A FAULT DETECTION METHOD USING MULTI-SCALE KERNEL PRINCIPAL COMPONENT ANALYSIS

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1 A FAUL DEECION MEHOD USING MULI-SCALE KERNEL PRINCIPAL COMPONEN ANALYSIS Xuemn an, Xaogang Deng, Shuang-Hua Yang College of Informaton and Control Engneerng,Chna Unversty of Petroleum Dongyng, Shandong 5706, P. R. Chna Department of Computer Scence, Loughborough Unversty LE 3U Loughborough, Lecestershre, UK Abstract: A fault detecton method based on mult-scale ernel prncpal component analyss (MSKPCA) s developed for nonlnear processes montorng. It ntegrates wavelet analyss and nonlnear transformaton usng ernel prncpal component analyss. Wavelet analyss can decompose measured sgnal nto approxmaton part and detal part at multple scales to capture tme-frequency nformaton, whle ernel prncpal component analyss s performed for nonlnear prncpal components at each scale by ernel functons. he combned method can smultaneously extract cross correlaton, auto correlaton, and nonlneartes from the data. Furthermore, a mult-scale prncpal component analyss smlarty factor s proposed for dentfyng fault pattern. Smulaton of a E benchmar process shows that the proposed method has a better performance compared wth the tradtonal PCA method n fault detecton and dagnoss. Key words: fault detecton, ernel functon, nonlnear prncpal component analyss, wavelet analyss,smlarty factor. INRODUCION he demands for mprovng process safety and producton qualty and reducng mantenance costs have stmulated the development of process montorng and fault dagnoss methods. Varous methods have been developed to detect abnormal events n the past three decades. hey can be classfed nto three maor categores: analytcal methods, nowledge-based methods, data-drven methods (Chang et al., 00). he frst two types of methods requre detaled process models and they can be appled to nformaton-rch processes. hey are not often used n large-scale ndustral process. Wth the development of modern computer control systems, large amounts of data can be collected whch reflect the physcal/chemcal prncples of the process. As a result, data-drven multvarate statstcal methods such as prncpal component analyss (PCA) and partal least square (PLS) have attracted great attenton from both academc researchers and process engneers. As a popular multvarate statstcal method, PCA has been successfully appled n many ndustry processes whch proects hgh-dmensonal, nosy and correlated data onto a lower-dmensonal subspace. PCA, however, s statc lnear transformaton n nature, whle most of ndustral processes have nonlnear characterstcs and process data have dynamc and mult-scale propertes. o overcome the drawbacs of PCA, many nonlnear PCA methods have been developed. Krammer (99) frst proposed a nonlnear PCA based on an autoassocatve neural networ wth a fve-layer structure. Dong and MacAvoy (995) presented an alternatve nonlnear PCA approach whch uses a prncpal curve algorthm and neural networ. he prncpal curve algorthm s used to generate nonlnear prncpal scores and neural networ to buld a nonlnear PCA model. Schölopf et al. (998) proposed a ernel PCA whch used ernel functons to complete nonlnear transformaton. Lee et al. (004) used ths method for nonlnear process montorng. Hden et al. (999) suggested non-lnear prncpal components analyss usng genetc programmng. Shao et al. (999) proposed a nonlnear PCA based upon an nput-tranng neural networ. Other nonlnear PCA methods were also developed by Ln et al. (000) and Foure et al. (000).

2 In order to extract dynamc nformaton from multscale process data, many approaches based on dynamc analyss and mult-scale analyss have been nvestgated. Ku et al (995) appled PCA to a tmelagged data matrx. Negz and Cnar (997) constructed a canoncal varate state space model to montor process. hs s called canoncal varate analyss (CVA). Wavelet analyss has good tmefrequency locaton and mult-resoluton property. Bash (998) proposed mult-scale prncpal component analyss (MSPCA) combned wth PCA to de-correlate the cross correlaton among dfferent varables, n whch wavelet analyss was used to capture auto-correlaton wthn ndvdual varable. Extenson of ths approach was made by Msra et al. (00) by ntroducng mult-scale fault dentfcaton. Lu et al. (003) utlzed wavelet analyss to extract quanttatve tme-frequency feature for on-lne process montorng. hs paper proposes a mult-scale ernel PCA (MSKPCA) method that uses ernel PCA (KPCA) to handle the nonlneartes and wavelet analyss to extract mult-scale dynamcs. he montorng charts of and Q at dfferent scales are constructed to detect faults. Once a fault s detected va MSKPCA, a mult-scale prncpal component smlarty factor s used for dentfyng the type of fault and dscoverng the fault source. he rest of the paper s organzed as follows. he concept of KPCA and wavelet analyss s ntroduced n Secton. Secton 3 gves the formulaton of MSKPCA and montorng strategy. Secton 4 explans the mult-scale prncpal component smlarty factor. In Secton 5 an example of the ennessee Eastman benchmar process s used to valdate the proposed montorng scheme. Fnally, conclusons are presented n Secton 6..KPCA AND WAVELE ANALYSIS. PCA and KPCA PCA s a powerful dmenson-reducng technque. It produces new varables that are uncorrelated wth each other and are lnear combnatons of orgnal varables. PCA can be mplemented by solvng an egenvalue problem. For a gven data matrx X whch represents m columns of measured varables at n rows of samplng ponts durng normal operaton condton, the covarance matrx of X s: cov( X) = X X/( n ) () If X has been autoscaled by subtractng the mean of each column and dvdng each column by ts standard devaton, the covarance matrx s the correlaton matrx. he loadng vector of PCA decomposton p, s the th egenvector of the covarance matrx and λ s the egenvalue correspondng to the egenvector p. cov( Xp ) = λ p () PCA decomposes the data matrx X as: X = t p + t p + + t p + E (3)... where t s a scores vector, p s a loadng vector, s the number of prncpal components(pcs) retaned n PC model, and E s a resdual matrx. It s not approprate to perform PCA decomposton n orgnal nput space wth strong nonlnear data structure snce PCA s a lnear transformaton. So KPCA s put forward to carry out nonlnear transformaton. he nonlnear data n orgnal nput space s mapped nto a lnear hgh-dmensonal space referred as feature space, n whch PCA s used to analyze the lnear correlaton. hs procedure s called KPCA. It utlzes a computatonally tractable soluton to complete nonlnear mappng by a smple ernel functon whle other nonlnear PCA methods resort to neural networ whch s complcated. It s assumed that Φ (.) s a nonlnear mappng functon that proects the data n nput space to feature space and the covarance matrx n feature space can be expressed as: n cov( Φ ( X)) = Φ( x ) Φ( x ) (4) n = where x (=,,n) s the th row of X. In the feature space the egenvalue problem must be solved for KPCA decomposton. cov( Φ ( X)) p = λ p (5) It should be ponted out that t s dffcult to solve the problem of Eq. (5) as Φ (.) can not be obtaned n most cases. However, there exst coeffcents α (=,, n) such that n p = α Φ( x ) = (6) Combnng Eqs.(4), (5) and (6), the followng equaton can be obtaned ( n ) λ α = Kα (7) where K s defned as [K] =(x,, x ) = < Φ( x ), Φ( x ) >. One can avod nonlnear mappng by defnng ernel functon (x, y) = < Φ( x), Φ( y) >.here are many ernel functons avalable. In ths paper ernel functon s selected as (x, y) = exp (- x y c), where c s a parameter specfed by user. Wth the use of ernel functon, the score vectors are expressed as: t =< p, Φ ( x) >= α <Φ( x ), Φ ( x ) > (8) = n For further detals, see the paper of Schölopf et al. (998).. Wavelet analyss Wavelet analyss s a powerful sgnal-processng tool that has been wdely used n many felds such as data compresson, mage processng and process montorng and dagnoss. In recent years, mult-scale methods based on wavelet analyss have demonstrated superor performance n fault detecton. (Lu et al, 003)

3 he theory of wavelet analyss mples that any fnte energy square ntegral functon f (t) can be ntegrated by proectng t down onto scalng functons φ, () t and wavelet functons ψ, () t, that s, f (t) can be decomposed at multple resolutons. Accordng to Mallat s theory of mult-resoluton analyss, a sgnal f (t) can be decomposed as (Mallat, 989; Lu et al, 003): ψ f() t = a φ () t + d () t,,,, Z = Z φ, () t φ = ( t ), Z (9) (0) ψ, () t = ψ ( t ), =,...,, Z () where s the scale factor, s the translaton factor, and s the coarsest scale. a, s called as approxmaton coeffcents or scalng coeffcents, whch are the proects of f(t) on scalng functons (see Eq.). d, s referred as detal coeffcents or wavelet coeffcents, whch are the proects of f(t) on wavelet functons (see Eq.3). a =< f(), t φ () t > (),, d =< f(), t ψ () t > (3),, Once approxmaton coeffcents and detal coeffcents are nown, the approxmaton sgnal A (t) and the detal sgnal D (t) can be reconstructed as A () t = a, φ, () t (4) Z D () t = d ψ (), t =,...,,, Z hen the orgnal sgnal f(t) can be represented as = + = f() t A () t D () t (5) (6) Accordng to above descrpton, wavelet analyss can be used to decompose the sgnal nto approxmaton part and detal part at multple scales, so sgnal can be analyzed at the tme-frequency doman rather than only at tme doman. he MSKPCA approach presented n Secton 3 uses ths property to develop the mult-scale analyss of process data for process montorng and fault dagnoss. 3. MSKPCA BASED PROCESS MONIORING SRAEGY he MSKPCA method of combng ernel PCA and wavelet analyss for process montorng s proposed n ths secton. hs method s motved from the dea that KPCA can de-correlate the cross correlaton among varables whle wavelet analyss can capture the nformaton at multple scales and de-correlate the auto correlaton among the ndvdual varable. he flow dagram of the proposed strategy s shown n Fg.. Wavelet analyss s frst used to analyze the measured data matrx X at multple scales, resultng n new data matrx D, D,, D, and A. It s necessary to pont out that the proper selecton of coarsest scale s mportant, whch s chosen as n ths paper. hen, KPCA s appled to each new data matrx to extract nformaton n lnear feature space for fault detecton. Fnally, mult-scale PCA smlarty factor s appled to dentfy fault pattern. measured data matrx X wavelet analyss A D D Fault detecton based on KPCA Fault detecton based on KPCA Fault detecton based on KPCA Fg.. Flow dagram of MSKPCA strategy 3. Fault detecton based on MSKPCA Fault dentfcaton based on mult-scale prncpal component smlarty factor method he MSKPCA-based fault detecton s smlar to PCA based methods n whch Hotellng s statstc and Q statstc are developed to determne f an abnormalty has occurred (Lee et al., 004). MSKPCA s statstc s the sum of the normalzed squared scores, whch montors the varaton wthn MSKPCA prncpal component subspace for data matrx D, D,, D, and A. It s expressed as Eq. (7). = [ t, t,... t ] Λ [ t, t,... t ] (7) where t s obtaned from Eq.(8) and Λ s a dagonal matrx of the egenvalues assocated wth the retaned PCs. Confdence lmt s computed by means of an F- dstrbuton as n ( ),n, α ~ F, n, α (8) n where n s the number of samples and s the number of PCs retaned n PC model,f (, n-,α ) s a F- dstrbuton wth degrees of freedom and n- and level of sgnfcance α. he MSKPCA s Q statstc, on the other hand, montors the varaton n the resdual subspace computed as n = = Q = t t (9) he confdence lmt for Q statstc can be calculated by dfferent means. In ths paper, t s computed by fttng a weghted χ dstrbuton as Q ~ h where g, h are the parameters that can be approxmated by statstcal nformaton of Q. α gχ (0) he steps of fault detecton based on MSKPCA are formulated as: Part I. Modellng procedure. Acqure and auto-scale measurement data matrx X at the normal operatng condton.. For each column n X, calculate the approxmate sgnal and detal sgnals by wavelet analyss. Obtan the new data matrx D, D D, and A.

4 3. For each data matrx, KPCA s appled and then MSKPCA model under the normal operatng condton s obtaned. 4. Calculate the montorng statstcs ( and Q) of the normal operatng data, and determne the confdence lmts of and Q statstcs. Part II. On-lne process montorng and fault detecton 5. Obtan new data for each varable and scale t. 6. Calculate the approxmate sgnal and detal sgnals for new data. 7. KPCA s carred out on approxmaton and detal where M ( D ) = L ( D ) Λ ( D ), H H H MX( D) = LX( D) ΛX( D ), =,,... M ( A ) = L ( A ) Λ ( A ), H H H M ( A ) = L ( A ) Λ ( A ) X X X (3) (4) and Λ s a dagonal matrx of the egenvalues assocated wth the retaned PCs. A unfed weghted smlarty factor S MSPCA s used when consderng both approxmatons and detals whch s shown as S = η S ( A ) + η S ( D ) (5) S ( D ) sgnals to compute the nonlnear scores. MSPCA MSPCA MSPCA 8. Calculate the montorng statstcs and determne η + MSPCA whether or Q exceeds ts confdence lmt. 3. Fault pattern dentfcaton based a mult-scale PCA smlarty factor In ths secton, a fault pattern dentfcaton strategy s presented to dentfy faults and locate fault sources. he method s based on a pattern recognton method that searches for the smlar faulty stuaton n a hstorcal database. Fault patterns of the database are defned by the correspondng fault data sets that reflect the characterstcs of the process. Once a fault s detected, the fault type can be detected by matchng the patterns n the fault database va a pattern recognton method. Krzanows(976) developed an effectve pattern recognton method to measure the smlarty of two datasets usng a PCA smlarty factor, S PCA. Consderng the varance explaned by each PC drecton, Snghal and Seborg(00) provded a modfed PCA smlarty factor S λ PCA, that weghts each prncpal component by square root of ts correspondng egenvalue. he modfed method made sgnfcant mprovement for fault dentfcaton. Based on ths, a new mult-scale PCA smlarty factor s proposed n ths paper to dentfy fault patterns. Consder a hstorcal dataset H and a snapshot dataset X that contan the same number of varables but not necessarly the same number of samples. If multscale PCA s appled on H and X respectvely, prncpal component subspace of D, D,, D, and A can be constructed for them. he PC subspaces for H are noted as L H (D ), L H (D ),, L H (D ), L H (A ), and the PC subspaces for X are noted as L X (D ), L X (D ),, L X (D ), L X (A ). he mult-scale PCA smlarty factor s defned as S ( D ) = MSPCA ( H( ) X( ) X( ) H( )), =,,..., trace M D M D M D M D trace( Λ ( D ) Λ ( D )) H H X X H H X X () S MSPCA ( A ) = trace( M ( A ) M ( A ) M ( A ) M ( A )) () trace( Λ ( A ) Λ ( A )) where η s weghtng factors between 0 and and the sum of all factors equals one. When the fault s revelled n the approxmaton, the smlarty factor for A s good enough for fault pattern recognton and the smlarty factor for D does not contrbute a lot to fault dentfcaton as t reflects the nose. But the smlarty factor for D should not be omtted because t can provde some mportant nformaton at detal sgnal. In ths paper weght factor η s gven a larger value whch s chosen as 0.8 n ths paper and η,..., η + are gven a smaller value. 4. CASE SUDY he proposed process montorng and fault dagnoss strategy s tested wth a smulated process, the wellnown ennessee Eastman (E) process. he practcablty of MSKPCA for process montorng and fault dagnoss can be demonstrated as follows. E process descrpton he ennessee Eastman process was frst ntroduced by Downs and Vogel (993) whch was wdely used as a benchmar process for comparson of the varous process montorng strateges and control schemes. E process model s from a realstc, standard model of an ndustral plant-wde chemcal operaton. A dagram of process s llustrated n Fg.. he process has manpulated varables, contnuous process measurements, and 9 composton measurements. Much wor has been done on ths process. MacAvoy and Ye (994) desgned the decentralzed PID control system for t. Rach and Cnar (995) used ths process to test fault dagnoss technque. Chang, et al (00) compared the effectveness of dfferent process montorng methods. A E smulator coded by FORRAN s provded by he smulaton code allows pre-programmed maor process faults. he tranng and testng datasets ncludng 5 varables for each fault can also be obtaned from smulaton code. More detals can be found n Chang, et al (00). Results and dscusson PCA and MSKPCA models are developed to detect fault. he number of PCs retaned n the model s determned by capturng 95% of data varance.

5 Fg.. A dagram of the ennessee Eastman process wo dfferent nds of process upsets are used to compare the performance of PCA and MSKPCA. All the faults are ntroduced at the samplng nstant 60 and 960 samplng data n total are collected. he frst fault s the fault IDV (4) whch nvolves a step change n reactor coolng water nlet temperature. From Chang et al. (00), when ths fault occurs, t s compensated by control loops. hs maes the fault detecton and dagnoss tass dffcult. he results of PCA and MSKPCA montorng can be dentfed n Fgs. 3 and 4. he varaton of the resdual space can be captured by PCA-based Q statstc, but PCAbased statstc s not able to provde clear ndcatons about abnormal stuaton. From Fg. 3, s value s fluctuatng around the confdence lmt, whle n Fg. 4, both two statstcs exceed the thresholds clearly after fault occurs. he second fault case IDV() s assocated wth valve stcng. he valve for stream 4 was fxed at the steady state poston. For ths fault, PCA method performs poorly and ts Q statstc detects the fault untl the samplng nstant 640, statstc exceeds the lmt untl about the samplng nstant 655 (see Fg. 5). Although MSKPCA s also far away from good detecton, t does behave better than PCA. statstc gves an alarm at about the samplng nstant 400 and two statstcs show a clear alarm at about the samplng nstant 560 (see Fg. 6). hs example clearly demonstrates the advantages of MSKPCA.. Once the fault s detected, MSPCA smlarty factor method s appled to dagnoss fault pattern for E process. he smlarty factors of fault IDV(4) and IDV() wth fault patterns are plotted n Fgs. 7 and 8. In Fg.7, t s showed that the 4th smlarty factor s the largest. hs ndcates that the detected fault may be the same as the 4th fault pattern. Smlarly, the st fault pattern s also ndcated correctly n Fg.8. hese fault dentfcaton results prove the valdty of the proposed approach. But ths method should not be overestmated, because sometmes fault can not be recognzed clearly or even msclassfed. Further studes are stll necessary. Fg.3 Fault detecton results of PCA wth 99% confdence lmts for fault IDV (4) Fg.4 Fault detecton results of MSKPCA wth 99% confdence lmts for fault IDV (4) Fg.5 Fault detecton results of PCA wth 99% confdence lmts for fault IDV () Fg.6. Fault detecton results of MSKPCA wth 99% confdence lmts for fault IDV ()

6 Fault pattern Fg.7. he bar plot of fault IDV(4) s smlarty factors Fault pattern Fg.8. he bar plot of fault IDV() s smlarty factors 5. CONCLUSIONS In ths paper, a new fault detecton and dentfcaton strategy based on wavelet analyss and ernel PCA has been formulated for supervsng mult-scale and nonlnear process. he proposed strategy used MSKPCA to montor the process whch effectvely captures cross correlaton, auto correlaton, and nonlneartes among the ndustral datasets. Once an abnormalty s detected, a mult-scale fault dentfcaton approach based on the mult-scale PCA smlarty factor s appled to dentfy the fault pattern. he applcaton on E process shows that the proposed strategy wors well. ACKNOWLEDGEMEN hs research was supported by 863 programme of Chna under Grant No. 004AA4050 REFERENCES Bash, B.R. (998). Multscale PCA wth applcatons to multvarate statstcal process montorng. AICHE ournal. 44(7), Chang, L. H., E. L. Russell, and R. D. Braatz (00). Fault detecton and dagnoss n ndustral systems. Sprnger: London. Dong, D., and.. McAvoy(996). Nonlnear prncpal component Analyss based on prncpal component curves and neural networs. Computers and Chemcal Engneerng, 0(), Downs,..,and E. F. Vogel(993). A plant-wde ndustral process control problem. Computers and Chemcal Engneerng, 7(3), Foure, S.H., and P. Vaal(000). Advanced process montorng usng an nonlnear multscale prncpal component analyss methodology. Computers and Chemcal Engneerng, 4(-7), Hden, H.G., M.. Wlls, M.. ham, and G.A. Montague(999). Non-lnear prncpal components analyss usng genetc programmng. Computers and Chemcal Engneerng, 3(3), Krammer, M.A.(99). Nonlnear prncpal component analyss usng autoassocatve neural networs. AICHE ournal, 37(), Krzanows, W..(979). Between groups comparson of prncpal component analyss. ournal of. Amercan. Statstcal. Assocaton., 74, Ku, W.F., R.H. Storer, and C. Gerogas(995). Dsturbance detecton and solaton by dynamc prncpal component analyss. Chemometrcs and Intellgent Laboratory Systems, 30(), Lee,.-M., C.K. Yoo, S.W. Cho, P.A. Vanrolleghem, and I.-B. Lee(004). Nonlnear process montorng usng ernel prncpal component analyss. Chemcal Engneerng Scence., 59(), Ln, W.L., Y. Qan, and X.X. L(000). Nonlnear dynamc prncpal component analyss for onlne process montorng and dagnoss. Computers and Chemcal Engneerng, 4(-7), Lu, N.Y., F.L. Wang., and F.R. Gao(003). Combnaton method of prncpal component and wavelet analyss for multvarate process montorng and fault dagnoss. Ind. Eng. Chem. Res., 4, MacAvoy,.., and N. Ye.(994). Base control for ennessee Eastman problem. Computers and Chemcal Engneerng, 8, Mallat, S.(989). A theory of multresoluton sgnal decomposton : he wavelet representaton. IEEE rans. Pattern Anal. Machne Intell. (7), Msra, M., H. Yue, S.. Qn, and C. Lng(00). Multvarate process montorng and fault dagnoss by mult-scale PCA. Computers and Chemcal Engneerng. 6(9), Negz, A.,and A. Cnar.(997). Statstcal montorng of multvarate dynamc processes wth state-space models. AIChE ournal, 43(8), Rach, A., and A. Cnar(996). Statstcal process montorng and dsturbance dagnoss n multvarate contnuous processes. AICHE ournal, 4(4), Schölopf, B., A.. Smola, and K. Muller(998). Nonlnear component analyss as a ernel egenvalue problem. Neural computaton, 0(5), Shao, R., F. a, E.B. Martn, and A.. Morrs(999). Wavelets and non-lnear prncpal components analyss for process montorng, Control Engneerng Practce,7(7),