Adaptive Multivariate Statistical Process Control for Monitoring Time-varying Processes 1

Transcription

1 Adapve Mulvarae Sascal Process Conrol for Monorng me-varyng Processes Sang Wook Cho *, Elane. Marn *, A. Julan Morrs *, and In-eum Lee + * Cenre of Process Analycs and Conrol echnology, School of Chemcal Engneerng and Advanced Maerals, Unversy of Newcasle upon yne, NE 7RU, UK + Deparmen of Chemcal Engneerng, Pohang Unversy of Scence and echnology, San 3 Hyoja Dong, Pohang, , KOREA Absrac: An adapve mulvarae sascal process monorng MSPC approach s descrbed for he monorng of a process whch ncurs changes n he operang condons. Sample-wse and block-wse recursve formulae for updang he mean and covarance marx are derved. y ulsng, he curren model and he updaed mean and covarance srucures, a new model s derved recursvely. ased on he updaed PCA represenaon, wo monorng mercs, Hoellng s and he Q-sasc are calculaed and her conrol lms are updaed. For more effcen model updang, forgeng facors, whch can change wh me, for he updang of he mean and covarance are consdered. Furhermore, he updang scheme proposed s robus n ha no only reduces he false alarm rae n he monorng chars bu n ha he model s nsensve o oulers. he adapve MSPC approach developed s appled o a mulvarae sac sysem and a connuous srred ank reacor process wh he resuls beng compared o hose from he applcaon of sac MSPC. he revsed approach s shown o be effecve for he monorng of processes where fas or slow changes are ncurred. Keywords: me-varyng process, Adapve PCA, Varable forgeng facor, Robus parameer esmaon Submed o Indusral & Engneerng Chemsry Research o whom all correspondence should be addressed. el: Fax: E-mal: e.b.marn@newcasle.ac.uk

2 . Inroducon Mulvarae sascal projecon mehods such as prncpal componen analyss PCA and paral leas squares PLS combned wh sascal monorng chars have been wdely appled for on-lne connuous and bach process monorng. -3 ypcally monorng chars based on Hoellng and he Q-sasc 4,5 are used o deec fauls wh he conrbuon plo 6, whch shows he conrbuon of each measured varable o he monorng sasc, beng adoped o asss n he denfcaon of a possble assgnable cause of he faul. Mulvarae sascal process conrol MSPC schemes based on PCA or PLS are formulaed on he assumpon ha he process varables are ndependen, dencally dsrbued and lnearly correlaed. Furhermore, f he underlyng dsrbuon of he process varables s mulvarae normal, hen he process can be descrbed oally by he mean and covarance of he varables. hus, due o he lmaons mposed by he underlyng assumpons, MSPC s usually performed under seady sae condons. Occasonally, problems assocaed wh he effcency of he process monorng scheme can arse because he underlyng assumpons of convenonal MSPC are volaed. For example due o non-lnear process behavour, he presence of me-dependency auocorrelaon, process dynamcs or changes n operang condons. o address hese lmaons, a number of alernave monorng schemes have been proposed such as dynamc, non-lnear or adapve MSPC. In hs paper, an adapve PCA based MSPC scheme s proposed o monor processes where operang condon changes are common. Indusral processes usually exhb me-varyng behavor due o nenonal dsurbances such as se-pon changes or unwaned dsurbances such as he ageng of equpmen, foulng and caalys degradaon. here have been several papers presened on adapve PCA and PLS. For example Wold 7 proposed a scheme whch ulzed he exponenally weghed movng concep for he updang of boh PCA and PLS models. However, one lmaon wh hs algorhm s ha requres all he hsorcal daa o be used n he updang of he model every me a new sample becomes avalable. Dayal and MacGregor 8 descrbed a recursve exponenally weghed PLS algorhm, usng he PLS kernel algorhm, based on recursvely updang he covarance marces as opposed o usng he oal hsorcal daa se. Qn 9 developed an alernave sample-wse and block-wse recursve PLS algorhm ha only ulzes he loadng marx and he regresson coeffcens beween he npu and oupu score vecor pars n he PLS model, nsead of he covarance marces, maeralzng n more rapd updang of he model. More recenly, L e al. proposed wo effcen learnng algorhms for recursve PCA for he monorng of me-varyng processes. he sudy repored n hs paper focuses on developng a new adapve monorng scheme for me-varyng processes wh a number of conrbuons beng presened. Frs sample-wse and bach-wse recursve formulae are derved for updang he sample mean, varance, and covarance pror

3 o an effcen and fas learnng algorhm beng proposed for calculang a new PCA represenaon hrough eher he sample or block-wse approach. Secondly, an overall model updang and process monorng mehod s presened. More specfcally a new calculaon procedure for he varable forgeng facors s developed for updang he mean and covarance. Fnally, a robus model updang scheme s descrbed o reduce he effec of oulers on he model updang procedure. he remander of hs paper s srucured as follows. Secon presens he recursve updang forms of he sample mean and covarance wh he focus n Secon 3 beng on an alernave and effcen way by whch o fnd he egenvalue egenvecor pars n he updaed covarance or correlaon marx. In Secon 4, an overall adapve sascal process monorng sraegy s proposed. wo specfc aspecs are consdered whch mpac on he monorng sascs, oulers, and he selecon of he forgeng facors one for he mean and he oher for he covarance. Fnally, n Secon 5, he resuls of he proposed mehodology are assessed hrough smulaed examples where a number of dfferen ypes of changes are consdered ncludng drf, se-pon changes, a ramp and a change n he underlyng dmensonaly of he process. Fnally a case sudy on a connuous srred ank reacor process s consdered.. Recursve Form of he Gaussan Densy Funcon of a Random Vecor In MSPC, measuremens obaned durng normal operang condons are regarded as beng colleced from a saonary Gaussan process,.e. x ~ N µ, Σ. he maxmum lkelhood esmae of µ and Σ s gven by samples. ˆ m n x µ and Σ ˆ S x m x m n, respecvely, based on n. Sample wse Updang When he process operang condons change eher gradually or abruply, he mean vecor and covarance marx wll no be consan and wll need o be updaed. Each me a sample or a sample block become avalable, boh he mean and covarance are updaed wh he degree of change n he model srucure beng dependen on he magnude of he forgeng facors. Consequenly he PCA represenaon should accordngly be updaed o allow he monorng of such a me-varyng process, x N µ, Σ. he esmaed mean vecor and covarance marx a me pon are gven n equaons ~ and, respecvely, : m α x α x + αx + L+ α + α + L+ α x

4 and β x m x m β S, he sample mean a me pon can be denoed more smply as a weghed sum of he sample mean a me pon, m -, and he sample a me pon, x : α α x + α m m 3 α α Lkewse an alernave form of he sample covarance s gven by: β ~ ~ β xx + β S S 4 β β where, ~ x x m s he mean-cenered sample vecor a me pon. As becomes large, Eqs. 3 and 4 can be furher smplfed o he followng forms: m 5 α x + αm and ~ ~ β xx + βs S 6 β dag ~ x ~ x + βd D, From Eq. 6, f only he sample varance s beng consdered, hen where D s a dagonal marx whose dagonal elemens are dencal o hose of S. Also, he correlaon marx s esmaed as: ~ D / R 7 / β D x ~ x β As becomes large, Eq. 7 can be smplfed: 3

5 4 / / ~ ~ + β β R D x x D R 8. lock wse Updang When a process changes slowly and he samplng me s small compared wh he process me consan, s neffcen o updae he model each me a new sample becomes avalable, as descrbed n he prevous secon, snce s compuaonally nensve. Insead he PCA represenaon can be updaed afer a block of samples have been colleced. he block-wse updang approach has wo aracve feaures, frs has a low compuaonal cos and secondly reduces he rsk of updang he model based on a false alarm, snce s possble o decde wheher a deeced alarm s false pror o updang he model. he block-wse approach s dencal o he sample-wse updang procedure excep ha a sample block s used o adap he model nsead of a sngle sample. he recursve equaons for he esmaed mean and covarance are gven by: + α α α α m X X m 9 and ~ ~ ~ ~ + β β β β n S X X X X S where m N R X s he h sample block m s he number of measuremens, [ ] N R L, n s he sample number n he h block, and ~ m X X. he recursve calculaon for he sample correlaon marx s gven by: / / ~ ~ + β β β β n R R D X X D R

6 where / ~ ~ / D X X D R s he correlaon marx of he curren sample block and ~ ~ D. D s a dagonal marx whose dagonal elemens are β dag X X + βd dencal o hose of S. 3. Prncpal Componen Analyss PCA Model Updang When a new sample or block becomes avalable, he egenvalues and egenvecors loadng vecors of he newly updaed correlaon marx are calculaed o oban a new PCA represenaon. A number of approaches have been proposed o calculae he egenvalues and egenvecors. One of he smples approaches s o perform a sngular value decomposon SVD on he curren correlaon marx. More recenly, L e al. proposed wo recursve PCA algorhms based on rank-one modfcaon and Lanczos rdagonalzaon, whch are compuaonally more effcen han SVD. he precedng mehods requre he sorage and updang of he correlaon marx o consruc an updaed PCA represenaon. When he number of measured varables s farly large, e.g. over a hundred, whch s common n many ndusral processes, he procedure s me consumng. Consequenly, a revsed and more effcen algorhm was proposed o updae he egenvalues and egenvecors and hs dea s exended o he bach-wse case. he block-wse updang procedure s frs consdered snce sample-wse updang s a specal case of he block-wse case. ased on he mean vecor a me pon -, m, he prncpal loadng marx m a P R and he dagonal marx of egenvalues a dag λ, λ, L, λ a a Λ R a me pon, he newly updaed values are obaned by usng hese values and he new sample block X a me pon. a s he approxmaed rank of R. Frs, he correlaon marx of X s calculaed, / ~ ~ / D X X R n D and SVD s appled gvng an approxmaon of R : P Λ P R he columns of m b P R are he prncpal egenvecors and b dag λ, λ, L, λ b b Λ R s he dagonal marx whose elemens are he sgnfcan egenvalues of R, where b s he approxmaed rank of R marx R can be approxmaed as:. Lkewse, he prevous correlaon 5

7 6 P Λ P R 3 Subsung Eqs. and 3 no Eq., R may be approxmaed as: ˆ + β β P Λ P P Λ P R 4 Combnng he wo erms on he rgh sde of Eq. 4, he equaon can be rewren as: [ ] ˆ K K P P Λ Λ P P R β β 5 where b a m b b a a + R βλ βλ λ β λ β p p p p K L L. he updaed loadng vecors and egenvalues can be obaned by performng SVD on K K : p p K K λ 6 Alernavely, he loadng vecors and egenvalues can be calculaed usng b a b a + + R K K nsead of m m R K K,.e. q q K K γ. Mulplyng boh sdes of hs relaonshp by K δ gves: q K K q K K δ γ δ 7 Comparng Eq. 7 wh Eq. 6, can be deduced ha λ γ and q p δk, where / λ δ o ensure δ q K p. Sample-wse updang s a specal case and hence K becomes:.5 ~ + R β λ β λ β a m a p x D p p K L 8 If > a + m n he sample-wse updang procedure or f b a m + > n he block-wse updang procedure, hen s compuaonally effcen o use K K, nsead of K K o calculae P and Λ be-

8 cause K K makes he egenvalue problem more smple and rapd o solve. I should, however, be noed ha here are wo sources of runcaon error n calculang P and Λ n he block-wse updang procedure. hese maeralse from dscardng he nsgnfcan egenvecors and egenvalues pars n Eqs and 3. In he case of he sample-wse updang procedure, runcaon error may occur when approxmang he correlaon marx as defned n Eq. 3. In he above updang algorhm, he egenvecors and egenvalues, whch are consdered o be nsgnfcan, are dscarded o approxmae and R. R herefore he dmensonaly of he wo marces R and R, a and b, requre o be deermned. In each recursve model updang sep, here are several ways o selec a and b 3 bu he approaches are dependen on he praccal applcaon. hree mehods can be consdered fxng he parameer a or b as consan values; reanng hose egenpars whose egenvalues are larger han a predefned hreshold, or 3 reanng hose egenpars such ha a specfed fracon of he oal sum of egenvalues s ncluded, e.g Adapve Process Performance Monorng 4.. Adapve Monorng Sascs In general, a PCA-based MSPC scheme ulses wo monorng sascs, Hoellng s : p Λ p x P Λ P x 9 where p p p Λ R s he leadng prncpal marx of Λ, and he Q-sasc: Q x I PP x whch are expressed n erms of he Mahalanobs and Eucldean dsances, respecvely. Hoellng s represens he dsance n he PCA model space, whereas he Q-sasc ndcaes a dsance from he model space. Under he assumpon of mulvarae normaly of he observaons and emporal ndependence, he δ % conrol lm for Hoellng s s calculaed by means of a lm α χ approxmaon,.e. χ p, whch s generally used n qualy conrol because of s smplcy. In hs case δ denoes he sgnfcance level. hs mehod s favourable snce requres only small compuaonal me n each model updang sep. Also he δ % conrol lm for he Q-sasc s gven by: 7

9 Q lm z θ h θ + + θh h / h θ δ θ a j j Λ k p kk where θ + for j,,3, h θθ3 3 θ, and z δ s he normal devae. 4 In adapve PCA monorng, he wo conrol lms requre o be calculaed every me he PCA model s updaed snce he number of prncpal componens, p and he dagonal marx whose elemens are he egenvalues of he covarance marx, Λ are me-varyng. 4.. Varable Forgeng Facor As presened n Eqs. and, calculaon of he weghed mean and covarance requres he weghng parameers ermed forgeng facors α and β, o be deermned. If boh forgeng facors are uny, hey are he maxmum lkelhood esmaes calculaed from all he daa. y usng a forgeng facor ha s less han one, prevous samples are auomacally weghed ou whou deleng daa from he process model. As he value ges close o uny, he process has a long-erm memory, ha s, he number of prevous samples ha have an effec on he curren model ncreases. o dae mos model updang approaches have used an emprcal consan forgeng facor. However he opmal value of he forgeng facor vares sgnfcanly dependng on he rae of process change. When he process changes rapdly, he updang rae should be hgh, whereas when he change s slow and hus he essenal process nformaon s vald for a long perod, should be low. However, s lkely ha he rae of process change or varaon n real processes vary wh me. If hs s he case, he forgeng facor should be deermned accordng o he underlyng objecve of he process monorng scheme. o cope wh hs suaon, nsead of usng a fxed forgeng facor, can be adjused accordng o he curren process condons. Dayal and MacGregor 8 used he algorhm developed by Forescue a al. 5 o adjus he forgeng facor n her recursve PLS algorhm. he same concep of he varable forgeng facor has been appled n recursve PCA. 6 In hs sudy, a new algorhm for adapng he forgeng facor s proposed. Fundamenally dffers from he prevous algorhm proposed by Forescue a al. 5 n wo aspecs. Frs, he varable forgeng facor used for updang he sample mean and he covarance marx can ake dfferen values, enablng greaer flexbly and generaly. Secondly, he wo forgeng facors drecly depend on he change n he mean and covarance srucures, unlke he prevous approaches n whch boh depended on Hoellng s and he Q-sasc. In he proposed updang algorhm, wo forgeng parameers α and β are used o updae he sample mean vecor and covarance or correlaon marx, respecvely. he forgeng facor α for updang he mean vecor s calculaed as: 8

10 [ exp{ m }] n nor α αmax αmax αmn k m where α max and α mn are he maxmum and mnmum forgeng values, respecvely, k and n are funcon parameers defned below, and m s he Eucldean vecor norm of he dfference beween wo consecuve mean vecors. Here, mnor s he averaged m obaned usng hsorcal daa as wll be explaned n Secon 4.4. Smlarly, he forgeng facor β for updang he covarance or correlaon marx s gven by: [ exp{ R }] n nor β βmax βmax βmn k R 3 where β max and β mn are he maxmum and mnmum forgeng values, respecvely, and R s he Eucldean marx norm of he dfference beween wo consecuve correlaon marces. here are four funcon parameers ha requre o be deermned, α max or β max, α mn or β mn, k, and n. he nfluence of hese four parameers on he forgeng facor value s shown n Fg.. he change n he paerns of he forgeng facor were nvesgaed accordng o he magnude of x x, nor where x m or R, by varyng one of he four parameers and fxng he oher hree parameers. hs procedure was repeaed eravely. he defaul values of he four funcon parameers were aken as α max.99, α mn. 9, k.693 and n. Here, he defaul k value,.693, corresponds o he case where α αmax + αmn when n and x x nor. Fgures a and b show he effec of he maxmum and mnmum values, respecvely, of α on he forgeng facor. oh α mn and α max affec he range of he change,.e. he forgeng facor vares whn he lms of α mn and α max. Also, α mn and α max regulae he maxmum and mnmum rae of model adapaon, respecvely. he parameers k and n conrol he sensvy of he change n α. he larger he value of k, he faser α decreases as x x ncreases Fg. c, ndcang ha he model can be updaed more rapdly by a change n he mean or covarance f k s large. In Fg. d, he races converge a,.945 snce k s se such ha α when x xnor. As n ncreases, α changes rapdly abou he pon,.945. In oher words, he model adapaon shows nor more rapd change resulng n greaer sensvy o x x abou uny. As a lmng case, α nor 9

11 becomes a bnary varable, α s equal o α max and α mn, as n becomes suffcenly large. hen, f x xnor >, α max, else, α s equal o α mn. Even hough hese parameers vary dependng on he process characerscs, here needs o be praccal gudance o selec hem, hus s proposed ha an emprcal parameer selecon procedure s adoped: Selec α max and α mn ypcally, α max.999 ~. 99 and α mn.95 ~. 9, Deermne k such ha α µ αmax αmn + αmn when x xnor, ndcang ha when he curren mean change s he same as he normal mean change.e. as per nomnal daa, α s se o a value beween α mn and α max. For example, α.5 αmax + αmn when µ.5 and α.67α α when µ.67. max mn 3 Selec n from beween and 3 by consderng he sensvy of α o x xnor. hs gudance s equally applcable n he deermnaon of β Robus Adapaon usng Oulyng Samples Oulyng observaons can be recorded durng he normal operaon of a process. here are wo sraeges when dealng wh oulers n adapve modellng, frs, oulers can be gnored and a model s held whou updang unl he nex sample or block s consdered, alernavely, a robus parameer esmaor can be used o updae he model usng he oulyng samples so ha he effec of oulyng samples on model updang can be suppressed. One common robus esmaor s he M-esmaor proposed by Huber. 7 hs approach has been used for regresson and dmensonaly reducon problems. 8,9 Mos robus mehods are used o denfy a relable model where a daa se conans oulers. herefore, modellng and ouler denfcaon are performed eravely unl a robus model s denfed snce he presence of oulers s unknown. However, n he adapve modellng approach, a relable PCA model wll have been developed n he prevous sep and hus may be possble o decde f a new sample s an ouler by examnng he monorng chars. herefore, he am s o denfy unusual measured varables n he new sample and correc for hem. Once an oulyng sample, x, s deeced by usng he monorng chars, s value s replaced wh a robus esmae, z, whch s hen used n he updang algorhm nsead of x o updae he curren PCA represenaon. Here, z s represened as a weghed x: z Wx 4

12 where W dag w, w, L, w s a wegh marx. he wegh of each varable s calculaed accordng m o s relably whch s a funcon of he varable resdual obaned usng he curren model: x ˆ k xk w + k 5 ck where c k s a unng parameer, whch conrols he sharpness of he wegh funcon he covarance of he resdual, and assumng Σ x I, k e a me pon,.e. Σ I P P Σ I P P I P P p c can be calculaed as k z pk e c α x w k. Consderng, where z α s a normal devae. As he varable resduals e k ncrease, he wegh w k decreases and hus he dfference beween he orgnal and correced values ncreases. Oulers may rgger a warnng sgnal n he monorng char n a smlar manner o ha of a faul. herefore here s a rsk ha a curren model adaps o a faul or a dsurbance as well as an ouler when a robus updang mehod s ulsed. Hence, s mporan o dscrmnae beween oulers and process fauls. Oulers are unlkely o be generaed consecuvely, whereas fauls and dsurbances las for a leas a ceran perod. In hs respec, f monorng chas show an ou-of-conrol sgnal for several consecuve samples, e.g., hree samples, a process s consdered o be abnormal. Oherwse, he curren model s updaed usng he robus updang mehod Overvew of he Monorng Procedure he overall sraegy for me-varyng monorng usng he proposed mehod s as follows: Off lne learnng: Oban he nal values of he sample mean m, he varance D, he loadng marx P and he egenvalue marx Λ based on he ranng daa block. Oban he wo conrol lms lm, and Q lm,, and he number of prncpal componens, p. 3 Selec α max, α mn, β and max β. max 4 Calculae mnor and Rnor. Dvde he ranng daa no wo secons and calculae he nal m, D, and R usng he frs ranng daa se. hen, wh a fxed α and β e.g. se boh o.96, whch corresponds o a value of α.67αmax +. 33α mn, when α max. 99 and α mn. 9, updae m, D, and R and sore m and R usng he second ranng daa se. Calculae he average values mnor and Rnor.

13 5 Deermne he model parameers: n and k. 6 Deermne he nal values of he wo forgeng facors α and β Here, boh are se o.99. In effec, hey are no very crcal n on-lne model adapaon. On lne learnng and monorng: A me pon, usng he prevous values of m, D, P, Λ, lm,, Qlm,, p Calculae m and and Q for a new sample x or block afer mean cenerng and auoscalng usng D. lm, If > or Q > Q lm, go o sep 3 else go o sep 6. 3 Check f he new sample s an ouler eher auomacally or manually. For nsance, f s > lm, s or Q s > Q lm, s s,.e. f hree consecuve ou-of conrol sgnals have been generaed, he new sample s no an ouler. Oherwse, s an oulyng sample. 4 If s an ouler, go o sep 5. Oherwse, consder he curren condon o be abnormal. ypcally he curren process condon canno be deermned by examnng one sample. In hs case, he model s reaned whou updang and he curren sample s sored. hen, f he process condon s proven o be normal subsequenly, he model can be updaed n a block-wse manner. 5 Calculae he robus esmaed value, z, of he new sample. 6 Recalculae 7 Calculae m and and Q based on z. D 8 Calculae P and Λ by performng SVD on K K or K K. 9 Fnd he number of prncpal componens o rean, p, and he conrol lms, and Q lm,. lm, 5. Applcaon Sudes 5.. Illusrave Examples Four examples are consdered o llusrae he updang procedures proposed n he paper. I should be noed ha he followng scenaros are assumed o be reflecve of normal process changes. he followng mulvarae sac process s consdered: x Ξ, where [,, L, ] R s a Gaussan random vecor and 3 Ξ R s a ransformaon marx. Here, ~ N,. and Ξ j ~ N,. herefore he measuremen vecor 3 x R follows a normal dsrbuon wh µ x and Σ x. ΞΞ. samples were generaed ha were reflecve of normal process operaon. Four

14 ypes of process change were smulaed, each comprsng samples. able summarses hese process changes and he smulaon condons mplemened o smulae hese changes. In hs sudy, a block-wse adapve PCA modellng approach was adoped wh a block sze of fve samples. ha s, he PCA model was updaed every fve me pons. For he forgeng facors o be adjusable, he followng parameers: α β. 99, α β. 9, k. 455 and n max max mn mn were seleced. y seng k.455, f he mean change s equal o he average values obaned from he ranng daa se, hen he forgeng facor akes he value.96. he frs sep was o develop he baselne model usng he off-lne learnng procedure descrbed prevously. he ranng samples were ulzed o denfy he nal PCA model. he frs samples were used o denfy he nal mean and covarance srucures and he remanng samples were used for he deermnaon of he parameers for he forgeng facor updang Eqs. and 3. Afer denfyng he model and he parameers, he procedure for on-lne monorng descrbed n Secon 4.4 was appled o he samples aken from he slowly drfng process. In he frs case sudy P, hree measured varables, x, x, and x, slowly drf away from he normal operang condons hereby smulang a slow process change such as caalys deacvaon. he slow drfs of magnude.,.5, and. are nroduced o he hree varables a me pons 3, 5, and 7, respecvely, and are connued unl he end of he smulaon. As shown n Fgs. a, b, and c, he change n he mean of varables x, x, and x s racked closely by he model updang procedure wh lmed me delay. Durng he smulaon, he number of prncpal componens s fxed a egh snce he rank of he covarance marx does no change over me Fg. d. he forgeng facor α for he sample mean m connues o decrease from me pon wh akng a consan low value of.9 from me pon 7 unl he end of he smulaon hence he model s updaed rapdly as a resul of he mean change n x, x, and x Fg. e. On he oher hand, he forgeng facor β flucuaes around.96 wh no dsnc changes beng evden snce he sample covarance marx does no change sgnfcanly durng hs perod Fg. f. Hoellng s and he Q-sasc for adapve PCA and sac PCA are compared n Fg. 3. he PCA model whou updang s no longer vald afer he process begns o change a me pon 3 and hus resuls n he process appearng o beng ou-of-conrol n boh monorng chars. In conras, he monorng chars for he adapve PCA model shows ha he process remans n-conrol afer he changes have been nroduced. 3

15 he second case sudy P reflecs wo dfferen ypes of changes n wo of he measured varables. oh of hem descrbe se-pon changes n he process by assumng ha he wo varables x and x are measured conrolled oupus. he se-pon of x changes from o + wh frs order dynamcs and a me consan of samplng me pons a me pon, whle ha of x changes from o wh frs order dynamcs and a me consan of 3 samplng me pons a me pon 7. Fgures 4a and b show he acual values and updaed means of varables x and x, respecvely wh Fg. 4c capurng he dfference beween he heorecal and esmaed mean. As me progresses afer a process change has occurred, he updaed sample mean approaches s heorecal value. Smlar o he frs case, he number of prncpal componens does no change snce he underlyng dmensonaly of he process s consan Fg. 4d. As llusraed n Fg. 4e and 4f, he value of he forgeng facor α decreases sharply a around me pons and 5 o updae he model accordng o he change n he mean values, whereas he value of he forgeng facor, β, flucuaes around.96 whou any sgnfcan change beng ncurred. As shown n Fg. 5, he Q-sasc for he adapve PCA model does no show any sgnfcan abnormal paerns represenng ou-of-conrol behavour unlke ha for he sac PCA model. I can be observed ha Hoellng s does no pck up he change snce he drecon of hs change s nearly orhogonal o he model subspace and hus he process change does no sgnfcanly ncrease he value of Hoellng s. In he hrd case sudy P3, a ramp sgnal s nroduced wh magnude. o he, elemens of he ransformaon marx from me pon unl he end of he smulaon,.e. Ξ Ξ +.. As expeced, he covarance of x changes wh me afer he change n he ransformaon marx has been nroduced. Fgures 6a and b represen he heorecal and updaed varance of x, respecvely, and Fgure 6c represens he dfference beween hem. As for he frs wo case sudes, he number of prncpal componens remans consan durng he smulaon as shown n Fg. 6d. he value of he forgeng facor, α, does no show any sgnfcan change snce he parameer change n Ξ does no affec he mean of he varables. In conras, he forgeng facor, β, s slghly lower han.96 hus he covarance marx s connuously updaed afer me pon Fgs. 6e and f. Even hough Hoellng s for adapve PCA shows several false alarms, whose frequency s sascally accepable, he adapve PCA model gves relable adapve monorng chars for boh Hoellng s and he Q-sasc, for he process showng he slow covarance change compared wh he resuls for he sac PCA model Fg. 7. he las sudy P4 smulaes a change n he underlyng dmensonaly of he process whch gves rse o a change n he number of prncpal componens reaned n he PCA model. A me pon 5, he dmensonaly of changes from o 4 and hus he rank of Ξ also ncreases by he order of four. 4

16 Fgs. 8a-c show he cumulave percen varance capured by he PCA model consruced based on he ranng daa se, he frs 5 es daa pons colleced before he process change, and he fnal 5 es daa pons aken afer he process change, respecvely. In each fgure, he number of black bars ndcaes he number of prncpal componens reaned, whch s seleced as ha when he cumulave varance aans a value larger han 9%. efore he process change, he number of prncpal componens for adapve PCA model s he same as ha for he PCA model obaned usng he ranng daa. However, afer he change n he underlyng dmensonaly of he process, he number of prncpal componens reaned ncreases by wo and hus becomes as observed n Fg. 8d. As ndcaed n Fg. 8f, β rapdly decreases o.93 a me pon 5 o updae he model as a resul of hs abrup process change. he monorng resuls wh adapve PCA are llusraed n Fgs. 9a and b and hose wh sac PCA are dsplayed n Fgs. 9c and d. he char for Hoellng s for adapve PCA has a smaller false alarm rae han ha for sac PCA. I can be observed ha he conrol lm n hs char ncreases as he number of prncpal componens reaned n he PCA model ncreases. he Q-sasc whou model updang produces ou-of-conrol sgnals jus afer me pon 5, whereas ha wh model updang s sll vald afer ha me excep ha here are some false alarms jus afer he process changes. hs s because he model updang procedure requres me unl suffcen samples are colleced from he newly changed process o descrbe he new condon. 5.. Smulaed CSR Process A nonsohermal CSR process, s consdered for he applcaon of he adapve process monorng algorhm. A schemac of he process s gven n Fg.. In he reacor, reacan A was premxed wh a E / R solven and hen was convered no produc wh rae r β k e C. he dynamc behavour of he process s descrbed by he mass balance of reacan A and he oal energy balance for he reacng sysem: r dc V F C C Vr 6 d d UA Vρ C p ρc p F c + H r Vr d + UA F ρ C 7 c c pc b where he hea ransfer coeffcen UA s emprcally represened as UA β UA af c. he concenraon of he reacan mxure s calculaed from C F C + F C / F + F. he oule emperaure and he a a s s a s oule concenraon C are conrolled by manpulang he nle coolan flow rae F c and he nle reacan flow rae F a, respecvely. he model parameers used n he CSR modellng and he con- 5

17 roller parameers of he wo PI conrollers are gven n able. All process npus and dsurbances are generaed usng frs order auoregressve models or snusodal sgnals. In addon, all measured varables are conamnaed wh whe Gaussan nose o descrbe measuremen nose able 3. o evaluae he performance of he new adapve mehod, wo knds of process change are consdered: a slow decrease n he reacon rae, and a se-pon change n he reacor emperaure. For each smulaon, he CSR process was run for 5 mn, wh he process change beng nroduced a me pon 3 mn. Nne process varables were measured every mnue able 3. he samples obaned for me pons - 3 mn were used o buld an nal PCA model and he subsequen samples 3-5 mn were used o updae he model on-lne. In he frs case, a slow drf n β r of magnude -3 mn - was nroduced a me pon 3 mn and connued unl he end of he smulaon. he decrease n he reacon rae resuls n a decrease n he reacan flow rae o manan he requred oule concenraon and he decrease n he coolan flow rae wll manan he oule emperaure. Fgs. a and b represen he acual values and her esmaed mean values, hrough he adapve mehod, of F a and F c, respecvely. he proposed adapve scheme was able o closely rack he me rajecory of he varables wh a me delay of 5 mn. Durng he drf, he forgeng facor α decreases from.96 o around.93 o ake accoun of he mean change n he process Fg. d. he process monorng mehods based on sac PCA and wo adapve PCA models wh and whou ouler correcon are compared n Fg.. he sac PCA model becomes nvald jus afer he process change occurs Fgs. a and d, whereas he adapve PCA model adaps he Hoellng s and Q sascs and her conrol lms, makng he curren model vald. here are several oulers caused by he measuremen error as depced n Fgs. b and e. he presence of oulers deeroraes he performance of he model updang procedure. y usng he proposed robus updang mehodology when an ouler occurs, he false alarm rae n he monorng chars could be sgnfcanly reduced, especally wh respec o he Q-sasc Fgs. c and f. Unlke he frs smulaon sudy, he second process change caused by he se-pon change s very fas and predcable. Snce he me of he process change s known, may be sraghforward o dscrmnae beween he alarms from he process change and hose from oulers or fauls. In hs case, he se-pon of he oule emperaure changed from 368 o 37 a me pon 3 mn. he sample mean values of F c and calculaed usng he recursve scheme are llusraed n Fgs. 3a and b. he number of prncpal componens for he adapve model s sx excep for several me pons where ncreases o seven Fg. 3c. As shown n Fg. 3d, he forgeng facor, α, drops rapdly from.96 o.9 jus afer he se-pon change and hen reurns o s normal value.96 afer me pon 35 because he esmaed mean values have converged o her new values. 6

18 he monorng resuls are shown n Fg. 4. For me pons 3 35 mn, he Q-sasc for all he mehods show ou-of-conrol sgnals due o he se-pon change. Wh he excepon of hs me perod, he adapve PCA model combned wh he ouler correcon mehod gave he mos relable monorng resuls Fg. 4f, makng possble o suppress he effec of oulers on he updang of he model and he conrol lms of he monorng sascs. 6. Conclusons A new adapve MSPC mehod for me-varyng process monorng has been proposed. ased on he newly defned weghed mean and covarance, recursve forms of he mean and covarance are defned n a sample-wse and block-wse manner. hen, by usng hese recursve formulas, an effcen mehod for fndng a PCA model recursvely s developed. In hs approach, a loadng marx s sored nsead of a full covarance marx for he nex model updae, whch means ha hs mehod has lower sorage requremens compared wh he prevous approaches descrbed n he leraure. Furhermore, several mporan ssues for adapve process monorng are addressed. Frs, wo dfferen varable forgeng facors for mean and covarance updang are consdered and her calculaon based on he recen mean or covarance change s proposed. Secondly, oulyng samples, whch are recorded que ofen n real processes, are pre-reaed by means of a robus esmaor such ha he model s nsensve o he oulers and hus s correcly updaed. ased on he overall process monorng sraegy presened n hs sudy, wo applcaon examples were consdered: a smple sac mulvarae sysem and a CSR process. he ably of he adapve model o keep rack of he changes n he varable mean and covarance was nvesgaed. Also, a change n he number of prncpal componens and he forgeng facors accordng o process changes were consdered. Chars of Hoellng s and he Q-sasc based on he adapve monorng sascs and her conrol lms were observed o be relable for he monorng of boh slowly and abruply changng processes. In parcular, he correcon for oulers usng he robus esmaon mehod resuls n no only reducng he number of false alarms sgnfcanly bu also n prevenng he curren model from beng updaed by oulers. Acknowledgemens Dr. Cho acknowledges he fnancal conrbuon of he Cenre for Process Analycs and Conrol echnology. References. Kresa, J.; MacGregor, J.F.; Marln,.E. Mulvarae sascal monorng of process operang performance. Canadan Journal of Chemcal Engneerng 99, 69,

19 . Wse,.M.; Gallagher, N.. he process chemomercs approach o process monorng and faul deecon. Journal of Process Conrol 996, 6, MacGregor, J.F.; Kour,. Sascal process conrol of mulvarae processes. Conrol Engneerng Pracce 995, 34, racy, N.D.; Young, J.C.; Mason, R.L. Mulvarae conrol chars for ndvdual observaons, Journal of Qualy echnology 99, 4, Jackson, J.E. A user s gude o prncpal componens; John Wley & Sons: New York, Mller, P.; Swanson, R.; Heckler, C. Conrbuon Plos: he mssng lnk n mulvarae qualy conrol, Fall Conf. of he ASQC and ASA, Mlwaukee, WI, Wold, S., Exponenally weghed movng prncpal componens analyss and projecons o laen srucures. Chemomercs and Inellgen Laboraory Sysems 994, 3, Dayal,.S.; MacGregor, J.F. Recursve exponenally weghed PLS and s applcaons o adapve conrol and predcon. Journal of Process Conrol 997, 73, Qn, S.J., Recursve PLS algorhms for adapve daa monorng. Compuers & Chemcal Engneerng 998,, L, W.; Yue, H.H.; Valle-Cervanes, S.; Qn, S.J. Recursve PCA for adapve process monorng. Journal of Process Conrol,, Felz, C.J.; Shau, J. J. H. Sascal process monorng usng an emprcal bayes mulvarae process conrol char. Qualy and Relably Engneerng Inernaonal, 7, Lu, X.; Chen,.; hornon, S.M. Egenspace updang for non saonary process and s applcaon o face recognon. Paern Recognon 3, 36, Hall, P.; Marshall, D.; Marn, R. Mergng and splng Egenspace models. IEEE ransacons on Paern Analyss and Machne Inellgence, 9, Jackson, J.E.; Mudholkar, G. Conrol procedures for resduals assocaed wh prncpal componen analyss. echnomercs 979,, Forescue,.R.; Kershenbaum, L.S.; Ydse,.E. Implemenaon of self unng regulaors wh varable forgeng facors. Auomaca 98, 7, Lane, S.; Marn, E..; Morrs, A.J.; Gower, P. Applcaon of exponenally weghed prncpal componen analyss for he monorng of a polymer flm manufacurng process. ransacons of he Insue of Measuremen and Conrol 3, 5, Huber, P.J. A robus esmaon of a locaon parameer. Ann. Mah. Sa. 964, 35, Huber, P.J. Robus sascs; John Wley & Sons: New York, Lang, Y.-Z.; Kvalhem, O.M. Robus mehods for mulvarae analyss - a uoral revew. Chemomercs and Inellgen Laboraory Sysems 996, 3,.. Yoon, S.; MacGregor, J.F. Faul dagnoss wh mulvarae sascal models par I: usng seady sae faul sgnaures. Journal of Process Conrol,,

20 . Cho, S. W.; Marn, E..; Morrs, A.J.; Lee, I.-. Faul deecon based on a maxmum lkelhood PCA mxure. Indusral and Engneerng Chemsry Research 5, n press. 9

21 Ls of Fgures Fgure Effec of funcon parameers on varable forgeng facor. a Effec of α max, b Effec Fgure of α mn, c Effec of k, and d Effec of p on α. he defaul values of he four parameers are α. max 99, α. 9 mn, k.693, and n. me-seres plos of varable mean and model parameers for case P. a Measured varables x, x, and x, b Esmaed mean mx, c Dfference beween heorecal and esmaed mean values µ x m x, d Number of PCs for adapve PCA, e Forgeng facor α, f Forgeng facor β. Fgure 3 Process monorng chars for case P. a Hoellng s for adapve PCA, b Q-sasc for adapve PCA, c Hoellng s for PCA, d Q-sasc for PCA. Fgure 4 me-seres plos of varable mean and model parameers for case P. a Measured varables x and x, b Esmaed mean mx, c Dfference beween heorecal and esmaed mean values µ x m x, d Number of PCs for adapve PCA, e Forgeng facor α, f Forgeng facor β. Fgure 5 Process monorng chars for case P. a Hoellng s for adapve PCA, b Q-sasc for adapve PCA, c Hoellng s for PCA, d Q-sasc for PCA. Fgure 6 me-seres plos of varable varance and model parameers for case P3. a heorecal sandard devaon of varables x, b Esmaed sandard devaon, c Dfference beween heorecal and esmaed sandard devaon, d Number of PCs for adapve PCA, e Forgeng facor α, f Forgeng facor β. Fgure 7 Process monorng chars for case P3. a Hoellng s for adapve PCA, b Q-sasc for adapve PCA, c Hoellng s for PCA, d Q-sasc for PCA. Fgure 8 Cumulave percen varance CPV capured by PCA model and he model parameers for case P4. a CPV capured by he PCA model for he ranng daa, b CPV capured by he PCA model for he es daa for me pons 5, c CPV capured by he PCA model for he es daa for me pons 5, d Number of PCs reaned for adapve PCA, e Forgeng facor α, f Forgeng facor β. Fgure 9 Process monorng chars for case P4. a Hoellng s for adapve PCA, b Q-sasc for adapve PCA, c Hoellng s for PCA, d Q-sasc for PCA. Fgure Nonsohermal CSR process and nne measured varables. Coolan emperaure, Reacan mxure emperaure, 3 Reacan A concenraon, 4 Solven concenraon, 5 Solven flow rae, 6 Solue flow rae, 7 Coolan flow rae, 8 Oule concenraon, and 9 Oule emperaure.

22 Fgure Fgure Fgure 3 Fgure 4 me-seres plos of varable mean and model parameers for he frs case sudy reacon rae decrease n he CSR process. a Measured value of F a and s esmaed mean, b Measured value of F c and s esmaed mean, c Number of PCs for adapve PCA, d Forgeng facors. Process monorng chars for PCA, adapve PCA, and robus adapve PCA. Hoellng s for a PCA, b Adapve PCA, c Robus adapve PCA; Q-sasc for d PCA, e Adapve PCA, f Robus adapve PCA. me-seres plos of varable mean and model parameers for he second case sudy se-pon change n he CSR process. a Measured value of F c and s esmaed mean, b Measured value of and s esmaed mean, c Number of PCs for adapve PCA, d Forgeng facors. Process monorng chars for PCA, adapve PCA, and robus adapve PCA. Hoellng s for a PCA, b Adapve PCA, c Robus adapve PCA; Q-sasc for d PCA, e Adapve PCA, f Robus adapve PCA.

23 forgeng facor a ncreasng α max forgeng facor b ncreasng α mn x / x nor x / x nor forgeng facor c ncreasng k forgeng facor d ncreasng p x / x nor x / x nor Fg.. Effec of funcon parameers on varable forgeng facor. a Effec of α max, b Effec of α mn, c Effec of k, and d Effec of p on α. he defaul values of he four parameers are α. max 99, α.9 mn, k.693, and n.

24 x µx - mx mx b mx 5 mx mx c a x x x µx -mx - µx -mx µx - -mx Number of PCs α β d e f Fg.. me-seres plos of varable mean and model parameers for case P a Measured varables x, x, and x, b Esmaed mean mx, c Dfference beween heorecal and esmaed mean values µ x m x, d Number of PCs for adapve PCA, e Forgeng facor α, f Forgeng facor β. 3

25 a 3 Adapve PCA c PCA b 5 Adapve PCA d PCA 5 Q Q Fg. 3. Process monorng chars for case P. a Hoellng s for adapve PCA, b Q-sasc for adapve PCA, c Hoellng s for PCA, d Q-sasc for PCA. 4

26 x mx µx - mx a x x b mx mx c µx -mx µx -mx Number of PCs α β d e f Fg. 4. me-seres plos of varable mean and model parameers for case P. a Measured varables x and x, b Esmaed mean mx, c Dfference beween heorecal and esmaed mean values µ x m x, d Number of PCs for adapve PCA, e Forgeng facor α, f Forgeng facor β. 5

27 a 5 Adapve PCA c 3 PCA b Adapve PCA d 8 PCA 6 Q Q Fg. 5. Process monorng chars for case P. a Hoellng s for adapve PCA, b Q-sasc for adapve PCA, c Hoellng s for PCA, d Q-sasc for PCA. 6

28 σx sx σx -sx b c.6.4. a Number of PCs α β d e f Fg. 6. me-seres plos of varable varance and model parameers for case P3. a heorecal sandard devaon of varables x σ x, b Esmaed sandard devaon s x, c Dfference beween heorecal and esmaed sandard devaon σ x s, d Number of PCs for adapve PCA, e Forgeng facor α, f Forgeng facor β. x 7

29 a 4 Adapve PCA 3 c PCA b Adapve PCA d 4 PCA 5 3 Q Q Fg. 7. Process monorng chars for case P3. a Hoellng s for adapve PCA, b Q-sasc for adapve PCA, c Hoellng s for PCA, d Q-sasc for PCA. 8

30 Fg. 8. Cumulave percen varance CPV capured by PCA model and he model parameers for case P4. a CPV capured by he PCA model for he ranng daa, b CPV capured by he PCA model for he es daa for me pons 5, c CPV capured by he PCA model for he es daa for me pons 5, d Number of PCs reaned for adapve PCA, e Forgeng facor α, f Forgeng facor β. 9

31 4 a Adapve PCA 4 c PCA b Adapve PCA 5 d PCA Q Q Fg. 9. Process monorng chars for case P4. a Hoellng s for adapve PCA, b Q-sasc for adapve PCA, c Hoellng s for PCA, d Q-sasc for PCA. 3

32 CC 3 C a 6 F a 8 C 9 4 C s 5 F s C c 7 F c Fg.. Nonsohermal CSR process and nne measured varables. Coolan emperaure, Reacan mxure emperaure, 3 Reacan A concenraon, 4 Solven concenraon, 5 Solven flow rae, 6 Solue flow rae, 7 Coolan flow rae, 8 Oule concenraon, and 9 Oule emperaure. 3

33 F a a F a mf a b Number of PCs c d F c 5 F C mf C Forgeng facor α β Fg.. me-seres plos of varable mean and model parameers for he frs case sudy reacon rae decrease n he CSR process. a Measured value of F a and s esmaed mean, b Measured value of F c and s esmaed mean, c Number of PCs for adapve PCA, d Forgeng facors. 3

34 a PCA d PCA 5 Q b APCA e APCA Q c Robus APCA f Robus APCA Q Fg.. Process monorng chars for PCA, adapve PCA, and robus adapve PCA. Hoellng s for a PCA, b Adapve PCA, c Robus adapve PCA; Q-sasc for d PCA, e Adapve PCA, f Robus adapve PCA. 33

35 F c 5 F C mf C Number of PCs m Forgeng facor α β Fg. 3. me-seres plos of varable mean and model parameers n he second case sudy se-pon change n he CSR process. a Measured value of F c and s esmaed mean, b Measured value of and s esmaed mean, c Number of PCs for adapve PCA, d Forgeng facors. 34

36 PCA 4 PCA Q APCA APCA Q Robus APCA Robus APCA Q Fg. 4. Process monorng chars for PCA, adapve PCA, and robus adapve PCA. Hoellng s for a PCA, b Adapve PCA, c Robus adapve PCA; Q-sasc for d PCA, e Adapve PCA, f Robus adapve PCA. 35

37 Ls of ables able able able 3 Four ypes of process changes Model parameers used n he CSR modellng and he PI conrol parameers Paerns of process npus and dsurbances and measuremen nose 36

38 Paerns * P x x +. 3 > 3 * x x.5 5 > 5 Scenaro Change n he mean unl he end of smulaon me * x x +. 7 > 7 * P x x + exp > 3 Change n he mean * x x exp 7 3 > 7 P3 Ξ Ξ +. > Change n he covarance P4 Ξ R Ξ R > 5 Change n he number of PCs able : Four ypes of process changes nroduced no he smulaon sudy 37

39 Noaon Parameers and Consans Value V Volume of reacon mxure n he ank m 3 ρ Densy of reacon mxure 6 g/m 3 ρ c Densy of coolan 6 g/m 3 C p Specfc hea capacy of he reacon mxure cal/gk C pc Specfc hea capacy of he coolan cal/gk H r Hea of reacon cal/kmol k Pre-exponenal knec consan mn - E/R Acvaon energy / Ideal gas consan 833 K K c emperaure conroller gan -.5 τ I Inegral me 5 K c C Concenraon conroller gan.485 τ I C Inegral me able : Model parameers used n he CSR modellng and he PI conrol parameers. 38

40 Inpu and Dsurbance Measuremens Varable Mean φ σ v Varable F s.9 m 3 /mn..9 - c K c 365 K C a. - β r C s.5-5 β UA F s Varable Mean a b σ e 6 F a σ v 7 F c. - C a 9. kmol/m C.5-5 C s.3 kmol/m able 3. Paerns of process npus and dsurbances and measuremen nose C a and C s are modelled as x a sn b + v and he remanng npus and dsurbances are modelled as x φx + v, where he process nose v ~ N, σ v nose e ~ N, σ. e. All measured varables are conamnaed wh whe Gaussan 39