UNIFYING PCA AND MULTISCALE APPROACHES TO FAULT DETECTION AND ISOLATION

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1 UNIFYING AND MULISCALE APPROACHES O FAUL DEECION AND ISOLAION Seongkyu Yoon an John F. MacGregor Dept. Chemical Engineering, McMaster University, Hamilton Ontario Canaa L8S 4L7 yoons@mcmaster.ca macgreg@mcmaster.ca Process signals represent the cumulative effects of many unerlying process phenomena. Multiresolution analysis is use to ecompose the cumulative process effects. he ecompose process measurements are rearrange accoring to their scales, an is applie to these multiscale ata to capture process variable correlations occurring at ifferent scales. Choosing an orthonormal mother wavelet allows each principal component to be a function of the process variables at only one scale level. he propose metho can ientify when a multiscale approach is neee. he conventional as well as MS moels are shown as the limiting cases of the propose moel. A proceure for both fault etection an isolation is presente. he propose metho is iscusse an illustrate in etail using simulate ata from a CSR system. A comparison stuy is one through Monte Carlo simulation. he propose metho significantly enhances FDI performance by using aitional scale information. Copyright 00 IFAC Keywors: Fault etection, Fault isolation, Fault, Scales,, Multiresolution analysis. INRODUCION Process measurement signals usually represent the cumulative effects of many unerlying process phenomena such as process ynamics, measurement noise, external isturbances, process egraation etc. Each effect manifests on a ifferent scale that is a reciprocal of frequency. Faults occurring at ifferent locations, times an frequencies are consiere one of the contributing events. In case of a complex fault, its signature is very likely to be confuse with multiscale nature of the process signals since the effect of the complex fault is propagate into other measurements through process ynamics an control actions. One can ecompose process signals such that all the contributing events are approximately iscriminate accoring to their scale contents an a fault is istinguishe from other events. he ecomposition is one with the multiresolution analysis via wavelet transformation which characterizes signal in the omain of time an scale (Mallat, 989; Daubechies, 990). Recently, multiscale (MS) has been formulate (Bakshi, 998). By the MS, one simultaneously extracts process correlations across ata an accounts for auto-correlation within sensor ata. his involves the ecomposition of variables on a selecte family of wavelets an the evelopment of separate moels at each scale. he moels at important scales are then combine in an efficient scale-recursive manner to yiel a moel for those scales of interest. he MS formulation makes it suitable to work with process signals having a multiscale nature. Fault etection has been emonstrate with the MS concept, but fault isolation has not been ealt with.

2 his paper presents an inirect usage of the process ynamics for FDI. A unifie framework that generalizes MS as well as is propose. An FDI proceure using the propose metho is presente. he propose metho is illustrate using a simulate ata from a CSR system with feeback control. he FDI performance of the propose metho is assesse through Monte Carlo simulation.. DECOMPOSIION OF A FINIE SIGNAL he ecomposition of signals accoring to their scale contents can be implemente with multi-channel filter banks built by cascaing two-channel banks. Since a two-channel filter bank splits a signal into a lowpass version, an a highpass version, this ecomposition is recursively applie on the lowpass version (Vetteri an Kovacevic, 995). his leas a hierachy of multiresolution ecomposition. In practice, the only ata that can be processe by an algorithm are iscrete. In the iscrete version of the multiresolution analysis, it is assume that the initial ata alreay represents an approximation at a certain scale that is relate to the sampling interval. By convention, this scale is fixe at j=0. his is the finest resolution, associate with the space V 0. hen, a finite number of ecomposition steps J leas to a coarsest resolution associate with V J. he input is ecompose into a very coarse resolution which exists in V J an ae etails which exist in the spaces W j, j =,, J,, where V J s are calle approximation spaces an W j s etail spaces. he action of a two channel analysis filter on an infinite signal column vector x is represente by using the analysis filter matrix, F a =[H H 0 ] where H i has the effect of filtering the signal by H 0 (z) an subsampling by (represente by the shift by in H 0 ) where i= for a high pass an i= for a low pass filter. he projections on the low-pass component are recursively ecompose to obtain coarser approximations. he projections on the high-pass component (wavelets at each scale) contain the finer etails. hus, x a (j) = H 0 x a (j-) an x (j) = H x a (j-) () where, x a (j) an x (j) are row vectors whose lengths are n/ j where n is a signal length. Assuming n is relatively large number, an the transformation matrix, W a is expresse as ( J ) Wa 678 = Diag { I, L , I, Fa } 4 4 Diag 44, 4 L J = [ H HH0 L HH ] 443 L H H 443 L H J J () () [ H H L H H0 ] { I, I, I, Fa } Diag { I Fa } F a = () (j) where H is the matrix containing wavelet filter coefficients corresponing to scale j an H (J) 0 is the matrix of scaling function filter coefficients at the coarsest scale. After J ecompositions, the transformation of x, or x (0) a is expresse in terms of x (j) a an x () a ; W (0) axa () () a ] = [ x x L x x (3) hus, W a x is the same size as the original ata sequence, but ue to the wavelet ecomposition, the eterministic component at lowest frequency is concentrate in a relatively small number of coefficients in x a (J), while the stochastic component in each variable is approximately ecorrelate an is sprea over all components in x a (), j =,, J, accoring to its power spectrum. Similarly, the synthesis filter matrix, F S =[G G 0 ] can be represente with its high pass an low pass components matrices, G an G 0. Base on it, the transformation matrix for the reconstruction W S is obtaine. herefore, the reconstructe signal is xˆ = W W x = W = G s a () () x () = G x + + G G0G 0 L J + G G 443 () s[ x () () x () 0Gx 0Gx x () + L + G + L + G0G 0 L J L x x + G0x 443 G a x a ] 0 xa (4) he filter bank necessarily operates over infinite signals (the matrix is infinite along both imensions) an the filters o not change with time. he application to finite lengths will generally involve either istortion at the bounary or the introuction of some reunancy. If a finite unitary matrix has the same block structure as the infinite unitary matrix, one can get a square unitary matrix for any size for a given filter set. It was shown how this problem coul be overcome in the case of two-channel orthogonal filter banks by using bounary filters (Herley, 995). 3. OF MULISCALE DAA As a ata sequence is transforme an expresse in terms of its contributions at ifferent scales, the same transformation can be applie to all the measurement ata. Let X be the observation matrix of imension, n m. All the columns of X can be ecompose into the etails at all levels an the approximation at the coarsest level. W W X = G s a () + G H ( j) () H X + G ( j) () X + L H () X + L + G H X + G0 H0 X = X + X + L+ X j + L+ XJ + XJ + (5)

3 H ( ) X H ( ) X... H J ( ) X H ( 0J ) X eigenvalues except for the one use in the previous step. X [ n, m] Bounary correcte Orthogonal Wavelet ransformation Inverse wavelet transformation he scores can be obtaine in two ways: One can use the block score accoring to the loaing selecte, or use X G an the loaings P G, ; G, = X G P G, Deflate resiuals; E G = X G - G, P G, () () G H X where each term represents scale contribution of the original ata matrix. All the terms have the same number of rows an columns. One can characterize the original ata by three parameters of variable(m), sample time(n), an scale(j). It is splitting all the scale contributions an laying the terms sie by sie to prouce a two-imensional matrix X G of size [ n ( J + ) m] (Fig. ). he X G is efine X G = VDiag(X) J+ (6) where V is the rearrange transformation matrix an Diag(X) J+ is a iagonal matrix whose iagonal component is X an off-iagonal terms are zero matrices with the same imension as X. Its imension becomes [(J+)n, (J+)m]. Once we choose an orthogonal mother wavelet, G 0 = H * 0, G = H * (J), then G 0 = (H (J) 0 ) (j), G = (H (j) ). V is simplifie in term of the analysis filter matrices. moel of multiscale ata is obtaine by applying on X G. Alternatively, one can obtain the moel by orering the principal components of all the scale blocks as follows; Perform regular on X j using algorithm to obtain P j an eigenvalues, λ j which contain all the eigenvalues of each sale block. Accoring to the eigenvalues magnitues, arrange all the corresponing loaings. Obtain the loaings of moel with the arrange block loaings. For example, when the largest eigenvalue is the largest one from j scale block, then the first loaings is; = 444 L PG, 0(, m) 0(, m) P 4443 b, j 0(, m) 444 L 0(, m) 4443 ( j ) m ( J + j) m hen, the secon loaing of moel is obtaine by selecting the loaing relate with the largest eigenvalue among the remaining... () () G H X G H X Fig.. Generation of multiscale ata G0 H0 X Return to step an repeat the proceure until one obtains all the loaings with all block loaings. One may capture the correlations among process variables as well as among scales. Due to the usage of an orthogonal mother wavelet, the moel inherits a few useful properties from the wavelet orthonormality an the relate filter features. Discrimination of process variations: Due to the orthogonality between scale blocks, the off-iagonal terms of X G X G have all zero values. he covariance matrix of X G X G becomes a block iagonal matrix. herefore, there will be no interactions between blocks, an the principal component will be a function of variables within only a scale block. Generalization of an MS: he iagonal terms of X G X G have non-zero values. heir block iagonal terms of X G X G are the same as those of iniviual blocks in MS, an the sum of the block iagonal terms becomes an ientity matrix. When one consiers the zero level of ecomposition, becomes. MS is a special case of without the information on the percents of process variations explaine by the principal components. hus the moel unifies as well as MS. Clustering of multiscale ata: When all the scale characteristics of ominant events in a process may not be shown in the orers of j S, but unevenly space from lower to higher scale, one may not nee to investigate all the ecompose scales. One then can cluster the several scales at which the similar correlations are shown. his will significantly reuce the number of scales to be analyze an simplify the multiscale analysis. Detrening: Once a multiscale ata base moel has been built, one can ientify a scale or a principal component which is not neee for process monitoring. One may further ientify a useless process variation at the ientifie scale. One can then remove one or all the principal components within the suspecte scale. Removing all the principal components within the scale is equal to a banwith filter. When a scale inclues both the useful an useless process variations, the base moeling metho provies enhance flexibility in hanling the

4 useless process variation for the process monitoring an obtaining a sensitive monitoring moel. 4. FAUL DEECION AND ISOLAION With the notion of, one has aitional information that is a scale contribution for fault analysis. One can estimate how much each scale contributes to a fault in terms of Hotelling s. he overall can be partitione into a contribution from each scale as follows: = j xg,( j ) i w( j ) i (7) i= A where j =,L, J +, pki w = k, p i = s ki is k -th t i variable weight in i -th latent variable, m is number of variables, an J is the number of scales. Once the contributing scale for a fault is ientifie, one can look at the iniviual variable contributions to the contributing scale. It can be expresse as follows: jk = x G,( j ) k w( j ) k. (8) he above equation is the scale variable contribution to relative to zero. he expression for contribution to a change in over some time interval is: jk = jk, t jk, t = x G,( j ) kw( j ) k Similar expressions can be obtaine for the scale contribution to the overall SPE given by, ( xg, ( j ) i xˆ G,( j ) i ) SPE = m j (9) i= where x ˆG, k = ( P APA ) k xg, k an the j-th variable contribution to the SPE at that scale SPE jk is SPE ( x ) jk = x G, ( j ) k ˆG,( j ) k (0) Eaxmine SPE j, j FS 5 Solvent CAS 3 Reactant 0 7 FA SP 6 CAA 4 M Reactor Outlet Cooling Water herefore, after a fault is etecte by the SPE or, one can inspect the scale contributions to the SPE or, an then the variable contributions to the contributing scales. Having ientifie the important scales an the variable contributions to these scales, one may isolate a fault with high accuracy. Figure outlines the FDI proceure base on the moel. 5. SIMULAION SUDY A nonisothermal continuous stirre-tank reactor (CSR) moel (Marlin, 995, page 90-9) is use for a case stuy (Fig. 3). he reaction is st orer (AÅB) an the reactor system involves heat transfer through cooling coils to remove the heat of reaction. he process has two fee streams (the solvent an the reactant A), one prouct stream, an a cooling water flow to the coils. he reactant (F A ) an the cooling water (F C ) flows control the reactor outlet concentration (C A ) an temperature (), respectively. However, in this application only the temperature controller is active. Measure process isturbances are the inlet concentrations (C AS, C AA ), the inlet temperature ( O ), the solvent flow (F S ), an the cooling water temperature ( C ). All of these isturbances are simulate to have first orer autoregressive variations uner both normal an fault conitions. In aition, unmeasure stochastic isturbances are also simulate as first orer autoregressive behaviors in the reaction rate (k) an heat transfer (UA) constants respectively. A negative correlation between F S an C AS is ae. In all stuies, the training ata was collecte for 00 minutes from the process uner routine operation when no faults were present. he ata collection interval S was 0 secons. C SP FC C 9 8 Fig. 3. Process flow iagram of CSR system CA New Observation No Yes SPE j or j > δucl No 5. Comparisons to an MS SPE or > δ UCL Yes No Yes Eaxmine SPE jk, jk Stop FDI Fig.. moel base FDI proceure he collecte ata was ecompose into 4 etail an one approximation blocks corresponing to S, 4 S, 8 S, 6 S, an the frequencies lower than 6 S. Deubechies-5 wavelet was use as a mother wavelet to generate the multiscale ata. hen, an MS/ moels were calculate with the

5 single-scale an multiscale ata, respectively. Figure 4 shows principal component loaings of the three moels. Each cell represents the effects of all 9 variables- C A,, C AS, C AA, F S, F A, O, C, an F C, respectively. he moel consists of 3 principal components. he first principal component explains the control action. It shows that the cooling water flow (F C ), the reactor outlet concentration (C A ) an the reactor outlet temperature () are positively correlate. he secon shows a negative correlation between the solvent flow (F S ) an the solvent concentration (C AS ). he thir one seems to be a combination of several effects. he moel reasonably escribes the relevant process correlations of the CSR system. Since the single-scale ata was iayically ecompose into 5 blocks of 4 etails an approximation, MS moel consists of moels of the 5 scale blocks. From left to right in the bottom of figure 4, each column correspons to the moel of the ecompose block of S, 4 S, 8 S, 6 S, an the lower frequencies than 6 S. he moels at all 5 scales show almost the same behavior as that of the single-scale moel. his means that the same process correlations existe over all scales. moel consists of 5 principal components since there are 5 ecompose blocks an each block has 3 principal components. Each principal component of the moel is expresse with 45 variables. It results from the fact that there are 5 ecompose blocks an each block is escribe with 9 process variables. However, the principal components of moel are actually escribe with 9 variables at only one scale ue to orthogonality among scale blocks. In the moel of figure 4, it is shown the st principal component is the function of variables at 4 th approximation block an escribes a positive correlation among the cooling water flow, the reactor outlet concentration, an the reactor outlet temperature. his correlation is base on the control action at the banwith beyon 6 S. he effect of the feeback control is most strongly shown at the frequency ban lower than 6 S. Similarly, the secon principal component inicates the similar correlation among the process variables at the frequency ban of 8 S. Information on the scale contents of the process correlations over the examine frequency bans is available only with the moel. he process correlations shown at ifferent scales with moel are almost the same as those shown by the regular moel. he first 5 principal components of the moel are very similar to the first principal component of the regular moel. he 6 th ~0 th principal components of MS blocks S 4S 8S Detail Increasing Scale 6S the moel in figure 4 appear corresponingly to the secon principal component of the regular moel. hus, one may not nee to apply multiscale analysis. In this case, becomes a very effective ata analysis tool. With the moel, one can juge if a multiscale analysis is neee. 5. Comparisons on FDI performance } App >6 S Fig. 4. Comparison of loaings for, MS an with b-5 an 4 scale levels Unstable contact or an ol instrument can cause an excessive oscillatory action on the measurement while its real value remains unchange. he inlet temperature ( O ) in the CSR moel is assume to show this behavior (Fig. 5(a)). he fault effect is not propagate into the other variables an is restricte to one frequency ban. It inicates that the precision egraation of O sensor has a single-scale effect. o examine the FDI performance on this fault, 500 sets of training an testing ata were generate with ranom sees an at each value of various FSRs (ratio of fault magnitue to square root of a signal variance). With each ata set, the type I an type II errors, an the average run lengths (ARL) for fault etection an isolation were calculate. he ype I error of MS is base on the threshol value ajuste with the Bonferroni rule. Figure 6(a) an (b) show the fault etection an the isolation ARLs for, MS an at various magnitues of the fault. he isolation ARL was calculate with the variable contributions assuming that the fault is etecte an ientifie. he /MS outperforms the regular. his is ue to the characteristic of the precision egraation whose effect is not sprea over wie frequency ranges. As a secon case, a sensor bias of the inlet temperature measurement ( O ) is simulate (Fig. 5(b)). All the other conitions are the same as the

6 above case. Figure 7(a) an 7(b) show the fault etection an the isolation ARLs for the sensor bias. he etection an isolation ARLs of /MS show almost the same results as that of the regular. his is ue to the characteristic of the sensor bias whose effect is sprea over wie frequency ranges. he simulation result confirms that the multiscale moel oes not significantly outperform the regular in such a case. 6. CONCLUSION In this stuy, that is of the multiscale ata has been propose as a moeling metho for ata showing multiscale features. When process signals represent the cumulative effects of many unerlying process phenomena an each of them manifests on a ifferent scale, one can explicitly ecompose those process effects over scale levels, an capture the process correlation among variables an scales. shows what scale ranges shoul be monitore since it gives scale information of significant process correlation. It also enables one to remove unnecessary effects of process events from further analysis, or to cluster several scale blocks showing similar correlation to efficiently summarize the multiscale process correlations. One can also confirm if the multiscale analysis is neee. he propose scheme unifies the existing methos since an MS are limiting cases of. thus not only conveys all the information contents shown in both MS an, but also reveals the relative significance of all the principal components over scales, which is unavailable with MS. Base on the propose metho, a proceure for both fault etection an isolation using the moel has been presente. Contributions of the various scales to the overall an SPE, an contributions of the variables to each scale have been propose as aitional tools for fault isolation. Due to the usage of scale information, FDI performance can be significantly enhance. FDI performance has been assesse through Monte Carlo simulation with the CSR system. When a fault occurs in only one frequency ban, multiscale methos will be more effective than the regular methos in etecting an isolating faults. However, it is questionable if the multiscale methos woul result in better performance when the fault effect is sprea over more than one frequency ban. hus, a monitoring metho that gives the best etection an isolation of faults will epen upon the fault characteristics. to Multivariate Statistical Process Monitoring, AIChE J., 44(7), Daubechies, I., (990). he wavelet transform, timefrequency localization an signal analysis, IEEE transactions on information theory, 36(5), Herley, C. (995), Bounary filters for finite-length signals an time-varying filter banks, IEEE transactions on circuit an systems II: Analog an Digital signal processing, 4(), p.0-4. Mallat, S. G., (989), A theory for multiresolution signal ecomposition: he wavelet representation, IEEE rans on pattern analysis an machine intelligence, (7), Marlin,.E., (995), Process Control: Designing Processes an Control systems for Dynamic Performances, McGraw-Hill, New York. Vetteri, M. an J. Kovacevic, (995), Wavelets an subban coing, Prentice Hall, Englewoo Cliffs, NJ. o (a) ime (min) o (b) ime (min) Fig. 5. Simulate faults on O sensor; (a) Precision egraation; (b) Sensor bias (Fault at 5 min uring [0 00]) Detection ARL 0 0 (a) MS Isolation ARL 0 0 (b) Fig. 6. FDI w.r.t. the precision egraation of O sensor; (a) etection ARL; (b) isolation ARL (99% CL) Detection ARL 0 0 (a) MS Isolation ARL 0 0 (b) REFERENCES Bakshi B., (998). Multiscale with Application Fig. 7. FDI w.r.t. O sensor bias; (a) etection ARL; (b) isolation ARL (99% CL)