Adaptive LQ Cascade Control of a Tubular Chemical Reactor

Transcription

1 MATEC Web of Confeene 7, () DOI:./ mateonf/ 7 CSCC Adaptive LQ Caade Contol of a Tubula Chemial Reato Dotal Pet, Vladimí obal and Jii Vojteek Toma ata Univeity in Zlin, Faulty of Applied Infomati, Nad Stanemi, 7 Zlin, Czeh Republi Abtat. The pape deal with adaptive LQ aade ontol deign of a tubula hemial eato with an exothemi oneutive eation. The ontol i pefomed in pimay and eonday ontol-loop whee the pimay ontolled output of the eato i the onentation of a main eation podut and the eonday output i the mean tempeatue of the eatant. A ommon ontol input i the oolant flow ate. The ontolle in the pimay ontol-loop i a nonlinea P-ontolle with the gain alulated uing imulated o meaued teady-tate haateiti of the eato. The ontolle in the eonday ontol-loop i a LQ adaptive ontolle. The popoed method i veified by ontol imulation. Intodution Tubula hemial eato (TCR) ae unit fequently ued in hemial induty, biotehnologie and ome othe. Fom the ytem theoy point of view, TCR belong to the la of nonlinea ditibuted paamete ytem. Thei mathematial model ae deibed by et of nonlinea patial diffeential equation (PDE). The method of modelling and imulation of uh poee ae deibed e.g. in [] o []. Detailed analyi of the peifi TCR i aied out fo example in []. Tubula hemial eato belong to hadly ontollable poee. Diffiultie aoiated with thei ontol ae given not only by thei nonlineaity but alo by the fat that onentation of eatant annot motly be meaued ontinuouly. Hee, the aade ontol take plae a a uitable and effetive ontol method. The aade ontol belong to moe omplex ontol tutue. It may be applied in uh ae whee moe output vaiable an be meaued and whee only one input vaiable i available to the ontol. Piniple of the aade ontol ae deibed e.g. in [], [] and []. In thi pape, the TCR ontol tategy i baed on the fat that onentation of omponent of eation taking plae in the eato depend on the eatant tempeatue. Moeove, the poedue aume that the eatant tempeatue i meaued at moe point along the eato fom whih i ubequently alulated mean eatant tempeatue. Then, in the aade ontol-loop, the onentation of a main podut of the eation i onideed a the pimay ontolled vaiable, and, the mean eatant tempeatue a the eonday ontolled vaiable. The oolant flow ate epeent a ommon ontol input. The pimay ontolle detemining the et point fo the eonday (inne) ontol-loop i deived a a nonlinea popotional ontolle uing the teady-tate haateiti of the eato. Sine the ontolled poe i nonlinea, a ontinuou-time adaptive ontolle i ued a the eonday ontolle. The poedue fo the adaptive ontol deign in the inne ontol-loop i baed on appoximation of a nonlinea model of the TCR by a ontinuou-time extenal linea model (CT ELM) with euively etimated paamete. In the poe of the paamete etimation, a oeponding delta model i ued, ee, e.g. [7], [8] and [9]. The eulting CT ontolle i deived on the bai of the LQ ontol theoy, ee, e.g. () and the polynomial method, ee, e.g. [] o []. The ontol i teted by imulation on the nonlinea model of the TCR with a oneutive exothemi eation. Model of the eato An ideal plug-flow tubula hemial eato with a imple exothemi oneutive eation A C in the liquid phae and with the ounteuent ooling i onideed. Heat loe and heat ondution along the metal wall of tube ae aumed to be negligible, but dynami of the metal wall of tube ae ignifiant. All denitie, heat apaitie, and heat tanfe oeffiient ae aumed to be ontant. Unde above aumption, the eato model an be deibed by five PDE in the fom A A v k A t z v ka k () t z T T Q U v ( T Tw) () t z ( ) d ( ) p k p k Coeponding autho: dotalp@fai.utb.z The Autho, publihed by EDP Siene. Thi i an open ae atile ditibuted unde the tem of the Ceative Common Attibution Liene. (

2 MATEC Web of Confeene 7, () DOI:./ mateonf/ 7 CSCC Tw du ( T Tw) t ( d d )( p) w du( TT w () T T ndu v ( T ) w T () t z ( d nd)( p) with initial ondition A(,) z A() z (,) (), z z T (,) (), z T z, Tw(,) z Tw() z T (,) (), z T z and bounday ondition (, t) ( t) A A (kmol/m ), (, t) ( t) (kmol/m ), T (, t) T ( t) (K), T ( L, t) T ( t) (K). Hee, t i the time, z i the axial pae vaiable, tand fo onentation, T fo tempeatue, v fo fluid veloitie, d fo diamete, fo denitie, p fo peifi heat apaitie, U fo heat tanfe oeffiient, n i the numbe of tube and L i the length of tube. The ubipt () tand fo the eatant mixtue, () w fo the metal wall of tube, () fo the oolant, and the upeipt () fo teady-tate value. The eation ate and heat of eation ae nonlinea funtion expeed a j j exp E j k k, j =, () RT L Q ( H ) k ( H ) k (7) A whee k ae pe-exponential fato, E ae ativation enegie, ( H ) ae eation enthalpie in the negative onideation and R i the ga ontant. The fluid veloitie ae alulated via the eatant and oolant flow ate a q q v, v (8) nd ( d nd) The paamete value with oepondent unit ued fo imulation ae given in Table. Fom the ytem engineeing point of view, A( L, t) Aout, ( L, t) out, T( L, t) Tout and T (, t) T ae the output vaiable and q () t, q () t, out A () t, T () t and TL() t ae the input vaiable. Among them, fo the ontol pupoe, motly the oolant flow ate an be taken into aount a the ontol vaiable, wheea othe input enteing into the poe an be aepted a ditubane. In thi pape, the mean eatant tempeatue n p T () t T ( z,) t (9) m p np p i onideed a the eonday (inne) ontolled output. Hee, z p efe to meauement point and n p i thei numbe. The onentation epeent the pimay ontolled output. Table. Paamete and input value. L = 8 m n = d =. m = 98 kg/m w = 78 kg/m = 998 kg/m U =.8 kj/m K d = m d =. m p =. kj/kg K pw =.7 kj/kg K p =.8 kj/kg K U =. kj/m K k =. / k =.8 8 / E /R = 77 K (-H ) =.8 kj/kmol E /R = 9 K (-H ) =.8 kj/kmol A.8 kmol/m kmol/m T K TL 9 K q. m / Fo olution of PDE, the finite diffeene method i employed. The poedue i baed on ubtitution of the pae inteval z, L by a et of diete node pointz i fo i =,, n, and, ubequently, by appoximation of deivative with epet to the pae vaiable in eah node point by finite diffeene. The poedue i in detail deibed in []. Contol objetive ai heme of the aade ontol i in Fig.. Hee, NPC tand fo the nonlinea popotional ontolle, AC fo the adaptive LQ ontolle. Figue. Caade ontol heme. The ontol objetive i to ahieve a onentation of the omponent a the pimay ontolled output nea to it maximum. A dependene of the onentation of on the mean eatant tempeatue i in Fig.. (kmol/m ) T (K) m Figue. Steady-tate haateiti. Inteval Inteval

3 MATEC Web of Confeene 7, () DOI:./ mateonf/ 7 CSCC Thee, the opeating inteval onit of two pat. In the fit inteval, the onentation ineae with ineaing eatant tempeatue, in the eond inteval it again deeae. The endpoint defining both inteval ae in Table. It an be een in Fig. that the maximum value of an be lightly highe than. kmol/m. Howeve, the maximum deied value of will be limited jut by. kmol/m. Table. Paamete and input value. Inteval Tempeatue Conentation 8.7 T m T m Fo pupoe of late appoximation, the mean tempeatue i tanfomed a Tm Tmmin () Tmmax Tmmin whee Tm min 8.7 and Tm max 8.7. The dependene of on i hown in Fig.. (kmol/m ) () Inteval Inteval Figue. Tanfomed teady-tate haateiti. The NPC deign The poedue of the NPC deign appea fom polynomial appoximation of tanfomed teady-tate haateiti hown in Fig.. Appoximate polynomial in both inteval ae tated inluding thei deivative in Table. Table. Appoximate polynomial and thei deivative. Inteva l Appoximate polynomial Now, a deied value of the mean eatant tempeatue in the output of the NPC an be omputed fo eah a w Kw( Tmmax Tmmin ) d w () d whee w w, w Tmw, and i a eletable gain oeffiient. CT and Delta extenal linea model Fo the ontol pupoe, the ontolled output and the ontol input ae defined a m m m u() t q () t q () t q, y() t T () t T () t T The CT ELM i popoed in the time domain on the bai of peliminay imulated tep epone in the fom of the eond ode diffeential equation yt () a yt () a yt () b ut () () and, in the omplex domain, a the tanfe funtion b G () a a. () Etablihing the opeato d T () whee i the fowad hift opeato and T i the ampling peiod, the delta ELM oeponding to () take the fom yt ( ) ayt ( ) a yt ( ) b ut ( ) () whee t i the diete time. When the ampling peiod i hotened, the delta opeato appoahe the deivative opeato, and the etimated paamete a, b eah the paamete a, b. Delta model paamete etimation Subtituting t k, equation () an be ewiten to the fom yk ( ) a yk ( ) a yk ( ) buk ( ) () Etablihing the egeion veto T ( ) y ( ) y ( ) u ( ) (7) whee yk ( ) yk ( ) yk ( ) (8) T the veto of delta model paamete T ( k ) a a b (9) i euively etimated by the leat quae method with exponential and dietional fogetting fom the ARX model, e.g. []. T yk ( ) ( k) ( k) ( k) () whee

4 MATEC Web of Confeene 7, () DOI:./ mateonf/ 7 CSCC yk ( ) yk ( ) yk ( ) yk ( ). () T 7 Adaptive LQ ontolle The eonday feedbak ontol loop i depited in Fig.. In the heme, w i a equene of tep efeene ignal a output fom the pimay ontolle, e i the taking eo, u i output of the eonday ontolle, and y i the ontolled output. The tanfe funtion G() of the CT ELM i given by (). Figue. Feedbak ontol loop. The ontolle deign i baed on the polynomial appoah and the LQ ontol theoy. Poedue fo deigning an be biefly deibed a follow: The tanfe funtion of the AC i in the fom q () Q () () p () whee q and p ae opime polynomial atifying the ondition of popene deg q ( ) deg p ( ). Suh ontol law i ought that minimize the quadati ot funtion { ( ) ( )} () S e t u t dt whee ut () i the ontolle output deivative and i the weighting oeffiient. A known, the poblem i olved by ontolle whoe polynomial ae given by a olution of the polynomial equation a () p () bq ()() d () () with a table polynomial d() on the ight ide, and, whee p () p () fo a tep input ignal. Now, the polynomial d() take the fom d () gn () () () whee g() i a moni fom of the polynomial h() given by petal fatoization a() a() b ()() b h () h(). () The eond polynomial n() enuing popene of the ontolle an be hoen a a eult of petal fatoization a () a() n () n(). (7) Fo G() with a () a a, polynomial h(), g() and n() take fom h () h h h h (8) g () g g g, gj hj h (9) n () n n () The polynomial d() ha the fom d() d d d d d () whee d gn, d gn gn, d gn gng () d gn gng, d gng Then, the eulting titly pope ontolle ha the tanfe funtion q qq Q () () ( p p) with paamete omputed a p d a a p d a a a b q = d a b q d b q d () The above poedue implie that the eonday ontolle paamete an be adjuted by ingle eletable paamete. 8 Simulation All imulation wee pefomed on nonlinea model of the TCR () () with paamete in Table.. The onentation i meaued in the peiod (). The aim of imulation i to how an effet of adjutable gain, an effet of the peiod and an effet of adjutable weighting paamete on ome ontol epone. At the tat of imulation, the P ontolle with a mall gain wa ued. Fo the -model paamete euive identifiation, the ampling peiod T =. wa hoen. In all ae, the efeene ignal w, the mean tempeatue T m and the onentation epone wee imulated. Stating value of the oolant flow ate wa alway hoen a q.8 m / and to it oeponding initial value.9 kmol/m and Tm 8.8 K. The deied value of ha been hoen a w. kmol/m. Effet of the paamete on above epone i evident Fig. 7. It an be een that an ineaing aeleate all ignal in the ontol loop. Howeve, it value i not unetited and it onvenient value hould be found expeimentally. An effet of the peiod an be een in Fig. 8. Although hotening lead to fate ontol epone, it length i not feely eletable but it i detemined by poibilitie of a meauement. The lat goup of imulation how an effet of eletable weight oeffiient on imulated epone. It an be een that it influene i little ignifiant. Thi fat i aued by mall tep in the equene of the efeene.

5 MATEC Web of Confeene 7, () DOI:./ mateonf/ 7 CSCC w (K) =. =. t () T m (K) 8 8 = = t () Figue. Sequene of tep efeene ( =, = ). Figue 9. Mean eatant tempeatue epone ( =., = ). T m (K) 8 =. =. 8 t () Figue. Mean eatant tempeatue epone ( =, = ). (kmol/m ) w = = t () Figue. Conentation epone ( =, = ). (kmol/m ) w =. =. t () Figue 7. Conentation epone ( =, = ). w (K) = = t () Figue. Sequene of tep efeene ( =., = ). w (K) = = t () Figue 8. Sequene of tep efeene ( =., = ). T m (K) 8 = = 8 t () Figue. Mean eatant tempeatue epone ( =., = ).

6 MATEC Web of Confeene 7, () DOI:./ mateonf/ 7 CSCC (kmol/m ) w = = t () Figue. Conentation epone ( =., = ). 9 Conluion The atile peent the aade ontol deign of a tubula hemial eato. The poedue i baed on a poibility of meauing the onentation of a main podut of a eation taking plae in the eato and meauing the tempeatue of the eatant at multiple point along the eato. The ontol i pefomed in the extenal (pimay) and inne (eonday) loed-loop whee the onentation of a main podut on the output of the eato i the pimay and the mean eatant tempeatue the eonday ontolled vaiable. A ommon ontol input i the oolant flow ate. The ontolle in the extenal ontol-loop i a diete nonlinea P-ontolle deived on the bai of imulated o meaued teady-tate haateiti of the eato. The ontolle in the inne ontol-loop i an adaptive LQ ontinuou-time ontolle. In it deivation, the euive paamete etimation of an extenal delta model of the eato, the polynomial appoah and the LQ ontol theoy ae applied. The peented method ha been teted by ompute imulation on the nonlinea model of the tubula hemial eato with a oneutive exothemi eation. Refeene. W.L. Luyben, Chemial eato deign and ontol (John Wiley & Son, Chihete, 7). D.E. Sebog, T.F. Edga, D.A. Mellihamp, Poe Dynami and Contol (John Wiley & Son, Chihete, 989). P. Dotál, Po. of nd Euopean Confeene on Modelling and Simulation, 87-9, Cypu (8).. J. F. Smut, Poe ontol fo patitione (OptiContol In., New Yok, ). M. King, Poe ontol: A patial appoah (John Wiley & Son, Chihete, ). R. Shmidt, Chemial poe deign and integation (John Wiley & Son, Chihete, ) 7. H. Ganie, L. Wang (ed.), Identifiation of ontinuou-time model fom ampled data (Spinge- Velag, London, 8) 8. S. Mukhopadhyay, A.G. Pata, G.P. Rao, Int. J. Cont., (99). 9. D.L. Steike, N.K. Sinha, Contol-theoy and adv. Tehnol. 9, (99). D. Henion, Po of IFAC Sympoium Robut Contol Deign, Pague (). J. Mikleš, M. Fika, Poe modellig, identifiation and ontol (STU Pe, atilava, ). V. obál, J. öhm, J. Fel, Digital elftuning ontolle (Spinge Velag, elin, )