A Study on the Convergence of Observer-Based Kinetics Estimators in Stirred Tank Bioreactors

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1 Pceedng f PE 94 (683~687) A tudy n the Cnvegence f Obeve-Baed Knetc Etmat n ted ank Beact R. Olvea, E. Feea, F. Olvea 3,. Fey de Azeved * Depatament de Engenhaa Químca, Faculdade de Engenhaa da Unvedade d Pt, Rua d Baga, 499 Pt Cdex, PORUGAL Unvedade d Mnh, Engenhaa Blógca, Lag d Paç, 47 Baga PORUGAL 3 Unvedade d Mnh, Electónca Indutal, Lag d Paç, 47 Baga PORUGAL ABRAC h pape devted t the tunng pblem f the "beve-baed knetc etmat" n ted tank beact. h algthm etmate the eactn knetc fm the n-lne knwledge f the tate vaable (ethe fm meauement by mean f tate beve), when the yeld ceffcent ae knwn. he elatn between the dynamc f cnvegence and the tunng pcedue expled. he methd pped mpe a ecnd-de dynamc t the cnvegence f the etmat. h appach wll be hwn t cmpae favuably wth a ple placement baed technque, n an applcatn t a bake' yeat fed-batch fementatn. INRODUCION w f the maj pblem lmtng the ue f mden cntl technque t bpcee ae the dffculty f mdellng the gwth knetc f mcganm and the lack f cheap and elable en f blgcal vaable. Mdel-baed tate beve and beve-baed paamete etmatn epeent ecent develpment whch may vecme uch dffculte. Batn and Dchan (99) pped a methdlgy f tate and paamete etmatn baed upn the cncept f a geneal dynamcal mdel f beact : dξ = Kϕξ ( ) Dξ+ FQ( ξ) () F the n-lne etmatn f eactn ate when the yeld ceffcent ae knwn and cntant, the pped beve-baed etmat expeed by ξ = KH( ξρ ) Dξ+ F Q Ω ( ξ ξ ) (a) ρ = KH( ξ) Γ ( ξ ξ ) (b) he eactn ate ae defned a ϕ(ξ)=h(ξ)ρ(t) t take advantage n any pble knwledge f the knetc mdel, whee H(ξ) an m knwn matx (functn f the tate) whle ρ(t) a vect f unknwn functn f the tate whch ae cndeed a cmpletely unknwn tme vayng paamete. A dffculty elated wth the applcatn f th methdlgy the tunng f the gan matce Ω and Γ whch ae degn paamete at the dpal f the ue f the cntl f the tablty and the tackng ppete f the algthm. h pblem dcued by Pmeleau and Pee (99) whch pped a ple placement baed tunng f the etmatn f the thee pecfc gwth ate nvlved n a Bake' yeat fed-batch fementatn. h pape devted t the tunng pblem f th etmat. An altenatve appach peented whch baed n the cncept f mpng that the etmated knetc fllw a ecnd de dynamc epne t the tue eactn knetc change, leavng fee t the ue the ettng f the natual ped f cllatn and f the dampng ceffcent. he matce f gan ae peented a functn f the ettng and f the ytem tate. Hence, they ae tme vaant but autmatcally adapted. ECOND-ORDER DYNAMIC BAED UNING It aumed that t uffcent t bae the knetc etmat n a ubet f equatn f the full tate pace mdel, pvded that they nvlve all the paamete that need t be etmated (th ubet f equatn dented by * Auth t whm all cepndence huld be addeed

2 the ndex ). In th cae the gan matce ae quae wth dmenn. In what fllw, a efmulated dynamcal mdel adpted by cndeng the tanfmatn ψ= K ξ, whch gve: d ψ = ϕξ D ψ + ( ) K ( F Q( ξ )) he etmat can then be ewtten a: (3) dψ ˆ = H ρ ˆ D ψ + K ˆ (F Q ) Ω ( ψ ψ ) (4a) ρ = H Γ ( ψ ψ) (4b) he dynamc f the bevatn e btaned by ubtactng eqn (4a) fm eqn (): d( ψ ψ) = H( ρ ) ρ + Ω ( ψ ψ) (5) If the matx Γ uch that [H(ξ)] Γ a cntant matx, then dffeentatng equatn (4b) gve d ρ H d ( ψ ψ) = Γ (6) Meve, f H(ξ) a dagnal matx, then cmbnng equatn (4b), (5), and (6), and ettng Ω=dag{-ω } and Γ=H(ξ) - dag{γ j } (whee γ j, ω R + ) the fllwng eult btaned: d ρ ρ + ζ + ρ = ρ =,... (7) 5. = ( γ h ) (8) ζ 5. = 5. ω ( γ h ) (9) whee h efe t the dagnal element f matx H(ξ). Eqn. (7), (8) and (9) hw that each paamete fllw a ecnd de dynamc epne t the tue paamete change wth a natual ped f cllatn f and a dampng ceffcent f ζ. Nevethele they ae functn f the ytem tate, and hence, they ae tme vaant. he applcatn f th methdlgy t the etmatn pblem f cmpletely unknwn eactn ate, pecfc eactn ate and pecfc gwth ate a fllw: ) cmpletely unknwn eactn ate In th cae we have: =M ρ(ξ)=ϕ(ξ) H(ξ)=I M Whch gve: = γ j -.5 ζ.=.5ω j γ j -.5 ) pecfc eactn ate In th cae we have: =M ρ(ξ)=α(ξ) H(ξ)=G(ξ)=dag{g j } whee g = j n~j ξ n mean multplcatn ve the cmpnent wth ndex n whch ae eactant n the eactn j. he eult : = (γ g ) -.5 ζ =.5ω (γ g ) -.5 ) pecfc gwth ate In th cae we have: =M ρ(ξ)=µ(ξ) H(ξ)=XI M whee X mean bma cncentatn. he eult : = (γ X) -.5 ζ.=.5ω (γ X) -.5 CAE UDY - BAKER YEA FED-BACH FERMENAION he pce mdel Yeat gwth chaacteed by thee metablc pathway: +C µ X +G (a) µ X + E + G (b) E e + C µ X + G (c) wth : gluce; C: xygen; X: bma; E: ethanl; G: cabn dxde and µ, µ, µ e : pecfc gwth ate f the thee pathway. Pathway (a), (b), and (c) efe epectvely t the epaty gwth n gluce (xydatve pathway), fementatve gwth n gluce (edutve pathway) and the epaty gwth n ethanl (xydatve pathway). he dynamc mdel f the fed-batch fement btaned fm a ma balance n the cmpnent, cndeng that the eact well mxed, the yeld ceffcent ae cntant and the dynamc f the ga phae can be neglected. he ma balance, n tem f

3 cncentatn, take the matx fm f the geneal dynamcal mdel (eqn. ): X X k k µ D n d E = k3 k 4 X D E µ + C k5 k 6 µ e C OR G k7 k8 k 9 G CR () whee D the dlutn ate and the k ae yeld ceffcent; n the ubtate cncentatn n the feed; OR the xygen tanfe ate and CR the cabn dxde tanfe ate. h dynamc mdel and the knetc mdel pped by nnletne and Käppel (986) wth mdfcatn made by Pmeleau and Pee (99) wee ued f mulatn pupe. It aumed that the bake' yeat fed-batch pce can nly be n an ethanl pductn tate n an ethanl cnumptn tate, meanng that the yeat can nly gw by tw pathway multaneuly: pathway a and b cepndng t ethanl pductn, and pathway a and c cepndng t ethanl cnumptn. he knetc etmat he "beve-baed knetc etmat" (eqn. 4) appled t tw patal mdel eflectng the tw pce tate mentned abve, takng the fm: dψ ˆ = X µ ˆ D ψ+ K ˆ (F Q ) Ω ( ψψ ) (a) dˆ µ = X Γ ( ψψ ˆ ) (b) wth: ψ = K ξ, ξ = C G, ( F Q ) = OR CR and wth the etmated pecfc gwth ate ˆµ wtchng between ([µ µ ] and [µ µ e ] ). he ue f eqn eque the n-lne knwledge f bma cncentatn. h acheved by mean f a "Luenbege-type aympttc beve" (Luenbege, 97) whch enable the n-lne etmatn f X, and E fm meauement f C and G: wth dzˆ = DZ ˆ + (F Q ) K K (F Q ) (3a) ξ ˆ = Ẑ+ K K ξ (3b) ξ = C G ξ ˆ [ ] = X E (F Q ) = OR CR ( F Q) = D n [ ] A uch, the etmatn pcedue cnt f tw tep, vz.- ) tate etmatn fm avalable pce meauement and ) pecfc gwth ate etmatn. he pce mdel and the knetc mdel adpted wee mplemented n a pce mulat (Fey de Azeved et al., 99; Pmenta and Fey de Azeved, 993) whch uppled th tw-tep etmatn algthm wth the mulated meaued vaable - C, G, CR, OR, n and F at amplng tme f 6 mnute. ecnd-de-dynamc baed tunng he gan matce f the cae f pecfc gwth ate ae gven by: γ = X ω m ζ = (4) whee and ζ ae the deed natual ped f cllatn and dampng ceffcent, and X m a mean value f bma etmate ve the tme nteval. A gven by eqn. 4, the γ paamete ae pecewe functn f bma,.e., γ eman cntant between meauement, beng adjuted at each amplng ped. he knetc etmat equatn (eqn. and 3) wee ntegated wth a but vaable-tep numecal ntegatn algthm (4th/5th de Runge-Kutta type embedded cheme due t Butche) emplyng alng the ntegatn lnea etmate f the elevant ampled vaable. Ple placement baed tunng he veall etmatn pcedue wa al caed ut, emplyng the Eule dcetzatn appach and tunng methd pped by Pmeleau and Pee (99). Bacally, th methd cnt n defnng tme tajecte f the gan paamete n de t mantan cntant the ptn f ple (n the dcete cmplex plan) f the dcete e ytem thughut the fementatn. he gan paamete ae gven by: (p ) γ = ω X p ( ) = (5) whee p the deed duble ple f the e ytem (<p <), X bma etmate, and the amplng ped (and al the ntegatn tep). Reult and Dcun Fgue, and 3 llutate the tunng pcedue pped n th wk. he eult n fg. ae btaned wth mla natual ped f cllatn ( =.) and dampng ceffcent (ζ =.5) f the thee cmpnent. he nfluence f and ζ n the dynamc f cnvegence can be aeed fm the plt n Fg. and 3. h nfluence n ageement wth the chaactetc f a typcal ecndde dynamc epne: deceang the epne becme fate and deceang ζ the epne becme me cllaty

4 h µ.35 IAE- µ IAE- µ :.56 µ :.344 µe :.48 :.6 µ :.4 µe :.47.3 µ µ t (h) µ h µ µ Fgue -pecfc gwth ate (full lne-tue; dted lneetmate) ung the nd de dynamc baed tunng (=., ζ=.5) Fgue 4 -pecfc gwth ate etmate (full lne-tue; dted lne-etmated) ung the ple placement baed tunng h =..5 h Fgue - etmate f dffeent dampng ceffcent (ζ) Fgue 5 - etmate f dffeent ple (p) h ζ =.8 Fgue 3 - etmate f dffeent f dffeent natual ped f cllatn () Fg. 4 and 5 llutate the ame applcatn wth the ple placement technque.. he bet eult ae btaned when the duble ple ae cle t ze (n gnfcant mpvement btaned when p<.). In fg. 4 the thee pecfc gwth ate (etmated v. 'tue') ae epeented f p=.. he nfmatn fm Fg. and 4 ugget that the ecnd de dynamc appach pduce bette eult than the ple placement methd. h cnfmed by the e ndexe emplyed (IAE ntegal f tme-weghted ablute e) whch ae, f the fme, an de f magntude lwe than the beved f the latte. he the pble advantage f the ecnd de tunng that the chce f paamete ha an ntutve ba nce th type f epne wdely beved n natual phenmena and t theetcal tudy well demnated. Futhe theetcal analy ut f cpe f th pape. Wk n pge whch am n patcula at etablhng the dman f valdty f the pcedue pped

5 REFERENCE Batn, G., Dchan, D. (99) 'On-Lne Etmatn and Adaptve Cntl f Beact', Eleve, Amtedam. Fey de Azeved,., Pmenta, P., Olvea, F., Feea, E. (99), tude n On-lne tate and Paamete Etmatn thugh A Real-me Pce mulat, n Kam, M. N. and tephanpul, J. (Ed.), Mdelng and Cntl f Btechncal Pcee, IFAC ymp. ee, pp , Pegamn Pe, N.Y. Luenbege, D.G. (97). An ntdutn t beve. IEEE an. Autmatc Cntl, vl. AC-6, pp Pmenta, P., Fey de Azeved,. (993) Real-me Dynamc mulatn f Nn-lnea MIMO ytem, Cmpute and Chem. Engng., 7, 343. Pmeleau, Y. and Pee, M. (99) 'Etmatn f Multple pecfc Gwth Rate n Bpcee', AIChE Junal, Vl. 36, n., 7-5. nnletne, B. and Käppel, O. (986) 'Gwth f acchamyce ceevae cntlled by t Lmted Repaty Capacty: Fmulatn and Vefcatn f a Hypthe', Btech. Beng., Vl. 8, June, pp ACKNOWLEDGEMEN h wk wa patally uppted by JNIC - Junta Nacnal de Invetgaçã Centífca e ecnlógca, unde cntact numbe BIC/636/9 and BD/4/9-IF and BD/476/9-RM. C CR D E F G g NOMENCLAURE dlved xygen cncentatn cabn dxde tanfe ate dlutn ate dlved ethanl cncentatn feed ate vect dlved cabn dxde cncentatn pduct f eactant cncentatn n eactn H(ξ) k K m n OR p Q n X X m α ϕ (m ) matx f functn f the tate yeld ceffcent yeld ceffcent matx numbe f eactn ate numbe f tate pace vaable xygen tanfe ate duble ple f the dcete e ytem ga emval ate vect numbe f paamete t etmate gluce cncentatn gluce feed cncentatn amplng ped bma cncentatn aveage value f bma cncentatn ve the amplng ped pecfc eactn ate vect eactn ate vect µ pecfc gwth ate vect µ^ vect f etmated pecfc gwth ate µ pecfc gwth ate f the fementatve gwth n gluce pathway µ e pecfc gwth ate f the epaty gwth n ethanl pathway µ pecfc gwth ate f the epaty gwth n gluce pathway ρ(t) vect f unknwn tme-vayng paamete ρ^ vect f etmated paamete natual ped f cllatn ω, γ dagnal element f Ω and Γ Ω, Γ gan matce ξ^ pedcted tate vect f cncentatn ξ vect f meaued cncentatn vect f nn-meaued cncentatn ξ ξ^ ζ etmated tate vect f nnmeaued cncentatn dampng ceffcent