Process Monitoring and Feedforward Control for Proactive Quality Improvement

Transcription

1 Inernionl Journl of Performbiliy Engineering Vol. 8, No. 6, November 0, pp RAMS Consulns Prined in Indi Process Monioring nd Feedforwrd Conrol for Procive Quliy Improvemen. Inroducion LIHUI SHI * nd KAILASH C. KAPUR Deprmen of Indusril nd Sysems Engineering, Universiy of Wshingon, Sele, WA 9895, U.S.A. (Received on Mrch 0, 0 nd revised on June 07, 0) Absrc: Process djusmen sregy is n imporn pr of he process improvemen mehods, which is lso clled engineering process conrol (EPC), nd i is ofen inegred wih sisicl process conrol (SPC) o improve he process conrol performnce. While feedbck conrol is used o compense for he oupu deviion, feedforwrd conrol is procive conrol sregy bsed on direc mesuremen of he disurbnce, nd i cs before he disurbnce ffecs he sysem. Feedforwrd conrol is usully combined wih feedbck conrol for vriion reducion. In his ricle, rionles for feedforwrd conrol re explined, nd new philosophy on is pplicion is given. The fesibiliy condiion for feedforwrd conrol pplicion illusred from new disurbnce decomposiion viewpoin, nd he vlidiy of some disurbnce models which work well for feedforwrd conrol is invesiged. Some relevn issues on process monioring, feedbck conrol nd feedforwrd conrol re discussed nd ddressed. Keywords: Sisicl process conrol, engineering process conrol, feedbck conrol, feedforwrd conrol, men squre error, disurbnce models Conrol is coninuous endevor o keep mesures of quliy s close s possible o heir rge vlues for indefinie periods of ime"[]. Process monioring nd process djusmens re wo differen echniques o chieve his gol, nd hey re pr of sisicl process conrol (SPC) nd engineering process conrol (EPC) respecively. Shewhr chrs, cumulive sum (CUSUM) chrs nd exponenilly weighed moving verge (EWMA) chrs re frequenly employed in SPC o find he specil cuses which should be furher removed or elimined. Unforunely, someimes he specil cuses re known bu cnno be economiclly removed, so compension mus be mde o preven process from wndering off he rge, clling for feedbck conrol nd feedforwrd conrol in EPC. SPC hs hisoricl foundion in prs indusry nd EPC hs origined in process indusry. SPC nd EPC were pplied by people wih differen echnicl bckgrounds for differen conrol objecives. Box nd Krmer [] discussed he inerfce beween SPC nd EPC, nd how hey cn be inegred efficienly. While feedbck conrol is recive process djusmen sregy bsed on he process oupu error, feedforwrd conrol is bsed on direc mesuremen of he disurbnce nd i cs before he disurbnce ffecs he sysem. Prevenion is beer hn recive correcion, especilly for he process wih lrge ineri. Feedforwrd conrol hs wide pplicions in chemicl, mechnicl nd mngemen indusries [3]. While feedbck conrol hs been exensively invesiged [-], [4], he exising reserch on feedforwrd conrol from sisicl perspecive is quie limied, hus furher invesigions on feedforwrd conrol re necessry nd worhwhile. * Corresponding uhor s emil: shilihui@uw.edu 60

2 60 Lihui Shi, nd Kilsh C. Kpur The res of his ricle is orgnized s follows. Secion illusres new perspecive on feedforwrd conrol pplicion bsed on disurbnce decomposiion, followed by wo rel life exmples. Secion 3 presens some bsic disurbnce models in SPC nd EPC lierure, nd proposes some disurbnce models h cn be djused by feedforwrd conrol. In Secion 4 we provide he feedbck nd feedforwrd conrol equions for hese disurbnce models, nd some numericl resuls for rndom sep chnge disurbnce model. Some discussions on relevn issues wih process monioring, feedbck conrol nd feedforwrd conrol re provided in Secion 5, which shed ligh on hese imporn quesions. Concluding remrks re given in Secion 6.. A New Perspecive on Feedforwrd Conrol Here we explin he bsic ides of feedbck conrol nd feedforwrd conrol using blck box in our conrol sysem, nd illusre feedforwrd conrol fesibiliy condiions from new perspecive, clled disurbnce decomposiion. () (b) Figure : Disurbnce Decomposiion for Feedforwrd Conrol Loop The fcors in feedbck conrol sysem cn be cegorized ino hree groups: conrol fcor X whose levels cn be djused, noise fcor e which is he source of vriion, nd n oupu vrible Y, wih rnsfer funcion Y = f ( X, e), s illusred by Figure (). We inerpre he fesibiliy condiion for feedforwrd conrol from disurbnce decomposiion viewpoin: if noise e cn be furher decomposed ino new observble bu unconrollble fcor B nd remining noise e ', wih new rnsfer funcion Y = f '( X, B, e '), hen feedforwrd conrol cn be pplied by djusing X bsed on B o mkey closer o he rge vlue, illusred by feedforwrd conrol loop on B in Figure (b). We cll B informive disurbnce vrible nd e' non-informive noise or bckground disurbnce. Since unmesured bckground disurbnces re lwys presen in ny conrol sysem, feedforwrd conrol is usully combined wih feedbck conrol o chieve he highes possible efficiency. How effecive his feedforwrd conrol cn be depends on he mgniude of he informive disurbnce vrible B h cn be ken ou from he noise e, nd we will explin his ler in deil. We firs discuss some rel exmples wih feedforwrd conrol from our disurbnce decomposiion viewpoin. In wer heer exmple where sem is used in he he exchnger o he he incoming cold wer o minin some rge emperure of he ho wer. Insed of only using he hermos on he ho wer oule for feedbck o djus he moun of sem, skillful operor could use simple feedforwrd sregy o minin he ho wer emperure on rge. The worker would compense for chnges

3 Process Monioring nd Feed Forwrd Conrol for Procive Quliy Improvemen 603 in inle cold wer emperure by monioring i nd in response o h, incresing or decresing he sem re o counerc he chnge in he emperure of he incoming ho wer. Here he cold wer emperure is our informive disurbnce vrible B nd i is very esy o observe nd mesure. Anoher exmple is he cruise conrol which enbles cr o minin sedy rod speed. When n uphill srech of rod is encounered, he cr slows down below he se speed; his speed error cuses he engine hrole o be opened furher, bringing he cr bck o is originl speed wih feedbck conrol. Feedforwrd conrol cn be pplied by mesuring he rod slope, upon encounering hill, hen i would open up he hrole by cerin moun uomiclly nd nicipe he exr lod. The cr does no hve o slow down ll for he correcion o come ino ply. Here he rod slope is he informive disurbnce vrible B. 3. Disurbnce Models The min objecive of SPC provides n ongoing check on he sbiliy of process by looking for he ssignble cuse which cuses deprure or deviion from he in conrol" process. In EPC, i is lwys desirble o djus process from ime o ime such h he process oupu is s close o he rge vlue s possible, nd o chieve he minimum men squre error (MMSE). The developmen nd implemenion of ny conrol, eiher for process monioring or djusmen, requires resonbly relisic represenion of disurbnce model z, i.e., he oupu deviion from rge h would occur if no djusmen cion were ken. 3. Bsic Sionry nd Non sionry Disurbnce Models In SPC, he simples nd mos fmilir disurbnce model for process in se of conrol is he whie noise series, z =, () { } where is sequence of iid rndom vribles h re normlly disribued wih men 0 nd vrince. The well known Deming's funnel experimen hs he similr ssumpion on he disurbnce model h is hs fixed men nd does no drif wy by iself. Then he bes conrol sregy for his process is jus o leve i lone" wihou ny djusmen, since ny conrol cion will emper" he process nd increse he vribiliy. I is known h in process indusries under EPC scope, more generl siuion for he disurbnce model is he sionry uo-correled process, such s (uoregressive) AR () model: z φz =, φ < () nd (uoregressive moving verge) ARMA (,) model: z,, φz = θ φ < θ < (3) McGregor [5] invesiged n ineresing modified Deming's funnel experimen in which he process men follows n uocorreled AR () model, nd showed h i clls for n cive djusmen rule insed of he no conrol sregy for vriion reducion. However, he ssumpion of sionriy migh be quesionble for disurbnces modeling in mny relisic conrol problems. As Deming [9] sys, no process, excep in rificil demonsrions by use of rndom numbers, is sedy nd unwvering", implies h he

4 604 Lihui Shi, nd Kilsh C. Kpur nonsionry disurbnce model is more relisic in EPC. Compred wih he sionry process, he nonsionry process does no hve fixed men. The inegred moving verge (IMA) model is he simples, mos common nd pproprie nonsionry disurbnce model [-], [6], [0]: z, z = θ θ < (4) In recen decdes, wih he fs developmen of some rdiionl nd modern high ech indusries, such s mnufcuring, semiconducor, compuer nd sofwre indusries, he processes in hese fields re becoming more nd more compliced nd re hybrids in nure. The disurbnce models of hese processes re ofen pproximely ddiive models of cerin bckground disurbnce [], such s he IMA, AR(), or ARMA(,) process, wih some ddiionl pr dded ogeher, like spike, susined men shif, rmp, n exponenil rise o new levels, ec. [], [6], [-3]. 3. Periodic Shif Disurbnce Models Periodic cycles very commonly exis in some high-vlue discree-pr mnufcuring processes. Moived by feedsock chnge problem discussed by Box nd Luceno [4], Shi nd Kpur [5] invesiged periodic vrince shifs disurbnce models, wih IMA, AR () nd ARMA(,) processes s bckground disurbnce respecively: z z = θ +, θ < z φz = +, 0< φ < z φz = θ +, 0< φ <, 0< θ < where is he shif h occurs periodiclly = T, T,3 T,... nd one-sep-hed esimor of ime wih errorε. Assume E( ) = E( ε ) = 0 Vr( ) =, Vr( ε ) = ε Usully we re concerned wih he vrince inflion esimion error ε (5) m = + ε is he (6) >, wih smll <. We cll hem model (), model () nd model (3) respecively. This periodic shif disurbnce model fis well wih relisic mnufcuring scenrio in which producion line works for producing very hin mellic films. In his process, feedsock meril is frequenly fed ino he producion line o produce he mellic films. Afer cerin number of mellic films re produced, lo is formed nd hen new lo is begins o form by subsequen mellic films. For such process, boh wihin-los vribiliy nd beween-los vribiliy exis. Wihin ech lo, he feedsock meril is relively homogeneous nd lck of uniformiy ppers beween differen los, which ffec he hickness of he producing mellic films. The wihin-lo vribiliy cn be modeled by n IMA, or AR(), or ARMA(,) process, nd s soon s enough bches of mellic film ws produced, relible prmeer esimes cn be mde. 3.3 Rndom Sep Chnge Disurbnce Models Process monioring of susined sep shif disurbnce models hs been discussed in some SPC lierure. Vnder Wiel [6] invesiged monioring n IMA process wih susined

5 Process Monioring nd Feed Forwrd Conrol for Procive Quliy Improvemen 605 level shif, z z = θ + D, wih D = 0, 0 < nd D, s = s s, where s is he unknown chnge-poin. Nembhrd nd Vlverde-Venur [6] proposed he cumulive score (Cuscore) sisics Q = Q sθ + e in heir Cuscore conrol chr o deec susined sep shif in he IMA disurbnce. Oher reserchers focused on he more chllenging disurbnce models subjec o rndom sep chnges insed of he susined shif, which re usully cused by vriions in he physicl condiions, such s he environmenl emperure nd rw meril quliies. Chen nd Elsyed [7] sudied using n EWMA esimor o monior he i.i.d normlly disribued process wih rndom sep chnges. In heir model, he rndom sep-chnge occurrence hs consn probbiliy p which is independen of he prior hisory of he process, wih size r = τ /, where is he sndrd deviion of he bckground norml process, ndτ is he sndrd deviion of he process men. They proposed n EWMA esimor wih closed-form expression for he opiml vlue of he weighed vrible λ, s funcion of he esimes ˆp nd ˆr derived from hisoricl d. Tsimyrzis nd Hwkins [8] invesiged he process monioring of men drif model of AR () process subjec o rndom sep chnges, in Byesin frmework. They supposed h he process men hs jump of size h occurs wih probbiliy p, nd ssumed h he prior informion bou he process men is vilble. Then ech ime when he new d comes, hey ge he poserior disribuion for he process men hrough Byes heorem o check if he men hs drifed or no. If here is no significn chnge, hey use his poserior s he prior for he nex sge. Their model is suible for some prcicl problems, for exmple, ool wer problems in which he wer incorpores rndom sep chnge (due, e.g., o ool chipping) s well s drif. They lso generlized he model wih ssigning prior disribuion o he size of he jumps, wih AR() process subjec o rndom sep chnges [3]. Since sep chnge is more difficul o see when buried in n IMA hn when buried in iid noise" [6] or sionry bckground noise, we furher generlize he bckground disurbnce o be he nonsionry IMA model subjec o rndom sep chnges, z z = θ + D where, N(0, ) wih probbiliy p D ~ (7) N(, ) wih probbiliy p nd is he sndrd deviion of he whie noise { }. Since we do no know he cul size of he sep shif, we ssume i is rndom wih cerin prior disribuion: π ( ) ~ ( µ, ) (8) where µ nd Boh µ nd N re he expeced men nd vriion of he sep shif size respecively. re ssumed o be known, considering h hey cn be deermined from previous engineering knowledge or experiences bou he process. Smll vlues of led o informive seings while lrge vlues indice priori ignornce for he size of he sep shif. The size of he sep shif cn be modeled by nd he unceriny of he

6 606 Lihui Shi, nd Kilsh C. Kpur sep shif cn be modeled by he sndrd deviion rio = /. r Figure : IMA Disurbnce Wihou nd Wih Rndom Sep Chnge Figure () shows n IMA disurbnce process of 00 d wih 0.8, nd Figure (b) shows he sme IMA disurbnce subjec o rndom sep chnge wih occurrence probbiliy 0.0. The shif size hs he prior disribuion ( ) ~ N(, ) p = π µ wih µ = nd = 3, so he sndrd deviion rio r = 3. There re 4 rndom sep chnges: 5.5 upwrd sep chnge in he 9h vlue of he IMA disurbnce z,.08 downwrd sep chnge in he 0h vlue, 3.7 downwrd sep chnge in he 50h vlue, nd 3.49 upwrd sep chnge in he 84h vlue. 4. Conrol Equions nd MSEO Resuls 4. Dynmic Model for Process Ineri Assume X is he level of djusmen (inpu) vrible, g is he process gin, i.e., he evenul chnge in he oupu y h is induced by uni djusmen X, hen i my ke some ime before he full effec of djusmen is experienced he process oupu, which cn be modeled by firs-order dynmic equion: Y = Y + g( ) X + C (9) where C is consn, 0 <. For every uni chnge in he inpu X, he proporion of he ol chnge h occurs in he firs ime inervl is. The lrger corresponds o slowly responding sysem wih greer ineri. We menioned erlier h feedbck conrol ends o hve low efficiency when process hs gre ineri, resuling in lrge dely beween he djusmen on he inpu nd is effec experienced on he oupu. When = 0, he full effec of n djusmen mde ime is relized he oupu y wihin he nex uni inervl, hen he dynmic model becomesy = gx + consn. This is clled he pure-gin model (lso he responsive sysem), nd we mke he simplified ssumpion h he sysem is responsive hroughou his pper. 4. Conrol Equions for Periodic Shif Models Suppose x = X X is he djusmen mde ime, e is he oupu error under conrol. For responsive sysem, he feedbck conrol equion for n IMA process wih θ =

7 Process Monioring nd Feed Forwrd Conrol for Procive Quliy Improvemen 607 θ is gx = Ge (0) where G (0 < G ) is he dmping fcor. When G = θ, i provides he minimum men squre error (MMSE) conrol []. Suppose z% = ( θ )( z + θ z + θ z +...) = θ z% + ( θ ) z () is he one-sep- hed EWMA forecs wih smoohing consnθ, 0 θ <, hen is MMSE conrol gx = Ge wih G = θ cn be wrien s gx = % z [9], nd he oupu error under feedbck conrol is jus he forecs error of EWMA forecs for he IMA disurbnce model. The feedbck conrol equions for he AR() nd ARMA(,) processes re given by Box nd Luceno[], nd Tsung e l. [8], respecively. Compred wih feedbck conrol, feedforwrd conrol is pplied o compense m, he shif esimed ime, nd i cn be wrien s gx = l( m, m,...) () where l is suible model-bsed funcion of he shifs esimed curren nd previous imes. The feedforwrd conrol equions for model (), models () nd (3) re proposed by Box nd Luceno [4], Shi nd Kpur [5] respecively, see Tble. Here B is he bckshif operor such h BX = X, z is n EWMA forecs wih smoohing consnθ, e% nd m% re EWMA forecss wih smoohing consnφ. Model () Model () Model (3) Tble : Feedbck nd Feedforwrd Conrol Equions for Models (), (), (3) Feedforwrd conrol equion gx = m = T, T,3 T,... Feedbck conrol equion gx = Ge, where G = gx = gx 0 G ei i= = ( % ) gx = φ ( e e% ) gx m m =,,3,... θ φ gx e e i = = φ φ i φb i= = ( % ) gx = ( φ θ )( e e% ) gx m m =,,3,... φ θ gx e e i = = ( φ θ) φ i φb i= The oupu men squre error (MSEO) is defined by lim MSEO = vr( ei ) (3) i= Wih he conrol equions in Tble, he closed-form MSEO formule for models (), (), (3) under feedbck conrol nd combined conrol (feedbck nd feedforwrd conrol) sregies, cn be derived in sric mhemicl proofs, nd he MSEO ble cn be found in Shi nd Kpur [5]. The derivion processes for he feedforwrd conrol equion nd

8 608 Lihui Shi, nd Kilsh C. Kpur MSEO formule re nonrivil, nd hey re minly deermined by enerining wih he oupu error relionship, see Shi nd Kpur [5] for deils. 4.3 Conrol Equions for Rndom Sep Chnge Models For model (4), we inroduce process djusmen procedure bsed on feedbck conrol nd n dded djusmen bsed on oupu errors monioring. We propose feedbck gx conrol = Ge, plus djusmens s soon s possible rndom sep chnge is deeced, i.e., when n oupu error e flls ouside of he 3 limis, wih conrol equion ( e 3 ), if e 3 gx = > (4) ( e + 3 ), if e 3 This dded djusmen is pplied o compense he possible rndom sep chnge, nd cn be considered s qusi-feedforwrd conrol, nd he overll combined conrol ws clled s qusi-feedbck feedforwrd conrol. The rionle for his qusi-feedforwrd conrol is due o he reson h he feedbck conrol equion is only for he IMA disurbnce, nd i is no sufficien when oher disurbnces re presen (rndom sep chnge here). The moun of he qusi-feedforwrd conrol is chosen o be he disnce beween oupu error e nd he3 limi on he sme side of he conrol chr. I is clled qusi-feedforwrd conrol, since i is bsed on neiher pure oupu error, nor he direc mesure of he rndom sep chnge; insed, i is bsed on he ouliers in sequence { e } h re ouside of he 3 limis under feedbck conrol, nd hese ouliers cll for furher remedil cions. Thus such dded djusmen goes beyond feedbck conrol nd does no mee he requiremen for feedforwrd conrol, so in some sense, i is in beween he rdiionl feedbck nd feedforwrd conrol cegories, which jusifies such nme. Box nd Krmer [] discussed similr process o feedbck conrol o show h he feedbck conrol does no necessrily concel he nure of he disurbnce h is being compensed. Here we go furher sep beyond hem by pplying n dded djusmen o coninue improving he process. 4.4 MSEO Resuls for Rndom Sep Chnge Models Now we presen some numericl resuls on he MSEO under combined conrol (qusi-feedbck feedforwrd conrol) in our rndom sep chnge model (4). Mos indusril ime series of IMA disurbnces hve 0.6 θ 0.8 [], nd we choose θ = 0.8 nd = in model (4). We consider wo occurrence probbiliies p = 0.0 nd p = 0.05 for he rndom sep chnge, nd choose some differen combinions for he prior disribuion of he sep chnge π ( ) ~ ( µ, ). We N invesige he expeced size of he sep shif µ from -0.3 o 0.3, for boh upwrd chnge nd downwrd chnge, nd he unceriny of he sep shif r from o 3. Due o symmery, we only need o include 0 µ 3 in our numericl resuls summry. We choose he smple size n = 00 in ech simuled disurbnce process nd mke 0,000 ierions for ech simuled disurbnce process in our simulion. We pply wo conrol mehods: feedbck conrol nd combined conrol, for he sme disurbnce process d, nd compre heir MSEOs resuls. In our numericl resuls

9 Process Monioring nd Feed Forwrd Conrol for Procive Quliy Improvemen 609 summry, we show boh he MSEO under combined conrol nd he improvemen percenge for he qusi-feedforwrd conrol, which is clculed by MSEOFB MSEOcomb (5) MSEOFB where MSEO is he MSEO under feedbck conrol nd FB MSEO is he MSEO comb under combined conrol. Tble : MSEOs under combined conrol for differen p, µ µ nd p = 0.0 p = (0.6%).30 (36.7%) 3.4 (6.%).4 (3.0%) 3.40 (.5%).06 (5.%) 3.40 (5.5%).0 (9.%).30 (9.8%).78 (.5%).3 (4.6%).63 (.%) 3.3 (.7%).8 (.0%).0 (5.%).50 (.0%).0 (.0%).7 (3.3%) (0.%).73 (9.3%) 0.6 (3.7%).4 (7.8%) 0.05 (0.%).3 (0.6%) in model (4) ( θ = 0.8 ) The numericl resuls re shown in Tble, noice h he improvemen percenge is given in he prenhesis followed by he MSEO under combined conrol. Since here re hree prmeers µ, nd p, we will fix ech wo ou of he hree prmeer vlues, nd compre he MSEO nd he improvemen percenge for differen vlues of he oher prmeer. Some useful conclusions cn be drwn from Tble re: () For he disurbnce model wih he sme prior disribuion of he rndom sep chnge N( µ, ), he higher occurrence probbiliy p will chieve lrger improvemen percenge for he qusi-feedforwrd conrol. However, he disurbnce wih lrger p will resul in lrger MSEO under combined conrol. () For he disurbnce model wih he sme expeced size of he rndom sep chnge µ nd he occurrence probbiliy p, he lrger unceriny r (equivlenly ) will chieve lrger improvemen percenge for he qusi-feedforwrd conrol. However, he disurbnce model wih lrger r will resul in lrger MSEO under combined conrol. (3) For he disurbnce model wih he sme unceriny of he rndom sep chnge r (equivlenly ) nd he occurrence probbiliy p, he lrger expeced rndom sep chnge size µ will chieve lrger improvemen percenge for he qusi-feedforwrd conrol. However, he disurbnce wih lrger µ will resul in lrger MSEO under combined conrol.

10 60 Lihui Shi, nd Kilsh C. Kpur All conclusions re consisen wih our inuiion: he lrger he µ, r or p, i.e., he more informion on he rndom sep chnge, which is he informive disurbnce vrible, cn be decomposed from he ol disurbnce, he more improvemen cn be chieved from feedforwrd conrol pplicion. However, he lrger nd more frequen rndom sep chnge lwys poenilly increses he process vribiliy, even fer djused by feedbck nd feedforwrd conrol, resuling in lrger MSEO. This explins he reson for he ineresing poin h lger improvemen percenge lwys corresponds o lrger MSEO under combined conrol. 5. Some Commens on Process Monioring nd Feedbck, Feedforwrd Conrol There re some relevn issues on SPC nd EPC h migh cuse misundersnding, confusion nd conroversies from reserchers nd prciioners. Now we discuss hese opics in greer deil nd furher illusre he rionles behind our new perspecives on he process monioring, feedbck nd feedforwrd conrol frmework. 5. When SPC is no Enough As we know, SPC provides n ongoing check on he sbiliy of process by using conrol chrs o idenify vriion which re due o specil cuses. However, he process sbiliy is no he only hing we need o cre for, since sble process migh no be cpble process, for exmple, if he process men is no on rge, or if he process vriion is oo lrge. Under such cses, we need o chnge he process so i cn mee he cusomer requiremens, insed of minining he sbiliy of his noncpble process. This clls for process djusmen sregies like feedbck conrol nd feedforwrd conrol o mke he process on rge nd reduce vriion. 5. When Feedforwrd Conrol Applicion is Possible? Koonz nd Brdspies [3] climed h even he mos enhusisic proponens of feedforwrd conrol dmi h, if inpu vribles re no known or unmesurble, he sysem will no work." Noice h heir inpu vribles jus correspond o our informive disurbnce vrible B. However, he idel ssumpion of bsolue ceriny on he informive disurbnce vribles hve prcicl limiions, nd someimes prmeer unceriny should be reed s ddiionl source of vribiliy, possibly due o poor undersnding of he process behvior. From our disurbnce decomposiion viewpoin, he bove semen of Koonz nd Brdspies [3] cn be relxed in some sense: s long s we hve some knowledge on he informive disurbnce vrible B, eiher complee or pril knowledge, feedforwrd conrol or les qusi-feedforwrd conrol is possible. For exmple, in model (4), wihou knowing he exc size of he rndom sep chnge ech ime, we mde he relisic ssumpion h wih some empiricl knowledge on he rndom sep chnge, is occurrence probbiliy p is known, nd is prior disribuion is normlly disribued wih known men nd vrince, following he ssumpion of Tsimyrzis nd Hwkins [3]. 5.3 Disurbnce Nure nd Chnge-poin Idenificion in EPC Box nd Krmer [] rgued h he feedbck conrol does no necessrily concel he nure of he disurbnce. However, someimes feedbck conrol does concel i. This is due o he objecive of EPC, nd we will explin i s follows. In SPC, when we monior sionry or nonsionry process, we cn eiher

11 Process Monioring nd Feed Forwrd Conrol for Procive Quliy Improvemen 6 monior he forecs errors hrough fiing ime series model, or monior he originl observions (disurbnce) direcly. For differen disurbnce models, no mehod is uniformly beer hn he oher. When here is no dvnge o using forecs errors, direcly monioring he rw d is much more convenien. However, when we need o djus such process in EPC, usully we cnno monior process wihou mking ny djusmen nd wi unil signl ppers; insed, we need o djus he process from he beginning (usully by feedbck conrol) nd ry o mke i on rge wih minimum vriion s much s possible, so we cn only observe he process under conrol nd he disurbnce cnno be direcly seen. When he size of he sep chnge is smll, or when here is only one chnge-poin, he signl is more likely o be hidden in he process fer djusmen. Therefore someimes he nure of he disurbnce migh be conceled by feedbck conrol. Forunely, process djusmen usully does no require he idenificion of he chnge-poin, for exmple, he EWMA forecs (feedbck conrol) filers ou he noise nd gives cler picure of how he rue men level vries [7]. In oher words, s long s he process under conrol hs smll vriion (i.e., MSEO), hen i is good conrol, nd usully we do no need o worry bou how he process looks like wihou conrol. However, in some priculr circumsnces we cn benefi subsnilly from he chnge-poin idenificion in EPC, nd his leds us o he nex discussion opic. 5.4 The Role of Process Monioring in he Inegrion of SPC nd EPC Process djusmen iself usully is insufficien if we hve high unceriny on he disurbnce model, for exmple, when he disurbnce hs muliple chnge-poins, he size of he sep chnge is unknown or under high unceriny. In such circumsnces, process djusmen needs o be combined wih process monioring o chieve higher efficiency. In model (4), we only know he occurrence probbiliy of he rndom sep chnge nd is prior disribuion, wihou knowing he exc chnge-poins locions. Conrol chr for process monioring cn be used s supplemenl ool for locing he possible chnge-poin, serving for he furher process improvemen funcion. This ide of process monioring serves for beer process djusmen" ws dvoced by Tucker [0], who rgued h he developmen of proper monioring funcion in he presence of feedforwrd nd/or feedbck conrol is (should be) cenrl reserch issue in he ques for coninuous quliy improvemen". Since in SPC, flse lrms nd filure of deecion lwys exis due o he ype I nd ype II errors, so similr problems exis in he inegrion of SPC nd EPC s well. Inuiively, wih lrger p, conrol ends o hve poenilly higher efficiency. 5.4 Disurbnce Decomposiion nd Cuscore Chrs µ nd r in our model (4), our qusi-feedbck feedforwrd While his disurbnce decomposiion viewpoin is relively new in EPC for feedforwrd conrol pplicion, similr ides cn be found on some disurbnce models in SPC. Someimes people hve experience on how he process will chnge so h he signl perns re niciped, nd Cuscore chr is devised o deec such niciped sysemic signls hidden in cerin noise for process monioring funcion. Box nd Luceno [] discussed vriey of differen signl nd noise combinions for Cuscore chr monioring. Very nurlly, he signl nd he noise moniored by Cuscore chr in SPC jus prllel he informive disurbnce vrible B, nd he bckground disurbnce e ' in EPC from our disurbnce decomposiion viewpoin.

12 6 Lihui Shi, nd Kilsh C. Kpur 5.5 Disurbnce Decomposiion nd Robus Prmeer Design One of our moivions on disurbnce decomposiion viewpoin in EPC comes from robus prmeer design. Now we explin how he rionle behind our feedforwrd conrol ide cn be unified in boh robus prmeer design nd EPC frmework. I is well known h robus prmeer design cn be successful only if he conrol fcors X re inercing wih he noise fcors e, which is ofen clled signl nd noise inercion". This implies h fer disurbnce decomposiion, he conrol fcors X should inerc wih he informive disurbnce vrible B s well s he noninformive noise e ', while B nd e' re independen of ech oher. I cn be proved h hese condiions re sisfied in our disurbnce decomposiion model s well. We only invesige model () here, since model () nd model (3) cn be similrly jusified. We show h X inercs wih nd he IMA noise. Since feedforwrd conrol is pplied o compense he effec of, so differen levels of cll for differen X ), wih conrol equion gx = m = ( + ε ), djusmen moun x (equivlenly hus X definiely inercs wih. Similrly feedbck conrol is inended o compense he IMA process, wih conrol equion gx = % z, where he EWMA forecs z% is funcion of he curren nd ll he previous z 's, so x (equivlenly X ) nd z re inercing o ech oher s well. Finlly, since model () is n ddiive model, so he decomposed informive disurbnce vrible nd he IMA noninformive noise re independen of ech oher. 6. Conclusions In his pper, procive feedforwrd conrol is recommended o be combined wih he rdiionl recive feedbck conrol for process improvemen, nd he fesibiliy condiion of feedforwrd conrol is illusred from disurbnce decomposiion viewpoin. Some disurbnce models, including he periodic shif models nd rndom sep chnge models re discussed, nd heir feedbck nd feedforwrd conrol equions re proposed. Some issues on process monioring, feedbck conrol nd feedforwrd conrol re discussed, which provide some insighs on he necessiy of feedforwrd conrol, he relionship beween process monioring nd process djusmen re invesiged, nd he rionle for our disurbnce decomposiion viewpoin is furher illusred. References [] Box, G. E. P., nd Luceno A. Sisicl Conrol by Monioring nd Feedbck Adjusmen, Wiley, New York, 997. [] Box, G. E. P., nd Krmer T. Sisicl Process Monioring nd Feedbck Adjusmen-A Discussion, Technomerics 99; 34(3): [3] Koonz, H., nd Brdspies R. W. Mnging Through Feedforwrd Conrol, Fuure-Direced View, Business Horizons 97; June: [4] Box, G. E. P., nd Jenkins G. M. Some Sisicl Aspecs of Adpive Opimizion nd Conrol, Journl of he Royl Sisicl Sociey, Series B, 96; 4():

13 Process Monioring nd Feed Forwrd Conrol for Procive Quliy Improvemen 63 [5] McGregor, J. F. A Differen View of he Funnel Experimen, Journl of Quliy Technology, 990; (4): [6] Vnder, Wiel S. A. Monioring Processs Th Wnder Using Inegred Moving Averge Models, Technomerics 996; 38(): [7] Luceno, A. Performnce of Discree Feedbck Adjusmen Schemes Wih Ded Bnd, Under Sionry Versus Nonsionry Sochsic Disurbnce, Technomerics 998; 40(3): [8] Tsung, F., Wu H., nd Nir V. N. On he Efficiency nd Robusness of Discree Proporionl -Inegrl Conrol Schemes, Technomerics 998; 40(3): 4-. [9] Deming, W. E. Ou of Crisis, Cmbridge, MA, 986. [0] Box, G. E. P., Jenkins G. M., nd Reinsel G. C. Time Series Anlysis, Forecsing nd Conrol, Prenice Hll (3rd ediion), NJ, 994. [] Del, Csillo E. Closed-Loop Disurbnce Idenificion nd Conroller Tuning for Discree Mnufcuring Processes, Technomerics 00; 44(5): [] Vnder, Wiel S. A., Tucker W. T., Fulin F. W., nd Dognksoy N. Algorihmic Sisicl Process Conrol: Conceps nd n Applicion, Technomerics 99; 34(3): [3] Tsimyrzis, P., nd Hwkins D.M. Byesin Srup Phse Men Monioring of n Auocorreled Process Th is Subjec o Rndom Sized Jumps, Technomerics 00; 5(4): [4] Box, G. E. P., nd Luceno A. Feedforwrd s Supplemen o Feedbck Adjusmen in Allowing for Feedsock Chnges, Journl of Applied Sisics 00; 9(8): [5] Shi, L., nd Kpur K. C., (0) A Synhesis of Feedbck nd Feedforwrd Conrol for Sionry nd Nonsionry Disurbnce Models, unpublished mnuscrip. [6] Nembhrd H. B., nd Vlverde-Venur R. Cuscore Sisics o Monior Nonsionry Sysem, Quliy nd Relibiliy Engineering Inernionl 007; 3(): [7] Chen, A., nd Elsyed E. Design nd Performnce Anlysis of he Exponenilly Weighed Moving Averge Esime for Processes Subjec o Rndom Sep Chnges, Technomerics 00; 44(4): [8] Tsimyrzis, P., nd Hwkins D.M. A Byesin Scheme o Deec Chnges in he Men of Shor-Run Process, Technomerics 005; 47(4): [9] Muh, J. F. Opiml Properies of Exponenilly Weighed Forecss of Time Series Wih Permnen nd Trnsiory Componens. Journl of he Americn Sisicl Associion 960; 55(90): [0] Tucker, W.T. Discussion of Sisicl Process Monioring nd Feedbck Adjusmen-A Discussion", Technomerics 99; 34(3): Lihui Shi is reserch scienis Webrends Inc. in Sele. He received his Ph.D. in Indusril nd Sysems Engineering in he College of Engineering he Universiy of Wshingon in 0. He received M.A. degree in Sisics from Nnki Universiy in 007, M.S. degree in Indusril nd Sysems Engineering from Universiy of Wshingon in 009, lso M.S. degree in Sisics from Universiy of Wshingon in 0. His reserch ineress include sisicl process conrol, process djusmen, relibiliy heory, design of experimens, ec.

14 64 Lihui Shi, nd Kilsh C. Kpur Kilsh C. Kpur is Professor of Indusril nd Sysems Engineering, College of Engineering, Universiy of Wshingon. He hs served s he Direcor of Indusril Engineering from He ws Professor nd he Direcor of he School of Indusril Engineering, Universiy of Oklhom from Prof. Kpur received he Ph.D. degree (969) in Indusril Engineering from he Universiy of Cliforni, Berkeley. He hs co-uhored he book Relibiliy in Engineering Design, John Wiley & Sons, New York, in 977. He is Fellow of ASQ nd IIE nd is regisered professionl engineer.