Open Loop Vibrational Control of a Recycled Bioprocess

Transcription

1 Proceedig of the 7th WSEAS Iteratioal Coferece o Automatio & Iformatio, Cavtat, Croatia, Jue 3-5, 6 (pp7-76) Ope Loop Vibratioal Cotrol of a Recycled Bioproce DA SELIŞEAU, EMIL PERE, HAI HAMDA, DORI POPESCU Departmet of Automatic Cotrol ad Mechatroic Uiverity of Craiova A.I. Cuza Str. o. 3, RO-585, Craiova ROMAIA SMS, UMR CRS 99, IRCAM, place Igor Straviky, 754 Pari FRACE Abtract: - hi paper deal with the vibratioal cotrol of a recycled bioproce, which take place ito a Cotiuou Stirred ak Bioreactor (CSB). he claical cotrol techique for bioreactor eceitate meauremet of tate ad/or diturbace. However, the bioprocee are ytem that require cotly ad difficult o-lie implemetatio for eor. Vibratioal cotrol i a o-claical ope-loop cotrol method, developed by Bellma, Betma ad Meerkov, ueful for the cae whe the itrumetatio i ot available or i very expeive. he vibratioal cotrol i applied by mea of zero mea parametric vibratio for hapig the repoe of a liear or oliear ytem. I thi work, the deig of a vibratioal cotrol for a CSB with recycle tream i preeted. umerical imulatio are coidered i order to illutrate the performace of the vibratioally cotrolled bioproce. Key-Word: - Ope-loop cotrol, Vibratioal cotrol, Liear time lag ytem, Biotechology Itroductio Feedback or feedforward cotrol techique ca be ued i order to achieve a deired chage i the behaviour of a ytem. he problem i that thee trategie require o-lie meauremet. I ome cae, the itrumetatio i ot available or i very expeive. hi i the cae of the bioidutry, where the cheap ad reliable itrumetatio i miig. A poibility to overcome thi problem i the applicatio of the o-called vibratioal cotrol. Vibratioal cotrol (VC) i a o-claical opeloop cotrol method propoed by Bellma, Betma ad Meerkov [], [3], [9]. he claical theory of VC for liear ytem wa developed by Meerkov i [9] ad exteded to oliear ytem by Bellma et al. [], [3]. Other igificat reult are obtaied by Lehma et al. [6], [7] for time lag ytem with bouded delay. Applicatio of the vibratioal cotrol theory ca be foud for: tabilizatio of plama, laer [8], chemical reactor [5], biotechological procee [], [], []. he VC techique i applied by ocillatig a acceible ytem compoet at low amplitude ad high frequecy. he amplitude ad the frequecy of the cotrol iput are cotat ad idepedet of the tate of the ytem, therefore thi techique i a form of ope-loop cotrol. he Sectio of thi paper deal with the preetatio of the model of bioproce that take place ito a CSB with recycle tream. he dyamic model, the equilibrium poit, the liearizatio aroud a equilibrium poit ad the phae portrait are aalyed i thi ectio. I Sectio 3, the VC theory for time-lag ytem i preeted. he vibratioal cotrol trategy i developed for the recycled bioproce, uig a liearized model. he exitece ad the choice of tabilizig vibratio, which eure the deired behaviour for the bioproce, are alo aalyed. Illutrative umerical imulatio are icluded. Cocludig remark are collected i Sectio 4. Model of the Recycled Bioproce A bioreactor i a tak i which everal biological reactio occur imultaeouly i a liquid medium. I idutry, the bioreactor operate i three mode: the cotiuou mode, the fed-batch mode ad the batch mode [], []. Bioreactor that operate i the cotiuou mode are uually kow a Cotiuou Stirred ak Bioreactor. I a CSB, the ubtrate (the utriet) are fed to the bioreactor cotiuouly ad a effluet tream i cotiuouly withdraw from the CSB uch that the culture volume i

2 Proceedig of the 7th WSEAS Iteratioal Coferece o Automatio & Iformatio, Cavtat, Croatia, Jue 3-5, 6 (pp7-76) cotat. Ofte, a part of the bioma i recycled. o recycle, the bioma mut be eparated from the ubtrate ad yield, the travel through pipe after eparatio. hi time of recycle itroduce delay i the tate ad complicate the dyamic. he beefit are that the recycle icreae the overall coverio ad reduce the cot. he dyamical tate-pace model of a biotechological proce i a CSB expree the ma balace of the compoet i the bioreactor [], []: = µ ( ξ, ξ ) ξ D ξ S i () = k µ ( ξ, ξ ) ξ D ξ + D () where ξ, ξ repreet the bioma ad the limitig ubtrate cocetratio [g/l]. S i i the ifluet ubtrate cocetratio ad D i o-called dilutio rate [h - ], i.e. the pecific volumetric outflow rate. I (), () µ i the pecific growth rate ad k > the yield coefficiet. he bioproce (), () i i fact a fermetatio proce, which uually occur i a bioreactor. I the CSB with recycle tream, a part of the bioma i recycled. If the recycle occur, the the bioproce model (), () mut be rewritte []: = µ ( ξ(, ξ ( ) ξ( Dξ ( + ( q) Dξ ( = kµ ( ξ(, ξ ( ) ξ( Dξ ( + F i (3) (4) he model (3), (4) i a prototype model for ome depollutio bioproce, like the activated ludge bioproce, which recycle a part of bioma [3]. I (3), (- q) D i the recycle flow rate. he cotat q varie from to, with zero correpodig to total recycle ad to o recycle. he cotat r i the recycle delay time ad F i i the iput flow. A compact repreetatio of the tatepace model (3), (4) i: ξ & = f (ξ) (5) ξ ξ ξ i the tate vector ad the fuctio f () i the oliear vector fuctio where = [ ] [ f ( ξ, ξ ), f ( ξ, ] f ( ξ ) = ξ ). he equilibrium tate of (3), (4) are of two type:. Wah-out equilibrium tate (E3), defied by: (E) = [ ξ ξ ] [ ] ξ F / D (6) = i hi equilibrium i i fact a tate whe the bacterial life ha diappeared.. Operatioal equilibrium tate (E), implicitly defied by: (E) µ ( ξ, ξ ) = qd kµ ( ξ, ξ ) ξ + Dξ = F i (7) Equilibria (E) correpod to the bioreactor wah-out, therefore oly equilibria (E) have a techological iteret. hee equilibria ca be attractive or repulive depedig o the particular form of µ ( ξ, ξ ). Oly thee equilibria have a practical iteret. Let aume that the pecific growth rate i: ξ µ ξ ξ = µ ξ = µ (8) (, ) ( ) K M + ξ + ξ / K i hi i the Haldae kietic model of the pecific growth rate []. K M i the Michaeli - Mete cotat, K i the ihibitio cotat ad µ the maxim pecific growth rate. ext, we uppoe that the form of the pecific growth rate i the Haldae kietic model (8) that take ito accout ubtrate ihibitio o the growth. he, from (7) we have two poibilitie for the equilibria (E): F D F D i ξ i ξ, (a) ξ = = ξ, = ; ξ = ξ, kqd kqd (9) F D F D i ξ i ξ, (b) ξ = =ξ, = ; ξ = ξ, kqd kqd () he cae (a) correpod to a table equilibrium poit (table ode). he cae (b) lead to a addle type for the equilibria (E) []. he phae plae correpodig to the ytem (3), (4) for the value of the proce parameter: µ 6 = h, K M = g / l, K i = g / l, k =, D = 3.6h, S i = g / l, F i = 4 h - g/l, q =.8, r =.5 h ad for differet iitial coditio i repreeted i Fig.. From thi picture it ca be ee that whe the ubtrate ihibitio appear, the proce ca exhibit utable or, maybe wore, the evolutio lead to wah-out teady-tate, for which the microbial life ha diappeared ad the reactor i topped. I thee ituatio, the bioproce require cotrol to tabilize the CSB. Alo, i may cae, the table equilibrium poit correpodig to (a) i ot techological operable (require a big iitial amout of bioma). Furthermore, a bigger value for the delay time ca iduce a wort behaviour.

3 Proceedig of the 7th WSEAS Iteratioal Coferece o Automatio & Iformatio, Cavtat, Croatia, Jue 3-5, 6 (pp7-76) 5 [g/l] wah-out ( λ A A ) det I = (6) Subtrate cocetratio 5 addle-poit C i table ode he root of equatio (5) decide the tability of the equilibria. For the operatioal equilibrium poit (b), which i iteretig from techological poit of view, after ome calculatio, from (6) we obtai that thi poit i utable (ee Fig. ) []. he delay time ifluece the tability propertie; therefore, thi equilibrium poit of the iitial ytem (5) i ideed utable. he goal of the cotrol trategy i to tabilize the equilibrium poit () Bioma cocetratio Fig.. Phae plae of the recycled bioproce [g/l] he cocluio i that for the CSB with recycle tream it i eceary to deig a cotrol trategy. For cotrol purpoe, it i eceary to fid the liear approximatio of the ytem (3), (4) or equivalet (5) aroud the equilibrium poit (E). he liear approximatio for give cotat iput D ad F i i: d ( ξ ξ ) = A ( ξ )( ξ( ξ ) + A ( ξ )( ξ( ξ ) ( () where the matrice A ad A are obtaied after traightforward calculatio for the pecific rate (8): µ ( ξ ) D γ A ( ξ ) = A ( ξ, ξ ) = () kµ ( ξ ) kγ D ( q) D A ( ξ ) = A ( ξ, ξ ) = (3) with γ =ξ dµ ( ξ ) ξ Let coider: = ξ µ ( K K / / K =ξ M i M +ξ x = ξ ξ = ξ ξ [ x x ] = [ ξ ξ ξ ] he the liear model () become: + ξ dx( = A + A (4) he characteritic equatio of the liearized ytem (4) aroud the equilibria (E) i a tracedetal equatio: ( λ A A e ) det I = (5) For a mall delay time r, the equatio (5) ca be approximated with the equatio: K i ) 3 Vibratioal Cotrol Deig 3. Problem tatemet A geeral theory of vibratioal cotrol wa developed by Bellma et. al. [], [3], who preeted the criteria for vibratioal tabilizability ad vibratioal cotrollability of liear ad oliear ytem. Coider a oliear ytem give by the equatio: x& = f ( x, α) (7) m with f : R R R, x R i a tate ad m α R i a parameter, i fact a vector that cotai the ytem parameter. Suppoe that for a fixed α = α the ytem (7) ha the equilibrium x = x (α). Let itroduce ow i (7) parametric vibratio accordig to the law: α ( t ) = α + g( (8) where α i a cotat vector ad g( i a almot periodic vector fuctio with average equal to zero (APAZ vector). he (7) become: x & = f ( x, α + g( ) (9) Defiitio []: A equilibrium poit x ( α ) of (7) i vibratioally tabilizable if for ay δ > there exit a APAZ vector g( uch that (9) ha a aymptotically table almot periodic olutio x ( characterized by: x x α ) < δ () x ( = ( lim x ( τ) dτ () = x Defiitio []: A equilibrium poit x ( α ) of (7) i totally vibratioally tabilizable if it i vibratioally tabilizable ad furthermore,

4 Proceedig of the 7th WSEAS Iteratioal Coferece o Automatio & Iformatio, Cavtat, Croatia, Jue 3-5, 6 (pp7-76) x ( = cot = x α ), t R ( Defiitio 3 []: A equilibrium x ( α ) i partially vibratioally tabilizable with repect to i- th compoet if for ay δ > exit a APAZ vector g( ad α = cot uch the ytem x & = f ( x, α + g( ) () ha a aymptotically table almot periodic olutio x (, the i-th compoet of which i characterized by: x x α ) < δ (3) i i ( he vibratioal tabilizability problem coit of fidig coditio for exitece of tabilizable vibratio. Meerkov ha demotrated ice 98 [9] that for liear ytem vibratioal tabilizability implie total tabilizability. If the matrix A of the liear ytem x & = Ax (4) i a oderogatory matrix, i.e. the miimal ad characteritic polyomial coicide, a ufficiet coditio of vibratioal tabilizability i: tr(a) < (5) Parametric vibratio ca tabilize the liear ytem oly if they have a o-called multiplicative form: g( = B(x (6) If B( i periodic, the the coditio (5) i alo eceary [9]. Oce the coditio for exitece of vibratioal tabilizability are ettled for a ytem, it i eceary to olve aother importat problem: fidig the pecific form of tabilizig vibratio. hi problem i referred a vibratioal cotrollability [3]. he VC theory wa exteded by Lehma et al. [6], [7] for time lag ytem with bouded delay. he obtaied reult demotrated the viability of VC techique a a poible alterative to feedback for time lag ytem. Let coider the liear time lag ytem: dx( = A + A (7) where x i the tate ad r the time delay. Coider alo the liear almot periodic ytem: dy where t = F y( (8) ε x y :[ r, ) R, F : R R, < ε <. + Defie the averaged equatio correpodig to (8) a: dz = A z(, (9) t z :[ r, ) R, A = lim F. ε Lemma [6]: Aume that all the root of the det( λ I Ae ) have ozero real part. he exit ε > uch that for ay < ε ε the trivial olutio of (8) ha the ame tability propertie a that of (9). Coider ow the ytem with vibratio: dx( = A + F( t, ε) + A (3) + + x with F : R R R periodic i the firt argumet zero average matrix (PAZ matrix) ad < ε <. Defiitio 4: he trivial olutio of the ytem (5) i aid to be totally vibratioally tabilizable if there exit a PAZ matrix F ( t, ε) uch that the trivial olutio of the ytem with vibratio (3) i aymptotically table. Remark : he ytem with vibratio (3) ca be iterpreted a the reult of itroductio of a parametric excitatio ito the matrix A. he problem of vibratioal tabilizatio for ytem (7) coit of fidig coditio for exitece of totally tabilizatio vibratio ad fidig the cocrete tabilizig vibratio. heorem [6]: Suppoe that: (i) there exit a PAZ matrix B( uch that the tate traitio matrix Φ (t,) of dx = B( (3) i almot periodic; (ii) the equatio det( λi A Ae ) =, (3) = lim Φ Ai ( τ,) Ai Φ( τ,) dτ, i =, ha all the root with the egative real part. he: (a) there exit ε > uch that for ay < ε ε the trivial olutio of the equatio dy t t = Φ, AΦ, y( ε ε (33) t t + Φ, AΦ, y( ε ε i aymptotically table.

5 Proceedig of the 7th WSEAS Iteratioal Coferece o Automatio & Iformatio, Cavtat, Croatia, Jue 3-5, 6 (pp7-76) (b) the trivial olutio of the ytem (7) i totally tabilizable by the vibratio: t F( t, ε ) = B (34) ε ε if ε ε. he heorem reduce the vibratioal tabilizatio problem for liear time lag ytem of form (7) to thi trategy: firt we iduce a aymptotic tability of the ytem (33) by the correct choice of the PAZ matrix B(, ad ecod we mut fid the value of ε (aalytically poible, but recommeded via umerical imulatio). 3. Deig of VC for the recycled bioproce he baic idea of vibratioal cotrolled CSB i to vibrate the flow rate ad i thi way to operate the bioreactor at average coverio rate which were previouly utable. By uig thi techique i poible to elimiate igificat expee aociated with feedback. Sice the vibratio deped oly o time ad ot o the value of tate, there o loger wa a eed to take meauremet of cocetratio. he deig of VC for CSB with recycle tream repreeted by the oliear model (3), (4) i baed o the liearized time lag ytem (), (), (3) (or equivalet (4)) aroud the operatioal poit (E). From Fig. it ca be ee that the equilibrium () of (E) i utable ad it i eceary cotrol i order to tabilize thi operatioal poit. he liear time lag ytem (4) i obviouly of the form (7). For the implemetatio of the vibratioal cotrol followig the methodology of the ubectio 3., we chooe the PAZ matrix B(: B ( = α co( (35) where α R. he form of the matrix B ugget that the cotrol actio coit i vibratig the iput flow rate, i.e. the ilet ubtrate rate. From (35), (), (3), the tate traitio matrix of (3), the averaged matrix A ad the averaged matrix A for our bioproce are (ee heorem ): x& ( x ( x ( = A + A x& ( x ( x ( t x ( + α co ε ε x ( (36) he ytem (36) i of the form (3) writte i the term of the CSB with B( from (35). he average cotrolled ytem i i thi cae the followig: z& ( = A z& ( z( + A z ( z( z ( (37) I cocluio, for α choe uch that all the root of the characteritic equatio (3): ) = det( λi A Ae are egative real part, the vibratio of the form (35) tabilize for ε ufficietly mall the trivial equilibrium of the liearized ytem (4). Remark : he vibratioal cotrol i applied i the cae of the CSB with recycle tream for a liearizatio aroud the operatioal poit, therefore the uique equilibrium of the liearized time lag ytem (4) i trivial (the origi). Simulatio reult. he imulatio value are furihed i the previou ectio. he time evolutio of the liearized CSB aroud the utable equilibrium i depicted i Fig.. he origi i utable i thi cae. Whe the VC i implemeted the imulatio of (36) for α = 5, ε =. lead to the time evolutio depicted i Fig. 3. he tate trajectorie go to the uique equilibrium poit i the cae of liear ytem - the origi. he average behaviour of the recycled bioproce with vibratio i decribed by (37). Fig. 4 how the averaged phae portrait of CSB. It ca be ee from the phae portrait that the addle-type of equilibrium from Fig. i tabilized. αi( ( q) D Φ( t,) = ; A = ; µ ( ξ, ) D γ + kµ ( ξ, ) α A =. kµ ( ξ, ) kγ D herefore, if the matrix B( i (35), the vibratioal cotrolled bioreactor i decribed by: Fig.. State trajectorie of iitial utable ytem

6 Proceedig of the 7th WSEAS Iteratioal Coferece o Automatio & Iformatio, Cavtat, Croatia, Jue 3-5, 6 (pp7-76) dyamic. hi fact together with phyical limitatio o the magitude ad the frequecy of vibratio i ome cae are the diadvatage of the techique. he major advatage i the ability to eure a deired behaviour whe the meauremet are ot o-lie. he reult ca be exteded i the future to the oliear time lag ytem cae. Fig. 3. ime evolutio of the cotrolled bioproce Fig. 4. Averaged phae portrait of the bioproce 4 Cocluio he reult reported i thi paper demotrate that i ome cae, the ope-loop vibratioal cotrol ca be ucceful ued. VC i i fact a form of highfrequecy cotrol techique (high-frequecy relative to the atural frequecy of the ytem). A compario betwee thi techique ad other' highfrequecy method - like lidig mode cotrol ad ditherig - ca be doe. A mai differece betwee VC ad thee method i that i vibratioal cae, a compoet of the ytem i vibrated idepedet of the tate; therefore the cotrol i a fuctio depedig oly o time. he practical egieerig VC problem ca be decribed a a three tep techique: firt it i eceary to fid the coditio for exitece of tabilizig vibratio, ecod to fid which parameter or compoet i phyically poible to vibrate ad fially to fid the parameter of vibratio that eure the deired repoe. he tudy doe i thi work how that vibratioal cotrol ca tabilize a previouly utable teady tate of a recycled bioproce. For the vibratioal techique to be effective, oe eed to have a accurate decriptio of ytem Referece: [] G. Bati, D. Dochai, O-lie Etimatio ad Adaptive Cotrol of Bioreactor, Elevier, 99. [] R.E. Bellma, J. Betma, S.M. Meerkov, Vibratioal Cotrol of oliear Sytem: Vibratioal Stabilizability, IEEE ra. A.C., vol. AC-3, o. 8, 986, pp [3] R.E. Bellma, J. Betma, S.M. Meerkov, Vibratioal Cotrol of oliear Sytem: Vibratioal Cotrollability ad raiet Behavior, IEEE r. A.C., vol. 3, o. 8, 986, pp [4] J. Betma, Vibratioal Cotrol of a Cla of oliear Sytem by oliear Multiplicative Vibratio, IEEE ra. A.C., vol. AC-3, o. 8, 987, pp [5] A. Ciar, J. Deg, S.M. Meerkov, X. Shu, Vibratioal Stabilizatio of a Chemical Reactor, IEEE r. A.C., vol. 3, o. 4, 987, pp [6] B. Lehma, J. Betma, Vibratioal Cotrol of Liear ime Lag Sytem with Arbitrarily Large but Bouded Delay, IEEE r. o A.C., vol. 37, o., 99, pp [7] B. Lehma, J. Betma, S.V. Luel, E.I. Verriet, Vibratioal Cotrol of oliear ime Lag Sytem with Bouded Delay: Averagig heory, Stabilizability, ad raiet Behavior, IEEE r. A.C., vol.39, o.5, 994, pp [8] S.M. Meerkov, G.I. Shapiro, Method of Vibratioal Cotrol i the Problem of Stabilizatio of Ioizatio-hermal Itability of a Powerful Cotiuou CO Laer, Autom. Remote Cotr., vol. 37, 976, pp [9] S.M. Meerkov, Priciple of Vibratioal Cotrol: heory ad Applicatio, IEEE ra. A.C., vol. AC-5, o. 4, 98, pp [] D. Selişteau, E. Petre, Vibratioal Cotrol of a Cla of Bioprocee, Cotrol Egieerig ad Applied Iformatic, vol.3, o.,, pp [] D. Selişteau, C. Ioete, D. Popecu, D. Şedrecu, O Vibratioal Cotrol of a Cla of oliear Sytem, 5 th It. Carpathia Cotr. Cof., Vol. I, 4, Polad, pp [] D. Selişteau, C. Ioete, Vibratioal Cotrol of a Bioreactor with Delay i the Recycle Stream, It. Cof. Q&A-R HEA, Cluj-apoca, Romaia, Vol.,, pp [3] D. Selişteau, E. Petre, E. Bobaşu, D. Popecu, Modellig ad O-lie Etimatio of Ukow Kietic for a Activated Sludge Bioproce, WSEAS ra. o Sytem, Iue, vol. 4, 5, pp