MIMO Integral-Action Anti-Windup Controller Design and Applications to Temperature Control in RTP Systems

Transcription

1 43rd IEEE Coferece o Deciio ad Cotrol December 4-7, 24 Atlati, Paradie Ilad, Bahama WeC9.2 MIMO Itegral-Actio Ati-Widup Cotroller Deig ad Applicatio to Temperature Cotrol i RTP Sytem A. N. Mete ad A. N. Güdeş Abtract A itegral-actio cotroller ythei i preeted, where the cotroller achieve cloed-loop tability ad zero teady-tate error due to tep-iput referece. Cloedloop tability i maitaied eve whe the itegrator i all of the cotroller chael are limited or completely witched off to protect agait itegrator widup. The propoed method i applied to temperature cotrol i Rapid Thermal Proceig ytem, where critical requiremet are impoed o trackig of temperature profile. I. INTRODUCTION We coider cotroller ythei for liear, time-ivariat (LTI) multi-iput multi-output (MIMO) plat ubject to iput aturatio. Our goal i to achieve cloed-loop tability ad aymptotic trackig of tep-iput referece with zero teady-tate error. To achieve robut trackig, the cotroller i deiged to have itegral-actio. The performace of itegral-actio cotroller deped o the ytem operatig i a liear rage. They uffer eriou lo of performace due to a pheomeo called itegral widup, which occur whe the actuator i the cotrol-loop aturate (ee e.g. [6]). Limitatio of actuator put a upper boud o the amplitude of the cotrol igal. Actuator reach their aturatio limit whe a referece igal requirig a cotrol effort beyod that boud i applied to the ytem. The itegrator i the cotroller cotiue to itegrate the error although the iput i cotraied, ad oce the iput come out of the aturatio limit, the iitial coditio that the itegrator ha built up to caue a large traiet repoe ad eriou performace degradatio. Effect of itegral widup ca rage from large overhoot i traiet repoe to lo of tability. Therefore, itegralactio cotroller hould be modified by takig actuator limitatio ito accout. Mot method of dealig with the effect of itegrator widup are baed o the idea of turig off the itegrator whe the iput reache a limit ad reettig the itegrator tate. Numerou ati-widup modificatio have bee propoed, may of which ca be coidered a oberver-baed modificatio (ee e.g., []). A ythei procedure for deigig oberver-baed cotroller gai i aticipatio of widup ca be foud i []. There are everal way to deig LTI itegral-actio cotroller (ee e.g. [2], [2], [8], [9], [5] for decetralized ad cetralized itegral-actio cotroller ythei). The implet cotroller that achieve itegral-actio i i the Thi work wa upported by the NSF Grat ECS The author are with Electrical ad Computer Egieerig, Uiverity of Califoria, Davi, CA amete@ece.ucdavi.edu & gude@ece.ucdavi.edu /4/$2. 24 IEEE 259 proportioal+itegral+derivative (PID) form. Cloed-loop tability ca be achieved uig PID-cotroller oly for certai clae of plat, while may other caot be tabilized uig PID-cotroller. Stadard oberver-baed itegralactio cotroller deig apply Liear Quadratic Regulator (LQR) or pole-placemet method to a augmeted plat model, which iclude the tate of the plat ad the y tate of the itegrator [3], [], [5]. A augmeted fullorder oberver-baed deig reult i a ( + y )-th order cotroller. The itegrator caot be completely witched off ice the gai matrix correpodig to the itegrator tate are part of the feedback loop that eure tability. Itead of turig off the itegral-actio completely to protect agait itegrator widup, the itegrator are limited o that the iput doe ot reach the aturatio limit. I thi paper, we propoe a itegral-actio ythei procedure baed o addig itegral-actio oto the ytem, where a iitially deiged tabilizig cotroller i already preet i the feedback loop. Thi i achieved i two tage: A iitial tabilizig cotroller i deiged for the origial plat uig ay deired method (LQR, H, etc.) ad it doe ot have itegral-actio. The a PID-cotroller i deiged for a table ytem aociated with the plat. Thee two block are cofigured o that the fial cotroller achieve cloed-loop tability ad itegral-actio. Furthermore, all itegral-actio cotroller ca be obtaied from thi cotroller by icluio of a free cotroller parameter. If a full-order oberver-baed cotroller i choe for the iitial tabilizig tage, the oly the plat tate are etimated ad the tate feedback gai are oly aociated with the plat tate without augmetig. The trafer-fuctio of the fial cotroller oce the PID block i added o i ( +).To protect agait widup due to the itegrator i the PID block cotaiig the itegral-actio, thi ecod block ca be completely witched off without affectig cloed-loop tability or it ca be limited. The propoed itegral-actio cotroller deig i applied to temperature cotrol i Rapid Thermal Proceig (RTP) ytem (ee e.g. [4], [3] for modellig ad cotrol of RTP). The recet techology of RTP i itegrated circuit maufacturig i a fat ad efficiet multi-chamber iglewafer techology that ue a much maller chamber tha a batch proce. Sigle-wafer proceig achieve more uiform film thicke for larger wafer. Durig a RTP proce, it i crucial to maitai uiform temperature o the wafer urface at all time ice mall temperature variatio ca lead to large variatio i reactio rate [3]. The beefit of RTP caot be realized without meetig the triget

2 temperature uiformity pecificatio. Fat trackig cotrol law that achieve ear uiform patial temperature ditributio acro a emicoductor wafer durig both traiet ad quai teady-tate phae of the proce eed to be developed for RTP ytem, which are iheretly oliear dyamic procee with actuator aturatio. The paper i orgaized a follow: The problem decriptio ad defiitio are i Sectio II. The mai reult are i Sectio III: Sectio III-A explai the propoed cotroller ythei. Theorem 3. tate that all itegralactio cotroller ca be realized i two tage, where the itegral-actio i achieved uig a eparate PID block that ca be witched off or limited without affectig cloedloop tability. Oe method of deigig the PID tage i the ytematic procedure i Propoitio 3.. Sectio III-B give a review of itegral-actio deig uig a augmeted plat model. I Sectio IV, the reult are applied to a liearized MIMO RTP ytem model ubject to lamp iput aturatio oliearity a decribed i [4]. The propoed ytematic ythei approach i compared with the tadard oberverbaed cotroller deig with augmeted tate feedback. Although we dicu cotiuou-time ytem here, all reult alo apply to dicrete-time ytem with appropriate modificatio. The followig otatio i ued: Let CI, IR deote complex ad real umber. The exteded cloed right-half complex plae i U = { CI Re() } { }; R p deote real proper ratioal fuctio of ; S R p i the table ubet with o pole i U; M(S) i the et of matrice with etrie i S ; I i the idetity matrix. The H -orm of M() M(S) i M := up U σ(m()), where σ i the maximum igular value ad U i the boudary of U. We drop () i trafer matrice uch a G() wheever thi caue o cofuio. We ue coprime factorizatio over S ; i.e., for G R y u p, G = N g Dg deote a right-coprime-factorizatio (RCF), where N g S y u, D g S u u, det D g ( ) ; G = D g Ñ g deote a left-coprime-factorizatio (LCF), where Ñg S y u, Dg S y y, det D g ( ). II. PROBLEM DESCRIPTION Coider the LTI, MIMO uity-feedback ytem Sy(G, Ĉ) i Fig. ; G R p y u ad Ĉ R p u y are the plat ad the cotroller trafer-fuctio, repectively. Aume that the feedback ytem i well-poed, the plat ad cotroller have o utable hidde-mode, ad the plat G R y u p i full ormal rak. Let H er = (I y + GĈ) = I y GĈ(I y + GĈ) I y GH wr deote the (iput-error) trafer-fuctio from r to e. Defiitio 2.: i) The ytem Sy(G, Ĉ) i aid to be table iff the cloed-loop trafer-fuctio from (r, v) to (y, w) i table. ii) The cotroller Ĉ i aid to tabilize r e Ĉ v w G y Fig.. Uity-Feedback Sytem Sy(G, Ĉ). 259 G iff Ĉ i proper ad Sy(G, Ĉ) i table. iii) The table ytem Sy(G, Ĉ) i aid to have itegral-actio iff H er ha blockig zero at the origi. Suppoe that Sy(G, Ĉ) i table ad that tep referece iput r are applied to the ytem. The teady-tate error due to all tep iput goe to zero if ad oly if H er () =, i.e., the ytem ha itegral-actio. Let G = N g Dg = D g Ñ g, Ĉ = N cr Dcr = D cl N cl be ay RCF ad LCF of the plat ad the cotroller. The Ĉ tabilize G if ad oly if M L i () equivaletly, M R, i uimodular [5], [7]: D g D cr + ÑgN cr =: M L, D cl D g + N cl N g =: M R () The H er ca be writte a i (2), ad equivaletly, (3): H er =(I y + GĈ) = D cr M L g, (2) H er = I y GĈ(I y + GĈ) = I y N g M R N cl. (3) By Defiitio 2., Sy(G, Ĉ) ha itegral-actio iff H er () = (D cr M L D g )() =. Defiitio 2.2: The cotroller Ĉ = N cr Dcr i aid to be a itegral-actio cotroller iff Ĉ tabilize G ad the deomiator matrix D cr for ay RCF of Ĉ ha blockig zero at the origi, i.e., D cr () =. By Defiitio 2.2 ad (2), if Ĉ = N crdcr i a itegralactio cotroller, the Sy(G, Ĉ) ha itegral-actio. Obviouly, D cr () = i ufficiet but ot eceary for H er () = (D cr M D L g )() =. IfG ha pole at =, rak D g () < y ; hece, the ytem may achieve itegralactio eve if D cr ().IfG ha o pole at =, the Sy(G, Ĉ) ha itegral-actio if ad oly if Ĉ = N crdcr i a itegral-actio cotroller, i.e., D cr () =. Lemma 2. give the eceary coditio impoed o G due to the itegral-actio requiremet: Lemma 2.: Let G R y u p. If the ytem Sy(G, Ĉ) ha itegral-actio, the i) (ormal) rakg = y u ; ii) G ha o tramiio zero at the origi. Proof: The tability of Sy(G, Ĉ) implie H er() = I y GH wr () =, i.e., GH wr () = I y. Therefore, (ormal) rak(gh wr ) = y mi{rakg, rakh wr } implie y rakg mi{ y, u }. By (3), H er () = implie N g ()M R ()N cl() = I y ; hece, rakn g () = rakn cl () = y. III. MAIN RESULTS The implet itegral-actio cotroller are i PID form. We coider a realizable form of proper PID-cotroller, where K p,k i,k d IR u y are called the proportioal, the itegral, ad the derivative cotat, repectively [5]: C pid = K p + K i + K d τ d +, (4) To implemet the derivative term, a pole i typically added to the derivative term (with τ d > ) o that C pid i (4) i proper. The itegral-actio i C pid i preet whe K i. The cotroller i (4) i i proportioal+itegral (PI) form

3 C pi = K p + K i / whe K d =, itegral+derivative (ID) form C id = K i / + K d /(τ d +) whe K p =, pure itegral (I) form C i = K i / whe K p = K d =. Although PID-cotroller are imple ad low order, ome (utable) plat G are ot tabilizable with ay C pid. Sice the plat coidered here are ot retricted to be table, exitece of tabilizig PID-cotroller i ot guarateed. Propoitio 3. how that table ytem ca be tabilized uig PID-cotroller, with K i, if ad oly if G ha o tramiio zero at the origi, ad propoe a method of electig the cotat K p,k i,k d. Propoitio 3.: Let N g S y u, (ormal) rakn g = y u. i) There exit tabilizig PID-cotroller with ozero itegral cotat K i IR u y if ad oly if rakn g () = y. ii) Suppoe rakn g () = y. Let N g () I IR u y be ay right-ivere of N g (). Chooe ay ˆK p, ˆK d IR u y, τ d >. With Ĉpd defied a i (5) below, let K p = ρ ˆK p, K d = ρ ˆK d, K i = ρn g () I, where ρ IR i ay poitive cotat atifyig (6): Ĉ pd := ˆK p + ˆK d τ d + ; (5) <ρ< N g ()Ĉpd + N g()n g () I I. (6) The N g i tabilized by the PID-cotroller i (7): C pid = ρ ˆK p + ρn g() I + ρ ˆK d τ d +. (7) I (7), ˆKd =give a PI-cotroller; ˆKp =give a ID-cotroller; ˆKd = ˆK p =give a pure I-cotroller. Proof : For ay poitive a IR, defie Z M(S) a Z := C pid + a =(K p + K d τ d + ) + a + K i + a. (8) The C pid = Z( +a I y ) i a RCF of C pid. Sice C pid tabilize N g, the matrix M defied by (9) i uimodular: M := + a I y + N g Z = ( ) I y + N g C pid + a I y (9) By (9), a, rakm() = rak(a N g ()K i )= y mi{rakn g (), rakk i } mi{ y, u } = y implie rakn g () = y. The there exit a right ivere N g () I IR u y, i.e., N g ()N g () I = I y. Defie ˆM a ˆM = + a + ρ M = + ρ I + N gc pid + ρ. () I (), add ad ubtract ˆM = I y + ρ + ρ ( N g ρ +ρ I y to obtai Ĉpd+ N g()n g () I I y ). () Sice ρ +ρ = ρ, for ay ρ> atifyig (6), we have ρ + ρ ( N g Ĉpd + N g()n g () I I y ) <. (2) By (2), ˆM i uimodular; equivaletly, M i uimodular ice a, ρ > ; therefore C pid tabilize N g A. Two tage itegral-actio deig Stadard itegral-actio deig are geerally baed o augmetig the plat to iclude the itegrator tate i tate-feedback. Thi approach i briefly reviewed i Sectio III-B. I thi ectio, we propoe a approach that doe ot ivolve augmetig the plat tate. Theorem 3. tate that ay itegral-actio cotroller for G = N g Dg ca be expreed a the um of two block: The firt i ay tabilizig cotroller C g = Ỹ X, deiged uig ay method, ad doe ot have itegral-actio. The ecod i Ỹ C pid, where C pid i ay PID-cotroller that tabilize N g ; it ca be deiged uig ay method icludig the procedure i Propoitio 3.. Thi ecod block provide itegral-actio ad eve if it i witched-off, the cloedloop remai table due to C g till remaiig i the loop. Theorem 3.: Let the plat be G R y u p, (ormal) rakg = y u ; let G have o tramiio zero at =. Let G = D g Ñ g be ay LCF, G = N g Dg be ay RCF. Chooe ay cotroller Cg o R u y p that tabilize G. There exit a LCF Cg o = Ñ c of Cg o that atifie D c D c D g + ÑcN g = I u. (3) Let C pid be ay PID-cotroller tabilizig N g with K i. The Ĉ i a itegral-actio cotroller for G if ad oly if Ĉ = ( D c QÑg) ( Ñc + Q D g + C pid ), (4) where Q S u y atifie det( D c QÑg)( ). We prove that ay Ĉ i (4) i a itegral-actio cotroller for G; a detailed proof that all itegral-actio cotroller are i the form give by (4) ca be foud i [8]: Proof : Defie Z M(S) a i (8). Sice C pid tabilize N g, the matrix M i (9) i uimodular. Let Cg o = N c Dc be a RCF of Cg o that atifie D g D c + ÑgN c = I y. (5) The all tabilizig cotroller C g for G ca be expreed a C g = Ỹ X = XY, where Ỹ := ( D c QÑg), X := ( Ñ c + Q D g ), Y := (D c N g Q), X =(N c + D g Q), (6) ad Q S u y atifie det( D c QÑg)( ) [5], [7]. Uig Ỹ X = XY ad (9), Ĉ i (4) ca be writte a: Ĉ = Ỹ ( X + C pid ) = Ỹ X + Ỹ (ỸD g + XN g )C pid = Ỹ X(Iy + N g C pid )+ D g C pid = XY ( ) I y + N g C pid + Dg C pid = (X + D g C pid + a M Y ) Y M( + a I y ). Therefore, a RCF Ĉ = N crdcr for the cotroller Ĉ i give by N cr Dcr =(X + D g ZM Y )( + a M Y ), (7) where N cr,d cr M(S) ad D cr i biproper. It follow by (9), (5) ad ÑgD g = D g N g that M L = D g D cr + Ñ g N cr = D g ( + a M Y )+Ñg(X + D g ZM Y )=

4 D g ( + a M Y +N g ZM Y )+ÑgX = D g MM Y + Ñ g X = D g Y + ÑgX = I y i uimodular. The ytem Sy(G, Ĉ) i table ice M L i () i uimodular. Sice D cr () = ( + a M Y ) = =, D cr ha blockig zero at =. By Defiitio 2.2, ay Ĉ = N crdcr give by (4) i therefore a itegral-actio cotroller for G. The block diagram of Sy(G, Ĉ), with Ĉ a i (4), i i Fig. 2. For Q =, the itegral-actio cotroller Ĉ become Cˆ o = Cg o + D c C pid. (8) The cotroller Ĉ i (4) i implified for table plat a follow: Let G S y u, rakg() = y u. Let C pid, with K i, be a PID-cotroller that tabilize G. The Ĉ i a itegral-actio cotroller for G if ad oly if Ĉ = (I QG) ( Q + C pid ), where Q S u y atifie det(i QG)( ). The parametrizatio give i (4) ca alo be obtaied uig a tate-pace repreetatio (A, B, C, D) of G R y u p, where A IR, (A, B) i tabilizable ad (C, A) i detectable. Let K IR u ad L IR y be uch that F L ad F K defied i (9) are table: F L := (I A+LC), F K := (I A+BK). (9) The uig G = N g D g = D g Ñ g, where D g = I KF K B,N g =(C DK)F K B + D, D g = I CF L L, Ñ g = CF L (B LD)+D, (2) a cotroller Cg o = D c Ñ c = N c Dc i give by D c = I + KF L (B LD), Ñ c = KF L L, D c = I +(C DK)F K L, N c = KF K L, Cg o = K ( I A + BK + L(C DK) ) L. (2) With the omial full-order oberver-baed cotroller Cg o i (2), the expreio for all itegral-actio cotroller i (4) of Theorem 3. become Ĉ =[I + KF L(B LD) Q(CF L (B LD)+D)] [ KF L L+Q(I CF L L)+C pid ], where Q S u y i uch that det(i Q( )D). With Q =, the cotroller Co ˆ i (8) i expreed a Cˆ o = K(I A + BK + L(C DK)) L + D c C pid. (22) The block diagram of Sy(G, Ĉ), with Co g a i (2), i i Fig. 3. The PID block for N g =[(C DK)F K B + D] ca be deiged uig Propoitio 3.: Sice G ha o tramiio zero at =, N g () ha a right-ivere r e C pid N g () I = [ D (C DK)(A BK) B ] I. (23) The pair (A a,b a ) i tabilizable if ad oly if (A, B) i tabilizable ad G ha o tramiio zero at the origi. The expreio (6) i implified a follow: Sice F K M(S), F K () = ( A + BK) A tate-feedback K a =[K x K ξ ] i the determied for exit. By (23), [N g () N g ()]N g () I the augmeted ytem i (25), ad the reultig (+ y )-th =[(C DK)F K B+D D+(C DK)(A BK) B]N g () I = (C DK)(A BK) order oberver-baed cotroller i called [(I(A Ĉa, give by BK) I) + I]BN g () I = (C DK)(A Ĉ a = K a [I A a + B a K a + L a (C a DK a )] L a, BK) [(I(A BK) I) I(A BK) ]BN g () I = (26) 2593 Ñ c D g D c Q Ñg v w u y G Fig. 2. The ytem Sy(G, Ĉ) with itegral-actio cotroller Ĉ. r e I Ki C pid K d τ d + K p v Q w u K G L C (I A) ˆx D B Fig. 3. Sy(G, Ĉ) uig full-order oberver-baed cotroller Co g. (C DK)(A BK) (I A + BK) BN g () I implie (N g ()N g () I I) = (C DK)(A BK) F K BN g () I. The boud i (6) for ρ> become ρ< N g ()Ĉpd +(C DK)(A BK) F K BN g () I. (24) The N g i tabilized by the PID cotroller C pid i (7). B. Itegral-actio deig baed o plat augmetatio We briefly review the well-kow full-order oberverbaed itegral-actio cotroller ythei, where the itegrator tate are alo icluded i tate-feedback [5], [4]. The tate-feedback matrix ca be deiged uig pole-placemet or LQR. Let (A, B, C, D) be a tate-pace repreetatio of G R y u p, where A IR, (A, B) i tabilizable, (C, A) i detectable. Let L IR y be uch that F L S. Defie the ( + y )-th order augmeted ytem a [ ] [ ] A B A a :=,B C a :=,C D a := [ C ]. (25) y

5 [ L where L a := I y ]. The block diagram of Sy(G, Ĉa) with the ( + y )-th order Ĉa i i Fig. 4. The augmeted ytem i tabilized uig K a ; if K ξ =, the deig doe ot guaratee tability with K x actig aloe. The itegrator caot be take out of ervice completely. IV. APPLICATIONS TO RTP SYSTEMS We apply the itegral-actio cotroller deig procedure of Sectio III-A to temperature cotrol i RTP ytem. We ue the liearized model decribed i [4], with three tadard tugte haloge lamp a actuator, ad three temperature eor. Let x = [x x 2 x 3 ] T deote the temperature. The liearized MIMO ytem ha a tatepace repreetatio (A, B, C, D), where A = , B = , C = I 3, D = We deig a itegral-actio cotroller followig the twotage procedure of Sectio III-A; a full-order oberver-baed tabilizig cotroller i choe for the firt tage ad a PIcotroller for the ecod block. We chooe to deig the iitial cotroller Cg o a i (2). The etimator i deiged here by pole placemet, with the pole of F L located at { 3, 4, 5}. We ue LQR to fid a tate-feedback gai K IR 3 3 uch that F K i table. We chooe the tate weightig matrix ˆQ = 2I 3 ad the cotrol weightig matrix ˆR = I 3. The L, K IR 3 3 are L = K = , (27). (28) The cotroller C o g = K(I A+BK+L(C DK)) L = ( ij ) /d, where d =( +4.69)( +5.92)( +6.89), i a tabilizig cotroller for the plat G ad it doe ot have itegral-actio. We follow Propoitio 3. to deig r e I ξ K ξ Ĉ a v Kx w u G L C (I A) ˆx D B y the PID block for N g i (2) of the RCF G = N g Dg. Chooig ˆK p = 5I 3 ad ˆK d =, the iequality (24) become < ρ < N g ()Ĉpd +(C DK)(A BK) F K BN g () I = Chooe ρ =.3; with N g () = , C pi = ρ( ˆK p N g() ) = 4.5I N g(). Fially, with Q =, the itegral-actio cotroller Co ˆ = Cg o + D c C pi = (h ij ) / ˆd, where ˆd = d, i a fourth-order cotroller due to the third-order oberver-baed cotroller Cg o i the firt tage ad the firt-order PI block i the ecod tage. We alo deig a tadard oberver-baed cotroller Ĉa a i Sectio III-B uig the augmeted plat decriptio (25), with =3, y =3. We ue the ame tate-etimator gai L a i (27) ad chooe the augmeted tate weightig Q a = diag[2i 3, I 3 ], the cotrol weightig R = I 3. The tate feedback K a = [K x K ξ ] i foud uig LQR. The itegral-actio cotroller Ĉ a i computed uig (26). Fig. 5 how the cloed-loop tep repoe of the three temperature idividually for Sy(G, C ˆo ) ad Sy(G, Ĉa), with uit tep applied at each of the three iput. The tep repoe characteritic are very imilar for all three temperature. The repoe of Sy(G, C ˆo ) (olid lie) diplayed o overhoot ad fat rie time (le tha ec.) i the abece of actuator aturatio. Temperature uiformity o the wafer urface i alo maitaied. The repoe for Sy(G, Ĉa) are lower ad have 22% overhoot. Step repoe of Sy(G, C ˆo ) ad Sy(G, Ĉa) whe the itegrator are tured off due to actuator aturatio are how i Fig. 6. Saturatio oliearitie icluded i the cotrol loop aturate at ±.3. The tep repoe of both ytem are lowed dow becaue of the actuator aturatio, with Sy(G, C ˆo ) diplayig fater repoe. Although itegral-actio i ot available due to aturatio, the teady-tate error are egligibly mall for both ytem. Aother ati-widup Y Fig. 4. Sy(G, Ĉa) with augmeted oberver-baed cotroller Fig. 5. Step repoe of Sy(G, ˆ C o ) ad Sy(G, Ĉa).

6 techique limit the itegral value whe aturatio occur. The tep repoe of the two ytem are i Fig. 7; the ati-widup gai (K aw ) of itegrator limitig feedback loop for Sy(G, C ˆo ) ad Sy(G, Ĉa) are elected a.8 ad.2, repectively. Slightly fater repoe are obtaied compared to thoe i Fig. 6, with zero teady-tate error. I the ytem Sy(G, C ˆo ), where Ĉo = Cg o + D c C pi, the etire PI block C pi ca be tured off whe the ytem ha iput aturatio. Sice Cg o deiged to tabilize G i till active whe C pi i et to zero, the cloed-loop ytem i till table. The tep repoe of Sy(G, Cg o ) uder actuator aturatio i i Fig. 8. The repoe without the PI block i reaoably fat ad the teady-tate error i mall, although it i ot zero due to the abece of itegral-actio. Thee imulatio reult idicate that Sy(G, Ĉo) geerally ha better tep repoe tha Sy(G, C a ) i the abece of actuator aturatio. The ytem diplayed very imilar characteritic uder actuator aturatio: Repoe for both ytem lowed dow ad temperature uiformity o wafer urface wa ot maitaied; the third chael how approximately 2 ecod more delay tha the other two. Y Fig. 6. Y Step repoe of Sy(G, ˆ C o ), Sy(G, Ĉa) without itegrator. REFERENCES [] K. J. Åtröm, L. Rudqwit, Itegrator widup ad how to avoid it, Proc. America Cotrol Coferece, pp , 989. [2] P. J. Campo, M. Morari, Achievable cloed-loop propertie of ytem uder decetralized cotrol: Coditio ivolvig the teadytate gai, IEEE Tra. Automat. Cotr., Vol. 39, pp , 994. [3] T. F. Edgar, S. Butler, W. J. Campbell, C. Pfeiffer, C. Bode, S. B. Hwag, K. S. Balakriha, ad J. Hah, Automatic cotrol i microelectroic maufacturig: Practice, challege ad poibilitie, Automatica, 36:, pp , 2. [4] G. F. Frakli, J. D. Powell, ad A. Emami-Naeii, Feedback Cotrol of Dyamic Sytem, 4th ed., Pretice Hall, 22. [5] G. C. Goodwi, S. F. Graebe, ad M. E. Salgado, Cotrol Sytem Deig, Pretice Hall, 2. [6] J. C. Doyle, R. S. Smith, D. F. E, Cotrol of plat with iput aturatio oliearitie, Proc. America Cotrol Cof., pp , 987. [7] A. N. Güdeş, C. A. Deoer, Algebraic Theory of Liear Feedback Sytem with Full ad Decetralized Compeator, Lect. Note i Cotrol ad Iform. Sciece, 42, Spriger, 99. [8] A. N. Güdeş, M. G. Kabuli, Parametrizatio of tabilizig cotroller with itegral actio, IEEE Tra. Automatic Cotrol, 44:, pp. 6-9, 999. [9] A. N. Güdeş, M. G. Kabuli, Reliable decetralized itegral-actio cotroller deig, IEEE Tra. Automatic Cotrol, Vol. 46, pp , 2. [] N. Kapoor, A. R. Teel, P. Daoutidi, A ati-widup deaig for liear ytem with iput aturatio, Automatica, Vol. 34, No. 5, pp , 998. [] J. M. Maciejowki, Multivariable Feedback Deig, Addio-Weley, 989. [2] M. Morari, E. Zafiriou, Robut Proce Cotrol, Pretice Hall, 989. [3] K. Ogata, Moder Cotrol Egieerig, 3rd editio, Pretice Hall, New Jerey, 997. [4] J. D. Stuber, I. Trachteberg, T. F. Edgar, Deig ad modelig of rapid thermal proceig ytem, IEEE Tra. Semicod. Maufacturig, :3, pp , 998. [5] M. Vidyaagar, Cotrol Sytem Sythei: A Factorizatio Approach, Cambridge, MA: M.I.T. Pre, Fig. 7. Step repoe of Sy(G, C ˆo ), Sy(G, Ĉa) with limited itegrator. Y Fig. 8. Step repoe of Sy(G, Cg o ) with aturated iput.