Temperature Distribution Control of Reactor Furnace by State Space Method using FEM Modeling

Transcription

1 Mmois of th Faculty of Egiig, Okayama ivsity, Vol. 4, pp , Jauay 008 Tmpatu Distibutio Cotol of Racto Fuac by Stat Spac Mthod usig FEM Modlig Tadafumi NOTS Divisio of Elctoic ad Ifomatio Systm Egiig Gaduat School of Natual Scic ad Tchology Okayama ivsity 3-1-1, Tsushima-Naka Okayama, Masami KONISHI Divisio of Idustial Iovatio Scic Gaduat School of Natual Scic ad Tchology Okayama ivsity 3-1-1, Tsushima-Naka Okayama, Ju IMAI Divisio of Idustial Iovatio Scic Gaduat School of Natual Scic ad Tchology Okayama ivsity 3-1-1, Tsushima-Naka Okayama, (Rcivd Dcmb 1, 007 Th opatios of a acto fuac such as a blast fuac still dpd o th xpics ad ituitios of skilld opatos du to th complxity of ital fuac phoma ad high tmpatu. Th is a gat dmad fo stabl opatios ud th cicumstacs of a dcas i th umb of skilld opatos ad difficulty i tchology ihitac. This pap aims to costuct mathmatical dsciptio fo cotol of a acto fuac. Futh, th lia-quadatic-gaussia cotol systm fo a acto fuac is poposd, which stimats i fuac tmpatu distibutio usig masud data a fuac wall. Tmpatu distibutio of a fuac is cotolld basd o th stimatd i tmpatu distibutio chagig of bouday coditios of a fuac. Th pacticability of th poposd cotol mthod was chckd though umical xpimts. 1 INTRODCTION A acto fuac such as a blast fuac hav b playig vital ols i stl idusty. Th pfomac of a fuac has b impovd makably accompayig lagmt of fuac facilitis ad xpadig poductio. Howv, th opatios of a fuac still dpd o th xpics ad ituitios of skilld opatos bcaus of th complxity of ital fuac phoma ad high tmpatu. Th a vaious otsu@ct.lc.okayama-u.ac.jp factos govig such as gas flow, chmical actios, bu-though ad aastomoss of io os ad movmt of filligs i a fuac. Stabl opatios of blast fuac a stogly quid fom difficultis i fuac gulatio. Rctly, th occud a gat dmad fo automatd opatios accompaid with dcas i th umb of skilld opatos ad difficulty i tchology ihitac. I this sach, w pusu th costuctio of cotol systm fo a acto fuac that stimats ad This wok is subjctd to copyight. All ights a svd by this autho/authos. 79

2 Tadafumi NOTS t al. MEM.FAC.ENG.OKA.NI. Vol. 4 cotols tmpatu distibutio i th acto fuac. As mtiod, a acto fuac has complx phoma such as chmical actios, high pssu ad high tmpatu itally. Vaiabls oly a th fuac wall ca b masud ad ca b usd fo fuac cotol. Bcaus a acto fuac is hug siz ad th ital situatio of th fuac ca t b chagd dictly. Thfo, cotol systm that stimatd i tmpatu distibutio of a fuac fom th masud data a th fuac wall ad cotolld tmpatu distibutio of a fuac basd o th stimatd i tmpatu distibutio chagig of bouday coditios of a fuac was costuctd. Fist, th acto fuac simulato is costuctd, which tats gas flow ad tmpatu distibutio i th fuac. Th umically simulatd fuac has istumtatios oly a th fuac wall ad cotol iputs ca b opatd oly at boudais. This sach caid out by usig th acto fuac simulato. To mak th cotol systm dsig, stat spac modl is divd by applyig fiit lmt mthod (FEM to th acto fuac simulato. Though this pocdu, lia cotol thoy is applid fo dsig of cotol systm. Futh, lia-quadatic-gaussia (LQG cotol is adoptd i od to dal with th poblm dsciptio. Dtails a dscibd i th followig sctios. I th followig, acto fuac modl will b costuctd. Two-dimsioal acto fuac modl is show i Fig., wh i1 i V i1 ad V i a vlocity of blowig gass fom tuys. Also, od umb (i, j is assigd to od poit as show i Fig.. Futhmo, fatus of acto fuac modl a show Tabl.1. Istumtatios a st up as show i Fig.3, wh is gas flow mt ad is solid thmomt. Ths istumtatios a st up alog th outsid of acto fuac modl ad th top of solid lay. Suppos that pssu distibutio is P (x, y, t, gas flow distibutio is V (x, y, t = (x, y, ti + V (x, y, tj, (1 gas tmpatu distibutio is T g (x, y, t ad solid tmpatu distibutio is T s (x, y, t i acto fuac modl. Ths vaiabls a disctizd by usig od umb (i, j ad tim as show i Tabl.. Racto fuac modl calculats th disct valus show i Tabl. by applyig MAC mthod ad fiitdiffc mthod[1] to govig quatio of P, V, T g ad T s i th followig subsctio. REACTOR FRNACE MODEL Blast fuacs a usd wh pig ios a mad fom io os. Schmatic of a blast fuac is show i Fig.1. Th hight of a blast fuac is about 40 mts ad th diamt of it is about 0 mts. Io os ad coks a chagd altatly fom upp pat of a blast fuac. Moov, th a multipl tuys at th bottom of a blast fuac ad blasts of hot ai of about 100 dgs C a blow though ths tuys. Coks bu ad cabo mooxid is gatd by th blasts of hot ai. Io os a ducd to pig ios i about 8 hous by th actio with th cabo mooxid ad pig ios accumulat at th bottom of a blast fuac. O th oth had, th blasts of hot ai a dischagd fom upp pat of a blast fuac. As w hav mtiod abov, a blast fuac has complx phoma such as chmical actios, high pssu ad high tmpatu itally. Thfo, istumtatios of a blast fuac ca b locatd oly a th fuac wall. Fig. 1: Schmatic of a blast fuac Tabl. 1: Fatus of acto fuac modl od umbs of i 13 od umbs of j 1 od umb (i,j of outlt (6,1 (7,1 (8,1 od umb (i,j of ilt (4,1 (10,1 (1,3 (13,3 80

3 Jauay 008 Tmpatu Distibutio Cotol of Racto Fuac by Stat Spac Mthod usig FEM Modlig +,-./- #$% &* $' ( t = + 1 R + V ( + P (f 1 + f + V (3 01./-!" # & $%' #$% & $%' 01./- ( V t = V + 1 R + V V ( V + V P (f 1 + f + V V ( #$ $' #( $' & & 01./- 01./ Fig. : Two-dimsioal acto fuac modl Bouday coditios hold Fuac bottom ad wall = V = 0, P = 0 : omal vcto (5 Outlt i,1 = i,0, V i,1 = V i,0, P i,1 = 0 (i = 6, 7, 8 (6 Ilt 1,3 = i1, 13,3 = i, V 4,1 = V i1, V 10,1 = V i (7 Fig. 3: Istumtatios wh P is pssu distibutio, V is gas flow distibutio, R is Ryolds umb, f 1 ad f a cofficits of th Euga quatio, i1, i, V i1 ad V i a vlocity of blowig gass fom tuys. Tabl. : Vaiabls of od umb (i,j at V (x, y, t i,j, V i,j P (x, y, t Pi,j T g (x, y, t Tg i,j T s (x, y, t Ts i,j.1 Gas Flow Modl[, 3, 4, 5] Assumig that pssu ad gas flow a dscibd by th quatio of cotiuity ad th Navi-Stoks quatio addd th Euga quatio[6, 7], w gt + V = 0 (. Gas Tmpatu Modl[, 3, 4, 5] Assumig that gas tmpatu is dscibd by gy quatio addd hat tasf btw solid ad gas[6], w gt = t V + 1 ( T g RP + T g α(t g T s (8 Bouday coditios hold Fuac bottom = 0 (9 Wall ad outlt = a(t g T out : omal vcto (10 81

4 Tadafumi NOTS t al. MEM.FAC.ENG.OKA.NI. Vol. 4 Ilt Tg 1,3 = T g i1, Tg 13,3 = T g i, Tg 4,1 = T g i3, Tg 10,1 = T g i4 (11 wh T g is gas tmpatu distibutio, T s is solid tmpatu distibutio, P is Padtl umb, α ad a a cofficits of hat tasf, T out is xtal tmpatu, T g i1, T g i, T g i3 ad T g i4 a tmpatu of blowig gass fom tuys. spac modl is divd, lia cotol thoy[9, 10, 11] bcoms availabl. Fist, liaizd quatios a divd by usig (- (16 ad dsid distibutio. Th, dscipto systm is divd by applyig FEM to th liaizd quatios. Nxt, stat quatio is divd by applyig galizd ivs matix to th dscipto systm. Fially, output quatio is costuctd by istumt placmt show i Fig.3. Though this pocdu, stat spac modl is costuctd..3 Solid Tmpatu Modl[, 3, 4, 5] Assumig that solid tmpatu is dscibd by hat coductio quatio addd actio hat of matial ad hat tasf btw gas ad solid[8], w gt T s t = β ( T s + T s Bouday coditios hold Fuac bottom γ(t s T g + f 3 (, V, T g (1 T s = 0 (13 Top of solid lay Wall T s = b up(t s T g (14 T s = b(t s T out : omal vcto (15 wh β, γ, b up ad b a cofficits of hat tasf, f 3 is actio hat of matial. f 3 is th followig f 3 (, V, T g = Q 1 D p xp{ E/(R T g } 1 + xp{(t g 1000/K 1 } Q + V (16 wh D p is diamt of matial paticl, Q 1, Q, E, R ad K 1 a costat. 3 DERIVATION OF STATE SPACE MODEL I this sctio, stat spac modl will b costuctd basd o (-(16. Stat spac modl psts chaactistic of dsid distibutio ighbohood ad cosists of stat quatio ad output quatio. If stat 3.1 Liaizatio of Dsid Distibutio Nighbohood[1] A stady stat solutio of (-(16 is show i Fig.4-7 ud th coditio of appdix A. Suppos that a stady stat solutio show i Fig.4-7 is dsid distibutio (P V T g, T s ad costat vlocity of blowig gass fom tuys fo dsid distibutio is (Ũi1, Ũ i Ṽi1, Ṽ i. Dfiig th ptubatio vaiabls ( P,, V, T g, T s ad ( i1, i, V i1, V i as P = P + P, = +, V = V + V T g = T g + T g, T s = T s + T s (17 i1 = Ũi1 + i1,, i = Ũi + i V i1 = Ṽi1 + V i1, V i = Ṽi + V i (18 ad istig ito (-(16, w gt th liaizd quatios t V t + V ( = + + V + V + 1 ( R + P = 0 (19 f 1 + V f V f V (0 + V + V ( V V = + V V + V + 1 ( V R + V + V P f 1 V + V f V V f (1 + V + V 8

5 Jauay 008 Tmpatu Distibutio Cotol of Racto Fuac by Stat Spac Mthod usig FEM Modlig Bouday coditios hold T g t Fuac bottom = V = 0, Outlt P = 0 : omal vcto i,1 = i,0, V i,1 = V i,0 ( P i,1 = 0 (i = 6, 7, 8 (3 Ilt 1,3 = i1, 13,3 = i V 4,1 = V i1, V 10,1 = V i (4 ( Tg = + T g + T g V + V T g + 1 RP ( T g + T g Bouday coditios hold T s t Fuac bottom T g Wall ad outlt Ilt α( T g T s (5 = 0 (6 T g = a T g : omal vcto (7 Tg 1,3 = 0, Tg 13,3 = 0, Tg 4,1 = 0, Tg 10,1 = 0 (8 ( T s = β + T s + T g =T g + = V V =V + γ( T s T g T g =T g = V =V V T g =T g T g (9 V =V = Bouday coditios hold Fuac bottom T s Top of solid lay T s Wall = 0 (30 = b up ( T s T g (31 T s = b T s (3 Fig. 4: Stady distibutio of pssu (t = 5[ ]! " #! " Fig. 6: Stady distibutio of gas tmpatu (t = 90[ ] Fig. 5: Stady distibutio of gas flow (t = 6[ ]! " #! " Fig. 7: Stady distibutio of solid tmpatu (t = 80[ ] 3. Applicatio of FEM to Liaizd Equatios[13] I this subsctio, FEM is applid to th liaizd quatios. Dfiig positios of ptubatio vaiabls ad tiagula lmts of FEM a show i Fig.8-10, 83

6 Tadafumi NOTS t al. MEM.FAC.ENG.OKA.NI. Vol. 4 wh is dfiig positio of ptubatio vaiabl ad ptubatio vaiabls of, V ad T g a dfid at th sam positio. A typical tiagula lmt i Fig.8-10 is show i Fig.11, wh 4, 5 ad 6 a middl poits btw sids. Ptubatio vaiabls pstd as (33 a dfid at od poits ad middl poits btw sids as show i Fig.11. O th oth had, ptubatio vaiabls of P ad dsid valus pstd as (34 a dfid at od poits. {ϕ (t} T = [ ] {ϕ V (t} T = [ V1 V V3 V4 V5 V6 ] { ϕ Tg (t } T = [ T g1 Tg Tg3 Tg4 Tg5 Tg6] {ϕ Ts (t} T = [ T s1 T s T s3 T s4 T s5 T s6 ] {ϕ P (t} T = [ P 1 P P 3 ] {ϕ } T = [ 1 3] {ϕ V } T = [V1 V V3] { T ϕtg} = [T g1 Tg Tg3] (33 {ϕ Ts } T = [T s1 T s T s3] (34 Th appoximatios of, V, T g, T s, P,, V, T g ad T s i lmt a dfid as = [M] {ϕ (t} V = [M] {ϕ V (t} T g = [M] { ϕ Tg (t } T s = [M] {ϕ Ts (t} (35 P = [L] {ϕ P (t} = [L] {ϕ } V = [L] {ϕ V } T g = [L] { ϕ Tg } T s = [L] {ϕ Ts } (36 wh M ad L a [M] = [(L 1 1L 1 (L 1L (L 3 1L 3 [L] = [L 1 L L 3 ] L 1 = A 1, L = A, L 3 = A 3 A 1 = 1 A 3 = 1 1 x y 1 x y 1 x 3 y 3 1 x 1 y 1 1 x y 1 x y 4L 1 L 4L L 3 4L 3 L 1 ], A = 1, = 1 1 x 1 y 1 1 x y 1 x 3 y 3 1 x 1 y 1 1 x y 1 x 3 y 3 (37 Fom (19-(3, (35 ad (36, th appoximatios a obtaid by slctig a st of tst fuctios M ad L ad by quiig that ( [L] T ( [M] T t + V + V 1 ( R +f + V dxdy = 0 ( P + V V +f + V + V + + f 1 V dxdy = 0 (39 ( [M] T V + V t + V + V V + V V + P 1 ( V R + V + f 1 V +f + V V +f + V + V V dxdy = 0 (40 ( [M] T Tg + t + + V + V T g 1 ( T g RP +α( T g T s ( [M] T Ts t +γ( T s T g T g=t g = V =V T g + T g dxdy = 0 (41 ( T s β T g=t g = V =V + T s V T g=t g = V =V V T g dxdy = 0 (4 wh patial itgal a usd fo (38, th svth tm of (39 ad (40, th sixth tm of (41 ad th 84

7 Jauay 008 Tmpatu Distibutio Cotol of Racto Fuac by Stat Spac Mthod usig FEM Modlig scod tm of (4. Th 4-6th tms of (4 a appoximatd by T g =T g = [L] = V =V V V Tg=T g1 = 1 V =V 1 T g =T g = [L] = V =V T g=t g1 = 1 V =V 1 V T g =T g = [L] = V =V T g=t g1 = 1 V =V 1 Tg=T g = V =V T g=t g = V =V T g=t g = V =V V Tg=T g3 = 3 V =V 3 T g=t g3 = 3 V =V 3 T g=t g3 = 3 V =V 3 T T T (43 Fom (38-(43, cosid all tiagula lmts i Fig.8-10 ad bouday coditios, that is i = 0 (i : umb at wall V i = 0 (i : umb at wall P i = 0 (i = 6 = i1, 30 = i, V 3 = V i1, V 34 = V i T g 6 = 0, T g 30 = 0, T g 3 = 0, T g 34 = 0 (44 ad suppos that stat vcto x : (99 1 ad cotol vcto u : (4 1 a x = [x 1 x ] T (99 1 x 1 = [x 11 x 1 x 31 x 41 ] (88 1 x = [ P 1 P 3 P 1 ] (11 1 x 11 = [ T g1 T g5 T g7 T g8 T g9 T g31 T g33 T g35 ] (31 1 x 1 = [ T s1 T s5 ] (5 1 x 31 = [ x 41 = [ V 3 V 7 V 8 V ] (16 1 V 7 V 8 V 9 ] (16 1 u = [ i1 i V i1 V i ] T, (4 1 (45 w gt Eẋ = Ax + Bu : E(99 99, A(99 99, B(99 8 [ ] [ ] x 1 E 11 0 x =, E = x 0 0 [ ] [ ] A 11 A 1 B 1 A =, B = (46 A 1 A B wh A = 0. Sic E is t osigula matix, (46 is t calld stat quatio usd widly i cotol dsig. Equatio (46 is calld dscipto systm[14]. Dscipto systm is costuctd bcaus th a t tmpoal difftiatio of P i (19-(1. If dscipto systm of (46 is impuls cotollabl ad fiit dyamics stabilizabl, it is possibl to dsig cotol systm[14]. But, quatio (46 is t impuls cotollabl. So, i th followig subsctio, w will tasfom dscipto systm of (46 ito stat quatio by usig galizd ivs matix[15] so that cotol systm ca b dsigd. Fig. 8: Ptubatio vaiabls of P abls of Fig. 9: Ptubatio vai- 3.3 Applicatio of Galizd Ivs Matix to Dscipto Systm[15] Fom (46, w gt A x = A 1 x 1 B u (47 Sic A = 0, galizd ivs matix A of A is 0. So, fom (46 ad (47 w hav that x = A ( A 1 x 1 B u = 0 (48 ad istig ito (46, w gt th stat quatio x 1 = E 1 11 A 11 x 1 + E 1 11 B 1 u (49 85

8 Tadafumi NOTS t al. MEM.FAC.ENG.OKA.NI. Vol. 4 Fig. 10: Ptubatio vaiabls of T s LQG cotol systm of acto fuac modl is show i Fig.1. LQG cotol systm is costuctd by (50, wh K is optimal gulato gai, L is kalma filt gai, ˆx is stimatd ital stat valu of acto fuac modl. I th followig, flow of cotol is show. Stp1 I stat vaiabls of acto fuac modl a stimatd by kalma filt ad th masud data of istumtatios show i Fig.3. Stp Basd o optimal gulato ad i stat vaiabls stimatd i Stp1, vlocity of blowig gass fom tuys is modifid i od to match iitial distibutio of acto fuac modl to dsid o. wh vlocity of blowig gass fom tuys is ( i1, i, V i1, V i, iitial distibutio is (P (x, y, 0, V (x, y, 0, T g (x, y, 0, T s (x, y, 0, dsid distibutio is (P (x, y V (x, y T g (x, y, T s (x, y ad (P V T g, T s is a stady stat solutio of (-(16. Fig. 11: A typical tiagula lmt _[`Qa R^b SXWXY Z[\]XYS^SXWXY + E cd$f "#$ F! "#$ q D B CA / / / -, NO! "#$ PQRSQT VRWXY ad suppos that E 11 1 A 11 is A ad E 11 1 B 1 is B ad cosid istumt placmt show i Fig.3, w gt th stat spac modl ẋ = Ax + Bu : A(88 88, B(88 8 y = Cx : C(13 88 x = x 1 T (13 1 u = [ i1 i V i1 V i ] T (4 1 I JKL I JKM &( * '(% ghf Od$f "#$ kldf"df Od$f ml!o! "#$ H G? < =>; ol!omop"fold ml Od$f Fig. 1: LQG optimal cotol systm ifoj"f "#$ : foj"# $#"fl y = [ T s1 T s6 T s10 T s11 T s15 T s1 T s5 3 V 3 ] T (13 1 (50 Sic (50 is stabilizabl ad dtctabl[9], lia cotol thoy bcoms availabl. 4 CONTROL SYSTEM OF REACTOR FRNACE LQG cotol[10] is adoptd i od to dalt with poblm dsciptio i xt sctio. Th dtaild xplaatio of LQG cotol is giv i appdix B. 5 NMERICAL EXPERIMENT I this sach, tmpatu distibutio cotol of acto fuac modl is aimd by usig acto fuac modl costuctd i sctio. I this sctio, umical sults of LQG cotol a show. Poblm dsciptio is show i th followig. Vlocity ( i1, i, V i1, V i of blowig gass fom tuys is dcidd so that iitial distibutio (P (x, y, 0, V (x, y, 0, T g (x, y, 0, T s (x, y, 0 i Fig ca b cotolld to dsid distibutio (P V T g T s i Fig

9 Jauay 008 Tmpatu Distibutio Cotol of Racto Fuac by Stat Spac Mthod usig FEM Modlig Iitial tmpatu distibutio i Fig.15 ad 16 is high tha dsid o i Fig.6 ad 7 totally. Suppos that Q, W, R ad V of appdix B a Q = W = diag( lmts {}}{ 10,, 10 3 lmts {}}{ 0.1,, 0.1 : (88 R = V = diag(1, 1, 1, 1 : (4 4 Sic th puposs of this sach is to stimat i tmpatu valus of acto fuac modl ad to match tmpatu distibutio of acto fuac modl to dsid o, two objctiv fuctios a dfid as Fig. 13: Iitial distibutio of pssu Fig. 14: Iitial distibutio of gas flow ˆf(t = 35 i=1 i 6,30,3,34 T gi (t ˆT gi (t + 5 i=1 T si (t ˆT si (t (51!" #!"! " #! " wh T gi (t ad T si (t a tmpatu valus of acto fuac modl at od poits of Fig.9 ad 10, ˆT gi (t ad ˆT si (t a stimatd tmpatu valus by LQG cotol at abov poits. Equatio (51 psts o stimato of tmpatu valus at abov poits. Fig. 15: Iitial distibutio of gas tmpatu Fig. 16: Iitial distibutio of matial tmpatu f(t = 13 1 i=1 j=1 T g i,j T g i,j i=1 j=1 T s i,j T s i,j (5 wh Tg i,j ad T s i,j a dsid tmpatu valus at (i, j. Equatio (5 psts o of tmpatu valus at all od poits. Th vlocity chags of blowig gass fom tuys a show i Fig.17 ad 18. Th chags of th two objctiv fuctios a show i Fig.19. Fom Fig.17 ad 18, th vlocity of blowig gass fom tuys is opatd i about 15[-]. Wh gass blow fom tuys as show i Fig.17 ad 18, th two objctiv fuctios dcas with tim. Th optimizd tmpatu distibutio is show i Fig.0 ad 1. So, i tmpatu valus of acto fuac modl a stimatd ad tmpatu distibutio appoachs th dsid o by usig LQG cotol systm. 6 CONCLSION Fig. 17: i1 ad V i1 I this pap, th acto fuac simulato was costuctd, which could calculat gas tmpatu, solid tmpatu, gas flow ad pssu distibutio i th fuac. Th simulato had istumtatios oly a th fuac wall ad cotol iputs at oly boud- 87

10 Tadafumi NOTS t al. MEM.FAC.ENG.OKA.NI. Vol. 4 applid fo dsig of cotol systm. Futh, LQG cotol was adoptd i od to dal with th poblm dsciptio. W cofimd that tmpatu distibutio of th simulato was stimatd ad cotolld by LQG cotol systm. I th futu, it will b cssay to impov stimatd i tmpatu valus ad gulat tmpatu distibutio to th dsid o i a shot tim. Fo th pupos, w will adopt oth lia cotol thois such as H cotol ad H cotol. Fig. 18: i ad V i REFERENCES [1] T. Kawamua: Fluid aalysis I, Asakua shot (1996, [] T. Shibuta: Mast Thsis. Okayama iv., (003 [3] K. Ishimau: Mast Thsis. Okayama iv., (005 [4] T. Takda: Gaduatio Thsis. Okayama iv., (006 [5] K. Takatai, T. Iada ad K. Takata: ISIJ Itatioal, (001, Fig. 19: Chags of ˆf(t ad f(t [6] I. Imai: Fluid Dyamics, Shokabo (1973, [7] S. Egu: Chmical Egiig Pogss, 48- (195, ! " #! " Fig. 0: Th optimizd distibutio of gas tmpatu(t = 50[ ]! " #! " Fig. 1: Th optimizd distibutio of matial tmpatu(t = 50[ ] ais. To mak th cotol systm dsig, stat spac modl was divd by applyig FEM to th simulato. Though this pocdu, lia cotol thoy could b [8] M. Ii ad Y. Ii: Patial Difftial Equatio, Asakua shot (1983, [9] K. Z. Liu: Lia Robust Cotol, Cooasha (00, [10] S. Hoso: Systm ad Cotol, Ohmsha (1997, [11] A. Fujimoi: Robust Cotol, Cooasha (001, [1] O. M. Aamo ad M. Kstic: Flow Cotol by Fdback, Spig (003, [13] G. Yagawa: Bgiig FEM of Flow ad Hat Tasf, Baifuka (1983, [14] T. Katayama: Optimal Cotol of Lia Systm, Kidai Kagaku sha (1999, [15] H. Kimua: Lia Algba, ivsity of Tokyo Pss (003,

11 Jauay 008 Tmpatu Distibutio Cotol of Racto Fuac by Stat Spac Mthod usig FEM Modlig Appdix A I this sach, fiit-diffc mthod is usd to calculat gas flow ad tmpatu distibutio of acto fuac modl. Flow chat of th calculatio is show i Fig.. Th gas flow is calculatd by MAC mthod. Ths calculatios a cotiud util th covgc coditios a fulfilld. Th covgc coditios a dfid as f P = max P (x, y, P P (x, y, P 1 < ( P = 1, f = max (x, y, V (x, y, V 1 < ( V = 1, f V = max V (x, y, V V (x, y, V 1 < ( V = 1, f Tg = max T g (x, y, 10 Tg T g (x, y, 10( Tg 1 < 0.1 ( Tg = 1, f Ts = max T s (x, y, 10 Ts T s (x, y, 10( Ts 1 < 0.1 ( Ts = 1, (53 Fig. 4: Coditio of f fo dsid distibutio Fig. 5: Coditio of D p fo dsid distibutio Tabl. 3: Coditio 1 of dsid distibutio dt 0.01 b up 1 dx dy 0.1 T out 0 R 100 Q P 0.5 Q 1000 α γ 10 K 1 50 β 0.01 E a b 1 R 8.31 Tabl. 4: Coditio of dsid distibutio Th coditios a show i Fig.3-5, Tabl.3 ad 4 i od to gt a stady stat solutio of Fig.4-7 i subsctio 3.1. Racto fuac modl calculats th disct valus i Tabl. at dt(= 0.01 itvals ud th coditios of Tabl.3. i1, V i1, V i 0.1 i -0.1 T g i1, T g i, T g i3, T g i4 100 i,j 0 V i,j 0 0 (xcpt ilt Tg i,j 0 0 (xcpt ilt Ts i,j ; 5 7 A < 34 : => G B F E CD JKJLM ( % $ & ' # O 1 0 -,. / + HI N " *! Fig. 3: Coditio of f 1 fo dsid distibutio Fig. : Flow chat to calculat mathmatical modl Appdix B I LQG cotol, stat spac modl is assumd to b subjctd to distubacs tig additivly i quatio. Thus, w hav ẋ = Ax + Bu + w : A(, B(m y = Cx + v : C(l (54 wh w is a distubac ad v is masumt ois. w ad v a assumd to b ucolatd Gaussia 89

12 Tadafumi NOTS t al. MEM.FAC.ENG.OKA.NI. Vol. 4 stochastic pocsss with zo mas ad covaiac E{w(t} = 0 E{v(t} = 0 {[ ] w(t [ ] } E w(τ T v(τ T v(t [ ] W 0 = δ(t τ (55 0 V spctivly, wh W ad V a costat matics. E{ } dots th xpctatio opato ad δ dots th dlta fuctio. Th LQG cotol poblm is to fid th fdback u that miimizs th cost fuctioal J = E { lim T 1 T ( x T Qx + u T Ru dt T t 0 } (56 wh Q ad R a costat wightig matics satisfyig Q = Q T 0 ad R = R T > 0. Th poblm has a uiqu solutio giv by u = K ˆx, K = R 1 B T P ˆx = Aˆx + Bu + L(y C ˆx, L = SC T V 1, ˆx(t 0 = 0 (57 wh P ad S uiquly solv th algbaic Riccati quatios A T P + P A + Q P BR 1 B T P = 0, AS + SA T + W SC T V 1 CS = 0. (58 W show that th xists a admissibl LQG cotol systm, if th followig two coditios hold (A.1 (A,B ad (A T,C T a stabilizabl, that is ak[si A B] =, ak[si A T C T ] =, R[s] 0, s C (59 (A. ( Q,A ad ( W,A T a dtctabl, that is ak [ ] [ si A =, ak Q si A T W ] =, R[s] 0, s C (60 90