INVESTIGATION OF DYNAMIC MULTIVARIATE PROCESS MONITORING. Lei Xie, Shu-qing Wang, Jian-ming Zhang

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1 IVESIGAIO OF DYAMIC MULIVARIAE PROCESS MOIORIG Le Xe, Sh-qng Wang, Jan-mng Zhang atonal Key Laboratory of Indstral Control echnology, Zheang Unversty, Hangzho 37, People s Repblc of Chna Abstract: Chemcal process varables are always drven by random nose and dstrbances he closed-loop control yelds process measrements that are ato & cross correlated he nflence of ato & cross correlatons on statstcal process control (SPC) s nvestgated n detal It s revealed both ato and cross correlatons among the varables wll case nexpected false alarms Dynamc PCA and ARMA-PCA are demonstrated to be neffcent to remove the nflences of ato & cross correlatons Sbspace dentfcaton based PCA (SI-PCA) s proposed to mprove the montorng of dynamc processes hrogh state space modellng, SI-PCA can remove the ato & cross correlatons effcently and avod nexpected false alarms he applcaton n ennessee Eastman challenge process llstrates the advantages of the proposed approach Copyrght 5 IFAC Keyword: Mltvarate statstcal processes control (MSPC); Sbspace dentfcaton, False alarms rate (FAR), Dynamc processes IRODUCIO Wth the advent of mproved nstrmentaton and atomaton, chemcal processes now prodce large volmes of nformaton whch are hghly correlated Several mltvarate statstcal methods have been developed to dentfy the correlatons between varables and create a redced set of varables n the orthogonal axes that captre most of the varablty n the collected nformaton One of most poplar MSPC methods s prncpal component analyss (PCA), whch also has been appled to chemcal processes (omos 995, Wse 996, Cnar 999, Venatasbramanan 3) However, PCA s based on the assmpton that the process varables are ndependent and dentcally normally dstrbted (d), e, statonary or ncorrelated n tme (Jacson 99, K 995) In practce, ths assmpton s always volated, as chemcal process varables are drven by random nose and dstrbances De to the feedbac control, the mpact of dstrbances propagates throgh to both the npt and otpt varables hs the varables wll move arond the steady state and exhbt some degrees of ato-correlaton and cross-correlaton In order to montor the process dynamcs, PCA has been extended to nclde the tme-seres strctres of varables (K 995, egz 997, Callao 3, Smoglo ) Among these extensons, dynamc PCA (DPCA) by K et al (995) s wdely adopted and can be treated as a mltvarate AR-le tme seres modellng approach Althogh applcatons n ennessee Eastman (K 995) and some batch processes montorng (Chen ) have demonstrated the effcences of dynamc PCA, t s proved recently that dynamc PCA can not elmnate the ato & cross correlatons of varables (Krger 4) If a dynamc PCA s sed, the score varables wll be ato and cross-correlated even when the process varables are nether ato nor cross correlated In other words, dynamc PCA wll always ndce the dynamcs nto the score varables In order to overcome ths qeston, Krger et al nvolved ARMA flters to remove the ato correlatons from the PCA scores Bt nfortnately, the cross correlatons stll reman n the fltered score varables he contrbtons of ths paper are as follows Frst, the nflences of ato & cross correlatons of process varables are nvestgated throgh a nmercal experment It s revealed that the presence of ato

2 and cross correlatons wll both case the false alarms Secondly, crterons to determne whether varables are ato or cross correlated are ntrodced he ARMA flterng approach sggested by Krger s neffcent to redce the cross correlatons of PCA scores hrdly, a sbspace dentfcaton modellng approach combned wth PCA s proposed to remove the entre dynamcs from the score varables A novel nformaton based crtera s presented to determne the order of relatve state-space model Forthly, the effectveness of proposed approach s demonstrated sng the ennessee Eastman process Fnally, some remars and conclsons are presented MOIORIG CROSS CORRELAED PROCESS VARIABLES PCA, Dynamc PCA and ARMA-PCA Gven the measrements, normal operatng process data are collected and pt n a two-dmensonal data n matrx X R * wth samples and n varables PCA decomposes the data matrx X n terms of r lnear prncpal components wth r n : ~ ~ X = XPP + XPP, () = P + E n r where P R * r and = X * P R * are defned as the prncpal component loadngs and scores, respectvely P ~ contans the retaned prncpal component drectons E s the resdal matrx When PCA s appled to montor a process, and SPE statstcs are commonly sed A more detaled analyss of PCA refers to Jacson (99) Dynamc PCA arranges the process varables to form an atoregressve (AR) strctre: ( d )*( d + ) n X [ X X X R, () where = d ] X s an agmented set of varables, representng an AR model strctre of order d and the sbscrpts,, d refer to the bacshfts appled PCA s appled to X and the correspondng and SPE statstcs can be obtaned It s demonstrated by Krger (4) that the retaned scores varables are ato-correlated rrespectve of whether the process varables are ato-correlated or not In order to overcome the defcences of dynamc PCA, Krger et al (4) ncorporate ARMA flters n the PCA analyss (ARMA-PCA) he ARMA-PCA method apples tradtonal PCA method to orgnal data matrx X frst and r ARMA flters are dentfed to remove the ato correlatons of each score varable Althogh the ato correlatons are effcently elmnated, the cross correlatons stll exst and the fltered scores are not ndependent yet Performance nvestgaton of ARMA-PCA ype I error rate or false alarms rate refers to the percentage of statstcs volatng ts confdence bond when montorng normal operatng process For PCA, the deal type I error rate for statstc s eqal to the sgnfcant levelα In practce, false alarms rate s one of the most mportant parameters to determne oo hgh false alarms rate wll reslt n poor acceptance among shop-floor personnel whle too low rate wll mae the montorng system nsenstve to potental process or sensor falts he nflence of ato-correlaton on process montorng reslts has been well stded n Krger (4) In ths secton, t wll be llstrated that not only ato- bt also cross- correlaton wll reslt n hgher false alarms rate n montorng chart In order to stdy the cross correlaton effect solely, t s assmed that PCA score varables has been fltered wth the PCA-ARMA approach presented n Krger (4) And there are 3 fltered scores have the followng descrpton: t = 3w t = w (3) 3 t = w, where w (,) s a seqence of normally dstrbted vales wth zero mean and nt varance he fltered score varables are all shfted seqences of w It s clear that the fltered scores are only cross correlated In order to stdy the nflence of cross correlaton on the statstc, samples are generated wth (3) he frst samples are selected as reference data and the remanng samples served as testng data he montorng chart s gven n Fg (sold lne represents the 99% control lmt) and shows that the nmber of volatons exceeds % sgnfcantly arond the th sample Consectve false alarms also appear arond the 5th samples whch wll mslead the process engneers Samples Fg False alarms cased by cross correlatons among PCA score varables Frther more, becase ARMA-PCA approach gnores the cross correlatons of score varables, ths wll lead to ARMA models wth hgher orders than necessary For example, n the example stded by Krger (4), the nderlyng model order s 4 whle the AR orders selected are all larger than 4 except for the last score varable

3 3 SUBSPACE IDEIFICAIO BASED PCA 3 Ato & cross correlaton coeffcents Consder a set of random tme seres = [ r ], the ato & cross covarance coeffcent s defned as: ρ ( τ ) =, ( t), )), ( t), ( t)) ( t), ( t)), r, (4) ( t), )) = ( ( t) )( ) ), τ > τ t= + ( t), ( t ( τ ))), τ < = ( t), t= (5) where ( t), )) s the cross covarance between and, s the nmber of τ, samples For mlt-normally dstrbted random varables (ndependent), ρ, ( τ ) = whenτ he ato & cross correlaton coeffcents can be defned n a smlar manner In the followng sectons, t s assmed process s statonary and varables are zero-centred, so the correlaton coeffcent eqals the covarance coeffcent and both of them are denoted as ACFs he followng two theorems are ntrodced to determne whether varables are ato or cross-correlated: heorem : Sppose s a Gass whte nose seqence (so ndependent) and >> M hen the ato correlaton coeffcent ρ, ( τ ) s approxmately normally dstrbted, e ρ ( τ) ~ (,), τ M, heorem : Sppose s a Gass whte nose seqence and ndependent of and >> M hen the cross correlaton coeffcent ρ, ( τ ) s approxmately normally dstrbted, e ρ, τ M M ( τ) ~ ρ, ( m) ρ, m= M ( m) (,), he detaled proof of theorem and refers to Lng (999, Secton 66), the condton of theorem can be relaxed as beng the lnear combnaton of tme shfted Gass whte noses 3 Sbspace dentfcaton based PCA (SI-PCA) As mentoned above, Krger s ARMA-PCA approach gnores the cross correlatons of score varables In order to descrbe the ato & cross correlatons smltaneosly, one shold consder that the ARMA approach can be extended to mltvarate case and a mltvarate tme seres model shold be establshed However, the mltvarate ARMA model s mch more dffclt to analyze becase all the coeffcents n nvarate model wll become matrces and the model complexty wll grow rapdly as the model order ncreases (George 994) An alternatve approach to analyze mltvarate tme seres s state-space modellng As arged by Lng () that Generally speang, t s preferable to wor wth state-space models n the mltvarate case, snce the model strctre complexty s easer to deal wth and any lnear model strctre (ARX, ARMA etc) can be represented by state-space model (Lng 999) A state-space model for tme seres s gven by: z = + Az + Ke, (6) t = Cz + e where z R t nz* R r* s the correspondng tme seres, s the vector of state varable, A, C are the system matrces, K s the Kalman gan ote that there are no npts n ths model he resdals r* of state-space model e R, whch are assmed d, are employed to process montorng nstead of correlated scores Sbspace dentfcaton (SI) algorthms have been wdely adopted to dentfy the state-space model from npt-otpt data becase t does not need teratve, nonlnear optmzaton as maxmm lelhood method and SI s very easy to mplement (Overschee 996) For sbspace algorthms, t s crcal to estmate the state z whch s defned as a lnear combnaton of past otpts, p = [ t t t d] (7), z = Jp where d s the nmber of lags as mentoned n dynamc PCA Once J s determned, the z can be calclated by Eq (6) and the state-space matrces can be estmated va lnear sqares regresson Dfferent approach to calclate J dstngshes varos dervatons of sbspace algorthms ncldng CVA, 4SID and PLS etc More detals abot sbspace dentfcaton algorthms refer to Lng (999) and Overschee(996)

4 o determne the order of system n Eq (6), a nmber of approaches have been proposed For nstance, 4SID determnes the system order by checng the snglar vales Aae nformaton crteron (AIC) s also employed to determne the model order atomatcally In ths context, however, the prpose of modellng s to remove the ato & cross correlatons from the score varables as mch as possble, e, redce the ACFs ( ) e ρ, τ of as close to zero as possble whenτ o the end, the followng Aae le nformaton crteron s sggested: q AICx = log( V ( nz)) + (8) r r M V ( nz) = ρ, ( τ), Mr (9) = = τ= M τ where q s the nmber of estmated parameters n Eq (6), s the length of modellng data and q M = 3*nz s the maxmm tme lag s nclded to avod over-fttng he system order nz s then selected to mnmze the AICx obectve fncton Once the state-space model s dentfed and the resdal e s generated, we have the followng statstc: r( ) = e Se e ~ Fα, r, r, () ( r) where Se = ee s the correlaton matrx = of e ote that not all scores are necessarly nclded n the state-model If a score s ndependent on tself and other scores, t shold be exclded to redce the complexty of state-space model In ths staton, the statstc wll become: eˆ ˆ e r ( ) = Set ~ Fα, r, r t t ( r), () ˆ ˆ where Set = e e s the correlaton = t t ~ matrx of ê and t ê s the resdal of ato & ~ cross correlated scores tˆ and t s the ndependent part Remar: One cold also apply the sggested state-space modellng approach to the orgnal process varables frst and then establsh a PCA model However, ths mght change the lnear relatonshp among the varables and case the varable redcton by PCA neffcent Frthermore, f a set of process varables are drven by same dynamc processes, eg, the reactor may have several temperatre sensors on dfferent locatons, applyng PCA frst wll reslt fewer dynamc scores In secton 4, by the applcaton n E process, t s revealed that PCA does not only concentrate the varaton nformaton bt also the dynamc nformaton n the scores and the comptaton cost to establsh the state-space model s therefore less demandng 33Dynamc process montorng framewor based on SI-PCA he procedres of off-lne and on-lne montorng sng SI-PCA are as follows: Off-lne: develop the normal operatng condton model (OC) Collect an operatng data set drng normal operaton X Apply PCA to X and obtan the score tˆ varables t = ~ he nmber of components can t be determned by cross valdaton or other crtera he ndependence of exclded prncpal components relatve to SPE statstc shold also be checed If dynamcs exst, state-space model can also be employed 3 Sbspace dentfcaton method n secton 3 s employed to remove the dynamcs of tˆ he confdence nterval of statstc s determned based ~ on the resdal ê, t On-lne montorng: For new observaton, obtan the score vales va t = P x Apply the dentfed sbspace model to calclate the resdal ê 3 Determne the and SPE statstcs and compare wth the confdence ntervals 4 APPLICAIO SUDIES Snce the ntrodcton n 993 by Downs and Vogel, ennessee Eastman (E) process has been wdely stded n the lteratres (K 995, Rcer 996, Chang, Kano, Rssel ) he E model ncldes 5 process nts: a reactor, a condenser, a vapor-lqd separator, a recycle compressor and a prodct strpper here are 4 measrements and manplated varables he process s open-loop nstable and reqres reglatory controllers E process ncldes programmed dstrbances ncldng composton step change n reactants, random react coolng water nlet temperatre random dstrbance and valve stcng etc Detaled descrpton abot the operaton of the E process can be fond n Downs (993)

5 In ths paper, Rcer s E smlator based on Matlab 65 was sed to generate the data set he closed-loop control strategy of Rcer (996) was also adopted he reference dataset to constrct OC model ncldes samples of contnosly measred varables whch were recorded at h nterval It s somewhat cmbersome to plot the ACFs of all the varables In Fg, only the ato correlaton coeffcents of varables are llstrated It can be seen that ten varables are strongly ato correlated ncldng feed A (), reactor pressre (7), prge rate (), separator temperatre (), separator pressre (3), strpper pressre (6), strpper temperatre (8), compressor power (), reactor otlet coolant temperatre (), separator otlet coolant temperatre () Var Var Var 3 Var Var Var Var 4 Var Var Var Var 5 Var PCA was frst appled to the OC data and 5 scores whch explan 876% of total varaton were retaned he ato correlaton coeffcents of scores are llstrated n Fg3 As descrbed n secton 3, not only the varaton nformaton bt also the dynamcs nformaton are concentrated n the frst 5 scores Based on the crteron presented n secton 3, a 5-order state-space model was dentfed to remove the dynamcs effcently Fg4 confrms that resdals are no longer ato or cross correlated In Var 4 Var Var Var Var Var Var Var Var Var Fg Ato correlaton plots of E process varables 5-5 PC PC 6 PC PC PC 7 PC PC PC 8 PC PC PC PC PC PC Fg3 Ato correlaton plots of PCA scores for E process 5-5 PC 5 e e e 3 e 4 e e -5 5 e -5 5 contrast, the ARMA-PCA had to se as hgh as order flters to remove the ato correlatons of the frst two scores o demonstrate the capablty of SI-PCA to detect abnormal behavor, excessve varaton of the reactor coolng water temperatre was smlated (dstrbance type ) to generate a falt dataset he falt dataset also contans samples and the falt was nected after the sample he control chart s presented n Fg5 and the data arond the control lmt are zoomed n Fg6 for clear llstraton he last vales of statstc prodce an excessve nmber of volatons So SI-PCA detects the ot-of-control staton correctly 5 COCLUSIOS hs paper stdes the nflences of process dynamcs on false alarms rates of statstcal process control It s shown that the presence of cross correlaton wll also case false alarms Applcatons llstrate that two mproved PCA methods, dynamc PCA and ARMA-PCA, do not remove the effects of tme-seres strctres of varables Motvated by above facts, a sbspace dentfcaton approach based PCA (SI-PCA) s proposed to remove the ato & cross correlatons of score varables smltaneosly Aae nformaton le crteron s ntrodced to determne the state-space model Applcaton n E process falt detecton reveals that the SI-PCA s effcent n removng the ato & cross correlaton among varables and detectng the process abnormal behavor hs paper focs on the behavor of statstc and the nflence of ato & cross correlatons on SPE statstc s also worth frther stdy ACKOWLEDGEMES hs proect s spported by atonal Hgh-ech Program of Chna nder grant o AA43 e e Fg4 ACFs of resdals by SI-PCA (E process) e 5

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