On-line State Observation and Reaction Rates Estimation in a Baker s Yeast Cultivation Process

Transcription

1 Cap 6. On-lin Sa Obvain and acin a Eimain in a Ba Ya Culivain c Cap 6 On-lin Sa Obvain and acin a Eimain in a Ba Ya Culivain c Abac. In i pap algim f a bvain and inic imain a dvlpd and applid a ba ya fd-bac culivain pc. An impan dign cndiin wa p numb f quid n-lin maumn a lw a pibl. vall imain cm aim a imain f 3 a vaiabl and 3 pcific gw a quiing n-lin maumn f dilvd xygn, dilvd cabn dixid and ff-ga analyi. A ga dal f anin i givn imain pblm f acin inic. In i pc a nw algim i ppd - Scnd Od Dynamic Eima (SODE- and cmpad an Obv Bad Eima (OBE. Sabiliy and dynamic f cnvgnc a iu ubjc f aild analyi. lain bwn numical implmnain and abiliy a al udid. I i wn a a dic-im fmulain p addiinal abiliy cnain. can b aily vcm by u f a bu vaiabl p ingain algim. I wa cncludd a OBE a w main diadvanag: i uning f dign paam mu b dn n a ial-and- bai, wil in SODE u can a nd d dynamic f cnvgnc fm imad inic u inic, and ii dynamic f cnvgnc f OBE a imvaying wil in ca f SODE i pn i im-invaian. i cap a bn ubmid f publicain: Olivia,., S. Fy d Azvd (998. On-lin Sa Obvain and acin a Eimain in a Ba Ya Culivain c. (ubmid 9

2 Cap 6. On-lin Sa Obvain and acin a Eimain in a Ba Ya Culivain c INODUCION w f maj pblm wic ind implmnain f advancd mniing and cnl md in biac a difficuly f mdlling gw inic f micganim and abnc f cap and liabl n capabl f pviding dic al im maumn f a vaiabl. dign and implmnain f Sfwa Sn pvid a uiabl anw cp wi lac f inumnal n. Sfwa Sn a algim f n-lin imain f a vaiabl and paam wic a n mauabl in al im, fm lad maumn wic a m aily accibl. mamaical dcipin f micganim gw inic i a ciical iu in bipc mdlling. Qui fn inic mdl a bad n unucud and nnggad cll mdl. Unfunaly, in many ca, uc mdl a n accua nug lv pblm in udy. ciical iu i lad idnificain f inic paam. aam idnificain qui a caful and xpniv xpimnal planing. A uc, i a cla incniv dvlp algim f a imain and paam imain wil aviding nwldg f undlying inic mdl. In pn pap, acin inic imain fm lad n-lin maumn i a cnal iu. In i pc a nw algim - Scnd Od Dynamic Eima (SODE- i ppd, wic aim a imping d dynamic f cnvgnc f imad inic cpnding u inic. ppi f SODE a udid and cmpad Obv-Bad Eima (OBE ppd by Dcain and Bain (99. Sabiliy and dynamic f cnvgnc a ubjc f aild analyi. lain bwn numical implmnain and abiliy i al udid. baviu f algim i cafully analyd by applicain a ba' ya fd-bac culivain pc. A gnal imain cm i ppd, w 3 a vaiabl (bima, gluc and anl cncnain in b and 3 inic (pcific gw a lad gluc xidain, gluc fmnain, and anl xidain a imad, uing nly n-lin maumn f a vaiabl (cncnain f dilvd xygn and f dilvd cabn dixid and ff-ga analyi. GENEAL FAMEOK Bain and Dcain (99 ppd a mdlgy f a and paam imain bad upn a gnal dynamical mdl f id-an biac: d[ KM ([ D[ F Q ( 9

3 Cap 6. On-lin Sa Obvain and acin a Eimain in a Ba Ya Culivain c w [ i a vc ( f n cmpnn cncnain, K an num yild cfficin maix, D diluin a, F fd a vc wi dim(f=n and Q gau uflw a vc wi dim(q=n. On quin f maj cncn i dign f Sfwa Sn aviding inic mdlling. In qn. ( acin a M([ w dfind a: M ([ ([ U ([ i=,...,m ( i i i w i ([ i a nwn funcin f a wil U i( [ i an unnwn funcin f a. O m gnally: M[ ( H( [U[ ( (3 wi H([ an mu maix f nwn funcin f a and U([ a vc f unnwn funcin f a. agy i in in H([ nly pi nwldg gading inic and n cnid U([ a a cmplly unnwn "im-vaying" paam wic can b imad n-lin ug u f paam ima. In fllwing cin 3 algim a pnd f a bvain and inic imain. algim w dvlpd auming gnal ucu f dynamical mdl (. y a la ud in cin 3 f digning a cmpl a imain and inic imain cm f a ba ya culivain pc.. Obv-Bad Eima (OBE F imain f acin a fm n-lin nwldg f a vaiabl, wn yild cfficin a nwn and cnan, Bain and Dcain (99 ppd an bv-bad ima wic i xpd by: Obv-Bad Eima (OBE d[ du KH( [U D[ F Q :([ [ (4a [ [ [ KH( *( (4b w : and * a qua nun maic wic a dign paam a dipal f u f cnl f abiliy and acing ppi f algim. 9

4 Cap 6. On-lin Sa Obvain and acin a Eimain in a Ba Ya Culivain c baic ida i u a a bv (4a ima an bvain ([ [ wic i uppd flc mimac bwn U[ ( and U[ ( and n u i a divn fc in updaing law (4b... Sabiliy analyi. cninuu ym can b baind dfining bvain [ [ [ and acing U U U and ubacing qn. ( by qn. (4a: de wi AE B (5 E > [ A : > KH( [ [ * B du dynamic f ym a lina im-vaying (LV du pnc f a vaiabl in maix A. glbal abiliy f ym (5 i nud if diubanc vc B i bundd and if unfcd ym i xpnnially abl. A BIBS (Bndd Inpu Bundd Sa analyi f dynamic mdl ( (Dcain and Bain, 99 giv a if: C. diluin a i bundd blw: D d min D( (6 C. fd a a bundd : F d i( F( i (7 C3. Eac acin invlv a la n acan a i ni a caaly n an aucaaly. n a vaiabl [ a piiv and bundd f all. If addiinally C4. U ([ i a diffniabl funcin f [. n bundn f diubanc vc B i nud. On and, if: C5. : i a nun cnan maix wi all i ignvalu aving icly al pa. 93

5 Cap 6. On-lin Sa Obvain and acin a Eimain in a Ba Ya Culivain c C6. * i a nun cnan maix uc a maix :* *: i ngaiv dfini. C7. KH( [ i a pinly xciing maix. n unfcd ym i xpnnially abl and f, pubd ym (5 i glbally abl... uning f dign paam. A ad abv : and * a qua nun maic wic a dign paam a dipal f u f cnl f abiliy and acing ppi f algim, u playing a dciiv ll n pfmanc f ima. A cmmn cic i a: : diag^ Z ` * diag^j i ` i=,...,n (8 i w Z i and J i a un icly piiv al cnan. i i cic cndiin C4 and C5 a aumaically vifid and uning pcdu duc calibain by ial and f un cala cnan...3 ducd-d Obv-Bad Eima. bv-bad ima (4 i bad n full dynamical mdl f pc. In pacic i i n alway ncay. I i fn ufficin dign ima fm a ub f a quain pvidd y invlv all paam wic nd b imad. In paicula, und fllwing aumpin: A. a =m paam wic nd b imad A. i a ub f m quain f full a pac mdl a invlv all m paam wic nd b imad: d[ a K H( [ U ([ D[ F Q (9 a a a a A3. In qn. 9 K a i a mum full-an maix and by cniding anfmain: \ K a [ a ( n qn. (9 can b win a: d\ H( [ U ([ D\ K ( F Q a a a ( 94

6 Cap 6. On-lin Sa Obvain and acin a Eimain in a Ba Ya Culivain c Bad n i ducd-d fmulad pc mdl, bv-bad ima can b win a: ducd-d Obv-Bad Eima d\ du HU D\ K ( F Q :(\ \ (a H * (\ \ a a a (b. Scnd Od Dynamic Eima (SODE Scnd Od Dynamic Eima i a vaian f ducd d Obv-Bad Eima. y diff lly n way g in updaing law f U i ad (qn. (b. I can b applid und aumpin A ug A3 and addiinally: A4. acin a can b dfind by qn. (: M ([ ([ U ([ i=,...,m ( i i i maning a H([ i a mum diagnal maix. cnd d dynamic bad ima i win a fllw: Scnd Od Dynamic Eima (SODE d\ du HU D\ K ( F Q : (\ \ (3a * H (\ \ a a a (3b w : and * a qua mum maic wic a in ca f Obv-Bad Eima, a dign paam f cnl f abiliy and dynamic f cnvgnc. Dfining :=diag(z i and *=diag(j i, qn. (3 can b dcupld, giving: d\ du i i U D\ U Z (\ \ (4a i i i i i i i i J i (\ i \ i (4b 95

7 Cap 6. On-lin Sa Obvain and acin a Eimain in a Ba Ya Culivain c wi i=,...,m.. Sabiliy analyi. ym f qn. (4 i a cnd d lina im vaian (LV ym: d \ i Ui Zi J ii i \ i Ui du i (5 wi \ \ \ i i i U U U i i i w dui/ i cnidd a an xnal pin diubanc. I i a andad ul f BIBO abiliy y a a LV ym pubd by an xnal diubanc i glbally abl if unpubd ym i unifmly aympically abl and diubanc vc i bundd (Nanda and Annawamy, 989. Cndiin C ug C4 au bundn f dui/ (Bain and Dcain, 99. Sill, i main b pfd a unfcd ym i unifmly aympically abl. ad i fd appndix w i pf i pnd.... Dynamic f cnvgnc and uning. Diffniaing qn. (4b giv d d d U J \ J \ (6 Cmbining qn (6, (4b and fi quain in (5 i fllw a J d d U U J a ( U U (7 wi a( givn by qn. (8. a ( Z( d (8a Supping a m ( I [ ([ d (8b 96

8 Cap 6. On-lin Sa Obvain and acin a Eimain in a Ba Ya Culivain c i nwn n-lin cupd by an H( (in pacic n-lin nwldg f I([ qui an appximain im divaiv d/, and f, imad valu I ([ i alway cupd by appximain H(: I ([ I ([ H( (9 an dfining J and Z( by: J (a ( ( ( ( (b Z ] J I [ ] J I [ H n qn. (7 bcm: d d ( U ] U ( U U bing ]( lad did valu ] d and H( in fllwing way: ]( ] d ( H ] d ¹ ( Hnc, cncluin i an a ac U ( cnvg i' u valu U( wi a cnd d dynamic pn wi cnan naual pid f cillain and imvaying damping cfficin ](. Nic a dfining Z( by qn. (b impli a cndiin C7 wic a a Z( mu b alway lag n -I([, i vifid if ] d i alway lag n appximain H(. Nic al a a givn by qn. ( i i quivaln a a ]( mu b piiv f all...3 Numical implmnain and abiliy. numical implmnain f qn. (4 qui a dic-im fmulain. wic fm cninuu-im quain dic im vin lad pcific abiliy pblm in wic ingain p play an impan l. On f m ppula appac i Eul diciain f cninuu im quain. In i cin, implicain f uing an Eul dciiain in abiliy f SODE a analyd. A fwad Eul diciain f qn. (4 wi Z( and J givn by qn. ( ul in fllwing dic-im quain: \ ( ( D \ U U Z \ \ (3a 97

9 Cap 6. On-lin Sa Obvain and acin a Eimain in a Ba Ya Culivain c U ( \ \ (3b U wi Z ] (4 dic ym i a fllw ( \ Z \ U (5a U \ U ( U U (5b wic i quivaln E AE B (6 wi E \ U A ] U U dic-im ym (6 i lina im-invaian (LI. unfcd ym i xpnnially abl (and nc upu f (6 i bundd if ingnvalu f maix A ay inid uni cicl. ingnvalu f maix A a givn by O O ] ] ] ] (7a (7b Ca - ]<. ingnvalu a w cmplx cnjuga numb O O ] i ] ] i ] (8a (8b Hnc, abiliy cndiin appli 98

10 Cap 6. On-lin Sa Obvain and acin a Eimain in a Ba Ya Culivain c O O (9 wic i quivaln ] (3 Ca - ]=. ingnvalu a dubl givn by O O (3 lading fllwing abiliy cndiin (3 Ca 3 - ]>. ingnvalu a w diffn al numb givn by qn. (7. abiliy cndiin i a fllw: ] ] (33 A givn by qn. (3, (3 and (33 ang allwd f ingain p i bundd and cndiind by cn and ]. Nic al a icin a am if analyi wuld av bn caid u fm dic vin f qn. (..3 Lunbg-yp aympic bv Eqn. ( can b dividd in w paiin: fi n includ quain lad maud a vaiabl ([ ; cnd paiin includ quain lad nn-maud a vaiabl ([. dynamic mdl i win a: d[ d[ K M ([ D[ F Q (34a K M[ ( D[ F Q (34b w K (a full an maix, K, F, F, Q, Q cpnd diviin f K, F and Q ac paiin. Bad n anfmain: Z [ KK [ (35 99

11 Cap 6. On-lin Sa Obvain and acin a Eimain in a Ba Ya Culivain c and n qn. (34, Lunbg-yp aympic bv (Lunbg, 97 i win a: dz Z Lunbg-yp Aympic Obv DZ ( F Q K K ( F Q (36a K K [ [ (36b Nic a numb f maud a vaiabl mu b qual numb f unnwn acin a in vc M([. 3 SAE OBSEVAION AND KINEICS ESIMAION IN A BAKE S YEAS FED-BACH CULIVAION OCESS In cin gnal-u a bvain and inic imain algim w pnd. y a ging b nw applid a ba ya fd-bac culivain pc. bjciv i dvlp a a bvain and a pcific gw a imain cm quiing a minimum numb f aily accibl n-lin maumn. dicuin will flw ug fllwing pic: Gnal dynamical mdl ucu f a ba ya fd-bac culivain pc. i mdl will b ba n wic imain algim a divd. Samn f imain pblm. 3 Divain f Lunbg-yp aympic bv, ducd d OBE and SODE f cn n-lin maumn vall imain cm will pmi imain f 3 a vaiabl (bima, gluc and anl cncnain in b, 3 pcific gw a (lad gluc xidain, gluc fmnain and anl xidain uing maumn fm a vaiabl (dilvd xygn and dilvd cabn dixid and ff-ga analyi (xygn anf a and cabn dixid anf a. 3. Dynamic mdl f a ba ya fd-bac culivain pc Ya gw i caacizd by fllwing acin cm (Snnlin and Käppli, 986:

12 Cap 6. On-lin Sa Obvain and acin a Eimain in a Ba Ya Culivain c S S E + C X + G (piay gw n gluc (37a X + E + G (fmnaiv gw n gluc (37b + C X + G (piay gw n anl (37c w S i gluc, C i xygn, X i bima, E i anl and G i cabn dixid., and a pcific gw a wic flc capaciy f ya xpli diffn caablic paway f ngy and baic maial uc. dynamical mdl f fd-bac fmn i baind fm a ma balanc n cmpnn, cniding a ac i wll mixd, yild cfficin a cnan and dynamic f ga pa can b nglcd. ma balanc, in m f cncnain, a win a: dx ds de dc dg ( DX (38a DS ( S ( X (38b in DE ( X (38c 3 4 DC O ( X (38d 5 6 DG C ( X ( and addiinal quain: dv F DV (39 w i a yild cfficin, O i xygn anf a (dfind a * O La( C C w L a i ma anf cfficin and C * quilibium cncnain f dilvd xygn, C i cabn dixid anf a (dfind a C KVKL ag, V i luin vlum in ac, F i inpu fd a and D i diluin a (dfind a D=F/V. Equain (38 a maix fm: d X S E C G X S DS in X D E C O G C (4

13 Cap 6. On-lin Sa Obvain and acin a Eimain in a Ba Ya Culivain c wic a ucu f gnal dynamic mdl ( wi n=5 a pac vaiabl and m=3 acin a: [ X S E C G K > F DS O Q > C@ A uc, algim pnd in cin can b aigfwad applid dynamical mdl (4. 3. Eimain pblm a 3 pcific gw a invlvd in dynamical mdl (4. a ad a 3 unnwn pc vaiabl a mu b imad uing OBE SODE. A uc, funcin H([ and U([ in qn. (3 a dfind in fllwing way: H( diag( X ( [ U [ Sinc a 3 unnwn inic, applicain f Lunbg-yp aympic bv (qn. 36 mdl (4 qui n-lin maumn f 3 a vaiabl. F am an, applicain f ducd-d OBE and f SODE mu b bad n =3 a pac quain. Sinc E, C and G a a vaiabl m aily accibl n-lin, fm pacical pin f viw, i i impan dign an imain cm bad n i f maumn. Unfunaly, accding yild cfficin valu, E, C and G a linaly dpndn, bing cpnding yild maix ill-cndiind (mlau and i, 99. Sinc invin f i maix i quid in all algim dicud pviuly, numical implmnain wuld ul in xmly niiv algim numical, u aving pfmanc damaically dgadd. lv i pblm mlau and i (99 uggd a fmulain f mdl (4. i fmulain i bad n diviin f cmpl pc mdl (4 in w paial mdl: (i pi-fmnaiv paial mdl ( cpnding anl pducin a f pc and (ii piaiv paial mdl ( cpnding anl cnumpin a f pc. fmnaiv paial mdl ( i ad a:

14 Cap 6. On-lin Sa Obvain and acin a Eimain in a Ba Ya Culivain c 3 d X S E C G X D X S E C G DS O C in (4 and piaiv paial mdl ( a: d X S E C G X D X S E C G DS O C in (4 Nic a y av idnical ucu wi n=5 a pac vaiabl bu nly m== acin a. nly diffnc bwn m i flcd n way pcific gw a vc and yild cfficin maix a ad. F pifmnaiv ( paial mdl y a: K and f piaiv paial mdl (: K Lunbg bv, ducd-d OBE and SODE mu nw b applid b paial mdl, uling w "paial" algim wic mu b alnaivly ud in accdanc wi acual pc a: anl pducin anl cnumpin. ucc f uc an imain cm dpnd upn cin capabiliy f cc pc a fllwd by u f pp f quain. aniin bwn pc a can b cd by aniin bwn piiv and ngaiv valu f pcific gw a ima lad anl cnumpin ( pducin ( (mlau and i,99. F xampl, if acual pc a i anl cnumpin and if la ima i ngaiv n pc a a wicd fm anl cnumpin ( anl pducin (. abv mnind paial mdl av nly w pcific gw a invlvd. i a vy impan implicain a nly w maud a vaiabl a quid. cnvninc vaiabl C and G w cn. vaiabl a aily accibl n-lin and numical pblm mnind pviuly n lng xi. A

15 Cap 6. On-lin Sa Obvain and acin a Eimain in a Ba Ya Culivain c uing n- uc, imain cm will pvid ima f X, S, E, lin maumn f C, G, O, C and F. 3.3 Ovall a bvain and inic imain cm, and 3.3. Applying Lunbg-yp aympic bv. applicain f Lunbg-yp aympic bv (qn. 36 b paial mdl (4 and (4 ul in w paial algim wi idnical ucu givn by fllwing f quain: Z Z Z d 3 X S E Z Z Z D 3 Z Z Z 3 KK DS C G in K K O C (43a (43b diffnc bwn paial algim a flcd in way maic K and K a dfind. diffnc a cmpild in abl I. abl I Diffnc bwn paial Lunbg-yp bv pi-fmnaiv a( piaiv a ( K K (44a K K (45a K K 7 8 (44b K K 8 9 (45b 3.3. Applying ducd-d OBE. applicain f ducd-d OBE (qn. ( b paial mdl (4 and (4, uing a pac quain f C and G, ul in w paial algim wi idnical ucu givn by fllwing f quain: \ K a C G (46a 4

16 Cap 6. On-lin Sa Obvain and acin a Eimain in a Ba Ya Culivain c d\ d O X D\ Ka (\ \ C : (46b X * (\ \ (46c w "paial" OBE diff n f pcific gw a ima, n yild cfficin maix K a, and n gain maic : and *. diffnc a cmpild in abl II. abl II Diffnc bwn paial ducd-d OBE pi-fmnaiv a ( piaiv a ( (47a (48a K a K a (47b Ka K a (48b : : Z : : Z Z (47c Z (48c 3 * * J J (47d * * J 3 J (48d Applying cnd-d dynamic ima. applicain f SODE (qn. (3 b paial mdl (4 and (4, uing a pac quain f C and G, ul in w paial algim wi idnical ucu givn by fllwing f quain: \ d\ d K a C G (49a O X D\ Ka (( \ \ C : (49b X * (\ \ (49c 5

17 Cap 6. On-lin Sa Obvain and acin a Eimain in a Ba Ya Culivain c 6 A in ca f ducd-d OBE, w "paial" SODE diff n f pcific gw a ima, n yild cfficin maix K a, and n gain maic :( and *. diffnc a cmpild in abl III. abl III Diffnc bwn paial SODE pi-fmnaiv a ( piaiv a ( (5a (5a K K a a (5b K K a a (5b : : ( ( ( ( X X I ] I ] (5c : : ( ( ( ( 3 3 X X I ] I ] (5c * * (5d * * 3 (5d Nic a in qn. (5c and (5c, I(X i givn by: I( X X X X (5 w f ingain p, X + and X bima cncnain a im inanc + and pcivly Swicing mcanim bwn paial algim. uccful applicain f paial algim mnind abv qui an n-lin cin agy f cun pc a: anl pducin ( anl cnumpin (. mlau and i (99 cncludd a i cin mcanim culd b bad n aniin bwn piiv and ngaiv valu f pcific gw a ima lad anl (cnumpin- pducin-. F xampl, if acual pc

18 Cap 6. On-lin Sa Obvain and acin a Eimain in a Ba Ya Culivain c a i anl cnumpin and if la ima i ngaiv n pc a a wicd fm anl cnumpin ( anl pducin (. A uc, vall imain cm will cni n p: Ingain f "paial" Lunbg-yp bv (qn. (43 cpndn acual pc a. Ingain f "paial" ducd-d OBE (qn. (46 paial SODE (qn. (49 cpndn acual pc a. 3 Cc if aniin bwn pc a a ccud by ccing ignal f pcific gw a ima lad anl. Fm pacical pin f viw, aniin bwn "paial" algim can b alid ju by wicing bwn qn. (44a-b/(45a-b and, bwn (47a-c/(48a-c wn ducd-d OBE i ud bwn (5a-c/(5a-c wn SODE i ud. 4 ESULS AND DISCUSSION pfmanc f imain algim will b analyd a aid f a imulain xpimn. pc dynamical mdl (4 wa implmnd n a pc imula (imna al., 993 auming inic mdl ppd by Snnlin and Käppli (986 ( valu f inic paam ud a lid in abl IV. imulain xpimn wa mad und fllwing iniial cndiin: X(=. g/l, S(=. g/l, E(=.5 g/l, C(=.66 g/l, G(=.8 g/l, V(=3.5 L abl IV - Kinic paam(an fm Snnlin and Käppli (986 aam Valu q,max 3.5 g gluc/(g bima q c,max.56 g O /(g bima q,max.36 g anl/(g bima K. g L - K i. g L - K. g L - K c. g L - valu f K L a, K v and C * w aumd -,. and.7 g/l pcivly. valu f yild cfficin a lid in abl V. 7

19 Cap 6. On-lin Sa Obvain and acin a Eimain in a Ba Ya Culivain c abl IV - Yild cfficin (an fm Snnlin and Käppli (986 Cfficin (Cfficin - Valu Y.49 g bima/g gluc Y.5 g bima/g gluc Y 3. g bima/g anl Y 4.7 g bima/g anl Y c 5. g bima/g O Y c 6.64 g bima/g O Y g 7.8 g bima/g CO Y g 8. g bima/g CO Y g 9. g bima/g CO A fmnain un f 8 u i aumd. In Fig. ud fd a pfil and cpnding b vlum vluin a wn. pfil f gau uflw a, f a vaiabl, and pcific gw a a wn in Fig., 3 and 4 pcivly. F V % Sin ( 8 Fig.. Inpu fd a F(. -.5 L/, gluc cncnain n fd Sin (5 g/l and vlum V (. -. L 8

20 Cap 6. On-lin Sa Obvain and acin a Eimain in a Ba Ya Culivain c % O C ( 8 Fig.. Gau anf a: Oxygn anf a O (. 7. g L - - and cabn dixid anf a C (.. g L - - C X % G G E S ( 8 Fig. 3. Sa pac vaiabl: bima X (. 3. g/l, gluc S (. -.5 g/l, anl E (. -. g/l, xygn C (. -.7 g/l and cabn dixid G (. -.3 g/l. i inpu fd a f Fig. and wi iniial cndiin mnind abv wic bwn pi-fmnaiv and piaiv caablic a ccud 6 im. Fig. 4 includ a p a ul diinguiing w diffn pc a: pifmnaiv ( wi anl pducin, and piaiv ( wi anl cnumpin. pin wn pc a wic ccu a mad wi l a, b, c, d,, f, and g. 9

21 Cap 6. On-lin Sa Obvain and acin a Eimain in a Ba Ya Culivain c a b c d f g 8 ( Fig. 4. Spcific gw a pfil uing inic mdl ppd by Snnlin and Käppli (986, wi inic paam lid in abl IV. i imulain xpimn upplid imain algim wi lvan maud vaiabl: C, G, O, C, Sin, V and F a a ampling a f 6 minu. baviu f imain cm dvlpd in pviu cin will nw b analyd a aid f i imulain xpimn. A uc, cuv wn in Fig.4 a an a u pcific gw a pfil. vaiabl C, G, O, C, and F in Fig., and 3 a cnidd a pc n-lin maumn, bing upplid imain algim a a ampling a f 6 minu. fllwing figu illua u f ducd-d OBE and f SODE. Eac f m includ a ul imila n f Fig. 4 diinguiing w diffn pc a: pi-fmnaiv ( wi anl pducin, and piaiv ( wi anl cnumpin. pin wn wic bwn pc a ccu a mad wi l a, b, c, d,, f and g. u pcific gw a f Fig. 4 a pnd by d lin wil pciv ima a pnd by full lin. accuacy f ima can b ad by IAE indx (IAE - ingal f im-wigd ablu givn in lgnd f pcific gw a. Fig. 5, 6 and 7 w ul pducd by ducd-d OBE f diffn uning (dfind uiically. quain w ingad wi a bu vaiabl p ingain algim (4/5 d ung-kua yp mbddd cm du Buc mplying alng ingain lina ima f lvan ampld vaiabl.

22 Cap 6. On-lin Sa Obvain and acin a Eimain in a Ba Ya Culivain c Z = Z = Z 3 = J = J = J = IAE: a b c d f g 8 ( Fig. 5. Spcific gw a ima (full lin and u (d lin givn by ducd-d OBE ingad wi an 4/5 d vaiabl p ingain uin Z = Z = Z 3 = J = J = J = IAE: a b c d f g 8 ( Fig. 6. Spcific gw a ima (full lin and u (d lin givn by ducd-d OBE ingad wi an 4/5 d vaiabl p ingain uin.

23 Cap 6. On-lin Sa Obvain and acin a Eimain in a Ba Ya Culivain c Z = Z = Z = J 3 = J = J = 3 IAE: a b c d f g 8 ( Fig. 7. Spcific gw a ima (full lin and u (d lin givn by ducd-d OBE ingad wi an 4/5 d vaiabl p ingain uin. A caful analyi f pl val a dynamic f cnvgnc i im-vaying, i.., pn bcm incaingly fa and cillay a un appac nd. ul yild by SODE a dpicd in Fig (8-4. quain w ingad wi a bu vaiabl p ingain algim imila n ud f ducd-d OBE. influnc f ] can b ad fm pl in Fig (8- w i p cnan a.5 u wil ] aum valu.5,.5,. and.5 pcivly. influnc f can b ad fm pl in Fig ( and (-4 w ] i p cnan a. wil aum.5,.,.5 and. u pcivly.

24 Cap 6. On-lin Sa Obvain and acin a Eimain in a Ba Ya Culivain c = = =.5 3 = = =.5 ] ] ] 3 IAE: a b c d f g 8 ( Fig. 8. Spcific gw a ima (full lin and u (d lin givn by SODE ingad wi an 4/5 d vaiabl p ingain uin = = =.5 3 = = =.5 ] ] ] 3 IAE: a b c d f g 8 ( Fig. 9. Spcific gw a ima (full lin and u (d lin givn by SODE ingad wi an 4/5 d vaiabl p ingain uin. 3

25 Cap 6. On-lin Sa Obvain and acin a Eimain in a Ba Ya Culivain c = = =.5 3 = = =. ] ] ] 3 IAE: a b c d f g 8 ( Fig.. Spcific gw a ima (full lin and u (d lin givn by SODE ingad wi an 4/5 d vaiabl p ingain uin = = =.5 3 = = =.5 ] ] ] 3 IAE: a b c d f g 8 ( Fig.. Spcific gw a ima (full lin and u (d lin givn by SODE ingad wi an 4/5 d vaiabl p ingain uin. 4

26 Cap 6. On-lin Sa Obvain and acin a Eimain in a Ba Ya Culivain c = = 3 =. = = =. ] ] ] 3 IAE: a b c d f g 8 ( Fig.. Spcific gw a ima (full lin and u (d lin givn by SODE ingad wi an 4/5 d vaiabl p ingain uin = = 3 =.5 = = =. ] ] ] 3 IAE: a b c d f g 8 ( Fig. 3. Spcific gw a ima (full lin and u (d lin givn by SODE ingad wi an 4/5 d vaiabl p ingain uin. 5

27 Cap 6. On-lin Sa Obvain and acin a Eimain in a Ba Ya Culivain c = = 3 =. = = =. ] ] ] 3 IAE: a b c d f g 8 ( Fig. 4. Spcific gw a imad (full lin and u (d lin givn by SODE ingad wi an 4/5 d vaiabl p ingain uin. A givn by pl in Fig. (8-4 caaciic f dynamic f cnvgnc f imad valu u valu appa b in agmn wi a ypical cnd-d dynamical pn. I i wn a dcaing pduc fa pn, wil dcaing ] pduc m cillay pn. Fum, fm pl in Fig (8- i can b cncludd a ]= cniu fni bwn cillay and nn cillay baviu. In cin (..3 numical implmnain f SODE wa dicud. I wa wn a a fwad Eul diciain f cninuu quain p abiliy pblm. In paicula, lain (qn. (3, (3 and (33 w divd wic dfin abl inval f ingain p in lain a pcific uning. fllwing Fig. amp illua baviu f SODE wi an Eul diciain, wn abiliy limi a dibyd. ingain p and ampling im w aumd b 6 minu. ul in Fig (5-7 w baind wi ]=.75. A givn by qn. (3, if =. n mu b lag n.7. ul baind wi =.7, =.75 and =.85 u a wn in Fig. (5, (6 and (7 pcivly. 6

28 Cap 6. On-lin Sa Obvain and acin a Eimain in a Ba Ya Culivain c = = 3 =.7 = = =.75 ] ] ] 3 IAE: a b c d f g 8 ( Fig. 5. Spcific gw a ima (full lin and u (d lin givn by SODE uing Eul diciain = = 3 =.75 = = =.75 ] ] ] 3 IAE: a b c d f g 8 ( Fig. 6. Spcific gw a ima (full lin and u (d lin givn by SODE wi Eul diciain. 7

29 Cap 6. On-lin Sa Obvain and acin a Eimain in a Ba Ya Culivain c = = 3 =.85 = = =.75 ] ] ] 3 IAE: a b c d f g 8 ( Fig. 7. Spcific gw a ima (full lin and u (d lin givn by SODE wi Eul diciain. i =.7 imad cuv divg, and algim i u unabl. Incaing.75 divgnc ppd bing bvd. Nvl, pin cillain a xibid ugging a ima pa na abiliy limi. F =.85 a nmal upu i baind. ul in Fig (8- w baind wi ]=.. A givn by qn. (3, if =. n mu b lag n.. ul baind wi =.95, =. and =.5 a wn in Fig. (8, (9 and ( pcivly. i =.95 algim val ilf unabl. Incaing. divgnc ppd bing bvd. Nvl, pn xibi ccainally v wic i n caaciic f a cnd-d pn wi ]=. i ugg a ima pa na abiliy limi. F =.5 v i ducd ugging a ima pa fa fm abiliy limi. 8

30 Cap 6. On-lin Sa Obvain and acin a Eimain in a Ba Ya Culivain c = = 3 =.95 = = =. ] ] ] 3 IAE: a b c d f g 8 ( Fig. 8. Spcific gw a ima (full lin and u (d lin givn by SODE wi Eul diciain = = 3 =. = = =. ] ] ] 3 IAE: a b c d f g 8 ( Fig. 9. Spcific gw a ima (full lin and u (d lin givn by SODE wi Eul diciain. 9

31 Cap 6. On-lin Sa Obvain and acin a Eimain in a Ba Ya Culivain c = = 3 =.5 = = =. ] ] ] 3 IAE: a b c d f g 8 ( Fig.. Spcific gw a ima (full lin and u (d lin givn by SODE wi Eul diciain.

32 Cap 6. On-lin Sa Obvain and acin a Eimain in a Ba Ya Culivain c AENDIX: f f Unifm Aympic Sabiliy f SODE E Sym L' fi cnid fllwing fmulad ym and cncna u anin unfcd ym : dy d U a ( y J y du U (A-a (A-b wic can b baind fm ym (5 by cniding anfmain y ( implify pn analyi indx "i" will b mid, w / \ a ( Z( d (A- Cing fllwing candida Lyapunv funcin: V( y, y U J U (A3 w im divaiv alng luin f (A- i givn by: dv ay ( E QE ( (A-4 w E ( > y Q a ( n i fllw a if a ( n Q( i piiv mi-dfini. Hnc quilibium a E= i unifmly abl (Nanda and Annawamy, 989. Supping a cndiin C ug C3 ld, cndiin und wic a ( a ad a: C5. ([ i a diffniabl funcin f [, wic man a i bundd C6. ([ i bundd a fllw: d ( d min C7. Z( i [ max d Sill, inc Q( i piiv mi-dfini and im-vaying, i can n b cncludd a ym (A- i unifmly aympically abl. In lin bllw, xpnnial abiliy f (A- i pfd. A qualiaiv ulin f pf can b givn in w p: i in ym (A- y( a aum a lag valu a

33 Cap 6. On-lin Sa Obvain and acin a Eimain in a Ba Ya Culivain c m inanc in vy inval [, + ], ii inc V i a givn by qn. (A-3, i impli a V( dca v vy inval f lng wic au unifm aympic abiliy. f. Supp a y ( E ( i D i w D Ingaing (A-a v im inval >, i fllw a G G G ³ U max ³ y ( y ( ( d a y ( d (A-5 w a max i maximum valu f a (. f y ( G U ( d ( Ga up y ( ³ G max, G (A-6 inc y ( i alway l n up y( in >, On and G G G ( d ( d ( ( d ³ ³ ³ U U U U G U ( G up U ( U ( > G ( d G U G U ³ (A-7 inc dianc bwn pin U( and U( i alway l n ac lng G U ³ d. Hnc, valuaing U fm (A-b, qn. (A-7 bcm G ( U d G U ( ³ b up y ( (A-8, G w b JG a max G. Fm iniial uppiin gading y (, i fllw a up y( in >, i alway l n D E ( G, and al U( i alway lag n D E (. Sinc de ( a ( y d min( J, d

34 Cap 6. On-lin Sa Obvain and acin a Eimain in a Ba Ya Culivain c n E ( E ( G. Hnc qn. (A-8 bcm y ( G G D b D E ( G (A-9 Cing G D G ( b (A- n qn. (A-9 bcm y ( G D E ( G (A- wic i a cnadicin f iniial aumpin gading y (. in i i f pfd. L' nw inga qn. (A-b v a im inval V( V( amax ³ y( d (A- and by Caucy-Scwaz inqualiy a max V( V( y( d ³ (A-3 Fu, by cniding a G ³ ³ ³ y( d y( d yd ¹ y ( de ( (A-4 w d amax, n cing G w av a y d D d E ³ ( ( (A-5 Hnc qn. (A-3 bcm ( ( max d d E ( V V a d D (A-6 3

35 Cap 6. On-lin Sa Obvain and acin a Eimain in a Ba Ya Culivain c bing l an D/d. Sinc E ( i alway lag n V(/max( J,, n cing min(, D / d w cnclud a V( dv( d( E V( d( E V( (A-7 w E amax D J d d ( max(, d (A-8 Sinc a max / d i alway l n and =D/(3d i a maximum pin f funcin f ( d( D d, w cnclud a E D 3 7 J max(, 8, (A-9 and, nc, final cncluin can b an a ima (A- i unifmly aympically abl. 4

36 Cap 6. On-lin Sa Obvain and acin a Eimain in a Ba Ya Culivain c NOMENCLAUE C C D E F G H([ K i O Q S S in X xygn cncnain cabn dixid anf a diluin a anl cncnain ma fd a vc cabn dixid cncnain (mu maix f nwn funcin f a yild cfficin maix yild cfficin Oxygn anf a a f ma mvl in gau fm vc gluc cncnain gluc cncnain in fd ampling pid bima cncnain G l M acin a vc [ imad a vc pcific gw a f fmnaiv gw n gluc paway pcific gw a f piay gw n anl paway pcific gw a f piay gw n gluc paway [ imad vc f nnmaud a vaiabl vc f imad pcific gw a :, * gain maic i naual pid f cillain pcific gw a vc U vc f imaiv f U( U( vc f cmplly unnwn im-vaying paam Z i, J i diagnal lmn f : and * ] i damping cfficin [ a pac vc [ maud a pac vc [ nn-maud a pac vc Mamaical nain up min max diag{.} upmun minimum maximum diagnal maix 5

37 Cap 6. On-lin Sa Obvain and acin a Eimain in a Ba Ya Culivain c EFEENCES Bain, G. and D. Dcain (99 On-Lin Eimain and Adapiv Cnl f Biac, Elvi, Amdam. mlau, Y. and M. i (99 Eimain f Mulipl Spcific Gw a in Bipc, AICE Junal, 36(, pp mlau, Y. (99, "Mdéliain cnôl d'un pcédé fd-bac d culu d lvu à pain",. D. i, Ecl lycniqu d Mnéal, Canada. Snnlin, B. and O. Käppli (986 Gw f Saccamyc cviia i cnlld by i Limid piay Capaciy: Fmulain and Vificain f a Hypi, Bic. Bing., 8, pp Lunbg, D.G. (97 An induin bv, IEEE an. Aumaic Cnl, AC-6, pp Fy d Azvd, S.,. imna, F. Olivia, E. Fia (99 "Sudi n On-Lin Sa and aam Eimain ug a al-im c Simula, in Kaim, M. N. and Spanpul, J. (Ed., Mdling and Cnl f Bicnical c, IFAC Symp. Si, , gamn N.Y. Nanda, K.S. and A.M. Annawany (989 "Sabl Adapiv Sym," nic Hall, Englwd Cliff 6