03 Nieliniowe uklady operacyjne (2)

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]HELHJLQDSL FLRZHOXESUGRZH

]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]JO GX QD Z\NRQ\ZDQH RSHUDFMH QLHOLQLRZH

PR*HP\Z\Uy*QLüQDVW SXMFHJUXS\XNáDGRZH

XNáDG\NV]WDáWXMFHIXQNFMHSU]HG]LDáDPLSURVWROLQLRZH

XNáDG\SRUyZQXMFH

XNáDG\ORJDU\WPLF]QHLZ\NáDGQLF]H

XNáDG\PQR*FHLG]LHOFH

XNáDG\SRW JXMFHLSLHUZLDVWNXMFH

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]JO GX QD QLH]E\W GX*

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

WPR*QDUHDOL]RZDüMHG\QLHQLHNWyUHRSHUDFMHSRGVWDZRZH
 0HWRGDSRUHGQLD

:PHWRG]LHWHMJHQHURZDQIXQNFM Z\UD*DP\SU]H]LQQHIXQNFMH

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

SHZQ\PL NROHMQ\PL HWDSDPL Z\QLNDMF\PL ] SU]\M WHJR 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 QDVW SQLH JHQHURZDQD MHVW IXQNFMD

DSURNV\PXMFD 'RELHUDMF IXQNFM  DSURNV\PXMF QDOH*\ XZ]JO GQLü

EáGSU]\EOL*HQLDMDNLáDWZRüMHMUHDOL]DFML

:SUDNW\FHMDNRIXQNFMHDSURNV\PXMFHVWRVXMHVL QDMF] FLHM

IXQNFM  SU]HG]LDáDPL SURVWROLQLRZ ZLHORPLDQ SRW JRZ\ RUD] IXQNFM

E GF

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\VW SXMFH Z ]DGDQHM IXQNFML D QDVW SQLH

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

GRGDZDQLDPQR*HQLD D*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]áRQXZS WOLVSU] *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

,

,...,

,

,

REM W\MHVWS WOVSU] *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]ZURWQHJRRSHUDFM G]LHOHQLDOLF]QLNDSU]H]PLDQRZQLN

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

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

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

]HZQ WU]Q\FK QS]PLDQWHPSHUDWXU\QDSL ü]DVLODMF\FKLWS RUD]RG

URG]DMX]DVWRVRZDQ\FKSREXG]H

-HGQ\P]SDUDPHWUyZVWRVRZDQ\FKGRRFHQ\QLHOLQLRZHJRXNáDGX

RSHUDF\MQHJRMHVWZ\QLNRZ\VWDW\F]Q\EáGZ]JO GQ\RSHUDFML

( )

δ

0

-HVW

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

]DFLVNyZ ZHMFLRZ\FK XNáDGX QDSL FLDPL VWDá\PL R ZDUWRFLDFK

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

Z\]QDF]DQHVSU]\SU]\M FLXFLOHVSUHF\]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 Eá G]LH

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

ϕ

 Eá GX ID]\

DWDN*H F] VWRWOLZRü JUDQLF]Q

f

α

 SU]\ MHGQRSURFHQWRZ\P Eá G]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 QDSL FLD

Z\MFLRZHJR

u

O

 SURSRUFMRQDOQHJR GR LORF]\QX GZyFK ZHMFLRZ\FK

QDSL üVWHUXMF\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 QDSL FLH 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]QDNDFKQDSL ü

u

u

X

Y

i

-H*HOLQDSL FLDZHMFLRZH

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 QDSL FLD

ZHMFLRZHJR

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

SHZQ\PEá GHP

(

)

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]JO GQ\ L Z]JO GQ\ 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 Eá G]LH

DPSOLWXGRZ\P

f

ϕ

  F] VWRWOLZRü JUDQLF]QD SU]\ MHGQRSURFHQWRZ\P Eá G]LH ID]\

Eá G]LHZHNWRURZ\P 

6  ZLHONRV\JQDáRZD PDNV\PDOQD V]\ENRü QDUDVWDQLD QDSL FLD

Z\MFLRZHJRRNUHODVL SRGREQLHMDNGODXNáDGXOLQLRZHJR

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

ZHMFLDXNáDGXPQR*FHJRSU]\NáDGDVL QDSL FLDVWDáH9OXE9

SU]\ E

V

R

=

10

 ]D GUXJLH ZHMFLH SREXG]D VL  QDSL FLHP

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

LZ\]QDF]D VL  F] VWRWOLZRFL SU]\ RNUHORQ\FK Eá GDFK 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 QDSL FLD

Z\MFLRZHJR

( )

S

du

t

dt

O

=

max



 

ZPLHMVFHQDSL FLDVLQXVRLGDOQHJRVWRVXMHVL VWHURZDQLHV\PHWU\F]Q\P

QDSL FLHPSURVWRNWQ\PRZDUWRFLPL G]\V]F]\WRZHM9

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

RGSRZLHG]L U\V 

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

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

VWRVRZDQHVQDVW SXMFHPHWRG\

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\N ZV]\VWNLFKZ\PLHQLRQ\FKPHWRG



8NáDG\]PRGXODFMV]HURNRFLLDPSOLWXG\LPSXOVyZ

SURVWRNWQ\FK

:XNáDGDFKWHJRURG]DMXSU]HELHJSURVWRNWQ\MHVWPRGXORZDQ\

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

GRMHGQHJRQDSL FLDZHMFLRZHJRDDPSOLWXGDLPSXOVyZGRGUXJLHJR

QDSL FLDZHMFLRZHJR2WU]\PDQ\ZWHQVSRVyESU]HELHJ]PRGXORZDQ\

FKDUDNWHU\]XMHVL W\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\LPSXOVyZD SU]HELHJL

F]DVRZHE XSURV]F]RQ\VFKHPDWLGHRZ\

'RNRPSDUDWRUDVGRSURZDG]DQHQDSL FLD

u

X

RUD]V\PHWU\F]QH

QDSL FLHWUyMNWQH

( )

u t

T

RDPSOLWXG]LH

E

R

RNUHVLH

T

L]HURZHMZDUWRFL

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

( )

u t

S

U\VD VWHUXMF\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

,

VQDSL FLDPLZ\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]XMHPRGXODFM DPSOLWXG\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 QDSL FLD QD Z\MFLX ILOWUX

GROQRSU]HSXVWRZHJRZ\QRVL

X

7

7

X

2U

<

=











 

3RGVWDZLDMF  GR  RWU]\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]DVDG G]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

 

3RGVWDZLDMF  GR  RWU]\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]JO GQLDMF  RWU]\PXMHP\



u

u u

E

O

X

Y

R

=



D

3U]\

E

R

9 ]DNUHV\ QDSL ü 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 QDVW SXMFH

SDUDPHWU\

(

)

δ

µ





   



 

=

<



 







I

6

G%

N+]

9

V



8NáDG\]]DVWRVRZDQLHPNZDGUDWRUyZ

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

JHQHUXMFHIXQNFM NZDGUDWRZ W]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

RGQDSL FLDEUDPND(UyGáR

u

GS



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

WRIXQNFMHVXPRZDQLDLRGHMPRZDQLDVSURZDG]DM VL GRQDMSURVWV]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]PLDQQDSL FLD

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\D VFKHPDWXNáDGX

E FKDUDNWHU\VW\NL

1DSL FLH

u

X

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

T T

1

2

,

PR*HE\ü

GRGDWQLH OXE XMHPQH QDWRPLDVW QDSL FLH

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 QDSL FLH 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 QDSL FLHP

u

Y

 (UyGáD SUGRZHJR PR*QD

ZSHZQ\P]DNUHVLH]PLDQWHJRQDSL FLDRSLVDüSU]\EOL*RQ]DOH*QRFL

OLQLRZ>@

i

I

g u

O

m

Y

= +

0



 

3RGVWDZLDMF]DOH*QRü  GR  RWU]\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*QRFL  Z\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 QDSL FLD

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]JO GHPV\JQDáX

u

X

DQLHMHVW]UyZQRZD*RQHZ]JO GHPV\JQDáX

u

Y



-H*HOLZ\UD*HQLH  SU]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\P Z]JO GHP

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 QDSL FLH Z\MFLRZH

u

R

2

  MHVW UyZQH ]HUX 6WG

XNáDGWHQQD]\ZDVL PRGXODWRUHPSRGZyMQLH]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*QLFRZHQDSL FLD

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\D VFKHPDWLGHRZ\E FKDUDNWHU\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]JO GQLHQLHP

RGSRZLHGQLFKQDSL üUy*QLFRZ\FKLSUGyZZREZRGDFKHPLWHUyZSR

LFKSRGVWDZLHQLXGRUyZQDQLD  RWU]\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\]ZL NV]\ü]DNUHV\QDSL üVWHUXMF\FKSU]\]DFKRZDQLXZD*QRFL

Z\UD*HQLD  NRQLHF]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*FHJR U\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*FHJRD FKDUDNWHU\VW\NDSU]HGL

SROLQHDU\]DFMLE XNáDG*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 QDSL FLH

u

G

VWHUXMFHSRMHG\QF]\XNáDGUy*QLFRZ\

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

FKDUDNWHU\VW\NL SUGRZR  QDSL FLRZH 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]\

SRPLQL FLX 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*QRFL  L  RWU]\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\MFLRZHQDSL FLHUy*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

QDSL FLDZ\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

W PHWRG RWU]\PXMHVL V]HURNRSDVPRZHGRNáDGQHXNáDG\PQR*FHR

SDUDPHWUDFK

(

)

δ

0

0 5

2

=

,

%

LF] VWRWOLZRFLJUDQLF]QHM

f

dB

3

 ]DOH*QHM

QDMF] FLHMRG]HZQ WU]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

=DVDG G]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=F R]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]XMFUyZQDQLD  RWU]\PXMHP\

u

K u u

E

u

O

X

Y

R

Z

=



2



 

=DNáDGDMF*H K

→ ∞

RWU]\PXMHP\]DOH*QRüGODXNáDGX]DPNQL WHJR

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

WHJRIDNWXUyZQDQLD  PR*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]XMHRSHUDFM PQR*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

=

=

,

QDSRGVWDZLHUyZQDQLD  RWU]\PXMHP\

u

E

u

u

u

O

R

Z

X

X

=

<

;

0



 

8NáDG UHDOL]XMH RSHUDFM  G]LHOHQLD QDSL ü ZHMFLRZ\FK

u u

Z

X



1DSL FLH

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

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

:\WZRU]RQ\ Z W\P WRU]H V\JQDá

u

 NWyU\ MHVW SURSRUFMRQDOQ\ GR

QDSL FLDZHMFLRZHJR

u

Z

VWHUXMHZHMFLHPRGZUDFDMF\PZ]PDFQLDF]D

Z\MFLRZHJR:2

'ODXNáDGX]RWZDUWS WORWU]\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]DPNQL WHJRRSLVDQDMHVW]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*FHJRZS WOLXMHPQHJRVSU] *HQLD]ZURWQHJR U\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]RUX  RWU]\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\GRNZDGUDWX U\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 S WOD 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 S WOL VSU] *HQLD ]ZURWQHJR

background image



]DWHPMHMSDUDPHWU\QLHPDMSUDNW\F]QHJRZSá\ZXQDSUDF XNáDGXZ

QRUPDOQ\FKZDUXQNDFK

 'ZXSRáyZNRZ\SURVWRZQLNSUHF\]\MQ\

8NáDG GZXSRáyZNRZHJR SURVWRZQLND UHDOL]XMH RSHUDFM

Z\WZDU]DQLDEH]Z]JO GQHMZDUWRFL U\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\ QDSL FLHP ZHMFLRZ\P

u

X

 RUD]

IDOSURVWRNWQ ]Z\MFLDNRPSDUDWRUD RSRODU\]DFMLQDSL FLD]JRGQHM

]SRODU\]DFMV\JQDáXZHMFLRZHJR

 8NáDGSU]HWZRUQLNDZDUWRFLVNXWHF]QHM

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

XNáDGSRGQRV]F\GRNZDGUDWXILOWUGROQRSU]HSXVWRZ\ GODX UHGQLHQLD

NZDGUDWXQDSL FLD RUD]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]QHQDSL FLHWUyMNWQH

( )

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]NáDGDMFIXQNFM VLQ

ω

t

ZV]HUHJSRW JRZ\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*HQLH  SU]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*FHJR U\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\

1DSL FLHQDZ\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

XQRUPRZDQHJRQDSL FLDZHMFLRZHJR

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

QDSL FLHQRUPXMFH

1DSL FLH ZHMFLRZH MHVW ]DZV]H XQLSRODUQH GRGDWQLH ZWHG\

U

R

>

0

OXEXMHPQH

U

R

<

0

1DSL FLHZ\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 S WOL 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

F XNáDG]NRPSHQVDFM

F] VWRWOLZRFLRZ

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

]DOH*QRFL(EHUVD0ROOD SU]\

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 GX RGVHWHNS$GRNLONX

background image



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

RSHUDF\MQH PDáH ZHMFLRZH SUG\ SRODU\]DFML RUD] PDáH QDSL FLH

QLH]UyZQRZD*HQLD PR*HP\RWU]\PDüFKDUDNWHU\VW\N ORJDU\WPLF]QZ

]DNUHVLHQDSL üZHMFLRZ\FKREHMPXMF\FKQDZHWGHNDG

=PLDQDWUDQ]\VWRUDQSQQDWUDQ]\VWRUSQS U\VE SR]ZDODQD

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

u

I

<

0.

 8PLHV]F]HQLH

WUDQ]\VWRUDZS WOLVSU] *HQLD]ZURWQHJRZ]PDFQLDF]DSRZRGXMHZ]URVW

Z]PRFQLHQLD QDSL FLRZHJR S WOL L PR*H E\ü SU]\F]\Q QLHVWDELOQRFL

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

QDSL FLRZH S WOL 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\QDQDSL FLHZ\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 QDSL ü 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

 QDSL FLH

u

A

 PR*QD

Z\]QDF]\ü]G]LHOQLNDQDSL FLD

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]XMFXNáDGUyZQD  RWU]\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



1DSL FLH

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 QDSL FLH

ϕ

T

 PR*QD VNRPSHQVRZDü SU]H]

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

VWDELOL]XMF\PWHPSHUDWXU WUDQ]\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

& 

 

]DSHZQLDSHáQNRPSHQVDFM WHPSHUDWXURZ

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*RQGRNáDGQRFLIXQNFM ORJDU\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

ZHMFLRZHQDSL FLHUy*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\IXQNFM DSURNV\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

QDSL FLHZ\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]XMFXNáDGUyZQD  RWU]\PXMHP\

u

i

K

K

R R

O

WR

C

=

+ +

1

0



 

3RQLHZD*

K

R R

C

>> +

1

0

WRZ\UD*HQLH  SU]\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 QDSL ü ZHMFLRZ\FK 5R]V]HU]HQLH ]DNUHVX

G\QDPLF]QHJR QDSL ü ZHMFLRZ\FK X]\VNXMH VL  SU]H] ]ZL NV]HQLH

OLF]E\SDUUy*QLFRZ\FK n 

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

XNáDGPRQROLW\F]Q\61 7H[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 QDSL ü 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

,

QDSL FLDQRUPXMFH

U

U

W

E

=

ln 10





:W\SRZ\FKUR]ZL]DQLDFKQDSL FLH

u

I

PR*HPLHüGRZROQ\]QDN

QDWRPLDVW QDSL FLH 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\V Z]PDFQLDF]RSHUDF\MQ\VSHáQLD URO  OLQLRZHJR SU]HWZRUQLND

i

C

QDQDSL FLH

u

O



&KDUDNWHU\VW\N SU]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\NL U\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]QLNLQDSL FLD

,GHDOQ\ RJUDQLF]QLN QDSL FLD  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]JO GQH ZDUWRFL SURJyZ

RJUDQLF]DQLD V UyZQH W]Q U

U

I

I

max

min

= −

 RJUDQLF]QLN MHVW QD]\ZDQ\

V\PHWU\F]Q\P U\VE 

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

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

RJUDQLF]DQLD OLQLHSU]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]QLNyZQDSL FLDD QLHV\PHWU\F]QHJR

E V\PHWU\F]QHJR

3URVW\XNáDGRJUDQLF]QLNDGLRGRZHJRSU]HGVWDZLRQRQDU\V

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

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 QDSL FLHP

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\D VFKHPDWE FKDUDNWHU\VW\NDGLRG\LSU]\M WDMHM

DSURNV\PDFMDF FKDUDNWHU\VW\NDRJUDQLF]QLND

'ODPDá\FKQDSL üZHMFLRZ\FKRE\GZLHGLRG\VVSRODU\]RZDQH

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

QDSL FLRZ\

u

R

R

R

u

m u

O

L

L

I

I

=

+

=



 

: ]DNUHVLH GRGDWQLFK QDSL ü ZHMFLRZ\FK 

u

I

>

0

 GLRGD

D

1

]RVWDMH ZáF]RQD JG\ QDSL FLH ZHMFLRZH RVLJQLH ZDUWRü

U

U

D

1

+



2GSRZLDGD WR ZHMFLRZHPX QDSL FLX 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 QDSL FLD ZHMFLRZHJR u U

I

I

>

max

,

QDVW SXMHRJUDQLF]DQLHQDSL FLDZ\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\FKQDSL ü

u

I

<

0



2JUDQLF]QLNLQDSL FLDVF] 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=HQHUDD VFKHPDWE FKDUDNWHU\VW\ND

( )

X

I X

2

,

=

: XNáDG]LH QDVW SXMH REXVWURQQH RJUDQLF]HQLH SU]HELHJX

Z\MFLRZHJRGRZDUWRFL

(

)

U

U

Z

D

+

JG]LH

U

Z

MHVWQDSL FLHP=HQHUDD

U

D

QDSL FLHPSURJRZ\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\D QLHRGZUDFDMF\E RGZUDFDMF\

background image



:Sá\Z\ QDSL FLD SU]HZRG]HQLD GLRG\

U

D

 L MHM UH]\VWDQFML

G\QDPLF]QHM

r

d

 ]RVWDá\ ]UHGXNRZDQH G]L NL G]LDáDQLX XMHPQHJR

VSU] *HQLD]ZURWQHJRREHMPXMFHJRGLRG ZáF]RQZWRUZ]PRFQLHQLD

:XNáDG]LHQLHRGZUDFDMF\P U\VD GOD

u

I

<

0

GLRGD 'MHVW

Z\áF]RQD ]DWHP

u

O

=

0

 3RQLHZD* S WOD 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 QDSL FLX

ZHMFLRZ\PZ]PDFQLDF]MHVWSU]HVWHURZDQ\LMHJRQDSL FLHZ\MFLRZH

MHVW UyZQH QDSL FLX RJUDQLF]HQLD

u

U

O

O

=

min

 'OD

u

D

I

>

0

GLRGD

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

XNáDGXZ\QRVL

k

u

u

R

R

f i

O

I

=

= +

1

2

1



 

1DSL FLH SURJRZH GLRG\

U

D

 PR*H E\ü WUDNWRZDQH MDNR

Z\MFLRZH QDSL FLH 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 REM WHJR S WO

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

ZL F 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 ]ZL NV]D F]DV Z\MFLD ] ]DNUHVX RJUDQLF]DQLD GR

]DNUHVXSU]HQRV]HQLD PDáDF] VWRWOLZRüSUDF\ 

:DG\WHMQLHSRVLDGDRGZUDFDMF\XNáDGSURJRZ\SU]HGVWDZLRQ\

QDU\VE:XNáDG]LHW\PZ]DNUHVLHSU]HQRV]HQLD W]QGOD

u

I

>

0

GLRGD 'MHVWZáF]RQDZVNXWHNF]HJRS WODVSU] *HQLD]ZURWQHJRMHVW

]DPNQL WD'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

 

1DSL FLHSURJRZHGLRG\

U

D

LMHMUH]\VWDQFMDG\QDPLF]QD

r

d

VVLOQLH

]UHGXNRZDQHLPRJE\üSRPLQL WH: ]DNUHVLH RJUDQLF]HQLD W]Q GOD

u

I

<

0

GLRGD

D

MHVWZ\áF]RQDDGLRGD

D

SU]HZRG]L]DSRELHJDMF

SU]HVWHURZDQLXZ]PDFQLDF]D:WHG\QDSL FLDVUyZQH

background image



u

U

u

U

K

O

D

I

D

'

'

'

'

,

=

=

:HMFLRZH QDSL FLH Uy*QLFRZH u

I

'

 SU]HGRVWDMH VL  QD Z\MFLH

SU]H]UH]\VWDQFM VSU] *HQLD]ZURWQHJR

R

2

GDMFQDSL FLHEá GX

u

U

K

O

D

≈ −

'

0



 

2GZUDFDMF\ XNáDG SURJRZ\ MHVW ZL F 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]LHQDSL FLHZHMFLRZH

u

I

MHVWGRGDWQLHWRQDSL FLH

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\D VFKHPDWE FKDUDNWHU\VW\NLVWDW\F]QHZ

SU]\SDGNX K

→ ∞

background image



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

]DP\NDS WO XMHPQHJRVSU] *HQLD]ZURWQHJRRUy*Q\FKSDUDPHWUDFK

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

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

→ ∞

 RWU]\PXMHP\ QDVW SXMFH

Z\UD*HQLHRNUHODMFHQDSL FLDQDZ\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]HVXQL W\RSHZQZDUWRüSURJRZ U

p

Z]JO GHPSRF]WNXXNáDGX

ZVSyáU] GQ\FK7DNLHHOHPHQWDUQHXNáDG\SURJRZHRFKDUDNWHU\VW\NDFK

MDN QD U\V OHF] SU]HVXQL W\FK Z]JO GHP 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 QDSL FLHP SURJRZ\P

(

)

U

R R

U

P

R

R

= −

1



background image



X

2%

X

2$

1DSL FLH 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

1DSL FLH 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]HMFLRZHF VFKHPDWEORNRZ\



8NáDG\ UHDOL]XMFH ZDUWRü EH]Z]JO GQ SUHF\]\MQH

SURVWRZQLNLGZXSRáyZNRZH

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

Z\]QDF]DQLDZDUWRFLEH]Z]JO GQHM QD]\ZDQHJRUyZQLH*SUHF\]\MQ\P

SURVWRZQLNLHPGZXSRáyZNRZ\P RSLVXMH]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]JO GQHMD VWDáDD E VWDáDD 

background image



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

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

QDMF] FLHM QD GLRGRZ\FK XNáDGDFK SURVWRZQLF]\FK REM W\FK S WO

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

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 QDSL FLH 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]JO GQ

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]JO GQHM

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



 

3RGVWDZLDMF  GR  RWU]\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\QDSL FLHZ\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\N SU]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 QDSL ü 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]\VWRUyZ D

LMHJRFKDUDNWHU\VW\NL EF

background image



'LRG\

D

D

n

1

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

QDSL FLH

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

MHVWQDSL FLHPSURJRZ\PGLRG\

3U]\XSRU]GNRZDQ\FKQDSL FLDFK

U

Pk

SRODU\]XMF\FKGLRG\

′ < ′ < < ′

U

U

U

P

P

Pn

1

2

...

 

Z\SDGNRZDFKDUDNWHU\VW\NDSUGRZRQDSL FLRZD

( )

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 ZNO VáD 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

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

SRG]LDOH QDSL FLD 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

QDSL üSURJRZ\FK]DQDFK\OHQLHSRV]F]HJyOQ\FKRGFLQNyZSRSU]H]

GREyUZDUWRFLNRQGXNWDQFML



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

SURVWROLQLRZ\PL

 5HDOL]DFMDIXQNFMLSDUDEROLF]QHM SDUDERODNZDGUDWRZD

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 1DSL FLH QD Z\MFLX Z]PDFQLDF]D :



 MHVW

RGZUyFRQH Z ID]LH R 

R

 Z]JO GHP QDSL FLD ZHMFLRZHJR G]L NL

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

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

R

R

11

12

,

 QDSL FLHP 

u

I

  ]D

GUDELQNL]UH]\VWRUDPL

R

R

21

22

,

QDSL FLHP

u

I

:WHQVSRVyEX]\VNXMH

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

ZHMFLRZ\P PQLHMV]\P RG QDSL FLD SU]HZRG]HQLD GLRG QDSL FLH

Z\MFLRZH MHVW UyZQH ]HUX : PLDU  Z]URVWX QDSL FLD ZHMFLRZHJR

PL G]\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 ]ZL NV]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 QDSL FLD ZHMFLRZHJR GRáF]DQH V UyZQROHJOH

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

R

R

21

22

,

8NáDGXPR*OLZLDUHDOL]DFM IXQNFMLNZDGUDWRZ\FKZ]DNUHVLH

QDSL ü ZHMFLRZ\FK RG   9 GR   9 SU]\ SRSXODUQ\FK

Z]PDFQLDF]DFK RSHUDF\MQ\FK  ] Eá GHP PQLHMV]\P RG

±

   SU]\

PDNV\PDOQHMZDUWRFLQDSL FLD

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:PLDU Z]URVWXQDSL FLD

u

I

GRáF]DMVL UyZQROHJOH

GRUH]\VWRUD

R

2

NROHMQR

5



L

5



 DZPLDU ]PQLHMV]DQLDWHJRQDSL FLD

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 QDSL FLD Z\MFLRZHJR PR*H E\ü XVWDODQD Z V]HURNLP

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

ZLHORNURWQRFLQDSL FLD

U

D

DPDNV\PDOQDOLQLRZRFLZ]PDFQLDF]D

RSHUDF\MQHJR



.203$5$725<



0RQROLW\F]QHNRPSDUDWRU\QDSL FLD

)XQNFMD XNáDGX SRUyZQXMFHJR SROHJD QD SRUyZQDQLX

ZHMFLRZHJR V\JQDáX DQDORJRZHJR  QDSL FLRZHJR 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

MHVWQDSL FLHPRGQLHVLHQLD

:UyGXNáDGyZSRUyZQXMF\FKZ\Uy*QLDVL GHWHNWRU\ 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

QDSL FLDPL 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\

SRPL G]\ ZDUWRFLDPL

U

i

U

R

R

1

2

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

HOHPHQW\ SRUyZQXMFH QDMF] FLHM V VWRVRZDQH VFDORQH NRPSDUDWRU\

QDSL FLD VSHFMDOQLH SU]H]QDF]RQH GR WHJR FHOX L E GFH RGPLDQ

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

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\MFLX E\áMDNQDMPQLHMV]\3DUDPHWU\WHV]DOH*QH

RGVLHELHDZL*HMHLORF]\QZ]PRFQLHQLDLV]HURNRFLSDVPD

= W\FK Z]JO GyZ NRPSDUDWRU\ QDSL FLD ]EXGRZDQH V SRGREQLH

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

RGZ]PDFQLDF]\QDSL FLHZ\MFLRZHNRPSDUDWRUyZSU]\MPXMHMHG\QLH

GZLHVNUDMQHZDUWRFLDSRQLHZD*SUDFXMEH]ZHZQ WU]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\QDSL üQDZ\MFLXNRPSDUDWRUyZPXV]

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

ZVSyáSUDFRZDü QS

U

U

OL

OH

,

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

Z]JO GyZZNRPSDUDWRUDFKV]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]JO GQLDVL UyZQLH*IDNW*HLFKQDSL FLDZ\MFLRZHSU]\MPXMW\ONR

GZLHVNUDMQHZDUWRFLFRR]QDF]D*HF]DVRGSRZLHG]L OXERSy(QLHQLH

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

Z]JO GX NRPSDUDWRU\ SUDFXM EH] NRPSHQVDFML F] VWRWOLZRFLRZHM

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

V]\ENRü]PLDQ\QDSL FLDZ\MFLRZHJRR]QDF]DQD

S

UOM



.RPSDUDWRU\ QDSL FLD V SURMHNWRZDQH QD V]HUV]\  QL*

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

3RVW SXMH VL  WDN QDZHW ZWHG\ JG\ SRFLJD WR ]D VRE ]ZL NV]HQLH

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

]PQLHMV]HQLHUH]\VWDQFMLZHMFLRZHM]ZL NV]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]HMFLHQDSL FLDZ\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 QDSL FLD 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

QDSL FLDE\áXNáDG

µ

$RSUDFRZDQ\ZUSU]H]ILUP )DLUFKLOG

6HPLFRGXFWRU -HJR VWUXNWXUD XNáDGRZD MHVW SRGREQD GR VWUXNWXU\

W\SRZHJR

Z]PDFQLDF]D

RSHUDF\MQHJR

SU]\

XZ]JO GQLHQLX

Z\PLHQLRQ\FK QD ZVW SLH 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 QDSL FLD

]DVWRVRZDQR GRGDWNRZH ]DELHJL Z FHOX SROHSV]HQLD SDUDPHWUX

S

UOM



8]\VNDQRWRG]L NLZSURZDG]HQLXGRGDWNRZ\FKVWRSQLZ]PDFQLDMF\FK

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

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

GRGDWQLHJR VSU] *HQLD ]ZURWQHJR Z FHOX ]ZL NV]HQLD V]\ENRFL

SU]HU]XWX ZNRPSDUDWRUDFKUHJHQHUDF\MQ\FK 

6]\ENLHNRPSDUDWRU\QDSL FLDFKDUDNWHU\]XMVL PR*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

]DSDPL WDQLHQDZ\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=DVDG EXGRZ\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]DPNQL FLHNOXF]D3

L ]DSDPL WDQLH 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 ZL NV]\ RG SUGX

(UyGáD,





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

QDSL FLD 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

$]DNUHVLHQDSL üZHMFLRZ\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



1DSL FLH

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]DQL FLD 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]QHJR 9 QDZHM FLH

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



 GR SDU\ WUDQ]\VWRURZHM

( ,

).

T T

5

9

1DVW SXMHSRZUyWGRQRUPDOQHMSUDF\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 ]ZL NV]HQLH

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

(

)

T

T

T

T

3

2

11

10

,

.

 'LRG\ =HQHUD '



 L '



 ]DSHZQLDM SU]HVXQL FLH

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



L '



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

]ZURWQH W\SXUHJHQHUDF\MQHJR ZSU]HU]XWQLNRZHMF] FLNRPSDUDWRUD

SDUD7



7



]ZL NV]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 QDSL ü 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 ] ]HZQ WU]Q\P GRGDWQLP VSU] *HQLHP

]ZURWQ\PW\SXQDSL FLRZRV]HUHJRZHJR U\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]]QDMGXMHVL ZVWDQLH

QDV\FHQLD OXE RJUDQLF]DQLD  XWU]\PXMF SUDZLH VWDáH QDSL FLH

Z\MFLRZHRGSRZLHGQLR

U

i

U

O

O

max

min

'ODXNáDGX]U\VDSU]\

EUDNXVSU] *HQLD]ZURWQHJR RGáF]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\PD VFKHPDW

E FKDUDNWHU\VW\NDZ]PDFQLDF]DRSHUDF\MQHJRF FKDUDNWHU\VW\NLXNáDGX

G FKDUDNWHU\VW\NDW\SXKLVWHUH]\

:]PRFQLHQLH XNáDGX ] ]DPNQL W S WO  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\NLVWDQLHVL EDUG]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 QDSL FLHP

( )

( )

u t

u t

1

2

OXE

 1DSL FLH 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]HMFLRZDNRPSDUDWRUDRGZUDFDMFHJRF FKDUDNWHU\VW\NDSU]HMFLRZD

NRPSDUDWRUD QLHRGZUDFDMFHJR G XNáDG ]H VWDELOL]DFM SR]LRPyZ

SU]HMFLRZ\FK

D .RPSDUDWRURGZUDFDMF\

8NáDG ] U\VD MHVW VWHURZDQ\ QDSL FLHP

( )

X W





SU]\

X

FRQVW





=

=

8

:XNáDG]LHW\PQDSL FLDSURJRZHZ\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\N SU]HMFLRZWHJRXNáDGX
E .RPSDUDWRUQLHRGZUDFDMF\

8NáDG

]

U\VD

MHVW

VWHURZDQ\

QDSL FLHP

( )

u t

u

U

2

1

1

SU]\

FRQVW

=

=

1DSL FLDSURJRZHSU]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]JO GQLü 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]\ZDUWRüQDSL FLDZHMFLRZHJRPLHFLVL Z

]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\QDSL FLD QSSRGZyMQ\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\D VFKHPDWXNáDGXE FKDUDNWHU\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  QDSL FLH

8

2PD[

V 3U]\

ZDUWRFLDFK V\JQDáX

u

I

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

Z\VW SXMHQDSL FLH

U

O min

.


Wyszukiwarka

Podobne podstrony:
03 Nieliniowe uklady operacyjne
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