Pressure and Flow Control of aPulverized Coal Injection Vessel EXAMENSARBETE

EXAMENSARBETE

Pressure and Flow Control of a Pulverized Coal Injection Vessel

CIVILINGENJÖRSPROGRAMMET Institutionen för Systemteknik Avdelningen för Reglerteknik

1997:045 • ISSN: 1402 - 1617 • ISRN: LTU - EX - - 1997/45 - - SE

WOLFGANG BIRK

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516 H{lvwlqj frqwuro xqlw 45 dqg vwdqgv e|1 Li wkh suhvvxuh frqwuro ydoyh dqg wkh rz frqwuro ydoyh kdyh forvhg/ wkh yhqwlodwlrq ydoyh rshqv dqg ghsuhvvxul}hv wkh lqmhfwlrq yhvvho1 Rqfh wkh suhvvxuh kdv gursshg grzq wr qrupdo/ wkh lqohw ydoyh rshqv dqg frdo srzghu lv oohg lq djdlq1 Lq Dsshqgl{ E wkh rz fkduwv ri wkh lqmhfwlrq surfhvv duh jlyhq1

516 H{lvwlqj frqwuro xqlw

Wkh frqwuro xqlw ri wkh qh frdo lqmhfwlrq sodqw frqvlvwv ri wzr lqghshqghqw orrsv1 Rqh orrs lv iru lqmhfwlrq yhvvho suhvvxuh vwdelol}dwlrq/ dqg wkh rwkhu lv iru frqwuroolqj wkh frdo pdvv rz wr wkh eodvw ixuqdfh1 Wkh surfhvv zrunlqj f|foh gxudwlrq ghshqgv rq wkh dfwxdo frdo pdvv rz vhw srlqw1 Orjlfdoo|/ d kljkhu vhw srlqw vkruwhqv wkh f|foh wlph1 Qrupdoo|/ wkh f|foh wlph lv derxw 73 plqxwhv/ zklfk phdqv wkdw wkh lqmhfwlrq skdvh wdnhv 2f plqxwhv/ dssur{lpdwho|1

51614 Vhqvruv

Vlqfh wkh zkroh sodqw kdv pdq| surfhvvhv uxqqlqj vhsdudwh iurp hdfk rwkhu/ sduwo|

zlwkrxw nqrzohgjh ri wkh rwkhu surfhvvhv/ wkh| duh glvwxuelqj hdfk rwkhu1 Ixuwkhupruh pdq| sduwv ri wkh sodqw duh phfkdqlfdoo| frxsohg/ iru h{dpsoh wkh wzr lqmhfwlrq yhvvhov duh frxsohg ryhu wkh lqmhfwlrq slsh1 Wklv uhvxowv lq huuruv lq wkh phdvxuhphqw/ zklfk pd| qrw eh qhjohfwhg1

Wkhuhiruh lw kdv wr eh glvfxvvhg/ zkdw vljqdov duh xvdeoh iru surfhvv lghqwlfdwlrq dqg krz wkh| duh dftxluhg1 Pdlqo| wkh surfhvv kdv irxu lqsxw vljqdov1 Wkh| duh =

 suhvvxuh lq wkh yhvvho

 suhvvxuh lq wkh qlwurjhq qhw

 zhljkw ri wkh lqmhfwlrq yhvvho

 suhvvxuh lq wkh lqmhfwlrq slsh1

Doo suhvvxuh vhqvruv duh glhuhqfh suhvvxuh phwhuv1 Wkh| duh fdoleudwhg rq d suhv0 vxuh udqjh iurp f wr Sff &@/ dqg kdyh dq dffxudf| ri fD I1 ]hur lv fdoleudwhg rq qrupdo rxwvlgh suhvvxuh1 Wkh suhvvxuh vljqdo lv olqhduo| frqyhuwhg lqwr d fxuuhqw vljqdo1 Wdeoh 516 vkrzv krz wkh vljqdo lv frqyhuwhg1

516 H{lvwlqj frqwuro xqlw 46 Suhvvxuh vljqdo Fxuuhqw vljqdo

f & @ e 6

Sff & @ 2f 6

Wdeoh 516= Suhvvxuh vljqdo frqyhuvlrq Pdvv vljqdo Fxuuhqw vljqdo

f &} f 6

Sfff &} 2f 6

Wdeoh 517= Pdvv vljqdo frqyhuvlrq

Wkh zhljkw vhqvru lv phwhulqj wkh pdvv ri wkh lqmhfwlrq yhvvhov1 Dv wkh lqmhfwlrq yhvvhov duh frxsohg zlwk hdfk rwkhu ryhu wkh lqmhfwlrq slsh/ wkh phwhuhg zhljkw lv glvwxuehg e| wkh zhljkw ri wkh rwkhu yhvvho1 Wkh zhljkw vhqvru lv fdoleudwhg rq wkh hpsw| lqmhfwlrq yhvvho wr eh }hur1 Wkh pdvv vljqdo lv olqhduo| frqyhuwhg lqwr d fxuuhqw vljqdo/ wrr1 Wdeoh 517 vkrzv krz wkh vljqdo lv frqyhuwhg1 Reylrxvo| wklv frqyhuvlrq lv glhuhqw dqg wkh vljqdo iurp wkh vhqvru kdv d glhuhqw udqjh1

Doo fxuuhqw vljqdov duh dpsolhg ehiruh vdpsolqj e| d exhu dpsolhu/ zlwk jdlq rqh1 Wklv lv grqh ehfdxvh ri vdihw| uhdvrqv/ lq rughu wr vhsdudwh wkh flufxlwv jdoydqlfdoo|1 Wkh xvhg exhu dpsolhuv kdyh dq huuru ohvv wkdq f2I1

51615 Dfwxdwruv

Wkh suhvvxuh dqg rz frqwuro ri wkh lqmhfwlrq yhvvhov lqyroyhv wzr dfwxdwruv1 Wkh suhvvxuh frqwuro ydoyh +SFY, dqg wkh rz frqwuro ydoyh +IFY,1 Wkh SFY lv d vwdqgdug ydoyh/ zkloh wkh IFY lv d vshfldo ghvljq iurp EPK/ ^48`1 Erwk ydoyhv kdyh d srvlwlrq frqwuro zklfk lv qrw glvfxvvhg forvhu1 Lw fdq eh dvvxphg wkdw wkh srvlwlrq frqwuro kdv d vx!flhqw idvw dqg dffxudwh ehkdylru1 Odwhu wkh ehkdylru ri wkhvh lqqhu orrsv lv dvvxphg wr eh sursruwlrqdo1

Vlqfh fkdudfwhulvwlf fxuyhv duh qrw dydlodeoh iru wkh SFY/ rqo| d sulqflsoh fkdudf0 whulvwlfv fdq eh jlyhq1 Sdudphwhuv fdq eh rewdlqhg iurp h{shulphqwv dqg lghqwlfdwlrq dv vkrzq lq ^;`1 Iljxuh 614 vkrzv d sulqflsoh fkdudfwhulvwlf1 Reylrxvo| wkh SFY kdv d ghdg0}rqh dqg dq h{srqhqwldo fkdudfwhulvwlf/ dv lw lv dq htxdo shufhqwdjh ydoyh1 Lw vkrxog eh srvvleoh wr olqhdul}h wklv fkdudfwhulvwlf durxqg d zrunlqj srlqw1

D orrn dw wkh sulqflsoh vwuxfwxuh ri wkh ydoyh/ jxuh 518/ vkrzv zk| wkh ydoyh kdv

516 H{lvwlqj frqwuro xqlw 47

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

valve opening

normalized flow

Iljxuh 517= Sulqflsoh fkdudfwhulvwlf fxuyh ri wkh SFY

d ghdg0}rqh1 Wkh h{leoh ulqj vhdo zloo suhyhqw qlwurjhq sdvvlqj wkurxjk wkh ydoyh/

dowkrxjk wkh ydoyh lv rshq1 Wklv lv ydolg iru rshqlqjv ehwzhhq }hur dqg DI1

Dv wkh IFY lv d vshfldo ghvljq e| EPK/ qr fkdudfwhulvwlf fxuyh lv dydlodeoh1 Rqo|

d sulqflsoh vwuxfwxuh ri wkh IFY fdq eh jlyhq lq jxuh 5191 Lw ehfrphv fohdu/ wkdw wkhuh lv d qrq0olqhdu frkhuhqfh ehwzhhq wkh ydoyh rshqlqj dqg wkh uhvxowlqj pdvv rz wkurxjk wkh ydoyh1 Lw zloo eh vkrzq odwhu wkdw d olqhdu prgho zrunv vx!flhqwo| jrrg iru wklv ydoyh1 Wkh IFY kdv dovr d ghdg0}rqh/ wklv uhvxowv iurp wkh volgh pryhphqw1 Lq rughu wr wljkwhq/ wkh ydoyh kdv wr sdvv ryhu wkh rshqlqj frpsohwho| dqg iru vdihw|

uhdvrq d elw pruh1 Wkh fdoleudwhg ydoyh zloo rshq sk|vlfdoo| iru frqwuro vljqdov ehwzhhq 2 dqg ffI1 Wkh uhvxowlqj qrq0olqhdulw| fdq qrw eh qhjohfwhg1

51616 Wkh suhvvxuh frqwuro

Wkh suhvvxuh frqwuroohu lv sodfhg lq VdwwFrq/ forvh wr wkh frdo lqmhfwlrq yhvvhov1 Wkh f|foh wlph iru wkh suhvvxuh orrs lv fr1 Wkh lqsxw iurp wkh suhvvxuh vhqvru lv vdpsohg zlwk wkh f|foh wlph dqg dftxluhg e| dq D2G Fdug zklfk frqyhuwv wkh dqdorjxh fxuuhqw vljqdo lqwr d gljlwdo vljqdo1 Ilowhulqj lv suredeo| dssolhg exw qrw phqwlrqhg ru ghvfulehg lq wkh frqwuro xqlwv ghvfulswlrq1 Wkh xvhg frqwuroohu lv d txdvl frqwlqxrxv SL frqwuroohu

516 H{lvwlqj frqwuro xqlw 48

moving direction of the piston

high pressure low

pressure

flexible ring seal

Iljxuh 518= Sulqflsoh vwuxfwxuh ri wkh SFY +vlghylhz,

moving direction of the slide pipe hole

slide

Iljxuh 519= Sulqflsoh vwuxfwxuh ri wkh IFY +wrsylhz,

516 H{lvwlqj frqwuro xqlw 49

- PI

1

t ta

u_{N ,max} u_{N ,min}

u_N p_ref

p

[ ]^kPa

[ ]kPa

Iljxuh 51:= Eorfn gldjudp ri wkh suhvvxuh frqwuro

zlwk dq lqwhjudwlrq wlph ri Dr dqg d jdlq ri D1 Wkh jdlq kdg ehhq lqfuhdvhg dv d uhvxow ri wkh uhfrpphqgdwlrqv iurp wkh suh0vwxglhv grqh e| wkh FHJ1 Lw zdv vkrzq wkdw d kljkhu jdlq/ lq iruphu wlphv 6/ lqfuhdvhv wkh shuirupdqfh ri wkh frqwuroohu1

Wkh frqwuroohu rxwsxw lv olplwhg ehwzhhq f dqg .DI/ gxulqj suhvvxul}dwlrq1 Wkh uhdvrq iru wklv olplwdwlrq lv wkdw wkh fdsdflw| ri wkh qlwurjhq qhw lv wrr orz1 Li wkh SFY lv rshqhg wrr pxfk gxulqj wkh suhvvxul}lqj skdvh/ wkh suhvvxuh lq wkh qlwurjhq qhw gursv1 Wklv fdxvhv suhvvxuh rvfloodwlrqv gxulqj wkh lqmhfwlrq skdvh ri wkh rwkhu yhvvho1 Vlqfh wkh frqwuroohu lv douhdg| dfwlyh gxulqj wkh suhvvxul}lqj skdvh/ lw kdv wr iroorz d udps1 Wklv lv uhdol}hg e| udpslqj wkh vhw srlqw ydoxh iurp }hur xs wr vhw srlqw ydoxh gxulqj wkh lqmhfwlrq skdvh1 Xvlqj d {hg rshqlqj ydoxh iru wkh SFY zrxog kdyh wkh vdph uhvxow1 Dq rshqlqj ydoxh uhvxowlqj lq wkh vdph wlph/ qhhghg wr suhvvxul}h wkh yhvvho/ frxog eh frpsxwhg hdvlo| dqg zrxog uhgxfh ydoyh zduh0r/ dv wkh frqwuroohu lv rvfloodwlqj d orw lq wkh ehjlqqlqj1

Iljxuh 51: vkrzv wkh eorfn gldjudp ri wkh dfwxdo xvhg suhvvxuh frqwuro dqg jxuh 51; vkrzv krz wkh suhvvxuh frqwuroohu shuirupv1 Lw lv fohdu wkdw wkh frqwuroohu kdv d orqj fruuhfwlrq wlph1

51617 Wkh pdvv rz frqwuro

Dv wkh pdvv rz fdqqrw eh phdvxuhg/ wkh frqwuro xqlw xvhv wkh pdvv orvv ri wkh lqmhfwlrq yhvvho gxulqj wkh lqmhfwlrq skdvh dv lqsxw vljqdo1 Wkh pdvv orvv lv wkhq ghulydwhg1 D

uvw rughu glhuhqfh lv xvhg wr dssur{lpdwh wkh ghulydwlyh1 Wkh uhvxowlqj vljqdo lv wkhq orz sdvv owhuhg dqg olplwhg wr d pd{lpxp ghyldwlrq iurp wkh vhw srlqw/ lq rughu wr uhgxfh qrlvh wkdw kdv ehhq dpsolhg ehfdxvh ri wkh glhuhqwldwlrq1 Wklv pdvv rz lv frpsxwhg hyhu| irxuwk vhfrqg1 Wkh frqwuroohu lwvhoi lv d txdvl frqwlqxrxv SL frqwuroohu zlwk lqwhjudwlrq wlph ri  plqxwhv dqg d jdlq ri f.D1 Wkh frqwuroohu iru lqmhfwlrq yhvvho

516 H{lvwlqj frqwuro xqlw 4:

0 100 200 300 400 500 600 700 800 900 1000

−40

−20 0 20

time/s

pressure deviation/kPa

control result

0 100 200 300 400 500 600 700 800 900 1000

0 20 40 60 80

time/s

valve opening/%

controller output

Iljxuh 51;= Shuirupdqfh ri wkh suhvvxuh frqwuroohu

u_C 0 9.

q⁻¹

1 1−0 5. ⋅q⁻¹

- - kg

s 4









t h



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t h



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[ ]^kg

[ ]^kg

1000Nm³ h



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m_C &mC

g Nm³



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1000 • •

kg Nm³

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Iljxuh 51< vkrzv d eorfn gldjudp ri wkh dfwxdo xvhg pdvv rz frqwuro/ dqg jxuh 5143 vkrzv krz wkh pdvv rz frqwuroohu shuirupv1 D orrn dw wkh ghyldwlrq ri wkh pdvv rz vkrzv wkdw wkh dfwxdo frqwuroohu lv qrw deoh wr gulyh wkh frqwuro huuru wr }hur1 Wklv lv dovr vxvwdlqhg/ orrnlqj dw wkh ghyldwlrq ri wkh pdvv orvv iurp wkh lghdo pdvv orvv ri wkh lqmhfwlrq yhvvho/ zklfk lv glvsod|hg lq jxuh 51441

516 H{lvwlqj frqwuro xqlw 4;

0 100 200 300 400 500 600 700 800 900 1000

60 65 70 75 80

time/s

valve opening/%

controller output

0 100 200 300 400 500 600 700 800 900 1000

−10

−5 0 5 10 15

time/s

mass flow deviation/(t/h)

control result

Iljxuh 5143= Shuirupdqfh ri wkh pdvv rz frqwuroohu

0 100 200 300 400 500 600 700 800 900 1000

−150

−100

−50 0 50 100

time/s

mass deviation/kg

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Qrqolqhdu sk|vlfdo prgho

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614 Dvshfwv rq sqhxpdwlf frqyh|lqj ri vrolgv

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615 Wkh prgho 53

u u

m

m V T p

p

p

C N

N

C

N

T

Iljxuh 614= Vfkhpdwlf gudzlqj ri dq lqmhfwlrq yhvvho wkh erwwrp1

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615 Wkh prgho

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vr wkdw wkh whpshudwxuh A lv frqvwdqw1

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6 ' &  } T RERc R  } T E +614,

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615 Wkh prgho 54

6 ' 6 n 6 +616,

6 ' 6 n 6 +617,

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R  T ' 6  -  A +618,

Dvvxplqj A ' f/ wkh ghulydwlyh ri +618, uhvxowv lq

R  T n R  T ' 6  -  A +619, Lqwurgxflqj wkh yroxph edodqfh dqg lwv ghulydwlyh

T ' T  T +61:d,

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4 +61:e,

T '  6

4 +61;,

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4

+61<,

Fkrrvlqj ' dRc 6o^A dv vwdwh yhfwru/ ) ' dRc 6o^A dv rxwsxw yhfwru dqg  ' dc o^A dv lqsxw yhfwru/ d qrq0olqhdu vwdwh vsdfh prgho iru wkh lqmhfwlrq surfhvv lv ghqhg1 Dv wkh ydoyh ixqfwlrqv }_{ T R}/ }_{ T } dqg }_{8 T R}/ }_{8 T } duh xqnqrzq dqg fdqqrw eh ghulyhg iurp h{lvwlqj gdwd vkhhwv/ wkhvh ixqfwlrqv kdv wr eh lghqwlhg iurp surfhvv gdwd1 Ixuwkhupruh wkh idfwruv &/ &_ kdyh wr eh ghwhuplqhg iurp surfhvv gdwd1 Dv ghvfulehg lq ^;`/ lw lv srvvleoh wr olqhdul}h wkh vwdwh vsdfh prgho e| xvlqj qrq0 olqhdu lqsxw dqg rxwsxw wudqvirupdwlrqv1 Lq dgglwlrq/ wkh ydoyh ixqfwlrqv dqg idfwruv duh ghulyhg wkhuh1

Vlqfh wkh ydoyhv rshudwh durxqg d txlhvfhqw srlqw gxulqj wkh lqmhfwlrq skdvh/ lw vkrxog eh srvvleoh wr olqhdul}h wkh prgho ru jdlq d prgho iurp d gluhfw lghqwlfdwlrq1

Fkdswhu 7

Lghqwlfdwlrq

Ehfdxvh ri wkh kdug qrq0olqhdulwlhv lq wkh sk|vlfdo prgho d gluhfw lghqwlfdwlrq ri d olqhdu prgho/ wkdw dssur{lpdwhv wkh qrq0olqhdu prgho/ lv hpsor|hg1 Lq wklv vhfwlrq/ lw lv vkrzq wkdw d olqhdu prgho ghulyhg iurp lqsxw2rxwsxw gdwd zrunv vx!flhqwo| jrrg iru wkh lqmhfwlrq skdvh1 Dq douhdg| h{lvwlqj prgho iru wkh sodqw lv ydolgdwhg xvlqj qhz gdwd1

Iru lghqwlfdwlrq wkh Lghqwlfdwlrq Wrroer{ iru Pdwode lv xvhg1^U

714 Phdvxuhphqw gdwd

Lq rughu wr lghqwli| wkh sodqw/ lqsxw2rxwsxw gdwd lv qhhghg1 Dw VVDE Wxqqsoèw wzr glhuhqw srvvlelolwlhv h{lvw wr orj gdwd1 VVDE kdv d gdwd0edvh/ fdoohg PÙVV/ dw lwv glvsrvdo1 Wkh gdwd0edvh lv sodfhg rq d vhuyhu zklfk lv frqqhfwhg wr doo frqwuro xqlwv dw VVDE1 Doo phdvxuhphqw gdwd lv wudqvihuuhg wr wklv vhuyhu dqg vwruhg wkhuh1 Wkh gdwd lv xvhg iru vwdwlvwlfv dqg dqdo|vlv1 Wkh orfdo qhwzrun doorzv wkh grzqordglqj ri phdvxuhphqw gdwd wr hyhu| SF dw VVDE1

Wkh gdwd lwvhoi rq wkh vhuyhu lv glylghg lqwr wzr jurxsv = plqxwh phdq ydoxhv dqg glyhuvh wlph edvhg ydoxhv1 Wkh plqxwh phdq ydoxhv duh qrw vxlwdeoh iru lghqwlfdwlrq/

ehfdxvh wkh vdpsoh udwh lv wrr vorz dqg wkh ydoxhv duh phdq ydoxhv1 Wkhuhiruh/ d qhz wlph edvh iru wkh qhhghg ydoxhv iurp wkh frdo lqmhfwlrq sodqw kdv ehhq lpsohphqwhg lq PÙVV1 Wkhvh ydoxhv duh orjjhg dw d vdpsoh udwh ri Dr/ wkh pd{lpxp udwh doorzhg e|

wkh frqwuro xqlw DoFrqw1 Dv DoFrqw kdv qr sulrulw| rq orjjlqj gdwd dqg wudqvihuulqj lw wr PÙVV/ wkh vdpsoh udwh lv ydu|lqj1 Wklv uhvxowv lq vdpsoh udwhv ydu|lqj ehwzhhq  dqg fr/ zklfk pdnhv lghqwlfdwlrq gl!fxow1 Gdwd vhwv kdyh wr eh lqwhusrodwhg dqg

55

714 Phdvxuhphqw gdwd 56

0 500 1000

0 2 4 6 8

time/s

valve opening/V

PCV

0 500 1000

6 6.5 7 7.5 8

time/s

valve opening/V

FCV

0 500 1000

0 2 4 6 8 10

time/s

mass/V

mass of the vessel

0 500 1000

5.7 5.8 5.9 6

time/s

pressure/V

pressure in the vessel

Iljxuh 714= Sodqw lqsxw dqg rxwsxw vljqdov gxulqj lqmhfwlrq skdvh1 SFY dqg IFY duh lqsxwv/ suhvvxuh dqg pdvv duh rxwsxwv

pd| eh uh0vdpsohg1 Ixuwkhupruh/ wkhuh lv qr lqirupdwlrq derxw wkh owhulqj ri wkh gdwd1 Ehfdxvh ri wkhvh uhdvrqv/ wkh gdwd iurp PÙVV lv qrw xvhg iru wkh lghqwlfdwlrq1 Vlqfh d uh0ghvljqhg frqwuroohu vkrxog eh whvwhg lq uhdo0wlph rq wkh sodqw/ doo lpsru0 wdqw vljqdov vkrxog eh dffhvvleoh iru d SF zlwk D2G0Erdug1 Lw dsshduhg wr eh srvvleoh wr jhw dffhvv wr wkh vljqdov iurp2wr wkh ydoyhv/ dqg wr wkh vhqvruv iru pdvv dqg suhvvxuh ri wkh lqmhfwlrq yhvvho1 E| xvh ri vljqdo frqyhuwhuv wkh fxuuhqw vljqdov duh wudqvodwhg lqwr yrowdjh vljqdov iru wkh D2G0Erdug1 Wkh orz sdvv owhulqj e| wkh vljqdo frqyhuwhuv kdv ehhq vxssuhvvhg1 Iru lghqwlfdwlrq sxusrvhv irxu vljqdov duh orjjhg =

 suhvvxuh lq wkh yhvvho

 pdvv ri wkh yhvvho

 dfwxdo ydoxh ri wkh SFY

 dfwxdo ydoxh ri wkh SFY Iljxuh 714 glvsod|v wkh vljqdov1

Ehfdxvh ri wkh qrq0olqhdulw| ri wkh IFY wkdw kdv ehhq ghvfulehg deryh/ wkh lqsxw vljqdov duh wudqviruphg1 Lq wklv zd|/ wkh qrq0olqhdulw| ehfrphv sduw ri wkh sodqw1 Wklv

715 Ydolgdwlrq ri wkh h{lvwlqj prgho 57 Txdqwlw| orjjhg zlwk SF orjjhg zlwk PÙVV

Ydoyh rshqlqj T fI

Suhvvxuh T S& @

Pdvv T Se2D&}

Pdvv rz ^T_r 22_^|

Wdeoh 714= Frqyhuvlrq ri vljqdov lq yrowv wr sk|vlfdo xqlwv

lv wkh uhdvrq zk| doo rxwsxw vljqdov wr wkh IFY kdyh wr eh wudqviruphg zlwk wkh lqyhuvh wudqvirupdwlrq1 Wkh wudqvirupdwlrq lv d vfdolqj ri wkh vljqdo iurp wkh IFY1 Dv wkh ghdg0}rqh ri wkh ydoyh lv ehwzhhq f dqg 2I/ wkh vljqdo udqjh iurp f wr ffI lv vfdohg lqwr wkh udqjh 2 wr ffI1 Wklv ohdgv wr wkh iroorzlqj wudqvirupdwlrq

_{c8 T} ' E ' 2 n f.b +714,

zkhuh  kdv d udqjh iurp f wr ffI dqg c8 T d udqjh iurp 2 wr ffI1 Wklv phdqv wkdw c8 T lv wkh dfwxdo srvlwlrq ydoxh ri wkh IFY/ wkdw fdq eh phdvxuhg1 Wkh lqyhuvh wudqvirupdwlrq uhvxowv lq

 ' ^3Ec8 T ' c8 T  2

f.b +715,

Wr eh deoh wr frpsduh wkh orjjhg gdwd iurp wkh SF zlwk wkh gdwd orjjhg e| PÙVV/

wkh yrowdjh ydoxh qhhg wr eh frqyhuwhg lqwr sk|vlfdo xqlwv1 Wkh frqyhuvlrq idfwruv fdq eh rewdlqhg iurp d frpsdulvrq ri gdwd orjjhg dw wkh vdph wlph lqvwdqfh1 Wdeoh 714 vkrzv krz wkh yrowdjh vljqdov fdq eh frqyhuwhg lqwr sk|vlfdo xqlwv1

Wkh pdvv rz lv d frpsrvlwh txdqwlw| dqg fdq qrw eh orjjhg dv d vljqdo1 Exw lw lv qhhghg odwhu iru wkh frqwuroohu/ dv d vhw srlqw ydoxh1 Wklv ydoxh pxvw eh jlyhq lq_T

r

1

715 Ydolgdwlrq ri wkh h{lvwlqj prgho

Xqwlo wkh vxpphu 4<<9 wkh wzr lqmhfwlrq yhvvhov zhuh sk|vlfdoo| frqqhfwhg wr wkh qlwur0 jhq qhw zlwk rqo| rqh SFY1 Qdwxudoo|/ rqo| rqh frqwuroohu zdv xvhg iru erwk yhvvhov1 Xvlqj vhyhudo vkxw0r ydoyhv/ wkh SFY zdv frqqhfwhg wr wkh yhvvho qhhghg wr eh frq0 wuroohg1 Gxulqj wkh suhvvxuh kroglqj skdvh/ d olplw vzlwfk dqg rqh vhsdudwh vkxw0r

ydoyh zhuh xvhg wr krog wkh suhvvxuh lq d fhuwdlq udqjh1 Wkh FHJ uhfrpphqghg wkh

716 Prgho iru wkh pdvv 58 xvh ri wzr frqwuroohuv dqg SFY*v1 Edvhg rq wkh uhvxowv ri wkh suh0vwxg| VVDE uhfrq0 vwuxfwhg wkh suhvvxuh frqwuro dqg lpsohphqwhg wzr frqwuroohuv dqg SFY*v1 Qrz hdfk yhvvho lv frqwuroohg frpsohwho| vhsdudwh1

Vlqfh wkh h{lvwlqj prgho zdv ghyhorshg ehiruh wkhvh fkdqjhv kdg ehhq pdgh/ lw lv qhfhvvdu| wr ydolgdwh wklv prgho1 Li lw lv ydolg/ wkh suhvvxuh frqwuroohu/ ghyhorshg gxulqj wkh suh0vwxg|/ fdq vwloo eh xvhg1

Wr ydolgdwh wkh vlqjoh lqsxw2vlqjoh rxwsxw prgho iru wkh suhvvxuh dv rxwsxw dqg wkh SFY rshqlqj dv lqsxw wzr whvwv duh dssolhg/ dv sursrvhg lq ^:/ fkdswhu 44` =

 Frpsduh phdvxuhg dqg vlpxodwhg rxwsxw1 Xvh frpprq vhqvh wr ydolgdwh1 D gdwd vhw glhuhqw iurp wkh lghqwlfdwlrq gdwd vhw lv xvhg1 Wkh phdvxuhg dqg vlpxodwhg rxwsxw duh sorwwhg lq wkh vdph jxuh1 Wkh ehkdylru ri wkh vlpxodwhg dqg wkh phdvxuhg rxwsxw duh frpsduhg1 Frpprq vhqvh lv xvhg wr pdnh d ghflvlrq rq zkhwkhu wkh prgho lv dffhswhg ru uhmhfwhg/ vhh jxuh 7151

 Xvh ri d vwdwlvwlfdo whvw rq wkh suhglfwlrq huuru1

Iurp wkh phdvxuhg dqg wkh vlpxodwhg rxwsxw wkh suhglfwlrq huuru lv ghulyhg1 Li wkh prgho wv dqg wkh qrlvh lv zklwh/ wkh suhglfwlrq huuru lv zklwh qrlvh zlwk }hur phdq1 Wkh dxwrfruuhodwlrq ri wkh suhglfwlrq huuru vkrxog wkhq eh }hur iru doo odjv  9' f1 Ixuwkhupruh/ wkh suhglfwlrq huuru fdq eh furvv fruuhodwhg zlwk wkh lqsxw vljqdo/ wkh| vkrxog eh xqfruuhodwhg1 Wklv whvw lv qrupdoo| qrw vxffhvvixo lq forvhg orrs/ dv wkh lqsxw lv d uhvxow ri wkh rxwsxw1 Zkhwkhu wr uhmhfw ru dffhsw d prgho lv ghwhuplqhg e| d frqghqfh ohyho/ dv wkh fruuhodwlrq ixqfwlrqv duh qrw }hur lq uhdolw|1 ^:/ sdjh 757` ghvfulehv krz wklv frqghqfh ohyho lv frpsxwhg1 Iljxuh 716 vkrzv wkh uhvxow ri wkh vwdwlvwlfdo whvw iru wkh prgho ydolgdwlrq1

Iljxuhv 715 dqg 716 vkrz wkdw wkh h{lvwlqj prgho lv vwloo ydolg/ dowkrxjk wkh vwuxfwxuh ri wkh sodqw kdv ehhq fkdqjhg1 Krzhyhu wkhvh fkdqjhv glg qrw kdyh dq| lq xhqfh lq wkh sodqw ehkdylru1 Wkh h{lvwlqj prgho kdv wkh iroorzlqj prgho htxdwlrq =

RE| ' fffS    fDSS^^3

  fHHDH^^3 fe^³²  E| +716,

716 Prgho iru wkh pdvv

Iru wkh ghvljq ri wzr vhsdudwh vlqjoh lqsxw2vlqjoh rxwsxw v|vwhpv/ rqh frqwuroolqj wkh suhvvxuh dqg wkh rwkhu frqwuroolqj wkh pdvv rz/ d prgho iru wkh pdvv lv ghulyhg1 Dv

716 Prgho iru wkh pdvv 59

50 100 150 200 250 300 350 400 450

−0.15

−0.1

−0.05 0 0.05 0.1

time/s

pressure deviation/V

Iljxuh 715= Frpsdulvrq ri phdvxuhg rxwsxw +gdvkhg0grwwhg, dqg vlpxodwhg prgho rxwsxw +vrolg,

−100 −80 −60 −40 −20 0 20 40 60 80 100

−0.1

−0.05 0 0.05 0.1

Autocorrelation of residuals for pressure

−100 −80 −60 −40 −20 0 20 40 60 80 100

−0.2

−0.1 0 0.1 0.2

Samples

Cross correlation for PCV and pressure residuals

Iljxuh 716= Uhvxow ri wkh vwdwlvwlfdo whvw1 Fruuhodwlrq ydoxhv +vrolg,/ frqghqfh ohyho +grwwhg,

716 Prgho iru wkh pdvv 5:

0 50 100 150 200 250 300 350 400 450

−3

−2.5

−2

−1.5

−1

−0.5 0

time/s

mass/V

Iljxuh 717= Frpsdulvrq ri phdvxuhg pdvv +gdvkhg0grwwhg, dqg vlpxodwhg pdvv +vrolg, lqsxw wkh IFY dqg dv rxwsxw wkh pdvv ri wkh yhvvho duh fkrvhq1 Xvlqj d ohdvw vtxduh phwkrg iru wkh lghqwlfdwlrq jlyhv wkh ehvw uhvxow1 Wkh uhvxowlqj DU[0prgho kdv wkh iroorzlqj htxdwlrq =

6E| ' ffff^^3

  fbbbbe^^3  E| +717,

Reylrxvo| wkh prgho lv ri uvw rughu dqg kdv d sroh forvh wr wkh vwdelolw| erughu1 Wklv vhhpv wr eh orjlfdoo| fruuhfw dv iurp wkh nqrzohgjh ri wkh sodqw wkh pdvv ri wkh yhvvho lv dq lqwhjudo ri wkh pdvv rz rxw ri wkh lqmhfwlrq yhvvho1 Wklv ohdgv wr wkh uhvxow wkdw wkh IFY wrjhwkhu zlwk lwv srvlwlrq frqwuroohu fdq eh dvvxphg wr kdyh d sursruwlrqdo ehkdylru1 Dvvxplqj wkh vwuxfwxuh ri wkh prgho ehlqj pruh frpsoh{/ grhv qrw jlyh d ehwwhu lghqwlfdwlrq uhvxow1

Wkh prgho iurp htxdwlrq +717, lv qrz ydolgdwhg lq d zd| vlplodu wr wkdw iru wkh prgho lq wkh suhylrxv vhfwlrq1 Wkh uhvxow ri wkh frpprq vhqvh whvw lv glvsod|hg lq

jxuh 717 dqg 7181 Iurp wkh uhvxowv ri wkh vwdwlvwlfdo whvw rqh pljkw uhmhfw wkh prgho1 Exw dv ghvfulehg lq ^:/ fkdswhu 44` wkh vwdwlvwlfdo whvwv gholyhu qr devroxwh ghjuhh/ wkdw ghflghv zkhwkhu d prgho kdv wr eh uhmhfwhg ru dffhswhg1 Lq wkh vhtxho/ wkh lghqwlhg prgho lv dffhswhg1

717 PLPR prgho 5;

−100 −80 −60 −40 −20 0 20 40 60 80 100

−0.5 0 0.5

Samples

Cross correlation of mass residuals and FCV

−100−1 −80 −60 −40 −20 0 20 40 60 80 100

−0.5 0 0.5 1

Autocorrelation of mass residuals

Iljxuh 718= Uhvxow ri wkh vwdwlvwlfdo whvw1 Fruuhodwlrq ydoxhv +vrolg, dqg frqghqfh ohyho +grwwhg,

717 PLPR prgho

Lq rughu wr gr d sdudphwulf hvwlpdwlrq ri wkh prgho/ d prgho vwuxfwxuh dqg lwv rughu kdv wr eh fkrvhq1 Vlqfh wkh vrxjkw prgho kdv wzr lqsxwv dqg rxwsxwv dqg d ohdvw vtxduh phwkrg lv xvhg dw uvw/ wkh prgho vwuxfwxuh iru wkh PLPR fdvh lv qrz suhvhqwhg1 Wkh lqsxw/ rxwsxw dqg htxdwlrq huuru yhfwruv duh ghqhg dv

E| ' 5 7 ^E|

E|

6

8 +718d,

)E| ' 5 7 RE|

6E|

6

8 +718e,

0E| ' 5 7 0^E|

02E|

6

8 +718f,

dqg dffruglqj wr ^8`/ wkh DU[0surfhvv uhvxowv lq

E^^3  )E| ' E^^3  E| n 0E| +719,

717 PLPR prgho 5<

zkhuh ^^3 lv wkh edfn0vkliw rshudwru dqg

E^^3 ' 5

7  n @^^^3n    n @_?@^^3?@ @_2^^3n    n @_2?@^^3?@

@2^^3n    n @2?@^^3?@  n @22^^3n    n @22?@^^3?@

6 8+71:d,

E^^3 ' 5

7 K^^^3n    n K_?K^^3?K K_2^^3n    n K_2?K^^3?K K2^^3n    n K2?K^^3?K K2^^3n    n K2?K^^3?K

6

8 +71:e,

Wkh rughu ri wkh prgho lv ghqhg e| wkh sdudphwhuv ?@ dqg ?K/ zklfk kdyh wr eh ghwhuplqhg lq dgydqfh1 Wklv phdqv wkdw hljkw sro|qrpldov kdyh wr eh hvwlpdwhg1 Wkhvh hljkw sro|qrpldov ghvfuleh vl{ lqghshqghqw DU[0surfhvvhv1

Wkh glvdgydqwdjh ri wklv prgho vwuxfwxuh iru wkh PLPR fdvh lv wkdw doo sro|qrpldov lq E^^3 dqg wkrvh lq E^^3 kdyh wkh vdph rughu1 Wklv phdqv wkdw doo vl{ DU[0 surfhvvhv duh dvvxphg wr eh ri wkh vdph rughu1 Li sroh0}hur fdqfhoodwlrqv lq wkh lghqwlhg DU[0prgho dsshdu/ wkh rughu ri wklv surfhvv lv fkrvhq wr kljk1 D prgho rughu uhgxfwlrq lv wkhq srvvleoh1

Lq ^:/ fkdswhu :` wkh ohdvw0vtxduhv phwkrg lv dssolhg wr wkh sureohp ri sdudphwhu hvwlpdwlrq ri VLVR DU[0surfhvv1 Wkh htxdwlrqv fdq eh uhzulwwhq iru wkh PLPR fdvh1 Htxdwlrq +719, ehfrphv

)E| ' )E|^AX n 0E|c +71;,

e| lqwurgxflqj wkh phdvxuhphqw gdwd yhfwru )E| dqg wkh sdudphwhu pdwul{ X dv

)E|^A ' 

)^AE|       )^AE|  ?@ ^AE|      ^AE|  ?K

+71<d, X '

5

7 E@^c @2_   E@c @2_?@ EKc K2_   EKc K2_?K E@2c @22_   E@2c @22_?@ EK2c K22_   EK2c K22_?K

6

8 +71<e,

Wkh hvwlpdwhg sdudphwhu pdwul{ X wkdw plqlpl}hv _^ S^

|' 0²E| fdq eh frpsxwhg iurp

X '

% 

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+7143, Dq dojrulwkp wr vroyh wklv sureohp lv douhdg| lpsohphqwhg lq wkh Lghqwlfdwlrq Wrroer{1 Wkh phdvxuhg gdwd lv lpsruwhg lqwr Pdwode1 Vhsdudwh lghqwlfdwlrq dqg^U ydolgdwlrq vhwv duh xvhg1 Dv wkh prgho rughu kdv wr eh ghqhg iru wkh dojrulwkp/ d

717 PLPR prgho 63

50 100 150 200 250 300 350 400 450

−0.1

−0.08

−0.06

−0.04

−0.02 0 0.02 0.04 0.06

time/s

pressure deviation/V

Iljxuh 719= Phdvxuhg suhvvxuh +gdvkhg0grwwhg, dqg vlpxodwhg suhvvxuh +vrolg, fkrlfh ri ?@ ' 2c ?K ' 2 vhhpv wr eh dssursuldwh iru wkh uvw wuldo1 Wklv fkrlfh lv edvhg rq wkh rughu ri wkh h{lvwlqj prgho1 Vhyhudo lghqwlfdwlrq uxqv kdyh ehhq pdgh/

ydu|lqj wkh sdudphwhuv ?@ dqg ?K1 Wkh ydolgdwlrq ri hdfk prgho surriv/ wkdw wkh uvw fkrlfh zlwk ?@ ' 2c ?K ' 2 shuirupv ehvw1

Wkh ydolgdwlrq frqvlvwv ri wkh wzr whvwv ghvfulehg lq wkh suhylrxv vhfwlrq1 Wkh qxpehu ri sorwv lqfuhdvhv h{wuhpho| frpsduhg wr wkh VLVR fdvh/ wkhuhiruh rqo| wkh frpprq vhqvh whvw lv glvsod|hg khuh1 Wkh sorwv iurp vwdwlvwlfdo whvw fdq eh irxqg lq Dsshqgl{ F1 Iljxuh 719 dqg 71: vkrz wkh uhvxowv ri wkh frpprq vhqvh whvw1 Lw lv hdv|

wr vhh wkdw wkh prgho lv shuiruplqj zhoo1 Wkh uhvxowlqj prgho sdudphwhuv duh

E^^3 ' 5

7   fH2^^3 fDH2^³² fffSf^^3 fffSe^³²

ffeS^^3n ff.H^³²   feSS^^3 fDS^³² 6

8+7144d,

E^^3 ' 5

7 ffff^^3n ffff^³² ffffe^^3 fffS^³²

ffffb^^3n ffff^³² ffff^^3 fffe.^³² 6

8  +7144e,

Wkh htxdwlrq huuru 0E| lv qhjohfwhg1 Wklv phdqv wkdw wkh hvwlpdwhg prgho lv dvvxphg wr pdwfk uhdolw|1 Dv wkh vwdwh vsdfh uhsuhvhqwdwlrq kdv dgydqwdjhv iru revhuyhu dqg

718 Vwdwh vsdfh prgho dqg prgho rughu uhgxfwlrq 64

50 100 150 200 250 300 350 400 450

−3

−2.5

−2

−1.5

−1

−0.5 0

time/s

mass/V

Iljxuh 71:= Phdvxuhg pdvv +gdvkhg0grwwhg, dqg vlpxodwhg pdvv +vrolg,

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0 200 400 600 800 1000 1200

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