03 Nieliniowe uklady operacyjne

background image

1,(/,1,2:(8.à$'<23(5$&<-1(

./$6<),.$&-$

,

0(72'<

*(1(5$&-,

)81.&-,

1,(/,1,2:<&+

=DGDQLHP QLHOLQLRZ\FK XNáDGyZ RSHUDF\MQ\FK MHVW UHDOL]DFMD

RNUHORQ\FK QLHOLQLRZ\FK RSHUDFML IXQNF\MQ\FK

( )

y

f x

=

IXQNFML

MHGQHM ]PLHQQHM OXE

(

)

y

f x

x

n

=

1,

...

IXQNFML ZLHOX ]PLHQQ\FK 3U]H]

]PLHQQHQLH]DOH*QHQDOH*\UR]XPLHüSU]HELHJLQDSLFLRZHOXESUGRZH

]DGDZDQH Z SRVWDFL XQRUPRZDQHM MDNR ZLHONRFL EH]Z\PLDURZH

1LHOLQLRZHXNáDG\RSHUDF\MQHVQD]\ZDQHWDN*HJHQHUDWRUDPLIXQNFML

QLHOLQLRZ\FKOXEQLHOLQLRZ\PLSU]HWZRUQLNDPLIXQNF\MQ\PL'RNRQXMF

LFK NODV\ILNDFML ]H Z]JOGX QD Z\NRQ\ZDQH RSHUDFMH QLHOLQLRZH

PR*HP\Z\Uy*QLüQDVWSXMFHJUXS\XNáDGRZH

XNáDG\NV]WDáWXMFHIXQNFMHSU]HG]LDáDPLSURVWROLQLRZH

XNáDG\SRUyZQXMFH

XNáDG\ORJDU\WPLF]QHLZ\NáDGQLF]H

XNáDG\PQR*FHLG]LHOFH

XNáDG\SRWJXMFHLSLHUZLDVWNXMFH

XNáDG\ZLHORIXQNF\MQH

LQQHXNáDG\VSHFMDOQH

0HWRG\JHQHUDFMLIXQNFMLQLHOLQLRZ\FKPR*QD]JUXEV]DSRG]LHOLü

QDWU]\SRGVWDZRZHJUXS\
0HWRGDEH]SRUHGQLD

0HWRGD WD SROHJD QD EH]SRUHGQLP Z\NRU]\VWDQLX QLHOLQLRZ\FK

FKDUDNWHU\VW\N HOHPHQWyZ HOHNWURQLF]Q\FK 6WRVXMF W PHWRG

RWU]\PXMHP\QDMSURVWV]HXNáDG\RSHUDF\MQHRGX*HMV]\ENRFLG]LDáDQLD

L GRNáDGQRFL UHDOL]RZDQHM RSHUDFML =H Z]JOGX QD QLH]E\W GX*

Uy*QRURGQRü FKDUDNWHU\VW\N HOHPHQWyZ SyáSU]HZRGQLNRZ\FK PHWRG

WPR*QDUHDOL]RZDüMHG\QLHQLHNWyUHRSHUDFMHSRGVWDZRZH
0HWRGDSRUHGQLD

:PHWRG]LHWHMJHQHURZDQIXQNFMZ\UD*DP\SU]H]LQQHIXQNFMH

áDWZH GR JHQHUDFML )XQNFMD GRFHORZD MHVW Z\WZDU]DQD SRUHGQLR

SHZQ\PL NROHMQ\PL HWDSDPL Z\QLNDMF\PL ] SU]\MWHJR VSRVREX MHM

SU]HGVWDZLHQLD 0HWRGD WD VWZDU]D V]HURNLH PR*OLZRFL JHQHUDFML

background image

IXQNFML QLHOLQLRZ\FK L MHVW GRPLQXMF PHWRG UHDOL]DFML XNáDGyZ

RSHUDF\MQ\FK
0HWRGDDSURNV\PDF\MQD

: PHWRG]LH WHM GRNRQXMH VL DSURNV\PDFML OXE LQWHUSRODFML

]DGDQHM IXQNFML QLHOLQLRZHM D QDVWSQLH JHQHURZDQD MHVW IXQNFMD

DSURNV\PXMFD 'RELHUDMF IXQNFM DSURNV\PXMF QDOH*\ XZ]JOGQLü

EáGSU]\EOL*HQLDMDNLáDWZRüMHMUHDOL]DFML

:SUDNW\FHMDNRIXQNFMHDSURNV\PXMFHVWRVXMHVLQDMF]FLHM

IXQNFM SU]HG]LDáDPL SURVWROLQLRZ ZLHORPLDQ SRWJRZ\ RUD] IXQNFM

EGF

OLQLRZ

NRPELQDFM

FKDUDNWHU\VW\N

SU]HMFLRZ\FK

WUDQ]\VWRURZ\FKSDUUy*QLFRZ\FK

=DOH*QLHRGZ\MFLRZHMSRVWDFLIXQNFMLQLHOLQLRZHMQDSRGVWDZLH

NWyUHMSRV]XNXMHP\VWUXNWXU\XNáDGXPR*HP\GRNRQDüLQQHJRSRG]LDáX

PHWRGJHQHUDFMLIXQNFMLQLHOLQLRZ\FK

0HWRGDIXQNFMLMDZQHM

=DGDQD IXQNFMD QLHOLQLRZD ]DSLVDQD MHVW Z SRVWDFL MDZQHM L MHVW

UHDOL]RZDQD NROHMQ\PL HWDSDPL QDMSLHUZ JHQHURZDQH V HOHPHQWDUQH

IXQNFMH QLHOLQLRZH Z\VWSXMFH Z ]DGDQHM IXQNFML D QDVWSQLH

Z\NRQ\ZDQH V NROHMQR RSHUDFMH QD IXQNFMDFK HOHPHQWDUQ\FK QS

GRGDZDQLDPQR*HQLDD*GRX]\VNDQLDIXQNFMLGRFHORZHM
0HWRGDIXQNFMLRGZURWQHM

-H*HOL GDQ\ MHVW F]áRQ QLHOLQLRZ\ JHQHUXMF\ PRQRWRQLF]Q

IXQNFM

( )

x

f y

=

WR IXQNFM RGZURWQ

( )

y

f

x

=

1

X]\VNXMHP\ SU]H]

XPLHV]F]HQLHWHJRF]áRQXZSWOLVSU]*HQLD]ZURWQHJR

I \

.

L

I X

2

=

+

\ I

[

=

[

X

,

5 L

L

X

2

X

2

D

E

.

+

5\V:]PDFQLDF]RSHUDF\MQ\]QLHOLQLRZ\PVSU]*HQLHP]ZURWQ\P

:XNáDG]LHSU]HGVWDZLRQ\PQDU\VDRWU]\PXMHP\

( )

x

f y

y

K

=

( )

Gdy K

→ ∞

=

y

f

x

1

background image

3RGREQLH GOD XNáDGX ] U\VE ]DNáDGDMF *H Z]PDFQLDF]

RSHUDF\MQ\MHVWLGHDOQ\

(

)

K

R

in

→ ∞

= ∞

,

RWU]\PXMHP\

( )

i

i

u

R

f u

I

O

1

2

= −

= −

;

6WG

( )

u

R f

u

O

I

= −

1

0HWRGDIXQNFMLXZLNáDQHM

0HWRG W PR*QD VWRVRZDü ZyZF]DV JG\ ]DGDQ IXQNFM

(

)

y

f x x

x

k

=

1

2

,

,...,

PR*QDSU]HGVWDZLüZSRVWDFL

(

)

y

g x x

x

y

k

=

1

2

0

,

,...,

,

=DVDG JHQHUDFML IXQNFML QLHOLQLRZHM PHWRG IXQNFML XZLNáDQHM

]LOXVWURZDQRQDU\V

[

[

[

N

[

N

+

J

\ J [ [

[ \

N

=

.

..

5\V

*HQHUDFMD

IXQNFML

QLHOLQLRZHM PHWRG IXQNFML

XZLNáDQHM

&]áRQ QLHOLQLRZ\ JHQHUXMF\ IXQNFM

(

)

k

+

1

]PLHQQ\FK

(

)

y

g x x

x

x

y

k

k

=

+

1

2

1

,

,...,

,

,

REMW\MHVWSWOVSU]*HQLD]ZURWQHJRZWDNL

VSRVyE*H

[

\

N

+

=

0HWRGD WD MHVW V]F]HJyOQLH X*\WHF]QD SU]\ JHQHUDFML Uy*QHJR

URG]DMX IXQNFML Z\PLHUQ\FK SR]ZDOD ERZLHP ]UHDOL]RZDü ]D SRPRF

VSU]*HQLD]ZURWQHJRRSHUDFMG]LHOHQLDOLF]QLNDSU]H]PLDQRZQLN

5]HF]\ZLVWH XNáDG\ UHDOL]XM QLHOLQLRZH RSHUDFMH Z VSRVyE

SU]\EOL*RQ\]SHZQ\PEáGHP%áGWHQ]DOH*\RGURG]DMXUHDOL]RZDQHM

RSHUDFML NRQNUHWQHJR UR]ZL]DQLD XNáDGRZHJR RG F]\QQLNyZ

]HZQWU]Q\FKQS]PLDQWHPSHUDWXU\QDSLü]DVLODMF\FKLWSRUD]RG

URG]DMX]DVWRVRZDQ\FKSREXG]H

-HGQ\P]SDUDPHWUyZVWRVRZDQ\FKGRRFHQ\QLHOLQLRZHJRXNáDGX

RSHUDF\MQHJRMHVWZ\QLNRZ\VWDW\F]Q\EáGZ]JOGQ\RSHUDFML

( )

δ

0

-HVW

RQ UR]XPLDQ\ MDNR EáG PDNV\PDOQ\ Z\VWSXMF\ SU]\ VWHURZDQLX

]DFLVNyZ ZHMFLRZ\FK XNáDGX QDSLFLDPL VWDá\PL R ZDUWRFLDFK

]DZDUW\FK Z FDá\P GRSXV]F]DOQ\P LFK SU]HG]LDOH %áG\ G\QDPLF]QH

Z\]QDF]DQHVSU]\SU]\MFLXFLOHVSUHF\]RZDQ\FKGODGDQHJRXNáDGX

SREXG]H 2SLV\ZDQH V RQH ]D]Z\F]DM ]D SRPRF JyUQHM

F]VWRWOLZRFL JUDQLF]QHM 2ERN X*\ZDQHM Z XNáDGDFK OLQLRZ\FK

background image

F]VWRWOLZRFL JUDQLF]QHM

f

dB

3

SU]\ GHF\EHORZ\P EáG]LH

DPSOLWXGRZ\P GHILQLXMH VL F]VWRWOLZRü JUDQLF]Q f

ϕ

EáGX ID]\

DWDN*H F]VWRWOLZRü JUDQLF]Q

f

α

SU]\ MHGQRSURFHQWRZ\P EáG]LH

DPSOLWXGRZ\P

5y*QRURGQRüLV]HURNL]DNUHV]DVWRVRZDQLHOLQLRZ\FKXNáDGyZ

RSHUDF\MQ\FK XWUXGQLD LFK V\VWHPDW\F]QH L Z\F]HUSXMFH RPyZLHQLH

'ODWHJR Z GDOV]\FK UR]ZD*DQLDFK ]RVWDQ RPyZLRQH SRGVWDZRZH

QLHOLQLRZHXNáDG\RSHUDF\MQH]QDMGXMFHQDMV]HUV]H]DVWRVRZDQLD

$1$/2*2:(8.à$'<012)&(

3DUDPHWU\LPHWRG\UHDOL]DFMLXNáDGyZPQR*F\FK

8NáDG\ PQR*FH RGJU\ZDM ]QDF]Q URO ZUyG QLHOLQLRZ\FK

XNáDGyZ RSHUDF\MQ\FK ,FK ]DGDQLHP MHVW Z\WZRU]HQLH QDSLFLD

Z\MFLRZHJR

u

O

SURSRUFMRQDOQHJR GR LORF]\QX GZyFK ZHMFLRZ\FK

QDSLüVWHUXMF\FK

u

u

X

Y

i

u

k u u

u u

E

O

m

X

Y

X

Y

R

=

=

SU]\F]\P

k

E

m

R

=

1 /

MHVWVWDáVNDORZDQLD

E

R

QRUPXMFH QDSLFLH RGQLHVLHQLD ]D]Z\F]DM UyZQH

9OXE9

8NáDG

PQR*F\

QD]\ZDP\

F]WHURNZDGUDQWRZ\P

F]WHURüZLDUWNRZ\P MH*HOL GRNRQXMH RSHUDFML PQR*HQLD SU]\

GRZROQ\FK]QDNDFKQDSLü

u

u

X

Y

i

-H*HOLQDSLFLDZHMFLRZH

u

u

X

Y

i

PRJ E\ü W\ONR MHGQRELHJXQRZH XQLSRODUQH WR WDNL XNáDG PQR*F\

QD]\ZDP\

MHGQRNZDGUDQWRZ\P

:

SRUHGQLFK

XNáDGDFK

GZXNZDGUDQWRZ\FK XVWDORQD MHVW ELHJXQRZRü W\ONR MHGQHJR QDSLFLD

ZHMFLRZHJR

5]HF]\ZLVW\ XNáDG PQR*F\ UHDOL]XMH RSHUDFM PQR*HQLD ]

SHZQ\PEáGHP

(

)

u

u

u

u

u

O

X

Y

X

Y

=

+ =

+

E

E

R

R

1

0

δ

JG]LH

,

δ

0

R]QDF]DM RGSRZLHGQLR EH]Z]JOGQ\ L Z]JOGQ\ EáG

PQR*HQLD

3DUDPHWU\G\QDPLF]QHWDNLHMDN

background image

f

dB

3

PDáRV\JQDáRZDWU]\GHF\EHORZDF]VWRWOLZRüJUDQLF]QD

f

α

F]VWRWOLZRü JUDQLF]QD SU]\ MHGQRSURFHQWRZ\P EáG]LH

DPSOLWXGRZ\P

f

ϕ

F]VWRWOLZRü JUDQLF]QD SU]\ MHGQRSURFHQWRZ\P EáG]LH ID]\

EáG]LHZHNWRURZ\P

6 ZLHONRV\JQDáRZD PDNV\PDOQD V]\ENRü QDUDVWDQLD QDSLFLD

Z\MFLRZHJRRNUHODVLSRGREQLHMDNGODXNáDGXOLQLRZHJR

: FHOX Z\]QDF]HQLD F]VWRWOLZRFL JUDQLF]Q\FK GR MHGQHJR

ZHMFLDXNáDGXPQR*FHJRSU]\NáDGDVLQDSLFLDVWDáH9OXE9

SU]\ E

V

R

=

10

]D GUXJLH ZHMFLH SREXG]D VL QDSLFLHP

VLQXVRLGDOQ\P VWDQGDUGRZR DPSOLWXGD SREXG]HQLD Z\QRVL 9

LZ\]QDF]D VL F]VWRWOLZRFL SU]\ RNUHORQ\FK EáGDFK DPSOLWXG\ OXE

ID]\

X

2

> @

9

W

X

2

9

W

5\V

2GSRZLHG(

XNáDGX

PQR*FHJR

QD

VWDQGDUGRZH

SREXG]HQLH

ZLHONRV\JQDáRZH

WHVWXMFHSDUDPHWU6

'OD Z\]QDF]HQLD PDNV\PDOQHM V]\ENRFL QDUDVWDQLD QDSLFLD

Z\MFLRZHJR

( )

S

du

t

dt

O

=

max

ZPLHMVFHQDSLFLDVLQXVRLGDOQHJRVWRVXMHVLVWHURZDQLHV\PHWU\F]Q\P

QDSLFLHPSURVWRNWQ\PRZDUWRFLPLG]\V]F]\WRZHM9

3DUDPHWU 6 Z\]QDF]D VL QD SRGVWDZLH SU]HELHJX F]DVRZHJR

RGSRZLHG]LU\V

: DQDORJRZHM WHFKQLFH VFDORQHM XNáDG\ PQR*FH PRJ E\ü

UHDOL]RZDQH NLONRPD Uy*Q\PL PHWRGDPL VSRUyG NWyU\FK QDMF]FLHM

VWRVRZDQHVQDVWSXMFHPHWRG\

PHWRGDPRGXODFMLV]HURNRFLLDPSOLWXG\LPSXOVyZSURVWRNWQ\FK

PHWRGD]]DVWRVRZDQLHPRSHUDFMLORJDU\WPLF]QHMLZ\NáDGQLF]HM

PHWRGD]]DVWRVRZDQLHPNZDGUDWRUyZ

PHWRGD

Z\NRU]\VWXMFD

XNáDG\

Uy*QLFRZH

R

]PLHQQHM

WUDQVNRQGXNWDQFML QD]\ZDQD UyZQLH* PHWRG VWHURZDQHJR SRG]LDáX

SUGX

background image

1DMEDUG]LHM

UR]SRZV]HFKQLRQ

L

RG]QDF]DMF

VL

QDMNRU]\VWQLHMV]\PL FHFKDPL X*\WNRZ\PL MHVW PHWRGD Z\NRU]\VWXMFD

XNáDG\ Uy*QLFRZH R ]PLHQQHM WUDQVNRQGXNWDQFML 1L*HM SRGDP\ NUyWN

FKDUDNWHU\VW\NZV]\VWNLFKZ\PLHQLRQ\FKPHWRG

8NáDG\]PRGXODFMV]HURNRFLLDPSOLWXG\LPSXOVyZ

SURVWRNWQ\FK

:XNáDGDFKWHJRURG]DMXSU]HELHJSURVWRNWQ\MHVWPRGXORZDQ\

GZXNURWQLH Z WDNL VSRVyE *H V]HURNRü LPSXOVyZ MHVW SURSRUFMRQDOQD

GRMHGQHJRQDSLFLDZHMFLRZHJRDDPSOLWXGDLPSXOVyZGRGUXJLHJR

QDSLFLDZHMFLRZHJR2WU]\PDQ\ZWHQVSRVyESU]HELHJ]PRGXORZDQ\

FKDUDNWHU\]XMHVLW\P*HMHJRUHGQLDZDUWRüMHVWSURSRUFMRQDOQDGR

LORF]\QXREXV\JQDáyZZHMFLRZ\FK

u

u

X

Y

i

1D U\VXQNX SU]HGVWDZLRQR XSURV]F]RQ\ VFKHPDW LGHRZ\

W\SRZHJR XNáDGX UHDOL]XMFHJR RSHUDFM PQR*HQLD PHWRG GZXNURWQHM

PRGXODFML

D

E

X

7

(

5

(

5

X

;

W

L

X

5

<

X

5

<

W

5

5

5

X

<

5

5

3

7

7

X W

6

X

;

7

7

7

&


+

X W

7

.


+

5

5

L

&

X

2U

.


+

5\V0QR*HQLHPHWRGPRGXODFMLV]HURNRFLLDPSOLWXG\LPSXOVyZDSU]HELHJL

F]DVRZHEXSURV]F]RQ\VFKHPDWLGHRZ\

'RNRPSDUDWRUDVGRSURZDG]DQHQDSLFLD

u

X

RUD]V\PHWU\F]QH

QDSLFLHWUyMNWQH

( )

u t

T

RDPSOLWXG]LH

E

R

RNUHVLH

T

L]HURZHMZDUWRFL

UHGQLHM 1D Z\MFLX NRPSDUDWRUD RWU]\PXMHP\ SU]HELHJ

( )

u t

S

U\VDVWHUXMF\SU]HáF]QLNLHPHOHNWURQLF]Q\P3SU]\F]\P

background image

( )

u t

U

u

u

U

u

u

S

S

X

T

S

X

T

=

>

<

max

min

,

,

dla

dla

]D

U

U

S

S

max

min

,

VQDSLFLDPLZ\MFLRZ\PLNRPSDUDWRUD

:RSDUFLXRU\VDRWU]\PXMHP\]DOH*QRü

u

E

T

T

X

R

=

2

1

1

JG]LH

T T

1

MHVWZVSyáF]\QQLNLHPZ\SHáQLHQLDLPSXOVXSURVWRN WQHJR

:HMFLRZ\Z]PDFQLDF]RSHUDF\MQ\VWHURZDQ\GUXJLPQDSL FLHP

u

Y

UHDOL]XMHPRGXODFMDPSOLWXG\LPSXOVyZSURVWRNWQ\FK$PSOLWXGD

SUGX

i

2

QD RGZUDFDMF\P ZHMFLX VXPXMF\P Z]PDFQLDF]D Z ILOWU]H

GROQRSU]HSXVWRZ\PZ\QRVL

u

R

Y

GOD SyáRNUHVX

T

2

RUD]

u

R

Y

GOD

SyáRNUHVX

T

1

=DWHP UHGQLD ZDUWRü QDSLFLD QD Z\MFLX ILOWUX

GROQRSU]HSXVWRZHJRZ\QRVL

X

7

7

X

2U

<

=





3RGVWDZLDMFGRRWU]\PXMHP\

X

X X

(

2U

;

<

5

=

'RNáDGQRüRSHUDFMLPQR*HQLDUHDOL]RZDQHMWPHWRGMHVWGX*D

=DVDGQLF] ZDG MHVW QDWRPLDVW RJUDQLF]RQ\ ]DNUHV F]VWRWOLZRFL

Z\QLNDMF\ ] ZDUXQNX SRSUDZQHM PRGXODFML f T

f

g

g

<<

1 (

JyUQD

F]VWRWOLZRü JUDQLF]QD ZLGPD V\JQDáyZ ZHMFLRZ\FK : SUDNW\FH

XNáDG\ WH VWRVXMH VL GR PQR*HQLD V\JQDáyZ ZROQR]PLHQQ\FK

(

)

f

kHz S

V ms

dB

3

1

0 7

<

<

,

,

/

8NáDG\]]DVWRVRZDQLHPRSHUDFMLORJDU\WPLF]QHML

Z\NáDGQLF]HM

1D U\VD SU]HGVWDZLRQR VFKHPDW IXQNFMRQDOQ\ XNáDGX

PQR*FHJR Z\NRU]\VWXMFHJR RSHUDFMH ORJDU\WPLF]QH L Z\NáDGQLF]H

'ODWHJRXNáDGXRWU]\PXMHP\

E

u

E

u

E

u u

E

u

R

X

R

Y

R

X

Y

R

O

exp ln

ln

+



 =

=

SU]\F]\P

u

u

X

Y

,

>

0

background image

Σ

N

X
(

(

;

5

OQ

N

X

(

(

<

5

OQ

X

;

X

<

N

(

N

(

+
+

D

OQ

X X

(

; <

5

(

5

H[S

X

X X

(

2

; <

5

=

Σ

Σ

Σ

E
X

;

X

<

(

5

+

+

+

+

+

′ =

′ ′

X

X X

(

2

; <

5

X

;

X

<

X

2

5\V 0QR*HQLH ] Z\NRU]\VWDQLHP RSHUDFML ORJDU\WPLF]QHM L Z\NáDGQLF]HM D

VFKHPDW IXQNFMRQDOQ\ XNáDGX MHGQRNZDGUDQWRZHJR E ]DVDGD G]LDáDQLD

XNáDGXF]WHURNZDGUDQWRZHJR

1DU\VESRGDQR]DVDGG]LDáDQLDF]WHURNZDGUDQWRZHJRXNáDGX

PQR*FHJR Z\NRU]\VWXMFHJR XNáDG MHGQRNZDGUDQWRZ\ RUD] WU]\

GRGDWNRZHVXPDWRU\

:XNáDG]LHW\P

u

u u

E

E

O

X

Y

R

R

'

'

'

;

=

>

0

u

E

u

u

E

u

X

R

X

Y

R

Y

'

'

=

>

=

>

0

0

3RGVWDZLDMFGRRWU]\PXMHP\

(

)(

)

u

E

u

E

u

E

E

u

u

u u

E

O

R

X

R

Y

R

R

X

Y

X

Y

R

'

=

=

+

6XPDWRUZ\MFLRZ\UHDOL]XMHRSHUDFM

u

u

E

u

u

O

O

R

X

Y

=

+

+

'

8Z]JOGQLDMFRWU]\PXMHP\

u

u u

E

O

X

Y

R

=

D

3U]\

E

R

9 ]DNUHV\ QDSLü ZHMFLRZ\FK PLHV]F] VL Z

SU]HG]LDáDFKRG9GR9

background image

&]WHURNZDGUDQWRZH XNáDG\ PQR*FH Z\NRU]\VWXMFH RSHUDFMH

ORJDU\WPLF]QRZ\NáDGQLF]H FKDUDNWHU\]XM VL XPLDUNRZDQ\P

VWRSQLHP ]áR*RQRFL XNáDGRZHM ]DSHZQLDMF RULHQWDF\MQLH QDVWSXMFH

SDUDPHWU\

(

)

δ

µ

=

<

I

6

G%

N+]

9

V

8NáDG\]]DVWRVRZDQLHPNZDGUDWRUyZ

: PHWRG]LH WHM Z\NRU]\VWXMH VL QLHOLQLRZH SU]HWZRUQLNL

JHQHUXMFHIXQNFMNZDGUDWRZW]ZNZDGUDWRU\ y a x

=

2

0QR*HQLHV\JQDáyZ

u

u

X

Y

,

SU]HELHJD]JRGQLH]UHODFM

(

)

(

)

a u

u

a u

u

a u u

u

X

Y

X

Y

X

Y

O

+

=

=

2

2

4

6WRVXMF

E

a

R

=

1 4

RWU]\PXMHP\

u

u u

E

O

X

Y

R

=

0HWRGD Z\PDJD VWRVRZDQLD NZDGUDWRUyZ RUD] XNáDGyZ VXPXM F\FK L

RGHMPXMF\FK 3UHF\]\MQH NZDGUDWRU\ PR*QD ]UHDOL]RZDü ]D SRPRF

VLHFL ]HVWDZLRQHM ] GLRG SyáSU]HZRGQLNRZ\FK L UH]\VWRUyZ

DSURNV\PXMFHM

FKDUDNWHU\VW\N

SDUDEROLF]Q

RGFLQNDPL

SURVWROLQLRZ\PL

3U]\ X*\FLX WHM PHWRG\ PR*OLZH MHVW ]UHDOL]RZDQLH

V]HURNRSDVPRZ\FK

XNáDGyZ

PQR*F\FK

R

SDUDPHWUDFK

δ

µ

<

<

<

I

6

G%

0+]

9 V

0QR*HQLH V\JQDáyZ ] ]DVWRVRZDQLHP NZDGUDWRUyZ MHVW

V]F]HJyOQLH

DWUDNF\MQH

SU]\

Z\NRU]\VW\ZDQLX

WUDQ]\VWRUyZ

XQLSRODUQ\FK QS Z WHFKQRORJLL &026 Z NWyU\FK Z\VW SXMH

NZDGUDWRZD]DOH*QRüSUGXGUHQX

i

D

RGQDSLFLDEUDPND(UyGáR

u

GS

-H*HOLSRQDGWRXNáDGPQR*F\MHVWUHDOL]RZDQ\ZW]ZWU\ELHSUGRZ\P

WRIXQNFMHVXPRZDQLDLRGHMPRZDQLDVSURZDG]DM VLGRQDMSURVWV]HJR

VXPRZDQLD SUGyZ Z]áRZ\FK 6WZDU]D WR PR*OLZRü UHDOL]DFML

V]HURNRSDVPRZ\FKQLH]E\WVNRPSOLNRZDQ\FKXNáDGyZPQR *F\FK

5y*QLFRZHXNáDG\PQR*FHR]PLHQQHMWUDQVNRQGXNWDQFML

8NáDGGZXüZLDUWNRZ\

1D U\VD SU]HGVWDZLRQR VFKHPDW ELSRODUQHJR Z]PDFQLDF]D

Uy*QLFRZHJR Z NWyU\P Z\GDMQRü (UyGáD SUGRZHJR Z REZRG]LH

HPLWHUyZPR*QDUHJXORZDüZSHZQ\P]DNUHVLH]PLDQQDSLFLD

u

Y

background image

D

X

&

5

&

5

&

2EFL *HQLH

X

&

X

5

8

&&

X

;

X

<

7

7

7

8

((

5

(

L X

2

<

,

&

,

&

X

5

L

2

L

L

2

2

>

X

;

E

5\V1DMSURVWV]\XNáDGPQR*F\GZXüZLDUWNRZ\DVFKHPDWXNáDGX

EFKDUDNWHU\VW\NL

1DSLFLH

u

X

VWHUXMFHV\PHWU\F]QLHED]\WUDQ]\VWRUyZ

T T

1

2

,

PR*HE\ü

GRGDWQLH OXE XMHPQH QDWRPLDVW QDSLFLH

u

Y

PR*H E\ü W\ONR GRGDWQLH

DE\ ]DSHZQLü GRGDWQL SUG

i

O

7DN VNRQVWUXRZDQ\ Z]PDFQLDF] PR*H

E\ü

QDMSURVWV]\P

XNáDGHP

PQR*F\P

GZXüZLDUWNRZ\P

GZXNZDGUDQWRZ\P SRQLHZD* MHJR FKDUDNWHU\VW\NL V SRáR*RQH Z

SLHUZV]HMLWU]HFLHMüZLDUWFHXNáDGXZVSyáU]GQ\FKu u

X

R

,

2

U\VE

:\NRU]\VWXMF]QDQH]DOH*QRFLGODSDU\Uy*QLFRZHM

i

i

i

u

u

u

i

I

u

i

I

u

i

i

i

i

E

E

O

BE

BE

X

E

ES

BE

T

E

ES

BE

T

C

E

C

E

1

2

1

2

1

1

2

2

1

0

1

2

0

2

1

1

+

=

=

=



 −



=



 −



=

=

exp

,

exp

,

ϕ

ϕ

α

α

PR*QD SU]HGVWDZLü Z\MFLRZ\ SUG Uy*QLFRZ\

i

i

i

R

C

C

2

1

2

=

]QDQ

]DOH*QRFL

i

i tgh

u

R

O

X

T

2

2

=

ϕ

:\MFLRZH Uy*QLFRZH QDSLFLH u

u

u

R

C

C

2

2

1

=

SU]\ SHáQHM

V\PHWULLXNáDGX

R

R

R

C

C

C

1

2

=

=

Z\QRVL

background image

u

i R tgh

u

R

O

C

X

T

2

2

=

ϕ

:\GDMQRü VWHURZDQHJR QDSLFLHP

u

Y

(UyGáD SUGRZHJR PR*QD

ZSHZQ\P]DNUHVLH]PLDQWHJRQDSLFLDRSLVDüSU]\EOL*RQ]DOH*QRFL

OLQLRZ>@

i

I

g u

O

m

Y

= +

0

3RGVWDZLDMF]DOH*QRüGRRWU]\PXMHP\

u

I

g u

R tgh

u

I R

u

g R

u u

R

m

Y

C

X

T

C

X

T

m

C

X

Y

T

2

0

0

2

2

2

=

+

+

(

)

ϕ

ϕ

ϕ

:]DOH*QRFLZ\NRU]\VWDQRSU]\EOL*HQLH

tgh

u

u

u

X

T

X

T

X

T

2

2

2

ϕ

ϕ

ϕ

<<

dla

3LHUZV]\ F]áRQ Z\UD*HQLD Z\UD*D Z]PRFQLHQLH QDSLFLD

u

X

QDWRPLDVWGUXJLPQR*HQLH

u

u

X

Y

*G\

u

X

=

0

WRUyZQLH*

u

R

2

0

=

QDWRPLDVWSU]\

u

Y

=

0

ZRJyOQ\PSU]\SDGNX

u

R

2

0

.

7DNLH]DFKRZDQLH

XNáDGXR]QDF]D*HMHJRZ\MFLHMHVW]UyZQRZD*RQHZ]JOGHPV\JQDáX

u

X

DQLHMHVW]UyZQRZD*RQHZ]JOGHPV\JQDáX

u

Y

-H*HOLZ\UD*HQLHSU]HGVWDZLP\ZSRVWDFL

u

R g u

R

C

m

X

2

JG]LH

′ =

+



g

I

g

u

m

m

Y

T

0

ϕ

WR XNáDG PR*H E\ü WUDNWRZDQ\ MDNR

Z]PDFQLDF]V\JQDáXUy*QLFRZHJR

u

X

R]PLHQQHMWUDQVNRQGXNWDQFML

g

m

PRGXORZDQHM SU]H] V\JQDá

u

Y

8NáDG QD]\ZD VL PRGXODWRUHP

SRMHG\QF]R]UyZQRZD*RQ\PZ]JOGHP

u

X

8NáDGF]WHURüZLDUWNRZ\

1D U\V SU]HGVWDZLRQR XNáDG PQR*F\ F]WHURüZLDUWNRZ\

F]WHURNZDGUDQWRZ\ Z NWyU\P RED V\JQDá\ VWHUXMFH

u

u

X

Y

,

PDM

V\PHWU\F]QHZHMFLDUy*QLFRZH3U]\]HURZHMZDUWRFLNWyUHJRNROZLHN

V\JQDáX VWHUXMFHJR QDSLFLH Z\MFLRZH

u

R

2

MHVW UyZQH ]HUX 6WG

XNáDGWHQQD]\ZDVLPRGXODWRUHPSRGZyMQLH]UyZQRZD*RQ\P

3DU\ Uy*QLFRZH

T

T

T

T

1

2

3

4

,

VWHURZDQH V V\JQDáHP

u

X

ZSU]HFLZID]LHGODWHJRREDWHXNáDG\Z\WZDU]DMQDUH]\VWDQFMDFK

NROHNWRURZ\FKVNáDGRZHUy*QLFRZHQDSLFLD

u

R

2

RSU]HFLZQ\FKID]DFK

background image

7

7

7

7

7

7

L

L

L

L

L

L

L L

+

L L

+

X

5

X

;

X

<

8

&&

5

&

5

&

,

D

E

X

5

X

;

8

<

=

8

<

,

>

8

8

<

,,

<

,

>

8

<

,,,

<

8

8

<

,9

<

,,,

<

5\V8NáDGPQR*F\F]WHURüZLDUWNRZ\DVFKHPDWLGHRZ\EFKDUDNWHU\VW\NL

:\SDGNRZ\SUGUy*QLFRZ\XNáDGXZ\QRVL

(

) (

) (

) (

)

i

i

i

i

i

i

i

i

i

R

2

1

3

2

4

1

2

4

3

= +

+

= −

:\UD*DMF ] NROHL SUG\ Uy*QLFRZH SRV]F]HJyOQ\FK SDU

Uy*QLFRZ\FK SU]\ SRPRF\ ]DOH*QRFL ] XZ]JOGQLHQLHP

RGSRZLHGQLFKQDSLüUy*QLFRZ\FKLSUGyZZREZRGDFKHPLWHUyZSR

LFKSRGVWDZLHQLXGRUyZQDQLDRWU]\PXMHP\

(

)

i

i tgh

u

i tgh

u

i

i

tgh

u

I tgh

u

tgh

u

I

u u

R

X

T

X

T

X

T

X

T

X

T

T

X

Y

2

5

6

5

6

0

0

2

2

2

2

2

2

4

=



 −



 =



 =

=





 ≈

ϕ

ϕ

ϕ

ϕ

ϕ

ϕ

3RGREQLHMDNZXNáDG]LHGZXüZLDUWNRZ\PUyZQLH*Z]DOH*QRFL

SU]\EOL*HQLH MHVW VáXV]QH W\ONR Z SU]\SDGNX JG\

u

u

X

Y

T

,

<<

2

ϕ

$E\]ZLNV]\ü]DNUHV\QDSLüVWHUXMF\FKSU]\]DFKRZDQLXZD*QRFL

Z\UD*HQLDNRQLHF]QDMHVWOLQHDU\]DFMDXNáDGX

background image

8NáDGF]WHURüZLDUWNRZ\]SU]HWZRUQLNLHP*LOEHUWD

/LQHDU\]DFM FKDUDNWHU\VW\N XNáDGX PQR*FHJR X]\VNXMH VL QD

GZDVSRVRE\

SU]H]]DVWRVRZDQLHZXNáDG]LHW]ZSU]HWZRUQLND*LOEHUWD

SU]H]]DVWRVRZDQLHGX*\FKUH]\VWDQFMLZREZRGDFKHPLWHUyZ

:áF]HQLH Z V]HUHJ ] V\JQDáHP VWHUXMF\P L ]áF]DPL ED]D

HPLWHU WUDQ]\VWRUyZ SDU\ Uy*QLFRZHM GX*\FK UH]\VWDQFML V]HUHJRZ\FK

SRZRGXMH ]QDF]QH ]PQLHMV]HQLH ZSá\ZX QLHOLQLRZRFL FKDUDNWHU\VW\N

ZHMFLRZ\FKWUDQ]\VWRUyZDOHUyZQLH*SRZRGXMH]QDF]QH]PQLHMV]HQLH

VWURPRFLFKDUDNWHU\VW\NXNáDGXPQR*FHJRU\VD>@

8NáDG

]OLQHDU\]RZDQ\

D

8NáDG EH] OLQHDU\]DFML

X

;

X

5

E

5

&0

5

&

5

&

'

'

7

7

L

L

L

$

L

%

L

L

FRQVW

$

%

+ =

,

L L

2

= +

X

*

8

&&

5\V/LQHDU\]DFMDFKDUDNWHU\VW\NXNáDGXPQR*FHJRDFKDUDNWHU\VW\NDSU]HGL

SROLQHDU\]DFMLEXNáDG*LOEHUWD

6SRVyE OLQHDU\]DFML SU]\ X*\FLX SU]HWZRUQLND *LOEHUWD

SU]HGVWDZLRQR QD U\VE 1D LGHQW\F]Q\FK GLRGDFK

D D

1

2

,

]UHDOL]RZDQ\FK MDNR WUDQ]\VWRU\ Z SRáF]HQLX GLRGRZ\P SRG

ZSá\ZHP SUGyZ VWHUXMF\FK

i

i

A

B

,

MHVW Z\WZDU]DQH QDSLFLH

u

G

VWHUXMFHSRMHG\QF]\XNáDGUy*QLFRZ\

-H*HOL ]áF]D GLRGRZH L ]áF]D HPLWHURZH PDM LGHQW\F]QH

FKDUDNWHU\VW\NL SUGRZR QDSLFLRZH WR SU]\ VSHáQLHQLX ZDUXQNyZ

i

i

const

i

i

const

A

B

1

2

+ =

+ =

ZXNáDG]LH]DFKRG]LSURSRUFMD

i

i

i

i

A

B

2

1

=

background image

=OLQHDU\]RZDQ\

XNáDG

PQR*F\

F]WHURüZLDUWNRZ\

]

]DVWRVRZDQLHP SU]HWZRUQLND *LOEHUWD L RSRUQLNyZ HPLWHURZ\FK

SU]HGVWDZLRQRQDU\V

7

7

7

7

L

L

L

L

X

*

X

5

'

'

L

,

L

$

2

;

=

5

&0

5

&

5

&

8

&&

7

7

5

<

,

L

2

<

,

L

2

<

+

L

<

,

,

X

<

7

7

5

;

L

,

L

%

2

;

=

+

L

;

,

,

X

;

5\V /LQHDU\]DFMD XNáDGX F]WHURüZLDUWNRZHJR ] ]DVWRVRZDQLHP SU]HWZRUQLND

*LOEHUWDLRSRUQLNyZHPLWHURZ\FK

=DNáDGDMF *H ZVSyáF]\QQLNL Z]PRFQLHQLD SUGRZHJR

β

0

]DVWRVRZDQ\FK WUDQ]\VWRUyZ V GX*H ZLG]LP\ *H GLRG\

D D

1

2

,

SU]HWZRUQLND *LOEHUWD V VWHURZDQH SUGDPL NROHNWRURZ\PL SDU\

Uy*QLFRZHM

T

T

7

8

3DU\ Uy*QLFRZH ]DVLODQH V SU]H] GZD (UyGáD

SUGRZH Z REZRGDFK HPLWHUyZ FR SR]ZROLáR QD ]DVWRVRZDQLH

SRMHG\QF]\FK UH]\VWRUyZ OLQHDU\]XMF\FK

R

R

X

Y

,

3U]\ GX*\FK

ZDUWRFLDFK UH]\VWDQFML

R

R

X

Y

,

PR*QD SRPLQü QLHOLQLRZH UH]\VWDQFMH

ZHMFLRZH WUDQ]\VWRUyZ SDU Uy*QLFRZ\FK WDN *H SUG\

i

i

X

Y

,

V Z

SU]\EOL*HQLXUyZQH

L

X
5

L

X
5

;

;

;

<

<

<

background image

3U]HWZRUQLN *LOEHUWD VWHUXMH GZLH SDU\ Uy*QLFRZH D SU]\

SRPLQLFLX SUGyZ ED] ]JRGQLH ] R]QDF]HQLDPX QD U\V

RWU]\PXMHP\

I

i

I

i

i

i

i

i

X

X

01

01

2

1

3

4


+

= =

'ODSDUUy*QLFRZ\FK

T

T

T

T

1

2

3

4

,

RWU]\PXMHP\]DOH*QRFL

i

i

I

i

i

i

I

i

Y

Y

1

2

01

3

4

01

+ =

+

+ =

:\NRU]\VWXMF]DOH*QRFLLRWU]\PXMHP\

I

i

I

i

I

i

i

I

i

I

i

I

i

I

i

i

I

i

X

Y

Y

X

Y

Y

01

01

2

02

3

02

01

01

1

02

4

02

2

2

=

+

=

+

=

+

=



1D SRGVWDZLH SURSRUFML PR*HP\ RNUHOLü Z\MFLRZ\ SUG

Uy*QLFRZ\LZ\MFLRZHQDSLFLHUy*QLFRZHZSRVWDFL

(

)

i

i

i

i

i

i i

I

R

X

Y

2

1

3

2

4

01

2

= + −

+

=

(

)

u

i

R

R

I

i i

k u u

R

R

C

C

X

Y

m

X

Y

2

2

01

2

=

=

=

:\NRU]\VWXMF ]DOH*QRFL L Z\]QDF]DP\ VWDá

PQR*HQLD

(

)

k

R

I

R R

m

C

X

Y

=

2

01

1D

U\V

SU]HGVWDZLRQR

W\SRZH

FKDUDNWHU\VW\NL

]OLQHDU\]RZDQHJRXNáDGXPQR*FHJRZNWyU\PVWDáDPQR*HQLDZ\QRVL

9

D OLQLRZRü FKDUDNWHU\VW\N MHVW ]DSHZQLRQD SU]\ ]PLDQDFK

QDSLFLDZ\MFLRZHJRLREXV\JQDáyZZHMFLRZ\FKZ]DNUHVLH

±

9

9

9

9

9

+

9

+

9

9

9

X

<

=

X

<

X

;

X

;

X

2

X

X X

9

2

; <

=

5\V7\SRZH

FKDUDNWHU\VW\NLXNáDGX

PQR*FHJR]OLQHDU\]RZDQHJR

background image

6SRUyG Z\PLHQLRQ\FK PHWRG UHDOL]DFML XNáDGyZ PQR*F\FK

PHWRGDVWHURZDQHJRSRG]LDáXSUGXMHVWQDMEDUG]LHMUR]SRZV]HFKQLRQD

DSU]\W\PSURVWDGRUHDOL]DFMLZWHFKQLFHXNáDGyZVFDORQ\FK6WRVXM F

WPHWRGRWU]\PXMHVLV]HURNRSDVPRZHGRNáDGQHXNáDG\PQR*FHR

SDUDPHWUDFK

(

)

δ

0

0 5

2

=

,

%

LF]VWRWOLZRFLJUDQLF]QHM

f

dB

3

]DOH*QHM

QDMF]FLHMRG]HZQWU]QLHGRáF]RQ\FKUH]\VWDQFML

R

C

GRFKRG]FHMGR

0+]

8QLZHUVDOQHXNáDG\PQR*FH

8QLZHUVDOQH XNáDG\ PQR*FH SRVLDGDM GRGDWNRZH PR*OLZRFL

Z\NRQ\ZDQLD LQQ\FK QLHOLQLRZ\FK RSHUDFML WDNLFK MDN QS G]LHOHQLH

SLHUZLDVWNRZDQLHLWS

=DVDGG]LDáDQLDW\FKXNáDGyZLOXVWUXMVFKHPDW\IXQNFMRQDOQHQD

U\V

D

E

X

;

X

;

X

<

X

<

X

=

X

;

X

;

X

;

X

;

X

<

X

<

X

<

X

<

X

=

X

=

X

=

X

,

X

,

&]WHUR

NZDGUDQWRZ\

XNáDG

PQR*F\

&]WHUR

NZDGUDQWRZ\

XNáDG

PQR*F\

L

L

′ =

X

X X

(

2

; <

5

X

.

X X

(

=

; <

5

+

=

X

..

X X

(

X

2

=

; <

5

=

=



X

. X

= =

=

X

.X

2

,

=

:2

X

.X

2

,

=

X

..

X X

(

X

2

=

; <

5

=

=



5

5

L

.


+

:2

.

=

+

.

+

X

=

F

X

;

X

<

:<

;

;

<

<

=

=

X

2

5\V 8QLZHUVDOQH XNáDG\ PQR*FH D ] QLHV\PHWU\F]Q\P ZHMFLHP = E ]

V\PHWU\F]Q\PZHMFLHP=FR]QDF]HQLHVFKHPDWRZHXNáDGXE

'ODXNáDGX]U\VDSU]\]DáR*HQLX*HZ]PDFQLDF]RSHUDF\MQ\

:2 SRVLDGD SDUDPHWU\ ]EOL*RQH GR XNáDGX LGHDOQHJR VáXV]QH V

]DOH*QRFL

background image

i

u

u

R

u

u u

E

R

u

u

K

Z

X

Y

R

O

=

=

+

= −

5R]ZL]XMFUyZQDQLDRWU]\PXMHP\

u

K u u

E

u

O

X

Y

R

Z

=



2

=DNáDGDMF*H K

→ ∞

RWU]\PXMHP\]DOH*QRüGODXNáDGX]DPNQLWHJR

u u

E u

X

Y

R

Z

=

8NáDG ] U\VD FKDUDNWHU\]XMH VL GZRPD ZHMFLDPL

V\PHWU\F]Q\PLLMHGQ\PZHMFLHPQLHV\PHWU\F]Q\P'ODSRGNUHOHQLD

WHJRIDNWXUyZQDQLDPR*HP\]DSLVDüZSRVWDFL

(

)(

)

u

u

u

u

E u

X

X

Y

Y

R

Z

1

2

1

2

=

3U]\SRáF]HQLXZ\MFLDZ]PDFQLDF]D]ZHMFLHP=W]Q

u

u

Z

O

=

XNáDG

UHDOL]XMHRSHUDFMPQR*HQLD

u

u u

E

O

X

Y

R

=

àF]F Z\MFLH Z]PDFQLDF]D ] ZHMFLHP < W]Q

u

u

Y

O

=

(

)

u

u

u

Y

O

Y

1

2

0

=

=

,

QDSRGVWDZLHUyZQDQLDRWU]\PXMHP\

u

E

u

u

u

O

R

Z

X

X

=

<

;

0

8NáDG UHDOL]XMH RSHUDFM G]LHOHQLD QDSLü ZHMFLRZ\FK

u u

Z

X

1DSLFLH

u

X

PXVL E\ü XMHPQH ERZLHP Z SU]HFLZQ\P Z\SDGNX

(

)

u

X

>

0

ZXNáDG]LHSRZVWDMHGRGDWQLHVSU]*HQLH]ZURWQH

.ROHMQ\P SU]\NáDGHP XQLZHUVDOQRFL XNáDGX MHVW UHDOL]DFMD

RSHUDFML SLHUZLDVWNRZDQLD SU]\ SRáF]HQLX

u

u

u

X

Y

O

= −

=

: W\P

SU]\SDGNX

u

E u

u

O

R

Z

Z

= −

<

;

0

8NáDG ] V\PHWU\F]Q\P ZHMFLHP WRUX = U\VE FHFKXMH VL

ZLNV] XQLZHUVDOQRFL Z VWRVXQNX GR SRSU]HGQLHJR UR]ZL]DQLD

:\WZRU]RQ\ Z W\P WRU]H V\JQDá

u

NWyU\ MHVW SURSRUFMRQDOQ\ GR

QDSLFLDZHMFLRZHJR

u

Z

VWHUXMHZHMFLHPRGZUDFDMF\PZ]PDFQLDF]D

Z\MFLRZHJR:2

'ODXNáDGX]RWZDUWSWORWU]\PXMHP\

u

K u

K K

u u

E

u

O

I

Z

X

Y

R

Z

=

=



background image

JG]LH

K

Z

Z]PRFQLHQLHWRUX=

3U]\ K

→ ∞

SUDFDXNáDGX]DPNQLWHJRRSLVDQDMHVW]DOH*QRFL

(

)(

)

u

u

u

u

E

u

u

X

X

Y

Y

R

Z

Z

1

2

1

2

1

2

=

(

)

5R]ZL]DQLHWR]QDOD]áR]DVWRVRZDQLHZPRQROLW\F]Q\PXNáDG]LH

VFDORQ\P $' 8NáDG WHQ PR*H RSUyF] Z\PLHQLRQ\FK ZF]HQLHM

RSHUDFMLUHDOL]RZDüZLHOHLQQ\FKIXQNFMLMDNQSZ]PDFQLDQLHQDSL FLD

]DXWRPDW\F]QUHJXODFMZ]PRFQLHQLD

3U]\NáDG\]DVWRVRZDXNáDGyZPQR*F\FK

8NáDG\G]LHOFH

2SHUDFMD G]LHOHQLD PR*H E\ü ]UHDOL]RZDQD SU]H] XPLHV]F]HQLH

XNáDGXPQR*FHJRZSWOLXMHPQHJRVSU]*HQLD]ZURWQHJRU\V

5

.

+

5

5

5 5

=

X

=

X

X

2

;

=

X

<

>

5\V3RGVWDZRZ\

XNáDGG]LHOF\

: XNáDG]LH ] U\V ]DNáDGDMF SRPLMDOQLH PDá UH]\VWDQFM

Z\MFLRZXNáDGXPQR*FHJRRWU]\PXMHP\

u

R

R

R

u u

E

R

R

R

u

K

Z

X

Y

R

X

2

1

2

1

1

2

+

+

+

= −

6WG

u

u

R

R

E

u

u

E

u

R

R

K R

X

O

R

Z

Y

R

Y

=

= −

+

+

2

1

1

2

1

1

1

3U]\ K

→ ∞

RWU]\PXMHP\

u

u

R

R

E

u

u

O

X

R

Z

Y

=

≈ −

2

1

:GRW\FKF]DVRZ\FKUR]ZD*DQLDFKSU]\MPRZDOLP\LGHDOQ\XNáDG

PQR*F\ -H*HOL VWDW\F]Q\ EáG RSHUDFML PQR*HQLD Z\QRVL

δ

0

F]\OL

background image

Z\QLNRSHUDFMLPQR*HQLDMHVWUyZQ\

(

)

u u

E

X

Y

R

+

δ

0

ZWHG\ZPLHMVFH

Z]RUXRWU]\PXMHP\

u

u

R

R

E

u

u

E

u

E

u

R

R

K R

R

R

E

u

u

E

u

X

O

R

Z

Y

R

Y

R

Y

R

Z

Y

R

Y

=

= −

+



+

+

≈ −

+

2

1

0

1

2

1

2

1

0

1

1

δ

δ

-DN Z\QLND ] ]DOH*QRFL EáG XNáDGX G]LHOFHJR URQLH

V]\ENRZUD]]H]PQLHMV]DQLHPV\JQDáX

u

Y

8NáDGSLHUZLDVWNXMF\

8NáDG SLHUZLDVWNXMF\ PR*QD ]UHDOL]RZDü PHWRG IXQNFML

RGZURWQHM VWRVXMF Z REZRG]LH VSU]*HQLD ]ZURWQHJR Z]PDFQLDF]D

RSHUDF\MQHJRXNáDGSRGQRV]F\GRNZDGUDWXU\V

']LDáDQLHXNáDGXRSLVXMHUyZQDQLH

u

R

R

R

u

E

R

R

R

u

K

Z

X

R

X

2

1

2

2

1

1

2

+

+

+

= −

VNGGOD

u

Z

<

0

RWU]\PXMHP\

u

u

R

R

E u

u

u

R

R

K R

R

R

E u

O

X

R

Z

X

Z

R

Z

=

= −

+

+



 ≈ −

2

1

1

2

1

2

1

1

5

.

+

5

5

X

=

<

'

5

/

X

2

X

X

;

<

=

5\V8NáDG

SLHUZLDVWNXMF\

: XNáDGDFK SUDNW\F]Q\FK SWOD VSU]*HQLD ]ZURWQHJR MHVW

]DP\NDQD ]D SRUHGQLFWZHP GLRG\ ' =DSRELHJD WR V\WXDFML JG\ Z

SU]\SDGNX QLHZáDFLZHM SRODU\]DFML V\JQDáX ZHMFLRZHJR

u

Z

Z]PDFQLDF] MHVW ZSURZDG]DQ\ Z QDV\FHQLH -HJR QRUPDOQH G]LDáDQLH

PR*QD ZyZF]DV SU]\ZUyFLü MHG\QLH SR UR]ZDUFLX S WOL VSU]*HQLD

]ZURWQHJR 3RQDGWR GLRGD ]QDMGXMH VL Z SWOL VSU]*HQLD ]ZURWQHJR

background image

]DWHPMHMSDUDPHWU\QLHPDMSUDNW\F]QHJRZSá\ZXQDSUDFXNáDGXZ

QRUPDOQ\FKZDUXQNDFK

'ZXSRáyZNRZ\SURVWRZQLNSUHF\]\MQ\

8NáDG GZXSRáyZNRZHJR SURVWRZQLND UHDOL]XMH RSHUDFM

Z\WZDU]DQLDEH]Z]JOGQHMZDUWRFLU\V

X

;

W

5\V

'ZXSRáyZNRZ\

SURVWRZQLN

SUHF\]\MQ\

X

;

X

<

X

X

2

=

=

&


+

X

2

W

X

<

8

2PD[

8

2PLQ

W

8NáDG PQR*F\ MHVW VWHURZDQ\ QDSLFLHP ZHMFLRZ\P

u

X

RUD]

IDOSURVWRNWQ]Z\MFLDNRPSDUDWRUDRSRODU\]DFMLQDSLFLD]JRGQHM

]SRODU\]DFMV\JQDáXZHMFLRZHJR

8NáDGSU]HWZRUQLNDZDUWRFLVNXWHF]QHM

: FHOX Z\]QDF]HQLD ZDUWRFL VNXWHF]QHM QDOH*\ Z\NRU]\VWDü

XNáDGSRGQRV]F\GRNZDGUDWXILOWUGROQRSU]HSXVWRZ\GODX UHGQLHQLD

NZDGUDWXQDSLFLDRUD]XNáDGSLHUZLDVWNXMF\U\V

5

5

5

5

&

5 5

5

5 5

=

X

(

2

5

X

2

.


+

.


+

X

(

,

5

X

,

5\V8NáDGSU]HWZRUQLNDZDUWRFLVNXWHF]QHM X

5

5 5&

X GW

2

,

7

=

background image

3U]HWZRUQLNSU]HELHJXWUyMNWQHJRQDVLQXVRLGDOQ\

6\PHWU\F]QHQDSLFLHWUyMNWQH

( )

u t

T

RDPSOLWXG]LH

U

T

L]HURZHM

ZDUWRFLUHGQLHM

( )

( )

(

)

u t

U

t

k

T

k

T

= −

1

2

π

ω

π

GOD

(

)

(

)

2

1

2

2

1

2

k

t

k

+

π

ω

π

NOLF]ED FDáNRZLWD PR*H E\ü

Z\NRU]\VWDQHGRXIRUPRZDQLDSU]HELHJXVLQXVRLGDOQHJR

5R]NáDGDMFIXQNFMVLQ

ω

t

ZV]HUHJSRWJRZ\RWU]\PXMHP\

( ) ( )

U

t

U

t

t

t

m

m

sin

!

!

...

ω

ω

ω

ω

=

+

3

5

3

5

-H*HOLV]HUHJWHQRJUDQLF]\P\GR GZyFKSLHUZV]\FKZ\UD]yZWR

]DZDUWRü KDUPRQLF]Q\FK SU]HELHJX XNV]WDáWRZDQHJR ] SU]HELHJX

WUyMNWQHJR Z\QRVL h

2 2

. %

/HSV]\ UH]XOWDW RWU]\PXMH VL VWRVXMF

SU]\EOL*HQLH

( )

u

U

t

U

t

t

O

m

m

=

sin

,

ω

ω

ω

3

6 81

3U]\ WHM DSURNV\PDFML PLQLPDOQD ]DZDUWRü KDUPRQLF]Q\FK

Z\QRVL

h

min

, %

0 4

8NáDGUHDOL]XMF\Z\UD*HQLHSU]HGVWDZLRQRQDU\V

X

(

7

5

8

(

5

7

5



5

5

5

X

2

.


+

X

(

7

5

X W

7

W

X W

7

8

7

8

7

5\V8NáDGNV]WDáWRZDQLDSU]HELHJXVLQXVRLGDOQHJR

8NáDGGHWHNWRUDID]RZHJR

'HWHNWRU ID]RF]Xá\ PDMF\ GX*H ]QDF]HQLH SUDNW\F]QH QS Z

XNáDGDFK 3// GHWHNWRUDFK )0 PR*H E\ü áDWZR ]UHDOL]RZDQ\ SU]\

X*\FLXXNáDGXPQR*FHJRU\V

background image

5

5

&

5

X

=

X

8 8

(

2

P

P

5

= −

VLQ

ϕ

.


+

8

W

P

FRV

8

W

P

VLQ

Ω +

ϕ

5\V'HWHNWRUID]RF]Xá\

1DSLFLHQDZ\MFLXXNáDGXPQR*FHJRZ\QRVL

(

)

[

]

u

U

U

E

t

Z

m

m

R

=

+

+

1

2

2

2

sin

sin

ϕ

ϕ

6NáDGRZD ZLHONLHM F]VWRWOLZRFL ]RVWDMH RGILOWURZDQD SU]H]

GROQRSU]HSXVWRZ\ILOWUZ\MFLRZ\=DWHP

u

U

U

E

O

m

m

R

= −

1

2

2

sin

ϕ

8.à$'</2*$5<70,&=1(,:<.à$'1,&=(

8NáDG\ORJDU\WPLF]QH

=DGDQLHP XNáDGX ORJDU\WPLF]QHJR MHVW Z\WZRU]HQLH QDSL FLD

Z\MFLRZHJR R ZDUWRFL SURSRUFMRQDOQHM GR ORJDU\WPX ZDUWRFL

XQRUPRZDQHJRQDSLFLDZHMFLRZHJR

u

k

u

U

k

u

U

O

D

I

R

E

I

R

= −

= −

lg

ln

JG]LH

k

k

D

E

,

VWDáHVNDORZDQLD

(

)

k

k

D

E

=

ln 10

U

R

QDSLFLHQRUPXMFH

1DSLFLH ZHMFLRZH MHVW ]DZV]H XQLSRODUQH GRGDWQLH ZWHG\

U

R

>

0

OXEXMHPQH

U

R

<

0

1DSLFLHZ\MFLRZHPR*HPLHüGRZROQ

SRODU\]DFM

background image

8NáDGSRGVWDZRZ\

)XQNFM ORJDU\WPLF]Q PR*QD ]UHDOL]RZDü XPLHV]F]DMF

HOHPHQW R FKDUDNWHU\VW\FH Z\NáDGQLF]HM Z SWOL XMHPQHJR VSU]*HQLD

]ZURWQHJRZ]PDFQLDF]DRSHUDF\MQHJR

1D U\V SU]HGVWDZLRQR QDMSURVWV]H UHDOL]DFMH XNáDGyZ

ORJDU\WPLF]Q\FK]Z\NRU]\VWDQLHPWUDQ]\VWRUyZELSRODUQ\FK

5

L

&

X

,

>

X

8

X

, 5

2

7

,

(6

= −



OQ

D

E

.


+

L

&

X

%(

5

L

&

X

,

<

X

8

X

, 5

2

7

,

(6

=



OQ

.


+

L

&

X

%(

F

5

X

,

5

(

&

N

X

2

.


+

5\V3URVWHXNáDG\

ORJDU\WPLF]QH

D]WUDQ]\VWRUHPQSQ

E]WUDQ]\VWRUHPSQS

FXNáDG]NRPSHQVDFM

F]VWRWOLZRFLRZ

3U]\MPXMF *H SUG NROHNWRURZ\ WUDQ]\VWRUD MHVW RSLVDQ\

]DOH*QRFL(EHUVD0ROODSU]\

U

BC

=

0

I

I

U

C

ES

BE

T

=



exp

ϕ

WR SU]\ ]DáR*HQLX LGHDOQHJR Z]PDFQLDF]D RSHUDF\MQHJR GOD XNáDGX ]

WUDQ]\VWRUHPQSQRWU]\PXMHP\

u

u

i

I

U

u

I

R

O

BE

T

C

ES

T

I

ES

= −

= −

= −

ϕ

ln

ln

1

OXE

u

u

I

R

O

T

I

ES

= −

(

ln

) lg

ϕ

10

1

SU]\F]\P

u

I

>

0

2WU]\PDQH ]DOH*QRFL PDM WDN VDP SRVWDü MDN

Z\UD*HQLH

: WHPSHUDWXU]H 7 .

R

& QDFK\OHQLH FKDUDNWHU\VW\NL

Z\QRVL

≈ −

ϕ

7

OQ

P9 GHN

'OD WUDQ]\VWRUyZ SODQDUQ\FK SU]\

U

BE

T

>

4

ϕ

]DOH*QRü

MHVWVSHáQLRQDZV]HURNLP]DNUHVLH]PLDQSU GXRGVHWHNS$GRNLONX

background image

P$ 6WRVXMF ]DWHP RGSRZLHGQLH WUDQ]\VWRU\ RUD] Z]PDFQLDF]H

RSHUDF\MQH PDáH ZHMFLRZH SUG\ SRODU\]DFML RUD] PDáH QDSLFLH

QLH]UyZQRZD*HQLDPR*HP\RWU]\PDüFKDUDNWHU\VW\NORJDU\WPLF]QZ

]DNUHVLHQDSLüZHMFLRZ\FKREHMPXMF\FKQDZHWGHNDG

=PLDQDWUDQ]\VWRUDQSQQDWUDQ]\VWRUSQSU\VESR]ZDODQD

UHDOL]DFM FKDUDNWHU\VW\NL ORJDU\WPLF]QHM GOD

u

I

<

0.

8PLHV]F]HQLH

WUDQ]\VWRUDZSWOLVSU]*HQLD]ZURWQHJRZ]PDFQLDF]DSRZRGXMHZ]URVW

Z]PRFQLHQLD QDSLFLRZHJR SWOL L PR*H E\ü SU]\F]\Q QLHVWDELOQRFL

XNáDGXORJDU\WPLF]QHJR$E\WHPX]DSRELHF]PQLHMV]DVL Z]PRFQLHQLH

QDSLFLRZH SWOL SU]H] ZáF]HQLH UH]\VWRUD

R

E

GR REZRGX HPLWHUD

U\VF D SRQDGWR ZáF]D VL GRGDWNRZ SRMHPQRü

C

k

GOD

VNRPSHQVRZDQLDZSá\ZXSRMHPQRFLZHMFLRZHMZ]PDFQLDF]D*áyZQ

ZDG SURVWHJR XNáDGX ORJDU\WPLF]QHJR MHVW VLOQD ]DOH *QRü MHJR

FKDUDNWHU\VW\NL VWDW\F]QHM RG WHPSHUDWXU\ VSRZRGRZDQHM ]PLDQDPL

ϕ

T

ES

I

RUD]

8NáDG\]NRPSHQVDFMWHUPLF]Q

:Sá\Z]PLDQWHPSHUDWXU\QDQDSLFLHZ\MFLRZHXNáDGXPR*QD

GRü VNXWHF]QLH VNRPSHQVRZDü VWRVXMF XNáDG SU]HGVWDZLRQ\ QD

U\V

5

(

7

7

L

&

L

&

X

%(

X

%(

5

5

5

5

&

N

$

X

,

X

2

X

$

.


+

5

5

&

N

8

5

.


+

5\V8NáDGORJDU\WPLF]Q\]NRPSHQVDFMWHPSHUDWXU\

=DOH*QRFL SUGyZ NROHNWRUyZ RG QDSLü ED]D HPLWHU

WUDQ]\VWRUyZGDQHVSU]\EOL*RQ\PLUyZQDQLDPL(EHUVD0ROOD

i

I

u

C

ES

BE

T

1

1

1

=

exp(

)

ϕ

D

i

I

u

C

ES

BE

T

2

2

2

=

exp(

/

)

ϕ

E

background image

=DNáDGDMFWDNLHVDPHSUG\QDV\FHQLDWUDQ]\VWRUyZ

I

I

ES

ES

1

2

=

RWU]\PXMHVL

i

i

u

u

u

C

C

BE

BE

T

A

T

1

2

1

2

=

=



exp

exp

ϕ

ϕ

3RPLMDMF SUG ED]\ WUDQ]\VWRUD

T

2

QDSLFLH

u

A

PR*QD

Z\]QDF]\ü]G]LHOQLNDQDSLFLD

R R

3

4

,

u

u

R

R

R

A

O

=

+

4

3

4

3U]\MPXMF *H ZHMFLRZH SUG\ SRODU\]DFML Z]PDFQLDF]\

RSHUDF\MQ\FKVSRPLMDOQLHPDáHRWU]\PXMHP\

i

u

R

i

U

R

C

I

C

R

1

1

2

1

=

=

;

'

5R]ZL]XMFXNáDGUyZQDRWU]\PXMHP\

u

U

R

R

R

R

u

U

O

T

I

R

= −

+





1

3

4

1

1

ln

'

*G\

ϕ

T

R

R R

U R R

=

=

=

26

15 7

1

3

4

1

1

P9

9WR

,

'

[ ]

[ ]

u

u

u

u

O

I

I

O

= −

log

,

SU]\F]\P

9

9

1DFK\OHQLHFKDUDNWHU\VW\NLORJDU\WPLF]QHMZ\QRVLZWHG\9GHN

u

I

1DSLFLH

u

O

MHVWQLH]DOH*QHRGSUGXQDV\FHQLD

,

(6

WUDQ]\VWRUyZ

QDWRPLDVWQDGDO]DOH*\RG

ϕ

T

W]QQDFK\OHQLHFKDUDNWHU\VW\NL]PLHQLD

VL]WHPSHUDWXU

u

u

T

T

O

O

T

T

=

=

ϕ

ϕ

=PLDQD WD Z\QRVL

R

&

7 . 7 .

:Sá\Z WHPSHUDWXU\ QD QDSLFLH

ϕ

T

PR*QD VNRPSHQVRZDü SU]H]

XPLHV]F]HQLH SDU\ WUDQ]\VWRUyZ Z XNáDG]LH VFDORQ\P ] WHUPRVWDWHP

VWDELOL]XMF\PWHPSHUDWXUWUDQ]\VWRUyZ

,QQ\P VSRVREHP NRPSHQVDFML ZSá\ZX ]PLDQ QDSL FLD

ϕ

T

MHVW

X]DOH*QLHQLHUH]\VWDQFML

R

4

RGWHPSHUDWXU\DQDORJLF]QLHGR]PLDQ

ϕ

T

=DOH*QRüWHPSHUDWXURZD

5 5

7

=

R

&

]DSHZQLDSHáQNRPSHQVDFMWHPSHUDWXURZ

background image

,VWRWQ\PSDUDPHWUHPXNáDGXORJDU\WPLF]QHJRFKDUDNWHU\]XM F\P

V]\ENRü MHJR G]LDáDQLD MHVW PDáRV\JQDáRZD WU]\GHF\EHORZD

F]VWRWOLZRü JUDQLF]QD

f

dB

3

0R*QD SU]\Mü *H W\SRZH XNáDG\

UHDOL]RZDQHPHWRGSU]HGVWDZLRQQDU\V]DSHZQLDM

f

dB

3

0+]

6]HURNRSDVPRZHXNáDG\ORJDU\WPLF]QH

6]HURNRSDVPRZH XNáDG\ ORJDU\WPLF]QH UHDOL]RZDQH V SU]H]

RGSRZLHGQLH ]HVWDZLHQLH SDU Uy*QLFRZ\FK WUDQ]\VWRUyZ ELSRODUQ\FK

>@8NáDG\WDNLHJHQHUXMIXQNFM

( )

y

b

a x

j

j

n

j

=

=

1

tanh

DSURNV\PXMF]]DáR*RQGRNáDGQRFLIXQNFMORJDU\WPLF]Q

y

y

a

b

x

a

= +

lg

JG]LH[\V]PLHQQ\PLXQRUPRZDQ\PL

8SURV]F]RQ\ VFKHPDW LGHRZ\ XNáDGX JHQHUXMFHJR IXQNFM R SRVWDFL

SU]HGVWDZLRQRQDU\V

L

Q$

L

Q%

7

Q$

7

Q%

E ,

Q

X

D X

5Q

Q ,

=

L

:$

L

:%

$

%

5

&

5

&

5

5

X

+

X

X

2

8

&&

8

((

.


+

L

$

L

%

7

$

7

%

E ,

X

D X

5

,

=

L

$

L

%

7

$

7

%

E ,

X

D X

5

,

=

5\V6]HURNRSDVPRZ\XNáDGORJDU\WPXMF\

'OD ND*GHM SDU\ Uy*QLFRZHM ]H (UyGáHP VWDáRSUGRZ\P , Z

REZRG]LHHPLWHUyZ]DFKRG]L]ZL]HN

i

I

u

R

R

T

=



α

ϕ

tanh

2

JG]LH

i

R

Uy*QLFRZ\SUGNROHNWRURZ\

u

R

ZHMFLRZHQDSLFLHUy*QLFRZH

background image

'OD QSDUUy*QLFRZ\FKZ\QLNRZ\SUGUy*QLFRZ\

i

WR

PR*HP\ RSLVDü

]DOH*QRFL

i

i

i

i

I

b

a

u

WR

WA

WB

Rj

j

n

j

j

n

j

I

T

=

=

=



=

=

1

1

2

α

ϕ

tanh

:SURZDG]DMF]PLHQQHXQRUPRZDQH

y

i

I

x

u

WR

I

T

=

=

α

ϕ

;

2

RWU]\PXMHP\IXQNFMDSURNV\PXMF

: XNáDG]LH QD U\VD Z\MFLRZ\ Z]PDFQLDF] RSHUDF\MQ\

VSHáQLD URO SU]HWZRUQLND SU]HWZDU]DMFHJR SUG Uy*QLFRZ\

i

WR

QD

QDSLFLHZ\MFLRZH

u

O

=DSLVXMFUyZQDQLDELODQVyZSUGyZGODZ]áyZ$L%PDP\

U

u

R

u

u

R

i

U

u

R

u

R

i

CC

C

O

WA

CC

C

WB

=

+

=

+

+

+

0

0

5R]ZL]XMFXNáDGUyZQDRWU]\PXMHP\

u

i

K

K

R R

O

WR

C

=

+ +

1

0

3RQLHZD*

K

R R

C

>> +

1

0

WRZ\UD*HQLHSU]\MPXMHSRVWDü

u

i

R

O

WR

=

0

JG]LH

i

WR

MHVWRSLVDQ\]DOH*QRFL

1D U\V SU]HGVWDZLRQR Z\NUHV IXQNFML SU]\MPXMF

SDUDPHWU\

n

a

b

a

=

= =

=

2

1

0 178

1

1

2

,

,

,

RUD]QDV]NLFRZDQR FKDUDNWHU\

VW\N y

x

=

tanh ,

RGSRZLDGDMFSU]\SDGNRZL

n

=

1.

-HGQD SDUD Uy*QLFRZD SU]\EOL*D FKDUDNWHU\VW\N ORJDU\WPLF]Q Z GRü

ZVNLP ]DNUHVLH QDSLü ZHMFLRZ\FK 5R]V]HU]HQLH ]DNUHVX

G\QDPLF]QHJR QDSLü ZHMFLRZ\FK X]\VNXMH VL SU]H] ]ZLNV]HQLH

OLF]E\SDUUy*QLFRZ\FK n

3U]\NáDGHPXNáDGXG]LDáDMFHJRZJSRZ\*V]HM]DVDG\PR*HE\ü

XNáDGPRQROLW\F]Q\617H[DV,QVWUXPHQWV=DZLHUDRQZMHGQHM

REXGRZLH GZD XNáDG\ ORJDU\WPLF]QH ] NWyU\FK ND *G\ VNáDGD VL ]

F]WHUHFKSDUUy*QLFRZ\FK n &KDUDNWHU\VW\ND ORJDU\WPLF]QD MHVW

UHDOL]RZDQD ] GRNáDGQRFL

±

G% Z ]DNUHVLH QDSLü ZHMFLRZ\FK

RNG%-HJRF]VWRWOLZRüJUDQLF]QDZ\QRVL

I

G%

=

0+]

background image

\ WJK[ E WJK

[

=

+

\ WJK[

=

Q

=

D E [

+

OJ

[

X

,

7

=

ϕ

E

=

E

=

E

=

Q

=

Q

=

Q

=

\

5\V 3U]\NáDG\ FKDUDNWHU\VW\N VWDW\F]Q\FK V]HURNRSDVPRZHJR XNáDGX

ORJDU\WPLF]QHJR]U\V

8NáDG\Z\NáDGQLF]H

8NáDG Z\NáDGQLF]\ JHQHUXMH IXQNFM RGZURWQ GR IXQNFML

ORJDU\WPLF]QHM

(

)

u

k

k

u U

O

w

u U

w

I

E

I

W

=

=

10

exp

JG]LH

k

w

VWDáDVNDORZDQLD

U

U

W

E

,

QDSLFLDQRUPXMFH

U

U

W

E

=

ln 10

:W\SRZ\FKUR]ZL]DQLDFKQDSLFLH

u

I

PR*HPLHüGRZROQ\]QDN

QDWRPLDVW QDSLFLH Z\MFLRZH

u

O

MHVW ]DZV]H XQLSRODUQH

(

)

u

k

u

O

w

I

=

=

,

GOD

0

2GZURWQD GR ORJDU\WPLF]QHM IXQNFMD Z\NáDGQLF]D PR *H E\ü

]UHDOL]RZDQD]JRGQLH]]DVDGIXQNFMLRGZURWQHM:SURVW\PXNáDG]LH

U\VZ]PDFQLDF]RSHUDF\MQ\VSHáQLD URO OLQLRZHJR SU]HWZRUQLND

i

C

QDQDSLFLH

u

O

&KDUDNWHU\VW\NSU]HMFLRZRSLVXMH]DOH*QRü

(

)

u

I

R

u

O

ES

I

T

=

1

exp

ϕ

SU]\F]\P

u

I

<

0

background image

X

,

X

2

L

&

L

&

5

X

%(

.

+

5\V3URVW\XNáDGZ\NáDGQLF]\

:DG SURVWHJR XNáDGX Z\NáDGQLF]HJR MHVW VLOQD ]DOH *QRü MHJR

FKDUDNWHU\VW\NLSU]HMFLRZHMRGWHPSHUDWXU\:SRGREQ\VSRVyEMDNZ

SU]\SDGNX XNáDGX ORJDU\WPLF]QHJR PR*HP\ VNRQVWUXRZDü VFKHPDW

XNáDGX Z\NáDGQLF]HJR ] NRPSHQVDFM ]PLDQ WHUPLF]Q\FK

FKDUDNWHU\VW\NLU\V

&KDUDNWHU\VW\ND SU]HMFLRZD XNáDGX ] U\V MHVW RSLVDQD

]DOH*QRFL

u

U

R

R

u

R

R

R

O

R

I

T

=

+



2

1

4

3

4

exp

ϕ

*G\

ϕ

7

5

8 5

5

5
5

=

=

=

P9

9

WRu

O

u

I

=

10

SU]\ F]\P

[ ] [ ]

X

X

,

2

9

9

5

(

7

7

L

&

L

&

X

%(

X

%(

5

5

5

X

,

5

8

5

5

&

N

X

2

.


+

&

N

5

.


+

5\V8NáDGZ\NáDGQLF]\]NRPSHQVDFM]PLDQWHUPLF]Q\FK

background image

8.à$'<.6=7$à78-&()81.&-(35=('=,$à$0,
352672/,1,2:(

2JUDQLF]QLNLQDSLFLD

,GHDOQ\ RJUDQLF]QLN QDSLFLD Z\WZDU]D IXQNFM

( )

u

f u

O

I

=

R

SRVWDFL

u

U

u

U

m u

U

u

U

U

u

U

O

O

I

I

I

I

I

I

O

I

I

=

<

≤ ≤

>



min

min

min

max

max

max

,

,

,

GOD

GOD

GOD

,GHDOQ FKDUDNWHU\VW\N VWDW\F]Q XNáDGX SU]HGVWDZLRQR QD

U\VD : SU]\SDGNX JG\ EH]Z]JOGQH ZDUWRFL SURJyZ

RJUDQLF]DQLD V UyZQH W]Q U

U

I

I

max

min

= −

RJUDQLF]QLN MHVW QD]\ZDQ\

V\PHWU\F]Q\PU\VE

&KDUDNWHU\VW\NDU]HF]\ZLVWHJRRJUDQLF]QLNDZ\ND]XMHRGFK\OHQLD

RG SU]HELHJX LGHDOQHJR JáyZQLH Z RWRF]HQLX SURJyZ L Z ]DNUHVLH

RJUDQLF]DQLDOLQLHSU]HU\ZDQHQDU\V

X

2

X

,

QDFK\OHQLH P

X

2

X

,

8

,PLQ

8

2PD[

QDFK\OHQLHP

D

E

8

,PD[

8

2 PLQ

8

,PD[

8

2PD[

8

,PD[

8

2PD[

5\V&KDUDNWHU\VW\NLRJUDQLF]QLNyZQDSLFLDDQLHV\PHWU\F]QHJR

EV\PHWU\F]QHJR

3URVW\XNáDGRJUDQLF]QLNDGLRGRZHJRSU]HGVWDZLRQRQDU\V

: XNáDG]LH W\P SURJL RJUDQLF]DQLD Z\]QDF]DM QDSLFLD

8

L

8

SRODU\]XMFH GLRG\ Z NLHUXQNX ]DSRURZ\P &KDUDNWHU\VW\N VWDW\F]Q

RJUDQLF]QLND Z\]QDF]\P\ SU]\ DSURNV\PDFML FKDUDNWHU\VW\NL GLRG\

OLQL áDPDQ MDN QD U\VE JG]LH

U

D

MHVW SURJRZ\P QDSLFLHP

GLRG\D

r

d

MHVWUH]\VWDQFMG\QDPLF]QZ]DNUHVLHSU]HZRG]HQLD

background image

5

X

2

X

,

8 8

P

'

+

QDFK\OHQLH P

5

5 5

/

=

+

QDFK\OHQLH

U 5

G

QDFK\OHQLH

U 5

G

+

8 8

P

'

8 8

'

+

+

8

8

'

D

F

'

'

5

/

8

8

X

,

X

2

U

G

L

'

8

'

QDFK\OHQLH

,

X

6

'

7

H[S

ϕ

E

X

'

5\V2JUDQLF]QLNGLRGRZ\DVFKHPDWEFKDUDNWHU\VW\NDGLRG\LSU]\MWDMHM

DSURNV\PDFMDFFKDUDNWHU\VW\NDRJUDQLF]QLND

'ODPDá\FKQDSLüZHMFLRZ\FKRE\GZLHGLRG\VVSRODU\]RZDQH

]DSRURZR L XNáDG ]DFKRZXMH VL MDN OLQLRZ\ UH]\VWDQF\MQ\ G]LHOQLN

QDSLFLRZ\

u

R

R

R

u

m u

O

L

L

I

I

=

+

=

: ]DNUHVLH GRGDWQLFK QDSLü ZHMFLRZ\FK

u

I

>

0

GLRGD

D

1

]RVWDMH ZáF]RQD JG\ QDSLFLH ZHMFLRZH RVLJQLH ZDUWRü

U

U

D

1

+

2GSRZLDGD WR ZHMFLRZHPX QDSLFLX SURJRZHPX ]JRGQLH ]

]DOH*QRFL

U

U

U

R

R

R

U

m U

O

D

L

L

I

I

max

max

max

=

+

=

+

=

1

3U]\ GDOV]\P Z]URFLH QDSLFLD ZHMFLRZHJR u U

I

I

>

max

,

QDVWSXMHRJUDQLF]DQLHQDSLFLDZ\MFLRZHJR

u

O

:]DNUHVLHRJUDQLF]DQLDQDFK\OHQLHFKDUDNWHU\VW\NLZ\QRVL

(

)

R r

R

R r

r

R

r

R R

L

d

L

d

d

d

L

+

<<

,

,

GOD

background image

:DQDORJLF]Q\VSRVyEPR*QDZ\]QDF]\üSDUDPHWU\RJUDQLF]QLND

GODXMHPQ\FKQDSLü

u

I

<

0

2JUDQLF]QLNLQDSLFLDVF]VWRNRQVWUXRZDQHSU]\X*\FLXGLRGL

Z]PDFQLDF]\RSHUDF\MQ\FK3URVW\XNáDGV\PHWU\F]QHJRRJUDQLF]QLND]

GLRGDPL=HQHUDSU]HGVWDZLRQRQDU\V

X

,

5

5

5

X

2

'

'


+

.

D

E

X

2

QDFK\OHQLH

8

8

=

'

+

+

8

8

=

'

5

5

QDFK\OHQLH

+

U U

5

]

G

X

,

5\V6\PHWU\F]Q\RJUDQLF]QLN]GLRGDPL=HQHUDDVFKHPDWEFKDUDNWHU\VW\ND

( )

X

I X

2

,

=

: XNáDG]LH QDVWSXMH REXVWURQQH RJUDQLF]HQLH SU]HELHJX

Z\MFLRZHJRGRZDUWRFL

(

)

U

U

Z

D

+

JG]LH

U

Z

MHVWQDSLFLHP=HQHUDD

U

D

QDSLFLHPSURJRZ\PGLRG

D

D

1

2

L

1DFK\OHQLHFKDUDNWHU\VW\NLZ

]DNUHVLH SU]HQRV]HQLD GLRG\

D

D

1

2

L

QLH SU]HZRG] Z\QRVL

R

R

2

1

/

]D Z ]DNUHVLH RJUDQLF]DQLD

(

)

+

U U

5

]

G

SU]\ F]\P

U U

]

G

V

UH]\VWDQFMDPL G\QDPLF]Q\PL GLRG Z ]DNUHVLH QDSL FLD =HQHUD L SU]\

SRODU\]DFMLZNLHUXQNXSU]HZRG]HQLD

8NáDG\SURJRZH

8NáDGSURJRZ\UHDOL]XMHIXQNFM

(

)

y

mx

= ±

max

, 0

OXE

(

)

y

mx

= ±

min

, 0

W]QSU]HELHJQDZ\MFLXMHVWOLQLRZIXQNFMV\JQDáXZHMFLRZHJRSU]\

MHGQHMMHJRSRODU\]DFML]DMHVWUyZQ\]HUXSU]\RGZURWQHMSRODU\]DFML

WHJR V\JQDáX 0R*OLZH FKDUDNWHU\VW\NL VWDW\F]QH LGHDOQHJR XNáDGX

SURJRZHJRSU]HGVWDZLRQRQDU\V

background image

[

\

$

QDFK\OHQLH P

\

P[

$

=

PLQ

D

[

\

%

QDFK\OHQLH P

\

P[

%

=

PD[

E

[

\

&

QDFK\OHQLH P

\

P[

&

=

PD[

F

\

\

&

$

= −

[

\

'

QDFK\OHQLH P

\

P[

\

'

%

=

= −

PLQ

G

P

<

P

>

5\V0R*OLZHFKDUDNWHU\VW\NLLGHDOQHJRXNáDGXSURJRZHJR

1DMSURVWV]H UHDOL]DFMH GLRGRZH XNáDGyZ SURJRZ\FK ]ZDQ\FK

SUHF\]\MQ\PL SURVWRZQLNDPL MHGQRSRáyZNRZ\PL OXE VXSHUGLRGDPL

SU]HGVWDZLRQRQDU\V

X

,

X

2

X

2

5

5

5

X

,

'

Q N

5 5

I L

=

= +

X

,

X

2

X

2

5

5

5

X

,

'

X

2

X

2

X

,

X

2

X

2

8

.

'

Q N

5 5

I L

=

= −

'

8

'

8

'

+



5

5

X

2

D

E

.


+

.


+

X

,

'

3

N

5

5

I L

= +

X

2

X

,

'

3

N

5

5

I L

= −

X

2

X

2

X

2

X

,

8

'

X

2

X

2

X

2

X

2

8

2PLQ

8

.

'

5\V'LRGRZ\XNáDGSURJRZ\DQLHRGZUDFDMF\ERGZUDFDMF\

background image

:Sá\Z\ QDSLFLD SU]HZRG]HQLD GLRG\

U

D

L MHM UH]\VWDQFML

G\QDPLF]QHM

r

d

]RVWDá\ ]UHGXNRZDQH G]LNL G]LDáDQLX XMHPQHJR

VSU]*HQLD]ZURWQHJRREHMPXMFHJRGLRGZáF]RQZWRUZ]PRFQLHQLD

:XNáDG]LHQLHRGZUDFDMF\PU\VDGOD

u

I

<

0

GLRGD 'MHVW

Z\áF]RQD ]DWHP

u

O

=

0

3RQLHZD* SWOD VSU]*HQLD ]ZURWQHJR MHVW

SU]HUZDQD]DWHP]DNUHVSU]HMFLRZ\ZáF]DQLDGLRG\MHVWEDUG]RPDá\

UyZQ\

U

K

D

/

1DZHW SU]\ EDUG]R PDá\P XMHPQ\P QDSLFLX

ZHMFLRZ\PZ]PDFQLDF]MHVWSU]HVWHURZDQ\LMHJRQDSLFLHZ\MFLRZH

MHVW UyZQH QDSLFLX RJUDQLF]HQLD

u

U

O

O

=

min

'OD

u

D

I

>

0

GLRGD

SU]HZRG]L ]DP\NDMF SWO VSU]*HQLD ]ZURWQHJR D Z]PRFQLHQLH

XNáDGXZ\QRVL

k

u

u

R

R

f i

O

I

=

= +

1

2

1

1DSLFLH SURJRZH GLRG\

U

D

PR*H E\ü WUDNWRZDQH MDNR

Z\MFLRZH QDSLFLH QLH]UyZQRZD*HQLD NWyUH SR VSURZDG]HQLX GR

ZHMFLD Z]PDFQLDF]D RGSRZLDGD ZDUWRFL

U

U

K

S

D

0

=

/

U\VD

5HGXNFMDZSá\ZXUH]\VWDQFMLG\QDPLF]QHMGLRG\

r

d

MHVWDQDORJLF]QDMDN

UHGXNFMD ZSá\ZX UH]\VWDQFML Z\MFLRZHM Z]PDFQLDF]D REMWHJR SWO

XMHPQHJR VSU]*HQLD ]ZURWQHJR 1LHRGZUDFDMF\ XNáDG SURJRZ\ MHVW

ZLF UyZQRZD*Q\ ÄVXSHUGLRG]LH´ D

p

R SDUDPHWUDFK V]F]WNRZ\FK

U

K

r

K

D

d

/

,

/

0

0

RUD]Z]PRFQLHQLX k

R R

f i

= +

1

2

1

2SLVDQ\ XNáDG SRVLDGD W ZDG *H Z ]DNUHVLH RJUDQLF]DQLD

Z]PDFQLDF] RSHUDF\MQ\ MHVW SU]HVWHURZDQ\ XMHPQ\P QDSL FLHP

ZHMFLRZ\P FR ]ZLNV]D F]DV Z\MFLD ] ]DNUHVX RJUDQLF]DQLD GR

]DNUHVXSU]HQRV]HQLDPDáDF]VWRWOLZRüSUDF\

:DG\WHMQLHSRVLDGDRGZUDFDMF\XNáDGSURJRZ\SU]HGVWDZLRQ\

QDU\VE:XNáDG]LHW\PZ]DNUHVLHSU]HQRV]HQLDW]QGOD

u

I

>

0

GLRGD 'MHVWZáF]RQDZVNXWHNF]HJRSWODVSU]*HQLD]ZURWQHJRMHVW

]DPNQLWD'ODRGFLQNRZROLQLRZHMDSURNV\PDFMLFKDUDNWHU\VW\NLGLRG\

RWU]\PXMHP\

u

R

R

u

U

K

R

R

K

R

R

r

R R

R

R

u

O

I

D

d

I

= −

+

+





+

+

+



≈ −

2

1

1

2

2

1

1

2

2

1

1

1

1

1

1

1DSLFLHSURJRZHGLRG\

U

D

LMHMUH]\VWDQFMDG\QDPLF]QD

r

d

VVLOQLH

]UHGXNRZDQHLPRJE\üSRPLQLWH: ]DNUHVLH RJUDQLF]HQLD W]Q GOD

u

I

<

0

GLRGD

D

MHVWZ\áF]RQDDGLRGD

D

SU]HZRG]L]DSRELHJDMF

SU]HVWHURZDQLXZ]PDFQLDF]D:WHG\QDSLFLDVUyZQH

background image

u

U

u

U

K

O

D

I

D

'

'

'

'

,

=

=

:HMFLRZH QDSLFLH Uy*QLFRZH u

I

'

SU]HGRVWDMH VL QD Z\MFLH

SU]H]UH]\VWDQFMVSU]*HQLD]ZURWQHJR

R

2

GDMFQDSLFLHEáGX

u

U

K

O

D

≈ −

'

0

2GZUDFDMF\ XNáDG SURJRZ\ MHVW ZLF UyZQRZD*Q\

ÄVXSHUGLRG]LH´ D

p

RSDUDPHWUDFKV]F]WNRZ\FK

(

)

1

0

1

2

+

R R U

K

D

r K

d

0

LZ]PRFQLHQLX k

R R

f i

= −

2

1

1D U\V SU]HGVWDZLRQR GZXZ\MFLRZ\ XNáDG SURJRZ\

SUHF\]\MQ\SURVWRZQLNMHGQRSRáyZNRZ\] RGG]LHOQ\PL Z\M FLDPL GOD

SyáRNUHVX GRGDWQLHJR L XMHPQHJR > @ R SRGREQ\FK ZáD FLZRFLDFK

MDNLHSRVLDGDXNáDG]U\VE

-H*HOLZXNáDG]LHQDSLFLHZHMFLRZH

u

I

MHVWGRGDWQLHWRQDSLFLH

Z\MFLRZHZ]PDFQLDF]DMHVWXMHPQH

(

)

u

K u

O

I

= −

LGODWHJRSU]HZRG]L

GLRGD

D

1

]DGLRGD

D

2

MHVWVSRODU\]RZDQD]DSRURZRLQLHSU]HZRG]L

*G\

u

u

I

O

<

>

0

0

,

WR

LSU]HZRG]LGLRGD

D

2

QDWRPLDVWGLRGD

D

1

MHVW

Z\áF]RQD

X

,

X

.X

2

,

= − ′

5

$

5

%

X

2%

X

2$

D

X

2$

1DFK\OHQLH

5

5

$

X

,

X

2%

1DFK\OHQLH

X

,

E

5

5

%

5

X

,

5

$

5

%

$

%

'

'

X

'

X

'

.

+

5\V'ZXZ\MFLRZ\XNáDGSURJRZ\DVFKHPDWEFKDUDNWHU\VW\NLVWDW\F]QHZ

SU]\SDGNX K

→ ∞

background image

.D*GD ] GLRG ]DOH*QLH RG SRODU\]DFML QDSLFLD ZHMFLRZHJR

]DP\NDSWOXMHPQHJRVSU]*HQLD]ZURWQHJRRUy*Q\FKSDUDPHWUDFK

:\NRU]\VWXMF NRQFHSFM ÄEH]SUGRZHJR ]ZDUFLD´ QD ZHMFLX

Z]PDFQLDF]D RSHUDF\MQHJR SU]\ K

→ ∞

RWU]\PXMHP\ QDVWSXMFH

Z\UD*HQLHRNUHODMFHQDSLFLDQDZ\MFLDFKSURVWRZQLND

u

R

R

u

u

u

OA

A

I

I

I

=

>




2

1

0

0

0

SU]\

SU]\

u

R

R

u

u

u

OB

B

I

I

I

=

>




2

1

0

0

0

SU]\

SU]\

%DUG]LHMRJyOQ\PUR]ZL]DQLHPLGHDOQ\FKHOHPHQWyZSURJRZ\FK

V XNáDG\ NWyU\FK FKDUDNWHU\VW\NL SRVLDGDM SXQNW ]DáDPDQLD

SU]HVXQLW\RSHZQZDUWRüSURJRZ U

p

Z]JOGHPSRF]WNXXNáDGX

ZVSyáU]GQ\FK7DNLHHOHPHQWDUQHXNáDG\SURJRZHRFKDUDNWHU\VW\NDFK

MDN QD U\V OHF] SU]HVXQLW\FK Z]JOGHP SRF]WNX XNáDGX

ZVSyáU]GQ\FK R GRZROQ GRGDWQL OXE XMHPQ ZDUWRü SURJRZ V

F]VWRZ\NRU]\VW\ZDQHGRV\QWH]\FKDUDNWHU\VW\NRGFLQNDPLOLQLRZ\FK

1D U\V SU]HGVWDZLRQR XNáDG ] UHJXORZDQ\P SURJLHP

FKDUDNWHU\VW\N SU]HMFLRZ\FK NWyU\ Uy*QL VL RG XNáDGX ] U\V

MHG\QLHQLHZLHONPRG\ILNDFMREZRGXZHMFLRZHJR>@

&KDUDNWHU\VW\NLSU]HMFLRZHXNáDGXRSLVXM]DOH*QRFL

u

R

R

u

U

u

U

u

U

OA

A

I

P

I

P

I

P

=

>




2

1

0

(

)

GOD

GOD

u

R

R

u

U

u

U

u

U

OB

B

I

P

I

P

I

P

=

<




2

1

0

(

)

GOD

GOD

SU]\ F]\P

U

P

MHVW ZHMFLRZ\P QDSLFLHP SURJRZ\P

(

)

U

R R

U

P

R

R

= −

1

background image

X

2%

X

2$

1DSLFLH SURJRZH

8

5

5

8

3

5

5

= −

D

5

$

5

%

$

%

'

'

.


+

X

2$

1DFK\OHQLH 5

5

$

X

,

8

5

<

8

5

=

8

5

>

X

2

8

3

X

,

8

3

P

1DSLFLH SURJRZH

F

X

,

5

5

5

8

5

X

2%

1DFK\OHQLH 5

5

%

X

,

8

5

<

8

5

=

8

5

>

E

5\V 8NáDG SURJRZ\ ] UHJXODFM SURJX FKDUDNWHU\VW\N D VFKHPDW XNáDGX E

FKDUDNWHU\VW\NLSU]HMFLRZHFVFKHPDWEORNRZ\

8NáDG\ UHDOL]XMFH ZDUWRü EH]Z]JOGQ SUHF\]\MQH

SURVWRZQLNLGZXSRáyZNRZH

&KDUDNWHU\VW\NL SU]HMFLRZH LGHDOQHJR XNáDGX RSHUDF\MQHJR GR

Z\]QDF]DQLDZDUWRFLEH]Z]JOGQHMQD]\ZDQHJRUyZQLH*SUHF\]\MQ\P

SURVWRZQLNLHPGZXSRáyZNRZ\PRSLVXMH]DOH *QRü

X

D X

2

,

=

SU]\ F]\P VWDáD

a

PR*H E\ü GRGDWQLD OXE XMHPQD QS a

= ±

1

MDN QD

U\V

X

2

X

,

X

X

2

,

= +

X

X

2

,

= −

X

X

2

,

=

D

E

X

2

X

,

( )

X

X

2

,

= − +

( )

X

X

2

,

= − −

X

X

2

,

= −

5\V&KDUDNWHU\VW\NLSU]HMFLRZHXNáDGXRSHUDF\MQHJRGRZ\]QDF]DQLDZDUWRFL

EH]Z]JOGQHMDVWDáDD EVWDáDD

background image

,VWQLHMH ZLHOH PR*OLZ\FK UR]ZL]D XNáDGyZ Z\]QDF]DMF\FK

ZDUWRü EH]Z]JOGQ SU]\ F]\P ]DVDGD LFK G]LDáDQLD RSLHUD VL

QDMF]FLHM QD GLRGRZ\FK XNáDGDFK SURVWRZQLF]\FK REMW\FK SWO

XMHPQHJR VSU]*HQLD ]ZURWQHJR Z FHOX ]UHGXNRZDQLD ZSá\ZX QDSLü

SU]HZRG]HQLDLUH]\VWDQFMLG\QDPLF]Q\FKX*\W\FKGLRG

=DVDGD WD MHVW Z\NRU]\VWDQD Z SURVW\P XNáDG]LH SUHF\]\MQHJR

SURVWRZQLNDGZXSRáyZNRZHJRQDU\VZNWyU\P

u

u

u

K

I

R

O

=

= ′

/

.

*G\ K

→ ∞

,

WR

u

u

I

R

=

]D

u

R i

u

R i

R

L

O

L

L

=

=

RUD]

:\QLNDVWG*H

(

)

X

5 5 X

2

/

,

=

SU]\F]\P

a

R

R

L

=

/

:DG WHJR UR]ZL]DQLD MHVW WR *H QDSLFLH Z\MFLRZH

u

O

QLH PD

ZVSyOQHJRSXQNWX]PDVXNáDGX

X

,

X

2

5

/

X

2

.

+

L

/

5

5\V3UHF\]\MQ\

SURVWRZQLNGZXSRáyZNRZ\

,QQ\ SU]\NáDG XNáDGX Z\]QDF]DMFHJR ZDUWRü EH]Z]JOGQ

SU]HGVWDZLRQRQDU\V>@

D5

5

5

5

5

:

X

2

.

+

X

,

X

2$

5

5

5

$

5

%

$

%

'

'

.


+

5

5

:

5\V8NáDGRSHUDF\MQ\GRZ\]QDF]DQLDZDUWRFLEH]Z]JOGQHM

background image

=DZLHUD RQ XNáDG SURJRZ\ ]H Z]PDFQLDF]HP RSHUDF\MQ\P :

LGHQW\F]Q\ MDN RSLVDQ\ ZF]HQLHM QD U\V RUD] VXPDWRU

]UHDOL]RZDQ\QDZ]PDFQLDF]X:

8NáDGSURJRZ\UHDOL]XMHIXQNFM

(

)

u

R

R u

u

u

OA

A

I

I

I

=

>


2

1

0

0

0

/

,

]DZ]PDFQLDF]VXPXMF\:

Z\NRQXMHRSHUDFM

u

R

R

u

R

R

u

O

I

OA

= −

5

3

5

4

3RGVWDZLDMFGRRWU]\PXMHP\

u

R

R

R R

R

R

u

u

R

R

u

u

O

A

I

I

I

I

=



>



2

5

1

4

5

3

5

3

0

0

,

,

'RELHUDMFUH]\VWDQFMHZXNáDG]LH

R

R

a

R

R

R R

a

A

5

3

2

5

1

4

1

2

= ≥

=

;

(

)

QSR

R

R

R

R

a

R

A

1

2

3

4

5

2

=

=

=

=

=

/

,

RWU]\PXMHP\QDSLFLHZ\MFLRZH

+

D X

,

SU]\

X

,

>

RUD]

D X

,

SU]\

X

,

<

F]\OL

(

)

u

a u

R

R

u

R

R

O

I

I

=

=

5

5

,

0HWRG\UHDOL]DFMLFKDUDNWHU\VW\NDSURNV\PRZDQ\FK

RGFLQNDPLSURVWROLQLRZ\PL

:SUDNW\FHF]VWR]DFKRG]LSRWU]HEDZ\WZDU]DQLD]DGDQHMIXQNFML

QLHOLQLRZHM PHWRG DSURNV\PDF\MQ SU]\ IXQNFML DSURNV\PXMFHM

RGFLQNDPLOLQLRZ\PL'RZROQFKDUDNWHU\VW\NSU]HMFLRZRGFLQNDPL

OLQLRZ PR*QD RWU]\PDü Z Z\QLNX VXSHUSR]\FML RNUHORQHM OLF]E\

HOHPHQWDUQ\FK XNáDGyZ SURJRZ\FK R RGSRZLHGQLR GREUDQ\FK

QDFK\OHQLDFK OLQLRZHJR RGFLQND

(

)

m

m

1

2

,

, ...

RUD] SXQNWyZ ]DáDPDQLD

XVWDORQ\FK ]D SRPRF QDSLü SURJRZ\FK

U

U

P

P

1

2

,

, ...

OXE SUGyZ

SURJRZ\FK

I

I

P

P

1

2

,

, ...

ZSU]\SDGNXJG\XNáDGUHDOL]RZDQ\MHVWZWU\ELH

SUGRZ\P

background image

1DU\VSRGDQRSU]\NáDGUHDOL]DFMLFKDUDNWHU\VW\NLX]\VNDQHM

ZZ\QLNXVXPRZDQLDF]WHUHFKHOHPHQWDUQ\FKXNáDGyZSURJRZ\FK

X

2

X

,

8

3

8

3

8

3

8

3

P

P

P P

+

(

)

+

P P

(

)

u

u

U

m

O

Oi

i

Pi

i

=

=

1

4

,

5\V3U]\NáDGUHDOL]DFML

FKDUDNWHU\VW\NLRGFLQNDPL

OLQLRZHM

(OHPHQWDUQHXNáDG\SURJRZHPRJE\üUHDOL]RZDQHZRSDUFLXR

Z]PDFQLDF]H RSHUDF\MQH FR MHGQDN Z SU]\SDGNX Z\WZDU]DQLD

ZLHORRGFLQNRZHM FKDUDNWHU\VW\NL SU]HMFLRZHM VNXWNXMH GRü GX*

]áR*RQRFL XNáDGX L ]QDF]Q\P RJUDQLF]HQLHP PDNV\PDOQHM

F]VWRWOLZRFL SUDF\ 1D U\V SU]HGVWDZLRQR SURVW\ SU]\NáDG

UHDOL]DFML Z\SXNáHM FKDUDNWHU\VW\NL SU]HMFLRZHM RGFLQNDPL OLQLRZHM

( )

u

f u

O

I

=

,

XWZRU]RQHM]DSRPRF VLHFL ]áR*RQHM ] UH]\VWRUyZ L GLRG

>@

X

,

'

'

8

3

5

*

=

D

X

X

2

8

3

L

L

L

L

*

*

*

'

Q

8

3Q

L

Q

*

Q

F

L

8

3

8

3

8

3

X

* * *

+

+

* * * *

+

+

+

* *

+

*

E

8

3

8

3

8

3

P

*

* * * *

=

+

+

+

P

*

* *

=

+

X

2

X

,

P

*

* * *

=

+

+

5\V*HQHUDWRUIXQNFMLQLHOLQLRZHM]áR*RQ\]NOXF]\GLRGRZ\FKLUH]\VWRUyZD

LMHJRFKDUDNWHU\VW\NLEF

background image

'LRG\

D

D

n

1

SHáQL URO NOXF]\ ZáF]DQ\FK ] FKZLO JG\

QDSLFLH

u

u

iR

I

= −

SU]HNURF]\ ZDUWRü SURJRZ

U

Pk

Z NWHM JDá]L

F]\OLZWHG\JG\

u

U

U

U

k

n

Pk

Pk

D

= ′ +

=

,

, ...,

GOD

1

JG]LH

U

D

MHVWQDSLFLHPSURJRZ\PGLRG\

3U]\XSRU]GNRZDQ\FKQDSLFLDFK

U

Pk

SRODU\]XMF\FKGLRG\

′ < ′ < < ′

U

U

U

P

P

Pn

1

2

...

Z\SDGNRZDFKDUDNWHU\VW\NDSUGRZRQDSLFLRZD

( )

i

g u

=

UyZQROHJOH

SRáF]RQ\FKn JDá]LMHVWRSLVDQD]DOH*QRFL

(

)

[

]

i

G u

G

u

U

k

n

k

Pk

=

+

=

0

1

0

max

,

-HVW WR ZNOVáD FKDUDNWHU\VW\ND SU]HG]LDáDPL SURVWROLQLRZD R

SU]HELHJXSRND]DQ\PQDU\VE

=DOH*QRü

( )

u

f u

O

I

=

ZXNáDG]LHQDU\VDZ\QLND]SRG]LDáX

QDSLFLD ZHMFLRZHJR SU]H] QLHOLQLRZ\ G]LHOQLN R ]PLHQLDMF\P VL

SRG]LDOH QDSLFLD SU]\ SU]HNUDF]DQLX NROHMQ\FK ZDUWRFL SURJRZ\FK

1DFK\OHQLDNROHMQ\FKRGFLQNyZFKDUDNWHU\VW\NLZ\QRV]

(

)

m

G

G

G

u

U

m

G

G

G

G

U

u

U

m

G

G

G

G

G

U

u

U

O

P

P

O

P

k

k

P k

O

Pk

1

0

1

2

0

1

1

2

0

1

1

1

=

+

=

+

+

=

+

+

+ +

<

+

GOD

GOD

GOD

-HVW WR Z\SXNáD FKDUDNWHU\VW\ND SU]HG]LDáDPL SURVWROLQLRZD

U\VF NWyUHM SXQNW\ ]DáDPDQLD PRJ E\ü XVWDZLDQH ]D SRPRF

QDSLüSURJRZ\FK]DQDFK\OHQLHSRV]F]HJyOQ\FKRGFLQNyZSRSU]H]

GREyUZDUWRFLNRQGXNWDQFML

3U]\NáDG\UHDOL]DFMLIXQNFMLDSURNV\PRZDQ\FKRGFLQNDPL

SURVWROLQLRZ\PL

5HDOL]DFMDIXQNFMLSDUDEROLF]QHMSDUDERODNZDGUDWRZD

6FKHPDWLGHRZ\SU]HNV]WDáWQLNDSU]HGVWDZLRQRQDU\V>@

background image

&KDUDNWHU\VW\ND SU]HMFLRZD XNáDGX MHVW UHDOL]RZDQD PHWRG

DSURNV\PDF\MQ SU]\ F]\P DSURNV\PDFMD MHVW F]WHURRGFLQNRZD Z

WU]HFLHM L F]ZDUWHM üZLDUWFH 1DSLFLH QD Z\MFLX Z]PDFQLDF]D :

MHVW

RGZUyFRQH Z ID]LH R

R

Z]JOGHP QDSLFLD ZHMFLRZHJR G]LNL

F]HPX X]\VNXMHP\ V\PHWU\F]QH VWHURZDQLH GUDELQHN GLRGRZR

UH]\VWDQF\MQ\FK GUDELQNL ] UH]\VWRUDPL

R

R

11

12

,

QDSLFLHP

u

I

]D

GUDELQNL]UH]\VWRUDPL

R

R

21

22

,

QDSLFLHP

u

I

:WHQVSRVyEX]\VNXMH

VL UHDOL]DFM REX JDá]L SDUDEROL NZDGUDWRZHM 3U]\ QDSLFLX

ZHMFLRZ\P PQLHMV]\P RG QDSLFLD SU]HZRG]HQLD GLRG QDSLFLH

Z\MFLRZH MHVW UyZQH ]HUX : PLDU Z]URVWX QDSLFLD ZHMFLRZHJR

PLG]\ZHMFLHZ]PDFQLDF]DRSHUDF\MQHJR:

DZHMFLHFDáHJRXNáDGX

V GRáF]DQH UyZQROHJOH NROHMQH UH]\VWRU\ 5 Z GUDELQFH GLRGRZR

UH]\VWDQF\MQHM ] UH]\VWRUDPL

R

R

11

12

,

FR SRZRGXMH ]PQLHMV]HQLH

UH]\VWDQFML JDá]L VSU]*HQLD ]ZURWQHJR L ]ZLNV]HQLH Z]PRFQLHQLD

XNáDGX

5

5

5

5

5

5

X

,

5

5

8

D

5

5

5

5

.


+

:

5

X

2

.


+

:

5

/

5\V6FKHPDWLGHRZ\GZXüZLDUWNRZHJRSU]HNV]WDáWQLNDSDUDEROLF]QHJR

: PLDU ]PQLHMV]DQLD QDSLFLD ZHMFLRZHJR GRáF]DQH V UyZQROHJOH

NROHMQRUH]\VWRU\5ZGUDELQFHGLRGRZRUH]\VWDQF\MQHM]UH]\VWRUDPL

R

R

21

22

,

8NáDGXPR*OLZLDUHDOL]DFMIXQNFMLNZDGUDWRZ\FKZ]DNUHVLH

QDSLü ZHMFLRZ\FK RG 9 GR 9 SU]\ SRSXODUQ\FK

Z]PDFQLDF]DFK RSHUDF\MQ\FK ] EáGHP PQLHMV]\P RG

±

SU]\

PDNV\PDOQHMZDUWRFLQDSLFLD

u

O

background image

3U]HNV]WDáWQLNSU]HELHJyZWUyMNWQ\FKQDVLQXVRLGDOQH

6FKHPDWLGHRZ\XNáDGXSU]HGVWDZLRQRQDU\V

&KDUDNWHU\VW\ND

SU]HMFLRZD

MHVW

UHDOL]RZDQD

PHWRG

DSURNV\PDF\MQ SU]\ DSURNV\PDFML WUyMRGFLQNRZHM Z SLHUZV]HM L

WU]HFLHMüZLDUWFH:PLDUZ]URVWXQDSLFLD

u

I

GRáF]DMVLUyZQROHJOH

GRUH]\VWRUD

R

2

NROHMQR

5

L

5

DZPLDU]PQLHMV]DQLDWHJRQDSLFLD

NROHMQR

5

L

5

SRZRGXMF]PQLHMV]HQLHUH]\VWDQFMLJDá]LVSU]*HQLD

]ZURWQHJRLZ]PRFQLHQLDXNáDGX

8NáDGXPR*OLZLDSU]HNV]WDáFHQLHIDOLWUyMNWQHMZVLQXVRLGDOQR

ZVSyáF]\QQLNX]DZDUWRFLKDUPRQLF]Q\FKPQLHMV]\PQL*

5

X

,

+

8

&&

8

((

X

2

5

'

'

+

8

&&

5

5

5

5

5

5

5

5

'

'

8

((

5

/

.


+

5\V6FKHPDW

LGHRZ\SU]HNV]WDáWQLND

SU]HELHJyZWUyMNWQ\FK

QDVLQXVRLGDOQH

$PSOLWXGD QDSLFLD Z\MFLRZHJR PR*H E\ü XVWDODQD Z V]HURNLP

]DNUHVLH]PLHQQRFLSU]\F]\PMHMPLQLPDOQDZDUWRüRJUDQLF]RQDMHVW

ZLHORNURWQRFLQDSLFLD

U

D

DPDNV\PDOQDOLQLRZRFLZ]PDFQLDF]D

RSHUDF\MQHJR

.203$5$725<

0RQROLW\F]QHNRPSDUDWRU\QDSLFLD

)XQNFMD XNáDGX SRUyZQXMFHJR SROHJD QD SRUyZQDQLX

ZHMFLRZHJR V\JQDáX DQDORJRZHJR QDSLFLRZHJR OXE SUGRZHJR

]V\JQDáHPRGQLHVLHQLD

background image

,GHDOQD FKDUDNWHU\VW\ND XNáDGX SRUyZQXMFHJR U\VD MHVW

RSLVDQD]DOH*QRFL

U

U

U

U

U

U

O

O

R

R

O

R

=

<

=
>




min

max

GOD X

GOD X

GOD X

,

,

,

0

JG]LH

U

R

MHVWQDSLFLHPRGQLHVLHQLD

:UyGXNáDGyZSRUyZQXMF\FKZ\Uy*QLDVLGHWHNWRU\ SU]HMFLD

SU]H]]HUR

U

R

=

0

RUD]G\VNU\PLQDWRU\SURJRZH

(

)

U

R

0

SU]\F]\P

PRJ WR E\ü XNáDG\ RGZUDFDMFH OXE QLHRGZUDFDMFH ] KLVWHUH] OXE

EH]KLVWHUH]\

X

2

D

X

,

8

5

8

2 PLQ

8

2PD[

X

2

E

X

,

8

5

8

2 PLQ

8

2PD[

8

,

8

,PLQ

8

,PD[

5\V &KDUDNWHU\VW\NL

( )

X

I X

2

,

=

XNáDGX SRUyZQXMFHJR D LGHDOQHJR E

U]HF]\ZLVWHJR

'R XNáDGyZ SRUyZQXMF\FK ]DOLF]D VL UyZQLH* G\VNU\PLQDWRU\

RNLHQNRZH Z NWyU\FK V\JQDá ZHMFLRZ\ MHVW SRUyZQ\ZDOQ\ ] GZRPD

QDSLFLDPL RGQLHVLHQLD

U

i

U

R

R

1

2

3U]\ X*\FLX G\VNU\PLQDWRUD

RNLHQNRZHJR PR*HP\ VWZLHUG]Lü F]\ V\JQDá ZHMFLRZ\ MHVW ]DZDUW\

SRPLG]\ ZDUWRFLDPL

U

i

U

R

R

1

2

F]\ SR]D W\P REV]DUHP -DNR

HOHPHQW\ SRUyZQXMFH QDMF]FLHM V VWRVRZDQH VFDORQH NRPSDUDWRU\

QDSLFLD VSHFMDOQLH SU]H]QDF]RQH GR WHJR FHOX L EGFH RGPLDQ

Z\VSHFMDOL]RZDQ\FKZ]PDFQLDF]\RSHUDF\MQ\FK.RPSDUDWRU\QDSLFLD

SRZLQQ\ FKDUDNWHU\]RZDü VL GX*\P Z]PRFQLHQLHP RUD] V]HURNLP

SDVPHP SU]HQRV]RQ\FK F]VWRWOLZRFL DE\ F]DV RGSRZLHG]L F]DV

]PLDQ\VWDQXQDZ\MFLXE\áMDNQDMPQLHMV]\3DUDPHWU\WHV]DOH*QH

RGVLHELHDZL*HMHLORF]\QZ]PRFQLHQLDLV]HURNRFLSDVPD

= W\FK Z]JOGyZ NRPSDUDWRU\ QDSLFLD ]EXGRZDQH V SRGREQLH

MDNV]HURNRSDVPRZHZ]PDFQLDF]HRSHUDF\MQH-HGQDN*HZRGUy*QLHQLX

RGZ]PDFQLDF]\QDSLFLHZ\MFLRZHNRPSDUDWRUyZSU]\MPXMHMHG\QLH

GZLHVNUDMQHZDUWRFLDSRQLHZD*SUDFXMEH]ZHZQWU]QHJRVSU]*HQLD

]ZURWQHJR V VWDOH SU]HVWHURZ\ZDQH 'ODWHJR NRPSDUDWRU\ V WDN

SURMHNWRZDQHDE\Z\HOLPLQRZDüNRQVHNZHQFMHZ\QLNDMFH]G]LDáDQLD

background image

Z ZDUXQNDFK SU]HVWHURZDQLD WM DE\ SU]HFLZG]LDáDü QDV\FHQLX

WUDQ]\VWRUyZ3RQDGWRSR]LRP\QDSLüQDZ\MFLXNRPSDUDWRUyZPXV]

E\ü GRVWRVRZDQH GR Z\PDJD XNáDGyZ F\IURZ\FK ] NWyU\PL EG

ZVSyáSUDFRZDüQS

U

U

OL

OH

,

ZORJLFH77/OXE&026=SRZ\*V]\FK

Z]JOGyZZNRPSDUDWRUDFKV]D]Z\F]DMVWRVRZDQHGRGDWNRZHV]\ENLH

HOHPHQW\RJUDQLF]DMFHDSR]LRP\RJUDQLF]DQLD

U

U

O

O

max

min

,

VVWDáHL

GRSDVRZ\ZDQH GR X*\WHM ORJLNL 3U]\ SURMHNWRZDQLX NRPSDUDWRUyZ

XZ]JOGQLDVLUyZQLH*IDNW*HLFKQDSLFLDZ\MFLRZHSU]\MPXMW\ONR

GZLHVNUDMQHZDUWRFLFRR]QDF]D*HF]DVRGSRZLHG]LOXERSy(QLHQLH

SURSDJDF\MQH PD ZLNV]H ]QDF]HQLH QL* V]HURNRü SDVPD = WHJR

Z]JOGX NRPSDUDWRU\ SUDFXM EH] NRPSHQVDFML F]VWRWOLZRFLRZHM

-HGQ\P ] QDMZD*QLHMV]\FK SDUDPHWUyZ NRPSDUDWRUD MHVW PDNV\PDOQD

V]\ENRü]PLDQ\QDSLFLDZ\MFLRZHJRR]QDF]DQD

S

UOM

.RPSDUDWRU\ QDSLFLD V SURMHNWRZDQH QD V]HUV]\ QL*

Z]PDFQLDF]H RSHUDF\MQH ]DNUHV ZHMFLRZHJR V\JQDáX ZVSyOQHJR

3RVWSXMH VL WDN QDZHW ZWHG\ JG\ SRFLJD WR ]D VRE ]ZLNV]HQLH

ZHMFLRZHJR QDSLFLD QLH]UyZQRZD*HQLD L MHJR GU\IWX FLHSOQHJR

]PQLHMV]HQLHUH]\VWDQFMLZHMFLRZHM]ZLNV]HQLHZHMFLRZ\FKSUGyZ

SRODU\]XMF\FK RUD] Z\VWSLHQLH QLHVWDELOQRFL Z OLQLRZ\P ]DNUHVLH

SUDF\

: U]HF]\ZLVW\P NRPSDUDWRU]H FKDUDNWHU\VW\ND

( )

u

f u

O

I

=

]DZLHUD SHZLHQ SU]HG]LDá QLHRNUHORQRFL

U

I

Z\QLNDMF\ ]H

VNRF]RQHM ZDUWRFL Z]PRFQLHQLD NRPSDUDWRUD NWyUHPX RGSRZLDGD

SU]HMFLHQDSLFLDZ\MFLRZHJRRGSR]LRPXORJLF]QHJRÄ´GRSR]LRPX

ORJLF]QHJRÄ´U\VE

6]HURNRüSU]HG]LDáX

U

I

Z\UD*RQDZ PLOLZROWDFK MHVW F]VWR

QD]\ZDQDF]XáRFLOXEUR]G]LHOF]RFLNRPSDUDWRUD

3RQLHZD* NRPSDUDWRU QDSLFLD MHVW V]F]HJyOQ RGPLDQ

Z]PDFQLDF]D RSHUDF\MQHJR ]DWHP MHJR ZáDFLZRFL PRJ E\ü

FKDUDNWHU\]RZDQH ]D SRPRF SDUDPHWUyZ ]EOL*RQ\FK GR SDUDPHWUyZ

Z]PDFQLDF]D 5y*QLFH Z GHILQLFMDFK QLHNWyU\FK SDUDPHWUyZ Z\QLNDM

SU]HGH ZV]\VWNLP ] F\IURZHJR FKDUDNWHUX V\JQDáX Z\MFLRZHJR

NRPSDUDWRUD

.RPSDUDWRU\ RJyOQLH G]LHOL VL QD QLHUHJHQHUDF\MQH L QD

UHJHQHUDF\MQH

3LHUZV]\P PRQROLW\F]Q\P QLHUHJHQHUDF\MQ\P NRPSDUDWRUHP

QDSLFLDE\áXNáDG

µ

$RSUDFRZDQ\ZUSU]H]ILUP)DLUFKLOG

6HPLFRGXFWRU -HJR VWUXNWXUD XNáDGRZD MHVW SRGREQD GR VWUXNWXU\

W\SRZHJR

Z]PDFQLDF]D

RSHUDF\MQHJR

SU]\

XZ]JOGQLHQLX

Z\PLHQLRQ\FK QD ZVWSLH XZDJ =DSHZQLD RQ F]DV RSy(QLHQLD QV

UR]G]LHOF]RüP9LPDNV\PDOQ\SUGSRODU\]DFML

µ

$

background image

: Sy(QLHMV]\FK UR]ZL]DQLDFK V]\ENLFK NRPSDUDWRUyZ QDSLFLD

]DVWRVRZDQR GRGDWNRZH ]DELHJL Z FHOX SROHSV]HQLD SDUDPHWUX

S

UOM

8]\VNDQRWRG]LNLZSURZDG]HQLXGRGDWNRZ\FKVWRSQLZ]PDFQLDMF\FK

SU]HGUy*QLFRZ\PVWRSQLHPZHMFLRZ\PREFLQDQLXDPSOLWXG\V\JQDáX

]D SRPRF GLRG 6FKRWWN\¶HJR RUD] ]DVWRVRZDQLX ZHZQWU]QHJR

GRGDWQLHJR VSU]*HQLD ]ZURWQHJR Z FHOX ]ZLNV]HQLD V]\ENRFL

SU]HU]XWXZNRPSDUDWRUDFKUHJHQHUDF\MQ\FK

6]\ENLHNRPSDUDWRU\QDSLFLDFKDUDNWHU\]XMVLPR*OLZRFLLFK

VWURERZDQLDOXE]DWU]DVNLZDQLD6WURERZDQLHSROHJDQDZ\PXV]DQLXQD

Z\MFLX NRPSDUDWRUD RNUHORQHJR VWDQX ORJLF]QHJR QLH]DOH*QLH RG

]PLDQ V\JQDáX ZHMFLRZHJR 1DWRPLDVW ]DWU]DVNLZDQLH SR]ZDOD QD

]DSDPLWDQLHQDZ\MFLXNRPSDUDWRUDVWDQXORJLF]QHJRNWyU\LVWQLDáZ

FKZLOL JG\ V\JQDá ]DWU]DVNXMF\ UR]SRF]á RGG]LDá\ZDQLH 6\JQDá

VWUREXMF\ MHVW ZSURZDG]DQ\ QDMF]FLHM GR ORJLF]Q\FK XNáDGyZ

Z\MFLRZ\FK NRPSDUDWRUD D V\JQDá ]DWU]DVNXMF\ GR VWRSQLD

ZHMFLRZHJR=DVDGEXGRZ\SURVWHJRXNáDGX]DWU]DVNRZHJRV]\ENLHJR

NRPSDUDWRUDUHJHQHUDF\MQHJRSU]HGVWDZLRQRQDU\V

+

8

&&

X

2

X

2

5

5

7

7

7

7

3

,

,

,

>

8

((

6\JQDá

]DWU]DVNXMF\

X

,

X

,

5\V=DVDGDEXGRZ\

XNáDGX]DWU]DVNXMFHJR

V]\ENLHJRNRPSDUDWRUD

'RNROHNWRUyZ7

L7

WZRU]F\FKZHMFLRZ\VWRSLHUy*QLFRZ\

F]VWR ND*GH ] ZHMü NRPSDUDWRUD SRSU]HG]D ZWyUQLN HPLWHURZ\

SU]\áF]RQD MHVW ED]DPL SDUD WUDQ]\VWRUyZ 7

L 7

Z WDNL VSRVyE *H

ZUD]]UH]\VWRUDPL 5

L5

WZRU]V\PHWU\F]Q\SU]HU]XWQLNELVWDELOQ\

background image

'RSURZDG]HQLHV\JQDáX]DWU]DVNXMFHJRSRZRGXMH]DPNQLFLHNOXF]D3

L ]DSDPLWDQLH VWDQX Z\MFLRZHJR Z SU]HU]XWQLNX ]DWU]DVNRZ\P

]WUDQ]\VWRUDPL7

L7

$E\VWDQ\QLHXOHJá\]PLDQLHZF]DVLHWUZDQLD

V\JQDáX ]DWU]DVNXMFHJR SUG (UyGáD ,

PXVL E\ü ZLNV]\ RG SUGX

(UyGáD,

3U]\NáDGHP V]\ENLHJR L ]DUD]HP GRNáDGQHJR NRPSDUDWRUD

QDSLFLD MHVW PRQROLW\F]Q\ XNáDG $P ILUP\ $GYDQFHG 0LFUR

'HYLFHV =DSHZQLD RQ RSy(QLHQLH SURSDJDF\MQH QV UR]G]LHOF]Rü

P9 SU]\ ZHMFLRZ\P SUG]LH SRODU\]XMF\P

µ

$ ZHMFLRZ\P

SUG]LHQLH]UyZQRZD*HQLD

µ

$]DNUHVLHQDSLüZHMFLRZ\FK

±

9L

F]DVLH]DWU]DVNLZDQLDQV8SURV]F]RQ\VFKHPDWXNáDGXSU]HGVWDZLRQR

QDU\V

5

+

8

&&

7

'

'

5

7

'

7

8

%

7

'

X

2

X

2

5

7

7

7

7

7

7

7

7

8

((

8

((

:H V\JQDáX

]DWU]DVNXMFHJR

X

,

X

,

8

%

,

,

8

((

5

1DSLFLH

SRODU\]XMFH

5\V8SURV]F]RQ\

VFKHPDWNRPSDUDWRUD

$P

8NáDG PD GZD Z\MFLD NRPSOHPHQWDUQH W\SX ÄRWZDUW\ HPLWHU´

Z\PDJDMFH

]DVWRVRZDQLD

UH]\VWRUyZ

REFL*DMF\FK

(

)

5 5

=

N

: NROHNWRUDFK ZHMFLRZHM SDU\ Uy*QLFRZHM

(

)

T T

3

11

,

XPLHV]F]RQRZSRáF]HQLXNU]\*RZ\PGZLHSDU\WUDQ]\VWRURZH

(

)

7 7

L

(

)

7 7

WZRU]FH SU]HU]XWQLN : VWDQLH ]DWU]DQLFLD QLVNL

VWDQORJLF]Q\9QDZHMFLX]DWU]DVNXMF\P (UyGáRSUGX ,

MHVW

background image

SU]HáF]RQHGRSDU\

(

)

T T

4

8

,

LV\JQDáZHMFLRZ\QLHZSá\ZDQDXVWDORQH

ZF]HQLHMVWDQ\Z\MFLRZHNRPSDUDWRUD

'RSLHURSRGDQLHZ\VRNLHJRVWDQXORJLF]QHJR9QDZHM FLH

]DWU]DVNXMFH SRZRGXMH SU]HáF]HQLH SUGX ,

GR SDU\ WUDQ]\VWRURZHM

( ,

).

T T

5

9

1DVWSXMHSRZUyWGRQRUPDOQHMSUDF\NRPSDUDWRUDSU]\NWyUHM

VWDQ\ ORJLF]QH QD Z\MFLDFK ]DOH* RG ZHMFLRZHJR V\JQDáX

Uy*QLFRZHJR : FHOX ]PLQLPDOL]RZDQLD ZSá\ZX SRMHPQR FL

REFL*DMF\FK NROHNWRU\ Uy*QLFRZHM SDU\ ZHMFLRZHM ]ZLNV]HQLH

V]\ENRFL G]LDáDQLD ]DVWRVRZDQR NDVNRGRZ EXGRZ WHJR VWRSQLD

(

)

T

T

T

T

3

2

11

10

,

.

'LRG\ =HQHUD '

L '

]DSHZQLDM SU]HVXQLFLH

SR]LRPyZ QDSLFLRZ\FK QD Z\MFLDFK 'LRG\ 6FKRWWN\¶HJR '

L '

RJUDQLF]DM ]DNUHV ]PLDQ QDSLFLD Z\MFLRZHJR 'RGDWQLH VSU]*HQLH

]ZURWQHW\SXUHJHQHUDF\MQHJRZSU]HU]XWQLNRZHMF] FLNRPSDUDWRUD

SDUD7

7

]ZLNV]DV]\ENRüSU]HáF]DQLDXNáDGX

.RPSDUDWRU\]KLVWHUH]

.RPSDUDWRU ] KLVWHUH] QD]\ZDQ\ SU]HU]XWQLNLHP 6FKPLWWD MHVW

XNáDGHPELVWDELOQ\PNWyUHJRFKDUDNWHU\VW\NDVWDW\F]QD

( )

u

f u

O

I

=

PD

NV]WDáW KLVWHUH]\ U\VG 5y*QLFD QDSLü SURJRZ\FK SU]HáF]DQLD

XNáDGXMHVWV]HURNRFLVWUHI\KLVWHUH]\

U

U

U

IH

I

I

=

max

min

.RPSDUDWRU ] KLVWHUH] PR*QD ]UHDOL]RZDü SU]H] ZSURZDG]HQLH

GRVWDWHF]QLHVLOQHJRGRGDWQLHJRVSU]*HQLD]ZURWQHJRZHZ]PDFQLDF]X

RVSU]*HQLXEH]SRUHGQLP

:Sá\Z GRGDWQLHJR VSU]*HQLD ]ZURWQHJR QD FKDUDNWHU\VW\N

VWDW\F]Q

( )

u

f u

O

I

=

UR]SDWU]\P\ QD SU]\NáDG]LH Z]PDFQLDF]D

RSHUDF\MQHJR SUDFXMFHJR ] ]HZQWU]Q\P GRGDWQLP VSU]*HQLHP

]ZURWQ\PW\SXQDSLFLRZRV]HUHJRZHJRU\VD

1D U\VE SU]HGVWDZLRQR FKDUDNWHU\VW\N VWDW\F]Q

Z]PDFQLDF]D RSHUDF\MQHJR SU]\ F]\P QDFK\OHQLH FKDUDNWHU\VW\NL Z

REV]DU]HDNW\ZQ\PRGSRZLDGDZ]PRFQLHQLXUy *QLFRZHPX

K

u

u

O

R

=

/

'OD

u

U

u

U

R

R

R

R

>

<

max

min

RUD]

Z]PDFQLDF]]QDMGXMHVLZVWDQLH

QDV\FHQLD OXE RJUDQLF]DQLD XWU]\PXMF SUDZLH VWDáH QDSLFLH

Z\MFLRZHRGSRZLHGQLR

U

i

U

O

O

max

min

'ODXNáDGX]U\VDSU]\

EUDNXVSU]*HQLD]ZURWQHJRRGáF]RQH5

QDFK\OHQLHFKDUDNWHU\VW\NL

VWDW\F]QHM

( )

u

f u

O

I

=

Z REV]DU]H DNW\ZQ\P Z\QRVL k

K

f 0

= −

background image

SRQLHZD*

u

u

I

R

= −

SRV]F]HJyOQH RGFLQNL FKDUDNWHU\VW\NL

RGSRZLDGDMFHMWHPXSU]\SDGNRZLVR]QDF]RQHF\IUQDU\VF

8

2PD[

X

2

8

,PLQ

G

X

,

8

2 PLQ

8

,PD[

X

2

X

,

F

8

,PD[

8

,PLQ

X

2

X

,

8

5 PD[

8

5 PLQ

8

2PD[

8

2PLQ

QDFK\OHQLH .

9

9

+

8

&&

8

((

2EV]DNW\Z 2EV]QDV\F

2EV]QDV\F

E

5

5

+

8

&&

8

((

X

5

5 5 5

=

X

,

D

X

X

+

X

2

.


+

5\V:]PDFQLDF]]GRGDWQLPVSU]*HQLHP]ZURWQ\PDVFKHPDW

EFKDUDNWHU\VW\NDZ]PDFQLDF]DRSHUDF\MQHJRFFKDUDNWHU\VW\NLXNáDGX

GFKDUDNWHU\VW\NDW\SXKLVWHUH]\

:]PRFQLHQLH XNáDGX ] ]DPNQLW SWO GRGDWQLHJR VSU]*HQLD

]ZURWQHJRZ\QRVL

k

K

K

f 0

1

= −

β

JG]LH

β

ZVSyáF]\QQLNVSU]*HQLD]ZURWQHJR

β =

+

R

R

R

1

1

2

D

-H*HOL ZH Z]PDFQLDF]X ]DVWRVXMHP\ VáDEH VSU] *HQLH ]ZURWQH

(

)

K

β <

1

WRURGNRZDF]üFKDUDNWHU\VW\NLVWDQLHVLEDUG]LHMVWURPD

RGFLQNL -HOL ZSURZDG]LP\ VLOQLHMV]H VSU]*HQLH ]ZURWQH *H

K

β

>

1

XNáDG QLHVWDELOQ\ WR Z FKZLOL JG\ XNáDG Z\MG]LH ] ]DNUHVX

QDV\FHQLD QDFK\OHQLH FKDUDNWHU\VW\NL VWDQLH VL GRGDWQLH RGFLQNL

background image

2WU]\PXMHP\XNáDGZNWyU\PSHZQDF]üFKDUDNWHU\VW\NL

( )

u

f u

O

I

=

MHVWIXQNFMWUyMZDUWRFLRZ

%DUG]LHMXQLZHUVDOQHUR]ZL]DQLHNRPSDUDWRUDSU]HGVWDZLRQRQD

U\VD .RPSDUDWRU MHVW VWHURZDQ\ ] MHGQHJR ZHM FLD QDSLFLHP

( )

( )

u t

u t

1

2

OXE

1DSLFLH QD GUXJLP ZHMFLX PD ZDUWRü VWDá L PR*H

E\üZ\NRU]\VWDQHGR]PLDQ\SR]LRPyZQDSL üSURJRZ\FKEH]ZSá\ZX

QDV]HURNRüKLVWHUH]\

5

5 5 5

=

X

X

X

,

5

D

.


+

8

2PD[

X

2

X

8

FRQVW

=

=

8

+

E

8

2 PLQ

8

PLQ

8

+

X

8

PD[

8

2 PD[

X

2

X 8

FRQVW

=

=

8

+

F

8

2 PLQ

8

PLQ

8

+

X

8

PD[

5

5 5 5

=

X

X

X

2

5

5

G

.


+

5\V .RPSDUDWRU ] KLVWHUH] D XNáDG SRGVWDZRZ\ E FKDUDNWHU\VW\ND

SU]HMFLRZDNRPSDUDWRUDRGZUDFDMFHJRFFKDUDNWHU\VW\NDSU]HMFLRZD

NRPSDUDWRUD QLHRGZUDFDMFHJR GXNáDG ]H VWDELOL]DFM SR]LRPyZ

SU]HMFLRZ\FK

D.RPSDUDWRURGZUDFDMF\

8NáDG ] U\VD MHVW VWHURZDQ\ QDSLFLHP

( )

X W

SU]\

X

FRQVW

=

=

8

:XNáDG]LHW\PQDSLFLDSURJRZHZ\QRV]

U

R

R

R

U

R

R

R

U

U

R

R

R

U

R

R

R

U

O

O

1

2

1

2

2

1

1

2

1

2

1

2

2

1

1

2

max

max

min

min

=

+

+

+

=

+

+

+



background image

1DU\VESRND]DQRFKDUDNWHU\VW\NSU]HMFLRZWHJRXNáDGX
E.RPSDUDWRUQLHRGZUDFDMF\

8NáDG

]

U\VD

MHVW

VWHURZDQ\

QDSLFLHP

( )

u t

u

U

2

1

1

SU]\

FRQVW

=

=

1DSLFLDSURJRZHSU]HáF]DQLDSU]HU]XWQLNDZ\QRV]

U

R

R

R

U

R

R

U

U

R

R

R

U

R

R

U

O

O

2

1

2

2

1

1

2

2

1

2

2

1

1

2

max

min

min

max

=

+

=

+



3R]LRP\Z\MFLRZHNRPSDUDWRUDPR*QDVWDELOL]RZDüSU]\X*\FLX

V\PHWU\F]Q\FK GLRG =HQHUD U\VF :H Z]RUDFK

QDOH*\ ZWHG\ XZ]JOGQLü QRZH ZDUWRFL SR]LRPyZ Z\MFLRZ\FK

NRPSDUDWRUD

U

U

U

U

O

O

Z

D

max

min

= −

=

+

'\VNU\PLQDWRU\RNLHQNRZH

6]F]HJyOQ RGPLDQ XNáDGX SRUyZQXMFHJR MHVW G\VNU\PLQDWRU

RNLHQNRZ\RNUHODMF\F]\ZDUWRüQDSLFLDZHMFLRZHJRPLHFLVLZ

]DGDQ\P SU]HG]LDOH '\VNU\PLQDWRU\ WHJR URG]DMX ]QDMGXM V]HURNLH

]DVWRVRZDQLH Z WHFKQLFH SRPLDURZHM L Z XNáDGDFK V\JQDOL]DF\MQ\FK

1DMáDWZLHM PR*QD ]EXGRZDü G\VNU\PLQDWRU RNLHQNRZ\ VWRVXM F GZD

VFDORQHNRPSDUDWRU\QDSLFLDQSSRGZyMQ\NRPSDUDWRU

µ

$

1D U\V SU]HGVWDZLRQR SURVW\ XNáDG G\VNU\PLQDWRUD

RNLHQNRZHJRLMHJRFKDUDNWHU\VW\N

( )

u

f u

O

I

=

8

2PLQ

8

2PD[

X

2

X

,

E

8

5

8

5

X

5

+

+

X

,

8

5

X

2

+

8

&&

D

5\V'\VNU\PLQDWRURNLHQNRZ\DVFKHPDWXNáDGXEFKDUDNWHU\VW\NDVWDW\F]QD

background image

-H*HOL V\JQDá ZHMFLRZ\ ]DZDUW\ MHVW Z SU]HG]LDOH RG

U

U

R

R

1

2

GR

WR QD Z\MFLX X]\VNXMH VL QDSLFLH

8

2PD[

V 3U]\

ZDUWRFLDFK V\JQDáX

u

I

OH*F\FK SR]D W\P SU]HG]LDáHP QD Z\MFLX

Z\VWSXMHQDSLFLH

U

O min

.


Wyszukiwarka

Podobne podstrony:
03 Nieliniowe uklady operacyjne (2)
03 Astronomiczne uklady wspolrzedn (2)
Wykład XIII Uniwersalne układy operacyjne
03 Astronomiczne uklady wspolrzedn (2)
2 WZMACNIACZE OPERACYJNE UKŁADY NIELINIOWE
wzm operacyjny - wyzysk, Inzynieria Materiałowa, I semestr, Elektrotechnika, elektrotechnika, Układy
[C] Badania Operacyjne Zadania (2009 03 01)
Politechnika Białostocka 03 Układy sterowania umożliwiające zmianę parametrów ruchu tłoka
wzmacniacz operacyjny - zastosowanie nieliniowe, Szkoła, Politechnika 1- 5 sem, chomikuj, 4 sem (gra
uklady rownan nieliniowych 0.12
XII Układy nieliniowe
03 Uklady sekwencyjne id 472117 (2)
11 Układy nieliniowe
lab6 uklady rownan nieliniowych
wzmacniacz operacyjny zastosowanie nieliniowe (3)
03 Plan Operacji

więcej podobnych podstron