Brunovsky Normal Form of Monod Kinetics Models and Growth Rate Control of a Fed-batch Cultivation Process

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1 Auoaion Bioauoaion uppl. 3 6 IN 3 45X Brunovsky Noral or of Monod Kinics Modls and Growh Ra Conrol of a d-bach Culivaion Procss Pavlov Y. Cnr of Biodical Enginring Prof. Ivan Daskalov Bulgarian Acady of cincs 05 Acad. G. Bonchv r. ofia 3 Bulgaria Phon: E-ail: yupavlov@clb.bas.bg yupavlov4@hoail.co uary: A ahaical hodology ha givs assisanc o dsign of fd-bach sabilizaion and conrol is prsnd. Th hodology is basd boh on Uiliy hory and opial Conrol hory. Th Uiliy hory dals wih h xprssd subjciv prfrncs and allows for h xpr prfrncs o b akn in considraion in coplx biochnological syss as criria for conrol and opiizaion. Th Conrol hory is usd for parars sabilizaion of a fd-bach culivaion procss. Th conrol is wrin basd on inforaion of h growh ra. Th siulaions show good fficincy of h conrol laws. Kywords: Opial Conrol Brunovsky Noral or Uiliy uncion d-bach rnaion.. INTRODUCTION Mahaical odls ar usd by nginrs o gain advanag hrough applicaions of odl basd procss dsign conrol and opiizaion. Thus building ahaically oivad and validad odls is a ky aciviy in bioprocsss nginring. Th spcific growh ra is on of h os iporan parar in biochnological culivaion procss. Th rlaionships bwn h ra of growh subsra concnraion and produc foraion ar crucial for onioring conrolling and opiizing hs procsss. Thrfor i is of iporanc o b abl o odl and conrol h spcific growh ra as a funcion of h bioass and subsras and vic vrsa [7]. I is h opinion of so sciniss ha wih opial conrol hory can b shown a fd-bach procss is likly o ouprfor boh a coninuous procss and a bach procss. Th probl is hn h drinaion of h bs fd ra of subsra in h bioracor as a funcion of h spcific growh ra and/or h bs praur o and h bs acidiy ph [7 0]. And sinc h 3

2 Auoaion Bioauoaion uppl. 3 6 IN 3 45X qualiy of h produc usually drioras wihin svral hours afr subsra dplion i is hnc of iporanc o harvs in i. Th aning of bs varis fro probl o probl [5 7]. Th coplxiy of h biochnological culivaion procsss du o h inhrn i varian propris and h lack of prcis asurn ak difficul h drinaion of h opial h bs procss parars. Th incopl inforaion is copnsad so is wih paricipaion of iprcis huan siaions. Th ncssiy of a rgr of pirical knowldg wih ahaical xacnss causs difficulis. Possibl approach for soluion of hs probls is h sochasic prograing and h Uiliy hory [3 5]. Th Uiliy hory dals wih h xprssd subjciv prfrncs. Possibl criria for h aning of bs can b an xpr dcision akr- DM uiliy funcion [3 5 9]. Th ai of his invsigaion is o donsra an opiizaion chniqu and h possibiliy o conrol opially h spcific growh ra of a biochnological culivaion procss. Th conrol dsign is basd on Monod-Wang odl in Brunovsky noral for. Th classical Monod for is a singular for of his odl [ ].. MODEL O THE ED-BATCH ERMENTATION PROCE Unsrucurd odls ak cll ass as a unifor qualiy wihou inrnal dynaic. Th racion ras dpnd only upon h acroscopic condiions in h liquid phas of h bioracor. Unsrucurd odls fail only whn inracllular dynaics us b considrd. Mahaical unsrucurd odls of fd-bach procss can b wrin basd on ass balanc quaions [0]: X = X X = k X o = = E == k X E 4

3 Auoaion Bioauoaion uppl. 3 6 IN 3 45X whr X prsns h concnraion of bioass [g/l]; h concnraion of subsra glucos [g/l]; - bioracor volu [l]; subsra fd ra [h - ]; 0 subsra concnraion in h fd [g/l]; ax - axiu spcific growh ra [h - ]; K sauraion consan [g/l]; k k yild cofficins [g/g] ra cofficin [-]; E h concnraion of hanol [g/l]. W prsrv h noaion U. for h DM uiliy funcion h criria for opiizaion. Th sys parars ar as follows: = 0.59 [h - ] K = [g/l] = 3 [ ] 0 = 00 [g/l] k = [ ] k = 3.79 [ ] ax = 0.9 [h - ] ax =.5 [l]. Th dynaics of in h so calld Monod-Wang odl is odld as a firs ordr lag procss wih ra consan in rspons o h dviaion in. Th fifh quaion dscribs h producion of hanol E. This quaion is quivaln dynaically o h firs on. Th donsraion is asy. W ipln a sipl ransforaion in h firs quaion: X = E. k Afr ha h firs and h fifh quaions bco quivaln. Th nw for of h non-linar kinic odl is: X = X X = k X o 3 = K s =. W shall us his for of h non-linar fd-bach kinic odl in h rs of h papr. Th iniial valus of h sa variabls ar: X i 0 = 0.99; i 0 = 0.0; i 0 = 0.; i 0 = UTILITY UNCTION AND DETERMINATION O THE BET GROWTH RATE Th coplxiy of h biochnological frnaion procsss aks difficul h drinaion of h opial procss parars. 5

4 Auoaion Bioauoaion uppl. 3 6 IN 3 45X Th incopl inforaion usually is copnsad wih h paricipaion of iprcis huan siaions. Our xprinc is ha h huan siaion of h procss parars of a culivaion procss conains uncrainy a h ra of [0 5] %. Hr is usd a ahaical approach for liinaion of h uncrainy in h DM s prfrncs basd boh on h Uiliy hory and on h ochasic prograing [9]. Th algorihic approach pris valuaion of h opial spcific growh ra of h fd-bach culivaion procss according o h DM poin of viw. W nd so ahaical forulaions. andard dscripion of h uiliy funcion applicaion is prsnd by ig.. Thr ar a variy of final rsuls ha ar consqunc of h xpr or DM s choic and aciviy. ig.. Uiliy funcion applicaion This aciviy is oivad by a DM objciv which possibly includs conoical social cological or ohr iporan procss characrisics. A uiliy funcion U. asssss ach of his final rsuls i i = n. Th DM judgn of h procss bhavior basd of h DM choic is asurd quaniaivly by h following forula [3 5]: U p = U i pi whr p = p p.. pi.. pn pi =. i i 4 6

5 Auoaion Bioauoaion uppl. 3 6 IN 3 45X W dno wih p i subjciv or objciv probabiliis which rflc h uncrainy of h final rsul. L Z b h s of alrnaivs Z = {spcific growh ras} = [0 0.6] and P b a convx subs of discr probabiliy disribuions ovr Z ig.. Th xpr prfrnc rlaion ovr P is xprssd hrough and his is also ru for hos ovr Z Z P. W know ha h uiliy funcion is dfind in h inrval scal in h proposd condiions [3 9]. A dcision suppor sys for subjciv uiliy U. valuaion is usd ig.. Th subjciv uiliy funcion U. is shown on ig. 3. Th uiliy U. is approxiad by a polynoial: 6 c i i = 0 i U = Th polynoial rprsnaion pris analyical drinaion of h drivaiv of h uiliy funcion and asy iplnaion in h opiizaion and conrol. Th uiliy funcion in his invsigaion is valuad wih 64 larning DM s answrs sufficin for a priary orinaion in h probl [9]. 5 ig.. Dcision suppor sys 7

6 Auoaion Bioauoaion uppl. 3 6 IN 3 45X ig. 3. Uiliy vrsus Growh ra This uiliy valuaion ahaical approach is discussd in dails in [9]. 4. BRUNOKY NORMAL ORM O MONOD-WANG MODEL TIME MINIMIZATION CONTROL AND TABILIZATION O THE GROWTH RATE W prsrv h noaion U. for h DM uiliy. Th conrol dsign of h fd-bach procss is basd on h nx subsidiary opial conrol probl: MaxUT in whr h variabl is h spcific growh ra [0 ax ] D [0 D ax ]. Hr U is an aggrgaion objciv funcion h uiliy funcion ig. 3 and D is h conrol inpu h diluion ra: ax U [0 X = X DX ax = k X o D = ] [0 T in ] D [0 D ax ] 6 8

7 Auoaion Bioauoaion uppl. 3 6 IN 3 45X Whn T in is sufficinly sall h opiizaion is in fac i iniizaion. Th diffrnial quaion in 6 dscribs a coninuous frnaion procss. Th odl pris xac linarizaion o h nx Brunovsky noral for Goursa as rgard o h diffrnial fors [ 8]: Y = Y Y Y = Y3 3 = W. Hr W dnos h conrol inpu of h odl 7. Th nw sa vcor Y Y Y 3 is: Y = u Y 3 3 = u 3 u Y = u u ku 3ku k 3 u X u X = o u X u 3 X u u 3 u ku u. Th drivaiv of h funcion Y 3 drins h inrconncion bwn W-odl 7 and D-odl 6. Th conrol dsign is a dsign basd on h Brunovsky noral for and applicaion of h Ponrjagin s axiu principl sp by sp for sufficinly sall i priods T [6 8]. Th opial conrol law has h analyical for [9]: 6 i T ky Dop = sign ici T ax D 9 i= whr : sign r = r > 0 sign r = 0 r 0. Th i inrval T can b h sp of discrizaion of h diffrnial quaion solvr. Th su in Eq. 0 is h drivaiv of h uiliy funcion U. I is clar ha h i-iniizaion conrol is drind fro h sign of h uiliy drivaiv. Thus h conrol 7 8 9

8 Auoaion Bioauoaion uppl. 3 6 IN 3 45X inpu is D = D ax or D = 0. Th soluion is a i-iniizaion conrol if h i priod T is sufficinly sall. Th conrol brings h sys back o h s poin for inial i in any cas of spcific growh ra dviaions. Th donsraion is shown in [8]. Th prvious soluion pris asy drinaion of h conrol sabilizaion of h fd-bach procss. Th conrol law is basd on h soluion of h nx opiizaion probl: MaxUT in whr h variabl is h spcific growh ra [0 ax ] [0 ax ]. Hr U is h uiliy funcion in ig. 3 and is h conrol inpu h subsra fd ra: ax U T in [0 ] [0 T ] [0 ax] X = X X = k X o = = ax in 0 Th conrol law of h fd-bach procss has h sa for 9 bcaus D is rplacd wih / in h fd-bach odl. Thus h fding ra aks = ax or =0 dpnding on D which aks D = D ax or D = 0. W conclud ha h conrol law 9 bring h sys o h opial poin opial growh ra wih a i iniizaion conrol saring fro any dviaion poin of h spcific growh ra ig. 4. Thus w dsign h nx conrol law:. A h inrval [0 ] h conrol is i-iniizaion conrol 9 whr = x 30 -ε ε > 0 x 30 = axu D is rplacd wih = γ ax γ>0 whn D = D ax Th choic of γ dpnds on h sp of h quaion solvr and is no a par of h opiizaion hr γ = 0.3;. A h inrval [ ] h conrol is = 0 =x 30 -ε = x 30 - o b scapd an ovrrgulaion; 0

9 Auoaion Bioauoaion uppl. 3 6 IN 3 45X 3. Afr his on h conrol is h conrol 9 wih = γ ax whn D = D ax charing conrol wih γ>0. Th dviaion of h fd-bach procss wih his conrol is shown on ig. 4. GROWTH RATE OPTIMAL CONTROL cond Ordr liding Mod Conrol pcific Growh Ra Equivaln liding Mod Conrol TIME [h] ig. 4. abilizaion of h fd-bach procss 0.8 ubsra Concnraion [g/l] TIME [h] ig. 5. ubsra concnraion in h bioracor Afr h sabilizaion h procss can b ainaind around h opial parars wih conrol in sliding od igs Possibl soluion in sliding od is alrnaion of as a funcion of h praur and h acidiy in h bioracor or alrnaion of [].

10 Auoaion Bioauoaion uppl. 3 6 IN 3 45X 5. OPTIMAL PROILE AND DETERMINATION O MOMENT AND MOMENT Th soluion dscribd in h prvious chapr is a charing conrol. In his chapr w drin a sooh conrol soluion for drinaion of h inrval [ ] - drinaion of h ons and. Hr is supposd ha i 0< 0< whr: x30 = = opial pon = x30 = 0.3. Th opiizaion probl is: in 0 d [0 X = X X = k X o = = ] [0 ] [0 ax ax ] = 0.3 A supplnary condiion is liinaion of any ovrrgulaion of. Th opial conrol law according h prvious soluion is:. A h i inrval [0 ] h conrol is = γ ax. This is provd in [8] γ = 0.3 his parar is no a par of h opiizaion;. A h inrval [ ] h conrol is = 0 o b scapd any ovrrgulaion. Th growh ra changs fro n o whr = n = and n <. W ffc his by drinaion of a anifold on h bas of h probl Eq. 4. Whn h sa vcor across ovr his anifold h conrol bcos = 0. Th on is h on of ovr crossing h anifold = d/d=0;

11 Auoaion Bioauoaion uppl. 3 6 IN 3 45X 3 3. Afr h on h conrol is = kx/o- whr X. is h quaniy of bioass in h bioracor if > ax w pos = ax and hr γ =. Th drinaion of on and on is basd on h nx opial conrol probl: ] [0 ] [ ] [0 in ax ax = = = = = = X k X d X Hr is supposd ha 0<. W propos h nx nurical soluion for drinaion of an approxiaion of on : - If 0> n is quals o zro = 0. In his cas h iniial condiions ar quivaln o hs of odl 3. An ovrrgulaion is possibl bcaus h procss is unconrollabl fro on 0 o on ; - If 0< n and 0< ; h on is h on a which h sa vcor of h diffrnial Eq. 3 across ovr h nx anifold: 0 xp ln xp = = f f f f f X Manifold 4. ln ln = X X f whr

12 Auoaion Bioauoaion uppl. 3 6 IN 3 45X Th on is drind approxialy as follows = f. Th soluion is shown on ig Growh Ra CONTROL TIME [h] ig. 6. Opial profil wih h anifold 4 Th opial profil of h fd-bach culivaion during h whol i priod is shown on ig. 7. This conrol law drins h sa opial soluion as h charing conrol 9. Growh Ra OPTIMAL PROILE CONTROL Parar Nois TIME [h] ig. 7. Opial profil of h growh ra 4

13 Auoaion Bioauoaion uppl. 3 6 IN 3 45X Th anifold 4 is drind on h bas of h opiizaion probls and 3 wih so siplificaion and approxiaion of h ingral. This is a nuric soluion. Th on is drind approxialy. If h procss do no rachs h valu a h calculad i his is a nurical soluion h sp is rpad iraivly. 6. CONCLUION In h papr is prsnd a conrol dsign basd boh on h Brunovsky noral for of h Monod-Wang kinics odl and on a charing opial conrol dsign. Th opial profil is drind providing agains h ovrrgulaion of h spcific growh ra. Th siulaion confirs h fac ha h sliding opial soluions ar robus soluions. Th valuaion of h xpr uiliy criria is an iraiv procss. This characrisic pris an iraiv nginr conrol dsign and asy corrcion of h conrol law in agrn wih so nw changs in h chnological condiions. Acknowldgns This work is parially suppord fro Naional cinc und Projc MI 505/005. REERENCE. Elkin. Rducion of Non-linar Conrol yss: A Diffrnial Goric Approach Mahaics and is Applicaions ol. 47 Handbound Kluwr Elyanov.. Korovin and A. Lvan. Highr-ordr liding Mods in Conrol yss Diffrnial Equaions ishburn P. Uiliy Thory for Dcision-Making Wily Nw York Gardnr R. W. hadwick Th G algorih for xac linarizaion o Brunovsky noral for IEEE Trans. Auo. Conrol

14 Auoaion Bioauoaion uppl. 3 6 IN 3 45X 5. Kny R. H. Raiffa Dcision wih Mulipl Objcivs: Prfrncs and alu Trad-offs Nw York Kroov.. Guran Mhods and Probls in h Opial Conrol Nauka Moscow Nlan R. Bioass Prforanc: Monioring and Conrol in Bio-pharacuical producion Thsis Wagningn Univrsiy Pavlov Y. K. Ljakova Equivaln Modls and Exac Linarizaion by h Opial Conrol of Monod Kinics Modls Bioauoaion Pavlov Y. ubjciv prfrncs valu and dcision: ochasic approxiaion approach Cops Rndus d L Acadi Bulgar ds cincs aniskis J. D. Lvisauskas R. iuis U. isur M Krisapsons Auoaion of Biochnological Procsss. Masurns Opiizaion and Conrol Zinan Riga 99.. Tzonkov. D. ilv I. ionov L. aklv Conrol of Biochnological Procsss Tchnika ofia 99.. Wang T. C. Moor D. Birdwll Applicaion of a Robus Mulivariabl Conrollr o Non-linar Bacrial Growh yss - In: Proc. of h 0 h IAC Congrss on Auoaic Conrol Munich