Takagi-Sugeno Fuzzy Observer for a Switching Bioprocess: Sector Nonlinearity Approach

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1 Takag-Sugen Fuzzy Obseve f a Swtchng Bpcess: Sect Nnlneaty Appach Enque J. Heea-López, Benadn Castll-Tled, Jesús Ramíez-Códva and Eugén C. Feea Cent de Investgacón y Asstenca en Tecnlgía y Dseñ del Estad de Jalsc Cent de Investgacón y de Estuds Avanzads del I.P.N, Undad Guadalajaa Insttute f Btechnlgy and Bengneeng, Unvesdade d Mnh, Baga, Méc, Ptugal 8. Intductn In a bpcess t s desed t pduce hgh amunts f bmass metabltes such as vtamns, antbtcs, and ethanl, amng thes. The measuement f blgcal paametes as the cell, by-pduct cncentatns and the specfc gwth ate ae essental t the successful mntng and cntl f bpcesses (Huch & Kshmt, 998). Adequate cntl f the fementatn pcess educes pductn csts and nceases the yeld whle at the same tme acheve the qualty f the desed pduct (amuna & Ramachanda, 999). Nevetheless, the lack f cheap and elable senss pvdng nlne measuements f the blgcal state vaables has hampeed the applcatn f autmatc cntl t bpcesses (Bastn & Dchan, 990). Ths stuatn encuages the seachng f new sftwae senss n bpcesses. A state bseve s used t ecnstuct, at least patally the state vaables f the pcess. Tw classes f state bseves sftwae senss f (b)chemcal pcesses can be fund n the lteatue (Dchan, 00). A fst class f bseves called asympttc bseves, s based n the dea that the uncetanty n bpcess mdels s lcated n the pcess knetcs mdels. A secnd class s based n the pefect knwledge f the mdel stuctue (Luenbege, Kalman bseves and nnlnea bseves). Dffeent applcatns f state bseves n bpcess ae epted n the lteatue (Cazzad & Lubenva, 995; Faza et al., 000; Guay & Zhang, 00; Lubenva et al., 00; Olvea et al., 00; Sh & Ca, 999; Vels et al., 007). Fuzzy lgc has becme ppula n the ecent yeas, due t the fact that t s pssble t add human epetse t the pcess. Nevetheless, n the case whee the nnlnea mdel and all the paametes f a pcess ae knwn, a fuzzy system may be used. A fst appach can be dne usng the Takag-Sugen fuzzy mdel (Takag & Sugen, 985), whee the cnsequent pat f the fuzzy ule s eplaced by lnea systems. Ths can be attaned, f eample, by lneazng the mdel aund peatnal pnts, gettng lcal lnea epesentatn f the nnlnea system.

2 56 New Develpments n Rbtcs, Autmatn and Cntl Anthe way t btan a mdel can be acheved usng the methd f sect nnlneates, whch allws the cnstuctn f an eact fuzzy mdel fm the gnal nnlnea system by means f lnea subsystems (Tanaka & Wang, 00). Fm ths eact mdel, fuzzy state bseves and fuzzy cntlles may be desgned based n the lnea subsystems. Dffeent fuzzy lgc applcatns t bpcesses can be fund n the scentfc lteatue (Genves et al., 999, Ascenc et al., 00; Kaakazu et al., 006). In ths chapte a Takag-Sugen fuzzy bseve based n sect nnlneates s ppsed and appled t a cntnuus nnlnea bake s yeast fementatn pcess. The bseve gans ae calculated usng lnea mat nequaltes. An nteestng featue f ths mdel s that t can be dvded n tw mdels: a esp-fementatve () mdel wth ethanl pductn and a espatve (R) mdel wth ethanl cnsumptn. The mdel can swtch t the R- mdel dependng n whethe the yeasts ae pducng cnsumng ethanl.. Fuzzy Systems Pelmnaes A nnlnea system may be epesented by lnea subsystems called Takag-Sugen, (fgue ). The Takag-Sugen fuzzy mdels ae used t epesent nnlnea dynamcs by means f a set f IF-THEN ules. The cnsequent pats f the ules ae lcal lnea systems btaned fm specfc nfmatn abut the gnal nnlnea plant. Fg. Takag-Sugen epesentatn f a nnlnea system The th ule f a cntnuus fuzzy mdel has the fllwng fm: Mdel Rule : If z (t) s φ and and z p (t) s φ p.

3 Takag-Sugen Fuzzy Obseve f a Swtchng Bpcess: Sect Nnlneaty Appach 57 { & t A t + Bu t THEN,,,...,, y t () C() t () whee φ j s a fuzzy set and s the numbe f ules n the fuzzy mdel; (t) R n s the state vect, u(t) R m s the nput vect, y(t) R q s the utput vect, A R nm, B R nm, and C R qn ae sutable matces, and z(t)[z (t),,z p (t)] s a knwn vect f pemse vaables whch may cncde patally depend n the state (t). Gven a pa f ((t), u(t)) and usng a sngletn fuzzfe, pduct nfeence and cente f gavty defuzzfe, the aggegated Takag-Sugen fuzzy mdel can be nfeed as: { } t &() h(()) zt At () + But (), yt () h(()) zt Ct (), () whee ( ()) h z t p Π j( zj() t ) j p Π j( zj() t ) j () f all t. The tem φ j (z j (t)) s the membeshp value f z j (t) n φ j. Snce p p ( ()) () Π z t 0 and Π z t > 0,,...,, j j j j j j hzt ( ) 0 and hzt ( ),,..., () f all t.. Sect Nnlneaty A nnlnea system may als be epesented by sects (Tanaka & Wang, 00). Cnsde a nnlnea system gven by & () t f( ()) t whee f(0) 0. A glbal sect s fund when & () t f((t)) [s s ](t), whee s (t) and s (t) ae lnes as shwn n fgue. A glbal sect guaantees an eact fuzzy epesentatn f the nnlnea mdel. Sme tmes t s dffcult t fnd glbal sects, n that case t s pssble t fnd a lcal sect bunded by the egn -a< (t) < a, as shwn n fgue.

4 58 New Develpments n Rbtcs, Autmatn and Cntl Fg. Glbal sect Fg. Lcal sect. Fuzzy Obseve The state f a system s nt always fully avalable, s t s necessay t use an bseve t ecnstuct, at least patally the states vaables f the pcess. Ths eques t satsfy the cndtn ( t t ˆ ) lm () () 0 t 0 (5) whee t ˆ dentes the state vect estmated by the fuzzy bseve. Thee ae tw cases f fuzzy bseves desgn dependng n whethe nt z(t) depends n the state vaables estmated by a fuzzy bseve (Tanaka & Wang, 00). Gven the Takag-Sugen fuzzy mdel (), the th ule f a cntnuus fuzzy bseve can be cnstucted as: Obseve Rule If z (t) s φ and and z p (t) s φ p. THEN ˆ & h (()){ ˆ() () ( () ˆ z t A t + Bu t + K y t y()), t } yt ˆ h( ) ˆ zt Ct. (6) whee K s the bseve gan and ŷ ( t ) s the fuzzy bseve utput f the th subsystem. If z(t) depends n the estmated state vaables, the bseve cnsequent pat takes the fllwng fm: THEN ˆ & h (()) ˆ { ˆ() () ( () ˆ z t A t + Bu t + K y t y()), t } yt ˆ h( ˆ) ˆ zt Ct. (7) It s pssble t calculate the bseve gans fm the slutn X, N f the fllwng nequaltes (Tanaka & Wang, 00).

5 Takag-Sugen Fuzzy Obseve f a Swtchng Bpcess: Sect Nnlneaty Appach 59 X > 0 T T T -A X - XA + C N + N C αx > 0, T T T T T T j j j j j j -A X - XA - A X - XA + C N + N C + C N + N C α X 0 < j st.. h h j (8) whee A s the state mat and C s the utput mat. The decay ate (α) s elated wth the bseve speed espnse. The nequaltes (8) can be cnveted t lnea mat nequaltes by means f Shu s cmplement (Baatz & VanAntwep, 000). The cndtn < j s.t. h h j Ø means that nequaltes (8) hlds f all < j eceptng h z(t) h j z(t)0 f all z(t). The bseve gan K and the cmmn pstve defnte mat P can be btaned by means f P X -, K X - N. (9) The fementatve mathematcal mdel wll nw be descbed.. Fementatn Mathematcal Mdel The Sacchamyces ceevsae yeast may gw n glucse fllwng thee metablc pathways (Snnletne & Käpell, 986)..- Odatve gwth n glucse, n pesence f ygen (O ) the glucse (S) s cnsumed t pduce bmass (X) and cabn dde (CO ). μ S + O X + CO (0) s.- Fementatve gwth n glucse, n absence f ygen the substate s used t pduce bmass, cabn dde and manly ethanl (E). μs S X + CO + E ().- Odatve gwth n Ethanl, the ethanl pduced by the fementatve pathway may be cnsumed n pesence f ygen pducng bmass and cabn dde. μe E+ O X + CO (). The Resp-Fementatve and Respatve Fementatn Mdels A cntnuus bake s yeast cultue can be epesented by the fllwng set f dffeental equatns

6 60 New Develpments n Rbtcs, Autmatn and Cntl s s e & ( μ + μ + μ ) D s s n & ( k μ k μ ) D + DS s e & ( k μ k μ ) D 5 s 6 e & ( k μ k μ ) D + OTR () whee the vaables f mdel () ae shwn n table. μ s Vaables Unts Bmass cncentatn g/l Glucse cncentatn g/l Ethanl cncentatn g/l Dsslved ygen cncentatn mgl D Dlutn ate /h S n Substate cncentatn feed g/l OTRK L a(c sat - ) Oygen tansfe ate mg/lh Specfc gwth ate (datve gwth n glucse) /h μ s Specfc gwth ate (fementatve gwth n glucse) /h μ e Specfc gwth ate (datve gwth n ethanl) /h k, k, k, k, k 5, k 6 yeld ceffcents Table. Vaables used n the bake s yeast mdel () The ygen tansfe ate s gven by OTR K L a(c sat - ) whch may be splt n tw tems, ne that s cnstant and anthe ne that depend n the dsslved ygen. -K L a () K L ac sat, (5) Pmeleau (990) suggested a efmulatn f mdel () usng tw patal mdels: a esp-fementatve patal mdel () wth ethanl pductn and a espatve patal mdel (R) wth ethanl cnsumptn. Wth ths efmulatn a splt pcess mdel s btaned, swtchng fm the patal mdel t the R patal mdel and vce vesa dependng n whethe the system s cnsumng pducng ethanl. T pecse these deas, cnsde a nnlnea system descbed by mdel (6-7), whch can be wtten as t & f( t ) + But + d,, (6) y() t h( ()) t, (7)

7 Takag-Sugen Fuzzy Obseve f a Swtchng Bpcess: Sect Nnlneaty Appach 6 whee f ((t)) descbe bth the and R patal mdels, namely f ( μ + μ ) D ( k μ k μ ) D s_ s_ : k μ D s_ s_ s_ 5 s_ L k μ D K a f (8) and f the R mdel f ( μ + μ ) D k μ D s_ R : k μ D s_ R e_ R e_ R 5 s_ R 6 e_ R ( k μ k μ ) D K a L fr, (9) The nput vect and the manpulated vaable ae gven by B [0, D, 0, 0] T, (0) u(t)s n () whee T s the nput vect tanspse. As aleady sad, OTR ate was dvded n tw tems, the fst ne -K L a () was ncluded n f (8) and f R matces (9); the secnd tem K L ac sat (5), s taken as a knwn and cnstant petubatn (d) gven by d [0, 0, 0, K L ac sat ] T () In the patal mdel, the metablc pathways datve gwth n glucse (0) and fementatve gwth n glucse () ae pesent, theefe ethanl s pduced. The specfc gwth ates f the patal mdel ae gven by: μ μ ma q s _ O K + O q q + K + ma ma s _ s c () In the R patal mdel, the pathway datve gwth n glucse (0) s als pesent; hweve, the specfc gwth ate s nw gven by:

8 6 New Develpments n Rbtcs, Autmatn and Cntl μ ma s_ R qs + () The datve gwth n ethanl () f the R patal mdel depends n, μ q e f qe < qe q e f qe qe e_r (5) whee q e q K ma e e Ke + K + (6) ma ma q O e e q qs K + + O O (7) The and R mdels cannt be enabled at the same tme. A cndtn f the tanstn between the R- patal mdels s gven by (Feea, 995) R f μ 0 s R f μe 0 (8) The paametes values and the ntal cndtns f the and R patal mdels ae gven n table ; a cmplete descptn f all paametes can be fund n (Feea, 995). The manpulated vaable u(t)s n was set as a squae sgnal as can be seen n fgue. Paamete Value Paamete Value.5 gs/gx h k gx / gs ma q s ma q e ma q 0.6 ge / gx h k gx / gs 0.56 go / g X h k - 0. gx / ge K 0. g/l k gx / ge K g/l k 5 -. gx / go C sat 7.0 mg/l k gx / go S n 0 g/l (0) 0. g/l K e 0. g/l (0) 0.0 g/l K s 0. g/l (0) 0.5 g/l

9 Takag-Sugen Fuzzy Obseve f a Swtchng Bpcess: Sect Nnlneaty Appach 6 Paamete Value Paamete Value K L a 00 /h (0) mg/l D 0. /h Table. Paametes values used n mdel () Fg. Input squae sgnal f the bake s yeast mdel. The Takag-Sugen Fuzzy Eact Mdel When the nnlnea dynamc mdel f the bake s yeast s knwn, as well as all the paametes, a fuzzy eact mdel can be deved fm the gven nnlnea mdel. Ths eques a sect nnlneaty appach (Tanaka & Wang, 00).. The Resp-Fementatve Fuzzy Eact Mdel T cnstuct the eact fuzzy mdel we need t epess the patal mdel as a nnlnea system (6-7). μ μ & + s_ s_ D & k μs_ k μs_ D D + 0 S & μ 0 n K 0 L ac k s_ 0 D 0 & 0 k5 μs_ 0 0 D KLa sat (9) y() t h( ()) t + h( ()) t (0) Substtutng the specfc ates () n the f ((t)) mat fm mdel (9), we btan the mat gven by (), f cnvenence called scheme f _I

10 6 New Develpments n Rbtcs, Autmatn and Cntl scheme f _I ma ma O ma O q c qc + qs D K + + ma ma O ma k q + kq 0 0 O c c kq D s K + f _ I + ma O ma kq c + kq s 0 D 0 K + + ma k q 0 0 D K a 5 O c L K + () Hweve; the mat f ((t)) may als be wtten as: Scheme f _II ma ma O ma q q D q 0 0 O c c s + + K ma ma O ma k O q c kq c kq s D + + _ K f II ma O ma kq kq D 0 c s K + + ma k q 0 0 D Ka 5 O c L K + () as Scheme f _III f _ III ma ma ma O 0 q D q q O c s c K + + K + ma ma ma O k O q c kq s D 0 k qc K + + K + ma ma O 0 kq D kq s c + K + ma k q 0 0 D K a 5 O c L K + () Althugh thee ae anthe pssble cmbnatns t wte the f ((t)) mat, wth these appaches we btan enugh nfmatn t pecse u pnt. Fm scheme f _I () tw nnlneates can be detected

11 Takag-Sugen Fuzzy Obseve f a Swtchng Bpcess: Sect Nnlneaty Appach 65 NL ; K + NL () + fm scheme f _II () als tw dffeent nnlneates can be bseved NL ; K + NL (5) + and fm scheme f _III () thee nnlneates ae pesent NL ; K + NL ; + NL (6) K+ Althugh wth each mat gven by (-) the eact fuzzy mdel can be bult, t wll take fu lnea subsystems ( ) f the scheme f _I and scheme f _II and eght lnea subsystems ( ) f scheme f _III. F cnvenence the nnlneates (5) and the scheme f _II () ae chsen t buld the eact fuzzy mdel, the easn wll be evdent n the net sectn whee the fuzzy bseve s cnstucted. The pemse vaable z () t s defned as NL z t f () ; K K + (7) Fm equatn (7) the mamum and mnmum values f the plt f (7) can be bseved. () z t can be btaned. In fgue 5 Fg. 5 Plt f the pemse vaable z () t

12 66 New Develpments n Rbtcs, Autmatn and Cntl The mamum and mnmum values f z () t n the ange (t) [0, 0.007] ae gven by mn z ( t) 0 a ma () t z t a () t (8) we defne the pemse vaable z () t as: NL z () t ; f - (9) + Fm equatn (9) the mamum and mnmum values f n fgue 6, z () t can be btaned, as shwn Fg. 6 Plt f the pemse vaable z () t The mamum and mnmum values f z () t n the ange (t) [0, 0] and (t) [0, ] ae gven by ma 50 t t () () z t b z ( t) mn 0 () t () t b (0) The membeshp functns ae bult fm these equatns () () z t z t a ; ( () t ) z () t z b j j j () whee the fllwng ppetes must hld

13 Takag-Sugen Fuzzy Obseve f a Swtchng Bpcess: Sect Nnlneaty Appach 67 ( z ( t) ) ( z ( t )) ( z ( t) ) ( z ( t )) + + () Slvng equatns ( and ) the fllwng membeshp functns ae btaned z ( t) a ( z () t ) ( z () t ) a a z ( t) b ( z () t ) ( z () t ) b b ( t) z + a a a ( t) z + b b b () Substtutng the mamum and mnmum values a, a, b and b n () we btan pssble cmbnatns t epess the lnea subsystems. A ma ma O ma q q 0 0 O c c a D q b s j ma ma O ma k q + kq a kq b D 0 0 O c c s j j, j, kq kq D ma O ma a b 0 c s j ma a 5 O c k q 0 0 D Ka L () The fuzzy ules f the patal mdel ae stated as: If z (t) s φ (z (t)) and z (t) s φ (z (t)) THEN & () t A () t + Bu() t + d If z (t) s φ (z (t)) and z (t) s φ (z (t)) THEN & () t A () t + Bu() t + d If z (t) s φ (z (t)) and z (t) s φ (z (t)) THEN & () t A () t + Bu() t + d If z (t) s φ (z (t)) and z (t) s φ (z (t)) THEN & () t A () t + Bu() t + d

14 68 New Develpments n Rbtcs, Autmatn and Cntl The aggegated mdel f the patal mdel s gven by { } () ( ()) ( ()) () () & t z t z t A t + Bu t + d j ( ()) ( ()) y ( t ) z t z t C ( t ),, j,. J j j j (5). The Respatve Fuzzy Eact Mdel The R patal eact mdel can als be bult fllwng the pcedue descbed n sectn.. We must be awae that the R mdel must be splt n tw mdels called R qe and R qe. As n schemes (-) seveal pssbltes may be fmulated t buld the R mdel, theefe a pssblty t epess the f R ((t)) mat (f Rqe and f Rqe ) can be wtten as: Scheme f Rqe_I f Rqe ma ma q K D q 0 0 e e s ( Ke + )( K + ) + ma 0 kq D 0 0 s + ma kq K 0 D 0 e e ( Ke + )( K + ) kq K kq 0 D Ka L ma ma 6 e e 5 s ( Ke )( K ) (6) Scheme f Rqe_I f Rqe ma ma ma O eq D e c qs O q s K + O ma 0 kq D 0 0 s + ma ma k eq k e q D 0 O c O s K + K s + O k + ma ma ma eq k q k eq 0 D KLa 6 O c 5 s 6 O s K + + O K s (7) T cnstuct the eact mdel f the R patal mdel we must use the nnlneates fm mdels (6-7). F mdel 6 we have that the fst nnlneaty s gven by Rqe NL 5 z () t ; f - Ke and -K (8) ( Ke + )( K + )

15 Takag-Sugen Fuzzy Obseve f a Swtchng Bpcess: Sect Nnlneaty Appach 69 whee the mamum and mnmum values ae gven by ( t ) ma () () z c t t mn 0. t t () () z t c (9) The emanng nnlneates fm mdels (6) and (7) wee the same aleady descbed by (7) and (9). The lnea subsystems f the R qe mdel can be btaned fm ma ma q Kc D q bj e e k s ma kq s bj D A R qe ma kq Kc D e e k ma ma 6 e e ck kq 5 s bj 0 jk, kq K D (50) and ma ma ma eq a D e 0 0 O q q c s O s b j O ma 0 k qs b D 0 0 j A ma ma R qe k e q a k e q b D 0 O c O s j O ma ma ma k 6 O eq a kq 5 s + k 6 O eq 0 c s b j D j, O (5) A geneal mdel t btan the ules f the R qe and R qe patal mdels s epessed as f R qe If z (t) s φ (z (t)) and z (t) s φ k (z (t)) )) Rqe Rqe jk THEN & () t A () t + Bu() t + d; jk, and f R qe If z (t) s φ (z (t)) and z (t) s φ j (z (t)) Rqe Rqe THEN & () t A () t + Bu() t + d; j, Fnally the aggegated mdel f the R qe and R qe patal mdels s epessed as: f R qe j

16 70 New Develpments n Rbtcs, Autmatn and Cntl { } Rqe () ( ()) ( ()) () () Rqe j k jk j k & t z t z t A t + Bu t + d Rqe ( ()) ( ()) y ( t ) z t z t C ( t ), j, k,. j k j k (5) and f R qe { } Rqe () ( ()) ( ()) () () Rqe j j j & t z t z t A t + Bu t + d Rqe ( ()) ( ()) y ( t ) z t z t C ( t ),, j,. j j (5) Althugh the pemse vaables f the patal mdels and R qe wee the same (7) and (9), they have dffeent behavs as they ae multpled by dffeent yeld ceffcents. The aggegated mdels (5) and (5-5) epesents eactly the nnlnea system () n the egn (t) [0, 0], (t) [0, ], (t) [0, 5] and (t) [0, 0.007] A cndtn f the tanstn between the R- patal mdels s gven by (8). 5. Fuzzy Obseve Nw that an eact fuzzy mdel f the nnlnea bake s yeast patal mdel has been btaned, a fuzzy bseve can nw be desgned. Fst f all we have t test the bsevablty mat f the btaned lnea subsystems. A lnea system s sad t be bsevable f f any unknwn ntal state (0) thee est a fnte t >0 such as the knwledge f the nput u and the utput y ve [0, t ] suffces t detemne unquely the ntal state (0). Othewse the equatn s unbsevable (Chen, 999). The pa (A,C) s bsevable f and nly f the bsevablty mat О [ C CA CA,,CA n- ] T n, (5) has full ank (ρ(o) n).e. s nnsngula. In sectn. we emak that the fuzzy eact mdel f the mdel may be bult fm thee schemes (amng many thes) namely f _I (), f _II () and f _III (). If we buld the fuzzy eact mdel f each scheme (-) and we test the bsevablty mat f these lnea subsystems; f eample (), we shuld fnd that (table ) Schemes bsevablty ank f C[0 0 ] f _I f _II f _III Table. Obsevablty mat f schemes (-)

17 Takag-Sugen Fuzzy Obseve f a Swtchng Bpcess: Sect Nnlneaty Appach 7 Fm table we may ntce that n full ank s acheved f f _I, theefe a full bseve cannt be bult f ths scheme. F schemes f _II and f _III almst full ank s acheved n evey lnea subsystem; hweve, f scheme f _III t wll take eght lnea subsystems t buld the fuzzy eact mdel, whle f scheme f _II nly fu lnea subsystems wll be needed. T avd buld cmplcated lnea systems, scheme f _II was chsen t cnstuct the Eact fuzzy bseve. Theefe befe cnstuctng a fuzzy eact mdel f an bseve a cntlle, t wll be advsable t analyze the way the pemse vaables ae chsen t avd lack f bsevablty cntllablty. The fllwng assumptns wee made t buld the fuzzy bseve: H. The nmnal values f the yeld ceffcents k, - k 6 ae cnstant and knwn. H. The ethanl, the dsslved ygen cncentatn and the OTR ae knwn. The pcedue t buld the eact fuzzy bseve s the same that was fllwed f the fuzzy eact mdel, althugh sme cnsdeatns must be taken nt accunt. An mptant cnsdeatn s elated wth the scheme () whee the pemse vaable (9) wll depend n the estmated state and, theefe the pemse vaable must be mdfed t: ˆ zˆ t f () ; ˆ + ˆ - (55) The same stuatn apples t the pemse vaable f mdel (6) Rqe z () t ; f - Ke and ˆ -K ( Ke + )( K + ˆ ) (56) The pemse vaable (7) emans unchanged. T guaantee full bsevablty ank (table ) the mnmum values f the pemse vaables ae mdfed t a b mn z t mn z t mn z t c 0. t t t t ( t) (57) Schemes Lnea subsystems bsevablty ank f _II f Rqe_I f Rqe_I Table Obsevablty mat f the lnea subsystems (, 50-5) The membeshp functns ae bult as befe; nevetheless, f (5-55) we have

18 7 New Develpments n Rbtcs, Autmatn and Cntl ( zˆ () t ) ( zˆ () t ) ( t) zˆ b b b zˆ c ( t) c c ( z ˆ () t ) ( z ˆ () t ) ( t) zˆ + b b b ( t) zˆ + c c c (58) The lnea subsystems gven by (), (50-5) ae used t bult the fuzzy bseve. A geneal ule t btan all the fuzzy ules f the, R qe and R qe patal mdels ae gven by: f If zˆ () t s φ (z (t)) and z (t) s φ j (z (t)) THEN ˆ ˆ ( ˆ) sat & j j ; L () t A t + Bu t + K y t y t + K ac j, f R qe f R qe If zˆ () t s φ j (z (t)) and zˆ () t s φ k (z (t)) & ˆ Rqe () t A Rqe ˆ t + Bu t Rqe sat + K y t y ˆ t + K ac ; jk, THEN ( ) jk If zˆ () t s φ (z (t)) and z (t) s φ j (z (t)) & ˆ Rqe () t A Rqe ˆ t + Bu t Rqe sat + K y t y ˆ t + K ac ; j, THEN ( ) j jk j L L The aggegated fuzzy bseves f the, R qe and R qe patal mdels ae gven by f { } () ( ()) ( ()) () () ( () ()) ˆ & t zˆ t z t A ˆ t + Bu t + K y t yˆ t + K ac sat j j j L j ( ()) ( ()) yˆ ( t) zˆ t z t Cˆ( t),, j,. j j (59) f R qe { } () ( ()) ( ()) () () ( () ()) & ˆ t zˆ t zˆ t A ˆ t + Bu t + K y t yˆ t + K ac Rqe Rqe Rqe sat j k jk jk L j k Rqe ( ()) ( ()) yˆ ( t) zˆ t zˆ t Cˆ( t), j, k,. j k j k (60) f R qe

19 Takag-Sugen Fuzzy Obseve f a Swtchng Bpcess: Sect Nnlneaty Appach 7 { } ( ()) ( ()) () () ( () ()) Rqe Rqe Rqe sat j j j L j ˆ & () t zˆ t z t A ˆ t + Bu t + K y t yˆ t + K ac Rqe ( ()) ( ()) yˆ ( t) zˆ t z t Cˆ( t),, j,. j j (6) 5. Fuzzy Obseve Smulatn The applcatn f the ppsed bseve scheme was smulated usng MATLAB TM. The fuzzy bseves wee tested usng the cntnuus and the R bake s yeast patal mdels gven abve. The nlet substate cncentatn was vaed between g/l and 0 g/l n de t fce the swtchng between the patal mdels. The patal mdels paametes wee gven n table. The decay ate (α) was set t ze. The estmated vaables wee the bmass and the substate, each bseved vaable was tested wth thee dffeent ntal cndtns, and g/l f bmass, and 0.0, 0.0 and 0.06 g/l f substate. The behav f the fuzzy bseve f bmass estmatn s shwn n fgue 7. The bseve cnveges aund the 0 hus f fementatn elapsed tme, almst n the R qe patal mdel. It can be ntced the dynamcs f the bake s yeast swtchng thugh the, R qe and R qe patal mdels. The bseve substate cnveges aund the 5 hus f fementatn elapsed tme (fgue 8), theefe the substate dynamcs s faste than the bmass. The bseve gans ae dsplayed n table 5 and wee calculated fm the nequaltes (8) thugh Lnea Mat Inequaltes. Fg. 7 Bmass bseve pefmance wth α 0 and ˆ (0), and g/l. Fg. 8 Substate bseve pefmance wth α 0 and ˆ (0) 0.0, 0.0 and 0.06 g/l.

20 7 New Develpments n Rbtcs, Autmatn and Cntl Gan K _ K _ K _ K _ K _ Rqe K _ Rqe K _ Rqe K _ Rqe K _ Rqe K _ Rqe K _ Rqe K _ Rqe Table 5. Obseve gans, wth α 0. Cmmn pstve defnte matces that guaantees glbal asympttc stablty (Tanaka & Wang, 00), wee fund f each patal mdel, namely P , P Rqe , P Rqe T mpve the bseve pefmance the decay ate at (α) was set t 0.. The behav f the fuzzy bseve f bmass estmatn s shwn n fgue 9. The bseve cnveges n abut 6 hus f fementatn elapsed tme, nw wthn the state. The bseve substate cnveges aund the 5 hus f fementatn elapsed tme (fgue 0). The bseve gans f α 0. ae dsplayed n table 6 and wee calculated usng the nequaltes gven by (8).

21 Takag-Sugen Fuzzy Obseve f a Swtchng Bpcess: Sect Nnlneaty Appach 75 Fg. 9 Bmass bseve pefmance wth α 0. and ˆ (0), and g/l. Fg. 0 Substate bseve pefmance wth α 0 and ˆ (0) 0.0, 0.0 and 0.06 g/l. Gan K _ K _ K _ K _ K _ Rqe K _ Rqe K _ Rqe K _ Rqe K _ Rqe K _ Rqe K _ Rqe K _ Rqe Table 6. Obseve gans, wth α 0..

22 76 New Develpments n Rbtcs, Autmatn and Cntl Cmmn pstve defnte matces that guaantees glbal asympttc stablty (Tanaka & Wang, 00), wee fund f each patal mdel, namely P ; P Rqe P Rqe Fm (58, 59 and 60) an eact fuzzy bseve f a nnlnea bake s yeast mdel was desgned. The fuzzy estmat had a satsfacty behav. A dffeent appach t cnstuct a fuzzy bseve usng the whle tem OTRK L a(c sat - ) as a knwn and cnstant petubatn was epted n (Heea, 007a). In ths case a patal bseve was cnstucted due that full ank n the bsevablty mat culd nt be acheved. ; 6. The Fuzzy Eact Mdel, (u(t)d). The cnstuctn f the fuzzy eact mdel f a cntnuus bake s yeast fementatn can becme qute cmple when the utput f the system s gven by u(t)d, f eample f the patal mdel μ μ & + s _ s _ & k μs _ k μs _ S n D K & μ 0 L ac k _ s & k5 μs _ 0 0 KLa sat (6) In ths case the nput mat s nt cnstant anyme, dependng nw n the vaables,,,. S we defne the new pemse vaable as

23 Takag-Sugen Fuzzy Obseve f a Swtchng Bpcess: Sect Nnlneaty Appach 77 z () t, z () t, z () t, z () t, (6) The new pemse vaables may be wtten as ( z () t ) ( z () t ) 5 ( z () t ) 6 ( z () t ) 7 z t d d d z t e e e z t f f f z t g g g ( z () t ) ( z () t ) 5 ( z () t ) 6 ( z () t ) 7 z t + d d d z t + e e e z t + f f f z t + b b b (6) whee the mamum and mnmum values f (6) ae dsplayed n table 7. Pemse vaable mamum mnmum z (t) d 0 d 0 z (t) e e 0 z (t) f 5 f 0 z (t) g g 0 Table 7. Mamum and mnmum values f z (t), z (t), z (t) and z (t) The nput mat can nw be wtten as [,,, ] B d e + S f g (65) lmn l m n n T The the pemse vaables ae stll gven by (7) and (9). A geneal ule t cnstuct all the fuzzy ules can be stated as If z (t) s φ (z (t)) and z (t) s φ j (z (t)) and z (t) s φ l (z (t)) and z (t) s φ 5m (z (t)) and z (t) s φ 6n (z (t)) and z (t) s φ 7 (z (t)) THEN & () t A () t + B u() t + d; jlmn, jlmn It must be emaked that 6 subsystems wuld be needed t cnstuct the patal mdel. Fnally the aggegated fuzzy system f the patal mdel s gven by jlmn

24 78 New Develpments n Rbtcs, Autmatn and Cntl { } 6 ψ _ jlmn jlmn & () t h (()) z t A () t + B u() t + d 6 y ( t) h ( z( t)) C( t), jlmn,. ψ _ (66) whee ψ _ + ( n ) + ( m ) + 8( l ) + 6( j ) + ( ), h ( z( t)) h ( z( t)) ( z ( t)) ( z ( t)) ( z ( t)) ψ _ ψ _ j l 5m( z ( t)) 6n( z ( t)) 7( z ( t)) (67) The R qe and R qe fuzzy eact mdel wee cnstucted fllwng the same ules and als 6 subsystems wee btaned f each patal mdel. As stated befe nw the eact fuzzy mdel gets qute cmple because t wll be necessay 9 subsystems t epesent the, R qe and R qe patal mdels. Fm the fuzzy eact mdel bult f the case eplaned a fuzzy bseve can als be bult, me detals ae epted n (Heea, 007b). A multple Takag- Sugen multple cntlle was desgned t fce the swtchng between the and the R bakng yeast patal mdels (Heea, 007c; Heea, 007d). The substate fuzzy cntlle tacked a squae efeence sgnal vaed between 0.0 g/l and 0.07 g/l. S n was set t 5 g/l.it s wth ntng that the cntlle was capable t fce the swtchng alng the patal mdels. 7. Cnclusn Based n the dea f splttng the bake s yeast mdel, a nvel TS fuzzy mdel was ppsed usng the sect nnlneates methd, gvng an eact epesentatn f the gnal nnlnea plant. Meve, an bseve f each patal mdel was cnstucted. It s wth ntng that the bseve was capable f swtchng alng the patal mdels, wthut pefmance degadatn. Theefe, the appach pesented hee may be cnsdeed a vald methd t desgn an bseve. Futue wk wll nclude the epemental valdatn f the fuzzy bseve and ptmal cntlles f fed-batch fementatn cultues. 8. Acknwledgments Ths wk has been suppted by the Mecan Cnsej Nacnal de Cenca y Tecnlgía (CONACyT), unde gants 658 and Refeences Ascenc, P.; Sbaba, D. & Azeved, S. (00). An adaptve fuzzy hybd state bseve f bpcesses. IEEE Tansactns n Fuzzy Systems, Vl., N. 5, pp. 6-65, ISSN

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