features into time-frequency information. Subsequently, a similarity measure is presented to compare process features for on-line process monitoring and fault diagnosis. Simulation results show that the proposed multivariate time-frequency process feature is effective in both fault detection and diagnosis, illustrating the potentials for real-world application.

1. Introduction tion plot was enhanced by Vedam and Venkatasubra- manian19 using signed digraphs. The combination of The demands for product quality and operation safety PCA with expert knowledge has also been reported for in the process industry have spurred the recent develop- fault diagnosis improvement for PCA-based methods.20,21 ment and application of process monitoring and fault All of the above methods can enhance the fault diagnosis methods. The traditional statistical process diagnosis of PCA-based approaches, provided that the monitoring approaches, founded on Shewhart control fault features are differentiable in the time domain. charts, are ill-suited for modern chemical processes. Certain process faults might show similar time-domain Multivariate statistical methods based on correlation models such as principal component analysis (PCA) and features because of an underlying correlation and/or the partial least-squares (PLS) have been widely used to employed closed-loop controls. In this case, it would be handle processes with large numbers of correlated desirable to extend the time-domain information into process variables.1-10 Process monitoring based on these the time-frequency domain to improve fault identifica- methods conducts statistical hypothesis tests on two in- tion. It is well-known that relations between process dices, the Hotelling T2 and Q statistics in principal com- conditions and frequency characteristics do, indeed, ponent and residual subspaces, respectively. These exist. Changes in the spectrum characteristics over an methods are effective in fault detection, but not diagnosis. operation period can provide useful information for fault Contribution plots,11 widely used as a diagnosis tool in diagnosis. It is, therefore, possible to utilize the frequency-domain information to improve PCA-based pro- PCA/PLS-based process monitoring, can only identify 22-27 a group of variables impacted by a process malfunction. cess monitoring and diagnosis. To improve fault diagnosis, many advances have been Wavelet transformation is a powerful tool for trans- reported. forming time-domain signals into the time-frequency According to Yoon and MacGregor,12 approaches for domain. Combinations of wavelet transforms and PCA fault detection and diagnosis might be classified into have begun to emerge recently. For example, process three categories: methods based on causal models, data can be calibrated first by a Harr wavelet transform methods based on knowledge, and methods based on (HWT) before PCA is applied.22 PCA was extended to multivariate statistics. Improved fault detection and wavelet-based multiscale PCA (MSPCA) by Bakshi,23 diagnosis performance have been obtained by combining to capture information in both the time and frequency methods of different categories, particularly, a statistical domains. This results in improvements in fault detec- method, for example, PCA, with other methods. Qin and tion. Using MSPCA, multiscale fault identification was his colleges13-15 have extended PCA for sensor fault proposed by Misra et al.27 to analyze contribution plots identification and reconstruction for both unidimen- at the corresponding scales. sional and multidimensional cases. The integration of This paper proposes to employ wavelet transforms to PCA and discriminant analysis was proposed by Raich improve PCA-based fault identification for cases where and Cinar16,17 to improve PCA’s ability in diagnosis. process faults with similar time-domain features can be PCA was combined with parity relation by Gertler et differentiated by their frequency information. The high- al.,18 to employ concepts of analytical redundancy for dimensional and correlated process data are first pre- improving isolation properties. The PCA-based contribu- processed by PCA to be projected onto the low-dimensional principal component subspace. The principal * To whom correspondence should be addressed. Tel.: +852- component scores are transformed into the time- 2358-7139. Fax: +852-2358-0054. E-mail: [email protected]. frequency domain, and decomposed into an approxima- † Hong Kong University of Science and Technology. tion signal and the corresponding detail signals of ‡ Northeastern University. different scales. The energy distribution over selected

- - scales is defined as a feature pattern to be used for ψ (t) ) 2 jψ(2 jt - k), j ) 1, ..., J; k ∈ Z (5) process monitoring and fault diagnosis. A similarity j,k measure, introduced to differentiate fault features from normal process features, is also employed for feature where j is the scale factor and k is the translation factor. mapping in diagnosis. The approximation coefficients a(J,k) and the detail coefficients b(j,k) can be computed by the Mallat algo- The remainder of this paper is organized as follows. 31,32 Section 2 briefly introduces the techniques of PCA and rithm. The approximation signal AJ(t) and the detail signals wavelet analysis. Section 3 first presents the definition ) of the process feature pattern to represent the multi- Dj(t)(j 1, 2, ..., J) are defined as variate time-frequency information, followed by the ) development of the proposed process monitoring and AJ(t) ∑a(J,k) φJ,k(t) (6) diagnosis approach. Application examples of the pro- k∈z posed scheme are demonstrated and discussed in section ) 4. Finally, conclusions are given in section 5. Dj(t) ∑b(j,k) ψj,k(t) (7) k∈Z 2. Background 28 2.1. Principal Component Analysis (PCA). PCA AJ(t) and Dj(t) are constructed from the approximation reduces the redundant information in a data block by coefficients a(J,k) and the detail coefficients b(j,k), projecting original process measurements onto a lower- respectively, representing the information on the cor- dimensional subspace defined by a few orthogonal latent responding scales. The original signal is actually the variables that contain most of the variance of the sum of AJ(t) and the Dj(t)’s original data. Let X ) (x1, x2, ..., xm)beann × m-dimensional data J ) + set. X can be decomposed as f(t) AJ(t) ∑Dj(t) (8) j)1 m ) T ) T X TP ∑tipi (1) Wavelet analysis uses the scale factor j instead of the i)1 frequency ω. In the decomposition procedures, the “frequency” is scaled down by 2j. Assuming that the where p is defined as the principal component loading i original signal has a frequency of 1, the initial space V0 vector and ti is the corresponding score vector. Less is ultimately decomposed into the low-frequency sub- important components, which mostly describe noise J space VJ with frequency band [0, 1/2 ] and the high- information in the data, can be abandoned without the frequency subspaces W with frequency band [1/2j, 8 j loss of significant information. By doing so, matrix X 1/2j-1]. This indicates that a time-domain signal can be can be reconstructed as the sum of an estimation value, decomposed by wavelet analysis into a sum of compo- ˆ X, and its residual, E nents of different frequency bands. The energy over certain frequency bands can represent certain process k ) ˆ+ ˆ) T information, and changes in the energy distribution over X X E, with X ∑tipi (2) scales can provide useful information for fault diagnosis. i)1

where k represents the number of principal components 3. Process Monitoring and Diagnosis Based on a Combination of PCA and Wavelet Analysis retained. The principal component score vectors, t1, ..., tk, span a lower-dimensional subspace used for further 3.1. Overview. Industrial processes, particularly analysis. chemical processes, are usually complicated, with many Process monitoring and fault diagnosis is conducted correlated process variables. It is desirable and compu- via statistical hypothesis tests on two indices, Hotelling tationally economical to reduce the process to a lower- T2 and Q statistics, in the principal component subspace dimensional space before applying a process monitoring and residual subspace, respectively. Details on these and diagnosis scheme. PCA-based process monitoring - methods can be found in many references.1 10 and diagnosis is enhanced by wavelet transform by 2.2. Wavelet Analysis. Wavelet analysis,29,30 a pow- extending the time-domain fault features into the time- erful signal-processing tool, can transform the time- frequency domain, for better fault interpretation and domain signals into the time-frequency domain. Under differentiation. the framework of multiresolution analysis, the original The question is then how to effectively extract quan- signal space V0 can be decomposed into a hierarchical titative feature patterns from wavelet coefficients of the set consisting of an approximation space VJ [the space principal component scores. As discussed before, the spanned by scale functions φJ,k(t), k ∈ Z] and detail time-domain signal can be decomposed as the sum of spaces Wj [the spaces spanned by wavelet functions its approximation and detail signals, each of which ψj,k(t), j ) 1, ..., J, k ∈ Z], where J is the coarsest scale, contains information in a corresponding frequency band. normally called the decomposition level. The energy of the original signal can also be decomposed According to the above description, a signal f(t) ∈ L2- as the sum of the energies of the approximation and (R) can be decomposed as follows details. The energy distribution over the selected frequency bands can be regarded as a quantitative feature. J Process features can then be defined in terms of the ) + f(t) ∑a(J,k)φJ,k(t) ∑∑b(j,k) ψj,k(t) (3) energy distributions of the principal components scores. k∈Z j)1k∈Z To compare two process features, a similarity measure ) -J -J - ∈ is subsequently introduced for process monitoring and φJ,k(t) 2 φ(2 t k), k Z (4) diagnosis. 中国科技论文在线 http://www.paper.edu.cn 4200 Ind. Eng. Chem. Res., Vol. 42, No. 18, 2003

The proposed method is limited to multivariate comparison of the two process features. For a process continuous processes with “time-invariant” normal pro- with only one principal component retained, the ex- cess features, that is, with time-invariant means and tracted process feature is the normalized energy distri- variances. Such processes have stable energy distribu- bution of the principal component score. The similarity tions at normal operation,33 which makes it possible to measure can be defined as conduct on-line process monitoring by comparing the current process feature with its historical behavior. For (¥ )T(¥ ) time-varying processes, such as those encountered in ) curr normal Scurr,normal || ||‚|| || (13) batch processes, the proposed method would have to be ¥curr ¥normal improved. This is one focus of our future research. 3.2. Quantitative Feature Extraction by Wavelet which, in fact, takes the form of the cosine of the vectors’ Analysis. For a signal, f(t) ∈ L2(R), in the time domain, angle. The energy distribution is a vector of positive the energy can be defined as elements; the similarity will fall within the range (0,1]. For a process with k principal components [t1, t2, ..., ∞ 2 ) tk], the similarity measure is defined as Ef ∫-∞f (t)dt (9) S ) g S + ‚‚‚ + g S As discussed in section 2.2, f(t) can be decomposed as curr,normal 1 t1,normal k tk,normal the sum of an approximation signal, AJ, and a set of detail signals, D (j ) 1, ..., J). A contains information T j J (¥t ) (¥normal) in frequency band [0, 1/2J], whereas D contains infor- ) i j St ,normal (14) j j-1 i ||¥ ||‚||¥ || mation in frequency band [1/2 , 1/2 ]. Normally, the ti normal finest-detail scale contains noise information. The ener- 24 gies of AJ and Dj take the form where N/2 1 2 ) + λi EA ∑ a(J,k m) J ) ) Nm)-N/2+1 gi , i 1, ..., k (10) k N/2 1 2 λ E ) ∑ b(j,k + m) ∑ j Dj j)1 Nm)-N/2+1

T where N is the window length. and λi represents the eigenvalues of X X in descending The energy distribution can be regarded as a feature order. This similarity measure also falls within the pattern that contains both time- and frequency-domain range (0,1]. The maximum similarity of 1 indicates an information. With a given decomposition level J, the exact match. The closer a similarity value is to 0, the normalized energy distribution, Eh f, is defined as a more unlikely two faults are to be similar. quantitative feature as follows It should be noted that the process data in the moving window might vary with time because of noise, resulting E ) [E , ..., E , E ] in small variations of the process features from window f D1 DJ AJ h ) || || (11) to window even under normal operation. A similarity Ef Ef/ Ef threshold, S*, is introduced to reduce the possibility of false alarms. With a similarity degree above the pre- Relations exist between a process fault and its energy defined threshold, the two process features are assumed distribution. For example, for a slowly drifting signal, to represent the same operating condition. The similar- the energy in the low-frequency subspaces, particularly ity threshold is set to allow 99% of normal data in the in the approximation subspace AJ, will be significant. training data set above the threshold. This corresponds High-frequency signal changes will result in changes to a level of significance R)0.01 considering the of the energy in the finer-detail subspaces. For a signal probability distribution of the similarity degree for with a cyclic disturbance, the energy in the correspond- normal data.34 The threshold set in this way can reduce ing scale will be significant. Analyzing changes in the the chances of false alarms, as it is insensitive to normal energy distribution can be useful for fault identification. process noises. It is, however, sensitive to the changes For a process having k principal components, t1, ..., of measurement noise patterns, because the energy tk, the process feature, ¥, is a set of the quantitative distribution covers up to the high-frequency bands, as feature patterns for each retained principal component illustrated in the second example in section 4. Fault diagnosis is, in fact, a procedure of pattern ¥ ) [Eh , Eh , ..., Eh ]T (12) t1 t2 tk matching, by comparing the current fault feature with the existing ones in the fault database. The current fault 3.3. On-line Process Monitoring and Fault Di- is assumed to match an existing fault in the database agnosis. As for existing statistical process monitoring if the similarity value is above the predetermined methods, a reference data set, X, must be collected from threshold. If the current fault feature matches none in the normal operation of the process for modeling. The the database, a new fault is deemed to have occurred. normal feature is then computed for comparison on-line To summarize, the proposed process monitoring and with the feature of the evolving process. A moving diagnosis scheme consists of modeling and on-line two- window is used for feature extraction and comparison step monitoring and diagnosis, as listed below: in the on-line implementation. Modeling Procedure. Process monitoring is conducted by comparing the 1. Select a proper wavelet function and determine the current process feature, ¥curr, with the normal process decomposition level, J, and the window length N (see feature, ¥normal. A similarity measure is defined for the discussion in section 4). 中国科技论文在线 http://www.paper.edu.cn Ind. Eng. Chem. Res., Vol. 42, No. 18, 2003 4201

2. Collect a period of process data under normal operation; develop the PCA model and calculate the principal component scores. 3. Form the normal process feature, ¥normal,byap- plying the wavelet transform to the principal component scores. ) 4. Form the fault features, ¥f(i) (i 1, ..., Nf), where Nf is the number of known faults, following steps 2 and 3 for fault operations. On-line Process Monitoring and Diagnosis. 5. Compute the principal component scores of the process data in the evolving moving windows. Apply the wavelet transform to the scores and calculate the current process feature, ¥curr. 6. Compare the current feature with the normal feature. If the similarity value, Scurr,normal, is above the predetermined threshold, S*, continuing the process monitoring by going back to step 5; otherwise, go to step Figure 1. Schematic of three-tank process. 7. 7. Compare the current process feature with the detail subspaces. The approximate subspace is not existing ones in the fault database. If the similarity with included, because the simulated faults of the three-tank an existing fault feature is above the threshold, the process have similar time responses. Inclusion of the diagnosis procedure ends. If the current process feature approximate subspace will result in a single large does not match any existing one, a new fault type has component in the energy feature vector and, hence, occurred. The database is updated by extending it with reduce the differentiability of those process faults. For the TE process, the energy distribution includes both the new fault feature, ¥curr, and a description of this new feature. the detail subspaces and the approximate subspace, [ED1, ED2, ED3, ED4, ED5, EA5]. The selection of the 4. Illustrative Examples frequency bands should consider the fault properties. In general, if the time-frequency features of two (or The proposed process monitoring and diagnosis ap- more) faults become difficult to identify because of their proach is tested with two processes, a coupled three- overwhelming high energy in a particular frequency tank process and the well-known Tennessee Eastman band, it is recommended that that particular frequency (TE) process. The first example demonstrates the ability band be temporarily omitted and the differentiation be of the proposed algorithm to differentiate faults with focused on comparing information in the remaining similar time-domain characteristics, whereas the second frequency bands. example focuses on the application of the proposed 4.1. Three-Tank Process. A three-tank system, as scheme to a more “complex” industrial process. shown in Figure 1, is used to demonstrate the proposed The proper wavelet function, decomposition level, and process monitoring and diagnosis. In this coupled moving window length have to be determined prior to system, the flow rates of the two pumps serve as the the application of the present method. The Daubechies manipulated variables in controlling the levels of tanks (dbN) wavelet is used in our work for its nice proper- 1 and 2, whereas the level of tank 3 is left to float to ties.33 The decomposition level is determined by the reflect the interaction between tanks 1 and 2. A decou- signal’s frequency bandwidth. The signals with abun- pling control strategy has been developed for the dant high-frequency information need larger numbers simultaneous closed-loop control of the levels of tanks of decomposition levels. For the two illustrated ex- 1 and 2. Two types of typical faults are simulated: amples, the decomposition level is set to be 5. The leakage faults caused by the opening of the the leakage moving window method results in new problems: de- valve, SLi (i )1, 2, 3), at the bottom of each tank and termination of the window length and the “border sensor failures caused by programming. Electrical white distortion” for finite-length signal. Because of dyadic noises are purposely introduced into all measurements down-sampling, the length of the wavelet coefficients in the simulations. is being reduced by a factor of 2j. To ensure the accuracy Five sensors are installed with this process: two flow of signal reconstruction, it is desirable to limit the sensors and three level sensors. They are sampled every minimum length of the coefficients in the coarsest level second. All process measurements are first normalized J J. With a minimum length of Lmin , the length of the with zero mean and unit variance. Three principal window for the original signals should be larger than components are retained after analysis based on the J J 23,27 35 2 × Lmin . Methods are also available to treat widely accepted cross-validation algorithm. The nor- border distortion.29,32 Simpler schemes based on signal mal process feature, extracted from the normal process extension at the boundaries are preferred as they data in the training data sets, is shown in Figure 2, involve the computation of fewer extra coefficients at where the energy distribution over five selected fre- each stage of the decomposition. Symmetric extension32 quency bands is plotted against the three retained is used here. This work focuses on the application of principal components. All together, nine faults are wavelet transforms, rather than on wavelet research introduced, as described in Table 1, with the process itself. Further information on wavelet function selection, features shown in Figure 3. Fault features are calcu- border distortion treatment, and so on can be found in lated from the data in the time span, (Tf - N/2, Tf + 29,32,33 the literature. N/2), where Tf is the fault occurrence time for the For the first example, the energy distribution is reference data sets in the fault database and N is the defined as [ED1, ED2, ED3, ED4, ED5], covering all of the moving window’s length. The similarity degrees between 中国科技论文在线 http://www.paper.edu.cn 4202 Ind. Eng. Chem. Res., Vol. 42, No. 18, 2003

The results obtained by the PCA-based contribution plot method are also presented for comparison in Figure 4 and Table 3. The contribution plots have difficulty in differentiating the first, fourth, seventh, and ninth faults, all of which actually lead to an increase in Q1 and a decreasing level in tank 1. Similarly, the contribution plots cannot differentiate the second, third, sixth, and eighth faults, all of which lead to an increase in Q2 and a decreasing level in tank 2. The bold values in Table 3 indicate the cases that are difficult to differentiate on the basis of their contribution plots. A comparison between Figures 3 and 4, or Tables 2 and 3, indicates that the proposed method outperforms the contribution plot, with the superiority of the proposed method lying in its time-frequency analysis. The use of the energy distribution to represent process features is effective for differentiating faults with similar time-domain charac- Figure 2. Normal process feature for the three-tank process. teristics. 4.2. Tennessee Eastman Process. The proposed Table 1. Simulated Nine Faults for the Three-Tank Process scheme was also applied to a well-known benchmark process, Tennessee Eastman (TE) industrial process, fault description of simulated fault which is widely used for comparison of the various 1 leakage of tank 1 process monitoring methods. The TE process was first 2 leakage of tank 2 introduced by Downs and Vogel.36 The MATLAB simu- 3 leakage of tank 3 37 4 leakage of tanks 1 and 2 lation procedure follows that of Ashish with the plant- 38 5 leakage of tanks 1 and 3 wide control strategy of McAvoy and Ye. 6 leakage of tanks 2 and 3 The TE process is schematically shown in Figure 5. 7 sensor failure of tank 1 The simulation for this paper is based on the operation 8 sensor failure of tank 2 9 sensor failure of tank 3 in the base case, corresponding to a 50/50 G/H product ratio. Among the 41 measurements and 12 manipulated pairs of process faults are calculated by eq 14. The variables, 52 are selected (all but the agitation speed of threshold of similarity is set to 0.975, corresponding to the reactor’s stirrer). The sampling interval is set to be the level of significance R)0.01, as discussed in section 3 min, the same as the recommended value in ref 34. 3. Table 2 shows the differentiability of the proposed Data collected from 96 h of normal operation are used monitoring and diagnosis method. to develop the PCA model. Six principal components are

Figure 3. Process features for the simulated nine faults (x axis is the process variable). 中国科技论文在线 http://www.paper.edu.cn Ind. Eng. Chem. Res., Vol. 42, No. 18, 2003 4203

Figure 4. Contribution plots for the simulated nine faults (x axis is the process variable).

Table 2. Similarity Degrees between Fault Features Represented by Energy Distribution for the Three-Tank Process fault fault 123456789 1 1.0000 0.9022 0.8803 0.9198 0.7967 0.9101 0.7184 0.9617 0.8938 2 - 1.0000 0.8742 0.9610 0.8722 0.8519 0.6343 0.9479 0.9519 3 -- 1.0000 0.8960 0.9275 0.8158 0.8351 0.9324 0.7917 4 --- 1.0000 0.8688 0.9014 0.6773 0.9614 0.9425 5 ---- 1.0000 0.7060 0.7864 0.8864 0.7458 6 ----- 1.0000 0.7429 0.8581 0.8162 7 ------1.0000 0.7137 0.4664 8 ------1.0000 0.9303 9 ------1.0000

Table 3. Similarity Degrees between Fault Features Represented by Contribution Rate to SPE for the Three-Tank Process fault fault 1 2 3 4 5 6 7 8 9 1 1.0000 0.0924 0.1031 0.9994 0.9549 0.2828 0.9969 0.0963 0.9983 2 - 1.0000 0.9997 0.0606 0.3781 0.9776 0.0210 1.0000 0.0386 3 -- 1.0000 0.0711 0.3893 0.9774 0.0310 0.9995 0.0488 4 --- 1.0000 0.9444 0.2531 0.9900 0.0646 0.9997 5 ---- 1.0000 0.5434 0.9292 0.3813 0.9364 6 ----- 1.0000 0.2164 0.9789 0.2326 7 ------1.0000 0.0251 0.9998 8 ------1.0000 0.0427 9 ------1.0000

retained, capturing major correlations in the process the 36th simulation hour. Each fault simulation oper- variables. The normal process feature is extracted from ates for about 48 h, and 960 measurements are collected the 96 h of training data. Additional data are generated for monitoring. with three different process upsets, IDV(1), IDV(4), and For normal operation, a “steady-state” condition, IDV(11). These three process faults are the typical cases measurements show little autocorrelation, most of which used in the work of Chiang et al.,34 where detailed can be reflected in the first few principal components. reviews and comparisons were conducted with various The energy distributions of the first few principal methods of process monitoring and fault diagnosis (PCA, components show high energy in the low-frequency DPCA, CVA, etc.) on the TE process. These three runs bands, whereas those of the remaining principal com- start with no faults, and the faults are introduced at ponents show higher energy in the higher-frequency 中国科技论文在线 http://www.paper.edu.cn 4204 Ind. Eng. Chem. Res., Vol. 42, No. 18, 2003

Figure 5. Tennessee Eastman process.

Figure 6. Normal process feature for the TE process. Figure 7. On-line monitoring for normal data.

bands, as illustrated in Figure 6. With closed-loop process feature. Figure 7 shows the similarity values control, most of upsets can introduce significant auto- of the normal process data in the training data set, correlation among process measurements. The process which will be used to set the threshold for on-line features represented by the proposed energy distribu- monitoring and diagnosis. As shown in Figure 7, the tion can change correspondingly in different manners. similarity varies with time because of process noise. The This is the basis of the proposed process monitoring threshold of similarity is set to eliminate any false method. alarms caused by the normal noise. There are 104 For on-line application, the minimum length of the moving windows for the normal training data, and the 5 coefficients in the coarsest level is set to be Lmin ) 8. threshold is set to be the second lowest similarity value, The moving window’s length is therefore 25 × 8 ) 256. which corresponds approximately to the level of signifi- The moving step is set to 16, so that the first 16 data in cance R)0.01, that is, S* ) 0.9382 is selected for the the old window are deleted and 16 new data are inserted following process monitoring and fault diagnosis. If the in the new window. Principal component scores for similarity value for the evolving window is less than S*, measurements in each moving window are subsequently an alarm will be given to indicate an abnormality. processed by wavelet analysis to capture the current Figure 8 shows the on-line monitoring charts for process feature, which is then compared to the normal faults IDV(1), IDV(4), and IDV(11). All three faults can 中国科技论文在线 http://www.paper.edu.cn Ind. Eng. Chem. Res., Vol. 42, No. 18, 2003 4205

Figure 8. On-line monitoring charts for faults IDV(1), IDV(4), Figure 9. On-line diagnosis of fault IDV(4). and IDV(11).

be detected by the proposed method, shortly after their proposed method performs better than the others, with occurrence at 36 h. Comparisons with the fault detection the lowest missed detection rates among all compared performance of other methods are conducted as in the methods. work of Chiang et al.,34 where the missed detection rate Figures 9 and 10 illustrate the diagnosis ability of the is used as the criterion. Table 4 gives the missed proposed method for faults IDV(4) and IDV(11), respec- detection rates of faults IDV(1), IDV(4), and IDV(11) for tively. Fault IDV(4) is caused by a step change in the PCA, DPCA, CVA, and the proposed multivariate time- reactor cooling water temperature, whereas fault IDV- frequency monitoring methods. According to Table 4, the (11) is caused by a random variation in the reactor 中国科技论文在线 http://www.paper.edu.cn

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Figure 11. Features of faults IDV(4) and IDV(11).

Table 4. Missed Detection Rates for Faults IDV(1), IDV(4), and IDV(11)a method statistics IDV(1) IDV(4) IDV(11) PCA T2 0.008 0.956 0.794 Q 0.003 0.038 0.356 DPCA T2 0.006 0.939 0.801 Q 0.005 0 0.193 2 CVA Ts 0.001 0.688 0.515 2 Tr 0 0 0.196 Q 0.003 0.975 0.669 proposed method similarity 0 0 0.133 a Compared with the results of Chiang et al.34

Table 5. Similarity Degrees between Features of Faults IDV(1), IDV(4), and IDV(11)

¥normal IDV(1) IDV(4) IDV(11)

¥normal 1.0000 0.2948 0.7327 0.8748 - Figure 10. On-line diagnosis of fault IDV(11). IDV(1) 1.0000 0.6674 0.4363 IDV(4) -- 1.0000 0.7426 cooling water temperature. The closed-loop control IDV(11) --- 1.0000 system makes these two faults look similar in time- domain behavior, which results in most existing fault whereas the energy distributions of fault IDV(11) weight diagnosis methods failing to differentiate them.34 Fig- to mid-frequency bands, a difference that can readily ures 9 and 10 show that the proposed time-frequency be explained by the different disturbance types. Table method can differentiate these two faults without any 5 shows the similarity degrees for the three selected problem, because they have different energy distribu- faults. tions, as shown in Figure 11. For fault IDV(4), the The on-line diagnosis charts, as shown in Figures 9 energy distributions weight to low-frequency bands, and 10, actually monitor the similarity degrees between 中国科技论文在线 http://www.paper.edu.cn

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