New Directions in Nonlinear Observer Design

5 downloads 0 Views 2MB Size Report
ho). ] yo. (2.12) bsh = Uh o yo. (2.13) where de 5 ?3 is the measured acceleration, and jo = jo V(ωo lh)V(ωo lh)so is the plumb-bob gravity where jo = [0, 0, gG]W.
Wklv lv sdjh l Sulqwhu= Rsdtxh wklv

Qhz Gluhfwlrqv lq Qrqolqhdu Revhuyhu Ghvljq Khqn Qlmphlmhu4 dqg Wkru L1 Irvvhq5 +Hgv1, 

Idfxow| ri Pdwkhpdwlfdo Vflhqfhv

2

Ghsduwphqw ri Hqjlqhhulqj F|ehuqhwlfv

Ghsw1 ri V|vwhpv/ Vljqdov dqg Frqwuro

Qruzhjldq X1 ri Vflhqfh dqg Whfkqrorj|

Xqlyhuvlw| ri Wzhqwh

Wurqgkhlp

Wkh Qhwkhuodqgv

Qruzd|

dqg

dqg

Idfxow| ri Phfkdqlfdo Hqjlqhhulqj

DEE Lqgxvwul DV

Hlqgkryhq Xqlyhuvlw| ri Whfkqrorj|

Pdulqh Glylvlrq/ Rvor

Wkh Qhwkhuodqgv

Qruzd|

f 4 u `

@

^{> |> } `

@ @ @ @ @

W

{ | }W ^yQ > yH > yG `W ^4 > ===> q `W ^4 > ===> q `W ^t4 > t5 > t6 > t7 `W ^y > y > y `

= Dqjxodu yhorflw| ri wkh erg| iudph uhodwlyh wr wkh orfdo iudph1 = Glvwdqfh iurp fhqwhu ri Hduwk wr wkh qdyljdwlrq v|vwhp lq Hduwk frruglqdwhv1 = Yhorflw| ri yhklfoh lq Hduwk frruglqdwhv1 = Qruwk/ Hdvw dqg Grzq yhorflwlhv1 = Yhfwru ri q svhxgrudqjh revhuydwlrqv1 = Yhfwru ri q ghowdudqjh revhuydwlrqv1 = Hxohu sdudphwhuv +xqlw txdwhuqlrqv,1

415 Prwlydwlrq Wkh ghyhorsphqw ri JSV whfkqrorj| wkh odvw wzr ghfdghv kdv kdg d juhdw lpsdfw rq qdyljdwlrq1 Iru orfdo duhd rshudwlrqv/ lw lv qrz srvvleoh wr dfklhyh phwhu dffxudf| e| xvlqj Glhuhqwldo JSV +GJSV,/ dqg vxe0ghflphwhu df0 fxudf| e| xvlqj Fduulhu Glhuhqwldo JSV +FGJSV,1 Wkh ghyhorsphqw ri zlgh duhd dxjphqwdwlrq v|vwhpv +ZDDV, lv h{shfwhg wr jlyh phwhu dffx0 udf| dfurvv hqwluh frqwlqhqwv1 Krzhyhu/ wkh lqwhjulw| dqg uholdelolw| ri JSV duh qrw rq d vx!flhqw ohyho1 Lqwhuihuhqfh rq gdwd olqnv/ dqg vhyhudo w|shv ri glvwxuedqfh fdxvlqj orvv ri orfn rq vdwhoolwhv/ duh frpprq sureohpv iru pdq| JSV dssolfdwlrqv1 Wkh odwwhu fdq eh d sureohp dw odwlwxghv ehwzhhq 93 dqg :3 ghjuhhv Qruwk/ zkhuh wkh vdwhoolwh fryhudjh lv orz wr ehjlq zlwk1 Lq dgglwlrq wr wkh wlph vshqw uhdtxlulqj vdwhoolwh vljqdov/ wkh dpeljxlw| lqwhjhuv pxvw dovr eh uhfdofxodwhg diwhu hdfk orvv ri orfn lq dssolfdwlrqv zkhuh FGJSV lv xvhg1 D vwudsgrzq Lqhuwldo Qdyljdwlrq V|vwhp +LQV, lv edvlfdoo| d foxvwhu ri dffhohurphwhuhv dqg j|urv nqrzq dv dq Lqhuwldo Phdvxuhphqw Xqlw +LPX,/ dqg d frpsxwhu1 Wkh LQV frpsxwhu fdofxodwhv srvlwlrq/ yhorflw|/ khdglqj dqg dwwlwxgh xvlqj wkh vwudsgrzq htxdwlrqv1 Wkh LQV lv vhoi0frqwdlqhg/ dqg grhv qrw ghshqg rq dq h{whuqdo vljqdo wr ixqfwlrq surshuo|1 Krzhyhu/ wkh orqj whup guliw uhvxowlqj iurp lqwhjudwlqj vpdoo phdvxuhphqw huuruv lq wkh j|urv dqg dffhohurphwhuv lv d pdmru sureohp/ dqg rqo| yhu| h{shqvlyh xqlwv fdq eh h{shfwhg wr kdyh vx!flhqw dffxudf| iru d orqj shulrg ri wlph zlwkrxw uh0lqlwldol}dwlrq1 Wkh frpsohphqwdu| surshuwlhv ri JSV dqg LQV/ vxjjhvw wkdw lqwhjudwlrq ri JSV zlwk orz0frvw LQV zloo jlyh dq drugdeoh/ kljko| uholdeoh/ dqg dffxudwh vroxwlrq1 Wkhuh duh vhyhudo ohyhov ri lqwhjudwlrq srvvleoh/ vhh Pd|ehfn ^7`1 Wkh prvw frpprq dssurdfkhv vkrzq lq Iljxuhv 4 dqg 5/ erwk vxewudfw wkh LQV srvlwlrq phdvxuhphqw iurp wkdw ri wkh JSV v|vwhp/ dqg ihhg wklv huuru vljqdo lqwr d Ndopdq owhu +NI,1 Wkh NI prgho lv xvxdoo| d Pdunry surfhvv ru lqwhjudwhg zklwh qrlvh iru hdfk ri wkh huuru vrxufhv lq wkh JSV dqg LQV v|vwhpv1 Wkh hvwlpdwhg LQV huuruv duh vxevhtxhqwo| hlwkhu vxewudfwhg iurp wkh LQV phdvxuhphqw wr jlyh wkh rxwsxw vljqdo +rshq0orrs ,/ ru ihg edfn wr wkh LQV vwudsgrzq frpsxwdwlrq surfhvvru +forvhg0orrs ,1 Zkhq wkh srvlwlrq phdvxuhphqwv iurp JSV duh orvw/ wkh NI lv qrw xsgdwhg/ dqg wkh LQV kdv

51

Qrqolqhdu Revhuyhu Ghvljq iru Lqwhjudwlrq ri GJSV dqg LQV

46:

GPS Receiver

GPS

GPS Measurements

P,V,q

Kalman filter

+ Kalman Filter Strapdown Equations

Gyros Accelerometers

P,V,q

Corrected P,V,q

+

IMU INS

ILJXUH 41 Rshq orrs lqwhjudwlrq ri LQV dqg JSV1

wr frdvw wkurxjk wkh rxwdjh xvlqj wkh odvw huuru xsgdwh iurp wkh owhu1 Wklv lv riwhq uhihuuhg wr dv xqfrxsohg lqwhjudwlrq lq wkh rshq0orrs fdvh/ dqg orrvh lqwhjudwlrq lq wkh forvhg0orrs fdvh1 Wkh owhu fdq eh ghvljqhg zlwk d vorz xsgdwh udwh gxh wr wkh vorz g|qdplfv ri wkh huuru prgho1 JSV uhfhlyhuv kdyh dq lqwhuqdo NI wkdw frpsxwhv srvlwlrq/ yhorflw| dqg wlph +SYW, vroxwlrqv/ dqg wkh rxwsxwv duh fruuhodwhg w|slfdoo| zlwk 443 vhfrqgv fruuhodwlrq wlph1 Lq rughu wr dyrlg sureohpv zlwk wkh vhfrqg NI/ wklv owhu qhhgv wr eh xsgdwhg vx!flhqwo| vorzhu1 Lq dgglwlrq/ wkh wlplqj dqg odjv ehwzhhq wkh wzr owhuv fdq eh gl!fxow wr suhglfw dqg frqwuro/ vhh Wd}duwhv dqg Pdun ^    > q`/ dqg {lv 5 ?6 lv wkh srvlwlrq ri vdwhoolwh l1 Wklv srvlwlrq f lv wkh vshhg ri oljkw/   lv wkh forfn eldv dqg %l lv wkh frpsrvlwh ri rwkhu vpdoohu svhxgrudqjh huuruv dqg zkhuh

lv wudqvplwwhg zlwk wkh qdyljdwlrq phvvdjh1

51

Qrqolqhdu Revhuyhu Ghvljq iru Lqwhjudwlrq ri GJSV dqg LQV

qrlvh1 Qrwh wkdw



lv frpprq wr doo

q

46
{3 lv dq hvwlpdwh ri {/ dqg { lv wkh srvlwlrq huuru1 Li zh l @

vwdfn

s

q phdvxuhphqwv wrjhwkhu lq d yhfwru/ zh rewdlq=

 @ {  W zkhuh { @ {W >  5 ?7>  5 ?q dqg 5 9

@ 997

+517,

4{ 4| 4} 4 5{ 5| 5} 4 1 1 1

1 1 1

1 1 1

1 1 1

q{ q| q} 4

6 : : : 8

5 ?q{7

+518,

{ fdq qrz eh irxqg iurp

{@ | |   4 W Qrwlfh wkdw/ wkh svhxgr0lqyhuvh  @ W   Khqfh/ e| xvlqj d NI iru lqvwdqfh/

{ dqg 

+519, ri

 dozd|v h{lvwv1

fdq eh frpsxwhg iurp d plql0

pxp ri irxu svhxgrudqjh phdvxuhphqwv1 Wkh glvwdqfh iurp wkh xvhu wr d vdwhoolwh lq ylhz lv dssur{lpdwho|

53> 333 np/ dqg  lv wkhuhiruh qrw yhu|

vhqvlwlyh wr srvlwlrq huuruv vlqfh l lv wkh olqh0ri0vljkw +ORV, xqlw yhfwru iurp wkh xvhu wr vdwhoolwh l= Wkxv/ zlwk rqh h{fhswlrq zklfk zloo eh dg0 guhvvhg odwhu/ zh fdq frqvlghu

 frqvwdqw wkurxjkrxw wklv fkdswhu/ hyhq

wkrxjk lw lv d ixqfwlrq ri erwk vdwhoolwh dqg xvhu srvlwlrqv1

Zkhq fdofxodwlqj srvlwlrq rqo|/ dqg qrw forfn eldv/ zh zloo xvh wkh iro0 orzlqj ORV pdwul{

5 9

 @ 997 Wkh svhxgr0lqyhuvh

4{ 4| 4} 5{ 5| 5} 1 1 1

1 1 1

1 1 1

q{ q| q}

6 : : : 8

5 ?q{6

+51:,

| ri  zloo dozd|v h{lvw1

Wkh ghowdudqjh htxdwlrq lv zulwwhq=

l

@ l+yvl  y, . fi  . %l

+51;,

473

51

Qrqolqhdu Revhuyhu Ghvljq iru Lqwhjudwlrq ri GJSV dqg LQV

zkhuh yvl lv wkh yhorflw| ri vdwhoolwh l/ i  lv wkh forfn guliw +ru iuhtxhqf| eldv,/ dqg %l lv wkh frpsrvlwh ri rwkhu vpdoohu ghowdudqjh huuruv dqg qrlvh1 y dqg i @ fi  fdq eh fdofxodwhg htxlydohqwo| wr +519, iurp zkhuh y @ ^yW >

y@ |

+51 ru gluhfwo| iurp y@ |+  vdw,

zkhuh y @ ^yW > i `W > dqg

vdw

+5143,

@ ^4yv4>    > qyvq`W

+5144,

lv wkh nqrzq vdwhoolwh yhorflw| frpsrqhqw ri wkh ghowdudqjh yhfwru1 Wkh pdlq irfxv ri wklv fkdswhu lv wkh LQV g|qdplfv/ dqg iru vlpsolflw| lw lv dvvxphg wkdw %l @ 3 dqg %l @ 31 Wkh forfn eldv dqg forfn guliw duh wkh rqo| JSV huuruv wkdw zloo eh hvwlpdwhg wrjhwkhu zlwk wkh LQV huuruv1 Pruh ghwdlov derxw JSV qdyljdwlrq surfhvvlqj fdq eh irxqg lq Sdunlqvrq dqg Vslonhu ^8`/ Fkdswhu 3> jG `W j

@

+5145, +5146,

o ,V+$lho ,so jo  V+$lh +5147,

51

Qrqolqhdu Revhuyhu Ghvljq iru Lqwhjudwlrq ri GJSV dqg LQV

474

sh 5 ?6 lv wkh srvlwlrq yhfwru iurp wkh fhqwhu ri wkh Hduwk wr wkh erg| o lv wkh urwdwlrq udwh ri wkh iudph ghfrpsrvhg lq wkh Hduwk0{hg iudph1 $lh o Hduwk ghfrpsrvhg lq wkh orfdo iudph/ dqg $ho lv wkh urwdwlrq udwh ri wkh orfdo iudph uhodwlyh wr wkh Hduwk0{hg iudph +gxh wr yhklfoh pryhphqw o dqg $hoo duh lq ryhu wkh Hduwk vxuidfh, ghfrpsrvhg lq wkh orfdo iudph1 $lh e jhqhudo vorz frpsduhg wr wkh dqjxodu yhorflw| $oe ri wkh erg| iudph uhodwlyh wr wkh QHG iudph1 Wkh| fdq wkhuhiruh/ hyhq wkrxjk wkh| duh ixqfwlrqv ri wkh olqhdu yhorflw| yo / eh dvvxphg nqrzq1 Ilqdoo|/ jG lv wkh dffhohudwlrq ri judylw|/ zklfk fdq eh hvwlpdwhg iurp wkh iroorzlqj htxdwlrq=

jG @ J4 +4  5k@u3 . 5hvlq5,  J5+4  6vlq5,  J6+4  6vlq5 . k@u3,frv5

+5148,

zkhuh J4> J5 / J6 / u3 dqg h duh srvlwlyh Hduwk prgho frqvwdqwv/ k lv dowlwxgh dqg  lv jhrghwlf odwlwxgh1 Iurp +5148, zh vhh wkdw wkh dffhohudwlrq ri judylw| ghfuhdvhv dv wkh khljkw lqfuhdvhv1 Wkh glhuhqwldo htxdwlrq iru wkh khljkw lv=

k‚ @ j

G

d

}

+5149,

Wkxv/ wkh LQV vwdqg0dorqh yhuwlfdo fkdqqho lv xqvwdeoh vlqfh wklv lv d sxuh grxeoh lqwhjudwru1 Lqwhjudwlrq zlwk JSV khljkw phdvxuhphqwv fdq eh xvhg wr vwdelol}h wkh yhuwlfdo fkdqqho1 Krzhyhu/ lq wkh hyhqw ri d JSV gursrxw/ wkh yhuwlfdo fkdqqho ri wkh LQV zrxog jr xqvwdeoh1 Wkhuhiruh/ d edurphwhu lv xvxdoo| hpsor|hg lq wkh lqwhjudwlrq owhu iru dssolfdwlrqv lq wkh dlu/ zkloh iru pdulqh dssolfdwlrqv d khljkw frqvwudlqw fdq eh xvhg1

615 Hduwk Iudph Uhsuhvhqwdwlrq Lw kdv ehhq vxjjhvwhg/ Zhl dqg Vfkzduw} ^45`/ wkdw dq Hduwk iudph phfkd0 ql}dwlrq fdq rhu vrph dgydqwdjhv ryhu wkh orfdo ohyho iudph1 Dq lqfuhdvh lq frpsxwdwlrqdo vshhg zdv irxqg/ zkloh wkh dffxudf| zdv irxqg wr eh frpsdudeoh wr wkdw ri d orfdo ohyho lpsohphqwdwlrq1 Rq wkh rwkhu kdqg/ dv vhhq iurp +5147,/ wkh judylw| htxdwlrqv duh udwkhu vlpsoh dqg gluhfwhg rqo| dorqj wkh yhuwlfdo d{lv lq wkh orfdo ohyho iudph1 Lq wkh Hduwk iudph/ krzhyhu/ gl!fxowlhv dulvh lq uhsuhvhqwlqj wkh Hduwk*v judylwdwlrqdo hog suhflvho| lq wkh qdyljdwlrq frpsxwhu/ vhh Wlwwhuwrq dqg Zhvwrq ^43`1 Judylw| zloo kdyh d frpsrqhqw dorqj doo wkuhh d{hv/ dqg doo wkuhh d{hv qhhg wr eh vwdelol}hg1 Wkh xvh ri edurphwhu gdwd wr vwdelol}h doo wkuhh d{hv zrxog uhtxluh pruh lqyhvwljdwlrq lqwr wkh sudfwlfdelolw| dqg ghwdlo ri lpsohphqwdwlrq/ vhh h1j1 Wdqj dqg Krzhoo ^;`1 Lq wklv fkdswhu/ zh zloo qrw ghdo zlwk wklv sureohp/ ehfdxvh JSV phdvxuhphqwv duh dvvxphg wr eh suhvhqw dw doo wlphv/ exw zh suhvhqw dq revhuyhu phfkdql}hg lq wkh h0iudph dv dq dowhuqdwlyh wr wkh v|vwhp lqwhjudwru1 Zh zloo wkhuhiruh frqvlghu wkh yhorflw| htxdwlrq ghfrp0

475

51

Qrqolqhdu Revhuyhu Ghvljq iru Lqwhjudwlrq ri GJSV dqg LQV

srvhg lq wkh Hduwk iudph/ wkdw lv=

yb sb

h h

U U d . U j  5V+$ ,y @y @

h o

o e

e

h o

o

h lh

+514:, +514;,

h

h

616 Dqjxodu Yhorflw| Htxdwlrqv Wkh urwdwlrq pdwul{ Uoe / zklfk jlyhv xv uroo/ slwfk/ dqg |dz dqjohv/ fdq eh

fdofxodwhg iurp

e, b oe @ Uoe V+$oe U

+514 t @ ^> %W `W >  5 ?> % 5 ?6

r

+5154,

Wkh prvw frpprq huuru uhsuhvhqwdwlrq xvhg iru xqlw txdwhuqlrqv lv t  t  t 5 K/ zkhuh wkh surgxfw lv ghqhg dv

t @t  t@



a %a

 

 %

  / a%a

%aW

a  V+%a,



 %



+5155,

 @ L dqg t  @U a W U= Zkhq wkh huuru lv }hur/ U  Wklv lv htxlydohqw wr U W ^ 4> 3> 3> 3` 1 Wkh g|qdplfv dvvrfldwhg zlwk wkh xqlw txdwhuqlrqv wdnhv wkh irup= tb @ lv



b %b



@

4 5



%W

^ L . V+%,`



e $le

5 4



%W ^ L  V+%,`



$loo

@

@

+5156,

Ilqdoo|/ zh qrwh wkdw wkh uhodwlrq ehwzhhq wkh urwdwlrq pdwul{ Uoe dqg t

Uoe +t, @ L6{6 . 5V+%, . 5V5 +%,

+5157,

7 Qrqolqhdu Revhuyhu Ghvljq Wkh pdlq wdvn ri wkh revhuyhuv lv wr hvwlpdwh wkh prvw lpsruwdqw huuruv ri dffhohurphwhuv dqg j|urv1 Lq dgglwlrq/ owhulqj ri JSV dqg LQV srvlwlrqv/

51

Qrqolqhdu Revhuyhu Ghvljq iru Lqwhjudwlrq ri GJSV dqg LQV

476

yhorflwlhv/ dqg dqjxodu yhorflw| phdvxuhphqwv duh dfklhyhg1 Wkh prvw grpl0 qdwlqj huuruv duh xvxdoo| eldvhv/ vfdoh0idfwru huuruv dqg plvdoljqphqw huuruv1 Wkh eldvhv dqg vfdoh0idfwru huuruv duh xvxdoo| suhfrpshqvdwhg wkurxjk fdo0 leudwlrq dqg wkhuprphwhu ihhgedfn/ dqg wkh plvdoljqphqw huuruv +zklfk lqfoxgh qrq0ruwkrjrqdolw| ri wkh vhqvru d{hv, duh pdgh dv vpdoo dv srvvl0 eoh gxulqj prxqwlqj ri wkh LPX1 Krzhyhu/ wkhuh duh vwloo uhvlgxdo huuruv wkdw qhhg wr eh hvwlpdwhg1 Lq wkh iroorzlqj/ wkh huuruv zloo eh prghohg lq d vwdqgdug zd| dv=

$

e le

de

zkhuh

@ ^L . +> ,$lpx` . e4 @ ^L . +> ,dlpx` . e5

5   @ +> !, @ 7 ! !

{

!  !

{|

|{ }{

|

}|

+5158, +5159,

6 ! ! 8  {} |}

+515:,

}

Khuh  @ ^{ > | > } `W duh wkuhh vfdoh idfwru huuruv dqg

! @ ^! > ! > ! ! > ! > ! ` duh vl{ vpdoo plvdoljqphqw dqjohv1 e4 5 ?6 dqg e5 5 ?6 uhsuhvhqw j|ur W

{|

{}

|{>

|}

}{

}|

dqg dffhohurphwhu eldvhv/ uhvshfwlyho|1 Qrwlfh wkdw doo huuru vljqdov duh gh0 frpsrvhg lq wkh erg| iudph1 Wkh pdjqlwxgh ri wkhvh huuruv duh lq jhqhudo gluhfwo| sursruwlrqdo wr wkh sulfh ri wkh LPX1 Wkh hhfw ri zklwh qrlvh rq vwdelolw| zloo qrw eh frqvlghuhg lq wklv fkds0 whu1 Krzhyhu/ doo uhvxowv fdq hdvlo| eh h{whqghg wr vkrz wkdw wkh vroxwlrqv zloo frqyhujh wr d edoo durxqg wkh ruljlq zkhq zklwh qrlvh lv frqvlghuhg1 Zlwkrxw orvv ri jhqhudolw|/ zh zloo wkhuhiruh uhvwulfw rxu dqdo|vhv wr wkh fdvh ri }hur qrlvh1

714 Dqjxodu Yhorflw| Revhuyhu Zkhq eldv/ vfdoh0idfwru dqg plvdoljqphqw huuruv duh lqfoxghg/ +5156, zulwwhq=



lv



  W 4  % @ 5 ^L . V+%,` ^+L . 4,$lpx . e4`   W 4  % +515;,  5 ^L  V+%,` $loo zkhuh 4 @ +> ,= Wkh j|ur huuru prghov duh dvvxphg wr eh ghvfulehg

b %b

e| wkh 4vw0rughu prghov=

b @ W4 4e4 . z4 b @ W5 4 . z5 b @ W6 4 . z6

e4

+515 5> 6,1 Wkh pdwulfhv Wl A 3

l @ 4> 5> 6, duh gldjrqdo pdwulfhv ri wlph frqvwdqwv/  @ ^{ > | > } `W wkuhh j|ur vfdoh idfwru huuruv/ dqg  @ ^{| > {} > |{ > |} > }{ > }| `W +

@

duh duh

ej|ur uhsuhvhqwv wkh eldvhv ri vl{ vpdoo j|ur plvdoljqphqw dqjohv1 e4 wkh j|urv1 Wkh qrqolqhdu revhuyhu ri Vdofxghdq ^9` lv h{whqghg wr lqfoxgh eldv dqg huuru xsgdwh odzv dffruglqj wr=



ab %ab





k

@ 54 ^aL .%aV+%a,` +L . a 4,$lpx . ea4 . N4%vjq+, W

l





W  45 ^aL %aV+%a,` $loo 4 4a ab 4 @ W e vjq+, 4 e4 . % 5

+5165, +5166,

4 ab @ W5 4 a . gldj+%, $lpx vjq+, 5 4 4 ba @ W a . +%, $lpx vjq+, 6  5

zkhuh

5 9 9 9 +%, @ 9 9 9 7

%4 3 3 3 3 %6

3 3 %5 3 %6 3

3 %4 3 %5 3 3

+5167, +5168,

6 : : : : : : 8

+5169,

Wkh huuru prgho lv irxqg e| frpelqlqj +515 %> e4> > , @ + 4> 3> 3> 3> 3,1 5

+516:, +516;, +516 3> 3> 3> 3, ri wkh huuru prgho +516:,+5173, duh h{srqhqwldoo| vwdeoh1 Surri1

Y

@

Frqvlghu wkh iroorzlqj O|dsxqry ixqfwlrq fdqglgdwh=

4 5

 W4 e 4 . 4  W  . 4  W  . 4 e 5

5

5



  4,5 . %W % + . 4,5 . %W %



li li

  3  ? 3

+5174,

51

Qrqolqhdu Revhuyhu Ghvljq iru Lqwhjudwlrq ri GJSV dqg LQV

478

Wkh wlph ghulydwlyh dorqj wkh wudmhfwru| ri +516:,+5173, lv=

Yb @ eW4 eb 4 . W b . W b .

 b  b

li   3 li  ? 3

4 W 4 @ eW4 W vjq+ , 4 e4  e4 % 5  W W5 4   45 W gldj+%, $lpx vjq+, W W6 4  45 W +%, $lpx vjq+, l 4 k 4  N4 %vjq+ , vjq+, . %W  4 $lpx . e 5 4 4 4 @ eW4 W W W6 4  %W N4% W W 4 e4   5  5 ? 3> ; e4> > > % 9@ 3

+5175,

zkhuh zh kdyh xvhg=

b . %W %b  3  4 $lpx @  W gldj+%,$lpx .  W +%, $lpx %W  Iurp wkh uhodwlrq 5 . %W %  4 lw lv vhhq wkdw dv % mm $ 4 h{srqhqwldoo|1 Wklv frpsohwhv wkh surri1

+5176, +5177,

$ 3 h{srqhqwldoo|/

715 Yhorflw| dqg Srvlwlrq Revhuyhuv Lq wklv vhfwlrq/ zh zloo vkrz krz wljkwo| lqwhjudwhg revhuyhuv iru srvlwlrq dqg yhorflw| fdq eh ghvljqhg1 Wkh revhuyhuv zloo eh xsgdwhg zlwk udz JSV

phdvxuhphqwv/ lq rughu wr wdnh dgydqwdjh ri JSV phdvxuhphqwv zkhq wkhuh lv ohvv wkdq irxu vdwhoolwhv dydlodeoh1 Zlwk ohvv wkdq irxu vdwhoolwhv dydlodeoh/ wkhvh revhuyhuv zloo vwloo shuirup ehwwhu wkdq wkh vwdqg0dorqh LQV1 Vhh h1j1 Ohzdqwrzlf} ^6` iru d vwxg| rq judfhixo ghjudgdwlrq ri dq lqwhjudwhg vroxwlrq zlwk ohvv wkdq irxu vdwhoolwhv lq ylhz1 Wkh vwdelolw| surriv ehorz zloo ri frxuvh qrw eh ydolg lq wklv fdvh1 Qrwh wkdw wkh udz JSV svhxgrudqjh dqg ghowdudqjh phdvxuhphqwv duh vfdoduv/ dqg dv vxfk qrw ghfrpsrvhg lq d uhihuhqfh iudph1 Krzhyhu/ wkh ORV xqlw yhfwruv wkdw jlyh wkh gluhfwlrq iurp wkh JSV uhfhlyhu dqwhqqd wr wkh vdwhoolwh dqwhqqdv/ duh hdvlhvw wr fdofxodwh lq wkh Hduwk iudph1 Wkh pdlq uhdvrq iru wklv lv wkdw wkh vdwhoolwh frruglqdwhv lq wkh qdyljdwlrq phvvdjh duh jlyhq lq wkh Hduwk iudph1 Wklv phdqv wkdw zh qhhg wr wudqvirup srvlwlrqv dqg yhorflwlhv wr wkh Hduwk iudph ehiruh wkh wudqvirupdwlrq wr svhxgrudqjhv dqg ghowdudqjhv fdq eh grqh1 Zh kdyh fkrvhq wr ghvljq wkuhh glhuhqw revhuyhuv zklfk duh phfkdql}hg lq wkh orfdo iudph/ Hduwk iudph/ dqg ORV0vxevsdfh ri wkh Hduwk iudph/ uhvshfwlyho|1

479

51

Qrqolqhdu Revhuyhu Ghvljq iru Lqwhjudwlrq ri GJSV dqg LQV

Orfdo Iudph Irupxodwlrq Wkh xsgdwh phfkdqlvp ri wkh revhuyhuv wdnhv wkh iroorzlqj irup=



s h 



@ +h ,| %



jsv 

h ,| @ + +h,|

t

&



+shv  sah,W +shv  sah ,  fa

jsv



t

 +shv  sah,W +shv  sah ,  fa

+5178,

iru srvlwlrq/ zkloh yhorflw| lv jlyhq e|



%



+Uho,W +h,| @ i +h,|

y o

&



jsv



 vdw  h Uhoyao  fia

+5179,

o duh wkh srvlwlrq dqg yhorflw| hvwlpdwlrq huuruv/ jsv dqg zkhuh s h dqg y jsv duh wkh svhxgrudqjh dqg ghowdudqjh yhfwruv phdvxuhg e| JSV/ vdw lv jlyhq e| +5144,/ h dqg h duh ghqhg lq +518, dqg +51:,/ dqg h | 5 ?7 lv wkh odvw urz ri h | 1 Wkh vxshuvfulsw h lv xvhg wr hpskdvl}h wkdw wkh ORV ao >  dqg i duh hvwlpdwhg yhfwruv duh ghfrpsrvhg lq wkh Hduwk iudph1 s ah > y Hduwk iudph srvlwlrq/ orfdo iudph yhorflw|/ uhfhlyhu forfn eldv/ dqg iuhtxhqf| eldv1 Wkh o0iudph yhorflw| dqg srvlwlrq xsgdwh htxdwlrqv +5145,+5146, zlwk eldv/ vfdoh0idfwru dqg plvdoljqphqw huuruv lqfoxghg duh zulwwhq=

 + ,

+ ,



a

yb o @ Uoe ^+L . 5 ,dlpx . e5 ` . jo 

sb @ U y eb 5 @ W7 4e5 . z7 4 b @ W 8  . z8 4 b @ W 9  . z9 4 ib @ w : i . z: b @ i . z; h

h o o

 @ + , @ ^ @^ e @ e @7 8 9



a

5V+$lho , . V+$hoo , yo

+517:, +517;, +517  >  { > | > } W duh wkuhh dffhohurphwhu vfdoh idfwru zkhuh 5 huuruv/ dqg  {| > {} > |{ > |} > }{ > }| W duh vl{ vpdoo dffhohurphwhu  dff uhsuhvhqwv wkh eldvhv ri wkh dffhohurph0 plvdoljqphqw huuruv1 5 whuv/  dqg i duh wkh vfdodu JSV forfn dqg iuhtxhqf| eldvhv/ dqg l 5 ?6 +l > > , dqg z: > z; 5 ? duh zklwh qrlvh1 l +l > > , duh odujh srvl0 wlyh ghqlwh wlph frqvwdqw pdwulfhv/ dqg w: A lv d odujh wlph frqvwdqw/ doo

`

W @7 8 9 3

z

51

Qrqolqhdu Revhuyhu Ghvljq iru Lqwhjudwlrq ri GJSV dqg LQV

47:

dvvxphg wr eh nqrzq1 Wkh iroorzlqj revhuyhu lv sursrvhg iru +517:,+5186,= k l  o o L a d o o y a5 jo  V $lh e V $ho ao 5 lpx e N5 y o Uho W s h +5187, h s a Uho y ao N6 s h +5188,

5 + ,. + , yab @ Ua + .  , . . . .+ , b @ . bea5 @ W7 4ea5 . +Ua oe,W yo +5189, 4 a oe ,W y ab @ W  . gldj+dlpx ,+U o +518:, 8 a ba @ W9 4a . W +dlpx,+Ua oe,Wyo +518;, bia @ w: 4ia . n7i .  +518 6, duh wzr 6  6 jdlq

pdwulfhv/ dqg n7 dqg n8 duh vfdodu jdlqv1 Wkh huuru g|qdplfv wdnhv wkh irup= k l o 5 a oe  5 dlpx e H L y  U e5 5 dlpx   +5194,  V $lho V $hoo yo  N5 yo  Uho W sh h h o h s  Uo y   N6 s  +5195, 4  5 W a o W o e +5196, 7 e5  Ue y  4 o W o ae y  W8   dlpx U  +5197, 4 W d a o W o  W +5198, 9  lpx Ue y

b @

. . ^+ .  , . ` 5 + ,. + , + , 

b @ b @ + , b@ gldj+ ,+ , b@

+ ,+ , 4  ib @ w : i  n7 i   b @ i  n8 

+5199, +519:,

@ Uoe  Ua oe= Qrz/ +5194,+519:, fdq eh zulwwhq= {b @ i +{> w, . j+w, k l   W >  5 > > >  i> zkhuh { @ y o > s h > e zkhuh H

5

i +{> w, @

9 9 9 9 9 9 9 9 9 9 7

k

l





+519;,

 5V+$lho , . V+$hoo , yo  N5yo  +Uho ,W sh Uho y o  N6 s h  4 a W7 e5  +Uoe,W yo  W8 4  gldj+dlpx ,+Ua oe,W yo W9 4   W +dlpx,+Ua oe,W yo w: 4i  n7i   i  n8 

a oe  5  5 dlpx . e U

6 : : : : : : : : : : 8

+519 w,

+51:4,

g Y +{+w,,  +{+w,, gw

+51:5,

zkhq m{m A  wkh uhodwlrq

krog doprvw hyhu|zkhuh1 Fkrrvh wkh qxpehu

I

 A 3 vxfk wkdw @

H iY +{,  j  H im{m  j

+51:6,

Wkhq=

d,

I

lv dq lqyduldqw vhw ri +51:4,1

e, Iru dq| vroxwlrq

{+w, ri wkh v|vwhp +51:4, zklfk pd| ghshqg rq lqlwldo frqglwlrqv/ surylghg wkdw wklv vroxwlrq ri +51:4, lv erxqghg rq ^3> W  ` =

51

Qrqolqhdu Re

vhuyhu Ghvljq iru Lqwhjudwlrq ri GJSV dqg LQV

Ohppd 515 +Dv|pswrwlf Vwdelolw| ri Irufhg V|vwhpv,

> w,> wkhuh h{lvw d vfdodu F 4 0vprrwk dqg d frqwlqxrv ixqfwlrq +{, A 3> vxfk wkdw iru wkh v|vwhp { b @ i +{

+L,

+LL,

Wkhuh h{lvw fodvv

Vxssrvh wkdw

ixqfwlrq

N ixqfwlrqv 4 5 vxfk wkdw ; { 4+m{m,  Y +{,  5+m{m,

47


w

+51:7,

> w, wkh iroorzlqj uhod0

Dorqj dq| vroxwlrq {+ , ri wkh v|vwhp { b @ i +{ wlrq krogv

g Y +{+w,,  +{+w,, gw

+51:8,

+LLL, olp vxs

m{m$4

m CYC{+{, m @ 3 +{,

+51:9,

Wkhq/

d,

Dq| vroxwlrq

{+w,

ri wkh v|vwhp

{b @ i +{> w, . j+w, zkhuh

e,

j+w, $ 3

;% A 3 m{3 m ? 

dv

w $ 4> whqgv wr }hur dv w $ 4=

 A 3 dqg d W  @ W +j+w,> %,  3  = m{+w> { ,m ? %= / wkhq ;w  W 3

wkhuh h{lvw d

Surri1 Jlyhq

j+w,=

+51::,

Vxssrvh wkdw

vxfk wkdw li

m{+w,m grhv qrw whqg wr }hur dv w $ 4= m{+w,m $ 4 ru m{+w,m  U=

Wkhq wkhuh duh wzr srvvleoh fdvhv= olp vxs

w$4

m{+w,m $ 4= Fkrrvh dq| % A 3= Wkhq Q , C {  5%

+51: Q ,

+51;3,

Gxh wr Ohppd 514 dqg +51:7,/ {+w, lv erxqghg1 Wkh frqwudglfwlrq suryhv wkdw m{+w,m  U1

5,


Wq $ .4 vxfk


== ;q4 Gxh wr +51:7, wkhuh h{lvw d

+51;4,

4 A 3 dqg d 5 A 3 vxfk wkdw

H im{m  5 j  H iY +{,  4j  H im{m  j

+51;5,

Fkrrvh

   CY +{,  4 +{, %@ plq pd{ 5 5 ?m{+w,m?U C{  5 ?m{+w,m?U dqg d W  @ W  +%,  3 vxfk wkdw ;w  W  = mj+w,m ? %= Djdlq xvlqj +51:;,/ zh kdyh wkdw ; w  W  / zkhq 5 ? m{+w,m ? U

4



   

CY +{, g Y +{+w,,  +{+w,, . j+w, gw C{    CY +{,  % ? +{+w,, .  C{ 

  45 +{+w,,

+51;6,

Wdnlqj dgydqwdjh ri Ohppd 514 dqg Uhpdun 514/ zh frqfoxgh wkdw wkhuh h{lvw W  A W  vxfk wkdw wkh vroxwlrq {+w, zloo ehorqj wr

H iY +{,  4j

+51;7,

Wklv lv d frqwudglfwlrq wr +51;4,+51;5,/ dqg wkxv m{+w,m $ 3 dv w $ 41 Wklv frpsohwhv wkh surri ri sduw d,1 6, Wr suryh e,/ lw lv qhfhvvdu| wr srlqw rxw wkdw gxh wr Vwhs 4/ li wkh lqlwldo

frqglwlrq {3 ehorqjv wr

H im{3m  j wkhq wkhuh h{lvw d U

{+w> {3 , 5 H im{m  U j ;{3 5 H im{3 m   j > ;w  3

vxfk wkdw +51;8,

51

Qrqolqhdu Revhuyhu Ghvljq iru Lqwhjudwlrq ri GJSV dqg LQV

Fkrrvh dq| vpdoo

5 A 3 vxfk wkdw

484

% A 3= Gxh wr +51:7, wkhuh h{lvwv d 4 A 3/ dqg d

H im{m  5j  H iY +{,  4 j  H im{m  %j  A 31 Wkhq iroorzlqj Vwhs 5/ 5 ? m{+w,m ? U zkhuh wkh uhodwlrq

Il{ dq|

W >

wkhuh h{lvwv d

4 g Y +{+w,,   +{+w,, gw 5

+51;9,

W

=

;w  +51;:,

lv ydolg1 Wkhq/

Y +{+w,, @ Y +{+W ,, .

]

w

W

Yb +{+ ,,g

 Y +{+W ,,  45 +w  W ,3 zkhuh

3 @

plq

5 ?m{+w,m?U

+51;;,

+{,=

Wkh lqhtxdolw| +51;;, lpsolhv wkdw dq| vroxwlrq {+w, vwduwlqj zlwk @ W  vkrxog uhdfk wkh vhw H im{m  5 j dw ohdvw dw wlph

H im{m  U j dw wlph w

W  @ W  . 5

pd{

m{mU

Y +{, 3 4

+51; w, . j+w, lv txdvl0htxl dv|pswrwlfdoo| vwdeoh lq wkh odujh/ dv ghqhg e| Odnvkplndqwkdp hw do1 ^5`1 Wkh qh{w wkhruhp frqvlghuv wkh frqyhujhqfh dqg vwdelolw| surshuwlhv ri wkh v|vwhp +519;,=

Wkhruhp 515 +THDV o0Iudph Revhuyhu,

Iru dq| vroxwlrq ri wkh o0

m m $ 3 dv $ 4 dqg THDV

iudph revhuyhu huuru htxdwlrq +519;,/ {+w, iroorzv iurp Uhpdun 5151

w

485

51

Qrqolqhdu

Revhuyhu Ghvljq iru Lqwhjudwlrq ri GJSV dqg LQV

{b i {

i{ {b i { 4 o,W yo . 4 +sh,W sh. 4 e5W e5 Y @ +y 5 5 5 4 4 4 W W 5 4 5    .   .  . i . 

Frqvlghu wkh xqirufhg v|vwhp @ + > w,/ zkhuh + > w, lv jlyhq e| +519 w,= Surri1

5

5

5

5

+51