Stability Analysis of EWMA Run-to-Run Controller Subjects to Stochastic Metrology Delay

Size: px

Start display at page:

Download "Stability Analysis of EWMA Run-to-Run Controller Subjects to Stochastic Metrology Delay"

Ilene Barnett
6 years ago
Views:

1 Preprins of he 8h IFAC World Congress Milano (Ialy) Augus 8 - Sepember, Sabiliy Analysis of EWMA Run-o-Run Conroller Subjecs o Sochasic Merology Delay Bing. Ai*, David Shan-Hill Wong**, Shi-Shang Jang**, Ying Zheng* * Deparmen of Conrol Science and Engineering, Huazhong Universiy of Science and echnology, Wuhan, Hubei, 4374, P.R. China (el: ; zyhidy@mail.hus.edu.cn) ** Deparmen of Chemical Engineering, Naional sing-hua Universiy, Hsin-Chu, aiwan ( ssjang@mx.nhu.edu.w) Absrac: In he semiconducor manufacuring bach processes, each sep is a complicaed physicochemical bach process; generally i is difficul o perform measuremens on-line. he effec of he merology delay on he sabiliy of he sysem is an imporan issue needs o be undersood. his paper invesigaes he sabiliy of sysems under exponenially weighed moving average (EWMA) runo-run conrol wih sochasic merology delay. Necessary and sufficien condiions for he sochasic sabiliy are esablished. Some numerical examples are provided o illusrae how o ge he sabiliy regions based on he proposed heorem. Keywords: run-o-run conrol, merology delay, sochasic sabiliy.. INRODUCION In he field of semiconducor manufacuring indusry, run-orun conrol is now widely acceped as a means for producion fabricaion faciliies o improve he efficiency and wasage in producion. I s a form of discree process and machine conrol in which he produc recipe wih respec o a paricular machine process is modified ex siu, i.e., beween machine runs, so as o minimize process drif, shif and variabiliy (Moyne e al. ()). wo of he more basic runo-run conrol algorihms used oday in he semiconducor manufacuring indusry are exponenially weighed moving average (EWMA) algorihm and predicor-correcor conroller (or double EWMA) algorihm. ime delay sysem has been exensively sudied in he las few years. For he sysem wih sochasic ime delay, as Ji and Chizec (99) poined ou ha many manufacuring process can be modeled by Marovian jump linear sysems. And he resuls of opimal conrol, robus conrol and sabiliy for such ind of sysem can be widely seen in he recen lieraures, such as Cosa and Marques (), Zhang e al. (3), Shi and Yu (9), and iao e al. (). However, he opimizaion problem, he robus conrol as well as he sabiliy problem for Marovian jump linear sysems usually change ino he problem of solving a se of linear marix inequaliies (LMIs). Grigoriadis (995), Geromel e al. (995), Ghaoui e al. (997) and Oliveria and Geromel (997) gave deailed algorihms o solve he LMIs. he pioneer wor on he sabiliy of EWMA run-o-run conroller wihou delay was carried ou by Ingolfsson and Sachs (993). Good and Qin () examined sabiliy bounds for he discoun facors of boh single inpu single oupu (SISO) and muliple inpus muliple oupus (MIMO) double EWMA run-o-run conrollers when here is planmodel mismach and delay beween produc manufacuring and produc merology. Few years laer, afer heir firs wor on sabiliy analysis of double EWMA run-o-run conrollers, Good and Qin (6) analyzed he sabiliy of MIMO EWMA run-o-run conroller wih merology delay by using he generalized Rouh-Hurwiz sabiliy crierion. Wu e al. (8) analyzed he influences of merology delay on boh he ransien and asympoic properies of he produc qualiy for he case when a linear sysem wih an iniial bias and a sochasic auoregressive moving average disurbance is under EWMA run-o-run conrol. All he aforemenioned wors on EWMA or double EWMA run-o-run conrol sysem are based on he assumpion ha he merology delay is fixed. However, he semiconducor manufacuring indusry is characerized by physically and chemically environmens maing measuremen in many of hese environmens difficul or ime-consuming, his combined wih he fac ha many process ools are no designed for he addiion of in siu sensor, resuled in measuremen aen less frequenly han every run, or a sochasic runs. he aim of his paper is o invesigae he sabiliy properies of he EWMA run-o-run conroller wih sochasic merology delay for SISO sysem. For case of presenaion, he remainder of he paper is organized as follows: in Secion, he sabiliy condiions are derived for EWMA run-o-run conrol sysems wih sochasic merology delay. Numerical examples are provided in Secion 3 o obain he sabiliy regions for he sysems subjec o differen merology delay. he conclusion remars are presened in Secion 4. Copyrigh by he Inernaional Federaion of Auomaic Conrol (IFAC) 354

2 Preprins of he 8h IFAC World Congress Milano (Ialy) Augus 8 - Sepember,. EWMA RUN-O-RUN CONROLLER WIH MEROLOGY DELAY. EWMA Run-o-Run Conroller Wih Fixed Merology Delay A ypical EWMA run-o-run conroller assumes a saic linear model beween conrol variable Y, and manipulaed variable u, i.e., Y = βu a () where β is he process gain beween he process inpu and oupu. a is he insananeous disurbance a run. Given he prediced model of he process Y = bu c () where b and c are model gain and offse parameers esimaed for he sysem, respecively. Boh of hese parameers are deermined a priori by a design of experimens procedure. If here is a fixed merology delay d a run, hen no informaion abou Y coming a he end of run, however, he conroller will always use he mos recen daa, hus, we a leas have Y d available for feedbac. By using he EWMA filer, he disurbance is esimaed o be aˆ ˆ = ω( Y d bu d) ( ω) a (3) where ω is a discoun facor beween zero and one. Assume he ransiion probabiliy marix of is P = [ p ij ]. ha is, jump from mode i o j, wih probabiliy p ij, which is defined by p = Prob( = j = i) (5) ij where i, j S, and p ij j= p ij =. hus, if he consrain condiions for merology delay are considered, hen he srucure of he ransiion probabiliy marix will be p p p p p P = (6) p p p p 3 p p p p p 3 p p p p p3 p Each row represens he ransiion probabiliies from a fixed sae o all he saes, he diagonal elemens are he probabiliies of merology sequences wih equal delays, he elemens below he diagonal indicae shorer delays, and he elemens above he diagonal are he probabiliies of encounering longer delays. Fig. illusraes hree saes ransiion diagram. From he figure, we can see ha i can jump from = and = o any saes, while i canno jump from = o =. A conrol law is used o deermine he conrol recipe for he nex run, i.e., aˆ u = (4) b where is he desired arge. Wihou loss of generaliy, in his paper, we assume =. Afer doing he well-nown bilinear ransformaion, we can ge sabiliy areas for he sysem ()-(4) by using Rouh- Hurwiz crierion.. ransiion Probabiliy Marix In an acual manufacuring plan, measuremen delay is a sochasic variable insead of being fixed. Le are variable merology delay sequence, assume ha, and aes values in S={,,,, }. Since merology delay canno h exceeds he run lengh, min(, ) in he producion run. Also, he conroller will always use he mos recen daa, hus, if we have Y - a run, bu here is no new informaion coming a run (here is longer delay a run ), hen we a leas have Y available for feedbac. his - means ha he delay can increase a mos a each run, i.e., Prob( ) =. Fig.. hree saes ransiion diagram.3 EWMA Run-o-Run Conroller Wih Sochasic Merology Delay We also assume he model and he prediced model of he process are linear ime-invarian discree-ime models as described by () and () respecively. Suppose ha here is a sochasic merology delay a run, hen a his ime, he EWMA run-o-run conroller wih sochasic merology delay can be used o esimae he disurbance, i.e., aˆ ˆ = ω( Y bu ) ( ) ω a (7) We also choose he conrol acion as (4). Combining (), (), (4) and (7), we have aˆ ˆ ˆ = ( ω) a ω( ξ) a (8) For he sysem wih sochasic ime delay, Rouh-Hurwiz crierion is no longer valid in obaining sabiliy region. 355

3 Preprins of he 8h IFAC World Congress Milano (Ialy) Augus 8 - Sepember, Le x = ˆ a be he sae of he sysem, and hen (8) can be denoed as x = ( ω) x ω( ξ) x (9) where S. Augmen he sae variable as = [ x x x x ] a run, hen he closed-loop sysem in (9) can be wrien as x ω ω( ξ) x x x x x () = x x x A(, ) i.e., = A(, ) () Remar: From () and (), i is clear ha he sochasic ime delay sysem described by (), (), (4) and (7) is ransformed ino a delay-free discree-ime sysem. he following heorem gives sufficien and necessary condiion o guaranee he sochasic sabiliy of sysem (). heorem: Sysem () is sochasically sable if and only if here exiss a posiive-definie marix Q(, ) > for S, saisfying he following marix inequaliies:, = L (, ) = p A (, ) Q (, ) A (, ) Q (, ) < Proof: () Sufficiency: consruc he sochasic Lyapunov funcion V(,, ) as follows: V(,, ) = Q(, ) (3) hen E[ ΔV(,, )] = EV [ (,, ) V(,, )] = = p A(, ) Q(, ) A(, ) Q(, ), [, (,) (, ) (,) (,)] = = p A Q A Q = L(, ) λ ( L(, )) β min (4) where λ ( (, )) min L denoes he minimal eigenvalue of L(, ), and β = inf{ λmin ( L(, )), S} >. From (4), we have β β EV [ (,,)] EV [ (,,)] EV [ (,,)] EV [ (,,)] EV [ (,, ) V(,, )] β Sum boh sides of (5), we obain lim E,, E[ V(,,)] = = β = Q(,) β < i.e., sysem () is sochasically sable. Necessiy: (5) (6) Define E [ Q (,, ) ] = E[ R(, ),, ] wih R(, ) >. I s obvious ha = λmax ( ) = = E[ R(, ),, ] R(, ) E[,, ]. Since he sysem is sochasic sable, i.e., lim E,, = < (7) = E [ Q (,, ) ] is bounded, and is limi can be denoed as E [ Q(, ) ] = lim E [ Q (,, ) ] Also = lim E[ R(, ),, ] = (8) E [ Q(, ),, ] = lim E [ Q (,, ) ] = lim E[ R(, ),, ] Equaion (9) subrac (8), we have = = (9) p A(, ) Q(, ) A(, ) Q(, ) = R(, ), () and 356

NUMERICAL EAMPLES In his secion, we firs give an example o exemplify how o calculae he ransiion probabiliy marix from Poisson disribuion.

4 Preprins of he 8h IFAC World Congress Milano (Ialy) Augus 8 - Sepember, =, = R(, ) Q(, ) p A(, ) Q(, ) A(, ) = L(, ) Since R(, ) >, i s clear ha L(, ) <. () Since L(, ) conains uncerainies of he sysem, i is differen o chec wheher () is feasible or no. o his end, we have he equivalen condiion for (), i.e., 3. NUMERICAL EAMPLES In his secion, we firs give an example o exemplify how o calculae he ransiion probabiliy marix from Poisson disribuion. hen, based on he heorem we go in Secion, several simulaion examples are provided o obain he sabiliy regions for he sysems subjec o differen merology delay P = However, from Fig. (b), we noice ha here are only 3 numbers of Poisson random numbers ae value in 4, few numbers of Poisson random numbers ae values in 3 compared wih hose ae values in,, and. In his siuaion, we should consider modes 3 and 4 are abnormal modes, hey should also be runcaed, and hus we only have hree normal modes S = {,, } for he resampled Poisson random numbers, he corresponding ransiion probabiliy marix can be calculaed, i.e., P = he calculaion of ransiion probabiliy marix from Poisson disribuion In his subsecion, we will ae Poisson disribuion for example o illusrae he relaionship beween originally Poisson numbers and resampled Poisson numbers. An example is provided o demonsrae he calculaion of ransiion probabiliy marix from resampled Poisson numbers. Le m be a random number generaed a he h run by a Poisson disribuion wih expecaion λ, and le are variable merology delay sequence. hen,, if m > ; m = () m, oherwise. h Also, if he merology delay of he run is longer han ha of he ( ) h run, hen he measured daa is oo old o use. A his ime, we should resample, hence he acual resampled disribuion is m, if m m ; = (3) m, oherwise. Fig. is he comparison of original and resampled Poisson random numbers wih he expecaion λ =. In Fig. (a), he original Poisson random numbers are shown, and i is clear ha here are eigh modes S = {,,,3,4,5,6,7}, i.e., he original Poisson disribuion cu-off afer mode 7. If we consider consrain condiions of equaions () and (3), hen he resampled Poisson random numbers can be obained as shown in Fig. (b). From Fig. (b), i can be noiced ha he ails of he original Poisson disribuion are runcaed, and he modes of he resampled Poisson random numbers is five, i.e., S = {,,,3, 4}. If S = {,,,3, 4} are chosen o calculae he ransiion probabiliy marix for resampled Poisson random numbers, hen, we have Fig.. Comparison of original and resampled Poisson numbers 3. Sabiliy regions for he sysems wih differen ransiion probabiliy marices he heorem we obained in Secion is based on Lyapunov s direc mehod. In his subsecion, we will compare he sabiliy regions obained by Lyapunov s direc mehod and Rouh-Hurwiz crierion for he sysems wih fixed merology delay. he sabiliy regions for he sysems wih sochasic merology delay can be go by Lyapunov s direc mehod. Wihou loss of generaliy, i will be only discusses he cases wih maximum delay lengh of wo runs, for he longer delay, i ll be sraighforward. For he sysem wihou delay, where S = {}, he ransiion probabiliy marix is denoed as P = [.], where he sabiliy region go by Lyapunov s direc mehod and Rouh- 357

Preprins of he 8h IFAC World Congress Milano (Ialy) Augus 8 - Sepember, Hurwiz crierion is given in Fig. 3.

sable for any discoun facor ω beween o. However, when ξ, ω is decreasing in ξ o eep he closed-loop sable...9. of P =... or P = [.]. From Fig. 5, we can.

5.3...6.3. Fig. 3. Sabiliy region for he sysem wihou delay he sabiliy regions for he sysem wih fixed and sochasic wo runs merology delay are shown in Fig. 4 and Fig. 5 respecively. In Fig.

4, i is concluded ha i can be arrived a comparable resuls as we obained for delay-free sysem (as shown in Fig. 3). In Fig. 5, if he ransiion probabiliy marix is... P =..., i.e., he sysem is delay-free sysem,.

.. is mos liely o subjec wo runs delay, has he smalles sabiliy region, and his sabiliy region has he same size wih he sabiliy region for he sysem wih fixed wo runs delay, i.e., P = [.

.. ha of P =... or P = [.], bu larger han ha... Fig. 4. Sabiliy region for he sysem wih fixed wo runs delay Fig. 5.

5 Preprins of he 8h IFAC World Congress Milano (Ialy) Augus 8 - Sepember, Hurwiz crierion is given in Fig. 3. From he figure, i is clear ha boh mehods arrive a he same sabiliy region and when he esimaed process gain is greaer han half of he rue process gain ( ξ < ), he sysem is guaraneed closedloop sable for any discoun facor ω beween o. However, when ξ, ω is decreasing in ξ o eep he closed-loop sable...9. of P =... or P = [.]. From Fig. 5, we can... also conclude ha he sysem ha is more liely o subjec..8. ime delay such as P =..3.5 has a smaller size of..3.6 sabiliy region compared wih he sysem ha is less liely o.8.. experience ime delay, for insance P = Fig. 3. Sabiliy region for he sysem wihou delay he sabiliy regions for he sysem wih fixed and sochasic wo runs merology delay are shown in Fig. 4 and Fig. 5 respecively. In Fig. 4, we also compared Lyapunov s direc mehod and Rouh-Hurwiz crierion in obaining he sabiliy region for he sysem wih fixed wo runs delay. From Fig. 4, i is concluded ha i can be arrived a comparable resuls as we obained for delay-free sysem (as shown in Fig. 3). In Fig. 5, if he ransiion probabiliy marix is... P =..., i.e., he sysem is delay-free sysem,... hen his case has he same sabiliy region as P = [.], which is he bigges sabiliy region. If he ransiion..9. probabiliy marix P =..., namely he sysem... is mos liely o subjec wo runs delay, has he smalles sabiliy region, and his sabiliy region has he same size wih he sabiliy region for he sysem wih fixed wo runs delay, i.e., P = [.], more deail discussion for his case will be illusraed in Fig. 6. In addiion, when... P =... or P = [.], i.e., he sysem subjecs o... one fixed delay, he size of sabiliy region will smaller han... ha of P =... or P = [.], bu larger han ha... Fig. 4. Sabiliy region for he sysem wih fixed wo runs delay Fig. 5. Comparison of sabiliy regions for he sysems subjec o sochasic merology delay wih maximum delay lengh of wo runs Fig. 6 shows he simulaion of he sysems wih differen ime delays. Comparisons of differen ime delays, i.e., =, = and =, show ha he size of sabiliy region will decrease wih he increase of ime delay. he sabiliy region for he sysems subjec o sochasic ime delay wih maximum delay lengh of one run, will be he same wih he sysem which exiss fixed one run delay, as long as 358

6 Preprins of he 8h IFAC World Congress Milano (Ialy) Augus 8 - Sepember,.. p =,i.e., for he sysems wih P = [], P =..,..9 P.9. =.. and P =.. have he same size of sabiliy region. When he maximum ime delay is wo runs, hen he same conclusion can be drawn, only if p =. I is also clear from he figure, he sysem is sable for any delays on condiions ha ξ <, and ω aes value beween o. Fig. 6. Comparison of sabiliy regions for he sysems wih differen ime delays 4. CONCLUSION In his paper, we sudied he sabiliy problems for boh sysems subjec o fixed and sochasic merology delay. Rouh-Hurwiz crierion is use for obaining he sabiliy regions of he sysem wih fixed merology delay; Lyapunov s direc mehod is adoped o derive he sufficien and necessary condiions of he sochasic sabiliy for he sysem subjec o sochasic merology delay. From he resul of numerical simulaion, i is nown ha Rouh-Hurwiz crierion and Lyapunov s direc mehod is equivalen in geing he sabiliy region of he sysem wih fixed merology delay, also wih he increase of merology delay, he size of sabiliy region will decrease for boh sysems wih fixed and sochasic merology delay. Moreover, he simulaion show ha when he esimaed process gain is greaer han half of he rue process gain, he sysem is guaraneed closed-loop sable for any discoun facor ω beween o. However, when model error is greaer han wo, he size of sabiliy region will decrease. I is worh menioning ha for he sysem wih any lengh of merology delay, he sabiliy region can be go by he heorem proposed in his paper. Moyne J., Casillo E.D., Hurwiz A.M. (). Run-o-Run conrol in semiconducor manufacuring. CRC Press, Florida,. Ji Y., Chizec H. J. (99). Jump linear quadraic Gaussian conrol in coninuous ime. IEEE rans. Auoma. Conrol, 37 (), Cosa O. L. V., Marques R.P. (). Robus H -conrol of discree-ime Marovian jump linear sysems. Inerna. J. Conrol, 73 (), -. Zhang L. Q., Huang B., Lam J. (3). H model reducion of Marovian jump linear sysems. Sys. Conrol Le., 5, 3-8. Zhang L. Q., Shi Y., Chen. W., Huang B. (5). A New Mehod for sabilizaion of newored conrol sysems wih random delays. IEEE rans. Auoma. Conrol, 5(8), Shi Y., Yu B. (9). Oupu feedbac sabilizaion of newored conrol sysems wih random delays modelled by Marov chains. IEEE rans. Auoma. Conrol, 54(7), iao L., Hassibi A., How J. P. (). Conrol wih random communicaion delays via a discree-ime jump sysem approach. In Proc. Amer. Conrol Conf., Grigoriadis K. M. (995). Opimal H model reducion via linear marix inequaliies: Coninuous- and discree-ime cases. Conrol Le., 6(5), Geromel J. C., Souza C. C. D., Selon R.E. (995). LMI numerical soluion for oupu feedbac sabilizaion. In Proc. IEEE Conf. Decision and Conrol, Ghaoui L. E., Ousry F., Airami M. (997). A cone complemenariy linearizaion algorihm for saic oupu-feedbac and relaed problems. IEEE rans. Auom. Conrol, 4(8), Oliveria M. C. D., Geromel J. C. (997). Numerical comparison of oupu feedbac design mehods. In Pro. Amer. Conrol Conf., 7-76, Albuquerque, NM. Ingolfsson A., Sachs E. (993). Sabiliy and sensiiviy of an EWMA conroller. J. Qualiy echnol., 5(4), Good R. P., Qin S. J. (). Sabiliy analysis of double EWMA run-o-run conrol wih merology delay. In Pro. Amer. Conrol Conf., 56-6, Anchorage, AK. Good R.P., Qin S.J. (6). On he sabiliy of MIMO EWMA run-o-run conrollers wih merology delay, IEEE rans. Semiconduc. Manufac., 9(), Wu M. F., Lin C. H., Wong D. S. H., Jang S.S., seng S.. (8). Performance analysis of EWMA conrollers subjec o merology delay, IEEE rans. Semiconduc. Manufac., (3), ACKNOWLEDGEMEN he auhors han he financial suppor for his wor from Chinese Naional Naural Science Foundaion (67475, 6346 and 6643). REFERENCES 359

Mean-square Stability Control for Networked Systems with Stochastic Time Delay

Mean-square Stability Control for Networked Systems with Stochastic Time Delay JOURNAL OF SIMULAION VOL. 5 NO. May 7 Mean-square Sabiliy Conrol for Newored Sysems wih Sochasic ime Delay YAO Hejun YUAN Fushun School of Mahemaics and Saisics Anyang Normal Universiy Anyang Henan. 455