One approach to adaptive control of a tubular chemical reactor

One App oach o Adap i e Con ol o a Tubula Chemical Reac o PETR DOSTÁL, VLADIMÍR BOBÁL, JIŘÍ VOJTĚŠEK, and ZDENĚK BABÍK Tomas Ba a Uni e si y in Zlin Depa men o P ocess Con ol, Cen e o Polyme Sys ems nam. T.G. Masa yka 5555, 760 01 Zlin CZECH REPUBLIC {dos alp, bobal, oj esek, babik}@ ai.u b.cz h p://www. ai.u b.cz/ Abs ac : – The pape deals wi h con inuous- ime adap i e con ol o a ubula chemical eac o wi h he coun e cu en cooling as a nonlinea single inpu – single ou pu p ocess. The mean eac an empe a u e and he ou pu eac an empe a u e a e chosen as he con olled ou pu s, and, he coolan low a e as he con ol inpu . The nonlinea model o he eac o is app oxima ed by an ex e nal linea model wi h a s uc u e chosen on he basis o con olled ou pu s s ep esponses. I s pa ame e s a e es ima ed ia co esponding del a model. The con ol sys em s uc u e wi h wo eedback con olle s is conside ed. The esul ing con olle s a e de i ed using polynomial app oach. The me hod is es ed on a ma hema ical model o he ubula chemical eac o . Key-Wo ds: – Nonlinea sys em, ubula chemical eac o , app oxima e linea model, pa ame e iden i ica ion, polynomial app oach, pole assignmen . 1 In oduc ion Tubula chemical eac o a e uni s equen ly used in chemical indus y. F om he sys em heo y poin o iew, ubula chemical eac o s belong o a class o nonlinea dis ibu ed pa ame e sys ems wi h ma hema ical models desc ibed by se s o nonlinea pa ial di e en ial equa ions (NPDRs). The me hods o modelling and simula ion o such p ocesses a e desc ibed e.g. in [1] – [5]. I is well known ha he con ol o chemical eac o s, and, ubula eac o s especially, o en ep esen s e y complex p oblem. The con ol p oblems a e due o he p ocess nonlinea i y, i s dis ibu ed na u e, and high sensi i i y o he s a e and ou pu a iables o inpu changes. E iden ly, he p ocess wi h such p ope ies is ha dly con ollable by con en ional con ol me hods, and, i s e ec i e con ol equi es applica ion some o ad anced me hods. He e, a ious e icien me hods may be used as he p edic i e con ol, e.g. [6], [7], [8], he obus con ol, e.g. [9], he uzzy nonlinea con ol, e.g. [10], he model e e ence con ol, e.g. [11], o nonlinea con ol, e.g. [12], [13] and [14]. Some o he s me hods a e desc ibed in [15]. One possible me hod o cope wi h his p oblem is using adap i e s a egies based on an app op ia e choice o a con inuous- ime ex e nal linea model (CT ELM) wi h ecu si ely es ima ed pa ame e s. These pa ame e s a e consequen ly used o pa allel upda ing o con olle ‘s pa ame e s. Some esul s ob ained in his ield we e p esen ed by au ho s o his pape e.g. in [16] and [17]. Fo he CT ELM pa ame e es ima ion, ei he he di ec me hod [18] and [19] o applica ion o an ex e nal del a model wi h he same s uc u e as he CT model can be used. The basics o del a models ha e been desc ibed in e.g. [20] and [21]. Al hough del a models belong in o disc e e models, hey do no ha e such disad an ageous p ope ies connec ed wi h sho ening o a sampling pe iod as disc e e z- models. In addi ion, pa ame e s o del a models can di ec ly be es ima ed om sampled signals. Mo eo e , i can be easily p o ed ha hese pa ame e s con e ge o pa ame e s o CT models o a su icien ly small sampling pe iod (compa ed o he dynamics o he con olled p ocess), as shown in [22]. This pape deals wi h con inuous- ime adap i e con ol o a ubula chemical eac o wi h a coun e cu en cooling as a nonlinea single inpu – single ou pu p ocess. Wi h espec o p ac ical possibili ies o a measu emen and con ol, he mean eac an empe a u e and he ou pu eac an empe a u e a e chosen as he con olled ou pu s, and, he coolan low a e as he con ol inpu . The nonlinea model o he eac o is app oxima ed by a CT ex e nal linea model wi h a s uc u e chosen on he basis o compu ed con olled ou pu s s ep esponses. The pa ame e s o he CT ELM hen a e es ima ed ia co esponding del a model. The con ol s uc u e wi h wo eedback con olle s is conside ed, e.g. [23]. The esul ing con olle s a e de i ed using he polynomial app oach [24] and he pole assignmen me hod (see, e.g. [25]). The me hod WSEAS TRANSACTIONS on FLUID MECHANICS Pe Dos ál, Vladimí Bobál, Jiří Voj ěšek, Zdeněk Babík E-ISSN: 2224-347X 13 Issue 1, Volume 7, Janua y 2012 is es ed on a ma hema ical model o a ubula chemical eac o . 2 Model o he Reac o An ideal plug- low ubula chemical eac o wi h a simple exo he mic consecu i e eac ion 12 kk ABC→→ in he liquid phase and wi h he coun e cu en cooling is conside ed. Hea losses and hea conduc ion along he me al walls o ubes a e assumed o be negligible, bu dynamics o he me al walls o ubes a e signi ican . All densi ies, hea capaci ies, and hea ans e coe icien s a e assumed o be cons an . Unde abo e assump ions, he eac o model can be desc ibed by i e PDRs in he o m 1 AA A cc kc z ∂∂ +=− ∂∂ (1) 12 BB AB cc kckc z ∂∂ +=− ∂∂ (2) 1 1 4() () () w p p TTQ U TT zcdc ρρ ∂∂ += − − ∂∂ (3) [ ] 11 22 21 22 4() ()() ( w w pw cw TdU T T dd c dU T T ρ ∂=−+ ∂− +− (4) 12 2 22 312 4() ()() cc cwc pc TT ndU TT z dnd c ρ ∂∂ −= − ∂∂ − (5) wi h ini ial condi ions (,0) () s AA cz cz=, ( ,0) ( ) s BB cz cz=, ( ,0) ( ) s Tz T z=, (,0) () s ww Tz Tz=, ( ,0) ( ) s cc Tz T z= and bounda y condi ions 0 (0, ) ( ) AA c c =(kmol/m3), 0 (0, ) ( ) BB c c =(kmol/m3), 0 (0, ) ( ) T T =(K), (,) () ccL TL T =(K). He e, is he ime, z is he axial space a iable, c a e concen a ions, T a e empe a u es, a e luid eloci ies, d a e diame e s, ρ a e densi ies, cp a e speci ic hea capaci ies, U a e hea ans e coe icien s, n1 is he numbe o ubes and L is he leng h o ubes. The subsc ip (⋅) s ands o he eac an mix u e, (⋅)w o he me al walls o ubes, (⋅)c o he coolan , and he supe sc ip (⋅)s o s eady-s a e alues. The eac ion a es and hea o eac ions a e nonlinea unc ions exp essed as 0exp j jj E kk R T − ⎛⎞ =⎜⎟ ⎝⎠ , j = 1, 2 (6) 11 2 2 ()() A B QHkcHkc=−Δ +−Δ (7) whe e k0 a e p e-exponen ial ac o s, E a e ac i a ion ene gies, () H −Δ a e in he nega i e conside ed eac ion en alpies, and R is he gas cons an . The luid eloci ies a e calcula ed ia he eac an and coolan low a es as 2 11 4 q nd π = , 22 312 4 () c c q dnd π =− (8) The pa ame e alues wi h co esponden uni s used o simula ions a e gi en in Table 1. Table 1. Used pa ame e alues L = 8 m n1 = 1200 d1 = 0.02 m d2 = 0.024 m d3 = 1 m ρ = 985 kg/m3 cp = 4.05 kJ/kg K ρw = 7800 kg/m3 cpw = 0.71 kJ/kg K ρc = 998 kg/m3 cpc = 4.18 kJ/kg K U1 = 2.8 kJ/m2s K U2 = 2.56 kJ/m2s K k10 = 5.61⋅1016 1/s k20 = 1.128⋅1018 1/s E1/R = 13477 K E2/R = 15290 K (-ΔH 1) = 5.8⋅104 kJ/kmol (-ΔH 2) = 1.8⋅104 kJ/kmol F om he sys em enginee ing poin o iew, ou (,) AA cL c=, ou (,) BB cL c=, ou (,) TL T= and ou (0, ) cc T T= a e he ou pu a iables, and, () q , ( ) c q , 0() A c , 0() T and ( ) cL T a e he inpu a iables. Among hem, o he con ol pu poses, mos ly he coolan low a e can be aken in o accoun as he con ol a iable, whe eas o he inpu s en e ing in o he p ocess can be accep ed as dis u bances. In his pape , he mean eac an empe a u e gi en by 0 1 () (,) L m T Tz dz L =∫ (9) and he eac an ou pu empe a u e ou () T a e conside ed as he con olled ou pu s. 3 Compu a ion Models Fo compu a ion o bo h s eady-s a e and dynamic cha ac e is ics, he ini e di e ences me hod is employed. The p ocedu e is based on subs i u ion o he space in e al 0,zL∈< > by a se o disc e e WSEAS TRANSACTIONS on FLUID MECHANICS Pe Dos ál, Vladimí Bobál, Jiří Voj ěšek, Zdeněk Babík E-ISSN: 2224-347X 14 Issue 1, Volume 7, Janua y 2012 node poin s { } i z o i = 1, … , n , and, subsequen ly, by app oxima ion o de i a i es wi h espec o he space a iable in each node poin by ini e di e ences. Two ypes o ini e di e ences a e applied, ei he he backwa d ini e di e ence 1 (,) ( ,) (,) (,) ( 1,) i ii zz yz yz yz zh yi yi h − = − ∂≈= ∂ −− = (10) o he o wa d ini e di e ence 1 (,)(,) (,) (1,) (,) i ii zz yz yz yz zh yi yi h + = − ∂≈= ∂ +− = . (11) He e, a unc ion ( , )yz is con inuously di e en iable in he in e al 0,L<> , and, hLn= is he disk e iza ion s ep. 3.1 Dynamic model Applying he subs i u ions (10), (11) in (1) – (5) and, omi ing he a gumen in pa en hesis, PDRs (1) – (5) a e app oxima ed by a se o ODRs in he o m [] 01 0 () () () ( 1) AAA dc i bkicibci d =− + + − (12) [] 102 0 () () () () () (1) BAB B dc i kic i b kic i d bc i =−+ + +− (13) 1020 2 () () ( ) () ( 1) () w dT i bQ i b b T i b T i d bT i =−+ +−+ + (14) [][] 34 () () () () () w w cw dT i bTi Ti bTi T i d =−+− (15) 56 5 6 () ()()(1) () ccc w dT m bbTmbTm d bT m =− + + + + + (16) o 1,...,in= and 1mni=−+, and, wi h ini ial condi ions (,0) () s AA ci ci=, (,0) () s BB ci ci=, (,0) () s Ti T i=, (,0) () s ww Ti Ti= and ( ,0) ( ) s cc Ti T i= o 1,...,in=. The bounda y condi ions en e in o Eqs. (12) – (14) and (16) o i = 1 . Now, nonlinea unc ions in Eqs. (12) – (16) ake he disc e e o m 0 () exp () j jj E ki k R Ti − ⎛⎞ =⎜⎟ ⎝⎠ , j = 1, 2 (17) 11 22 () ( ) () () ( ) () () A B Qi H kic i Hkici =−Δ + +−Δ (18) o i = 1, … , n. The pa ame e s b in Eqs. (12) – (16) a e calcula ed om o mulas 0 bh =, 1 1 () p bc ρ =, 1 21 4 () p U bdc ρ =, 11 322 21 4 ()() p w dU bdd c ρ =−, 22 422 21 4 ()() p w dU bdd c ρ =− (19) 5c bh =, 12 2 622 312 4 ()() p c ndU bdnd c ρ =−. He e, he o mulas o compu a ion o Tm and ou T ake he disc e e o m 1 1 () ( ,) n m i i T Tz n= =∑, ou () ( ,) n T Tz = (20) 3.2 S eady-s a e model Compu a ion o he s eady-s a e cha ac e is ics is necessa y no only o a s eady-s a e analysis bu he s eady s a e alues ( ) s yi also cons i u e ini ial condi ions in ODRs (12) – (16) (he e, y p esen s some o he a iable in he se (12) – (16)). The s eady-s a e model can simply be de i ed equa ing he ime de i a i es in (12) – (16) o ze o. Then, a e some algeb aic manipula ions, he s eady-s a e model akes he o m o di e ence equa ions 0 01 () ( 1) () ss AA s b ci ci bki =− + (21) 10 02 1 () () () ( 1) () ssss BAB s ci kici bci bki ⎡ ⎤ =+− ⎣ ⎦ + (22) 10 2 02 1 () () ( 1) () ssss w Ti bQi bTi bTi bb ⎡ ⎤ =+−+ ⎣ ⎦ + (23) 34 34 1 () () () sss w c Ti bTi bTi bb ⎡ ⎤ =+ ⎣ ⎦ + (24) WSEAS TRANSACTIONS on FLUID MECHANICS Pe Dos ál, Vladimí Bobál, Jiří Voj ěšek, Zdeněk Babík E-ISSN: 2224-347X 15 Issue 1, Volume 7, Janua y 2012 56 56 1 () ( 1) () sss ccw Tm bTm bTm bb ⎡⎤ =++ ⎣⎦ + (25) o 1,...,in= and 1mni=−+. Nonlinea unc ions acco dan wi h a s eady-s a e a e 0 () exp () j s jj s E ki k R Ti ⎛⎞ − =⎜⎟ ⎜⎟ ⎝⎠ , j = 1, 2 (26) 11 22 () ( ) () () ( ) () () sss A ss B Qi H kici Hkici =−Δ + +−Δ (27) Now, he o mulas o compu a ion T m and ou T ha e he o m 1 1() n ss m i i TTz n= =∑, ou () ss n TTz= (28) 3.3 S eady-s a e and dynamic cha ac e is ics Typical eac an empe a u e p o iles along he eac o ubes compu ed o 02.85 s A c=, 00 s B c=, 0323 s T=, 0293 s c T= and 0.15 s q= o a ious coolan low a es a e shown in Fig. 1. A p esence o a maximum on he eac an empe a u e p o iles is a common p ope y o many ubula eac o s wi h exo he mic eac ions. 012345678 320 330 340 350 360 Reac an empe a u e (K) z (m) 1 - qs c = 0.2 2 - qs c = 0.25 3 - qs c = 0.3 1 2 3 Fig. 1 Reac an empe a u e p o iles o a ious coolan low a es. A dependences o he eac an mean empe a u e and he eac an ou pu empe a u e on he coolan low a e is shown in Fig. 2. The o m o bo h cu es documen s a nonlinea ela ion be ween supposed con olled ou pu s and he coolan low a e which is conside ed as he con ol inpu . Dynamic cha ak e is ics we e compu ed in he neighbou hood o he chosen ope a ing poin 3 0.27 m /s s c q=, 334.44K s m T=, ou 326.10K s T= Fo he dynamic analysis and subsequen con ol pu poses, he con olled ou pu s a e de ined as 0.20 0.25 0.30 0.35 0.40 310 315 320 325 330 335 340 345 Ts m Ts ou Ts ou = 326.10 Tempe a u es (K) qs c (m3/s) qs c = 0.27 Ts m = 334.44 Fig. 2 Dependence o he eac an mean empe a u e on he coolan low a es. de ia ions om s eady alues 1 2 ou ou ou () () () () () () s mmm s y T T T y T T T =Δ = − =Δ = − . (29) Such o m is equen ly used in he con ol. The de ia ion o he coolan low a e is deno ed as () s cc c qq qΔ= −. (30) The esponses o bo h ou pu s o he coolan low a e s ep changes a e shown in Figs. 3, 4. 0 50 100 150 200 -12 -10 -8 -6 -4 -2 0 2 4 6 8 y1 (K) (s) 1 - Δqc = - 0.05 2 - Δqc = - 0.025 3 - Δqc = 0.025 4 - Δqc = 0.05 1 2 3 4 Fig. 3 Reac an mean empe a u e s ep esponses. 0 50 100 150 200 -10 -8 -6 -4 -2 0 2 4 y2 (K) (s) 1 - Δqc = - 0.05 2 - Δqc = - 0.025 3 - Δqc = 0.025 4 - Δqc = 0.05 1 2 3 4 Fig. 4 Reac an ou pu empe a u e s ep esponses. The abo e shown esponses demons a e mo e exp essi e nonlinea beha iou o he eac an ou pu empe a u e o inpu changes han he eac an mean empe a u e. This ac is e iden also om he gain alues compu ed as WSEAS TRANSACTIONS on FLUID MECHANICS Pe Dos ál, Vladimí Bobál, Jiří Voj ěšek, Zdeněk Babík E-ISSN: 2224-347X 16 Issue 1, Volume 7, Janua y 2012 () lim s c y gq →∞ =Δ (31) and p esen ed in Tab. 2. Tab. 2 Gains o a ious inpu hanges. Δqc - 0.025 - 0.05 0.025 0.05 Main eac an empe a u e gs -155.4 -166.2 -263.5 -205.1 Ou pu eac an empe a u e gs -69.6 -72.0 -194.3 -193.5 This ac p edica es be e p ope ies o he eac an mean empe a u e as he con olled ou pu han he eac an ou pu empe a u e. Mo eo e , he dynamics o he eac an ou pu empe a u e is slowe in compa ison wi h he dynamics o he eac an mean empe a u e. 4 CT and Del a ELM Fo he con ol pu poses, he con ol inpu a iable a e conside ed in he o m () () 10 s cc s c q q u q − = (32) This exp ession enables o ob ain con ol inpu and con olled ou pu a iables o app oxima ely he same magni ude. A choice o he CT ELM s uc u e does no s em om known s uc u e o he model (1) – (5) bu om a cha ac e o simula ed s ep esponses. I is well known ha in adap i e con ol a con olled p ocess o a highe o de can be app oxima ed by a linea model o a lowe o de wi h a iable pa ame e s. Taking in o accoun p o iles o cu es in Figs. 3 and 4 wi h ze o de i a i es in = 0, he second o de CT ELM has been chosen o bo h con olled ou pu s in he o m o he second o de linea di e en ial equa ion 100 () () () ()y ay a y bu ++ =   (33) whe e y = y1 o y = y2 , and, in he complex domain, as he ans e unc ion 0 210 () b Gs s as a =++ . (34) Es ablishing he δ ope a o 0 1q T δ − = (35) whe e q is he o wa d shi ope a o and T0 is he sampling pe iod, he del a ELM co esponding o (33) akes he o m 2100 () () () ()y a y a y bu δδ ′′ ′′′′′ ++= (36) whe e ′is he disc e e ime. When he sampling pe iod is sho ened, he del a ope a o app oaches he de i a i e ope a o , and, he es ima ed pa ame e s ,ab ′′ each he pa ame e s a, b o he CT model (33). 5 Del a Model Pa ame e Es ima ion Subs i u ing 2 k ′=−, equa ion (36) can be ew i en o he o m 210 0 (2) (2) (2) (2) yk a yk a yk buk δδ ′′ −+ −+ −= ′ =− (37) In he pape , he ecu si e iden i ica ion me hod wi h exponen ial and di ec ional o ge ing was used. Es ablishing he eg ession ec o () (1) (2) (2)(2) Tkykykuk δ δ −=− − − − −Φ(38) whe e 0 (1)(2) (2) yk yk yk T δ −− − −= (39) he ec o o del a model pa ame e s () 100 () Tkaab δ ′′′ =Θ (40) is ecu si ely es ima ed om he ARX model 2(2) ()(1)() T yk k k k δδ δε −= −+ΘΦ (41) whe e 2 2 0 () 2( 1) ( 2) (2) yk yk yk yk T δ −−+− −= (42) 6 Con olle Design The con ol sys em wi h wo eedback con olle s is depic ed in Fig. 5. - - e wu u 0 y R CT ELM Q Fig. 5. Con ol sys em wi h wo eedback con olle s In he scheme, w is he e e ence signal, deno es he load dis u bance, e he acking e o , u0 ou pu o con olle s, u he con ol inpu and y he WSEAS TRANSACTIONS on FLUID MECHANICS Pe Dos ál, Vladimí Bobál, Jiří Voj ěšek, Zdeněk Babík E-ISSN: 2224-347X 17 Issue 1, Volume 7, Janua y 2012 con olled ou pu . The ans e unc ion G(s) o he CT ELM is gi en by (34). The e e ence w and he dis u bance a e conside ed as s ep unc ions wi h ans o ms 0 () w Ws s =, 0 () Vs s = (43) The ans e unc ions o bo h con olle s a e in o ms () () () , () () () s qs Rs Qs p sps ==   (44) whe e q , and p  a e cop ime polynomials in s ul illing he condi ion o p ope ness deg deg p≤ and deg deg qp≤ . The con olle design desc ibed in his sec ion appea s om he polynomial app oach. The gene al equi emen s on he con ol sys em a e o mula ed as i s in e nal p ope ness and s ong s abili y (in addi ion o he con ol sys em s abili y, also he con olle s abili y is equi ed), asymp o ic acking o he e e ence and load dis u bance a enua ion. The p ocedu e o de i e admissible con olle s can be pe o med as ollows: T ans o ms o he basic signals in he closed-loop sys em ake ollowing o ms ( o simpli ica ion, he a gumen s is in some equa ions omi ed) [] () () () b Ys Ws pVs d =+  (45) [] 1 () ( ) () () E sapbqWsbpVs d =+ −   (46) [] () () () a Us Ws pVs d =+  (47) He e, [ ] () () () () () ()ds as ps bs s qs=++  (48) is he cha ac e is ic polynomial wi h oo s as poles o he closed-loop. Es ablishing he polynomial as () () () s s qs=+  (49) and subs i u ing (49) in o (48), he condi ion o he con ol sys em s abili y is ensu ed when polynomials p  and a e gi en by a solu ion o he polynomial Diophan ine equa ion () () ()() () asps bs s ds+=  (50) wi h a s able polynomial d on he igh side. Wi h ega d o he ans o ms (43), he asymp o ic acking and load dis u bance a enua ion a e p o ided by di isibili y o bo h e ms ap bq+  and p  in (46) by s. This condi ion is ul illed when polynomials p and q ha e o ms () () p ssps= , ( ) ( ) qs sqs= . (51) Subsequen ly, he ans e unc ions (44) ake o ms () () () qs Qs p s =, () () () s Rs s ps = (52) and, a s able polynomial p(s) in hei denomina o s ensu es he s abili y o con olle s. The con ol sys em sa is ies he condi ion o in e nal p ope ness when he ans e unc ions o all i s componen s a e p ope . Consequen ly, he deg ees o polynomials q and mus ul il hese inequali ies deg deg qp≤, deg deg 1 p≤+. (53) Now, he polynomial can be ew i en o he o m () () () s s sqs=+ . (54) Taking in o accoun he sol abili y o (50) and condi ions (53), he deg ees o polynomials in (50) and (52) can be easily de i ed as deg deg deg a==, deg deg 1qa=−, deg deg 1 p a≥−, deg 2degda≥. (55) Deno ing deg a = n, polynomials , and q ha e o ms 0 () n i i i s s = = ∑ , 0 () n i i i s s = = ∑ , 1 1 () n i i i qs qs− = =∑ (56) and, ela ions among hei coe icien s a e 00 =, iii q += o 1,...,in=. (57) Since by a solu ion o he polynomial equa ion (50) p o ides calcula ion o coe icien s i, unknown coe icien s i and qi can be ob ained by a choice o selec able coe icien s 0,1 i β ∈ such ha iii β =, (1 ) iii q β =− o 1,...,in=. (58) The coe icien s i β dis ibu e a weigh be ween nume a o s o ans e unc ions Q and R. Rema k: I 1 i β = o all i, he con ol sys em in Fig. 5 educes o he 1DOF con ol con igu a ion (Q = 0). I 0 i β = o all i, and, bo h e e ence and load dis u bance a e s ep unc ions, he con ol sys em co esponds o he 2DOF con ol con igu a ion. Fo he second o de model (34) wi h deg 2 a=, he con olle 's ans e unc ions ake speci ic o ms 21 0 2 210 0 () () () () () () ( ) qs qs q Qs ps s p s s s Rs sps s s p + == + ++ == + . (59) whe e 00 =, 111 β =, 222 β =, WSEAS TRANSACTIONS on FLUID MECHANICS Pe Dos ál, Vladimí Bobál, Jiří Voj ěšek, Zdeněk Babík E-ISSN: 2224-347X 18 Issue 1, Volume 7, Janua y 2012 111 (1 )q β =− , 222 (1 )q β =− . (60) The con olle pa ame e s hen esul om a solu ion o he polynomial equa ion (50) and depend upon coe icien s o he polynomial d. The nex p oblem he e is o ind a s able polynomial d ha enables o ob ain accep able s abilizing con olle s. In his pape , he polynomial d wi h oo s de e mining he closed-loop poles is chosen as 2 () ()( )ds ns s α =+ (61) whe e n is a s able polynomial ob ained by spec al ac o iza ion () () () ()asas nsns ∗∗ = (62) and α is he selec able pa ame e . No e ha a choice o d in he o m (61) p o ides he con ol o a good quali y o ape iodic con olled p ocesses. The coe icien s o n hen a e exp essed as 2 00 na=, 2 1100 22nana=+− (63) and, he con olle pa ame e s p0 and can be ob ained om solu ion o he ma ix equa ion 10 00 0 1000 00 00 000 ab ab b ⎛⎞ ⎜⎟ ⎜⎟ ⎜⎟ ⎜⎟ ⎝⎠ × 0 2 1 0 p ⎛⎞ ⎜⎟ ⎜⎟ ⎜⎟ ⎜⎟ ⎝⎠ = 31 20 1 0 da da d d − ⎛⎞ ⎜⎟ − ⎜⎟ ⎜⎟ ⎜⎟ ⎝⎠ (64) whe e 2 31 2 10 22 10100 2, 2 2, dn d nn dnndn αα α αα α =+ = ++ =+ = (65) Now, i ollows om he abo e in oduced p ocedu e ha uning o con olle s can be pe o med by a sui able choice o selec able pa ame e s β and α. The con olle pa ame e s and q can hen be ob ained om (60). The adap i e con ol sys em is shown in Fig. 6. Con olle Pa ame e es ima ion Con olled p ocess Compu a ion o con olle pa ame e s T 0 T 0 w u q, p y b, a - Fig. 6 Adap i e con ol scheme. 7 Con ol Simula ion Also he con ol simula ions we e pe o med in a neighbou hood o he ope a ing poin 3 0.27 m /s s c q=, 334.44K s m T=, ou 326.10K s T=. Fo he s a ( he adap a ion phase), he P con olle wi h a small gain was used in all simula ions. Wi h espec o mo e exp essi e nonlinea i y and slowe dynamics o he eac an ou pu empe a u e in compa ison wi h he eac an mean empe a u e, he changes o e e ences as well as he con ol unning ime in e als we e chosen di e en o bo h ou pu s. The e ec o he pole α on he con olled esponses is anspa en om Figs. 7 and 8. Fo bo h ou pu s, wo alues o α we e selec ed. The con ol simula ions show sensi i i y o con olled ou pu s o α. The highe alues o his pa ame e speed he con ol, howe e , hey p o ide g ea e o e shoo s (unde shoo s). O he he e no men io ed simula ions showed ha a ca eless selec ion o he pa ame e α can lead o con olled ou pu esponses o a poo quali y, o oscilla ions o e en o he con ol ins abili y. 0 100 200 300 400 500 600 700 800 -8 -6 -4 -2 0 2 4 6 y1 (K) (s) 1 - α = 0.075 2 - α = 0.2 1 2 1 2 w Fig. 7 Con olled ou pu y1 esponses: e ec o α (β1 = 1, β2 = 0.5). 0 200 400 600 800 1000 1200 -2 -1 0 1 2 y2 (K) (s) 1 - α = 0.05 2 - α = 0.025 w 1 2 1 21 2 Fig. 8 Con olled ou pu y2 esponses: e ec o α (β1 = 1, β2 = 0). Mo eo e , an inc easing α leads o highe alues and changes o he con ol inpu as shown in Fig. 9 and 10. This ac can be impo an in con ol o eal echnological p ocesses. The con olled ou pu y1 esponse o wo alues β2 is shown in Fig. 11. I can be seen ha an e ec o his pa ame e e is insigni ican . WSEAS TRANSACTIONS on FLUID MECHANICS Pe Dos ál, Vladimí Bobál, Jiří Voj ěšek, Zdeněk Babík E-ISSN: 2224-347X 19 Issue 1, Volume 7, Janua y 2012 0 100 200 300 400 500 600 700 800 0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.30 0.31 0.32 0.33 qc (m3/ s) (s) 1 - α = 0.075 2 - α = 0.2 1 1 1 2 2 2 Fig. 9 Coolan low a e esponses in con ol o eac an mean empe a u e – e ec o α (β1 = 1, β2 = 0.5). 0 200 400 600 800 1000 1200 0.22 0.24 0.26 0.28 0.30 qc (m3/ s) (s) 1 - α = 0.05 2 - α = 0.025 1 2 1 2 1 2 Fig. 10 Coolan low a e esponses in con ol o eac an ou pu empe a u e – e ec o α (β1 = 1, β2 = 0). 0 100 200 300 400 500 600 700 800 -6 -4 -2 0 2 4 y1 (K) (s) 1 - β2 = 0 2 - β2 = 1 w 1 2 Fig. 11 Con olled ou pu esponses: e ec o β2 (α = 0.1, β1 = 1). The con olled ou pu esponses documen ing an e ec o he pa ame e β1 a e in Figs. 12 and 13. In bo h cases, a highe alue o β1 esul s in g ea e o e shoo s (unde shoo s) whe eas i s in luence on he speed o con ol is inexp essi e. Co esponding con ol inpu esponses can be seen in Figs. 14 and 15. The e, an inc easing β1 leads o g ea e alues o inpu s, howe e , i can educe occu ed oscila ions, as shown in Fig. 15. O in e es , he e olu ion o es ima ed CT ELM pa ame e s in con ol o he eac an mean empe a u e is shown in Fig. 16. 0 100 200 300 400 500 600 700 800 -8 -6 -4 -2 0 2 4 6 y1 (K) (s) 1 - β1 = 0.2 2 - β1 = 1 w 1 2 1 2 Fig. 12 Con olled ou pu esponses: e ec o β1 (α = 0.15, β2 = 0). 0 200 400 600 800 1000 1200 1400 -2 -1 0 1 2 y2 (K) (s) 1 - β1 = 0 2 - β1 = 1 w 1 1 1 2 2 2 Fig. 13 Con olled ou pu esponses: e ec o β1 (α = 0.04, β2 = 0). 0 100 200 300 400 500 600 700 800 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.30 0.31 0.32 qc (m3/ s) (s) 1 - β1 = 0.2 2 - β1 = 1 1 1 1 2 2 2 Fig. 14 Coolan low a e esponses in con ol o eac an mean empe a u e – e ec o β1 (α = 0.15, β2 = 0). 0 200 400 600 800 1000 1200 1400 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.30 qc (m3/ s) (s) 1 - β1 = 0 2 - β1 = 1 1 11 2 2 2 Fig. 15 Coolan low a e esponses in con ol o eac an ou pu empe a u e – e ec o β1 (α = 0.15, β2 = 0). WSEAS TRANSACTIONS on FLUID MECHANICS Pe Dos ál, Vladimí Bobál, Jiří Voj ěšek, Zdeněk Babík E-ISSN: 2224-347X 20 Issue 1, Volume 7, Janua y 2012 0 100 200 300 400 500 600 700 800 -0.02 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 CT ELM pa ame e s (s) 1 - b0 2 - 10 x a0 3 - 10-1x a1 1 2 3 Fig. 16 CT ELM pa ame e e olu ion (α = 0.15, β1 = 1, β2 = 0). A p esence o an in eg a ing pa in he con olle enables ejec ion o a ious s ep dis u bances en e ing in o he p ocess. As an example, s ep dis u bances a enua ion o he ou pu y1 is p esen ed. S ep dis u bances 3 00.15 kmol/ m A cΔ= , 3 0.03 m /s qΔ=− and 02K TΔ= we e injec ed in o he nonlinea model o he eac o in imes 220s =, 440s = and 640s =. The con olle pa ame e s we e es ima ed only in he i s ( acking) in e al < 200 s. The au ho s' expe iences p o ed ha an u iliza ion o ecu si e iden i ica ion using he del a model a e eaching o a cons an e e ence and in p esence o s ep dis u bances dec eases he con ol quali y. F om his eason, du ing in e al ≥ 200 s, ixed pa ame e s we e used. The con olled ou pu esponses y1 a e shown in Fig. 17. 0 100 200 300 400 500 600 700 800 900 0 1 2 3 4 5 6 7 y1 (K) (s) w Fig. 17 Con olled ou pu in p esence o s ep dis u bances (α = 0.15, β1 = 0.5, β2 = 0). To illus a e an e ec o an addi i e andom dis u bance, he esul o he con olled ou pu y1 simula ion in a p esence o he andom signal 0 () () s A A c c=− is shown in Fig. 18. 8 Conclusions In his pape , one app oach o con inuous- ime adap i e con ol o he mean and ou pu eac an -0.04 -0.02 0.00 0.02 0.04 (kmol/ m3) 0 100 200 300 400 500 600 700 800 -2 -1 0 1 2 3 4 y1 (K) (s) w Fig. 18. Con olled ou pu in he p esence o andom dis u bance in 0 A c (α = 0.15). empe a u es in a ubula chemical eac o was p oposed. The con ol s a egy is based on he p elimina y s eady-s a e and dynamic analysis o he p ocess and on he assump ion o he empe a u e measu emen along he eac o . The p oposed algo i hm employs an al e na i e con inuous- ime ex e nal linea model wi h pa ame e s ob ained h ough ecu si e pa ame e es ima ion o a co esponding del a model. The con ol sys em s uc u e wi h wo eedback con olle s is conside ed. Resul ing con inuous- ime con olle s a e de i ed using he polynomial app oach and gi en by a solu ion o he polynomial Diophan ine equa ion. Tuning o hei pa ame e s is possible ia closed-loop pole assignmen . The p esen ed me hod has been es ed by compu e simula ion on he nonlinea model o he ubula chemical eac o wi h a consecu i e exo he mic eac ion. The simula ion esul s demons a e an applicabili y o he p esen ed con ol s a egy. Acknowledgmen s The au ho s wish o hank o he Minis y o Educa ion o he Czech Republic (MSM7088352101) o inancial suppo . This a icle was c ea ed wi h suppo o Ope a ional P og amme Resea ch and De elopmen o Inno a ions co- unded by he Eu opean Regional De elopmen Fund (ERDF) and na ional budge o Czech Republic wi hin he amewo k o he Cen e o Polyme Sys ems p ojec ( eg.numbe : CZ.1.05/2.1.00/03.0111). WSEAS TRANSACTIONS on FLUID MECHANICS Pe Dos ál, Vladimí Bobál, Jiří Voj ěšek, Zdeněk Babík E-ISSN: 2224-347X 21 Issue 1, Volume 7, Janua y 2012