Slovak University of Technology in Bratislava Institute of Information Engineering, Automation, and Mathematics PROCEEDINGS

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1 Slovak Univeity of Technology in Batilava Intitute of Infomation Engineeing, Automation, and Mathematic PROCEEDINGS of the 8 th Intenational Confeence on Poce Contol Hotel Titi, Tatanká Lomnica, Slovakia, June 7, ISBN Edito: M. Fika and M. Kvanica Dotál, P., Bobál, V., Vojtěšek, J.: Nonlinea Contol of a Chemical Reacto, Edito: Fika, M., Kvanica, M., In Poceeding of the 8th Intenational Confeence on Poce Contol, Tatanká Lomnica, Slovakia, 8,. Full pape online:

2 8th Intenational Confeence on Poce Contol June 7,, Tatanká Lomnica, Slovakia Le-We-,.pdf Nonlinea Contol of a Chemical Reacto P. Dotál, V. Bobál, J. Vojtěšek Toma Bata Univeity in Zlin, Faculty of Applied Infomatic nam. T.G. Maayka 5555, 7 Zlín, Czech Republic (Tel: ; dotalp@ fai.utb.cz. Abtact: The pape deal ith continuou-time nonlinea adaptive contol of a continuou tied tank eacto. The contol tategy i baed on an application of the contolle coniting of a linea and nonlinea pat. The tatic nonlinea pat i deived in the ay of a inveion and conecutive polynomial appoximation of a meaued o imulated input-output data. The deign of the dynamic linea pat i baed on appoximation of nonlinea element in the contol loop by a continuou-time extenal linea model ith paamete etimated uing a coeponding delta model. In the contol deign pocedue, the polynomial appoach ith the pole aignment method i ued. The nonlinea adaptive contol i teted by imulation on the nonlinea model of the CSTR ith a conecutive exothemic eaction.. INTRODUCTION Continuou tied tank eacto (CSTR ae unit fequently ued in chemical and biochemical induty. Fom the ytem theoy point of vie, CSTR belong to a cla of nonlinea ytem. Thei mathematical model ae decibed by et of nonlinea diffeential equation. Thei model ae deived and decibed in e.g. Ogunnaike and Ray (99, Schmidt (5 and Coiou (. It i ell knon that the contol of chemical eacto often epeent vey complex poblem. The contol poblem ae due to the poce nonlineaity and high enitivity of the tate and output vaiable to input change. In addition, the dynamic chaacteitic may exhibit a vaying ign of the gain in vaiou opeating point a ell a non-minimum phae behaviou. Evidently, the poce ith uch popetie i hadly contollable by conventional contol method, and, it effective contol equie application ome of advanced method. One poible method to cope ith thi poblem exploit a linea adaptive contolle ith paamete computed and eadjuted on the bai of ecuively etimated paamete of an appopiate choen continuou-time extenal linea model (CT ELM of the poce. Some eult obtained by thi method can be found in e.g. Dotál et al. (7 and Dotál et al. (9. An effective appoach to the contol of CSTR and imila pocee utilize vaiou method of the nonlinea contol (NC. Seveal modification of the NC theoy ae decibed in e.g. Atolfi et al. (8, Vincent and Gantham (997, Ioannou and Fidan ( o Zhang et al. (. Epecially, a lage cla of the NC method exploit lineaization of nonlinea plant, e.g. Huba and Ondea (9, an application of PID contolle, e.g. Tan et al. (, Bányáz and Keviczky ( o factoization of nonlinea model of the plant on linea and nonlinea pat, e.g. Nakamua et al. (, Valley et al. (9 and Chyi-Tong Chen et al. (. In thi pape, the CSTR contol tategy i baed on an application of the contolle coniting of a tatic nonlinea pat (SNP and dynamic linea pat (DLP. The tatic nonlinea pat i obtained fom imulated o meaued teady-tate chaacteitic of the CSTR, it inveion, polynomial appoximation, and, ubequently, it diffeentiation. On behalf of development of the linea pat, the SNP including the nonlinea model of the CSTR ae appoximated by a CT extenal linea model. Fo the CT ELM paamete etimation, an extenal delta model ith the ame tuctue a the CT model i ued (ee, e.g. Mukhopadhyay et al. (99, Goodin et al. ( and Steicke and Sinha (99. Then, the eulting CT contolle i deived uing the polynomial appoach and pole aignment method, e.g. Kučea (99. The imulation ae pefomed on a nonlinea model of the CSTR ith a conecutive exothemic eaction.. MODEL OF THE CSTR Conide a CSTR ith the fit ode conecutive exothemic k k eaction accoding to the cheme A B C and ith a pefectly mixed cooling jacket. Uing the uual implification, the model of the CSTR i decibed by fou nonlinea diffeential equation dc q q A = + k c + c A Af dt V V dcb q q = + k cb + kca + cb f dt V V dt h q AU h ( T T f ( T T c dt = ( ρc + V + V ( ρc p p ( ( ( 8

3 8th Intenational Confeence on Poce Contol June 7,, Tatanká Lomnica, Slovakia Le-We-,.pdf dtc qc AhU = ( Tcf Tc + ( T Tc dt V V ( ρc c c p c ith initial condition c ( = c, c ( = c, A B T ( = T and T ( = T. Hee, t i the time, c ae c c concentation, T ae tempeatue, V ae volume, ρ ae denitie, c p ae pecific heat capacitie, q ae volumetic flo ate, A h i the heat exchange uface aea and U i the heat tanfe coefficient. The ubcipt ae denoted (. fo the eactant mixtue, (. c fo the coolant, (. f fo feed (inlet value and the upecipt (. fo teady-tate value. The eaction ate and the eaction heat ae expeed a E j k exp j = k j, j =, (5 RT = h k c h k c ( h A + hee k ae pe-exponential facto, E ae activation enegie and h ae eaction entalpie. The value of all paamete, feed value and teady-tate value ae given in Table. Table. Paamete, inlet value and initial condition. V =. m V c =. m ρ = 985 kg m - ρ c = 998 kg m - k = 5.. min - k =.8. 8 min - h =.8. kj kmol - c Af =.85 kmol m - T f = K q =.8 m min - c A =.579 kmol m - T =.8 K B A c p =.5 kj kg - K - c pc =.8 kj kg - K - A h = 5.5 m U =.5 kj m - min - K - E / R = 77 K E / R = 59 K h =.. kj kmol - c Bf = kmol m - T cf = 9 K q c =.8 m min - c B =.975 kmol m - T c =.8 K In tem of the pactice, only the coolant flo ate can be taken into account a the contol input. A the contolled output, the eactant tempeatue i conideed. Fo the contol pupoe, the contol input and the contolled output ae defined a deviation fom teady value c ( ut ( = q( t q, y( t = T ( t T (7 c The dependence of the eactant tempeatue on the coolant flo ate in the teady-tate i in Fig.. In ubequent contol imulation, the opeating inteval fo q c ha been detemined a cmin c cmax q q ( t q (8 B T (K T max 5 T min q c min Opeating point (q c =.8, T =.8 q c max q c (m /min Fig.. Dependence of the eactant tempeatue on the coolant flo ate in the teady-tate.. CONTROLLER DESIGN A peviouly intoduced, the contolle conit of a tatic nonlinea pat and a dynamic linea pat a hon in Fig.. CONTROLLER e u u DLP SNP Fig.. The contolle cheme. The DLP ceate a linea dynamic elation beteen the tacking eo e(t and u ( t =Δ T ( t hich epeent a diffeence of the eactant tempeatue adequate to it deied value. Evidently, fo a ell popoed SNP, the limit elation lim u ( t = hold. t Then, the SNP geneate a tatic nonlinea elation betveen u and a coeponding incement (decement of the coolant flo ate.. Nonlinea pat of the contolle The SNP deivation appea fom a imulated o meaued teady-tate chaateitic. Fom the pupoe of a late polynomial appoximation, the coodinate on the gaph axi ae defined a c q q θ=, q cl cl hee q cl i the loe bound in the inteval cl c cu ξ= T T. (9 L q q q ( and, TL i the tempeatue coeponding to q cu. It can be ecommended to elect the inteval ( lightly longe than (8. In thi pape, loe and uppe value in (8 and ( ee choen qcl =., qcmin =., qc max =. and q cu =.. 9

4 8th Intenational Confeence on Poce Contol June 7,, Tatanká Lomnica, Slovakia Le-We-,.pdf In tem of the pactice, it can be uppoed that the meaued data ill be affected by meauement eo. The imulated teady-tate chaacteitic that coepond to eality i hon in Fig.. ξ (K θ ( Fig.. Simulated chaacteitic ξ = f (θ. Making the eplacement of coodinate, the invee of thi chaacteitic can be appoximated by a polynomial in the geneal fom n n θ= a + a ξ a ξ + a ξ. ( The invee chaacteitic accodant ith Fig. togethe ith the fouth ode appoximate polynomial i in Fig.. θ ( 7 5 θ = ξ+.8 ξ -. ξ +.5E-5 ξ ξ (K n Simulated (meaued] value Polynomial appoximation Fig.. Simulated and appoximated invee elation θ= f ( ξ. No, a diffeence of the coolant flo ate ut ( =Δ qc ( t in the output of the SNP can be computed fo each T a d θ ut ( =Δ qc( t = qcl u ( t ( dξ ξ( T The deivative of the appoximate polynomial i in Fig. 5.. CT extenal linea model of nonlinea element A tuctue of the CT ELM of the SNP in conjuction ith the CSTR nonlinea model a choen on the bai of tep epone imulated in a neighbouhood of the opeating point. The tep epone fo ome tep change of u ae hon in Fig.. Fo all epone, the gain of the yt ( SNP+CSTR ytem ha been computed a g = lim. t u n dθ / dξ (/ K d θ / d ξ = ξ -.8 ξ +.75E-5 ξ ξ (K Fig. 5. Deivative of θ ith epect to ξ u = -, - u = -, - u =, - u = g =.9 g =.8 g =. g = Fig.. Step epone of the SNP+CSTR. Taking into account pofile of cuve in Fig. ith zeo deivative fo t =, the econd ode CT ELM ha been choen in the fom of the econd ode linea diffeential equation yt ( + a yt ( + a yt ( = b u ( t ( o, in the tanfe function epeentation a Y( b G ( = = U a a. Delta extenal linea model ( + +. ( Etablihing the δ opeato q δ= (5 T hee q i the foad hift opeato and T i the ampling peiod, the delta ELM coeponding to ( take the fom δ yt ( + a δ yt ( + a yt ( = b u ( t ( hee t i the dicete time. When the ampling peiod i hotened, the delta opeato appoache the deivative opeato, and, the etimated paamete a, b of ( each the paamete a, b of the CT model ( a poved in e.g. Steicke and Sinha (99. Subtituting t = k, equation ( may be eiten to the fom δ yk ( + a δyk ( + a yk ( = bu ( k.(7

5 8th Intenational Confeence on Poce Contol June 7,, Tatanká Lomnica, Slovakia Le-We-,.pdf. Delta model paamete etimation Etablihing the egeion vecto ( Φ T δ ( k = δy( k y( k u( k (8 hee yk ( yk ( δyk ( = (9 T then, the vecto of delta model paamete ( Θ T ( k = a a b ( δ i ecuively etimated fom the ARX model hee T δ δ y( k = Θ ( k Φ ( k +ε( k ( δ yk ( yk ( + yk ( δ yk ( =. ( T The ecuive etimation of delta model paamete a pefomed ith the ampling inteval T =. min. Hee, the ecuive identification method ith exponential and diectional fogetting accoding to Rao and Unbehauen (5 and Bobál et al. (5 a ued..5 Linea pat of the contolle The DLP i ineted into the contol loop accoding to Fig e DLP u v CT ELM Fig. 7. Simplified cheme of the contol loop. In the cheme, i the efeence ignal, v i the ditubance, y i the contolled output and u i the input to the CT ELM. The tanfe function G of the CT ELM i given by (. Both the efeence and the ditubance v ae conideed to be tep function ith tanfom v W ( =, V ( = ( The tanfe function of the DLP i in the fom U ( q ( Q ( = E ( = p ( ( hee q and p ae polynomial in, and, deg q deg p. The contolle deign decibed in thi ection tem fom the polynomial appoach. Geneal condition equied to goven the contol ytem popetie ae fomulated a tong tability (in addition to the contol ytem tability, alo the tability of contolle i equied, intenal popene, aymptotic tacking of a tep efeence and tep ditubance attenuation. y It i ell knon fom the algebaic contol theoy that a contolle hich atifie above equiement i in the polynomial ing given by a olution of the polynomial (Diophantine equation a ( p ( + bq ( ( = d ( (5 ith a table polynomial d( on the ight ide. Fo tep input ignal and v, the polynomial p i in the fom p ( = p (. ( The degee of unknon polynomial in (5 and ( ae deg q = deg a, deg p = deg a, deg d = deg a. Then, fo the ELM (, the contolle tanfe function take the fom q ( q q q Q ( = = p ( ( p (7 In thi pape, the polynomial d ith oot detemining the cloed-loop pole i choen a d ( = n (( +α (8 hee n i a table polynomial obtained by pectal factoization a ( a( = n ( n( (9 and α i the electable paamete that can uually be choen by ay of imulation expeiment. Note that a choice of d in the fom (8 povide the contol of a good quality fo apeiodic contolled pocee. The polynomial n ha the fom ith coefficient n a =, n ( = + n+ n ( n = a + n a. ( The contolle paamete can be obtained fom olution of the matix equation hee p d a a b Ω q = d a a b q d b q d = α +α, =α ( d = n + α, d = α n + n +α. ( d n n d n Evidently, the contolle paamete can be adjuted by the electable paamete α. The complete adaptive contol ytem i hon in Fig. 8.. CONTROL SIMULATIONS The contol imulation ee pefomed in a neighbouhood of the opeating point ( q c =.8 m min -, T =.8 K.

6 8th Intenational Confeence on Poce Contol June 7,, Tatanká Lomnica, Slovakia Le-We-,.pdf Computation of DLP paamete q, p T CT ELM paamete etomation q c e u u DLP SNP CSTR - ξ T T q c min T T T - y fit (tacking inteval t < min. The expeience of autho of thi pape poved that an utilization of ecuive identification uing the delta model in the phae of a contant efeence and in a peence of tep ditubance deceae the contol quality. Fom thi eaon, duing inteval t min, fixed DLP paamete ee ued. The contolled output epone ae hon in Fig.. Fig. 8. Scheme of the nonlinea adaptive contol. Fo the tat (the adaptation phae, a P contolle ith a mall gain a ued in all imulation. The effect of the pole α on the contol epone i tanpaent fom Fig. 9, and. Hee, on the bai of pecomputed imulation, to value of α ee elected. The contol eult ho enitivity of the contolled output and the input ignal to α. Obviouly, caele election of thi paamete can lead to contolled output epone of a poo quality o even to untability. Futhe, a inceaing α lead to highe value and change of the input ignal α =.5 α =.5 8 Fig. 9. Nonlinea adaptive contol: Contolled output epone. q c (m / min q c α =.5 α =.5 8 Fig.. Nonlinea adaptive contol: Input to the CSTR. Contolle paamete.. q..8.. p q 5q.. 8 Fig.. Evolution of the DLP paamete duing contol. u (K α =.5 α =.5 8 Fig.. Nonlinea adaptive contol: Input ignal to the SNP. Evolution of the DLP paamete duing contol i hon in Fig.. A peence of the integating pat in the DLP enable ejection of vaiou tep ditubance enteing into the poce. Hee, tep ditubance Δ c Af =±.kmol m at time t = 5 min and t = 55 min ee injected into v v the CSTR. The DLP paamete ee etimated only in the 7 5 α =.5 α =.75 8 Fig.. Nonlinea adaptive contol: Step ditubance ejection. An influence of the SNP i evident fom the contol epone hon in Fig.. Hee, the tandad adaptive contol ithout the nonlinea pat of the contolle a ued. The imulation ha been pefomed unde the ame condition a by above peented cae. A confontation ith epone in Fig. 9 ho that an application of the nonlinea contol i uitable epecially fo geate change of the efeence ignal.

7 8th Intenational Confeence on Poce Contol June 7,, Tatanká Lomnica, Slovakia Le-We-,.pdf α =.5 8 Fig.. Adaptive contol ithout the SNP. 5. CONCLUSIONS In thi pape, one appoach to the nonlinea continuou-time adaptive contol of the eactant tempeatue in a continuou tied tank eacto a popoed. The contol tategy i baed on a factoization of a contolle into the linea and the nonlinea pat. A deign of the contolle nonlinea pat employ imulated o meaued teady-tate chaacteitic of the poce and thei additional modification. Then, the ytem coniting of the contolle nonlinea pat and a nonlinea model of the CSTR i appoximeted by a continuou time extenal linea model ith paamete obtained though ecuive paamete etimation of a coeponding delta model. The eulting continuou-time contolle linea pat i deived uing the polynomial appoach and given by a olution of a polynomial equation. Tuning of it paamete i poible via cloed-loop pole aignment. The peented method ha been teted by compute imulation on the nonlinea model of the CSTR ith a conecutive exothemic eaction. Simulation eult demontated an applicability of the peented contol tategy and it uefulne epecially fo geate change of input ignal in tongly nonlinea egion. It can be expected that the decibed contol tategy i alo uitable fo othe imila technological pocee uch a tubula chemical eacto. ACKNOWLEDGMENT The autho ih to thank to the Minity of Education of the Czech Republic (MSM7885 fo financial uppot. Thi aticle a ceated ith uppot of Opeational Pogamme Reeach and Development fo Innovation cofunded by the Euopean Regional Development Fund (ERDF and national budget of Czech Republic ithin the fameok of the Cente of Polyme Sytem poject (eg.numbe: CZ..5/../.. REFERENCES Atolfi, A., Kaagianni, D., and Otega, R. (8. Nonlinea and adaptive contol ith application, Spinge-Velag, London. Bányáz, C., and Keviczky, L. (. A Simple PID egulato applicable fo a cla of factoable nonlinea plant. Ameican Contol Confeence, Anchoage, Alaka, Bobál, V., Böhm, J., Fel, J., and Macháček, J. (5. Digital Self-tuning Contolle, Spinge Velag, Belin. Chen, Chyi-Tong, Chuang, Yao-Chen, and Hang, Chyi (. A Simple Nonlinea Contol Stategy fo Chemical Pocee. th Aian Contol Confeence, Bali, Indoneia, -7. Coiou, J.-P. (. Poce contol. Theoy and application, Spinge Velag, London. Dotál, P., Gazdoš, F., Bobál, V., and Vojtěšek, J. (7. Adaptive contol of a continuou tied tank eacto by to feedback contolle. 9th IFAC Wokhop Adaptation and Leaning in Contol and Signal Poceing ALCOSP'7, Saint Petebug, Ruia, P5- P5-. Dotál, P., Vojtěšek, J., and Bobál, V. (9. Simulation of adaptive contol of a continuou tied tank eacto. d Euopean Confeence on Modelling and Simulation, ECMS 9, Madid, 5-. Goodin, G.C., Gaebe, S.F., and Salgado, M.E. (. Contol Sytem Deign, Pentice Hall, Ne Jeey. Huba, M., and Ondea, M. (9. Simulation of nonlinea contol ytem epeented a genealized tanfe function. Euopean Contol Confeence 9, Budapet, Hungay, -9. Ioannou, P., and Fidan, B. (. Adaptive contol tutoial, SIAM, Philadelphia. Kučea, V. (99. Diophantine equation in contol A uvey. Automatica, 9, -75. Mukhopadhyay, S., Pata, A.G., and Rao, G.P. (99. Ne cla of dicete-time model fo continuo-time ytem. Intenational Jounal of Contol, 55, -87. Nakamua, M., Sugi, T., and Goto, S. (. Nonlinea epaation model and contol fo a complex poce ealized by conventional PID contolle hadae. th Aian Contol Confeence, Singapoe, Ogunnaike, B.A., and Ray, W.H. (99. Poce dynamic, modeling, and contol, Oxfod Univeity Pe, Ne Yok. Rao, G.P., and Unbehauen, H. (. Identification of continuou-time ytem. IEE Poc.-Contol Theoy Appl., 5, 85-. Schmidt, L.D. (5. The engineeing of chemical eaction, Oxfod Univeity Pe, Ne Yok. Steicke, D.L., and Sinha, N.K. (99. Identification of continuou-time ytem fom ample of input-output data uing the δ-opeato. Contol-Theoy and Advanced Technology, 9, -5. Tan, K.K., Fedou, R., and Huang, S. (. Cloed-loop automatic tuning of PID contolle fo nonlinea ytem. th Aian Contol Confeence, Singapoe, Valley, H., Neumaie, M., and Bu, M. (9. Anti-caual identification of Hammetein model. Euopean Contol Confeence 9, Budapet, Hungay, 7-7. Vincent, T.L., and Gantham, W.J. (997. Nonlinea and optimal contol ytem, John Wiley & Son, Ne Yok. Zhang, T., Ge, S.S., and Hang, C.C. (. Stable adaptive contol fo a cla of nonlinea ytem uing a modified Lyapunov function. IEEE Tanaction on Automatic Contol, 5(, 9-.