Jiongha Jin 1 Dpartmnt o Indstrial and Oprations Enginring, Th Univrsity o Michigan, Ann Arbor, MI 48109-2117 -mail: jhjin@mich.d Hairi Go Dpartmnt o Systms and Indstrial Enginring, Th Univrsity o Arizona, Tcson, AZ 85721-0020 Shiy Zho Dparmnt o Indstrial and Systms Enginring, Univrsity o Wisconsin, Madison, WI 53706 Statistical Procss Control Basd Sprvisory Gnralizd Prdictiv Control o Thin Film Dposition Procsss This papr prsnts a sprvisory gnralizd prdictiv control (GPC) by combining GPC with statistical procss control (SPC) or th control o th thin ilm dposition procss. In th sprvisd GPC, th dposition procss is dscribd as an ARMAX modl or ach prodction rn and GPC is applid to th in sit thicknss-snsing data or thicknss control. Sprvisory stratgis, dvlopd rom SPC tchniqs, ar sd to monitor procss changs and stimat th distrbanc magnitds dring prodction. Basd on th SPC monitoring rslts, dirnt sprvisory stratgis ar sd to rvis th distrbanc modls and th control law in th GPC to achiv a satisactory control prormanc. A cas stdy is providd to dmonstrat th dvlopd mthodology. DOI: 10.1115/1.2114912 Kywords: ARMAX modl, prdictiv control, statistical procss control, sprvisory stratgis, thin ilm dposition 1 Introdction In rcnt yars, th nd o thin ilm dposition has incrasd signiicantly in coating manactring procsss. Among varios coating tchniqs, high throghpt within a coating manactring procss can b achivd by dpositing a thin ilm into a larg ara o continos roll-by-roll lxibl sbstrats. Althogh th rcnt dvlopmnt o in sit snsing provids an opportnity or onlin masring o thin ilm thicknss, th complxity o th sorc sion, th hat dissipation, and th sorc consmption within th prodction rn, and th long dad tim prsnt in a continos procss, thicknss niormity control is still a critical challnging rsarch iss in procss dvlopmnt. This papr prsnts a sprvisory gnralizd prdictiv control GPC stratgy by combining th GPC with statistical procss control SPC or th thin ilm dposition procss control. An ovrviw o th thin ilm dposition procss is prsntd in Sc. 2, whr th thin ilm dposition chambr strctr, procss variabls, variation sorcs, crrnt control stratgy, and limitations ar discssd. In Sc. 3, th ramwork o th sprvisory GPC is proposd with th dtaild discssion on th procss modling, GPC dsign, SPC monitoring, and sprvisory stratgis. A cas stdy basd on prodction data is givn in Sc. 4. Finally, th smmary and rcommndations ar givn in Sc. 5. 2 Ovrviw o Charactristics o Thin Film Dposition Procsss 2.1 Thin Film Dposition Chambr Strctr. Th thin ilm dposition procss is normally condctd in a vacm chambr. A lxibl sbstrat is continosly transportd throgh svral dposition zons at a constant lx. With an appropriat dsign o ach sorc location, th sbstrat motion crats a controllabl lx proil at th sbstrat. Figr 1 shows th locations o sion sorcs in th chambr. Th sorc is illd into 1 To whom corrspondnc shold b addrssd. Contribtd by th Manactring Enginring Division o ASME or pblication in th JOURNAL OF MANUFACTURING SCIENCE AND ENGINEERING. Manscript rcivd Sptmbr 5, 2003; inal manscript rcivd Dcmbr 15, 2004. Rviw condctd by C. J. Li. an inslatd sorc containr, calld th sorc cll, with svral nozzls on th top o th sorc containr or vaporation. In gnral, tmpratr is a critical actor acting th sion rat and vaporation dposition in thin ilm dposition procsss. Th rqirmnt or a thin ilm dposition procss is to provid a high rat, simltanos dposition o matrials to th sbstrat srac and to orm a dns stoichiomtrical, wll-adhrnt ilm on th sbstrat. Larg ara dposition and ast sbstrat moving spd ar dsirabl in a practical manactring procss in ordr to achiv high prodction throghpt. Howvr, it lads to a challnging iss o rdcing th loss prodction bcas o ilm nonniormity as th dposition progrsss across both th width and lngth o th sbstrat. Th cass o ilm thicknss variations ar xtrmly complx, as thy ar actd by sorc dsign, hat dissipation to th srronding parts, and prodction conditions. 2.2 Prodct Qality Monitoring and Procss Control. For prodct qality control, th convntional SPC procdr is to condct a post-qality analysis by sing an o-lin masrmnt instrmnt to analyz th ilm layr strctr and thicknss. Bcas th major procss actors impacting th thin ilm qality ar sorc tmpratr and/or prssr, an appropriat control o ths actors is sd to compnsat or nanticipatd procss distrbancs dring prodction. Exampls o nanticipatd sorc variations ar cll matrial proprty, rbilding and rinstallation o sorc clls btwn rns, impritis in sorcs and dbris in a chambr, consmption o matrials ovr prodction tim, changs in charactristics o sorc hat dissipation to sbstrat and chambr, dgradation o sorc hatrs, tc. As a rslt, th ilm thicknss may vary vn whn th sorc tmpratr or prssr is maintaind at th sam targt stting points. In addition, it is xtrmly diiclt to modl th intraction o th sorc vaporation procss with othr procss variabls, sch as nvironmntal tmpratr and varios distrbancs. Ths, th compnsation o distrbanc cannot b achivd throgh ithr an atomatic dorward control or an o-lin prstting o th sorc tmpratr or prssr. In gnral, it is highly dsirabl or a manactring procss to hav th capability o ral-tim snsing and dtcting procss changs and o making a corrsponding procss adjstmnt to prvnt or rdc dctiv prodct loss ovr prodction. This Jornal o Manactring Scinc and Enginring FEBRUARY 2006, Vol. 128 / 315 Copyright 2006 by ASME
Fig. 1 Chambr strctr o thin ilm dposition procss iss is mor critical or a continos dynamic manactring procss whn thr is a nd to accommodat nanticipatd procss distrbanc, variability, or drits dring prodction tim. Th rcnt dvlopmnt o in sit thicknss snsors has providd a grat opportnity o achiving onlin monitoring and atomatic procss control or thin ilm dposition. 2.3 Procss Variabls and Procss Variations 2.3.1 Inpts and Otpts in a Procss Modl. In ordr to dvlop an atomatic dback controllr in a thin ilm procss, a procss modl is rqird to dscrib th rlationship o th in sit masrabl prodct qality and th controllabl procss variabls. In gnral, atr th procss stp variabls sch as chambr prssr, chambr nvironmntal tmpratr, and wb moving spd achiv thir targt vals, th sion sorc tmpratr is th most snsitiv procss variabl dirctly acting th physical sion proprtis dring th prodction rn. Th sorc tmpratr will b sd as th controllabl inpts o th procss modl. Th otpt o th procss modl is th dposition ilm thicknss. 2.3.2 Dad Tim Variation and Uncrtainty. Th thin ilm dposition procss is a continos manactring procss. In addition to considring th dynamic proprty o th sorc vaporization, th dad tim rom th tmpratr adjstmnt to th in sit thicknss masrmnt shold b considrd in th procss modling and controllr dsign. As shown in Fig. 1, th thicknss masrmnt is gnrally takn at th nd o th prodction lin. Th total systm dad tim inclds th dposition tim and th masrmnt dlay d to th tim ndd or th sbstrat to mov rom th sorc location to th thicknss masrmnt. Basd on th nominal moving spd o th sbstrat and th distanc btwn sorc and in sit snsors, th masrmntindcd dad tim can b calclatd or ach sorc cll. Howvr, th thicknss masrmnt will rlct th compondd ct o mltipl clls i on sorc has svral clls locatd at dirnt locations in a chambr, which wold lad to dirnt dad tims or dirnt clls. In addition, th sbstrat spd also varis arond th nominal stting val ovr a prodction rn, which acts both th ilm growth rats and sbstrat moving tim to th thicknss masrmnt. Th invitabl variations o ths actors will cas ncrtaintis in th dad tim and stimation rrors dring th prodction, which pos a challng in th dvlopmnt o a dback controllr. 2.3.3 Procss Distrbancs Within On Prodction Rn. Dring prodction, thr always xist signiicant and nanticipatd distrbancs lading to procss variation within th rn. For xampl, th tmpratr dissipation to th sbstrat and chambr and th consmption o sorc will lad to a ramp drit distrbanc ovr th prodction rn. A spik implsiv distrbanc is normally obsrvd bcas o rrors in snsor masrmnt or data acqisition systm. It is known that th control actions or a high-rqncy implsiv distrbanc and lowr rqncy linar drit distrbanc shold b dirnt. An inpropriat control action will likly cas a spitting problm in th procss. 2.4 Crrnt Control Stratgy and Its Limitations. Th crrnt thin ilm dposition control has two stps in ach rn: a manal procss control dring th initial start-p o prodction and an atomatic dback control dring th continos prodction. In ach nw prodction rn, a manal control is prormd to incras th tmpratr throgh svral stps. Atr th systm rspons i.., th ilm thicknss is clos to th targt point, an atomatic controllr is ngagd to d back th thicknss-snsing data in ordr to maintain th dpositd ilm thicknss at th targt. Th xisting controllr is a prdsignd tim-invariant controllr basd on a ixd modl rom o-lin idntiication. Althogh sch a controllr is simpl in its implmntation, dirnt distrbanc strctrs ar not considrd in th controllr dsign. 3 Sprvisory Prdictiv Control Stratgy A sprvisory control stratgy is proposd in this papr or thin ilm dposition procss control. Th ramwork o th sprvisory GPC is shown in Fig. 2. Dtaild discssions on ach modl ar providd in Scs. 3.1 3.3 3.1 Procss Modling Using ARMAX Modl. In th start-p priod o ach prodction rn, th tmpratr o th chambr is manally incrasd stp by stp, with ach stp having at last ight sampling intrvals. Thos inpts, togthr with th corr- Fig. 2 Th strctr o th sprvisory GPC 316 / Vol. 128, FEBRUARY 2006 Transactions o th ASME
Fig. 3 Initial prodction rn data o stp inpt and rspons sponding ilm thicknss rsponss, ar sd to modl ach rn o th thin ilm dposition procss. Th rn-to-rn variation is rlctd in th modl paramtr chang. An ARMAX n a,n b,n c,n d,d modl with distrbanc is sd to dscrib th thin ilm dposition procss, Aqy t = Bq t d + Cq t + Dq t 1 whr y t and t d ar th systm rspons o th ilm thicknss at tim t and control inpt o th corrsponding sorc tmpratr at tim t-d, rspctivly, and d is th dad tim o th procss, Aq, Bq, Cq, and Dq ar polynomial nctions with ordrs n a, n b, n c, and n d, rspctivly, and q 1 is th backshit oprator with q 1 y t =y t 1. Hr, Aq=1+a 1 q 1 +...+a na q n a, Bq =b 1 q 1 +...+b nb q n b, Cq=c 1 q 1 +...+c nc q n c, and Dq=1 +d 1 q 1 +...+d nd q n d. t rprsnts th distrbancs that ar possibly xisting in th procss as ithr a spik, a man shit, or a linar drit nction; t is an indpndnt idntically distribtd IID zro-man sqnc o modling rsidals with t N0, 2. Th modl paramtrs in Eq. 1 can b stimatd sing th last-sqars mthod i Dq is known. In th cas o nknown Dq, th two-stag mthods dscribd in Sc. 10.4 o Ljng 1 shold b sd. In or cas stdy discssd in Sc. 4, th ARX modl strctr with Dq=1 is stimatd by sing th last-sqars mthod. Eqation 1 can also b rwrittn as y t = Bq Aq t d + Cq Aq t + Dq Aq t 2 m Dnot Bq/Aq= j=0 g j q j m, Cq/Aq= j=0 g j q j, and m Dq/Aq= j=0 g j q j ; and m, m, and m ar th maximm ordr o ach corrsponding itm. Thy ar dtrmind basd on th impls rspons nction g s j rom ach inpt trm to th otpt y. That is, whn jm s, g s j 0 s=,,. An xampl o dtrmining m is givn in th cas stdy in Sc. 4. Eqation 2 can b rwrittn as y t = mint d 1,m j=0 g j t j d + mint 1,m j=0 g j t j + mint 1,m j=0 g j t j 3 dsign is th combind ct o svral sorcs. Morovr, th wb spd variation also lads to dad-tim variation. Withot accratly knowing th dad tim, it is diiclt to dsign a controllr sing th proportional, intgral, and drivativ PID or minimm varianc control stratgy. Ths, it is dsirabl to dsign a gnralizd prdictiv controllr to considr th possibl rang o mltipl dad-tim vals. Th scond rason o sing GPC is that th thin ilm dposition procss is gnrally a nonminimm phas systm dscribd by an ARMAX modl. Ths, signiicant challngs xist in th controllr dsign or PID or minimm varianc control stratgis. Th third rason o sing GPC is its capability o intgrating sprvisory stratgis or compnsating or nknown and tim-varying distrbanc pattrns: Dring th prodction, dirnt distrbanc pattrns sch as a man shit, a slow drit, or a spik may occr in th thin ilm dposition procss. An adaptiv control stratgy shold b dsignd to rlct th dirnc in distrbanc modls. In rcnt yars, rsarch achivmnts on sing an xponntial wighd moving avrag EWMA mthod to stimat th tim-varying man shit or slow linar drit hav dmonstratd grat sccss or a rn-to-rn controllr dsign 5,6. Howvr, most o ths mthodologis assm that th distrbancs xist dring all prodction tim with prdind distrbanc modl strctrs. Ths, EWMA modls ar sd to continosly stimat th nknown or tim-varying distrbanc modl paramtrs. Sachs t al. 7 invstigatd th rnto-rn controllr dsign by charactrizing th rn-to-rn variations as two dirnt mods: a slow mod and a ast mod. In contrast to othr modls, Sachs t al. assmd that a slow mod distrbanc xists ovr all prodction tim, which is stimatd by EWMA and continosly compnsatd in th controllr. Th compnsation o ast mod distrbanc is only condctd whn a larg 3.2 Gnralizd Prdictiv Control. A gnralizd prdictiv control GPC was irst proposd by Clark t al. 2, and th proprtis o GPC ar rthr stdid by Clark and Mohtadi 3 or a st o continos chmical procss control problms. Rcntly, GPC has also bn sd in a smicondctor control applications 4. Th irst rason to choos GPC or th thin ilm dposition procss control in this papr is th ncrtaintis in dadtim stimation. As dscribd in th procss ovrviw, th singl thicknss otpt y t is a compondd ct o svral sorc clls locatd at dirnt placs. Sinc thr is no snsor to masr ach cll sion, th dad tim cannot b dtrmind or ach cll sparatly. As a rslt, th dad tim sd in th controllr Fig. 4 AIC ndr dirnt modl strctrs Jornal o Manactring Scinc and Enginring FEBRUARY 2006, Vol. 128 / 317
Fig. 5 Comparison o th modling rslts with th ral procss rspons shit is dtctd sing an X-bar monitoring control chart. Thror, th SPC monitoring tchniq provids a sprvisory stratgy to onlin dtrmin th distrbanc modl strctr. Howvr, all o ths rn-to-rn control mthodologis ar dvlopd or a minimal varianc controllr, which is limitd by assming that th procss has a ixd dad tim. Thror, thy cannot b applid in th thin ilm dposition procss in which a larg and ncrtain dad tim xists. In this papr, or objctiv is to intgrat SPC with APC atomatic procss control to dvlop a sprvisory prdictiv controllr. Th nd o intgrating a sprvisory stratgy with a prdictiv controllr is mor critical than in a minimal varianc controllr bcas mor stps o distrbanc prdiction will b sd in th prdictiv controllr. A largr prdiction rror is sally gnratd whn th prdiction stps ar incrasd bcas o modling ncrtainty. Thror, accrat idntiication o th distrbanc modl strctr is xtrmly important in th dsign o a prdictiv controllr, spcially or a procss with larg dad tim. Considring all ths aspcts, a GPC stratgy is adoptd or th thin ilm dposition procss control. It is robst to th dad-tim variability and stimation ncrtainty, is ctiv in handling nonminimm phas systms, and has th ability to combin SPC with sprvisory stratgis. Th objctiv nction o th GPC control is dind as N+d J = i=d N qiŷ t+it y * t+i 2 + 2 ri t+i * whr y t+i is th rrnc or dsird targt otpt at tim t+i, qi and ri ar th wighting coicints, N is th sliding horizon or th otpt prdiction and inpt sris, and ŷ t+it is th ith 4 stp-ahad prdiction mad at tim t. This is obtaind rom Eq. 3 by taking th conditional xpctation i d ŷ t+it = Ey t+it = g j t+i d j + j=0 mint+i 1,m + g j t+i j + j=0 mint+i d 1,m j=i d+1 mint+i 1,m j=i g j t+i j g j t+i d j t+i j j=0,1,...,i 1 is th prdiction o th dtrministic distrbanc nction mad at tim t according to th idntiid distrbanc nction t at tim t. For th random rror t, th conditional xpctation ndr th crrnt tim t is E t+it =0 and E t it = t i i0. Ths, w hav E t+i 1 j=0 g j t+i j = t+i 1 j=i g j t+i j. Th basic concpts o th GPC stratgy can b smmarizd as ollows: At crrnt tim t, an optimal control law t is obtaind by solving th optimization problm dind in Eq. 4. By sing th sliding horizon window o ach o th N+1 stp prdictions, th control problm is simpliid rom a dynamic programing problm to a static optimization problm 8. In ordr to solv th prdictiv control problm, th objctiv nction in Eq. 4 is qivalntly rprsntd by a vctor and matrix orms as J = Ŷ Y * T QŶ Y * + U T 1 RU 1 6 whr Q=diagqN+d,qN+d 1,...,qd and R =diagrn,rn 1,...,r0 ar diagonal matrics o wighting coicints. Th vctor Ŷ rprsnts all N+1 stp prdictions in 5 Fig. 6 Impls and stp rspons o th thin ilm dposition procss modl 318 / Vol. 128, FEBRUARY 2006 Transactions o th ASME
Fig. 7 A simlatd control prormanc th sliding window, which is rprsntd basd on Eq. 5 as Ŷ = G 1 U 1 + G 0 U 0 + G F + G E In this rprsntation, vctor Ŷ =ŷ t+d+nt,ŷ t+d+n 1t,..., ŷ t+d+it,...,ŷ t+dt T rprsnts all th prdictd otpts basd on tim t within th sliding window. Corrsponding to Ŷ, th optimal op tr control inpt vctor U 1 is dnotd as U 1 = t+n, t+n 1,..., t+i,..., t T obtaind in Eq. 9. It shold b notd that only a singl t in Û op 1 will b sd to control th procss at tim t. Similarly, rom Eq. 5, th prviosly sd control inpt vctor is dnotd as U 0 = t 1,..., i,..., t mint 1,m T, which is ordrd backward rom tim t 1 to th maximm stps o m as dind in Eq. 3. Th ct o ths control inpts on th tr systm otpts is dtrmind by th dynamic systm mmory on th historical control inpts. Vctor Fˆ = t+d+nt, t+d+n 1t,..., t+d+n mint+d+n 1,m T is sd to rprsnt th dtrministic distrbancs rom tim t+d+n backward to th maximm stps o m as dind in Eq. 3. Vctor Ê = t, t 1,..., t mint 1,m d T is sd to dnot th random rsidals. Ths rsidals can b calclatd p to th crrnt tim t sing Eqs. 32, 34, and 35, which corrspond to a man shit, a linar drit, and a spik distrbanc, rspctivly. Ths qations will b discssd in Sc. 3.3.3. In th cas whr m d 0, w st E=0, which mans th historical random distrbancs do not inlnc th tr otpt. Corrsponding to ach o th vctors U 1, U 0, Fˆ, and Ê sd in Eq. 7, G 1, G 0, G, and G ar 7 sd to rlct th dirnt dynamic rspons wights, which ar dind basd on Eqs. 5 and 7 G 1 =g0 g 1 0 g 0 g N g N 1 0 0 g 0 N+1N+1 rlcts how th tr control inpt U 1 acts th otpt Ŷ; g N G 0 =gn+1 g N+2 g N+1 g mint 1,m +N g mint 1,m +N 1 g 1 g 2 g mint 1,m N+1mint 1,m rlcts th ct o th dynamic systm mmory o th historical control inpt U 0 on th otpts Ŷ; G =g0 g 1 0 g 0 g mind+t+n 1,m g mind+t+n 1,m 1 0 0 g mind+t+n 1,m N N+1mind+t+N 1,m +1 rlcts th ct o th dtrministic distrbanc F on th otpt Ŷ; = gn+d G g N+d 1 g N+d+1 g N+d g mint+d 1,m +N g mint+d 1,m +N 1 g d g d+1 g mint+d 1,m N+1max0,mint,m d+1 rlcts th ct o th dynamic systm mmory o th historical random rror E on th tr otpt Ŷ, which is ctiv only whn m d0 holds. It shold also b notd that whn th prodction tim is larg nogh, th dimnsion o th matrics G 0, G, and G will b constraind only by th maximm ordr o ach corrsponding impls nction as dind in Eq. 3. By solving th optimization problm with dj/du 1 =0, on has dj du 1 =2G 1 T QG 1 U 1 + G 0 U 0 + G F + G E Y * + RU 1 =0 8 Th optimal prdictiv control law U 1 op, is obtaind by solving Eq. 8 as U 1 op = G 1 T QG 1 + R 1 G 1 T QG 0 U 0 + G F + G E Y * Althogh th vctor o all tr inpts rom t to t+n is providd in U 1 op, only t will b sd to control th procss at tim t+1. At th nxt tim stp t+1, anothr optimization will b condctd basd on th nw obsrvation o y t+1, which is sd to obtain a nw optimal control vctor U 1 op = t+1, t+2,..., t+n+1. Th nwly obtaind t+1 will b sd as a control inpt at tim t+2. This stratgy is rpatd ntil th nd o th prodction rn. Rmark 1. Prdiction Horizon N Slction. Th principl in th 9 Jornal o Manactring Scinc and Enginring FEBRUARY 2006, Vol. 128 / 319
Fig. 8 Man-shit distrbanc nction and CUSUM monitoring control chart sliding horizon slction is to hav its lowr limit b qal to th dad tim o th systm. I th dad tim is stimatd with ncrtainty or th dad tim varis dring th prodction, thn th sliding horizon shold b st as th lowr bond o th dad-tim stimation. Th ppr limit o th sliding horizon shold b smallr than, or qal to, th sttling tim o th systm rsponss. Th sttling tim T 0 is dind as th tim o which th absolt val o th impls rspons o th procss is always blow a small thrshold val o 0 as tim gos to ininity. T 0 can b obtaind by invstigating th impls rspons o th modl as g j 0 j T 0 10 Th val o 0 is dtrmind by th modling accracy rqirmnt. An xampl o how to gt T 0 is givn in th cas stdy in Sc. 4. In addition, a smallr val or N shold b slctd i mor signiicant distrbancs or modl ncrtainty xist in th procss. Rmark 2. Wighting Coicints R Slction. Th paramtr R inlncs th procss stability and tracking rror. Corrct slction o an R val is important to rach a satisactory prormanc or controlling th nonminimm phas procsss. In gnral, a largr R val lads to a mor stabl systm, lss ovrshoot, slowr rspons, and largr tracking rrors. Ths, i a distrbanc xists, a smallr R val is prrrd ndr th constraints o stability and accptabl ovrshoot. In this way, a ast tracking prormanc can b achivd, spcially or a ast drit distrbanc. 3.3 Sprvisory Stratgis. Varios distrbancs may xist in th thin ilm dposition procss as dscribd in Sc. 2. Thr typical distrbanc pattrns man shit, linar drit or ramp, and spik will b stdid in th dvlopmnt o a sprvisory stratgy in th GPC dsign. 3.3.1 CUSUM Charts or Dtction and Estimation o Shit and Drit Distrbancs. A statistical cmlativ-sm CUSUM monitoring chart is sd to dtct and stimat th distrbanc t whn it is a man-shit or linar-drit nction. Th stimatd nction t is sd in Eq. 9 to rvis th tr control adjstmnt to compnsat or corrsponding distrbancs. In th ollowing sction, a bri rviw o th CUSUM chart is givn. A dtaild discssion o th CUSUM chart tchniq can b ond in Montgomry 9. A CUSUM chart is constrctd by positiv and ngativ CUSUM statistics, calclatd as S H t =max0,x t 0 x + K + S H t 1 11 S L t =max0, x 0 K x t + S L t 1 12 whr x t is th monitoring variabl, which has a man o x 0 ndr th in-control condition. Th starting vals o S H t and S L t ar S H 0=S L 0=0. K=/2 and is th man shit o x t to b dtctd. An ot-o-control point is indicatd at tim k i S H kh or S L kh, whr h is a dsign paramtr chosn as th control limit o a CUSUM chart. In gnral, th vals o h and K dtrmin CUSUM chart snsitivity to a distrbanc and inlnc th avrag rn lngth ARL 9. I an ot o control point is obsrvd at tim k, N + or N is th nmbr o consctiv sampls or which th ppr-sid CUSUM S H k or lowr-sid CUSUM S L k has had a positiv val. Ths, th tim at which th man shit is dtctd is s+1, and th occrrnc tim is s=k N + or an pward shit or s=k N or a downward shit. Th sm o man shits ovr N + or N stps is stimatd by Fig. 9 Comparison o control prormanc ndr man-shit distrbancs 320 / Vol. 128, FEBRUARY 2006 Transactions o th ASME
Fig. 10 Comparison o modl rsidal rrors ndr man shit i S H k h, N + x i = N + x 0 + K + S H k i=1 13 N i S L k h x i = N x 0 + K S L k i=1 In th dvlopd control stratgy, thr ar two dirnt CUSUM charts sd to monitor a man shit and a linar drit. 3.3.1.1 CUSUM chart or man shit distrbanc monitoring. In ordr to dtct and stimat th man shit, th monitoring variabl x t is dind as t as ollows: t = Aq Dq y t Bq Dq t d = Cq Dq t + t 14 This rlationship ollows dirctly rom Eq. 1. Using this, t can b dirctly calclatd or ach nwly obsrvd otpt y t and control inpt t d. Th ollowing hypothsis tst is sd with a CUSUM chart to dtct whthr thr xists a nonzro man shit o t : H 0 : t =E t = 0; whn t =0 H 1 : t =E t 0; whn t = 15 So, taking th xpctation o Eq. 14, w hav: t =E t = Cq n Dq t = I i t i 16 whr n is th maximm ordr o th invrs nction I i. This nction is obtaind by xpanding Cq/Dq with th backward oprator q 1. By sbstitting th condition o Eq. 15 into Eq. 16, th tst is H 0, t =0 H 1, t = minn s,n 17 whr N s is th nmbr o stps o th man shit occrring at tim t N s. Ths, i ithr S H kh or S L kh is dtctd in th CUSUM chart, th man shit can b stimatd basd on Eqs. 13 and 17 as i S H k h, i S L k h, N = N + K + S H k + j=1 N = N K + S L k j=1 minj,n minj,n 1 1 18 3.3.1.2 CUSUM chart or linar-drit distrbanc monitoring. Whn monitoring and stimating a linar-drit distrbanc, x t is dind as t as t = t t 1 19 From Eqs. 14 and 16, it can b sn that t = I 0 t + I 1 t 1 + + I n t n + t 20 t 1 = I 0 t 1 + I 1 t 2 + + I n t 1 n + t 1 21 Sbtracting Eq. 20 rom Eq. 21, w hav t = I 0 t t 1 + I 1 t 1 t 2 + + I n t n t 1 n + t t 1 22 Ths, th ollowing hypothsis tst is sd with th CUSUM chart to dtct whthr thr xists a nonzro slop o th linar drit nction t : H 0 : t =E t = 0; whn t t 1 =0 t 1 23 H 1 : t =E t 0; whn t t 1 = By taking th xpctation o Eq. 22 and sbstitting it into th condition o Eq. 23, w hav H 0, t =0 H 1, minn t =,n 24 whr N is th nmbr o stps o th linar drit occrring at tim t N. Ths, i ithr S H kh or S L kh is dtctd in th CUSUM chart, th slop o linar drit as i S H k h, i S L k h, N = N + K + S H k + j=1 N = N K + S L k j=1 minj,n minj,n 1 1 25 3.3.2 X-bar Chart or th Dtction and Estimation o a Spik Distrbanc. In a thin ilm dposition procss, a spik signal may b obsrvd rom th thicknss masrmnts. In practic, a spik signal is otn d to snsor rrors. Ths, it is dsirabl to dtct and rmov th snsing rrors rom th systm rspons, so that th controllr will not provid wrong dback in th procss control. For th thin ilm dposition procss modl, th rlationship btwn a singl spik rror at tim s and th tr systm otpt y t 0 withot spik rror can b modld as Jornal o Manactring Scinc and Enginring FEBRUARY 2006, Vol. 128 / 321
Fig. 11 CUSUM monitoring control chart or dtcting a drit y 0 t = y t t s 26 whr t s is th Kronckr nction, y t is th systm otpt masrmnt inclding th spik rror at tim s, and is th magnitd o th spik. So, whn thr is no procss chang t at t=s, w s rom Eq. 14 that t=s = Aq Dq y 0 t + t s Bq Dq t d = s + By taking th xpctation on Eq. 27, w hav 27 E t = whn t = s 28 0 othrwis From Eq. 28, An X-bar control chart can b dsignd to dtct a larg spik rror gratr than 3 2. Ths, th control limits o X-bar chart ar dind as UCL 2 = 3 2 CL = 0 LCL = 3 29 whr UCL LCL rprsnts th ppr lowr control limits, CL is th cntral lin o th control chart. Sinc a spik sally occrs only at on sampl intrval, a dcision rl is ndd to idntiy spik rrors, which is A spik occrs at tim s i s is idntiid as ot o control, bt all s +i i= b, b+1,..., 1,1,2,...,b ar idntiid as in control. Hr b is th dtction window dtrmining th monitoring rang. I an ot o control condition is dtctd, both th tim and magnitd o th spik will b providd to th GPC or rthr compnsation actions. Th xpctd magnitd o a spik is Et = whn t=s, and Et=0 whn ts. So, in a givn dtction window with lngth 2b, th stimatd magnitd o th spik is s+b t=s b,ts t ˆ = s 30 2b Ths, w s that this is an nbiasd stimator: Eˆ=Et s+b t/2b= 0=. E t=s b,ts 3.3.3 Control Law Rvision or Sprvisory Compnsation o Dtctd Distrbancs 3.3.3.1 Control law rvision ndr a man shit or linar drit distrbanc. A tim dlay always xists in dtcting a distrbanc sing a CUSUM chart. Assm that a man shit or drit occrs at tim indx s, which is dtctd at tim indx kks. Ths, a dtction dlay o k s is xprincd. In th nxt control law calclation or U t, tk, thr is a nd not only to rvis th tr prdiction o th distrbanc modl k+1k,..., k+d+nk basd on th dtctd distrbanc nction at k, bt also to rvis th distrbanc modl and rsidal rrors dring stk or s, s+1,..., k and s,..., k in ordr to rlct th ct o th distrbancs occrring at tim s. Whn a man shit actally occrs at tim s bt is dtctd at tim k, th rvisd dtrministic distrbanc modl sd in th control law o Eq. 9 is t = 0 t s 31 t s is calclatd by sing Eq. 18. Sbstitting t rom Eq. 31 into Eqs. 14 and 16, th rvisd rsidal is obtaind as t t s 32 t t = mint s,n i=1 s t k By sbstitting this rvisd t rom Eq. 32 and t rom Eq. 31 op into Eq. 9, th rvisd control law U 1 = t+n, t+n 1,..., t+i,..., t T can b obtaind, in which th singl t is sd to control th procss at tim t to compnsat th man shit. Similarly, whn a linar drit actally occrs at tim s bt is dtctd at tim k, th rvisd distrbanc modl sd in th control law o Eq. 9 is 0 t s t = 33 t s +1 t s and th rvisd rsidal is obtaind by sbstitting th t o Eq. 33 into Eqs. 14 and 16 = t t s t 34 t mint s,n i=1 t s +1 s t k By sbstitting this rvisd t o Eq. 34 and th t o Eq. 33 op into Eq. 9, th rvisd control law U 1 = t+n, t+n 1,..., t+i,..., t T can b obtaind, in which th singl t is sd to control th procss at tim t to compnsat th dtctd linar drit. 3.3.3.2 Control law rvision ndr a spik distrbanc. Ia spik d to a snsor rror is dtctd, thn it is dsirabl to rmov th spik data rom th calclation o th dback control. This can b achivd by rcalclating th rsidal sris t sd in th control law o Eq. 9 whn a spik is dtctd. Assm that a spik occrring at tim s has bn dtctd at tim k=s+b. In this cas, th rvisd t is obtaind basd on Eq. 27 as t = t t s 35 t t = s By sbstitting th rvisd t o Eq. 35 into Eq. 9, th rvisd control law U op 1 = t+n, t+n 1,..., t+i,..., t T can b obtaind. 322 / Vol. 128, FEBRUARY 2006 Transactions o th ASME
Fig. 12 Control prormanc comparison ndr drit distrbancs 4 A Cas Stdy 4.1 Thin Film Dposition Procss Modling. Ral prodction data rom a thin ilm dposition procss wr collctd or th cas stdy. In th data st, th stp inpt signals o th sorc tmpratr as shown in Fig. 3a wr applid dring th procss start-p priod, and th procss rspons is shown in Fig. 3b. Svnty data points wr collctd or th procss modling. Th sampling intrval sd in th data acqisition is 2 min. Basd on procss nginring knowldg, th thin ilm procss ndr th normal oprating condition is modld by an ARX modl with Cq=Dq=1. Th ordr o Aq, Bq, and dlay stp d, dnotd as n a,n b,d, will b qantitativly dtrmind sing an inormation-thortic approach basd on AIC Akaik s inormation critrion 10,11, i.., th smallst AIC val indicats th bst-it modl among all compard modl candidats. In this cas stdy, th possibl modl strctrs incld th ollowing 14 combinations: 11 d, 21 d, 31 d, 41 d, 22 d, 32 d, 42 d, 33 d, 43 d, 44 d, 55 d, 66 d, 77 d, and 88 d. Th possibl dad tims ar d=2,3,4,5,6. Figr 4 shows all AIC vals o ths 14 modls ndr ach dad tim. It is clar that th modl o 5 55modl strctr indx=11 has th smallst AIC val, which is considrd th bst-it modl among ths 14 modls. Atr slcting this bst-it modl, th statistical F tst is sd to rthr dtrmin whthr a lowr ordr modl n a 5,n b 5 can b sd. This statistical tsting is basd on whthr th sm o sqars o th modling rsidal rrors is signiicantly incrasd whn a lowr ordr is sd 12.AnF tst with th signiicanc lvl 5% is sd in th inal ordr dtrmination. Basd on th data st, th inal ARX modl strctr is 2 2 5 with th paramtrs as Aq=1 0.1297q 1 0.04919q 2, Bq=0.007376+0.004798q 1. Th rror trm sris is dtrmind with t N0,10 4. In ordr to vriy th modl accracy, Fig. 5a shows th comparison btwn th ral procss rspons and its on-stp-ahad prdiction, and Fig. 5b shows corrsponding rsidal rrors. It is clar that th idntiid modl has good track prormanc whn compard to th ral procss otpt. 4.2 Prdictiv Control. Th paramtrs sd in th GPC ar slctd Q=diag1,1,1,1, R=rdiag1,1,1,1, and r=10 7. Gnrally, th wight coicints Q o prdiction rror ar st to 1 and th wight coicints R o th control cost ar adjstd basd on th applications 8. In ordr or th systm to b abl to qickly compnsat or th procss chang, R is sally slctd as a rlativly small val ndr th controllr stability constraint. Hr, w slct a rasonabl val o r=10 7 basd on trial and rror. In ordr to calclat th optimal control inpt U op 1, th dimnsions o matrixs G 0, G, and G nd to b dtrmind. Ths dimnsions ar dtrmind by m, m, and m, rspctivly. An xampl o how to choos m basd on th systm stp rspons is illstratd in Fig. 6. In Fig. 6b, th sttling tim is dtrmind or th stp rspons to stay within ±1% 99% in this xampl o its inal stabl stp rspons 13. Ths, th thrshold 0 in Eq. 10 is qal to 1% o its inal stabl stp rspons. m is dtrmind by th nmbr o impls rsponss in Fig. 6a with thir absolt vals largr than 0. Th vals o m and m ar similarly dtrmind. Th width o sliding window is slctd as N=3. It is dtrmind basd on nginring knowldg o th dlay ncrtainty in th thin ilm procsss and m. In Eq. 9, th val o G 1 T QG 1 +R 1 G 1 T Q, which is dnotd as, is 135.1 105.1 61.5 39.9 0.315 134.9 105.0 61.5 = 0.17 0.414 134.9 105.1 0.073 0.17 0.315 135.1 A simlatd tracking control prormanc with th targt o 7.5 is shown in Fig. 7. From this igr, it can b sn that thr is a dlay o 10 sampls rom th starting o th prodction rn at sampl indx 1 to th procss otpt rspons. This 10-stp dlay coms rom a 5-stp dad tim atr th irst control inpt was addd at sampl indx 5. At t10, th otpt qickly achivs and maintains th targt val with a small variation. Th nois standard dviation sd in this simlation is 0.01, which is qal to th stimatd ˆ o th modling rsidal t. 4.3 Sprvisory Stratgis 4.3.1 Man-Shit Dtction and Compnsation. A CUSUM chart is dsignd to monitor a man shit o th procss. Th procss man shit is simlatd as = 1.5 = 0.015 with =0.01, which is addd to th procss at t30. Th dsign paramtrs o th CUSUM chart ar 0 =0, K=/2=0.0075, and h =5, =0.05. Figr 8a shows th man-shit distrbanc nction o t, and Fig. 8b shows th CUSUM monitoring control chart o t in th sprvisory GPC. It can b sn that atr th shit occrs at sampl 30, th irst ot-o-control point is dtctd at sampl k=37 with S L 37=0.0522h=0.05. Basd on Eq. 18 with n =1 and I o =1 in th modl, th shitd man at tim t =37 is stimatd as 37= K S L 37/N = 0.015 with N =7. Figr 8a shows th comparison o th stimatd mans shit and tr man shit ovr dirnt tim, in which th dottd lin o th stimatd man shit is vry clos to th dashd lin o th tr man shit xcpt or th dlay priod o th dtction. Atr dtcting th man shit, th sprvisory GPC will compnsat or it by sing th stimatd man shit. As shown in Fig. 9a, th controllr can qickly compnsat or th man-shit to bring th otpt back to th targt val. Howvr, withot compnsating or th man shit in th controllr, th man-shit distrbanc will lad to a man shit in th procss rspons as shown in Fig. 9b. It is also notd that th compnsation control is not lly applid to th systm ntil tim 42. This is d to th Jornal o Manactring Scinc and Enginring FEBRUARY 2006, Vol. 128 / 323
Fig. 13 Comparison o modl rrors ndr linar drit 5-stp dad tim atr th man shit is dtctd at t=37. Ths, rdcing th dad tim in th procss and having an arly dtction o th man shit will ctivly rdc th impact o th man-shit distrbanc. Th comparison o modl rrors dirncs btwn th onstp-ahad prdictions and th ral masrmnts o th procss ndr th man shit was also plottd in Fig. 10. 4.3.2 Drit Distrbanc Dtction and Compnsation. Todtct a linar drit distrbanc sing a CUSUM chart, w nd to monitor th random variabl t = t t 1, which has a standard dviation o = 2 =0.0141 ndr th in-control condition. Th simlatd drit has a slop o =1.5 =0.0212, which is addd to th procss at t30. Th paramtrs o th CUSUM chart ar dind as 0 =0, K=/2=0.0106, and h=4 =0.0566. Figr 11a shows th distrbanc signal o t, and Fig. 11b shows th CUSUM monitoring chart. It can b sn that th irst ot-ocontrol point is indicatd at tim k=33 with S L 33=0.0716 0.0566 and N =4. Similarly, basd on Eq. 25 with n =1 and I o =1 in th modl, th drit slop at tim t=33 is stimatd as 33= 0.0285. Figr 11a shows th comparison o th stimatd slop and th tr slop o th linar drit distrbanc ovr dirnt tim, in which th dottd lin o th stimatd slop is vry clos to th dashd lin o th tr slop xcpt or th dtction dlay priod. Th comparison o th control prormanc with and withot compnsation o th linar drit is shown in Fig. 12. As it can b sn rom Fig. 12a, atr a drit distrbanc was introdcd at t30, th procss otpt will xprinc a short priod o larg dviations rom th stting point. Howvr, atr dtcting th drit at t=33, th sprvisory GPC can qickly compnsat or th distrbancs at t38, bt th compnsation control is not lly applid to th systm ntil iv tim intrvals atr th drit is dtctd. Ths, an ctiv way to rdc th impact o distrbanc is to rdc th dad tim and dtct th drit arlir. Withot th sprvisory stratgis i.., no dtction o and compnsation or th drit, th controllr otpt has a linar drit as shown in Fig. 12b. Th comparison o modl rrors dirncs btwn th on-stp ahad prdictions and th ral masrmnts o th procss ndr a linar drit distrbanc was also plottd in Fig. 13. 4.3.3 Spik Distrbanc Dtction and Control Law Rvision. It is assmd that a spik, which lads to a 6% dviation in th otpt targt val, is introdcd to th systm at tim indx 30. Th control prormanc with th sprvision in GPC is shown in Fig. 14a. From this igr, it can b sn that a signiicant dviation rom th stting point occrs in th otpt rspons at data indx 35, which is d to th control raction or dback to th spik rror addd at t=30. Th iv-stp dlay is d to th dad tim o th procss. Undr th sprvision, an X-bar control chart is sd to dtct th spik distrbanc by sing th dcision rl dind in Sc. 3.3.2 b=2 is sd in th rl. Ths, th control chart dtcts th spik rror at k=37, which indicats a two-stp dlay atr th spik is shown on th rspons at t=35. Onc th spik is dtctd, a sprvisory corrctiv action is takn to rmov th spik cts sing Eq. 35. It can b sn that th otpt tracking rrors ar rdcd signiicantly at t=37 atr rmoving th spik in th control law calclation. Howvr, d to th two-stp dlay in th spik dtction, thr is no compnsation ort or th irst two tim indics t=35,36. Ths, a signiicant dviation rom th targt is obsrvd at t=35,36. Onc a spik distrbanc is conirmd, th sprvisory GPC will rmov th inlnc o th spik rom th procss and pll th otpt back to th normal condition qickly. By comparing Figs. 14a and 14b, it can b sn that withot sprvision th otpt rspons at indx 37 has a largr dviation than that with sprvision. Th ctivnss and importanc o having this spik rror rmoval atr dpnd on th dynamic charactristic o th procss modl. It will b mor dsirabl to compnsat or th spik rror i th impls rspons has a long mmory o th procss dynamics, which is qivalnt to hav th pols o th charactr- Fig. 14 Control prormanc comparison ndr th spik distrbanc 324 / Vol. 128, FEBRUARY 2006 Transactions o th ASME
Fig. 15 Control prormanc ndr a spik distrbanc or a longr mmory systm istic qation clos to 1. To illstrat this point, Fig. 15 givs a comparison which shows th ctivnss o th spik compnsation tchniq basd on anothr modl. Th modl paramtrs or this modl ar Aq=1 0.39q 1 0.126q 2, Bq 1 =0.003645+0.004168q 1, Cq=Dq=1, th dad tim o d=5, and th rror trm sris is gnratd rom t N0,10 4. Th pols o th charactristic qation ar 0.6 and 0.21. In this cas, a similar spik, which also lads to abot a 6% dviation in th otpt targt val, is addd as th snsor rror to dmonstrat th ctivnss o th spik rror compnsation. It can b sn rom Fig. 15 that withot th spik rmoval th otpt dviations rom th targt at indxs o 37 39 ar largr than that with th spik rmoval. 5 Conclsion Th control o a thin ilm dposition procss, which can b modld by a singl-inpt, singl-otpt SISO ARMAX linar dynamic modl, was invstigatd in this papr. Bcas o th inhrnt charactristics o th thin ilm dposition procss, a GPC was adoptd or th procss controllr dsign. A st o sprvisory stratgis is sd to obtain a satisactory control prormanc by compnsating or th dirnt typs o distrbancs. This papr mphasizs th importanc o dvloping sprvisory stratgis throgh monitoring th procss changs sing SPC tchniqs, and thn rvising th controllr paramtrs accordingly to achiv sprior control prormanc. Th intgration o SPC with atomatic procss control APC provids grat potntial or th dvlopmnt o ctiv controllrs in complx manactring procsss. Th cas stdis providd validat th ctivnss o th dvlopd sprvisory GPC stratgy. Thr ar svral opn isss to b invstigatd rthr. On is how to dtrmin th thrsholds or th SPC control charts. Bcas th SPC is sd to monitor a GPC-controlld procss, Typ I and Typ II rrors discssd in th convntional SPC litratr cannot b sd dirctly. Althogh som rsarch has bn don to invstigat th SPC monitoring or a PID-controlld procss 14, th SPC monitoring or th sprvisory GPC control, which is mor complx than th PID control stratgy, dsrvs som rthr attntion. Also, a catios control stratgy cold b intgratd to accommodat th stimation ncrtainty o th procss modl and distrbanc 15. Anothr opn iss worth invstigating is how to systmatically and simltanosly slct optimal GPC paramtrs.g., N, R, tc., SPC thrsholds, and altrnativ sprvisory stratgis. Crrntly, som trial-and-rror orts ar rqird to slct thos paramtrs 8. Anothr topic rlatd to thin ilm thicknss control is to xpand th rsarch prsntd in this papr rom SISO to mltipl-inpt mltipl otpt MIMO cass bcas thr ar mltipl matrial sorcs as wll as mltipl qality indics or a thin ilm dposition procss. Som arly works in prdictiv control basd on a stat spac modl can b considrd as a dirction or rthr xtnsion. Acknowldgmnt Th athors wold lik to thank th Associat Editor and th rviwrs or thir insightl commnts and sggstions, which hav signiicantly improvd th papr qality and radability. Th athors also gratlly acknowldg th inancial spport o th NSF Carr Award DMI-0133942 and NSF DMI-0330356, and AFOSR Grant F49620-03-1-0377. Rrncs 1 Ljng, L., 1999, Systm Idntiication Thory or th Usr, 2nd d., Prntic- Hall, Englwood Clis, NJ. 2 Clark, D. W., Mohtadi, C., and Ts, P. S., 1987 Gnralizd Prdictiv Control Part I. Th Basic Algorithm; Part II. Extnsions and Intrprtations, Atomatica, 232, pp. 137 160. 3 Clark, D. W., and Mohtadi, C., 1989, Proprtis o Gnralizd Prdictiv Control, Atomatica, 256, pp. 859 875. 4 L, L. L., Schapr, C. D., and Ho, W. K., 2002 Ral-Tim Prdictiv Control o Photorsist Film Thicknss Uniormity, IEEE Trans. Smicond. Man., 151, pp. 51 59. 5 Dl Castillo, E., and Hrwitz, A. M., 1997, Rn-to-Rn Procss Control: Litratr Rviw and Extnsions, J. Qality Tchnol., 29, pp. 184 196. 6 Moyn, J., Dl Castillo, E., and Hrwitz, A., 2000, Rn to Rn Procss Control in Smicondctor Manactring, CRC Prss, Boca Raton, FL. 7 Sachs, E., H, A., and Ingolsson, A., 1995, Rn by Rn Procss Control Combining SPC and Fdback Control, IEEE Trans. Smicond. Man., 81, pp. 26 44. 8 Astrom, K. J., and Wittnmark, B., 1995, Adaptiv Control, 2nd d., Addison- Wsly, Rading, MA. 9 Mongtmory, D., 2001, Introdction to Statistical Qality Control, 4th d., Wily, Nw York. 10 Ljng, L., and Sodrstrom, A., 1987, Thory and Practic o Rcrsiv Idntiication, MIT Prss, Cambridg, MA. 11 Aply, D., and Shi, J., 1999 An Ordr Downdating Algorithm or Tracking Systm Ordr and Paramtrs in Rcrsiv Last Sqars Idntiication, IEEE Trans. Signal Procss., 4711, pp. 3134 3137. 12 Box, G., Jnkins, G., and Rinsl, G., 1994, Tim Sris Analysis, Forcasting, and Control, 3rd Ed., Prntic-Hall, Englwood Clis, NJ. 13 Ko, B. C., and Golnaraghi, F., 2002, Atomatic Control Systms, 8th d. Wily, Nw York, p. 237. 14 Tsng, F., W, H., and Nair, V. N., 1998 On th Eicincy and Robstnss o Discrt Proportional-Intgral Control Schms, Tchnomtrics, 40, pp. 214 222. 15 Shi, J., and Aply, D. W., 1998, A Sboptimal N-Stp-Ahad Catios Controllr or Adaptiv Control Applications, ASME J. Dyn. Syst., Mas., Control, 120, pp. 419 423. Jornal o Manactring Scinc and Enginring FEBRUARY 2006, Vol. 128 / 325