SIMULATION OF CASCADE CONTROL OF A CONTINUOUS STIRRED TANK REACTOR

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1 SIMULATION OF CASCADE CONTROL OF A CONTINUOUS STIRRED TANK REACTOR Pet Dotál,, Vladmí obál,, ad Jří Vojtěše Cete of Polyme Sytem, Uvety Ittute, Toma ata Uvety Zl, Nad Ovcou 35, 7 Zl, Czech Republc. Depatmet of Poce Cotol, Faculty of Appled Ifomatc, Toma ata Uvety Zl, Nad Staem 5, 7 5 Zl, Czech Republc {dotalp;bobal;vojtee}@fa.utb.cz KEYWORDS Adaptve cotol, cacade cotol, cotuou ted ta eacto, cotol mulato. ASTRACT The pape peet mulato eult of the cacade cotol of a cotuou ted ta eacto. The cotol pefomed pmay ad ecoday cotol-loop whee the pmay cotolled output of the eacto the cocetato of a deed eacto poduct, ad, the ecoday output the eactat tempeatue. A commo cotol put the coolat flow ate. The cotolle the pmay cotol-loop a P-cotolle wth the ga calculated ug mulated o meaued teady-tate chaactetc of the eacto. The cotolle the ecoday cotol-loop a adaptve cotolle. The popoed method vefed by cotol mulato INTRODUCTION The cacade cotol method allow the cotol of pocee wth a ma ad ecoday cotolled vaable ad wth a gle cotol put. The method epecally ueful whe a ma cotolled output ca be meaued oly loge tme teval ad wth a addtoal output meauable hote tme peod. Pcple of the cacade cotol ae decbed e.g. (equette ; Mahoey et al. ; Sebog et al. 99; Smut ). Chemcal eacto ae typcal pocee utable fo a ue of the cacade cotol. I cae of o-othemal eacto, cocetato of the eacto poduct motly deped o a tempeatue of the eactat. Futhe, t ow that whle the eactat tempeatue ca be meaued almot cotuouly, cocetato ae uually meaued loge tme teval. The, the applcato of the cacade cotol method ca lead to good eult. I th pape, the cacade cotol decpto of a cotuou ted ta eacto (CSTR) wth eult of cotol mulato peeted. CSTRS ae appaatu wdely ued chemcal duty, botechologe, polyme maufactug, ad may othe. Fom the ytem theoy pot of vew, CSTR belog to a cla of olea ytem wth mathematcal model decbed by et of olea dffeetal equato a t ca be ee e.g. (Smth 5; Coou ). Hee, the cacade cotol-loop, the cocetato of a deed poduct of eacto codeed a the pmay cotolled vaable, ad, the eactat tempeatue a the ecoday cotolled vaable. The coolat flow ate epeet a commo cotol put. The pmay cotol vaable meaued dcete tme teval. The pmay cotolle detemg the et pot fo the ecoday (e) cotol-loop a dcete olea popotoal cotolle deved o the ba of teady-tate chaactetc of the eacto. Sce the cotolled poce olea, a cotuou-tme adaptve cotolle ued a the ecoday cotolle. The pocedue fo the adaptve cotol deg the e cotol-loop baed o appoxmato of the olea model of the CSTR by a cotuou-tme exteal lea model (CT ELM) wth ecuvely etmated paamete. I the poce of paamete etmato, the dect method by (Rao ad Ubehaue ); Gae ad Wag ) ued. The cotol loop wth two feedbac adaptve cotolle ued, ee, e.g. (Dotál et al. 7). The eultg cotolle ae deved by the pole placemet method, ee, e.g. (Gmble 993; Kučea 993; oga 99; Fal et al. ). The cacade cotol vefed by mulato o the olea model of the CSTR. NONLINEAR MODEL OF THE CSTR Code a CSTR wth exothemc eacto accodg to the cheme A, C ad wth a pefectly mxed coolg jacet. The deed poduct the compoet. Ug uual mplfcato, the model of the CSTR decbed by fou olea dffeetal equato dc q A = + ( c c ) () A Af A dt V dc q = + ( c f dt V c) () dt AU h ( f ) ( c ) dt ( ρcp) + V V( ρcp) (3) dtc qc AhU = ( Tcf Tc ) + ( T Tc ) dt Vc Vc( ρcp) c () whee = c = c (5) A Poceedg 9th Euopea Cofeece o Modellg ad Smulato ECMS Vale M. Mladeov, Peta Geogeva, Gha Spaov, Galdya Petova (Edto) ISN: / ISN: (CD)

2 ad, wth tal codto c () = c, c () = c, A T () = T ad T () = T. Hee, t tad fo the tme, c c c fo cocetato, T fo tempeatue, V fo volume, ρ fo dete, c p fo pecfc heat capacte, c q fo volumetc flow ate, fo eacto ate, A h the heat exchage uface aea ad U the heat tafe coeffcet. Subcpt deoted decbe the eactat mxtue, c the coolat, f the let value ad the upecpt teady-tate value. The eacto ate ad the eacto heat ae expeed a E j j = j exp, j =, (7) RT h = h + h () whee aee pe-expoetal facto, E ae actvato eege ad d h ae eacto ethalpe. The value off all paamete, let value ad the teady-tate value ae gve Table. Table : Paamete ad Ilet Value V =.7 m 3 V c =. m 3 ρ = 95 g m -3 ρ c = 99 g m -3 = 5.. m - =. m - h =.. J mol - c =.5 mol m -3 Af f T = 33 K A = THE CONTROL OJECTIVE = q =. m 3 m - A bac cheme of the cacade cotol Fg.. c p =.5 J g - K - c pc =. J g - K - A h = 5.5 m U = 3.5 JJ m - m - K - E / R = 35 K E / R = 5 K h =.. J mol - = mol m -3 c f T cf = 93 K A () ceag eactat tempeatue, the ecod ubteval t aga deceae. Smulato of the teady- tate chaactetc wa pefomed the teval 3K T 355K. The poduct cocetato eache a maxmum value hghe tha mol/m 3. Howeve, wth epect to ome followg pocedue, the maxmum deed d value c wll be lmted jut by max c w =mol/ 3 /m. c 3 (mol/m )...9. c ma ax w Iteval T Fgue : Steady-tate depedece of thee poduct cocetato o the eactat tempeatue. THE PC DESIGN Iteval The pocedue the deg of the olea PC appea fom polyomal appoxmato of thee teady-tatee chaactetc. The boudae of opeatg teval ae detemed a.733 c, 3 T the ft opeatg teval, ad c.33, T 355 the ecod opeatg teval. Fo pupoe of o appoxmato, the tempeatue tafomed a m T T ξ = max m T T, ξ, (9) m max whee T = 3K, T = 355K. The teady-tate chaactetc wth tafomed eactat tempeatue ca be ee Fg. 3. Fgue : Cacade cotol cheme. Hee, PC tad fo the olea popotoal cotolle, AC fo the adaptve cotolle ad CSTR R fo the eacto. The cotol objectve to acheve a cocetato of the compoet a the pmay cotolled output ea to t maxmum. A depedece of the cocetato of o the eactat tempeatue Fg.. A opeatgg teval cot of two pat. I the ft ubteval, the cocetato ceae wth c (mol/m3 )...9. c w max Iteval Iteval ξ () Fgue 3: Steady-tate depedece of thee poduct cocetato o tafomed eactat tempeatue. Futhe, the polyomal appoxmato of teady-tatee chaactetc above teval cludg the

3 devatve tae fom 3 c = 7.ξ +.9ξ +.77ξ +.7 () dc = ξ + 93ξ +.77 () dξ the ft opeatg teval, ad, 3 c = 3.39ξ 7.75ξ +.573ξ +.7 () dc 9.97ξ 5.757ξ.573 dξ = + (3) the ecod opeatg teval. Steady-tate chaactetc both teval togethe wth the appoxmato ae Fg. ad 5. c (mol/m3 ) Fgue : Steady-tate chaactetc the teval. c (mol/m3 )...9. Smulated data Appoxmato ξ () Smulated data Appoxmato ξ Fgue 5: Steady-tate chaactetc the teval. Now, a dffeece of the deed value of the eactat tempeatue the output of the PC ca be computed fo each c a U L dξ Δ Tw = Gw( T T ) Δcw () dc c whee G w a electable ga coeffcet. The devatve () calculated fom veo of () ad (3). ADAPTIVE CONTROLLER DESIGN The teady-tate depedece of the eactat tempeatue o the coolat flow ate ca be ee Fg.. It oleaty evdet. It hould be oted that the deed tempeatue value hall ot be fom the teval < T < Th equemet ca be fulflled by pogammg mea. Exteal Lea Model of the CSTR Fo the cotol pupoe, the cotolled output ad the cotol put ae defed a 3 T q c (m3 /m) Fgue : Steady-tate depedece of eactat tempeatue o the coolat flow ate. c c y() t = Δ T () t = T () t T, u() t = q () t q. (5) The CT ELM popoed the tme doma o the ba of pelmaly mulated tep epoe the fom of the ecod ode dffeetal equato yt () + a yt () + a yt () = b ut () () ad, the complex doma a the tafe fucto b G () =. (7) + a+ a CT ELM Paamete Etmato The method of the CT ELM paamete etmato ca be befly caed out a follow. Sce the devatve of both put ad output caot be dectly meaued, flteed vaable u f ad y f ae etablhed a output of flte c( σ ) u f ( t) = u( t) () c( σ ) y f ( t) = y( t) (9) whee σ = ddt the devatve opeato, c(σ) a table polyomal σ that fulfll the codto deg c( σ ) deg a( σ ). Note that the flte tme cotat mut be malle tha the tme cotat of the poce. Sce the latte ae uow at the begg of the etmato pocedue, t eceay to mae the flte tme cotat, elected a po, uffcetly mall. Wth egad to (7), the polyomal a tae the cocete fom a( σ) = σ + aσ + a, ad, the polyomal c ca be choe a c( σ) = σ + cσ + c. Subequetly, the value of the flteed vaable ca be computed dug the cotol by a oluto of () ad (9) ug ome tadad tegato method. It ca be ealy poved that the tafe behavo amog

4 flteed ad amog uflteed vaable ae equvalet. Flteed vaable cludg the devatve ca be ampled fom flte () ad (9) dcete tme teval t = T S, =,,,... whee T S the amplg peod. Now, the egeo vecto defed a Φ(tt ) = ( yf ( t ) y f ( t) uf ( t ) ) ( () ad, the vecto of paamete ca be etmated fom the ARX model T y ( t ) = Θ ( t ) Φ( t ) + e( t ). () Hee, the ecuve detfcato method wth expoetal ad dectoal fogettg wa ued accodg to (obál et al. 5). Cotolle Deg f T Θ ( t ) = a ( a b ) The cotol ytem wthh two feedbac cotolle depcted Fg.7. Fgue 7: Cotol Sytem. I the cheme, w the efeece gal w= Tw T, y deote the cotolled output, e the tacg eo ad u the cotolle output. The equece of efeece gal compoed of tepp fucto wth the tafom w W ( ) =. (3) The tafe fucto G epeet appoxmate tafe fucto the geeal fom (7). The tafe fucto of cotolle aee q ( Q () = ) p ( ), ( ) R () = () p ( ) whee q, ad p ae copme polyomal. The deg of cotolle decbed th ecto follow fomm the polyomal appoach. The devato decbed detal e.g. (Dotál et al. ). The eult off devato ca be ummazed a follow: The polyomal () ae gve by a oluto of the polyomal equato d( ( ) = a( ) p ( ) + b ()( () + q ()) (5) whee d a chaactetc polyomal wth oot a pole of the cloed-loop. Etablhg the polyomal t a t () = () + q () () () ad ubttutg () to (5), the polyomal p ad t ae gve by a oluto of the polyomal equato a () p () + b() )() t = d () (7) wth a table polyomal d o the ght de. Wth egad too the tafom (3), the aymptotc tacg povded fo polyomal p ad a q havg fom p ( ( ) = p( ), q () = q (). () Subequetly, the tafe fucto () tae fom q () () Q( ) =, R() = (9) p () p () It ca be ealy poved p that the degee of polyomal p q ad mut fulfll equalte degq deg p, deg deg p+. (3) Now, the polyomal t ca be e ewtte to the t fom t () = () + q(). (3) Subequetly, the degee off polyomal (7) ad (9) ca be ealy deved a deg t = deg = deg a,,degq = deg a. (3) deg p deg a, degg d dega The, deotg deg a =, polyomal t, ad q have fom ad, elato amog the coeffcet ae = t, + q = t fo =,...,. (3) Sce by a oluto of the polyomal equato e (7) povde calculato of coeffcet t, uow coeffcet ad a q ca bee obtaed by a choce of electable coeffcet β, uch that The coeffcet β dtbute a weght betwee umeato of tafe fucto Q ad R. Fo the ecod ode model (7) wth deg a =, the cotolle' tafe fucto tae pecfc fom whee = = β t, q = ( β ) t fo =,...,. (35) q () q + q Q( ) = = p () + p () + + R( ) = = p () ( + p ) t( ) = t, () =, q () = q (33) = = The cotolle paamete p the eult fom a oluto of thee polyomall equato (7) ad deped upo coeffcet of the polyomal d. The ext poblem heee to fd a table polyomal d that eable to obta acceptable tablzg cotollle.. (3) = t, = βt, = βt. (37) q = ( β ) t, q = ( β ) t

5 I th pape, the polyomal d wth oot detemg the cloed-loop pole choe a d () = ()( + α) (3) whee a table polyomal obtaed by pectal factozato a () a() = () () (39) ad α the electable paamete. The coeffcet of the ae expeed a a =, = a + a () ad, the cotolle paamete p ad t ca be obtaed fom oluto of the matx equato p d3 a a b t = d a () a b t d b t d whee d = + α, d = α + + α. () d d 3 = α + α, = α Now, t follow fom the above toduced pocedue that tug of cotolle ca be pefomed by a utable choce of electable paamete β ad α. The cotolle paamete ad q ca the be obtaed fom (37). SIMULATION RESULTS The goal of mulato to how the effect of electable paamete o the cotol coue. I th pape, the cotol mulato wee pefomed oly the ft opeatg teval. The mulato tated at the tatg pot c A = 77 mol/m 3, c = 3 mol/m 3, T = 33. K, T c = 3.5 K ad q c =. m 3 /m. I all mulato, the deed value c w = mol/m 3 ha bee choe. Fo the tat (the adaptato phae), the P cotolle wth a mall ga wa ued all mulato. A effect of the paamete G w o the cotol evdet fom Fg.. A ceag G w acceleate all gal the cotol loop. Howeve, t value ot uetcted ad t coveet value hould be foud expemetally. A effect of the peod t the ame opeatg teval ca be ee Fg. 3. Although hoteg t lead to fate cotol epoe, t legth detemed by meauemet poblte. The tedecy to ovehoot at mall meauemet peod ca be uppeed by electg a lowe ga G w. A fluece of the cloed- loop pole α o the cotol epoe ca be ee Fg.. Choog a hghe α value ca lead to ocllato. The lat goup of mulato Fg. 7-9 how a fluece of the paamete β o the cotol coue. Hee wee choe oly by h exteme value. It ca be ee that a favoable choce of othe paamete, the cotol qualty ca be mpoved jut by a utable opto of paamete β. T c (mol/m 3 ) G w =. G w = Fgue : Effect of G w : Refeece Sgal Coue (t =, α =., β = β = ) G w =. G w = Fgue 9: Effect of G w : Reactat Tempeatue Repoe (t =, α =., β = β = ).. c w G w =. G w = Fgue : Effect of G w : Cocetato Repoe (t =, α =., β = β = ). t = 5 t = Fgue : Effect of t : Refeece Sgal Coue (G w =., α =., β = β = ).

6 T c (mol/m 3 ) t = 5 t = Fgue : Effect of t : Reactat Tempeatue Repoe (G w =., α =., β = β = ).. c w t = 5 t = Fgue 3: Effect of t : Cocetato Repoe (G w =., α =., β = β = ). α =. α = Fgue : Effect of α: Refeece Sgal Coue (G w =.5, t =, β = β = ). c (mol/m 3 ) T Fgue : Effect of α: Cocetato Repoe (G w =., t =, β = β = ). c w α =. α = Fgue 7: Effect of β: Refeece Sgal Coue (G w =.5, t =, α =.5) β = β = β = β = Fgue : Effect of β: Reactat Tempeatue Repoe (G w =.5, t =, α =.5). T α =. α = Fgue 5: Effect of α: Reactat Tempeatue Repoe (G w =., t =, β = β = ). c (mol/m 3 ). c w β = β = Fgue 9: Effect of β: Cocetato Repoe (G w =.5, t =, α =.5).

7 CONCLUSIONS The pape deal wth the cacade cotol of a cotuou ted ta eacto. A eceay codto fo a ue of the peeted method meauemet of a ma poduct of the eacto tag place the eacto. The cotol pefomed the exteal ad e cloed-loop whee the cocetato of a ma poduct the pmay ad the eactat tempeatue the ecoday cotolled vaable. A commo cotol put the coolat flow ate. The cotolle the exteal cotol-loop a dcete olea P-cotolle deved o the ba of teadytate chaactetc of the eacto. The e cotolloop cot of two adaptve feedbac cotolle. Fo the devato, the ecuve paamete etmato, the polyomal appoach ad the pole placemet method ae appled. The pape cota umeou mulato documetg the fluece of each electable cotolle paamete o the cotol. ACKNOWLEDGEMENT Th atcle wa wtte wth uppot of Opeatoal Pogam Reeach ad Developmet fo Iovato cofuded by the Euopea Regoal Developmet Fud (ERDF) ad atoal budget of Czech Republc, wth the famewo of poject Cete of Polyme Sytem (eg. umbe: CZ..5/../3.). REFERENCES equette,. W.. Poce Dyamc: Modelg, Aaly, ad Smulato. Petce- Hall PTR, New Jeey. obál, V., J. öhm, J. Fel, ad J. Macháče. 5. Dgtal elf-tug cotolle, Spge Velag, el. oga, W.L. 99. Mode cotol theoy. New Jeey, Petce Hall, New Jeey. Coou, J.-P.. Poce cotol. Theoy ad applcato. Spge Velag, Lodo. Dotál, P., F. Gazdoš, V. obál, ad J. Vojtěše. 7. "Adaptve cotol of a cotuou ted ta eacto by two feedbac cotolle". I Poc. 9th IFAC Wohop Adaptato ad Leag Cotol ad Sgal Poceg ALCOSP'7, Sat Petebug, Rua, P5- P5-. Dotál, P., V. obál, J. Vojtěše, ad Z. abí.. Oe appoach to adaptve cotol of a tubula chemcal eacto. WSEAS Taacto o Flud Mechac, 7, 3-. Fal, G.F., J.D. Powell, ad A. Emam-Nae. Feedbac cotol of dyamc ytem. Petce Hall, New Jeey. Gae, H. ad L. Wag (ed.).. Idetfcato of cotuou-tme model fom ampled data. Spge- Velag, Lodo. Gmble, M.J Robut dutal cotol. Optmal deg appoach fo polyomal ytem. Petce Hall, Eglewood Clff. Kučea, V "Dophate equato cotol A uvey". Automatca, 9, Mahoey, D.P., W.Y. Svce, ad.r. Youg. () A Real- Tme Appoach to Poce Cotol. Joh Wley ad So, Chchete. Rao, G.P., ad H. Ubehaue.. Idetfcato of cotuou-tme ytem. IEE Poc.-Cotol Theoy Appl., 53, 5-.. Sebog,D.E., T.F. Edga, ad D.A. Mellchamp. 99. Poce dyamc ad cotol. Joh Wley ad So, Chchete. Smth, R. 5. Chemcal poce deg ad tegato. Joh Wley ad So, Chchete. Smut, J.F.. Poce cotol fo pacttoe. OptCotol, New Yo. AUTHOR IOGRAPHIES PETR DOSTÁL tuded at the Techcal Uvety of Padubce, Czech Republc, whee he obtaed h mate degee 9 ad PhD. degee Techcal Cybeetc 979. I the yea he became pofeo Poce Cotol. He ow head of the Depatmet of Poce Cotol, Faculty of Appled Ifomatc of the Toma ata Uvety Zlí. H eeach teet ae modellg ad mulato of cotuou-tme chemcal pocee, polyomal method, optmal ad adaptve cotol. You ca cotact hm o emal adde dotalp@fa.utb.cz. VLADIMÍR OÁL gaduated 9 fom the o Uvety of Techology, Czech Republc. He eceved h Ph.D. degee Techcal Cybeetc at Ittute of Techcal Cybeetc, Slova Academy of Scece, atlava, Slova Republc. He ow Pofeo at the Depatmet of Poce Cotol, Faculty of Appled Ifomatc of the Toma ata Uvety Zlí, Czech Republc. H eeach teet ae adaptve ad pedctve cotol, ytem detfcato, tme-delay ytem ad CAD fo automatc cotol ytem. You ca cotact hm o emal adde bobal@fa.utb.cz JIŘÍ VOJTĚŠEK tuded at Toma ata Uvety Zl, Czech Republc, whee he eceved h M.Sc. degee Automato ad cotol. I 7 he obtaed Ph.D. degee Techcal cybeetc at Toma ata Uvety Zl. He ow wo a a atat pofeo at the Depatmet of Poce Cotol, Faculty of Appled Ifomatc of the Toma ata Uvety Zl, Czech Republc. H eeach teet ae modelg ad mulato of cotuoutme chemcal pocee, polyomal method, optmal, adaptve ad olea cotol. You ca cotact hm o emal adde vojtee@fa.utb.cz