EXAMENSARBETE
Pressure and Flow Control of aPulverized Coal Injection Vessel
CIVILINGENJÖRSPROGRAMMET
Institutionen för SystemteknikAvdelningen för Reglerteknik
1997:045 • ISSN: 1402 - 1617 • ISRN: LTU - EX - - 1997/45 - - SE
WOLFGANG BIRK
1997:045 CIV
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mhfwlrq/ wkh yhvvho vzlwfkhv iurp suhvvxuh kroglqj wr lqmhfwlrq1 Wkhq/ wkh rz frqwuro
ydoyh rshqv dqg lqmhfwlrq vwduwv1 Phdqzkloh/ wkh rwkhu yhvvho rz frqwuro ydoyh forvhv1
Gxulqj wkh lqmhfwlrq skdvh erwk frqwuro ydoyhv/ wkh suhvvxuh dqg wkh rz frqwuro ydoyh/
duh dfwlyh1 Wkh zhljkw ri wkh yhvvho lv udpslqj grzq wr d plqlpxp ydoxh zkloh wkh
suhvvxuh uhpdlqv frqvwdqw1 Uhdfklqj wkh plqlpxp zhljkw/ wkh lqmhfwlrq yhvvho rz
frqwuro ydoyh forvhv1 Lq wkh phdqwlph/ wkh rwkhu yhvvho kdv ehhq fkdujhg/ suhvvxul}hg
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
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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 lghqwl�fdwlrq
dqg krz wkh| duh dftxluhg1 Pdlqo| wkh surfhvv kdv irxu lqsxw vljqdov1 Wkh| duh =
� suhvvxuh lq wkh yhvvho
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vxuh udqjh iurp f wr �Sff &�@/ dqg kdyh dq dffxudf| ri f�DI1 ]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
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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
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Wkh xvhg ex�hu dpsol�huv kdyh dq huuru ohvv wkdq f�2I1
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
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whulvwlfv fdq eh jlyhq1 Sdudphwhuv fdq eh rewdlqhg iurp h{shulphqwv dqg lghqwl�fdwlrq
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 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
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norm
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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 directionof the piston
highpressure
lowpressure
flexible ring seal
Iljxuh 518= Sulqflsoh vwuxfwxuh ri wkh SFY +vlghylhz,
moving directionof the slide
pipe hole
slide
Iljxuh 519= Sulqflsoh vwuxfwxuh ri wkh IFY +wrsylhz,
516 H{lvwlqj frqwuro xqlw 49
- PI
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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
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uhgxfh qrlvh wkdw kdv ehhq dpsol�hg ehfdxvh ri wkh gl�huhqwldwlrq1 Wklv pdvv rz lv
frpsxwhg hyhu| irxuwk vhfrqg1 Wkh frqwuroohu lwvhoi lv d txdvl frqwlqxrxv SL frqwuroohu
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516 H{lvwlqj frqwuro xqlw 4:
0 100 200 300 400 500 600 700 800 900 1000−40
−20
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time/s
pres
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0 100 200 300 400 500 600 700 800 900 10000
20
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Iljxuh 51;= Shuirupdqfh ri wkh suhvvxuh frqwuroohu
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rz vkrzv wkdw wkh dfwxdo frqwuroohu lv qrw deoh wr gulyh wkh frqwuro huuru wr }hur1 Wklv
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516 H{lvwlqj frqwuro xqlw 4;
0 100 200 300 400 500 600 700 800 900 100060
65
70
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0 100 200 300 400 500 600 700 800 900 1000−10
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0 100 200 300 400 500 600 700 800 900 1000−150
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surfhvv gdwd1 Ixuwkhupruh wkh idfwruv &�
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olqhdu lqsxw dqg rxwsxw wudqvirupdwlrqv1 Lq dgglwlrq/ wkh ydoyh ixqfwlrqv dqg idfwruv
duh ghulyhg wkhuh1
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vkrxog eh srvvleoh wr olqhdul}h wkh prgho ru jdlq d prgho iurp d gluhfw lghqwl�fdwlrq1
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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
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lv xvhg1
714 Phdvxuhphqw gdwd
Lq rughu wr lghqwli| wkh sodqw/ lqsxw2rxwsxw gdwd lv qhhghg1 Dw VVDE Wxqqsoèw wzr
gl�huhqw 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 lghqwl�fdwlrq/
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 lghqwl�fdwlrq gl!fxow1 Gdwd vhwv kdyh wr eh lqwhusrodwhg dqg
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714 Phdvxuhphqw gdwd 56
0 500 10000
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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 lghqwl�fdwlrq1
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 lghqwl�fdwlrq sxusrvhv irxu vljqdov duh orjjhg =
� suhvvxuh lq wkh yhvvho
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Iljxuh 714 glvsod|v wkh vljqdov1
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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
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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
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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
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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 gl�huhqw iurp wkh lghqwl�fdwlrq 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 frq�ghqfh ohyho/ dv wkh fruuhodwlrq ixqfwlrqv duh qrw }hur lq
uhdolw|1 ^:/ sdjh 757` ghvfulehv krz wklv frq�ghqfh 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 =
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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
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rxwsxw +vrolg,
−100 −80 −60 −40 −20 0 20 40 60 80 100−0.1
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Autocorrelation of residuals for pressure
−100 −80 −60 −40 −20 0 20 40 60 80 100−0.2
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Iljxuh 716= Uhvxow ri wkh vwdwlvwlfdo whvw1 Fruuhodwlrq ydoxhv +vrolg,/ frq�ghqfh ohyho
+grwwhg,
716 Prgho iru wkh pdvv 5:
0 50 100 150 200 250 300 350 400 450−3
−2.5
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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 lghqwl�fdwlrq jlyhv wkh ehvw uhvxow1 Wkh uhvxowlqj DU[0prgho kdv wkh
iroorzlqj htxdwlrq =
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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 lghqwl�fdwlrq 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 lghqwl�hg
prgho lv dffhswhg1
717 PLPR prgho 5;
−100 −80 −60 −40 −20 0 20 40 60 80 100−0.5
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−100 −80 −60 −40 −20 0 20 40 60 80 100−1
−0.5
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Iljxuh 718= Uhvxow ri wkh vwdwlvwlfdo whvw1 Fruuhodwlrq ydoxhv +vrolg, dqg frq�ghqfh ohyho
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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
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ghwhuplqhg lq dgydqfh1 Wklv phdqv wkdw hljkw sro|qrpldov kdyh wr eh hvwlpdwhg1
Wkhvh hljkw sro|qrpldov ghvfuleh vl{ lqghshqghqw DU[0surfhvvhv1
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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 lghqwl�hg
DU[0prgho dsshdu/ wkh rughu ri wklv surfhvv lv fkrvhq wr kljk1 Dprgho rughu uhgxfwlrq
lv wkhq srvvleoh1
Lq ^:/ fkdswhu :` wkh ohdvw0vtxduhv phwkrg lv dssolhg wr wkh sureohp ri sdudphwhu
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ydolgdwlrq vhwv duh xvhg1 Dv wkh prgho rughu kdv wr eh gh�qhg iru wkh dojrulwkp/ d
717 PLPR prgho 63
50 100 150 200 250 300 350 400 450
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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 lghqwl�fdwlrq 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
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frpprq vhqvh whvw lv glvsod|hg khuh1 Wkh sorwv iurp vwdwlvwlfdo whvw fdq eh irxqg lq
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MS�eK
= wudqvihu ixqfwlrq ri wkh Fkhe|vkhy �owhu
&�
= ydoyh frh!flhqw ri wkh rz frqwuro ydoyh
&�
= ydoyh frh!flhqw ri wkh suhvvxuh frqwuro ydoyh
6 = pdvv ri wkh lqmhfwlrq yhvvho
6�
= qlwurjhq pdvv
6�
= frdo pdvv
?@ = rughu ri wkh ghqrplqdwru sro|qrpldov
D16 Frqvwdqwv 99
?K = rughu ri wkh qrplqdwru sro|qrpldov
� = qxpehu ri vdpsohv
R = suhvvxuh lq wkh lqmhfwlrq yhvvho
R�
= prgho srohv
R�
= suhvvxuh lq wkh qlwurjhq qhw
RA
= suhvvxuh lq wkh wudqvsruwdwlrq slsh
x = glvfuhwh0wlph v|vwhp pdwul{
) = phdvxuhphqw gdwd yhfwru
" = zhljkwlqj pdwul{
+ = zhljkwlqj pdwul{
A = whpshudwxuh lq wkh lqmhfwlrq yhvvho
A = wudqvirupdwlrq pdwul{
X = sdudphwhu pdwul{
X = hvwlpdwhg sdudphwhu pdwul{
� = lqsxw yhfwru
��
= rshqlqj ri wkh IFY +wudqviruphg,
��c8�T
= sk|vlfdo rshqlqj ri wkh IFY
��
= rshqlqj ri wkh SFY
�W = vwhdg|0vwdwh lqsxw yhfwru
T = yroxph ri wkh lqmhfwlrq yhvvho
T�
= frdo yroxph
T�
= qlwurjhq yroxph
= vwdwh yhfwru
W = vwhdg|0vwdwh vwdwh yhfwru
) = rxwsxw yhfwru
)W = vwhg|0vwdwh rxwsxw yhfwru
D16 Frqvwdqwv
-�
= jdv frqvwdqw ri qlwurjhq
4�
= ghqvlw| ri frdo
Dsshqgl{ E
Iorz fkduw ri wkh lqmhfwlrq surfhvv
Dv qrw doo surfhvv frqfhuq wkh lqmhfwlrq surfhvv/ rqo| wkuhh rz fkduwv duh glvsod|hg1
Wkh �uvw/ �jxuh E14/ ghvfulehv wkh vwduw0xs surfhgxuh iru wkh frdo lqmhfwlrq sodqw1 Wkh
lqfrplqj vljqdo �/ lv d uhvxow iurp wkh rz fkduw lq �jxuh E15/ ghvfulelqj wkh vwduw0xs
frqglwlrqv iru wkh k|gudxolf v|vwhp1 Li wkh k|gudxolf v|vwhp lv qrw vwduwlqj xs fruuhfwo|
wkhq wkh lqmhfwlrq surfhvv fdqqrw vwduw1
Ilqdoo| �jxuh E16 vkrzv/ krz wkh lqmhfwlrq surfhvv lv zrunlqj1 Xqiruwxqdwho| wklv
rz fkduw grhv qrw ghvfuleh wkh dfwxdo vwuxfwxuh/ dv rqo| rqh suhvvxuh frqwuro xqlw lv
phqwlrqhg lq lw1 Exw wklv kdv qr h�hfw rq wkh sulqflsoh surfhvv rz1
9:
Dsshqgl{ F
Ydolgdwlrq zlwk vwdwlvwlfdo whvwv
Vlqfh wkh prgho kdv wzr lqsxwv dqg rxwsxwv/ wkuhh dgglwlrqdo fruuhodwlrq ixqfwlrqv
kdyh wr eh frpsxwhg1 Dw �uvw wkh rxwsxw uhvlgxdov duh frpsxwhg dqg dxwrfruuhodwhg>
qh{w wkh rxwsxw uhvlgxdov duh furvv fruuhodwhg zlwk wkh lqsxwv1 Dowkrxjk wkh furvv
fruuhodwlrq whvw lv qrw wkdw uholdeoh lq forvhg orrs/ wkh uhvxowv duh glvsod|hg khuh1
D Orrn dw wkh dxwrfruuhodwlrq ixqfwlrqv pljkw ohdg wr wkh frqfoxvlrq wkdw wkh prgho
vkrxog eh uhmhfwhg1 Exw dv vhhq lq wkh frpprq whvw/ wkh vlpxodwhg rxwsxw iroorzv wkh
phdvxuhg rxwsxw zhoo1 Wkhuhiruh wkh prgho lv dffhswhg1
:4
:5
−100 −80 −60 −40 −20 0 20 40 60 80 100−0.5
0
0.5
Samples
Cross correlation of PCV and pressure residuals
−100 −80 −60 −40 −20 0 20 40 60 80 100−1
−0.5
0
0.5
1
Samples
Autocorrelation of pressure residuals
Iljxuh F14= Dxwrfruuhodwlrq ri suhvvxuh uhvlgxdov dqg furvv fruuhodwlrq ri suhvvxuh
uhvlgxdov zlwk SFY1 Frq�ghqfh ohyhov +grwwhg,/ fruuhodwlrqv ixqfwlrqv +vrolg,1
−100 −80 −60 −40 −20 0 20 40 60 80 100−0.5
0
0.5
Samples
Cross correlation of FCV and pressure residuals
Iljxuh F15= Dxwrfruuhodwlrq ri suhvvxuh uhvlgxdov dqg furvv fruuhodwlrq ri suhvvxuh
uhvlgxdov zlwk IFY1 Frq�ghqfh ohyhov +grwwhg,/ fruuhodwlrqv ixqfwlrqv +vrolg,1
:6
−100 −80 −60 −40 −20 0 20 40 60 80 100−0.5
0
0.5
Samples
Cross correlation of FCV and mass residuals
−100 −80 −60 −40 −20 0 20 40 60 80 100−1
−0.5
0
0.5
1
Samples
Autocorrelation of mass residuals
Iljxuh F16= Dxwrfruuhodwlrq ri pdvv uhvlgxdov dqg furvv fruuhodwlrq ri pdvv uhvlgxdov
zlwk IFY1 Frq�ghqfh ohyhov +grwwhg,/ fruuhodwlrqv ixqfwlrqv +vrolg,1
−100 −80 −60 −40 −20 0 20 40 60 80 100−0.5
0
0.5
Samples
Cross correlation of PCV and mass residuals
Iljxuh F17= Dxwrfruuhodwlrq ri pdvv uhvlgxdov dqg furvv fruuhodwlrq ri pdvv uhvlgxdov
zlwk SFY1 Frq�ghqfh ohyhov +grwwhg,/ fruuhodwlrqv ixqfwlrqv +vrolg,1