Fault Detection and Isolation Using Concatenated Wavelet Transform ...

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 54, NO. 5, MAY 2006

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Fault Detection and Isolation Using Concatenated Wavelet Transform Variances and Discriminant Analysis G. D. Gonzalez, Life Senior Member, IEEE, R. Paut, A. Cipriano, Senior Member, IEEE, D. R. Miranda, and G. E. Ceballos

Abstract—A method for fault detection and isolation is developed using the concatenated variances of the continuous wavelet transform (CWT) of plant outputs. These concatenated variances are projected onto the principal component space corresponding to the covariance matrix of the concatenated variances. Fisher and quadratic discriminant analyses are then performed in this space to classify the concatenated sample CWT variances of outputs in a given time window. The sample variance is a variance estimator obtained by taking the displacement average of the squared wavelet transforms of the current outputs. This method provides an alternative to the multimodel approach used for fault detection and identification, especially when system inputs are unmeasured stochastic processes, as is assumed in the case of the mechanical system example. The performance of the method is assessed using matrices having the percentage of correct condition identification in the diagonal and the percentages misclassified conditions in the off-diagonal elements. Considerable performance improvements may be obtained due to concatenation—when two or more outputs are available—and to discriminant analysis, as compared with other wavelet variance methods. Index Terms—Discriminant analysis, fault detection, fault isolation, wavelet transform.

I. INTRODUCTION

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ETECTION consists of finding if there has been a change in the operating conditions of the system, e.g., if any fault has occurred. The localization of the fault, i.e., determining what has failed is called isolation, while determining the size or nature of the fault has been named fault identification [1]. When both input and output variables are measured in a plant or system, input–output models derived from this information may be used for fault detection and isolation (FDI) [2], [3] using a multimodel approach. In certain cases only output variables may be measured, because no input measurements are available (nonexistent sensors, impossibility of measurements due to the nature of the system). Even if input measurements were available it may happen that no reliable model has been yet developed. Heartbeat analysis to asses cardiac pathology [4] and analysis of seismic waves [5] are examples of such cases. Then the Manuscript received June 30, 2004; revised May 27, 2005. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Henrique Malvar. G. D. Gonzalez, R. Paut, D. R. Miranda, and G. E. Ceballos are with the Electrical Engineering Department, University of Chile, Santiago, Chile (e-mail: [email protected]; [email protected]; [email protected]; [email protected]). A. Cipriano is with the Electrical Engineering Department, Pontificia Universidad Católica de Chile, Santiago 22, Chile (e-mail: [email protected]). Digital Object Identifier 10.1109/TSP.2006.872608

inputs to the system are unmeasured disturbances, and FDI must rely only on output measurements [3]. Autoregressive (AR) or Kalman filter models may then be used in the multimodel approach, with each model representing the system under normal condition or each particular fault condition. Another approach is to use features extracted from the output variables which may characterize each normal or fault condition, such as features based on the continuous or discrete wavelet transform. In particular, the variance of the wavelet transform of measured signals is a feature that has been used for FDI as in [4] and [6]–[9]. In [4], the variance of the wavelet transform of human heart interbeat intervals is used to distinguish healthy patients from those affected by some forms of cardiac pathology. In [8], a method based on the CWT variance has been developed and tested in a simulated mechanical system for one normal and one fault condition. Here classification between normal and fault conditions is carried out in the space of principal components, onto which the CWT variances are projected. In such space Fisher’s discriminant analysis (FDA) [10] is used for classification. Also, advantages are shown to exist if two outputs are used. In [7], FDI in an oil refinery has been approached by characterizing patterns or templates based on the power of the wavelet transform—equivalent to the wavelet variance—of a plant output. For a given condition, the decision as to what condition is present is made by considering the smallest weighted distance between the output measurement CWT sample variance and the normal and fault templates. In [6], the wavelet variance has been used for the detection of flaws in structures with the aim of monitoring the conditions of a structure beyond its specified life time, e.g., in the case of aircraft structures. Again, no system model is employed in [9], where a wavelet-based method is used both to detect the appearance of abrupt failure of sensors in time, as well as to characterize the type of fault that has occurred. The different energy—equivalent to variance—at the different wavelet scales are compared before and after the fault detection and used to characterize the type of abrupt fault. The question of ergodicity is also important since in practice for online FDI only one measurement record of each measured variable can be used to obtain such statistics as the variance of the wavelet transforms. It has been shown [4] that the sample variance of the discrete wavelet transform is a consistent (ergodic) estimator in the case of white noise. Using outputs from a mechanical system, in [8] this property for a more general case of colored noise has been verified in the case of the CWT.

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As compared with the FDI methods using CWT variance found in [4], [6], [7], and [9], the method proposed here considers the combined characteristics of two or more outputs by means of concatenation, instead of dealing with single outputs. Another difference is that, in order to improve classification performance, statistical discrimination is incorporated in the method proposed here. By using the combined characteristics of two or more outputs it may be expected that better classification performances may be obtained with these methods, e.g., considering an additional output—perhaps blood pressure or some other measurement—combined with interbeat heart output in [4]. Further improvements might be possible if discriminant analysis were also used. The CWT has been employed here while in these quoted methods, except [6], the discrete wavelet transform (DWT) is used. However the method proposed here may be converted to DWT by dyadic discretization of scales and uniform discretization of displacement . As compared with the developments in [8], in this paper FDI considers several faults besides a normal condition, as well as the effect of an unmeasured input disturbance and the influence of measurement noise. Also, quadratic discriminant analysis (QDA) [10] has been included, as well as determination of the mean delay for fault detection for a given mean time between false alarms. In addition, some general insight is given concerning the benefits of concatenation. II. CHARACTERIZATION OF NORMAL AND FAULT CONDITIONS be a stochastic process corresponding to a system Let output, where is an element in some set of outcomes of a probability space [11]. In other words, a given measurement or realization of the system output is a time function corresponding . In many systems, and particularly in industrial to a given is available through meaplants, only one realization surement of an output variable in a given time interval. Thus certain statistics used for plant analysis or control must be estimated from this single measurement, in particular, the variance . The wavelet of the CWT of an output stochastic process transform of is [12]

(1)

If is wide sense stationary—or, in practice, at least may be considered so in a given time interval—it is easily seen that the expected value of (1) is zero, so that its variance is given . Furthermore, such variby the expected value of ance does not depend on the displacement , hence the variance of the wavelet transform may be denoted by (Ap’s—or their estimations—determined for pendix A). The different conditions serve as templates for classifying different operating conditions of a plant, such as those due to normal and from a single realizadifferent faults. In order to estimate , the average tion (measurement) of

(2)

Sample variances S (a;  ) (fine lines) are realizations of (a;  ) corresponding to different measurements (realizations) y (t;  ) of the random process y (t;  ) of output 1 for normal conditions. Thick solid line is variance V (a) = E fS (a;  )g, around which the sample Fig. 1. S

variances are distributed.

with respect to wavelet displacement of may be used. This average will be called the sample variance. The max2 is bounded by the length of meaimum displacement surement . This sample variance is an unbiased estimator of since using (2)

(3)

Hence different sample variance realizations for corresponding to different measurements —for a given plant operating condition—are distributed around , i.e., as shown in Fig. 1 for the case of output of the mechanical system. The variance of using (3) is (4) , an online estimate If this variance tends to zero as may be obtained through averaging corof of a responding to a single realization (measurement) system output using (2), with any desired precision for large of the CWT of output enough . Then the square is ergodic-in-the-mean [11]. From Table I it is verified norm of (4) decreases as increases, as determined that the by simulation for output of the mechanical system described in Appendix B for the case of normal condition (see also [8]). Therefore one would like to have as large as possible so that of —corresponding any realization or outcome of length —is very close to to a given measurement wavelet variance . A minimum value of may be . On the other defined so that of will usually be determined hand, the upper value is to be by practical considerations. For example, computed using a measurement of such length —and hence 2 2 —that the system characteristics are a window . approximately fixed (stationary). Then, in general

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TABLE I RELATIVE VARIANCE OF THE SAMPLE VARIANCE S (a) AS A FUNCTION OF WINDOW LENGTH L (s)

By (2) and (6), let the sample variance for output when the system is operating under condition be given by (7)

Fig. 2. Response of output y of the system of Fig. 9 to a change from normal F to fault F at t = 300 (s).

When a change occurs, the larger the window used to compute , the larger will be the delay in detecting the change and identifying the new condition, since data pertaining to the old condition may weigh unduly the computation while the window is progressing from the old to the new condition. A system may be subject to different operating conditions—in particular, normal and various fault operating conditions—and the problem is to decide which operating condition is present using measurements from the system outputs. Changes in operating conditions may be reflected in changes in different sets of outputs, depending on the operating conditions. be the output stochastic process corresponding Let to output when the system is in operating condition . Then, for a given realization , the measurement at output is . For example, Fig. 2 shows a realization of output of the mechanical system of Fig. 9 (Appendix B) when the to fault operating condition changes from normal condition caused by a step change in the upper damper condition parameter . As seen below, even though a certain output may not by itself be a good signal to detect and classify a given operating condition, it may help improve the classification abilities of other outputs when a combined signal characteristics approach is used. Let (5) be the template (variance) corresponding to system normal or fault condition corresponding to output , where in this case from (1) (6)

which is seen to have the form of a stochastic process of the real variable —or a random variable, given . Then for a given mea, the sample variance turns into a real number surement for a given value of scale . If only one measurement were used (e.g., as in online FDI), the problem would be to determine which condition is present and the set of templates using a single realization —or their estimations —for output . In other words, it must be found to what is equal. Variance for output and for operating condition may be estimated by the average (8) where

, of the sample variances obtained from the stochastic processes corresponding to operating condition and output . This is a case of repeated trials [11], where belongs to the Cartesian sets of outcomes corresponding to each product of the belongs to one of them. This is an unbiased trial, and each estimator of variance because by (3) and (8), (9) It should be expected that the larger , the more precise this estimate becomes, in the sense that the variance of decreases as increases. By (3) another unbiased estimator for is obtained by using the sample variance cor. Then for a relatively large time inresponding to terval—thus allowing a large value of —a small variance for may be obtained because of the ergodic property of with respect to . A. Representation in Euclidean Spaces In order to apply FDA or QDA [10] and principal component analysis, let the interval of the scale considered be discretized values so that turns into the into template row vector (10) Similarly, using variance estimator (8), let (11) (12)

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be estimated variance and sample variance vectors for output , when condition is present. Both (11) and (12) are unbiased estimators, since by (2), (3), and (7) (13) and from (9) (14) The set of discretized values were chosen equidistant as in [7]. If a dyadic scale discretization were made and an equidistant one for displacement , the DWT would be obtained [13]. Then the scale range is determined by discarding wavelet coefficients according to some criterion involving the difference between the reconstructed and the original function [7], [9], [14]. On the other hand, here a preliminary range of scales for the CWT is first chosen according to a compromise between the separation between the estimated CWT and —considered variances of the different cases for the mechanical system of Appendix B—and their standard deviation. The final choice of the scale range [1], [7] has been made according to the classification performance.

for all the 2) Several Operating Conditions: Let the operating conditions be placed together to form the matrix

1

(20) and let

B. Concatenation 1) Several Measurements: If two or more system outputs are available, FDI may be attempted by combining the variances corresponding to these outputs. Let the vectors (12) corresponding to the outputs when the system is sample in operating condition be concatenated in the variance vector (15)

(21) be a realization of . Therefore contains the sample variance vectors for the realizations for all operating conditions to and are included in the Euclidean space defined above. An estimate of the covariance of —obtained using the of —is given by centered version (22)

Similarly, using (14), let the estimated variance vector be (16) Let (17) where superscript denotes transposed, with the matrix realizations— to —of the sample variance containing vectors (15) for the case of operating condition . Each row of is a vector in an Euclidean space whose components are sample variances ranging from to , discretized scales, when the for all outputs and for all the is a realization for system is in operating condition . Then of (18) The covariance matrix of

Fig. 3. Pictorial example of a realization of concatenated variances and sample variances for the case of three operating conditions fj = 0; 1; 2g. The problem is to assign s (23) to one such condition using some distance or probabilistic criterion.

may be estimated by (19)

indicates centering, i.e., subtracting the estimated where the means from the elements of a realization of .

In addition, let the estimated variance vectors (16)—used as space. A pictorial reptemplates—be also included in this resentation of this is given in Fig. 3 for the case of three oper, and and two outputs and ating conditions: . Covariance matrix (22) of the concatenated variances contains not only correlations between the CWT variance at different scales for each output but also correlations between CWT variances of different outputs. 3) Test Sample Variance Vector: During a given time interval, let measurements of the system outputs be obtained. Let the sample variance vector corresponding to (15) obtained with these measurements be (23) The problem is now to determine to which of the 1 operating condition templates (16) is closest, in some sense, either in space to which and the belong (e.g., see the original Fig. 3) or after transformation to another space. The most direct way would be to compare the Euclidean distances between and the 1 template vectors . However, it may be advantageous to use some kind of weighted distance, for example, based on covariance matrices. This approach is taken in the FDA method

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—which are below line are projected below condition , so they would be classified as belonging to condition . provides In the case depicted in Fig. 4, the use of output better separation than . For this case, let be the point on and . Then it may be the horizontal axis equidistant from to the left seen that all sample variances belonging to ellipse are projected below . Hence, since their distance of line to is smaller than to , they will be wrongly classified as belonging to condition . Similarly, all points in to the right are wrongly classified as belonging to instead of . of But if both variances of and are used, then points in the plane may be projected to a line such as , on which and are the projections of and , while is equidistant to and . Using a similar reasoning as above, it may be seen that wrong classification is reduced, as compared with the cases of single outputs. Fig. 4. Example in which output y is not good for discriminating between and but helps to improve discrimination using only output y , cases when points are projected to a suitably selected axis .

G

G

L

[10] and in the simplified distance weighting method described below. Alternatively, this decision may be taken based on a probabilistic approach, also using the covariance matrices, as in the QDA method [10] described further on. In any case, once the decision as to which is the prevailing operating condition template , the fault location, its nature, and its size have been determined. Hence isolation and identification are equivalent as a consequence of the use of templates. Fig. 4 depicts the benefits of concatenation for the case of and . Points on two outputs and two operating conditions and the horizontal axis are CWT sample variances corresponding to output in operating condition and for different realizations. The mean variances for and are, respectively, and . Just for visualization purposes, only one scale has been selected. The same applies to the vertical axis for output and sample variances and with mean variances and for conditions and . on the plane result from Points concatenating the sample variances of both outputs for condition . Then mean variances in the concatenated case for condition and are points for . Points on and inside ellipse are sample variances having distances to —weighted by the corresponding covariance matrix—which are less than or equal to a given value , , and the same disand similarly for ellipse , condition in the case of output . tance When only CWT variances for output are used, there is a relatively large intersection of the projections of points of eland on the vertical axis. This means that many lipses and are subpoints belonging to operating conditions ject to confusion and consequently to wrong classification. For example, all points belonging to ellipse —and hence to condition —above line in Fig. 4 are projected to points in the than to , so at least these vertical axis that are closer to points would be wrongly classified as belonging to condition . Likewise, all points belonging to ellipse —and hence to

C. Principal Component Analysis (PCA) Both FDA and QDA require the inversion of the covariance matrices of estimated by (19). But it may happen that these matrices are badly conditioned, as in the example given below. In such a case—and also in order to benefits from the advantages provided by a reduced dimension space, space—principal component instead of dealing with the analysis (PCA) [10] may be employed. matrix of unit norm eigenvectors of Let be the be the corresponding eigencovariance matrix (22) and let values. Then a reduced space may be formed using the eigenvectors selected according to some criterion, i.e., those having be the resulting matrix the largest eigenvalues. Let formed by these eigenvectors. Then the projection of onto the reduced dimension orthonormal space is given by the matrix (24) row of , which are points in the Therefore, each 1 space corresponding to a realization , is prooriginal in the reduced principal component jected into a point [10]. space , where in the usual cases Then the covariance matrices in the reduced dimension principal component space, corresponding to the 1 operating matrices conditions, are the (25) where is centered. The usually drastic reduction in dimay avoid bad conditioning mensions caused because of covariance matrices (25). III. FDI METHODS A. Quadratic Discriminant Analysis (QDA) Let (26) (23) obtained from be the projected sample variance vector in a certain time interval on the a given measurement reduced PCA space.

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Under the assumption of gaussian distribution, covariance (25) in the PCA space and the respective means matrices —which are the projection of the variance templates (16) corresponding to each operating condition—determine the probabelongs to normal or the difbility that a given . The QDA discriminant ferent fault conditions function is [10]

(27) where is the probability of condition . Then, for , the prevailing condition is the one that gives the maximum of . Although the Gaussian assumption is not met because of the nonlinearity due to the squaring operation involved, good results have been obtained in the example considered below. B. Fisher Discriminant Analysis (FDA) As in the QDA method, PCA has also been used here, so FDA [10] is applied in the PCA space for FDI. However, in FDA no Gaussian distribution assumption is made. If the number to , this of classes to be discriminated upon was four, PCA space would be projected onto a three-dimensional space such that the ratio of the “between groups” (classes) sum of squares to the “within groups” sums of squares is maximum [10]. Intuitively, this operation is a compromise between having the means of the distributions as far as possible and their standard deviations as small as possible, thus facilitating classification. Classification in this three-dimensional space has been made according to the Mahalanobis distance [10] from the projected sample variance to the projected variance templates of each class . C. Simplified Distance Weighting (SDW) A simplified method has been used and applied to measurements in a distillation plant [7]. The discriminating function here is distance weighted by the inverse of the variance of the template at each scale. Since these variances appear in the diagonal given by (19), the distance from sample variance (23) of to template (16) is (28) This method has been extended here to several concatenated output variables, as in the case of QDA and FDA. Even though (19) may be ill conditioned, the diagonal matrix having as diagonal may not, in which case there is no need diagonal the for PCA. D. Detection Fault detection deals with finding if something has gone wrong in a system, e.g., that a fault has occurred [1]. Detection time is the time at which such an event—e.g., occurring at time —is detected. Clearly the detection delay is an important number. Since the system is subject to random disturbances, for different realizations of the random processes are most likely to be involved, different values of and of and will be random variables. obtained, i.e., in general Also, due to these random disturbances false detections may occur. In order to asses the detection performance of a detection

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method two indexes may be defined: (i) mean detection delay and (ii) mean time between false detections . When a fault occurs, there is a period of nonstationary conditions which is reflected in the characteristics of system outputs, . Although the new statistics are reached asymptote.g., ically, from a practical point they may be considered to attain final values after a period consisting of several (e.g., four) of the is a nonstationary largest time constant has elapsed. If stochastic process, its autocorrelation cannot be expressed as function in terms of only the difference between its arguments, as in (31) in Appendix A, and the nonstationary CWT variance . Then the sample average (2) is must be expressed as no longer an estimator of such variance, so estimator (8) cannot be used for estimating the CWT variance. Notwithstanding that the sample variance is not an estimator of the CWT variance during the transient period following the onset of a fault, it may be used in the detection detect the fault. As the window in which it being calculated—e.g., of length 100 (s)—moves through the transient period from the previous stationary condition to the new one, gradually data from the new operating condition enter integral (2) while data corresponding to the previous one get discarded (e.g., see Fig. 7). , let In order to estimate the -varying CWT variance

(29) where

. This is an unbiased estimator of because, using (29) (30)

For example, output in the case of Fig. 2 is nonstationary since parameter of damper 1 changes from 0.5 to 0.3 at (s). Its wavelet variance , estimated using (29), is shown in Fig. 7 parameterized by seven different scales in the set 1,2,3,4,5,6,7 and . It may be seen that the variance is independent of —except for leftover noise—before and after (s), where stationary conditions prevail, but depends on when considered in the complete interval [0, 600] or in intervals which include (s). IV. RESULTS A. Classification In order to test the proposed methods, the mechanical system of Appendix B was used. The system is acted upon by force , which is an unmeasured stochastic process having a stepwise changing mean, with steps between intervals larger than and the system settling time to input steps. Measurements of the position of masses 1 and 2 are disturbed by zero-mean white noise having standard deviations of 5% (normal) and 10% of the standard deviation of the corresponding output in normal operating conditions. The number of operating conditions—normal and three . The parameters subject to fault change faults—was from 0.5 to 0.3 from normal to fault condition as follows: and from 2 to 1 (N s/m) to (N s/m)

GONZALEZ et al.: FAULT DETECTION AND ISOLATION USING CONCATENATED WAVELET TRANSFORM VARIANCES

TABLE II MEAN PERCENTAGE OF HITS

IN CLASSIFYING USING QDA

OPERATING CONDITIONS

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TABLE IV MEAN PERCENTAGE OF HITS IN CLASSIFYING OPERATING CONDITIONS USING FDA

TABLE V PERFORMANCE MATRIX USING CONCATENATED SAMPLE VARIANCES USING FDA

TABLE III PERFORMANCE MATRIX USING CONCATENATED SAMPLE VARIANCES USING QDA

(N/m) . The step changes in the damper rates and clearly do not affect the mean values of outputs and once the transient period is over, but they will be affected by due to fault . Since the step change in the spring constant the CWT—calculated in a given time window—of a constant (in this case the mean values before an after the transient) is zero, this feature is not reflected in the methods proposed here. It is the change in the dynamic behavior of the system brought that permits the detection of this about by the change in change. The number of realizations for generating matrices . (17) was In this case of the example considered here, matrices (19) and (22) turned out to be ill conditioned, so projection onto a principal component space has been used. It was found that good results were obtained by using the eigenvectors corresponding to the eight largest eigenvalues of (22). 1) Classification Results Using QDA: The result of hits (correct classification) obtained for each fault or normal condition is shown in Table II. Clearly, the result using concatenated ) is by far variances (template and sample variances, case the best. It may be observed that if only CWT variances of (case ) are used, the result is quite bad in conditions , and . However, if these variances are concatenated (case ) with those of , a considerable improvement is obtained, as are used. compared with case when only those of output This is similar to the case shown in Fig. 4. are shown in more detail by The results for the best case means of a performance matrix in Table III. The percentage of hits—i.e., correct classifications—is given by the elements of its diagonal, while the percentage of mistaken identification is contained in the off diagonal elements. 2) Classification Results Using FDA: The result of correct classification (hits) obtained for each fault or normal condition is shown in Table IV. Again, the results using concatenated variare by far the best. Details of this are given in the perances formance matrix in Table V. Again, although results using only variances are unsatisfactory , when concatenated with

Fig. 5. Sample variances of output y projected in the space of the three Fisher axes show confusion between the F ; F ; F , and F cases. Visualization through axis rotation does not show better separation.

Fig. 6. Projection of sample variances in the concatenated case shows considerable improvement in the separation of normal (F ) and fault conditions (F F ) as compared with Fig. 5.

0

variances from , an important improvement is obtained. Comparison of Figs. 5 and 6 shows that there is a considerable improvement in separation of different operating condition sample variances in the Fisher space when concatenation is used, instead of using only output . 3) Classification Results Using SDW: The result of hits is shown in Table VI. Once more, the result using concatenated sample variances is much better. The results for the concatenated case are shown in more detail by means of the performance matrix of Table VII. The original simplified distance

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MEAN PERCENTAGE OF HITS

TABLE VI IN CLASSIFYING OPERATING CONDITIONS USING SDW

TABLE VII PERFORMANCE MATRIX USING CONCATENATED SAMPLE VARIANCES USING SDW

TABLE VIII COMPARISON OF CORRECT CLASSIFICATIONS (HITS) USING CONCATENATION IN SDW, FDA, AND QDA

weighting (SDW) method [7] corresponds to cases or , since it does not use concatenation. 4) Comparisons: Table VIII shows that QDA gives the best classification results, followed closely by FDA. The SDW method is clearly unsatisfactory, as compared with QDA and FDA. Even worse results are obtained for the original SDW method—which does not consider concatenation—as may be seen from Table VI in the or cases. In general, these results may be explained, on the one hand, considering that QDA and FDA use all the information provided by covariance matrices, while SDW uses only the information contained in the diagonal of these matrices. On the other hand, by using concatenation, the corresponding covariance matrices contain not only the correlations between CWT variances of single measurements but also the correlations between the CWT variances among the different measurements. The reasons given above in connection with Fig. 4 further explain the improved results obtained through concatenation.

Fig. 7. CWT variance of output y (t;  ) of the mechanical system for the set of scales f1; 2; . . . ; 7g—from bottom to top—when operation changes from normal to fault F at t = 300 (s).

Fig. 8. Distances to projected templates using FDA and filter show detection delay of 35 (s) when condition F turns into F at 300 (s).

mechanical system operating under normal conditions (condition ), a step was given to in the input mean value. During the transient, a false alarm lasting about 200 (s) happened, during was wrongly identified. which condition

B. Influence of Disturbances

C. Detection Delay

All the methods turned out to be practically insensitive to measurement noise as its standard deviation increased from the nominal case of 5% to 10%. When the unmeasured input mean value undergoes a step, the outputs experience a transient period, after which they attain new constant mean values, assuming the system is asymptotically stable, as in the case of the mechanical system. The fact that the CWT of a constant value is zero—so is its square—is the cause that after the transient the output CWTs computed in a time window of given length—e.g., 100 (s)—are the same as before the input step change. As a consequence, FDI is not affected once the stationary conditions are reestablished, as in the case of (fault ). For example, for the a change in spring constant

The mean delay for detection for a given mean time between false detections of 340 (s) has been determined. For all (s) ending at the methods, a moving window of length present time has been used for finding the sample variance. The discriminating function in each case is filtered using a first-order filter to find a compromise between mean detection delay and . Fig. 8 shows an example in the case of FDA detection of 340 (s)—and in the concatenated using filtering. For a case , which again gives the best results—FDA with 43 (s) mean detection delay is seen to be the best performer, followed by QDA with 53 (s) and SWD with 88 (s). After this transient is again wide-sense stationary, with a different period, CWT variance (e.g., see Fig. 7).

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APPENDIX B Fig. 9 shows the springs-masses-dampers system used in this paper for testing the FDI methods. Its fixed parameters are (Kg), (Kg), (N/m), (N/m). REFERENCES

Fig. 9.

Mechanical system used for testing FDI methods.

V. CONCLUSION Tests performed on a simulated mechanical system driven by an unmeasured stochastic process with step changing mean show that concatenation of the CWT variance templates for different outputs, as well as concatenation of the sample variance of the output measurements, considerably improve the overall performance in all three fault classification methods considered. Also improved is the mean delay to detection for a given mean time between false alarms. Other features that have contributed to the performance of the methods are the use of principal component analysis and statistical discriminant analysis. The methods developed here are applicable to systems whose outputs are stochastic processes which may be considered wide-sense stationary and moreover ergodic during long enough intervals so that fault isolation and identification may be achieved. Two or more system outputs should be available in order to benefit from the advantages provided by the concatenation of CWT variances. APPENDIX A From(1), istheoutputofalineartime-invariant(LTI) filterhavinganimpulseresponse and input [15]. Hence, letting be the autocorstochastic process—in general relation of the input nonstationary—to this LTI filter, from [11] the autocorrelation of its output may be written as

(31) where CWT variance of

. Hence, the (32)

depends on time displacement . On the other hand, if input to the filter is wide-sense stationary, its autocorrelation is time invariant (i.e., invariant), which implies that the autoof the filter output—in this case given by two correlation convolutions—is also time invariant [11]. Therefore (32) may now be written (33)

[1] J. Chen and R. J. Patton, Robust Model-Based Fault Diagnosis for Dynamic Systems. Norwood, MA: Kluwer Academic, 1999, pp. 2–3. [2] R. Isermann, “Process fault detection based on modeling and estimation methods—A survey,” Automatica, vol. 20, pp. 387–404, Jul. 1984. , “Trends in the application of model based fault detection and diag[3] nosis of technical processes,” in Proc. 13th IFAC Triennial World Congr., vol. 7f-01, San Francisco, CA, 1996, pp. 1–12. [4] C. Heneghan, S. B. Lowen, and M. C. Teich, “Analysis of spectral and wavelet-based measures used to assess cardiac pathology,” in Proc. 1999 IEEE Int. Conf. Acoust., Speech, Signal Process. (ICASSP), Phoenix, AZ. Paper SPTM-8.2. [5] M. Baseville and V. Nikiforov, Detection of Abrupt Changes—Theory and Applications. Englewood Cliffs, NJ: Prentice-Hall, 1993, pp. 9–11. [6] C. Boller, W. J. Staszewski, K. Worden, G. Manson, and G. R. Tomlinson, “Structure integrated sensing and related signal processing for condition-based maintenance,” presented at the Symp. Condition-Based Maintenance for Highly Engineered Systems, Università degli Studi di Pisa, Pisa, Italy, Sep. 25–27, 2000. [7] M. Daigugi, O. Kudo, and T. Wada, “Wavelet-based fault detection and identification in an oil refinery,” in Proc. 14th Triennial IFAC World Congr., Beijing, China, 1999, pp. 205–210. Paper P-7e-08-05. [8] G. D. Gonzalez, G. Ceballos, R. Paut, D. Miranda, and P. La Rosa, “Fault detection and identification through variance of wavelet transform of system outputs,” in Recent Advances in Intelligent Systems and Signal Processing, N. E. Mastorakis, C. Manikoupoulos, G. E. Antoniou, V. M. Mladenov, and I. F. Gonos, Eds. Athens, Greece: WSEAS Press, 2003, pp. 47–53. [9] J. Q. Zhang and Y. Yan, “A wavelet-based approach to abrupt fault detection and diagnosis in sensors,” IEEE Trans. Instrum. Meas., vol. 50, no. 5, pp. 1389–1396, Oct. 2001. [10] K. V. Mardia, J. T. Kent, and J. M. Bibby, Multivariate Analysis. New York: Academic, 1979, pp. 17, 31, 213–254, 300–320. [11] A. Papoulis and S. U. Pillai, Probability, Random Variables, and Stochastic Processes, 4th ed. New York: McGraw-Hill, 2002, pp. 46–71, 371–407, 399–401, 412–413, 523–532. [12] I. Daubechies, “Ten lectures on wavelets,” in CBMS-NSF Regional Conf. Series Appl. Math., vol. 61, 1992, p. 24. [13] M. Unser, “Wavelet theory demystified,” IEEE Trans. Signal Process., vol. 51, no. 2, pp. 470–483, Feb. 2003. [14] E. K. Lada, J.-C. Lu, and J. R. Wilson, “A wavelet-based procedure for process fault detection,” IEEE Trans. Semiconduct. Manufact., vol. 15, no. 1, pp. 79–90, Feb. 2002. [15] S. Mallat, A Wavelet Tour of Signal Processing, 2nd ed. New York: Academic, 1998, p. 79.

G. D. Gonzalez (M’68–SM’82–LSM’04) received the electrical engineer degree from the University of Chile in Santiago, Chile, in 1958. He received the M.Sc. degree in electrical engineering and the Ph.D. degree in computer, information, and control engineering from the University of Michigan, Ann Arbor, in 1959 and 1981, respectively. Since 1956, he has been with the University of Chile, where he currently is a Professor in the Department of Electrical Engineering. He is also currently with the Department of Mining Engineering, University of Chile. He visited the Julius Kruttshnitt Mineral Research Centre at the University of Queensland in Australia during 1985–1990, on behalf of which he has taught courses and consulted in automatic control in mineral processing plants. His research interests have included modeling, soft sensors, and fault detection and isolation, especially in problems arising from the mineral processing industry. His current research interests are in fault detection and isolation especially using wavelets, and modeling related to soft sensors. He has held several posts at the College of Engineering, University of Chile, including a member of the college academic evaluation committee, Dean of the college, and Chairman of the department. Prof. Gonzalez was the Founding President of the Chilean Automatic Control Society (1974–1982).

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R. Paut was born in Osorno, Chile, in 1977. He received the B.Sc. and M.Sc. degrees in electrical engineering from University of Chile, Santiago, in 2001 and 2005, respectively. He is currently with the Electrical Products Department, Chilean Superintendence of Electricity and Fuels. His research interests have mainly been in signal processing.

D. R. Miranda was born in Santiago, Chile, in 1979. He is currently pursuing the electrical engineer degree and master’s degree in electrical engineering from the Department of Electrical Engineering, University of Chile, Santiago. His research interests are in the areas of statistical signal processing and modeling of stochastic systems.

A. Cipriano (M’74–SM’95) received the electrical engineering and M.Sc. degrees from the Universidad de Chile, Santiago, in 1973 and 1974, respectively, and the Dr. Ing. degree from the Technische Universität München, Germany, in 1981. Since 1974, he has been with the Pontificia Universidad Católica de Chile, where he is currently Professor of Electrical Engineering. From 1998 to 2003, he was Dean of the College of Engineering at the same university. In 1990, he was a Visiting Professor at Universität Karlsruhe, Germany. He has been involved in the design and implementation of a variety of advanced control systems for mineral processing plants. His current research interests include fault detection and diagnosis, predictive control, and fuzzy systems. Dr. Cipriano was President of the Chilean Association of Automatic Control (1983–1985) and Chairman of the Chilean Joint Chapter of the IEEE Control Systems-Industrial Electronics Societies (1983). He was Cochairman of the 1994 IEEE International Symposium on Industrial Electronics and Chairman of the IFAC Technical Committee on Low Cost Automation (1996–1998).

G. E. Ceballos received the bachelor of engineering science degree, the electrical engineer degree, the diploma in economic project evaluation, and the M.Sc. degree in electrical engineering from the University of Chile, Santiago, in 1998, 2000, 2002, and 2004, respectively. From 1990 to 2004, he was a Consulting Engineer for professional matters in the College of Engineering, University of Chile. He has also consulted for a power plant and in project evaluation for the Chilean Ministry of Housing. His main research interests have been in system identification, soft sensors, and fault detection and identification.