A Simple Altitude Capture Avoiding TCAS Resolution Advisories During Level-Off Situations

HAL Id: hal-00998701

https://hal-enac.archives-ouvertes.fr/hal-00998701

Submitted on 2 Jun 2014

HAL

is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire

HAL, est

destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

A Simple Altitude Capture Avoiding TCAS Resolution Advisories During Level-Off Situations

Thierry Miquel, Philippe Louyot, Bernard Hasquenoph

To cite this version:

Thierry Miquel, Philippe Louyot, Bernard Hasquenoph. A Simple Altitude Capture Avoiding TCAS

Resolution Advisories During Level-Off Situations. AIAA GNC 2009, AIAA Guidance, Navigation

and Control Conference, Aug 2009, Chicago, United States. pp xxxx, �10.2514/6.2009-5986�. �hal-

00998701�

$PHULFDQ,QVWLWXWHRI$HURQDXWLFVDQG$VWURQDXWLFV

$6LPSOH$OWLWXGH&DSWXUH$YRLGLQJ7&$65HVROXWLRQ

$GYLVRULHV'XULQJ/HYHO2II6LWXDWLRQV

7KLHUU\0LTXHO

'LUHFWLRQGHV6HUYLFHVGHOD1DYLJDWLRQ$pULHQQH7RXORXVH)UDQFH 3KLOLSSH/RX\RW^‚

'LUHFWLRQGHV6HUYLFHVGHOD1DYLJDWLRQ$pULHQQH7RXORXVH)UDQFH DQG

%HUQDUG+DVTXHQRSK^Á (JLVDYLD7RXORXVH)UDQFH

7&$6 UHVROXWLRQ DGYLVRULHV JHQHUDWLRQ SURFHVV GXULQJ OHYHORII VLWXDWLRQ LV RYHUYLHZHG DQGLOOXVWUDWHGRQDUHDOVLWXDWLRQFDSWXUHGWKURXJKPRGH6UDGDUZLWK5$GRZQOLQNVHUYLFH ,W XQGHUOLQHV WKDW RQH FRPPRQ W\SH RI UHVROXWLRQ DGYLVRULHV LV WKDW ZKLFK LV LVVXHG ZKHQ DLUFUDIWDUHH[SHFWHGWROHYHORIIIHHWDSDUWDQGDUHFRQYHUJLQJDWWKHVDPHWLPH7KHVH UHVROXWLRQDGYLVRULHVRIWHQVXEVHTXHQWO\FODVVHGDVµRSHUDWLRQDOO\XQQHFHVVDU\¶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

, ,QWURGXFWLRQ

KH$LUERUQH&ROOLVLRQ$YRLGDQFH6\VWHP,,$&$6LVDQHVVHQWLDOFRPSRQHQWRIWKH$LU7UDIILF0DQDJHPHQW

$70V\VWHP$Q$&$6LVDQDLUFUDIWV\VWHPEDVHGRQVHFRQGDU\VXUYHLOODQFHUDGDUWUDQVSRQGHUVLJQDOVZKLFK RSHUDWHVLQGHSHQGHQWO\RIJURXQGEDVHGHTXLSPHQWWRSURYLGHDGYLFHWRWKHSLORWRQSRWHQWLDOFRQIOLFWLQJDLUFUDIWWKDW DUHHTXLSSHGZLWK665WUDQVSRQGHUV,WVHUYHVDVDODVWUHVRUWVDIHW\QHWLUUHVSHFWLYHRIDQ\VHSDUDWLRQVWDQGDUGV

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

3K'&LYLO$YLDWLRQ'HVLJQDQG2SHUDWLRQ(QJLQHHU6HSDUDWLRQDQG$OHUWLQJ6\VWHPV8QLWPLTXHO#FHQDIU

‚6HQLRUWHFKQLFDOH[SHUWRQ$6$6DQG$&$66HSDUDWLRQDQG$OHUWLQJ6\VWHPV8QLWORX\RW#FHQDIU

Á$LU7UDIILF&RQWUROH[SHUW$&$6VSHFLDOLVWKDVTXHQRSK#FHQDIU

Copyright © 2009 by DSNA. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

7

AIAA Guidance, Navigation, and Control Conference

10 - 13 August 2009, Chicago, Illinois

AIAA 2009-5986

Copyright © 2009 by Copyright © 2009 by DSNA. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

$PHULFDQ,QVWLWXWHRI$HURQDXWLFVDQG$VWURQDXWLFV

7KH $&$6 ,, PDQGDWH DSSOLHV ZRUOGZLGH ,&$2 IRU DOO FLYLO IL[HGZLQJ WXUELQHHQJLQHG DLUFUDIW ZLWK D PD[LPXPWDNHRIIPDVVH[FHHGLQJNJRUDPD[LPXPDSSURYHGSDVVHQJHUVHDWLQJFRQILJXUDWLRQRIPRUHWKDQ

&DUULDJHDQGRSHUDWLRQRIWKH$&$6,,FRPSOLDQWHTXLSPHQWLVPDQGDWRU\ZRUOGZLGHVLQFHVW-DQXDU\

7&$6 ,, YHUVLRQ LV WKHRQO\ HTXLSPHQW ZKLFK FRPSOLHV IXOO\ZLWK $&$6 ,, VWDQGDUGV DQG UHFRPPHQGHG SUDFWLFHV DV SXEOLVKHG E\ ,&$2 7KHUHIRUH 7&$6 ,, YHUVLRQ HTXLSDJH LV UHTXLUHG WR PHHW WKH $&$6 ,, PDQGDWH 7KLV YHUVLRQ RI 7&$6 ,, EHFDPH DYDLODEOH LQ :KHQ FRPSDUHG WR WKH IRUPHU YHUVLRQ D LW LPSURYHV7&$6FRPSDWLELOLW\ZLWKWKHDLUWUDIILFFRQWUROV\VWHP,WDOVRLQWURGXFHVFRPSDWLELOLW\ZLWK5960DQG RIIHUVLPSURYHPHQWVLQWKHYHUWLFDOWUDFNLQJRILQWUXGHUVDQGDILOWHULQJRIWKUHDWV EDVHGRQ DIRUHFDVWKRUL]RQWDO PLVVGLVWDQFH

5HVROXWLRQDGYLVRULHVDUHJHQHUDWHGZKHQ$&$6,,FRPSXWHVDULVNRIFROOLVLRQZLWKDQRWKHUDLUFUDIWZLWKLQWKH QH[WVHFRQGVRUOHVVGHSHQGLQJRQDOWLWXGHDQGZKHWKHUWKHDLUFUDIWLVIO\LQJOHYHORUQRW5$LVLVVXHGZKHQ DLUFUDIWDUHKRUL]RQWDOO\FRQYHUJLQJDQGDUHH[SHFWHGWREHDWWKHVDPHDOWLWXGHDWWKHVDPHWLPH2QHFRPPRQW\SH RI5$LVLVVXHGHYHQLIDLUFUDIWDUHFOHDUHGWROHYHORIIIHHWDSDUW7KHVHW\SHVRI5$VUHSUHVHQWVRILVVXHG 5$V WKH\ DUH RIWHQ FODVVLILHG DV µRSHUDWLRQDOO\ XQQHFHVVDU\¶ IURP DQ $7& VWDQGSRLQW DQG FDQ EH SHUFHLYHG DV GLVWXUELQJE\FRQWUROOHUVDQGE\SLORWV,QDGGLWLRQWKH\FDQLQGXFHDYHUWLFDOGHYLDWLRQRIWKHOHYHODLUFUDIWWKURXJK FRRUGLQDWHG5$

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

EHFDXVH 7&$6 DQWLFLSDWHV DFROOLVLRQ ULVN ZLWKLQ DUHPDLQLQJ WLPH OHVV WKDQVHFRQGV 7KLV RFFXUVGHVSLWHWKH SURYLVLRQRIIHHWIRUYHUWLFDOVHSDUDWLRQSODQQHGE\DLUWUDIILFFRQWURO,WLVZRUWKQRWLFLQJWKDWLQWKLVSDUWLFXODU H[DPSOH WKH LQWUXGHU DLUFUDIW GRHV QRW H[SHULHQFH DQ\ UHVROXWLRQ DGYLVRU\ RZLQJ WR D ORZHU DODUP WKUHVKROG DVVRFLDWHGZLWKOHYHOIOLJKW

-60 -40 -20 0 20

-120 -115 -110 -105 -100 -95

T rajectories in the horizontal plane [NM]

59200 59250 59300 59350 59400 59450

330 335 340 345 350 355 360 365

Altitude [FL] versus time [sec]

592000 59250 59300 59350 59400 59450

2 4 6 8 10 12 14

Horizontal range [NM] versus time [sec]

)LJ([DPSOHRIDFWXDOOHYHORIIVLWXDWLRQGXULQJZKLFKDUHVROXWLRQDGYLVRU\LVJHQHUDWHG

7KHSXUSRVHRIWKLVSDSHULVWRGHVLJQDVLPSOHDXWRSLORWDOWLWXGHFDSWXUHPRGHFDQEHVHWWRDYRLGUHVROXWLRQ DGYLVRULHVGXULQJVXFKOHYHORIIVLWXDWLRQV

7KHSDSHULVRUJDQL]HGDVIROORZVLQWKHQH[WVHFWLRQWKHUHVROXWLRQDGYLVRULHVJHQHUDWLRQSURFHVVGXULQJOHYHO RII VLWXDWLRQ LV RYHUYLHZHG DQG LOOXVWUDWHG RQ D UHDO VLWXDWLRQ FDSWXUHG WKURXJK PRGH 6 UDGDU ZLWK 5$ GRZQOLQN VHUYLFH 7KH GHVLJQ RI D VLPSOH DXWRSLORW DOWLWXGH FDSWXUH LV DGGUHVVHG LQ WKH WKLUG VHFWLRQ 7KLV VWDUWV E\ WKH SUHVHQWDWLRQRIDVLPSOHDOWLWXGHFDSWXUHPRGHO7KHVLPSOHDOWLWXGHFDSWXUHPRGHOLVEDVHGRQVHFRQGRUGHUOLQHDU

$PHULFDQ,QVWLWXWHRI$HURQDXWLFVDQG$VWURQDXWLFV

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⋅ZQUHPDLQV FRQVWDQWVRDVQRWWRPRGLI\WKHUHDOSDUWRIWKHSROHVRIWKHDXWRSLORWPRGH7KLVPHWKRGLVLOOXVWUDWHGWKURXJKWKH UHDOUDGDUHQFRXQWHUSUHYLRXVO\SUHVHQWHG)LQDOO\WKHFRQFOXVLRQXQGHUOLQHVWKDWIXUWKHUZRUNLVUHTXLUHGLQRUGHUWR H[WHQGWKHUHVXOWVWRWKHFDVHVZKHUHWKHLQWUXGHUDLUFUDIWLVQRWOHYHO

,, 2YHUYLHZRIUHVROXWLRQDGYLVRULHVJHQHUDWLRQSURFHVVGXULQJOHYHORIIVLWXDWLRQ

7KHSXUSRVHRIWKLVVHFWLRQLVWRRYHUYLHZWKHUHVROXWLRQDGYLVRULHVJHQHUDWLRQSURFHVVGXULQJOHYHORIIVLWXDWLRQ ,WLVLOOXVWUDWHGE\DUHDOVLWXDWLRQFDSWXUHGWKURXJKPRGH6UDGDUZLWK5$GRZQOLQNVHUYLFH,WLVDVVXPHGLQWKH IROORZLQJWKDWRZQDLUFUDIWZLOOOHYHORIIDIWHUDFOLPERUDGHVFHQWDQGWKDWWKHLQWUXGHUDLUFUDIWLVOHYHO

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

7LPHWRFRDOWLWXGHWHVWLVGHWHUPLQHGE\WKHYDOXHVRIRZQDQGLQWUXGHUDLUFUDIW¶VFXUUHQWYHUWLFDOUDWHV8QGHU WKHDVVXPSWLRQWKDWWKHLQWUXGHUDLUFUDIWLVOHYHOWKHFRPSXWHGWLPHWRFRDOWLWXGHτ₉KDVWKHIROORZLQJH[SUHVVLRQ ZKHUHKDQG9]GHVLJQDWHWKHDOWLWXGHDQGYHUWLFDOVSHHGRIRZQDLUFUDIWDQGKLFWKHFRQVWDQWDOWLWXGHRIWKHLQWUXGHU DLUFUDIW

¸¸¹·

¨¨©§ −

−

=

]

9 9 LF

K

τ K

,QWKHIROORZLQJWKLVUHODWLRQVKLSZLOOEHODEHOHGµ%RXQGDU\RIWKH5$]RQH¶ZKHQWKHFRPSXWHGWLPHWRFR DOWLWXGHτ₉LVHTXDOWRWKHWKUHVKROG6DWZKLFKDQ5$LVWULJJHUHG

7KHG\QDPLFRIWKHDOWLWXGHFDSWXUHPRGHRIWKHDXWRSLORWLVQRZFRQVLGHUHG/HWKFEHWKHVHOHFWHGDOWLWXGHE\

RZQ DLUFUDIW :KHQ WKH G\QDPLF RI WKH DOWLWXGH FDSWXUH PRGH LV VXSHULPSRVHG RQ WKH GRPDLQ9] K ZLWK WKH ERXQGDU\RIWKH5$]RQHWKHIROORZLQJILJXUHLVREWDLQHG,WKDVEHHQDVVXPHGWKDWWKHVHOHFWHGDOWLWXGHK_FLVKLJKHU WKDQWKHDOWLWXGHDWZKLFKWKHFDSWXUHVWDUWV,WLVZRUWKQRWLFLQJWKDWWKHVORSHRIWKHVWUDLJKWOLQHFRUUHVSRQGVWRWKH WKUHVKROGDWZKLFKDQ5$LVWULJJHUHG

KF KLF

τ9!6 τ9!6

'\QDPLF RIWKHDOWLWXGH

FDSWXUHPRGH

τ9 6 K

9] JHQHUDWHG5$

JHQHUDWHG5$

)LJ'\QDPLFRIWKHDOWLWXGHFDSWXUHPRGHVXSHULPSRVHGRQWKHGRPDLQZKHUHWKHFRPSXWHGWLPHWRFR DOWLWXGHWULJJHUVDQDODUP

)URPWKHSUHYLRXVILJXUHLWLVFOHDUWKDWLIWKHG\QDPLFRIWKHDOWLWXGHFDSWXUHPRGHHQWHUVWKHJUH\DUHDZKLFK GHOLPLWVWKHDUHDZKHUHDQDODUPLVWULJJHUHGWKHQDQ5$LVJHQHUDWHG,QRUGHUWRLOOXVWUDWHWKRVHUHODWLRQVKLSVWKH

$PHULFDQ,QVWLWXWHRI$HURQDXWLFVDQG$VWURQDXWLFV

IROORZLQJILJXUHSUHVHQWVWKHSORW9_]→K9_]IRUWKHFDSWXUHGHQFRXQWHUSUHVHQWHGGXULQJWKHLQWURGXFWLRQDVZHOODV WKHERXQGDU\RIWKH5$]RQHZKHUHWKUHVKROG6LVVHWWRVHFRQGVDQGLQWUXGHUIOLJKWOHYHOKLFLVVHWWR)/

-500 0 500 1000 1500 2000 2500 3000 3500 4000 4500

334 336 338 340 342 344 346 348 350 352

Domain (Vz[fpm] ; h[FL]) superimposed on the domain where the computed time to co-altitude triggers an alarm

$UHDZKHUHWKHGRPDLQ 9]KHQWHUVLQWKH

5$]RQH KF )/

$XWRSLORWG\QDPLFV

/LPLWRIWKH5$

]RQH

)LJ'RPDLQ9]KIRUWKHFDSWXUHGWUDMHFWRU\DQGERXQGDU\RIWKH5$]RQH

2QWKLVILJXUHLWLVFOHDUO\VHHQWKDWWKHDLUFUDIWDFKLHYHVDYHUWLFDOVSHHGRIDERXWISPDQGWKHQHQWHUVWKH 5$]RQHEHIRUHPRYLQJWRZDUGVWKHFOHDUHGDOWLWXGHZKLFKLV)/,QGHHGZKHQWKHDLUFUDIWFURVVHV)/

7&$6HVWLPDWHVWKHFRDOWLWXGHWLPHZLWKWKHLQWUXGHUWREHVHFRQGV)/−ISP×VHF )/DQG WKHQWULJJHUVDQ5$&RQVHTXHQWO\DZD\WRDYRLG5$VWULJJHULQJLVWRPRGLI\WKHG\QDPLFRIWKHDOWLWXGHFDSWXUH PRGHVRWKDWLWDOZD\VUHPDLQVRXWVLGHWKHDUHDZKHUHDQ5$LVJHQHUDWHG7KLVJHQHUDOLGHDZLOOEHLQYHVWLJDWHGLQ WKHIROORZLQJSDUDJUDSK

,,, 6LPSOHDOWLWXGHFDSWXUHDXWRSLORWGHVLJQ

$ $LUFUDIWPRGHO

$VIDUDVWKHDLUFUDIWG\QDPLFLVFRQVLGHUHGWKHIROORZLQJDVVXPSWLRQVDUHPDGH

‡ )ODWQRQURWDWLQJHDUWKVWDQGDUGDWPRVSKHUHDQGFRQVWDQWJUDYLW\

‡ 7KH DLUFUDIW LV UHJDUGHG DV D SRLQW PDVV WKH G\QDPLFV RI WKH PRYHPHQWV DURXQG LWV FHQWUH RI PDVV DUH LJQRUHG

‡ 7KHDLUFUDIWLVIO\LQJDVWUDLJKWOLQHDWFRQVWDQWHTXLYDOHQWDLUVSHHGDQGLVFRQWUROOHGWKURXJKORDGIDFWRUWKDW LVE\WKHUDWLREHWZHHQOLIWDQGZHLJKW

7DNLQJLQWRDFFRXQWWKHDERYHDVVXPSWLRQVWKHORQJLWXGLQDOSRLQWPDVVG\QDPLFHTXDWLRQVRIPRWLRQUHIHUULQJ WRWKH (DUWK IL[HG UHIHUHQFH IUDPH DUHWKH IROORZLQJ ZKHUH[VWDQG IRUWKHKRUL]RQWDOFRRUGLQDWHV DQGK IRU WKH YHUWLFDOFRRUGLQDWHVRIWKHDLUFUDIW9DIRUWUXHDLUVSHHG9HIRUHTXLYDOHQWDLUVSHHGJIRUJUDYLWDWLRQDODFFHOHUDWLRQȖ IRUIOLJKWSDWKDQJOHQ]IRUORDGIDFWRUDQGρDQGρKIRUWKHDLUGHQVLW\UHVSHFWLYHO\DWVHDOHYHODQGDWDOWLWXGHK

( ) ( )

( )

( )

°°

°

¯

°°°

®

­

−

⋅

=

⋅

=

⋅

=

γ γ

γ γ

FRV VLQ FRV

] D D D

9 Q J 9 K

9 [

( )

°

( )

¯

°®

­

=

⋅

=

H H D

9 9 K 9

ρ

ρ

$PHULFDQ,QVWLWXWHRI$HURQDXWLFVDQG$VWURQDXWLFV

% 'HVLJQRIWKHDOWLWXGHFDSWXUHPRGH

7KH SXUSRVH RI WKH DOWLWXGH FDSWXUH PRGH LV WR FRQVWUDLQ WKH DFWXDO DLUFUDIW DOWLWXGHK WR PRYH WRZDUGV WKH FRPPDQGHGDOWLWXGHK_F)LUVWDVVXPLQJDFRQVWDQWWUXHDLUVSHHG9_DDQGDIOLJKWSDWKDQJOHVPDOOHQRXJKVXFKWKDW VLQγ≈γWKHVHFRQGWLPHGHULYDWLYHRIWKHDOWLWXGHKZLWKUHVSHFWWRWKHWLPHLVREWDLQHGIURPWKHODVWWZRHTXDWLRQV RIDQGOHDGVWRDQH[SUHVVLRQZKHUHFRQWUROQ_]DSSHDUV

( )

(

γ

)

γ = ⋅ −FRV

⋅

≈9_D J Q_]

K

)HHGEDFNOLQHDUL]DWLRQLVWKHQXVHGWKHDOWLWXGHFDSWXUHPRGHFRQVWUDLQVWKHG\QDPLFRIWKHDFWXDODOWLWXGHKWR PRYHWRZDUGVWKHFRPPDQGHGDOWLWXGHK_FE\IROORZLQJDVHFRQGRUGHUVWDEOHGLIIHUHQWLDOHTXDWLRQZLWKXQGDPSHG QDWXUDOIUHTXHQF\ZQDQGGDPSLQJUDWLRP

(

_F

)

Q

Q K Z K K

Z P

K=−⋅ ⋅ ⋅− ⋅ −

&RPELQLQJDQGDQGWDNLQJLQWRDFFRXQWWKHDSSUR[LPDWLRQK≈9_D⋅γOHDGVWRWKHH[SUHVVLRQRIWKHFRQWURO Q_]

( ) ( )

J

K K Z 9 Z

Q_] ⋅P⋅ ^Q⋅ ^D⋅ + ^Q⋅ − ^F

−

=

FRV γ

γ

& 3DUDPHWHULGHQWLILFDWLRQ

7KH HVWLPDWLRQ RI WKH VORSH RI WKH DOWLWXGH FDSWXUH PRGH VHH RQ )LJ DERYH WKH DUURZ ODEHOHG µDXWRSLORW G\QDPLF¶IRUWKHUDGDUFDSWXUHGHQFRXQWHULVKHUHFRQVLGHUHG,WLVDVVXPHGWKDWWKHDOWLWXGHFDSWXUHPRGHGHVFULEHG LQWKHSUHYLRXVVHFWLRQFRUUHFWO\GHVFULEHGWKHDLUFUDIWEHKDYLRUGXULQJWKHDOWLWXGHFDSWXUH)LUVWLWLVZRUWKQRWLFLQJ WKDWWKHG\QDPLFRIWKHDXWRSLORWDOWLWXGHFDSWXUHPRGHFDQEHUHZULWWHQDVIROORZV

( )

⋅ ⋅ + − = +

⋅ _F

Q Q

K K Z K

K P

Z

,QWKHIROORZLQJLWLVDVVXPHGWKDWWKHDOWLWXGHFDSWXUHVWDUWVDURXQGDSRLQWZKHUHWKHVHFRQGGHULYDWLYHRIWKH DOWLWXGHKLVFORVHWR]HUR7KLVLVDVVXPHGWREHDURXQGWKHVZLWFKLQJDOWLWXGHZKHUHWKHDOWLWXGHFDSWXUHVWDUWV7KLV FRUUHVSRQGVWRWKHYHUWLFDOVHJPHQWLQWKHGRPDLQ9]KUHSUHVHQWHGLQ)LJ7KHQWKHG\QDPLFRIWKHDOWLWXGH FDSWXUHPRGHLVDSSUR[LPDWHGE\WKHIROORZLQJILUVWRUGHUG\QDPLF

( )

⋅ ⋅ + − ≈

F Q

K K Z K

P

/HW7VEHWKHWLPHDWZKLFK5$DSSHDUVDQGOHWWKHWLPHLQWHUYDO>7V7VVHFRQGV@EHWKHLGHQWLILFDWLRQZLQGRZ IRUWKHUDWLR⋅PZQ7KHVHFRQGVGXUDWLRQRIWKHLGHQWLILFDWLRQZLQGRZPD\DSSHDUTXLWHVPDOOEXWLVMXVWLILHGE\

WKH IDFW WKDW WKLV FRUUHVSRQGV WR WKH VWDQGDUG DLUFUHZ UHDFWLRQ WLPH IRU LQLWLDO 5$ $ ORQJHU GXUDWLRQ RI WKH LGHQWLILFDWLRQZLQGRZPD\FDSWXUHWKHG\QDPLFRIWKHPDQXDOFRQWUROODZUDWKHUWKDQWKHG\QDPLFRIWKHDOWLWXGH FDSWXUHPRGH,QRUGHUWRLGHQWLI\WKHUDWLR⋅PZ_QGHILQHGDVSDUDPHWHUSOHWεSEHWKHIROORZLQJTXDGUDWLFHUURU

( )

= ⋅

_¦ (

⋅

( )

+

( ( )

−

) )

⋅ Ÿ

=

N N N F

Q

K W K W K S Z S

S P

ε

7KHLGHQWLILFDWLRQRISDUDPHWHUSLVGRQHE\PLQLPL]LQJWKHTXDGUDWLFHUURUεSZLWKUHVSHFWWRS

( )

=

(

⋅

( )

+

( ( )

−

) )

⋅

( )

=Ö

∂

∂^ε_S^S

¦

_N ^{S KWN KWN KF KWN}

&RQVHTXHQWO\SDUDPHWHUSLVDSSUR[LPDWHGDVIROORZV

( )

( ) ( )

¦ ( )

¦

^{− ⋅}

−

=

N N

N N F N

W K

W K K W K

S

:KHQ WKLV LGHQWLILFDWLRQ SURFHVV LV DSSOLHG WR WKH UDGDU FDSWXUHG HQFRXQWHU SUHVHQWHG LQ WKH LQWURGXFWLRQ SDUDPHWHUS LV HYDOXDWHG WR EH DSSUR[LPDWHO\ VHFRQGV 'DPSLQJ UDWLRP KDV EHHQ VHW WR LQ RUGHU WR DYRLG RVFLOODWLRQVGXULQJWKHDOWLWXGHFDSWXUH$VSDUDPHWHUSKDVEHHQHYDOXDWHGWREHVHFRQGVDQGDVSHTXDOV⋅PZQ WKLVLPSOLHVWKDWWKHQDWXUDOIUHTXHQF\Z_QLVHYDOXDWHGWREHUDGVHF6LPXODWLQJWKHDOWLWXGHFDSWXUHWKDQNVWR DLUFUDIWPRGHODQGFRQWUROODZIURP7VWRWKHILQDOWLPHRIWKHWUDMHFWRU\OHDGVWRWKHIROORZLQJ9]KSORW

$PHULFDQ,QVWLWXWHRI$HURQDXWLFVDQG$VWURQDXWLFV

-500 0 500 1000 1500 2000 2500 3000 3500 4000 4500

334 336 338 340 342 344 346 348 350 352

Domain (Vz[fpm] ; h[FL]) superimposed on the domain where the computed time to co-altitude triggers an alarm

KF )/

/LPLWRIWKH5$

]RQH 7VVWDUWLQJWLPHRI

WKHVLPXODWHG WUDMHFWRU\

(UURUHW 6LPXODWHGDXWRSLORW

G\QDPLFV

)LJ'RPDLQ9_]KIRUWKHVLPXODWHGWUDMHFWRU\DQGERXQGDU\RIWKH5$]RQH

)URPWKHSUHYLRXVILJXUHLWLVFOHDUWKDWWKH5$LVUHSURGXFHGWKURXJKWKHVLPXODWLRQSURFHVVDQGWKDWWKHNH\

IDFWRUV ZKLFK HQDEOH WR DYRLG LVVXLQJ DQ 5$ LQ VXFK D VLWXDWLRQ DUH WKH DOWLWXGH DW ZKLFK WKH DXWRSLORW VZLWFKHV WRZDUGVWKHDOWLWXGHFDSWXUHPRGHDVZHOODVWKHSDUDPHWHUVRIWKHDOWLWXGHFDSWXUHPRGHLWVHOIQDPHO\WKHGDPSLQJ UDWLRPDQGWKHQDWXUDOIUHTXHQF\ZQ

,9 7XQLQJWKHDXWRSLORWDOWLWXGHFDSWXUHPRGHWRDYRLG5$GXULQJOHYHORII

$ &RPSXWDWLRQRIWKHWDQJHQWWRWKHERXQGDU\RIWKH5$]RQH

7KHVHFRQGRUGHUILOWHUPRGHOLQJWKHG\QDPLFRIWKHDOWLWXGHFDSWXUHPRGHFDQEHZULWWHQXQGHULWVVWDWHIRUP DVIROORZV

( )

[

$ [ Z P

$ Z K

K [ K

Q Q

F

⋅

−

Ÿ =

°°

¯

°°

®

­

»¼

« º

¬ ª

⋅

⋅

= −

»¼º

«¬ª −

=

Ö

Ö

/HWHW EH WKH GLIIHUHQFH EHWZHHQ WKH DOWLWXGH OLPLW RI WKH 5$ ]RQH VHH DQG WKH DOWLWXGH RI WKH DLUFUDIW GHILQHGE\DVUHSUHVHQWHGLQ)LJ

( )

W K 6 K

( )

W K

( )

W

H = _LF− ⋅ −

7KHHUURUHWFDQEHH[SUHVVHGDVDIXQFWLRQRIWKHVWDWHYHFWRU[DVIROORZV

( ) ( ) ( ( ) ) [ ] ^{( )}

( )

W K K

[

6

]

[

( )

W K

K W 6 K K K K W K W K 6 K K W

H F LF F

F LF F F

LF = − − ⋅

»¼º

«¬ª −

⋅

−

−

=

−

−

⋅

−

−

=

:KHQLVXVHGWKHWLPHGHULYDWLYHRIWKHHUURUHWKDVWKHIROORZLQJH[SUHVVLRQ

( )

W

[

6

]

[

( )

W

[

6

]

$ [

( )

W

[

6 Z

(

P 6 Z

) ]

[

( )

W

H =− ⋅ = ⋅ ⋅ = ⋅ Q ⋅ ⋅ ⋅ Q− ⋅

$VDFRQVHTXHQFHWKHYDOXHRIWKHVWDWHYHFWRU[_WWDQJHQWWRWKH5$]RQHVDWLVILHVWKHIROORZLQJVHWRIHTXDWLRQV

[ ]

( )

[ ] ^{( )}

_«¬^ª _»¼^º

= −

»⋅

¼

« º

¬ ª

−

⋅

⋅

⋅

⇔ ⋅

¯®

­

=

⋅

−

⋅

⋅

⋅

⋅

=

⋅

−

⇔ −

¯®

­

=

=

F W LF

Q W Q

Q Q

W F

LF K K

Z [ 6 P Z 6

6 [

Z 6 P Z 6

[ 6 K K H

H

'HQRWLQJE\9]WWKHYHUWLFDOVSHHGDWWKHWDQJHQWSRLQWDQGE\KWWKHDOWLWXGHDWWKDWSRLQWZHILQDOO\JHW

( )

( )

^»_¼

« º

¬ ª

⋅

⋅

⋅

⋅

⋅ − +

⋅

⋅

⋅

−

⋅

= −

»¼

« º

¬

ª −

=

⇔

»¼º

«¬ª −

»⋅

¼

« º

¬ ª

−

⋅

⋅

⋅

⋅

⋅ − +

⋅

⋅

⋅

−

= ⋅

Ö

Q Q Q

Q

F LF ]W

F W W

F LF Q

Q Q

Q W

Z 6

Z 6 P Z

6 P Z

6

K K 9

K [ K

K K Z

6

6 Z 6 P Z

6 P Z

[ 6

$PHULFDQ,QVWLWXWHRI$HURQDXWLFVDQG$VWURQDXWLFV

)URPWKHSUHYLRXVHTXDWLRQWKHUHODWLRQVKLSEHWZHHQWKHDOWLWXGHKWDQGWKHYHUWLFDOVSHHG9]WDWWKHSRLQWZKHUH WKHDOWLWXGHFDSWXUHG\QDPLFLVWDQJHQWWRWKHERXQGDU\RIWKH5$]RQHLVWKHIROORZLQJ

Q Q ]W

F W

Z 6

Z 6 P 9

K K

⋅

⋅

⋅

⋅

= −

−

,Q DGGLWLRQ DV IDU DV[_W EHORQJV WR WKH WUDMHFWRU\ GHILQHG E\ WKH GLIIHUHQWLDO HTXDWLRQ WKH VHW RI LQLWLDO FRQGLWLRQV[_WIRUZKLFKWKHDOWLWXGHFDSWXUHWUDMHFWRU\WDQJHQWVWKHERXQGDU\RIWKH5$]RQHLVJLYHQE\

( )

( ) ( )

_»

¼

« º

¬ ª

⋅

⋅

⋅

⋅

⋅ −

⋅

− + ⋅

⋅

⋅

⋅

−

⋅

= −

⇔

⋅

⋅

−

=

H[S

H[S

Q Q Q

Q

F W LF

W

W 6 Z

Z 6 W P

Z $ 6 P Z

6

K [ K

[ W

$

[

7KH H[SUHVVLRQ RI WKH PDWUL[H[S$⋅W FDQ EH IRXQG E\ XVLQJ WKH /DSODFH WUDQVIRUP ,Q WKH IROORZLQJ H[SUHVVLRQVUHSUHVHQWVWKH/DSODFHYDULDEOHDQG,WKHLGHQWLW\PDWUL[

(

⁻$^⋅W

)

⁼/⁻

[ (

V^⋅,⁺$

)

⁻

]

⁼_«¬^ª_D^D

^{( )} _{( )}

_W^W _D^D

^{( )} _{( )}

^W_W»¼^º

H[S

:KHUH

( ) ( )

( ) ( )

( ) ( ) _{( )}

( ) ( ) ( ) _{( )}

( ) ( )

( ) ( )

( )

°°

( )

°°

°°

°°

¯

°°°

°°

°°

°

®

­

−

=

−

=

⋅

⋅

− + ⋅

−

⋅

= ⋅

⋅

⋅

−

⋅

−

−

⋅

⋅ ⋅

−

=

⋅

−

=

⋅

⋅

−

⋅

−

⋅

−

⋅

= ⋅

⋅

⋅

−

− ⋅

−

⋅

− ⋅

=

P P

W Z P P

W W Z

D

W Z P P

P W Z Z

W D Z W D

W Z P P

Z

P W W Z

D

W Z P P

W W Z

D

Q Q

Q Q Q Q

Q Q

Q Q Q

φ φ

φ φ φ

φ

FRV VLQ

VLQ H[S VLQ

H[S VLQ

H[S VLQ

VLQ H[S VLQ

7KHXVHRIIRUPXODVLQDVVRFLDWLRQZLWKWKHYDOXHVRIWKHQDWXUDOIUHTXHQF\ZQDQGGDPSLQJUDWLRPRI WKH DOWLWXGH FDSWXUH PRGH GHULYHG LQ WKH SUHYLRXV VHFWLRQ OHDGV WR WKH IROORZLQJ ILJXUH 7KH DOWLWXGH FDSWXUH WUDMHFWRU\WDQJHQWVWRWKHERXQGDU\RIWKH5$]RQHDWWLPHW

-500 0 500 1000 1500 2000 2500 3000 3500 4000 4500

334 336 338 340 342 344 346 348 350 352

Domain (Vz[fpm] ; h[FL]) superimposed on the domain where the computed time to co-altitude triggers an alarm

KF )/

6LPXODWHGDXWRSLORWG\QDPLFV WDQJHQWWRWKH5$OLPLW]RQH

%RXQGDU\RIWKH 5$]RQH 6LPXODWHGDXWRSLORW

G\QDPLFV

)LJ6LPXODWHGDXWRSLORWG\QDPLFZKLFKWDQJHQWVWKHERXQGDU\RIWKH5$]RQH

% 7XQLQJWKHSDUDPHWHUVRIWKHDOWLWXGHFDSWXUHPRGH

,W LV FOHDU IURP )LJ WKDW WKH DFWXDO YDOXHV RI WKH QDWXUDO IUHTXHQF\Z_Q DQG GDPSLQJ UDWLRP RIWKH DOWLWXGH FDSWXUHPRGHGRQRWDOORZWRILQGVRPHLQLWLDOFRQGLWLRQVLHDOWLWXGHYHUWLFDOVSHHGDWZKLFKWKHDOWLWXGHFDSWXUH

$PHULFDQ,QVWLWXWHRI$HURQDXWLFVDQG$VWURQDXWLFV

PRGHVZLWFKHVRQZKLFKDYRLGDQ5$,QGHHGQRLQWHUVHFWLRQSRLQWH[LVWVEHWZHHQWKHDXWRSLORWG\QDPLFZKLFK WDQJHQWVWKHERXQGDU\RIWKH5$]RQHDQGWKHWUDMHFWRU\9]KZKLFKSUHFHGHVWKHVWDUWLQJRIWKHDOWLWXGHFDSWXUH LHWKHSRLQWV9_]KZKLFKDUHVLWXDWHGEHIRUHVWDUWLQJWLPH7_V

$VDFRQVHTXHQFHWKHDFWXDOYDOXHVRIWKHQDWXUDOIUHTXHQF\Z_QDQGGDPSLQJUDWLRPVKDOOEHPRGLILHGLQRUGHU WRVDWLVI\WKHIROORZLQJFRQVWUDLQWVILUVWO\ZQDQGPVKDOOEHFKRVHQVXFKWKDWWKHVHFRQGRUGHUFRQWUROV\VWHPLV DOZD\VRXWVLGHRIWKH5$]RQHDQGVHFRQGO\DQLQWHUVHFWLRQSRLQWVKDOOH[LVWEHWZHHQWKHDXWRSLORWDOWLWXGHFDSWXUH G\QDPLFDQGWKHWUDMHFWRU\9]KZKLFKSUHFHGHVWKHHQJDJHPHQWRIWKHDOWLWXGHFDSWXUHPRGHDQGZKLFKLVZKLFK LVVLWXDWHGEHIRUHVWDUWLQJWLPH7_V

,QRUGHUWRVDWLVI\ERWKFRQVWUDLQWVRQHRIWKHVLPSOHVWRSWLRQVFRQVLVWVLQFRQVLGHULQJWKHPD[LPXPYDOXHRIWKH YHUWLFDO VSHHG DFKLHYDEOH E\ WKH DLUFUDIW GHQRWHG9]PD[ 7KH VHFRQG WLPH GHULYDWLYH RI WKH DOWLWXGH LV WKHQ FRQVWUDLQHGWREH]HURDW9]PD[$FFRUGLQJWRWKLVOHDGVWRWKHORZHVWYDOXHRIWKHDOWLWXGHFDSWXUHRUWKHKLJKHVW YDOXHLIWKHDLUFUDIWLVGHVFHQGLQJEHIRUHWKHDOWLWXGHFDSWXUHZKLFKLVGHQRWHGK9]_PD[

( )

(

_PD[

) (

_PD[

)

_PD[

PD[

9]

Z K P 9]

K K 9]

K Z 9]

Z P

Q F F

Q

Q ⋅ ⋅

−

Ÿ =

−

⋅

−

⋅

⋅

⋅

−

=

7KHQWKHSRLQWK9]_PD[9]_PD[LVFRQVWUDLQHGWREHORQJWRWKHWUDMHFWRU\GHILQHGE\7KDWPHDQVWKDWDWLPH WHVKDOOH[LVWVXFKWKDWWKHIROORZLQJUHODWLRQVKLSKROGV

( ) ( )

_»

¼

« º

¬ ª

⋅

⋅

⋅

⋅

⋅ −

⋅

− + ⋅

⋅

⋅

⋅

−

⋅

= −

»»

¼ º

««

¬

ª ⋅ ⋅

−

PD[

PD[

H[S

Q H Q

Q Q

F LF

Q 6 Z

Z 6 W P

Z $ 6 P Z

6

K K 9]

Z 9]

P

7KHIROORZLQJILJXUHLOOXVWUDWHVWKHVHFRQGRUGHUILOWHUG\QDPLFZKLFKVDWLVILHV

KF KLF τ9 6

K

9] JHQHUDWHG5$

9]PD[

K9]PD[

$FWXDO9]

$OWLWXGHFDSWXUH

6HFRQGRUGHU ILOWHUG\QDPLFV

DXWRSLORW

)LJ6HFRQGRUGHUILOWHUG\QDPLFZKLFKWDQJHQWVWKHERXQGDU\RIWKH5$]RQH

$VIDUDVWKHDFWXDOYHUWLFDOVSHHGRIWKHDLUFUDIWGXULQJWKHFOLPERUGHVFHQWLVORZHUUHVSHFWLYHO\JUHDWHUWKDQ WKH PD[LPXP YHUWLFDO VSHHG9]_PD[ WKH XQLTXHQHVV RI WKH VROXWLRQ RI WKH GLIIHUHQWLDO HTXDWLRQ HQVXUHV WKDW DFWXDOG\QDPLFRIWKHDOWLWXGHFDSWXUHPRGHZLOOQHYHUHQWHULQVLGHWKHDUHDZKHUHDQ5$LVJHQHUDWHG

,Q DGGLWLRQ HTXDWLRQ LQGLFDWHV WKDW WKH DOWLWXGH FDSWXUH PRGH LV VZLWFKHG RQ DV IDU DV WKH IROORZLQJ LQHTXDOLW\KROGVZKHUHK_FLVWKHFRPPDQGHGDOWLWXGHDQG9]WKHDFWXDOYHUWLFDOVSHHGGXULQJFOLPERUGHVFHQW

Z 9]

K P K

Q

F ⋅ ⋅

<

−

& &DVHVWXG\

)RUWKHFDVHVWXG\LWZLOOEHDVVXPHGWKDWWKHSURGXFWP⋅ZQUHPDLQVFRQVWDQWLQRUGHUQRWWRPRGLI\WKHUHDOSDUW RIWKHSROHVRIWKHSUHYLRXVO\LGHQWLILHGDXWRSLORWPRGH7KHQWKHPD[LPXPYDOXHRIWKHYHUWLFDOVSHHGDFKLHYDEOH E\ WKH DLUFUDIW WKDW LV9]_PD[ LV VHW WR IHHWPLQ 6HWWLQJK_LF−K_F WR IHHW WKDW LV WKH YHUWLFDO VHSDUDWLRQ VWDQGDUGEHWZHHQDLUFUDIWSURYLGHGE\$LU7UDIILF&RQWUROWKHWKUHVKROG6WRVHFRQGVWKDWLVWKHXSSHUERXQGRI WKHWKUHVKROGXVHGE\7&$6DQGNHHSLQJWKHSURGXFWP⋅ZQWR⋅ UDGVHFWKHQDWXUDOIUHTXHQF\ZQ ZKLFKPDWFKHV HTXDWLRQ LVIRXQGWREHUDGVHFZKLFKLV DERXWORZHUWKDQWKHQDWXUDOIUHTXHQF\

ZKLFKKDVEHHQHVWLPDWHGLQWKHSUHYLRXVVHFWLRQWKDWLVUDGVHFZKHUHDVWKHGDPSLQJUDWLRPPRYHVIURP WRLQRUGHUWRNHHSFRQVWDQWWKHSURGXFWP⋅Z_Q

$PHULFDQ,QVWLWXWHRI$HURQDXWLFVDQG$VWURQDXWLFV

7KHIROORZLQJILJXUHSUHVHQWVWKHGRPDLQ9]K7KHVLPXODWLRQRIWKHFRQWUROODZFRXSOHGZLWKWKHDLUFUDIW PRGHOVWDUWVDW)/LQDFFRUGDQFHZLWK7KLVLVIHHWXQGHUWKHFOHDUHGIOLJKWOHYHO

KF )/

)/ %RXQGDU\RIWKH

5$]RQH

$GMXVWHGDXWRSLORW G\QDPLFV

6WDUWLQJWLPHRIWKH VLPXODWHGWUDMHFWRU\

)LJ$GMXVWHGDXWRSLORWG\QDPLFZKLFKWDQJHQWVWKHERXQGDU\RIWKH5$]RQH

1HYHUWKHOHVVWKHWLPHQHHGHGWRFDSWXUHWKHFOHDUHGIOLJKWOHYHOLVLQFUHDVHGDVLOOXVWUDWHGRQWKHIROORZLQJILJXUH ZKLFKSUHVHQWWKHUDGDUFDSWXUHGDQGVLPXODWHGYHUWLFDOSURILOHV,WFDQEHVHHQWKDWWKHVLPXODWHGYHUWLFDOSURILOHLV IHHWXQGHUWKHFOHDUHGIOLJKWOHYHODERXWVHFRQGVODWHUWKDQWKHUDGDUFDSWXUHGYHUWLFDOSURILOH'RWVUHSUHVHQW WKHLQVWDQWVZKHUHDQ5$LVLVVXHGRQERDUGWKHUDGDUFDSWXUHGWUDMHFWRU\ZKHUHDVWKHWKLQFRQWLQXRXVOLQHUHSUHVHQWV WKHDOWLWXGHFDSWXUHRIWKHWXQHGDXWRSLORWLWFDQEHVHHQWKDWQR5$DSSHDUV

)LJ5DGDUFDSWXUHGDQGVLPXODWHGYHUWLFDOSURILOH

8VLQJWKHDFWXDO7&$6YHUVLRQORJLFRQWKHVLPXODWHGYHUWLFDOSURILOHVKRZVWKDWD7UDIILFDGYLVRU\7$LV VWLOOLVVXHG%XWDVH[SHFWHGQR5$LVLVVXHG

9 &RQFOXVLRQ

,QWKLVSDSHUWKH7&$6UHVROXWLRQDGYLVRULHVJHQHUDWLRQSURFHVVGXULQJOHYHORIIVLWXDWLRQKDVEHHQRYHUYLHZHG DQGLOOXVWUDWHGRQDUHDOVLWXDWLRQFDSWXUHGWKURXJKPRGH6UDGDUZLWK5$GRZQOLQNVHUYLFH,WKDVEHHQHPSKDVL]HG WKDWRQHFRPPRQW\SHRIUHVROXWLRQDGYLVRULHVLVWKDWZKLFKLVLVVXHGZKHQDLUFUDIWDUHH[SHFWHGWROHYHORII IHHW DSDUW DQG DUH FRQYHUJLQJ DW WKH VDPH WLPH 7KHVH UHVROXWLRQ DGYLVRULHV RIWHQ VXEVHTXHQWO\ FODVVHG DV µRSHUDWLRQDOO\XQQHFHVVDU\¶IURPDQDLUWUDIILFFRQWUROVWDQGSRLQWFDQEHSHUFHLYHGDVGLVWXUELQJE\FRQWUROOHUVDQG

$PHULFDQ,QVWLWXWHRI$HURQDXWLFVDQG$VWURQDXWLFV

E\SLORWV7KLVLVVXHKDVEHHQWDFNOHGE\SURSRVLQJDPHWKRGWRSUHYHQW5$WULJJHULQJGXULQJOHYHORIIVLWXDWLRQ 7KLVPHWKRGLVEDVHGRQWKHVHWWLQJRIWKHDOWLWXGHDWZKLFKWKHDXWRSLORWVZLWFKHVWRZDUGVWKHDOWLWXGHFDSWXUHPRGH DQGRQDQLQWULQVLFSDUDPHWHURIWKHDXWRSLORWDOWLWXGHFDSWXUHPRGHQDPHO\LWVQDWXUDOIUHTXHQF\

$SSO\LQJWKHSURSRVHGPHWKRGWRWKHLOOXVWUDWLYHH[DPSOHSUHYHQWVWKHLVVXDQFHRIUHVROXWLRQDGYLVRULHVZKHQ DLUFUDIW DUH H[SHFWHG WR OHYHORII IHHW DSDUW 7KH PHWKRG ZKLFK KDV EHHQ GHYHORSHG LV RQH RI WKH VLPSOHVW RSWLRQVDQGDOORZVDOWLWXGHFDSWXUHZLWKRXWWULJJHULQJUHVROXWLRQDGYLVRULHVIRUYHUWLFDOVSHHGVXSWRIHHWPLQ 7KHWLPHQHHGHGE\WKHDLUFUDIWWRFDSWXUHWKHFOHDUHGIOLJKWOHYHOLVLQFUHDVHG

)XWXUHZRUNZLOOIRFXVRQLPSURYLQJWKHWLPHQHHGHGWRFDSWXUHWKHFOHDUHGIOLJKWOHYHOXQGHUWKHFRQVWUDLQWWR NHHSSRVLWLYHWKHGLVWDQFHEHWZHHQWKHDXWRSLORWDOWLWXGHFDSWXUHG\QDPLFDQGWKHERXQGDU\RIWKH5$]RQH$OVRWKH LQYHVWLJDWLRQRIWKHWXQLQJRIWKHDOWLWXGHFDSWXUHG\QDPLFLQRUGHUWRGHDOZLWKKLJKHUUDWHVRIFOLPEGHVFHQWVKRXOG EH DGGUHVVHG 7KH DOWLWXGH TXDQWL]DWLRQ DQG DOSKD ± EHWD DOWLWXGH WUDFNHU XVHG E\ 7&$6 VKRXOG EH WDNHQ LQWR DFFRXQW

,Q DGGLWLRQ H[WHQGLQJ WKH SURSRVHG PHWKRG WR WKH FDVHV ZKHUH WKH LQWUXGHU DLUFUDIW LV QRW OHYHO KDV WR EH FRQGXFWHG'RHVDQ\ERG\ZLVKWRFRQGXFWUHDOWLPHH[SHULPHQWV"

$FNQRZOHGJPHQWV

7KHDXWKRUVZLVKWRWKDQN-HDQ0DUF/RVFRVDQG&KULVWLDQ$YHQHDXIRUKHOSIXOLQSXWVDQGFRPPHQWV

5HIHUHQFHV

$YHQHDX&7&$67UDIILFDOHUWDQG&ROOLVLRQ$YRLGDQFH6\VWHPRYHU\HDUVRIZRUNE\'61$WRLPSURYH

$70VDIHW\5HYXH7HFKQLTXHGHOD'7,QRSS

KWWSZZZGVQDGWLDYLDWLRQFLYLOHJRXYIUDFWXDOLWHVUHYXHVJEUHYXHJEVLWHBHQLQWURGXFWLRQKWPO

(XURFRQWURO$&$63URJUDPPH5$VDQGIWOHYHORIIPDQRHXYUHV$&$6,,EXOOHWLQ1ƒ0DUFK

(VKHOE\0($LUFUDIWSHUIRUPDQFH7KHRU\DQG3UDFWLFH$,$$(GXFDWLRQ6HULHV

.KDOLO+.1RQOLQHDU6\VWHPV7KLUG(GLWLRQ3UHQWLFH+DOO

,QWHUQDWLRQDO&LYLO$YLDWLRQ2UJDQL]DWLRQ$QQH[9ROXPH,96XUYHLOODQFH5DGDUDQG&ROOLVLRQ$YRLGDQFH 6\VWHPV7KLUG(GLWLRQ-XO\

2JDWD.0RGHUQ&RQWURO(QJLQHHULQJWK(GLWLRQ3UHQWLFH+DOO