Красовский Н.Н. Теория управления движением

C

(

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c~

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~.

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,':':;

',,-:~

"

280

nOCTAHOBHA

H

PEIIIEHHE

I1POBJIEMbI

HABJIIO,IJ;EHHH

[rJI.

8

MHa'Ie

rOBOpH,

BLIIIOJIHHH

onmU.Jrta.ltbl-ly70

paapemaIOm;yIO

onepan;HIO

aap;a'IH

32.1

Ha

U8eMbl-lOM

CHrHaJIe

y

(,;)

= G

(,;)

X

(,;),

MLI

P;OJDK-

HLI

OOJI3aTeJIbHO

rrOJIy'IaTb

mO"{l-lO

Xi

(t).

TaKIIM

oopaaOM,

OrrTIIMaJIL-

HaH

paapemaIOm;aH

orrepa:D;IIH

aap;a'IlI

32.1

HBJIHeTCH

op;HOBpeMeHHO

paapemaIOm;eH

orrepa:U;lIeH

II

P;JIH

aap;a'IlI

31.1. B

caMOM

p;eJIe,

rrpep;-

rrOJIOmIlM,

'ITO

3'1'0

He

TaK.

ITyCTb,

CJIep;OBaTeJIbHO,

HeKOTopaH

orrepa-

n;IIH

<p~

[t,

z (,;)]

paapemaeT

aap;a'Iy

32.1, HO

He

yp;Oi3JIeTBOpHeT

ypaB-

HeHIIIO

(33.2).

Op;HaKo

eCJIII

orrepan;IIH

<p~

He

yp;OBJIeTBOpHeT

3TOMy

ypaBHeHIIIO,

TO

P;JIH

Hee

MomHO

rrop;oopaTb

BeKTOp

x

fJ

(t),

TaK

'ITO

<P~

[t,

{yfJ

(1:')

IXf

(t)}]

=

xf

(t) +

~,

(33.3)

rp;e

~

=1=

O.

3p;ecL

yfJ

(,;) -

IIp;eaJIbHLIH

ClIrHaJI,

omee"{alOUfuu

BLI-

opaHHoMY

cPaaoBoMy

BeKTopy

x

fJ

(t).

ClIrHaJI

y

(,;)

aaBIICIIT

0'1'

BeKTopa

X

(t)

JIIIHeHHO

(CM.

CTp.

266).

Ho

paa

3'1'0

TaK,

TO

cPaaOBoMy

COCTOHHIIIO

x

A

(t)

~

"Axa

(t)

oyp;eT

OTBe'IaTb

ClIrHaJI

yA

(-r)

=

"AyfJ

(t),

KaKOBa

OLI

HII

OLIJIa

CKaJIHp.HaH

rrOCTOHHHaH

"A.

CJIep;OBaTeJILHO,

lIa

(33.3),

y'IlI-

TMBaH

em;e

JIIIHeHHOCTb

onepa:U;1I11

<pf,

BLIBOP;IITCH

paBeHCTBO

<Pf

[t,

{yA

(,;) I

x~

(t)}]

=

x~

(t) +

"A~.

(33.4)

1IlICJIO

"A

MomHO

BLIollpaTb

rrpOllaBOJIbHO,

HO

Torp;a

(33.4)

OaHa'Ia-

eT,

'ITO

oIiIIIOKa

Wi

=

~"A

rrpll

BLI'IIICJIeHIIII

BeJIII'IIIHLI

Xi

(t)

orrepa:u;neH

III

<rf

[t, y (,;)]

Ha

IIp;eaJIbHOM

ClIrHaJIe

y

(,;)

MOiHeT

OKaaaTbCH

CKOJIb

yrop;Ho

OOJIbmoH.

CJIep;OBaTeJIbHO,

B

paCCMaTpllBaeMOM

CJIy'Iae

omllo-

".

ira

Wi (33.1)

He

OrpaHII'IeHa,

p;ame

eCJIII

L\

(,;) =

O.

::ha

omllOKa

no-

p;aBHO

He

MomeT

OLITb

OrpaHII'IeHHOH

B OOJIee

mllpOKOM

CJIyqae

JIIOOOH

nOMeXII

L\

(,;), CTeCHeHHOH

HepaBeHCTBOM

(32.7).

IlTaK,

MLI

rrpllmJIII

K

npoTIIBOpe'IIIIO

C

YCJIOBlIeM

(32.12),

cop;epmam;IIMCH

B

aap;a'Ie

32.1.

ITOJIy'IeHHOe

rrpOTIIBOpe'IlIe

p;OKaaMBaeT,

'ITO

onepan;IIH

<p~

[t, z (,;)],

He

yP;OBJIeTBOpHlOm;aH

ypaBHeHIIIO

(33.2),

He

MomeT

paapemaTb

aTy

aap;a'IY·

ITOJIeaHO

aaMeTIITb,

'ITO

p;OKaaaHHoe

yTBepmp;eHlIe

aHaJIOrll'IHO

TaK

HaaLIBaeMLIM

YCJIOBIIHM

l-lecMeUfel-ll-locmu, BBOP;IIMLIM B

BepOHTHOCTHOH

TeoplIlI

HaOJIIOp;eHIIH

CJIy'IaHHLIX

ClIrHaJIOB.

8TlI

YCJIOBIIH

CTaBHTCH

OOLI'IHO

Torp;a,

Korp;a

HeT

anpllopHLIX

CBep;eHIIH

0

BoaMomHOM

ypOB-

He

rrOJIeaHOrO

(y

Hac

-

IIp;eaJIbHoro)

ClIrHaJIa.

qllTaTeJIb

aaMeTIIJI,

KOHe'IHO,

'ITO

IIMeHHO

TaKOH

CJIyqaH

paCCMOTpeH

ap;eCb,

1I00

rrpep;LI-

p;ym;ee

p;OKaaaTeJIbCTBO

KaK

paa

II

OCHOBaHO

Ha

rrpep;rrOJIOmeHIIII,

'ITO

IIp;eaJIbHLIH

ClIrHaJI

y

('r)

MomeT

OTBe'IaTb

rrpOHaBOJIbHO

OOJIbmllM

aHa-

'IeHHHM

IWOpp;IIHaTM

Xi

(t).

TaK

KaK,

op;HaKO,

mlHeHHLIe

ypaBHeHIIH

p;BHmeHHH

cnpaBep;JIHBM

OOLI'IHO

JIHmb

rrpH

OrpaHII'IeHHLIX

aHa'IeHHHX

KOOpp;lIHaT,

Ham

BLIBOP;

lIMee'I'

HecKoJibKO

yCJIOBHLIH

CMLICJI.

TaKlIM

oopaaOM,

pemeHlIe

aap;a'IH

32.1

Hap;JIemHT

HCKaTb

TOJIbKO

cpep;H

orrepan;IIH,

KOTopLle

yp;OBJIeTBOpHIOT

yCJIOBIIIO

(33.2).

ITYCTb

I

§ 33]

PEIIIEHHE

3A,IJ;AQII OB

OrrTIIMAJIbHOM

HABJIIO,IJ;EHHH

281

t

cPHKClIpOBaHO.

PaCCMOTpHM

KaKyIO-HIIOY,L(b

TaKyIO

orrepa:U;IIIO

<Pi

[t, z (,;)] =

<P

[z ('t)]'

ITo

yCJIOBHHM

aap;a~IlI

paCCMaTpllBaeMaH

JIII-

I

HeHHaH

orrepan;lIH

<P

orrpep;eJIeHa

P;JIH JIlOOOH cPYHKD;lIl1 z

(,;)

(t -

11'

<

,;

< t), HBJIHIOm;eHCH

3JIeMeHTOM

BLIopaHHoro

cPYHK:U;HOHaJIbHOrO

rrpOCTpaHCTBa

9iJ[t-4J.,t] C

aap;aHHOH

B

HeM

HOpMOH

p [z (,;)],

KOTopaR

I

KaK

paa

II

cPHrYPHpyeT

B

HepaBeHCTBe

p

[L\

(,;)J

< 6 (32.7),

OrpaHH'III-

BaIOm;eM

nOMexy

L\

(,;).

HarroMHHM

Terrepb,

'ITO

B CBOIO

O'Iepep;b

orre-

pa:U;HR

<P

o:u;eHHBaeTCH

CBoeH

HOpMOH

p*

[<pJ

(CM.

§ 58). ,lJ.JIR

Hac

B

p;aJIbHeHmeM

.

Ba»~HO

CJIep;yIOm;ee

COOTHomeHlIe,

KOTopoe

BLITeKaeT

HerrOCpep;CTBeHHO

Ha

orrpep;eJIeHHR

rrOHRTHR

HOpMLI

p*

[<pJ:

.

sup'

<p[L\

(,;)11

=

6p*

[<pJ

rrpll

p

[L\

J <

6.

(33.5)

Ll

IToKameM,

nOJIbaYHCb

3TIIM,

'ITO

BepXHHH

rpaHb

MOp;yJIH

IWi I

omllOKH

Wi (33.1)

rrpH

BLI'IHCJIeHlIlI

Xi

(t)

orrepa:U;lIeH

<p?

=

<p

o:u;e-

HHBaeTCR

HOpMOH

p*

[<p).

B

caMOM

p;eJIe,

y'IIITMBaH

(33.2)

II.

JIIIHeH-

HOCTb

<p,

nOJIyqHM:

<p

[{z

(,;)1

Xi

(t))]

=<p

[{y

('t)1

Xi

(t))]

+<p

[L\

(,;)]

=Xi

(t)

+<p[L\ (,;)],

T.

e.

HMeeM

rrpOCTO

Wi =

<p

[L\

(,;»).

(33.6)

ITo

yCJIOBHRM

aap;aqll

rrOMexa

L\

(1:')

KaK

paa

rrpooeraeT

Bce

BoaMomHhIe

cPyHK:U;HH

L\

(,;),

CTeCHeHHhIeYCJIOBlIeM

p

[L\

(1:')J

<

6.

Ho

Torp;a

lIa

(33.5) H (33.6) H CJIep;yeT

HymHaH

HaMo:u;eHKa

sup

I

(Vi

I

6(>'

[<P].

(33.7)

Ll

TaKIIM

oopaaoM,

MhI

rrpHmJIII

K

rrepBoMy

BamHoMy

BhIBOP;Y.

IIycmb

lte1'>omopaJl onepaq,UJl

<p

[y

(,;) J

8aem

l-la

U8ea.ltblW.J,£

cuma-

.lte

{y

(,;) IXi (t)}

6e30uw6o"{ltoe

3l-la"{eltUe

Xi

(t).

Toe8a

mO"{l-lM

eepxTiJlJl

epal-lb Mo8y.ltJl

IWi ,

OUtu61'>U

Wi

npu

ebl"{uC.lt€ltUU eMu"{UTibl

Xi

(t)

amou

~e

onepaq,ueu

<p

[z

(1:')],

l-lO

y~e

eblnO.ltl-lJleMOU l-la pea.ltb7lOM cuel-la.lte

z

(,;),

c08ep~aUfeM

nOMexy

L\

(,;) (32.7), Onpe8MJlemCJl paeeTiCmeOM

(33.7). C.lte80eamMbl-lO,

ama

eepXl-lJlJl epal-lb

eMU'IUllbl

I

Wi

I

pael-la

npo-

u3ee8el-lUlO l-l0PMbl

p*

[<pJ

onepaq,uu

<p

[z (,;)J

tta

Ma1'>CUJr£a.ltbl-lO

e03:-

MO~l-lYlO

ltOpMy

P

[L\

(,;)J

noepeUtl-locmu

L\

(1:').

MTaK,

;a;JIR

pemeHHH

aap;aq:u:

32.1

Hap;JIemHT

HaHTII

orrepan;IIIO

<pUt,

y

(,;)1,

paapemaIOm;yIO

aap;a'Iy

31.1

II

HMeIOm;yIO

rrpll

Kamp;OM

pac-

CMaTpHBaeMOM

lma'IeHIIII

t

HallMeHbmyIO

BoaMomHyIO

HOpMy

p*

rep?].

qTOOLI

rrpOHJIJIlOCTpHpOBaTb

3TOT

BLIB

op;

,

BepHeMCH

CHOBa

K

aa-

p;a'Ie,

paccMoTpeHHoH

B KOHn;e § 32,

rp;e

BLIqllCJIHJIaCb

CKOpOCTb

~(t)

CBO-

OOP;HOH

TO'IKH

m,

p;BlImym;eHcR

rro

rrpHMoH.

OCTaHoBHMcH

em;e

paa

Ha

oocymp;eHHII

CJIyqaH,

Korp;a

rrOMexa

L\

(1:'),

cop;epmam;aRCH

B

pe-

aJIbHOM

CHrHaJIe

z

(,;)

=

~

(,;)

+

L\

(,;),

CTeCHeHa

HHTerpaJIbHhIM

'~'.-;'

c'

[rn.

R

rrOC'l'AHOBKA

II

PEillEHHE

IIPOBJIEMbI

HABJIIOAEHHII

282

HepaBeHCTBOM

(32,18), T.

e.

Korp;a

t

~

ILl(l')ld't-<o.

t-I;).

B:bIIIIe

yme

6bIJIO OTMelIeHO,

BnpolIeM,

6ea

TIU;aTeJIbHOrO

p;oKaaaTeJIb-

. 0

CTBa,

lITO

onTHMaJIbHaH

paapemarom;aH

onepaJJ;lIH

CP2

[t,

Z ('r)],

KO-

TopaH

BbIlIlICJIHeT

CKOpOCTb X

2

(t) =

~

(t),

HMeeT

BHP;

t

CP2[t,Z('t)]=-

;2

~

z('t)sgn('t-t+

~)d't=

i-iJ.

=

~

;2

~

i

£('t)

sgn

(1'

- t +

~)

d't-

i-I;).

t

_

;2

~

Ll

(1')

sgn

(1'

- t +

1)

d't =

xd

t

)

+

W2,

(33.8)

t-&

npnqeM

k

413

(33.9)

sup

I

:u21

=

{}2

•

~II

IIOCMOTplIM,

KaK

:no

corJIacyeTcH

Tenepb

C

HamHMH

06m;lIMlI

BbI-

Bop;aMH. )J;JIH

:nora

HaM

nplIp;eTCH

nOBTopHTb

CHOBa C

He60JIbmlIMH

YTOqHeHlIHMH

paCCYiKp;eHlIH

lIa

§ 32.

IIpemp;e

Bcero,

aaMeTIIM,

lITO

t"

onepaJJ;HH

cP~

(33.8) YP;OBJIeTBOpHeT YCJIOBlIro (32.22), a CJIep;OBaTeJIbHO,

}.

1I

ypaBHeHlIro

(33.2),

'1'.

e.

:na

onepaJJ;IIH,

p;eHCTBlITeJIbHO,

paapemaeT

aap;alIy

31.1

0

BbIlIHCJIeHlIH

CKOpOCTII

~

(t)

no

IIp;eaJIbHOMY

ClIrHaJIY

r

y

('r)

= £

(1'),

(t -

'0'

-<

l'

-<

t).· )J;aJIee, 'P;JIH

Toro

lIT06bI

orpaHlIlIe-

HHe

(32.18)

lIMeJIO

CMbICJI

HepaBeHCTBa,

CTeCHHIOm;ero

HOpMy

P

[Ll

(1')],

MbI

p;OJImHbI

CllIITaTh

nOMexy

Ll

(1')

:3JIeMeHTOM <PYHKJJ;lIoHaJIbHOrO

npo-

cTpaHcTBa

2

-&,

I].

Ho

06m;lIH

BlIP;

JIlIHeHHOH

onepaIJ;1I1I

cP

[z

(1')]

U

Hap; :3JIeMeHTaMH Z

(1')

lIa

npOCTpaHCTBa

2[i-&,

i]

aap;aeTCH

KaK

paa

<pop-

MyJIOH

(CM. § 59).

t

(33.10)

CP[z('t)] =

~.

Z('t) V

('t)d't,

i-iJ.

rp;e

v

(1')

_

orpaHlIlIeHHaH

HHTerplIpyeMaH

<PYHKIJ;lIH. 8'1'0

cOBepmeH-

HO

corJIacyeTcH

C

aanlIcbro

onepaIJ;lIlI

cP~

[t, Z

(1')],

<PlIrYPlIpYIOm;eH

B

§ 32,

rp;e,

CJIep;OBaTeJIbHO,

nplI

Kamp;OM

<pIIKclIpoBaHHoM

aHalIeHlIlI

t

. 4 ( {})

¢YHKIJ;IIH

V

(1')

Ha

(33.10) eCTb v (t,

1')

= -

{}2

sgn

l'

- t +

2"

•

BOJIee

Toro,

<PYHKIJ;lIH v

(1')

B (33.10) HBJIHeTCH :3JIeMeHTOM

npOCT-

paHcTBa

J!t

{v

(1'),

t

-,-

'0'

-<

l'

-<

t}

IIHTerplIpyeMbIx

orpaHHlIe~HbIx

0:

(:'. : :/

~~

....

§ 33]

PEillEHHE

3AAA'I1I

OB

OIITlIMAJIbHOM

HABJIIOAEHlIlI

283

<PYHKIJ;IIH V

(1'),

B KOTOpOM

HopMa

p*

[v] =

p*

[ep]

onpep;eJIeHa

paBeH-

CTBOM

p*

[ep]

=

sup

Iv

(1')1.

CJIep;OBaTeJIbHO,

HopMa

OnepaIJ;IIH

't'

ep~

[t, Z ('r)] (33.8) eCTb

p*

[cp~]

=

4/'0'2,

a

:3'1'0

H

oaHalIaeT

KaK

pa3,

lITO

06m;ee

COOTHomeHlIe

(33.7) CBOP;IITCH B

paCCMaTplIBaeMOM

CJIylIae

K

paBeHcTBy

sup

I

W2

I=

401'0'2,

KOTopoe

II

6bIJIO BbIBep;eHo B § 32

lIS

:3JIeMeHTapHbIX

coo6pameHHH.

.

II

pOp;OJImIIM

Tenepb

IISYlIeHlIe

CBOHCTB OnepaIJ;1I1I

cP~

[t,

Y ('r)].

IITaK,

nYCTb

BbI6paHbI

II

aa<pIIKclIpoBaHbI

HeKOTopbIe

SHalIeHIIH

i

II

t.

ClIrHaJIbI

y

(1'),

Z

(1')

II

nOMexy

Ll

(1')

(t -

'0'

-<

T

-<

t)

MbI

yme

YC-

JIOBIIJIIICb

paCCMaTplIBaTb

KaK

:3JIeMeHTbI y

(1')

IIJIII

«TOlIKII»

HeKO-

Toporo

npOCTpaHCTBa

fid[i-e.,

i]

{y

(T)},

rp;e

MeTplIKa

onpep;eJIeHa

HOpMOH

p.

PaccMoTplIM

MHomeCTBO

ffJ

{y

(1')}

Tex

<PYHKIJ;IIH y

(T)

lIS

fid,

KOTopbIe

MoryT

6bITb

peaJIlIaall;IIHMII

ClIrHaJIa

{y

(1')IXi

(t) = 1},

Hecym;ero

Xi

(t)

= 1. '

II

pOBeplIM

npemp;e

Bcero

CJIep;yrom;ee

yTBepmp;eUlIe.

JI e MM a 33.1. Onepaq,uJl,

cP

[y

(T)]

moa8a

u

mO.lLblW

moa8a paa-

peutaem

aa8a'l{,y

31.1,

1'i,oa8a

ama

onepaq,uJl,

y8oB/temeopJtem

paeeucmey

cP

[{y

(1')I

Xi (t) = 1}] = 1 (33.11)

ua

,/U060M

cuma.tte

{y

(T)!

Xi

(t)

= 1}.

Heo6xo8uMocmb yCJIOBIIH, (33.11)

OlIeBIIp;Ha,

TaK

I{aK

CJIep;yeT

cpa-

ay

lIa

onpep;eJIeHIIH

paapemarom;eH

OnepaIJ;lIlI

cP

II

lIS

onpep;eJIem'IH

ClIr-

HaJIa

y

(T)

-:-"'{y

(T)!Xi

(t) = 1}.

)J;OKameM

8ocmamO'l{,UOCmb

yCJIO-

BIIH

(33.11).

IITaK,

nycTb

HeKOTopaH

onepaIJ;IIH

cpo

yp;OBJIeTBopj{eT

yCJIOBIIIO (33.11).

HaM

HymHo

nOKaaaTh,

lITO :3Ta

onepaIJ;IIH

yp;OBJIeT-

BopHeT

yCJIOBlIIO

ep

[{y(1')/Xi (t)}] =

Xi

(t), (33.12)

KaKOBO

6bI

HII

6bIJIO

aHalIeHHe

Xi

(t).

IIOKameM

:3'1'0.

PaCCMOTplIM

Ka-

KOH-HII6YP;b

ClIrHaJI

{y

(1')

I

Xi

(t) =

'r]},

oTBelIarom;lIH

HeKoTopoMy

<paaOBoMy

BeKTopy

X (t).

II

yCTh

CHalIaJIa

'r]

=f=

O.

Torp;a

MomHO

no-

CTPOIITb

BeKTOp

x1i

(t) = X (t)/'r].

BbIUIe

MbI

yme

06pam;aJIlI

BHIIMaHlIe

Ha

TO, {ITO <paaOBbIH

BeKTOp

X

(t)

II

oTBelIarom;lIH

eMY

clIriIaJI

y

(1')

CBHaaHbI

p;pyr

C

p;pyroM

JIlIHeHHO

(CM.

CTp.

266). 110:3TOMy

<paaoBoMy

BeKTopy

x

1i

(t)

6yp;eT

OTBelIaTb

ClIrHaJI

ya

(T)

= Y

(T)/'r]

,

Hecym;IIH

xr

(t) =

1,

II,

CJIep;OBaTeJIbHO, COrJIaCHO (33.11)

6yp;eT

BbInOJIHHTbCH

'paBeHcTBo

cP

[y1i

(T)]

= 1.

OTcrop;a,

BBlIP;y

JIIIHeHHOCTlI OnepaIJ;1I1I

cP,

nOJIylIaeM

HymHoe

paBeHCTBO

,CP

[{y

(T)lxi

(t)

=

'r]}]

=

cP

['r]y1i

(T)]

=

'r]CP

[y1i

(T)]

=

'r].

(33.13)

TaKIIM

06paaoM,

MbI

npOBeplIJIlI,

lITO B

CJIYlIae

Xi

(t)

=

'r]

=f=

0

P;JIH

OnepaIJ;HII

cP

[y]

YCJIOBlIe (33.12), p;eHCTBIITeJIbHO, yP;OBJIeTBOpHeT-

CH.

Ho)no yCJIOBHe

yp;OaJIeTBOpHeTcH

II

npI{

Xi

(t)

=

'YJ

=

0,

lJT06:Ql

r-"

c:

\,~L:'

~,:",

[rJl.

8

IlOCTAHOBHA

1'1

PEmEHIim IlPOBJIEMbI

HABJIIO)J;EH1'1H

284

y6e.n;l'lTbCH B

aTOM,

paCCMOTp1'1M

HapJl.n;y C BeHTopOM x (t), y ROToporo

Xi (t) =

0,

RaRoR-H1'16y.n;b

BeRTOp

X.

(t) =

xa.

(t) + X (t),

r.n;e

xf

(t)

=

1.

Tor.n;a, 060SHa'IaJl

CRMBOJIaM1'1

Y.

(1:)

R

OTBe'IaIOm;Re

4JasOBhlM

BeRTopaM x.

(t)

R

xrx.

(t)

Mhl MomeM sarrRcaTb paBeHCTBa

ep

[ya.

(t)] = 1

=ep

[{y

(1:)

I

Xi

(t) =

O}]

+

ep

[ya.

('r)] =

1,

oTRy.n;a

ep

[{y

(1:)

I

Xi

(t)

=

O}]

==

O.

IhaR,

JIeMMa

33.1

.n;OHasaHa.

ya.

(1:)

CRrHaJIhl,

COOTBeTCTBeHHO,

R

ep

[Y.

(1:)]

.

R

CJIe.n;yeT,

'ITO'

CJIe.n;oBaTeJIbHO,

.n;JIJI

pemeHRJI sa.n;a'IR

32.1

HaM

Ha.n;JIemRT

HaRTR

orrepan;RIO

epO

[y

(1:)],

ROTOpaJl

y.n;OBJIeTBOpHeT

YCJIOBHIO

(33.11),

T.

e.

rrp1'1HRMaeT

Ha MHomeCTBe [jJ

{y}

SHa'IeHRe, paBHoe

e.n;RHRn;e,

H HMeeT rrpR

aTOM

HaRMeHbmyIO B03MomHylo

HOPMY·

HeTpy.n;HO

rrpOBep1'1Tb,

o.n;HaRO,

'ITO MHomeCTBO [jJ

{y}

JlBJIJleTCJI

MOC1f,UM

(CM.

§

61).

HarroMHRM, 'ITO MHomeCTBO [jJ

{y}

aJIeMeHTOB

y

(1:)

(t-1't,~:;;;><t)

HaShlBaeTCJI

rrJIOCHRM,

Ror.n;a «rrpJlMaJl»

yA

(1:)

= Ay(l)

(1:)

+(1 -

A)

y(2)

(1:)

(-00

< A< 00),

(33.14)

rrpoBeAeHHaH

'Iepes

JII06hle

)J;Be

«TO'IRID)

y(l)

(1:)

R

y(2)

(1:)

HS

[jJ, JIemRT

BceM1'1

CBORMR

TO'IRaMR

yA

(1:)

B [jJ. -

B HameM CJIy'Iae rrYCTb y(l)

(1:)

R

y(2)

(1:)

CyTb

.n;Ba

C1'1rHaJIa,

co.n;ep-

t",

mam;1'1XCJI

B [jJ

{y

(1:)},

T.

e.

y(l)

(1:)

R

y(2)

(1:)

CYTb

.n;Ba

1'1.n;eaJIbHhlX

CRr-

'"

Hami,

Hecym;RX Xi (t) =

1.

IIpR

aTOM

rryCTb CHrnaJI

yll)

(1:)

OTBe'IaeT

til

4JasoBoMY

BeHTopy

X(l)

(t), a CRrHaJI

y(2)

(1:)

- 4JasoBoMy

BeRTopy

A

X(2)

(t).

lis

JIRHeRHOcnI aToR

CBJl3R

CJIe.n;yeT,

'ITO BeHTopy x (t) =

!l

=

AX(l)

(t) +

(1

-

A)

X(2)

(t) 6YAeT OTBe'IaTb CHrnaJi

yA

(1:)

(33.14).

I'

,D;JIJI

rrocTpoeHHoro

HaMH 4JasoBoro BeHTopa x

A

(t), O'IeBHAHO, paBeH-

t:'

CTBO

xt

(t)

= 1 BhlrrOJIHJleTCJI, rroaToMy SaRJIIO'IaeM, 'ITO OTBe'IaIOm;HR

eMy

CHrHaJi

yA

(1:)

TaKme

JlBJIJleTCJI

HAeaJIbHhlM CHrHaJIOM, Hecym;HM

X~

(t) =

1.

Ho

aTO

R OSHa'IaeT, 'ITO

4JYHRn;HJI

yA

(1:)

COAepiKHTCJI

B Bhl-

.n;eJIeHHOH

COBoRyrrHocTH

fP

{y

(1:)},

T.

e.,

AeHCTBHTeJIbHO,

[jJ

{y

(1:)}

eCTb

rrJIOCROe

MHOiKeCTBO.

Ho

Terrepb Mhl BHAHM, 'ITO

CTOJIRHyJIHCb

C

TOR

iKe caMoR sa.n;a'IeR,

C

ROTOpOR

Mhl

yiKe

HMeJIH

.n;eJIo

B §

16

(CM.

cTp.100),

Ror.n;a pemaJIH

rrpo-

6JIeMy

MOMeHTOB.

,D;JIJI

rrepexo.n;a

0'1'

yrroMJlHyToH sa.n;a'IH B §

16

R 3a-

Aa'Ie, paccMaTpHBaeMoR SAeCb,

CJIe.n;yeT

JIHmb saMeHHTb

MHOiKeCTBO

[jJ

{h

(1:)},

(16.2),

(16.3)

Ha

MHomeCTBO

[jJ

{y

(1:)},

COCTOHm;ee

H3

CHr-

HaJIOB

{y

(1:)

IXi (t) =

1},

Hecym;HX Xi (t) =

1.

CJIe.n;oBaTeJIbHO,

H

pe-

meHHe

HameH

SaAa'IH Mhl

MOiKeM

rrOJIY'lHTb

cpasy,

rrOBTOpJlH pemeHHe

aHaJIOrH'lHOH SaAa'IH

HS

§

16,

rAe

CJIeAyeT JIHmb 3aMeHHTb

4JyHKn;HH

h(1:)

Ha

CRrnaJIbI y

(1:).

B 'IaCTHOCTH,

POJIb

MUHUMMbHOU

4JyHKn;RR

hO

(1:)

6y.n;eT

Terrepb

RrpaTb

MUHUMaJLbHbLU

CRrHaJI

yO

('r),

ROTOphlR

B cooTBeTcTBHH C

(16.7)

orrpe.n;eJIJleTCJI

R3

yCJIOBRJI

pO

= p

[~l

(t)]

=

min

p

[{~

(t)

Ix

1

(t) = 1}l. (3q.15)

l!

§ 33]

PEmEHRE 3A)J;A'IH

OB

OIlTHMAJIbHOM

HABJIlO)J;EH1'11'1

285

TaRRM rrYTeM IlpRXOARM R

CJIe.n;YIOm;eMY

OCHOBHOMy

rrpaBRJIy

.n;JIJI

pemeHRJI 3aAa'IR

32.1.

II

p a B R

JI

0 M R H R M a

1\

6 a

33.1.

/(JLJL

moeo

'lJ.m06bL

pemumb

8aaa'IJ.y

32.1,

cJLeayem

npu

1f,aJICaOM

8Ha'f,ertUU

i U t

rtaumu

cpeau cue-

rtMOe

{y

(1:)

I

Xi

(t)

= 1}

MUHUMMbrtbLU

cuenM

y~

(1:).

3aaa'IJ.a

UMeem

pemeHue

npu

ecex

paCCMampUeaeMbLX

8rta'f,ertU.R.x

i U t moeaa U

mOJLblW

moeaa,

'fi,oaaa

npu

ecex i U t

ebmOJLHJLemCJL

J-lepaeertcmeo

p~

= p

[y~]

>

O.

IIpu

amOM

U

mOJLb'fi,O

npu

amOM

yCJLoeUU

pa8pemaemCJL

ma'fi,JlCe

8aaa-

'lJ.a

31.1.

OnmUMMbJ-laJL

pa8pema1OUfaJL

onepa/fUJL

cp?

aJLJL

8aaa'IJ.U

32.1

UMeem

rtOPMY

p*

[ep~

[t,

zll

=

lIp~

U

ebLaeJLJI,.emCJL

cpeou

opyaux

JLUHeUHbLX

~

°( )

onepa/fuu

ep

ceoucmeOM

Ma'fi,CUMyMa:

rta

MUHUMaJLbHOM

CUUlaJLe

Yt

1:

ama

onepa/fUJL

aaem

Hau60JLbmUU

e08~tOJlCHQlU

pe8YJLbmam

no

cpaeJ-le-

HU10

co

eCeMU

opyauMu

onepa/fUJLMU

ep

cmou

JlCe

J-lOpMOU

p*

[ep]

=

lIp~,

m.

e.

.]

1

epnt,

y~

('t')] =

max

ep

[y~

(t)]

npu

p

[ep

=

-0

•

(33.16)

cp

Pt

3aMeTHM Terrepb, 'ITO B CJIy'Iae,

RorAa

B ypaBHeHHJlX

(32.5)

R

(32.6)

MaTpRIJ;hl A H G He 3aBRCJlT

0'1'

BpeMeHH, OrrTRMaJIbHaJl

pa3pe-

maIOm;aJl orrepan;HJI

ORaShlBaeTCH

TaKiKe 4JaRTH'IeCRR He

SaBHCJlm;eR

JlB-

HO

0'1'

BpeMeHR.

,D;JIJI

'1'01'0

'IT06hl

aTO

6hlJIO

JlCHO

H3

samlCR

orrepan;RR

ep~,

B TaRRX ,CJIy'IaHX HaAJIeiRRT IlpHMeHHTb rrpe06pasoBaHRe BpeMe-

HH

1:,

sarrRChlBaJl CHmaJIhl y

(1:)

H Z

(1:)

(t

-

1't

<

1:

<

t)

B 4JopMe

y (t

+~)

H z (t

+~)

(-1't

<'

<

0),

rAe,

CJIeAOBaTeJIbHO,

1:

= t

+-

+

~.

TorAa

B cJIY'lae IlOCTOJlHHhlX A H G .orrepan;HIO

ep~

[t,

z

(1:)]

CJIeAyeT RCKaTb B

BHAe

ep~

[z (t + , )

1.

IIpORJIJIIOCTpRpyeM rrpaBHJIO

33.1

Ha

pemeHRH paccMoTpeHHoH B

§

32

SaAa'IR 0 Bhl'lRCJIeHHR

CROpOCTR

TO'IRH m B CJIy'Iae,

RorAarrOMexa

~

(1:)

CTeCHeHa

HepaBeHCTBOM

(32.19).

Bhlme,

B §

32,

YiKe OTMe'IaJIOCb,

'ITO B

aTOM

CJIy'Iae CRrnaJIhl y

(1:),

Z

(1:)

R

rroMexy

~

(1:)

CJIeAyeT C'lR-

TaTb aJIeMeHTaMR rrpOCTpaHCTBa

yg

{'h

(1:),

t -

1't

<

1:

< t} HerrpephlB-

HhlX

4JyHRn;HH

h

(1:)

C

HOpMOR

p[h

(1:)]

=

max

Ih

(1:)

I.

Ho

B rrpOCTpaH-

'r

.

CTBe

yg[t-&.t]

06m;HR

BRA

JIHHeRHOH

orrepan;HH

ep

[h

(1:)]

orrpeAeJIJleTCJI

4JOpMyJIOR

(CM.

~

59)

t

ep[h(t)]

..

~

h'(t)dV(t),

t-~

rAe V

(1:)

eCTb

4JyHKJ:J;RJI

C OrpaHR'IeHHhlM RSMeHeHHeM.

IIpR

aTOM

HOp-

Ma

p*

[<p]

orrepan;HH

ep

orrpeAeJIeHa paBeHCTBOM

p.

[ep]

= val'

[V

(tm~.

.

...l.......-'-.

_.~.

~'_'.'

:

•.•••

~

••••••••

-:-

••

'

.

:.:.l..:..:....~~~:~:-.::-...

'.:.,

'.:.'

.:

:.

'-:.'.::

"'::"'.:.:;~.::

'-~'....

.·.:·:·

:···::.:::··._~:::~·:·::·:.:·::·.:~'.:·.:~i

c

286

rrOCTAHOBHA

M

PEIIlEHME

rrpOBJIEMbI

HABJIIO;LJ;EHMH

ern.

8

IhaK,

sa,n;alJa

CBO,n;MTCH

K

Haxom,n;eHHIO

,n;BYX

<PYHK~IIH

~

(t,

1')

.°

H V

2

(t,

1'),

KOTopbIe

onpe,n;eJIHIOT

OrrTMMaJIbHbIe

paspeIIlaIOm;lIe

orre-

pa~IIM

t

<p~

[t,

Z

(1')]

=

~

Z

('t')

dV~

(t,

't'),

t~

I

<Pg[t,z('t')] =

~

z('t')dVg(t,'t'),

I~

BbIlJIICJIHIOm;Me COOTBeTCTBeHHO

Koop,n;MHaTY

Xl

(t)

= £ (t) M

CKOpOCTb

X

2

(t)

=

~

(t) TOlJKM m C

HaMMeHbIIleH

BOSMomHOH

OIIllI6KOH

SUp

I

~i

I=

= min.

HaH,n;eM

CHalJaJIa

orrepa~IIIO

<pi· CJIe,n;YH

rrpaBIIJIY

33.1,

paCCMOT-

pMM

CMrHaJIbI

{y

(1')

I

Xl

(t)

= 1},

Hecym;lIe

Xl

(t)

= 1. B ,n;aHHOM

CJIy-

lJae

COrJIaCHO

(32.20)

IIMeeM:

{y

(1')\

Xl

(t) = 1} = 1 + X

2

(t)

(1'

- t). (33.17)

Terrepb

Ha,n;o

HaHTM

MIIHMMaJIbHbIH

ClIrHaJI

{yO

(1')

I

Xl

(t)

= 1}.

B COOTBeTCTBMII C

BbI60pOM

HOpMbI

P [h] :3T0 6y,n;eT

ClIrHaJI

(33.17),

,n;JIH

KOToporo

BeJIlIlJlIHa

max

11

+ X

2

(t)

(1'

- t) I (33.18)

"

6y,n;eT

HalIMeHbIIleH

rro

BceM

BOSMomHbIM

SHalJeHIIHM

X

2

(t).

MCKOMblH

MIIHIIMYM

paBeH,

OlJeBlI,n;HO,

e,n;IIHII~e,

T.

e.

B ,n;aHHOM

CJIylJae

p

[yO]

=

= 1.

O,n;HaKo

sa,n;alJa 0

MBHIIMyMe

BeJIlIlJlIHbI

(33.18),

T.

e.

sa,n;alJa

06

orrpe,n;eJIeHIIM

caMoro

ClIrHaJIa

yO

(1'),

peIIlaeTCH

s,n;eCb

He

o,n;HOSHalJHO

(ClIrHaJI

{yO

(1')/

Xi

(t) =

1}

He

e,n;IIHCTBeHeH (CM.

BbIIIle,

CTp.

278»!

MbI

BbI6epeM,

HarrplIMep,

X2

(t)

=

1/1}

II

rrOJIylJlIM

Tor,n;a

TaKOH

MII-

HIIMaJIbHbIH

ClIrHaJI:

.

T:-'-t

{yO('t') IXI (t) =

1}

= 1 +-{}-.

Terrepb

lIS

paBeHCTBa

p

[yO]

=

pO

= 1

II

lIS

p*

[<pO]

= 1/

p

O

SaKJIIOlJaeM,

'ITO

p*

[<p~]

= 1. ,n;JIH

rrOCTpoeHIIH

<P~,

T.

e.

,n;JIH

Haxom,n;e-

HIIH

<PYHK~1I1I

VI

(t,

1'),

OCTaJIOCb

JIMIIlb

BocrrOJIbSOBaTbCH

CBOHCTBOM

MaKclIMyMa

(33.16). A aTO

SHalJIIT,

lJTO

HaM

Ha,n;o

HaHTII

<PYHK~IIIO

, I

~

(t,

1')

C

rrOJIHbIM

MSMeHeHlIeM

var

[VI

(t,

1')

],-&,

paBHbIM

e,n;IIH~e,

J[

TaKyIO,

lJTO

I I

(1

+T:

{}

t)

dV~

(t,

't')

=

max

~

(1

+

T:

{}

t)dV

('r)

~

v

1-&

t-

[W~

var[V('t')]:_~=1,

§ 33]

PEIIlEHME

SA;LJ;AQII OB

OrrTBMAJIbHOMHABJIIO;LJ;EHllll

287

HeTpy,n;HO

c006paSlITb,

'ITO

YI\ailaHHOMY

yCJIOBlIIO

Yl~OBJIeTBOpReT

mOJ/,bl£O

<PYHH~IIH

~

(t, 't) = 0

rrpll

t -

'I:}

-<

't < t

11

V~

(t, t) = 1.

CJIep;OBaTeJIbHO,

oIIepa~IIH

<P~

IIMeeT

BII,n;

t

<P~[t,y('t')]=

~

]I('t')dV~(t,'t)=y(t)=xI(t).

1-&

Terrepb

aHaJIOrlIlJHbIM

06pasoM

MomHO

HaHTH

orrepa~IIIO

<pg.

IIPII

aTOM

MIIHIIMaJIbHbIM

ClIrHaJIOM

{yO

(1')

IX

2

(t) =

1}

OKaSbIBaeTCH

<PYHK~IIH{YO

(1')

IX

2

(t) =

1}

=

,;

- t + 1}/2,

HopMa

KOTOpOH

p[{yO

(,;)

IX

2

(t) =

1}]

=

1}/2.

B

OTJIlIlJlIe

0'1'

IIpe,n;bI,n;ym;ero

CJIyqaH

s,n;eCb

MIIHHMaJIbHbIH

ClIrRaJI

eOUNcmBeNert.

Y

CJIOBlIe

MaKclIMyMa

I t

~

('t'-t+

g)dVg(t,'t)=n~ax

\

('t'-t+

~)dV('t')

t-&

t-~

t 1

rrplI

val'

[V

('t')],_&

=

2{}

orrpe,n;eJIHeT

TaHme

e,n;IIHcTBeHHbIM

o6paSOM

<PYHH~lIIO

0

rrplI

't' = t -

1},

vg

(t, 't') = - +

rrplI

t

-1}

<'t' <t,

<Pdt,

y

(1')]

T [y (t) - Y(t

-;-1})]

T

[xI(t)

-

xI(t

-1})]

=

xd

t

).

I

o

rrplI';

= t,

II,

orr

CJIe,n;oBaTeJIbHO,

epa~IIIO

rrOJIylJaeM

TaKyIO

OrrTIIMaJIbHYIO

paspeIIlaIOIIJ;YIO

~

1 1

= =

TaKIIM

06pasoM,

MbI

BII,n;IIM,

lJTO

rrpOCTeHrrUle

OIIepa~1I1I

CPI

1I

CP2

(32.16),

HaH,n;eHHbIe

HaMM

B § 32, IICXO,n;H

lIS

aJIeMeHTapHblx

coo6-

pameHIIH,

KaK

pas

OKaSbIBaIOTCH

OrrTIIMaJIbHbIMII

paspeIIlaIOm;IIMII

orrepa~IIHMII

,n;JIH

paccMaTplIBaeMoH

sa,n;aTIH B

CJIylJae

OrpaHlIlJeHIIH

Ha

rroMexy

~

(,;),

sa,n;aHHoro

B

<popMe

HepaBeHCTBa

(32.19).

IIpIIBe):t;eM

em;e

O,n;IIH

rrpocToH

rrplIMep.

llpHMep

33.1.

PaccMO'1'pUM

ypaBHelUl:JI

MaJIbIX

KOJIe6aHHH

Ma'1'eMa'1'HqeCKOrO

MaJl'1'HHKa

OKOJIO YC'1'OHQHBOrO

IIOJIOiHeHHJl

paBHOBeCNJI

...

Xl

=

X2,

X2

= -

Xl·

(33.19)

IIpellIIOJIOmHM,

Q'1'O

B03MomHO

H3MepeHHe

yrJIOBOH

CKOpOC'1'H: x (t), KO'1'O-

2

poe

lIae'1'

IIoMexy

d

('r),

C'1'ecHeHHyIO

yCJIOBHeM

I

~

I

Ll

(T:)

I

dT:

<

O.

I-I>

~

....

.•..

.._~~..:..-:.._

...

-,,:,:!

",'::.

-.:.-.:.:...:

••

'~~;:..~..:....:.....:.:

..

~~'"

,':"

.;

..

:....

..::-:'.-

"::

....

::::.

<,:'::::.:.:}~;;~.

r ,

~._,.:.

[rn.

8

IIoc'rAHOBltA

II

PEillEHllE

IIPOBJIEMhI

HABJIIO,ll;EHllff

288

nYCT!.>

Tpe6yeTClI onpeAemlTb

YfOJIXl

(t) OTHJIOlleH.lfll MaJlTHI1I{a

OT

nepTJ1I{aJIR.

Bhl6epeM

iJ

=

2:rt.

TOfAa

II.MeeM

3a):(aQy

32.1 06

OllTII.MaJIbHOM

Ha6mOAeHJllI

CHCTeMbI

(33.19)

no

peaJIbHOMY CHfHaJIY Z ('t) =

X2

('t) +

!'J.

('t).

ComacHo

Ma-

TepRaJIy AaHHofo

napafpa¢a

3Ta 3aAaqa

cnOARTClI

K 3aAaqe 0 BaqllCJIeHHll

Xl

(t)

no

HAeaJIbHOMY

CllfHaJIY

y (t

+~)

=

X2

(t

+~)

= [0, 1]

[Xl

(t

+;)

]

(-

2:n

~

0),

X2

(t +

'::0)

CJIeAOBaTeJILHO,

Tpe6yeTClI

HaRTH

JIHHeRHyIO

onepa~HIO

npHqeM,

o

'PI

[y (t +b)] =

~

Y (t +

~)

V

O

m

d~

=

Xl

(t)

-2"

C HaHMeHbmeR

B03MO)HHOR

HOpMOR

p*

['P2]

=

p"

[vO

(~2]

=

max

IV

O

(~)

\,

. "

fAe

¢YHK~HlI

v

O

(~)

AOJIlliHa lIBJIlITbClI

3JIeMeHTOM

npocTpaHcTBa

.J{[-2"'0]

•

npn:

3TOM

cn:fHaJI y (t +

~)

HaAJIelliHT paccMaTpn:BaTb HaH

3JIeMeHT

npoCTpaHcTBa

2[-2",0]'

B

COOTBeTCTBHH

C

06~eR

npon;eAypoR

HaHAeM

CHaqaJIa cn:fHaJIbI

{y

(t +

~)

I

Xl

(t) = 1},

HeCY~lle

Xl

(t) = 1. nOJIy'lHM:

y (t

+~),

= [0, 1]

II

C~s

~

sin

~

II

[Xl

(t)J=

-Xl (t)

sin

~

+

X2

(t)

cos~.

. -

SIll

~

OOS

~

X2

(t)

".

CJIeAOBaTeJIbHO,

{y

(t +

~)

I

Xl

(t) =

1}

=

X2

(t) cos

~

-

sin~.

Tenepb

onpe-

I"

AeJIHM

MHHHMaJIbHhlH CHfHaJI

{yO

(t

+

~)

I

Xl

(t) = 1}.

)J;JIa

3TOfO

HaAo

HaRTn:

\,

MHHII.MyM

BeJIHqHHhl

ill,

no

BceM B03MO)I{HhlM

= _

sin

~,p

[yO]

=

4.

o

~

I

X2(t)

cos~

-

sin~

I

d~

-2"

X2

U).

OTCIOAa

CJIeAyeT,

'ITO

{yO

(t

+~)

I

Xl

(t) =

1}

=

113

yCJIOBHIl MilI(CHMYMa 3aHJIIOqaeM Tenepb,

qTO

¢YHHlJ

1H1

V

O

(~)

lIMBeT HOpMy P

[V

O

(~)]

= 1/4

II

nplI

3TOM

BhlnOJIHlleTCIl

yCJIOBHe

o 0

_

~

VO

(~)sin

~

d~

=

n;ax

.~

v

(~)

(-

sin~) d~

-2"

npR

v

(~)

~

1/4.

OTcIOAa HaxOAHM V

O

(~)

=

-4

1

sgn

[sin

~].

11TaH,

pellieRHe aaAaqn: 31.1 0 Ra-

6mOAeHlIH: .iHcTeMhl (33.19)

no

HAeaJIbH.oMY

CIlTHaJIY y (t +

~)

=

X2

(t

+~)

HMeeT

BRA

o

1

(.

'Pl[y(t+~)]

=-~

\,

X2(t+~)

sgn

[sin~l

d~=xt{t).

4 .

-2"

STa a{e

onepallHII

onpeAeJIlIeT

BeJTRQImy

Xl

(t)

no

peaJIbHoMy CHfRaJIY Z('t) =

=

X2

('t) +

6.

('t) H B

nOCJIeAHeM

CJIyqae 6YAeT AaBaTb OlllH6Ky

WI>

HOTopall

YAOBJIeTBOpJleT

HepaBeHcTBy I

wli

~

6/4.

(

~.

,':.',,":'

,,-~:-'

§ 34]

PEillEHllE

3A,ll;A

"Ill

0 HABJIIO,ll;EHHH

289

§ 34.

PemeH~e

sa.n;a'lH

0 HaOJIIO.n;eHHIl

PemeHHe

Sap;a'IH

31.1

°

Ha6JIIOp;eHllH

ROOpp;HHaT

Xi

(t)

<paSOBoro

BeRTopa

X (t)

no

Hp;eaJIhHoMy

CllrHaJIY

y

(L)

P;JIH

JIllHeHHOH

CHCTeMLI

(31.5),

(31.6)

6LIJIO nOJIYlJeHO B

npep;LI)l;ym;eM

naparpa<pe

nonyTHO

C

pe-

meHHeM

aap;a'IH

32.1,

TaR

RaR

TaM

BLIHCHHJIOCI.,

'ITO

OnTllMaJIhHaH

paspemaIOm;aH

onepan;HH

<p~

[t,

Z

(L)

1

P;JIH

sap;a'IlI

32.1

HBJIHeTCH OP;HO-

BpeMeHHO

paspemaIOm;eH

onepan;lIeH

<Pi

[t,

Y

(L)l

II

P;JIH

Sa)l;a'IH

31.1.

CJIep;OBaTeJII.HO, eCJIII

MLI

XOTHM

pemllTh

8TY

nOCJIep;HIOIO

sa)l;a'Iy,

HaM

p;OCTaTO'IHO sap;aTI.CH

RaRllM-HH6yP;I.

YCJIOBlleM

OnTHMaJII.HOCTll

(B<popMe

MHHllMyMa

RaROH-HH6YP;b

ROHRpeTHOH

HOPMbI

P*

[<pd)

H

u 0

HaRTll

onepan;llIO

<Pi'

OnTHMaJIhHYIO

pyROBOp;CTByHCh

npaBHJIOM

MU-

HUMaK,Ca

33.1.

PaSJIHqHLIM

HopMaM

p*

6YP;yT

COOTBeTCTBOBaTh

paSJIH'IHLIe

pemeHHH

<Pi

[t,

Y (-t)]' H

BLI60p

Toro

llJIII

HHoro

pemeHllH

<Pi

on-

pe)l;eJIHeTCH, HCXOP;H

H3

II060'IHLIX

co06pameHHH,

HanpllMep,

8TO

MoryT

6LITI.

JIH60

co06pameHHH,

CBHsaHHLIe

C yp;06CTBOM

TeXHll'IeC-

ROH

peaml3an;HH

onepan;1Il1

<Pi'

JIII60

co06pameHHH,

BLITeRaIOm;lle

liS

meJIaHHH

BCTpeTHTI.

HaHMeHI.mHe

BLI'IllCJIHTeJII.HhIe

TPYP;HOCTH

npll

OTLICRaHIIH

<Pi'

IIop;06HLIe

BonpOCI.I

BhIXOP;HT

sa

paMRH

p;aHHOH

RHll-

rll,

H

MLI

Ha

HHX

OCTaHaBJIHBaThCH

He

6yp;eM.

flTaR,

MLI

MomeM

C'Ill-

TaTI.,

'I1'O

06m;eepemeHHe

Sap;a'IH

31.1

HaMil

yme

OIIllcaHO

B §

33.3p;ecI.

em;e,

npaB)l;a,

OCTaeTCH

HeBLIHCHeHHhIM

OP;llH

llHTepeCHLIH

Bonpoc

'III-

CTO

MaTeMaTll'IeCROrO

xapaRTepa:

JII060e

JIll

pemeHHe

<Pi

Sap;a'IH

31.1

MomHO

nOJIy'IHTh

onllcaHHLIM

HaMIl

nyTeM,

llHa'Ie

rOBopH,

BCHRoe

JIH

pemeHHe

sap;a'IlI

31.1

B

CBOIO

O'Iepep;I.

oRameTCH

OnTllMaJIhHhIM

B

CMhICJIe

Sap;a'IH

32.1

IIpH

nop;xOp;Hm;eH

nop;rOHRe

R

HeMy

yCJIOBHH 8TOH

aap;aqH,

T.

e.

npH

nop;XOp;Hm;eM

nop;60pe

OrpaHll'IeHHH

Ha

!'J.

(L).

Op;-

HaRO

TaROH

Bonpoc

MhI

OCTaBHM B

CTopOHe,

OTCLIJIaH

IIHTepeCYIOm;e-

roCH

'IHTaTeJIH

R

CIIen;HaJIhHOH

JIHTepaType,

r)l;e

pas6HpaIOTcH

aHa-

JIOrH'IHhIe

Bonpochl

TeopHH

yilpaBJIeHHH

[89p;].

IIepeHp;eM

TenepI.

R

pemeHHIO

aap;atiH

31.2.

IIpH

aTOM

MLI

orpa-

HHqHMCH

JIllmI.

CJIYlJaeM

BOSMomHhIX

ynpaBJIeHHH

U (t).

Bce

He06-

XOp;HMhIe

HSMeHeHHH

B

sanHCHX,

ROTopLIenoTpe6yIOTcH

IIpll

IIepexop;e

R

0606m;eHHhIM

BOSMomHLIM

ynpaBJIeHHHM

dU

(t),

'IHTaTeJII.

MomeT

BhInOJIHHTI.

caMOCTOHTeJII.HO.

I:lTaR,

HaM

Hap;o

HaHTH

onepan;HH

<Pi

[t,

{y

(L),

U

(L)}J,

BOCCTa-

HaBJIllBaIOm;lle

Roopp;IIHaThI

Xi

(t)

<paSOBoro

BeRTopa

X

(t)

ynpaBJIHe-

Moil

CHCTeMLI

j;

= A (t) X +B (t) U

(34.1

)

no

CHrHaJIy

{y

(L),

U (L)},

TaR,

'ITo6hI

BLIIIOJIHHJIllCh

paBeHCTBa

<Pi

[t,

{y

(L),

U

(L)

I

Xi

(t)} 1 =

Xi

(t). (34.2)

10

H.

H. RpacoBcRlllt

'.",'

,':.".'o'.

:'.: '

,'

..

::,.'. :

;>:;

~

::,.

"-'Li..~~'

.1'·.·

.','

"".:'

; I

I I

II

'.

" .

....

:.':"

·

...

·.• i

C

[rJl.

8

nOCTAHOBHA

H

PEIDEHHE

npOBJIEMbI

HABJIIO)J;EHHH

290

TIoKameM,

'ITO

::lTO

pemeHHe JIerKO HaiiTII,

eCJIII

yme

nOCTpoeHa

paapemarom;aH

onepa~IIH

AJIH

COOTBeTcTByrom;eii

aaAaqll

31.1,

rAe

paccMaTpllBaeTcH TaKoii me CllrHaJI

y

(-r)

= G

(,;)

x

(,;)

II

Ta

me

CII-

cTeMa

(34.1),

HO

npll

yCJIOBIIII u (t) =

O.

IlTaK,

nYCTb

cP~

[t, Y (,;)] -

pemeHlle aaAaqll

31.1.

TaK

KaK

npll

U

(t)

=0 MIil IIMeeM

y

(,;)

-,-

G

(,;)

X [';,

t]

x

(t),

TO

onepa~IIH

{P~

YAOBJIeTBOpHeT

paBeHcTBy

(34.3)

q>~

[t,

G

(,;)

X [';,

t]

x

(t)]

=

Xi

(t),

KaKOB

6Iil HII 6IilJI

BeKTOp

x

(t).

3AeCb

X

[t,

to] -

cPYHAaMeHTaJIb-

HaH

MaTpII~a

CllCTeMIil

(34.1)

(npII

U

(t)

= 0).

Ho

JII060e ABllmeHlle

x

(,;)

CIICTeMIil

(34.1) onllCIilBaeTCH cPopMyJIoii

Homll

(5.6):

(34.4)

'Y"qIlTIilBaH JIIIHeiiHOCTb

onepa~1I11

cP~,

BIilBOAIIM

lIa (34.3)

II

(34.5)

CJIeAylom;ee

COOTHomeHHe:

"

cP~

[t,

Y ('r)] =

Xi

(t)

+

cP~

[t,

G

(,;)

~

H

[-r,

~]

u

(~)

a~

J.

(34.6)

t

OqeBIIAHO, MIil MomeM B BIilpameHIIH

"

cP~

[t,

G

(-r)

~

H [';,

t]

u

(~)

atl

t

npOHaBOJIbHIilM o6paaoM nepe060aHaqaTb nepeMeHHIile ,;

II

~

6ea

Ka-

KOrO-JIH60

HaMeHeHHH

npll

::lTOM

peayJIbTaTe

onepa~HH.

B

caMOM

AeJIe,

peayJIbTaT JIIIHeiiHOii

onepaIJ;1I11

HaA HeKOTopoii

cPYHK~lIeii

OqeBHAHIilM

06paaoM He

,aaBHCIIT

OT

Toro,

KaKoii

6yKBOii MIil 060aHaqaeM apryMeHT

::lToiicPYHK~III1.

Ho

TorAa MIil MomeM

COCTaBnTb

CJIeAyrom;ee

BhIpa-

meHHe:

CPt

[t,

{y

(-r),

u

(-r)}]

=

1;

=

cP~

[t,

Y

(-r)]

-

cP~

[t,

G

(t)

~

H

[~,

-rlu

(-r)

a-r],

(34.7)

t

p-

1:'

' •

'--'-

§ 34]

PEIDEHnE

3AAA'IH 0 HABJIroAEHHH

291

rAe,

CJIeAOBaTeJIbHp,

B nepBoM

CJIaraeMOM

onepa~HH

cP~

BIilnOJIHHeT-

CH

HaA

cPYHK~lIeii

y ('t) C apryMeHToM, 060aHa'leHHIilM 6YKBOii

-r,

a

BO

BTOpOM

CJIaraeMOM

::lTa

onepa~IIH

BIilnOJIHHeTCH

HaA

cPYHK~Heii

~

s

(t)

= G

(~) ~

H

[t,

-rl

u

(-r)

d-r,

(34.8)

t

apryMeHT KOTOpoii 060aHaqeH 6yKBOii

~.

BblpameHlle B

npaBoii

qaCTH

(34.7)

MomHO

paCCMaTpHBaTb

KaK

JIIIHeiiHyro

onepa~Hro

CPi

HaA

CHrHaJIOM

{y

(,;), u (,;n,

'ITO

H

060CHO-

BIilBaeT aannCb JIeBoii

'1aCTII

(34.

7)B

cPopMe

CPi

[t,

{y

(,;), u ('t)}).

AeHCTBIITeJIbHO,

AJIH

::lTOrO

AOCTaTOqHO

JIllmb npeAnOJIaraTb,

'ITO

HaMil

6IilJIa BIil6paHa TaKaH paapenrarom;aH

onepa'D;HH

cP~

AJIH

aaAaqH

31.1,

KOTopaH

nMeeT

CMIilCJI

H

Ha

cPYHKIJ;HHX

Y (,;) (34.5), II.

Ha

cPYHK~HHX

s

(~)

(34.8).

Bo

Bcex CJIyqaHX, C

KOTOpIilMH

MomHO

C'I'OJIKHyTbCH

npH paayMHoM

nOAXOAe

K np06JIeMe Ha6JIroAeHllH,

::lTO,

BcerAa

6YAeT

TaK. B

caMOM

AeJIe,

eCJIH

u

(-r)

- BoaMomHoe ynpaBJIeHlle,

TO

cPYHK-

~IIH

S

(~)

(34.8)

eCTb

cPYHK~HH

HenpepIilBHaH, H

nO::lTOMY

B

TaKOM

CJIY-

qae'

AOCTaTOqHO,

'1T06Iil

onepa~HH

cP~

lIMeJIa

CMIilCJI

AJIH

HenpepIilBHIilX

cPYHKIJ;lIii.

ECJIH

BoaMomHIil o606m;eHHIile ynpaBJIeHllH

au

(t),

TO

COOTBeTcTByrom;ee

BIilpameHlle

1;

G

(~) ~

H [t,

-rl

au

(-r)

t

6YAeT

cPYHKIJ;lIeii

Kyco'lHo-HenpepIilBHoii

II,

CJIeAOBaTeJIbHO,

Tor,n:a

AOCTaTO'lHO,

'1To6Iil

OnepaIJ;HH

cP~

lIMeJIa

CMIilCJI

AJIH

KyCOqHO-Henpe-

pIilBHIilX

cPYHKIJ;lIii.

CJIe,n:yeT

Ha6eraTb JIllmb

OAHOBpeMeHHoro

HCnOJIb-

aOBaHllH o606m;eHHIilX ynpaBJIeHllii

au

(t)

II

onepaIJ;Hii

cP~

B BIIAe

t

•

cpf

[t,

Y

(-r)]

=

~

y'

(-r)

av

(t,

-r),

t-~

TaK

KaK

Tor,n:a

cOBnaAeHlle TO'leK paapIilBa y

cPYHKIJ;lIii

V

II

U MomeT

,n:OCTaBJIHTb

HenpIlHTHOCTH.

IlTaK,

BLIpameHlle (34.7) onpeAeJIHeT JIHHeiiHyro

OnepaIJ;Hro

CPit

aap;aHHYro

Ha

MHomOOTBe

CHrHaJIOB

{y

(-r),

u (,;)}.

Ho

Terrepb

Ha

pa-

BeHCTBa

(34.6)

cpaay

CJIeJI;yeT,

'ITO

::lTa

OnepaIJ;HH

(34.7)

YAOBJIeTBO-

pHeT

YCJIOBllro

(34.2),

KaKOB

6Iil

HH

6LIJI

BeKTOp

x

(t).

TaKHM 06paaoM,

MIil, AeiicTBIITeJIbHO, rrOJIy'lllJIII pemeHlle

CPi

aaAa'l1l

31.2,

KOTopoe

orrpe,n:eJIHeTCH

paapemarom;eii onepaIJ;Heii

cpf

aa,n:a'lH

31

.1.

PaccMoTpHM npHMep.

to·

.:.

':,:::

>.,

,::-"::·::~;,::;··.::::·\,i;~i~;~.j;s·l

292

.....

: ,

.'

.....

",

.:

..

' '. :',::.':

;',.'

.:

·~~.~~~;_~:1~~:.:.;"~'·".':

','

.....

:,-:.~

..':

:'.":

'<.~:,~·2~

.:.~

.

:.:.:

.::':.:_:"-:_:j::::.~.wx.

r

\,',

::

~.'I

'.~.,'

:

[rJI.

8

IlOCTAHOBKA

H

PElllEHHE

IlPOBJIEMbI

HABJIIOP:EHHff

IIplI:\lep

34.1.

PaccMoTpRM ;n;BRmeIDle MaTeMaTRqeClWrO

MaHTHRRa

B oRpecT-

HOCTM

nepxHero

HeycTOHqRBOrO IIOJIomeHRJI

paBHOBeCHJI,

ynpaBJIJIeMOe

MOMeH-

'rOM

u (t). YpaBHeHHJI

;n;BHmeHRJI

B nepBOM npH6JIHmeHlUI

RMeIOT

BRA

Xl

=

X2'

X

2

=

Xl

+ u. (34.9)

IIpep;nOJIOmJ.l.M, ,tITO ;n;ocTynHbI

H3MepeHRIO

aHaqeHHJI

yrJIa

Xl

(t)

OTRJIOHeHRJI

MaJITHHKa

OT

BepTHKaJIR R aHatIeHRJI U (t) ynpaBJlJIIOm;erO

MOMeHTa.

Tpe6yeTcJI

llaiiTM yrJIOByIO

CROpOCTb

X!

(t).

CJIep;OBaTeJIbHO,

HMeeM

3ap;aqy

31.2

0 Ha6JIIO-

p;eIDlR

CHCTeMbI

(34.9)

no

CMrHaJIy

{y

(t

+

~),

u

(t

+6)}

(-it

<

~.:;;;;

0),

npM-

qeM

a;n;eCb

y (t +

~)

=

Xl

(t +

~)

=

Gx

(t +

~)

=

[1,0]

X (t +

~).

I1TaR, Tpe-

6yeTCJI HaHTH JIHHeiinyIO onepan;HID TaR, qT06bI

BbIlIOJIHJlJIOCb

paBeHCTBO

<P2

[{y (t +

~),

u (t +

6)}]

=

X2

(t).

J],JIJI

CMCTeMbI

(34.9)

npn

yCJIOBJ.l.H,

'tITO

U

==

0,

pemMM CHaqaJIa aap;atIy 31.1

o

Ha6JIlo;n;eIIMM

no

CllTHaJIY Y

(1:')

= Y (t +

~)

=

Gx

(t +

~)

(1:'

= t +

~).

3TO 03-

natIaeT,

tITO

.Ha;n;O

HaRTIl mmeHHyIO onepan;HIO

<P~

TaR, qT06hI

;n;JIJI

.Il.106030 BeRTopa

8

BbInOJIHJIJIOCb

paBeHCTBO

<P~

[GX

[t

+

~,

t]

8]

= 8

2

,

rp;e X _

!J>YH;n;aMeHTaJIbUaJI

MaTp:un;a

CHCTeMbI

(34.9).

B'

HameM cJIytIae

(--

it<

~

< 0).

GX

[t

+~,

t] 8

~

[1,

0]

ch

~

sh

~

'I

[

81

]

=

81

ch

~

+

82

sh

~

~,

"

I]

sh

~

ch

~

82

TaRIlM

06paaOM, ollepan;nIO

<P~

Ha;n;JIeHUIT

onpep;eJIJITb

Ila

yCJIOBMJI

<P~

[Sl

ch~

+

»f

If I

+ S2

sh

~]

= 8

2

,

\, !

IIpMMeM Tenepb

;n;JIJI

Ollpep;eJIeHHOCTR,

tITO

it

=

1,

M

6y;n;eM

paccMaTpRnaTb

CMrHaJI

y

(1:')

= Y (t +

~)

RaK

SJIeMeHT

IIpocTpaHcTBa 2t:!1,O]' Tor;n;a B

COOTBeT-

~I

'j

I

CTBIlH

co

CKaaaHIl.W.M

BbIme

;n;JIJI

pemeHIlJI aa;n;aqll 31.1

He06xo;n;HMO

HaRTIl

!J>YHK-

. ,

"

n;RIO

V

O

(~),

y;n;OBJIeTBOpJIIOm;yIO

paBeHcTBy

".

o

"

','

(34.10)

~

[slch~

+

82Sh~]

vO(~)d~=

82

-1

npH

yCJIOBRIl

o

(34.11)

~

[vO(~)]2d~

=

min

..

-1

3a;n;atIy

(34.10),

(34.11)

MomHO

pemllTb,

nOJIbaYJICb npaBHJIOM

33.1.

He

lIpllBOAIl

SToro pemeIDlJI, IWTopoe

CBO;n;IlTCJI

K TaKHM

me

npon;e;n;ypaM, RaKHe

paCCMOTpeHbI

"BbIIlIe, B §

33,

npllBe;n;eM

JIllmb

pe3YJIbTaT. ORa3bIBaeTCJI,

tITO

yO

(t

+

~)

=

sh

~

.Ii V

O

(£) =

-7,24

ch

~

+

14,73

sh~.

I1TaK, paapemaIOm;aJI onepan;RJI

<p~

AJIJI

aa;n;atIH

31.1

0

Ha6JIIO;n;eIDlH

CHCTeMbI

(34.9)

npR

u

==

0

110

CRrHaJIy y

(1:')

=

=y

(t+

~)

=

Gx

(t

+

~)

lIMeeT

BlI;n;

o

<P~

[y

(t

+ m=

<P~

[Xl

(t

+ m=

~

Xl

(t

+~)

[14,73sh

~

+

7,24ch~]

d~

= X2(t).

-1

Pl

\2.:~"')

§ 34]

PEillEHHE

SAp:A

~H

0

HA13JIIOP:EHHH

293

Tenepb,

nOJIbaYJICb

paBeHCTBOM

(34.7),

rAe

nOJI01RI1M

~

= t +

lj,

Haii:;n;eM

one-

pa~HIO

<J!2

[{y

(t

+

~),

u

(t

+ 6))],

(t

+

~

=

1:'),

paapemaIOm;yIO aaAatIY

31.2

o Ha6JIIOAeHlIH CllilTeMbI

(34.9).

I1MeeM:

epz[{y(t

+~),

u(t+~)}]=

11

=

<P~

[y

(t+

m-

<p~

[ G

~

H

[t

+lj, t +

~l

u tt

+~)

d~J

=

o

0011

=

~

Xl

(t

+

~)

VO

(~)

d~

-

~

VO

(~)

GH

[t + T], t +

~]

-1

-10

IIpo;n;e1IaB He06xo;n;HMbIe npe06paBOBaIDlJI, ROTopbIe

CBO;n;JITCJI

-

u (t +

~)

d~

dlj.

K llaMeHeHHIO

1I0pJIARa IIHTerplIpOBaHlIJI B

nOCJIe;n;HeM

;n;BOHHOM

lIHTerpaJIe, nOJIyqlIM OKOHqa-

'I'eJII,HO:

o

<P2

r{y

(t +

~),

u (t +

~)}]

=

~

Xl

(t

+~)

[14,73

sh

~

+ 7,24

ch~]

di;

+

-1

o

4-

~

u

(t

+

~)[(-7,37

~

+

0,96)

cll;

+

(-14,62

+

3,62

~)sh~]

d~=

X2

tt).

-1

BCIOP;Y

BbIIIIe

B

aTOM

H B

npep:bIp;YIIJ;HX

na

parpa4Jax

MbI

paCCMaT-

pHBaJIH

3ap;a'lY

0

HaOJIIOp;eHHH

Roopp;HHaT

Xi

(t)

4Ja30Boro

BeKTopa

X (t).

Op;HaRO

MorYT

BCTpeTHTbCH

CJIy'IaH,

Korp;a

3HaTI.

(ice

aTH

KO"

opp;HHaTbI

He

TpeoyeTcff,

HO

p;OCTaTO'lHO

BbI'lHCJIffTI.

JIHlllb

HeROTopbIe

4JYHKIJ;HH

Pi

It,

Xt

(t), ..., X

n

(t)] OT

HlU.

MI.I

OrpaHM'IHBaJIHCb

nORa

TOJII.KO

JIHHeHHbIMH

3ap;a'laMH.

IIoaToMy

H

TenepI.

OCTaHOBHMCff

JIHilib

ua

3ap;a'le

0

HaOJIIOp;eIDIH

JIHHeHHOH

<PYHKIJ;HH

rJ

= p

It,

X]

=

p'

(t) X (t),

(34.12)

rp;e

p (t) -

HeRoTopbIH

n-MepHbIH

BeRTOp.

<DOPMYJIHpOBKH

sap;a'l,

COOTBeTcTByIOIIJ;Me

3P;OOb

3ap;a'laM

0

HaOJIIOp:eIDIH

no

Hp;eaJIbHoMy

y ('r) = G ('r)· X ('r) H

peaJIbHoMy

Z ('r) = y

(1")

+

L1

(1')

CHrHaJIaM

(t - it

-<

l'

-<

t),

nOJIY'laIOTcff

O'leBHp;HOH

TpaHC<popMaIJ;HeH

3ap;a'l

31.1, 31.2 H 32.1.

IIpHBep;eM,

op:HaRO,

HX

p:JIH

nOJIHOTIiI

H3JIOmeIDIff.

3 a

p;

a

'I

a 34.1.

II

ycmb

8euJlCYUfuUCR,

o6'be1f,m

onUCbleaemCR,

ypae-

-neftue.u (32.5), npu'le.u 80cmynnbl

U3lttepeHUlO

3HaIteHUR,

U8ea.llbn080

CU8Ha.t£a

y

(1')

= G

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X

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(t

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-<

l'

-<

t). Tpe6yemcR,

Haumu

.IlU-

HeuHylO

onepalfulO

cp[t,

y (1')],

(fbl'lUCMtlOUfYlO

Be.tlU'lUHYrJ

(t)=p'

(t)

:x

(t)

·no

CU8Ha.llY

y

(1'),

ma1f,

'lmo

BblnO.llHMmcR,

paeeHcmeo

cp

.It,

{y

(1')

IrJ

(t)}]

=

p'

(t) x

(t).

3 a

p;

a

'I

a 34.2.

IIycmb

8euJlCYUfUUcR,

o6'be1f,m

onUCbZ8aemCR,

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neHueM

(32.5), npU'leM e npolfecce

8eUJlCeHUR,

U3MepMmCR,

pealibHblu

'tu8naJ£

Z

(1')

= G {1')x (1') +

L1

'(1')

(t - it

-<

l'

-<

t),

aBe'

nO/MXa

._

..

_.

,-~-.~.-._.~

'

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~,-~~:....,

.

-'-'-~

,

.:

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.'

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..

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..

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,.,0;'

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=::.'... :

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C

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294

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H

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HABJIIO,IJ;EHHII

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8

!1

('t')

cmecueua

oapaUUlJ,euuell£

(32.7). Tpe6yemc.f£

uaumu

onepa/.fUlO

<po

[t,

z

('t')

J,

1f,omopa.f£

yfJo8MmBOp.f£em

YC.lW8UJO

sup

I

<po

[t,

{z

('t')

I

p'

(t) x

(t)}]

-

p'

(t) x (t)

1=

z

=

min

sup

I<p[{z

('t')

I

p'

(t) x (t)}] -

p'

(t) x (t) I

<p

z

no

8Ce.M,

808l1£O:HCf£bZ.lft

peG./tU8a/.fU.f£.M,

z ('t) U

no

8Ce.M,

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one

pa-

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<p

[z ('t) ]. .

OlJ:eBlI,IJ;HO,

lJ:TO

npOOJIeMa

HaOmO,IJ;eHlIH

BeJIlIlJ:lIHhI

1)

(t) (34.12)

He

HBJIHeTCH,

no

cym;ecTBy,

HOBOH:

OHa

BCer,IJ;a

MomeT

OhITb

CBe,IJ;eHa R

lIaYlJ:eHHOH

yme

aa,IJ;alJ:e

0 HaOmO,IJ;eHlIlI

HeHOTopOH

KOOp,IJ;lIHaThI

Xi

(t)

BeRTopa

X (t).

AJIH

aToro

,IJ;OCTaTOlJ:HO

nplIMeHlITb

JIHHeHHOe

npeoo-

paaOBaHlIe

BeRTopa

X (t),

HOTopoe

npeBpam;aeT

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1)

(t) B

me-

JIaeMyIO

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HOBoro

!paaOBoro

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cpaay

CJIe-

,IJ;yeT,

lJ:TO

Bce

paccym,IJ;eHlIH

1I BhIBO,IJ;hI

Ha

§ 33 aBTOMaTHlJ:eCRH

pac-

npOCTpaHHIOTCH

Ha

'CJIYlJ:aH HaOJIIO,IJ;eHlIH

BeJIlIlJ:lIHhI

(34.12).

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aTOM

Ha,IJ;O

JIlImb

Bea,IJ;e

aaMeHlITb

CHrHaJIbI,

Hecym;lIe

BeJIlIlJ:lIHY

Xi

(t),

ClIrHaJIaMH,

Hecym;lIMH

BeJIlIlJ:HHY

1)

(t). B lJ:aCTHOCTlI,

npaBlIJIO

33.1

coxpaHHeTCH

npH

aTOM

,IJ;OCJIOBHO,

aa

TeM

lICRJIIOlJ:eHlIeM,

lJ:TO

BMec-

TO

ClIrHaJIOB

{y

('t) I

Xi

(t) = 1}

1I

{yO

('t) I Xi (t) = 1},

Hecym;HX

i

I'll

Xi

(t) = 1,

a,IJ;eCb

CJIe,IJ;yeT

paCCMaTplIBaTb

COOTBeTCTBeHHO

ClIrHaJIhI

1

11

{y

('t)lp'x

(t) =

1}

1I

{yO

('t)

Ip'x

(t) =

1},

Hecym;lIe

BeJIHlJ:lIHY

p'

(t)x

(t) = 1.

11'1:'

§ 35.

HaOJIIOAeHlle

RBa311JI11HeiiHOii

CIlCTeMbI

i!t'

BhIme

MhI

HeO,IJ;HORpaTHO OTMelJ:aJIH,

lJ:TO

HaCTOHm;aH

MOHorpa!pIIH

r,

I.:

"

IIOCBHm;eHa

lIaYlJ:eHlIIO

JIHHeHHhIX

,IJ;IIHaMIIlJ:eCRlIX

ClICTeM.

PaccMaT-

plIBaeMhIe

B

HeH

cnocoohI

pemeH'IIH

aa,IJ;alJ:

nplICIIOCOOJIeHhI

Cne~lIaJIb

HO

,IJ;JIH

TaKHX

CIlCTeM

II

onlIpaIOTCH

Ha

MaTeMaT~eCRlle

MeTO,IJ;hI

JIII-

HeHHoro

aHaJIlIaa.

8TO

aaMelJ:aHlIe

OTHOClITCH B

nOJIHOH

Mepe

H R aa,IJ;a-

lJ:aM 0 HaOJIIO,IJ;eHlIII. O,IJ;HaRO

npH

HaYlJ:eHlIlI

npOOJIeMhI

ynpaBJIeHlIH

,n;BlImeHHeM

MhI

yme

BH,n;eJIH,

lJ:TO

Hamil

MeTo,n;bI

MomHO

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npH-

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pemeHlIIO

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00

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CHCTe-

MaMlI.

AHaJIOrHlJ:HhIM

oopaaOM

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0

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R

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aa

paMRII

aToH

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MbI

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oy,n;eM

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HO

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no,IJ;Xo,n; R aa,n;alJ:e,

ROTOphIH,

RaR

II

B

aHaJIOr~

HOM CJIylJ:ae

npOOJIeMDI

ynpaBJIeHlIH

(CM. § 22),

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lI,n;eH MeTo,n;a

MaJIOrO

napaMeTpa.

AJIH

ynpom;eHlIH

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orpa-

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CJIyqaeM,

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OTCyTCTBy-

JOT.

IITaR,

paCCMOTpHM

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cHcTeMy,

oIUlChIB/leMYIO

t'

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.

g 35]

HAjj.fi:to,IJ;EHHE

it13A3HJtHHEntton

CiIC'I'EMbf

295

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ypaBHeHHeM

j;

==

A (t) X +

p.,!

(t, x). (35.1)

3,IJ;eCb X -

n-MepHhIH

BeRTop

!paaoBhIx

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Xi;

A

(t)

_

HenpephIBHaH

(n X

n)-MaTpHD;a;

! (t, x) -

HeJIHHeHHaH

!PYBRD;HH,

BenpephIBHaH

no

oooHM,aprYMeHTaM

H

aHaJIHTlIlJ:eCRaH

no

Xi

npH

Ram-

,IJ;OM

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aHa1J:eHlIH

t;

p.,

-

MaJIhIH

napaMeTp.

.

,D;JIH

CHCTeMDI (35.1)

onHmeM

pemeHlIe

aa,IJ;alJ:lI 31.1 0 HaOJIIO,IJ;eHlIH

ROOp,IJ;HHaT Xi (t)

!paaOBoro

BeRTopa

X (t)

no

lIlleaJIbHOMY

CHrHaJIy

y ('t) = G ('t) X

(T).

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rOBopH,

CHrHaJI

y

('t')

MomeT

H

HeJIHHeH-

BO

aaBHCeTb

OT

Xi ('t), o,n;BaRO

MDI

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npOCTOThI

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a.n;eCb

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(T)

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JIHHeHHOH

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TaR

RaR

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HeaHa'lHTeJIbHDIe

BaMeHeHHH,

ROTOphIe

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OY,IJ;eT

BBeCTH

B

HamH

paccym,IJ;eHHH

npH

rrepexo,n;e

R

HeJIHHeH-

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y ('t), OlJ:enH,IJ;HhI. .

UTaR,

HaM

Ba)J;JIemHT

HaHTH

onepa~HH

ei

[t,

y ('t)]

(Tenepb

yme

HeJIHHeHHhIe),

ROTOpDIe

y,IJ;OBJIeTBOpHIOTyCJIOBHHM

ei [t,

{y

('t) I

Xi

(t)}J = Xi (t).

(35,2)

8TH

onepa~HH

Ha,IJ;JIemHT

CTpOHTb

B BH)J;e

pH,IJ;OB

00

ei

(t,

Y

('t')J

=

2i

""keyi)

[t,

y

('t')J

k=-O

(3;5

3)

no

CTeneHHM

napaMeTpa

p.,.

B RalJ:eCTBe

nepBoro

lJ:JIeHa

e?

pfI)J;3. (35.3)

CJIe)J;yeT, O'leBH)J;Ho,

BDIoHpaTb

Ty

onepaD;HIO

<p~,

ROTopaH

pemaeT

aa-

.n;alJ:Y

0 HaOJIIO,IJ;eHHH CHCTeMDI (35.1)

npH

p.,

= 0,

H,

OTTaJIRHBafICb

OT

Hee,

HCRaTb

CJIe)J;YIOrn;He lJ:JIeHbI pH,IJ;a (35.3).

TaRHM

oopaaoM,

nYCTb

<pf

[t,

Y

(T)J

-

pemeHHe

aa,IJ;alJ:H

31.1

)J;JIH

CHCTeMhI

(35.1), HO

npH

p.,

= 0.

8TO

OaHalJ:aeT,

lJ:TO

)J;JIH

JIIOOOro

BeRTopa

s cnpaBe,IJ;JIHBO

paBeHCTBO

<pf

[t,

G

('t')

X ['t', t]

sJ

=

Si'

(35.4)

Cnocoo

onpe,IJ;eJIeHHH

onepa~HH

<P~

HaM

yme

HaBeCTeH

Ha

MaTepHaJIa

§§

33 H 34. B COOTBeTCTBHH C

aTHM

OY)J;eM ClJ:HTaTb

onepaD;HIO

<pf

yme

HaH,n;eHHoH.

BBe)J;eM

TenepbOOOaHalJ:eHlIe

X~l]

(t) =

<pf

[t,

Y ('t')J.

(35.5)

)J;BHmeHHIO

X ('t)

(t

- il' <

't'

< t)

RBaaHJIlIHeHHOH

CHCTeMbI

(35.1)

m,nOCTaBHM

,IJ;BHmeHlIe

X[I]

('t')

COOTBeTcTByIOm;eH

JIlIHeHHOH

CHCT€lMhI

dx[l]

('r) _ A

(r)

x[l]

('t')

-_

..

~------------,_.

~"

','

••.•

~-~...:.......~-'-,_

"',,'

••.

'"

•••••

,

't'

••

.!.....

"'(*:'t

',i

'1'

,

"I"

II

I

I~ll

, I

'"

1

h:::

"

ill

\

"I

C

t.:.<:',"

296

nOCTAHOBHA

H

PEWEHHE

IIPOBJIEMbI

HABJIIO,JJ;EHIHI

[rJI.

8

npH

Ha1JaJIbHOM

yCJIOBHH

X[l]

(t)

(35.5).

IIYCTb

Llxi

('r) = Xi ('r)

-'-

Xf1]

(t).

B

Rypce

p;HcPcPepeHIJ;HaJIbHhIX

ypaBHeHHH

p;ORaahIBaeTCH,

1JTO

p;BHme-

HHe

X ('t)

(t

- {} <'t

-<

t)

CHCTeMhI

(35.1)

MomHO

npep;CTaBHTb

B

BHp;e

pHp;a

no

CTeneHHM

MaJIOrO

napaMeTpa

f..l.

H BeJIH1JHH LlXi

(t):

X

(t)

= X ['t',

t]

(x

l1

]

(t)

+

Ax

(t»

+

00

+

~

f..l.kffi~m)

(tt

't',

Llx

(t»

,

(35.6)

1,=lm=O

rp;e

ffir - cPOPMhI

m-ro

nopHp;Ka

no

LlXi

(t).

HaBecTHo,

1JTO

pHP;

(35.6)

CXOP;HTCH B

oKpecTHocTH

p;BHmeHHH

X[l]

('t),

ecJIH

f..l.

H

Llxt

(t)

p;OCTa-

T01JHO

MaJIhI

H eCJIH

KpHBaH

X~l]

('t)

cop;epmHTcH

B OOJIaCTH

aHaJIHTH1J-

HOCTH

cPYHRIJ;HH f ('t,

x)

(CM.

[12·],

CTp.

56).

Ha

(35.6)

BhITeRaeT,

1JTO

y

('t')

= G

('t')

X ['t',

t]

(X[l]

(t)

+-

Llx

(t»

+

00 00

+ ~ ~

f..l.kG

('t')

ffikm)

(t,

't',

Ax

(t».

(35.7)

k=lm=O

IIOCKOJIbRY

OnepaIJ;HH

cP~

MhI

C1JHTaeM

HaBeCTHbIMH,

TO

MhI

MomeM

CTIHTaTb

HaBeCTHhIMH

H

BeJIH1JHHhI

Xf1]

(t).

CJIeP;OBaTeJIbHO,

P;JIH

onpe-

p;eJIeHHH

HCKOMhIX

BeJIH1JHH

Xi (t)

HaM

p;OCTaT01JHO

HaHTH

BeJIH1JHHhI

LlXi

(t).

II03TOMY

COCTaBHM

Tenepb

ypaBHeHHH

P;JIH

onpep;eJIeHHH

BeJIH1JHH !1Xi

(t).

HMeeM,

cPopMaJIbHO

Y1JHT:bIBaH (35.4), (35.7) H

CBOH-

CTBa

JIHHeHHOCTH

OnepaIJ;HH

cpr:

cpf

[t,

Y ('t')] =

Xf1]

(t)

+LlXi

(t)

+-

B

CHJIy

(35.5)

LlXi

(t)

PewaH

9TY

HRHp;eM:

00 00

+ ~

~

f..l.kCPf

[t;

G

(t)

ffikm)

(t,

't',

Ax

(t»].

k=lm=O

'

OROHlJaTeJIbHO

nOJIy1JaeM,

1JTO

00 00

= -

~ ~

f..l.,rcpf

[t,

G

('t')

ffiLm)

(t,

't', Llx

(t»]

(35.8)

k=lm=O

'

(i=1,

...

,n).

cHcTeMy

HeJIHHeHHhIX

aJIreOpRH1JeCKHX

ypaBHeHHH.

00

(35.9)

Axdt)

=

~

I-V'qid

t

).

"=1

CJIep;OBaTeJIbHO,

BeJIH1JHHbI

LlXt (t)

HMeIOT

TOT

me

nopHp;OK

MRJIOCTH.

r,"

~":1

§ ;J5]

HABJIIO,JJ;EHHE

HBA3HJIHHEi'l:HO:fI

CHCTEMbI

297

<iTo

H

f..l.,

H

n09TOMY

cPopMaJIbHOe

pemeHHe

ypaBHeHHH

(35.8)

npH

p;o-

CTaT01JHO

MaJIOM

I.t

HBJIHeTCH

p;eHCTBHTeJIbHbIM

pemeHHeM.

PHP;

(35.9)

CXOP;HTCH,

HOO

npH

MaJIhIX

!1Xi

(t)

CXOP;HTCH

pHP;

(35.8).

IfTaR,

Haxop;HM:

00

Xi

(t)

=

xf

1

l

(t)

+Llx,!

(t)

=

xfll

(t)

+

~

f..l.k

qi1c

(t)

k=l

(i

= 1,

...

,n).

(35.10)

TaRHM

oopaaOM,

pemeHHe

aap;a1JH 31.1 0

HaOJIIOp;eHHH

KBaaHJIH-

HeHHOH

CHCTeMhI

(35.1)

no

Hp;eaJIbHoMy

CHrHaJIy

y ('t) = G

('t)x

('t)

onpep;eJIHeTCH

pHP;OM,

CTOHIIJ;HM B

npaBoH

1JaCTH

paBeHCTBa

(35.10).

HaM

eIIJ;e

Hap;o

OhIJIO

OhI

nOKaaaTb,

1JTO

BeJIH1JHHbI

qik(t),

cPHrypHPYIO-

IIJ;He B (35.10),

MomHO

HCTOJIROBaTb

RaK

aHa1JeHHH

HeKOTophIX

onepa-

IIHH

e\'k}

[t,

y ('t)l,

BhInOJIHeHHhIX

Hap;

CHrHaJIOM

y ('t),

T.

e.

OCTaJIOCb

nOKaaaTb,

1JTO

paapeWaIOIIJ;He

OnepaIJ;HH

Si

[t,

Y ('t)] (35.10),

p;eHcTBH-

TeJIbHO,

HMeIOT

BHP; (35.3).

,l(JIH

9Toro

p;OCTaT01JHO

npOCJIep;HTb

cnocoo

BhI1JHCJIeHHH

BeJIH1JHH

qik (t).

Op;HaKo

MbI

He

oyp;eM

npOp;eJIhIBaTb

9TO

ap;eCb

H

He

oyp;eM,

CJIep;OBaTeJIbHO,

BhInHCbIBaTb

HBHhIH

BHP;

OnepaIJ;HH

sf')

[t,

11

('t)],

TaK

KaR

nOJIyTIaIOTCH

1JpeaBhI1JaHHO

rpOMoap;Ime

BhI-

pameHHH.

OTMeTHM

eIIJ;e B

aaKJII01JeHHe,

1JTO

npH

BhI1JHCJIeHHH

peweHHH

aap;a-

TIH

31.1 0

HaOJIIOP;eHHH

RBaaHJIHHeHHOH

CHCTeMbI

(35.1)

MbI

nonyTHo

yCTaHaBJIHBaeM

CJIep;YIOIIJ;HH

cPaKT:

ec.ltu

aaf)alf,a 31.1 0

l-la6.1tlOf)el-luu

no

uf)ea.ttbl-loMY

CU21-la.tty

y ('t) = G ('t) X ('t) paapeUlUMa

f).MI,

cucmeMbl

nep0020

npu6.1tuJlcel-lUR,

j;

= A

(t)x,

mo

ol-la

paapeUlUMa'npu f)ocmamo'l-

ItO

Ma.ltbZX

I.t

U

f).ltR,

ucxoal-lou

1l:0a8U.ltUl-leUl-lou

cucmeMbl (35.1)

f).ltR,

.lt106b1X

Xi

(t),

.lte:JICaufux

0

o6.1tacmu

al-la.ttumU'II-lOCmU

(/iYI-l1UiuU

f ('t,

x).

•••••

,_....:...:......'--u..:

.......

....L~>''''-,-'

__

••••

:

••.•

: __

.'_

•••

:.~

••.

.:

.••.••.

'toJll.~

c

\~:~~:'

rJl.aea9

ABOiICTBEHHOCTb MEJKAY YHPABJIEH:UEM

':U

HABJIIO,UEH:UEM

§

36.

CooTHomeHHe P;BoHcTBeHHocTH

B §

33

'1HTaTeJIh

BHp;eJI,

'ITO pemeHl'Ie Sap;a'lH

06

OIlTHMaJIhHOM

Ha-

6mop;eHHH

OKasaJIOCh

CXOP;HbIM

C pemeHHeM Sap;a'lH 06

OIlTHMaJIhHOM

yIlpaBJIeHHH, OIlHcaHRbIM B §

16,

17.

OTMe'leHHOe

CXOP;CTBO

B pemeHHH

9THX

sap;a'l nOKaSbIBaeT HX TecHyIO

CBHSh

p;pyr

C APyrOM.

MomHo

6bIJIO

6bI Bo06IIW C

caMoro

HaqaJIa

CBeCTH

OAHY

HS

HHX

K

APyroH.

OA-

HaKO MbI npep;IIO'lJIH npeAJIOmeHHbIH

cnoc06

HSJIOmeHHH,

'1T06bI

60-

JIee

peJIhecI>Ho

OTTeHHTb

CBHSh

paCCMaTpHBaeMbIX sap;a'l C

COOTBeT-

CTByIOm;HMH

MaTeMaTH'IeCKHMH np06JIeMaMH. :UTaK,

AO

CHX

nop

MbI

HSy'laJIH Sap;a'lH

06

ynpaBJIeHHH H 0 Ha6JIIOp;eHl'IH B

HSBeCTHOH

Mepe He-

!~!

saBHCHMO

p;pyroT

p;pyra.

Tenepb

MbI

p;OJImHbI

HCCJIeAOBaTh

Ty

CBHSb,

l

11

'

KOTopaH

HX

06'1>eAHHHeT.

8Ta

CBHSb

BbIpamaeTCH B

cI>opMe

npaBHJIa

~I

P;BoHcTBeHHocTH MemAY ynpaBJIeHl'IeM'H

Ha6JIIOP;eHHeM.

81'0 IIpaBHJIO

MomHO

BbIBeCTH,

cpaBHHBaH MaTepHaJI §

16,

17 H § 33,

op;HaRO

MbI

""',

Cp;eJIaeM

91'0

SAeCb

Henocpep;CTBeHHO H

3aHOBO.

IITaK,

BbIBOP;

npaBHJIa

':II!,

ABOHCTBeHHOCTH

II

COCTaBJIHeT

cOAepmaHHe HaCTOHm;ero naparpacI>a.

)11,

IIpemp;e

Bcero,

HaM nOHap;06HTCH nOHHTHe 0 COnpRJlCennbLX

ynpa-

"'I',

!~

6.l£ReMOU

U na6.tuoOaeMOU

cucmeMax.

TaKoe

CBOHCTBO

conpHmeHHOCTH

nOJIy'laeTcH B pesyJIbTaTe nepeHeceHlIH

yCJIOBHH

conpHmeHHOCTH, HS-

..

,

BeCTHoro

AJIH

CHCTeM

AHcI>cI>epeH~HaJIbHbIX

ypaBHeHHH, Ha lISY'laeMbIe

HaMH

06'1>eKTbI.

IIocJIep;Hero

yCJIOBHH

MbI

yme

RaCaJIHCb

B §

10.

,D;o-

rOBopHMCH 0

CJIep;YIOm;eM:

'1T06bI paSJIH'IaTh paCCMaTpHBaeMbIe p;aJIh-

rne

op;HOBpeMeHHO

p;Be

p;HHaMH'IeCKHe

CHCTeMbI,

H3

KOTOpbIX

op;Ha

yn-

paBJIHeTCH,a

p;pyraH

Ha6JIIOp;aeTCH,

6yp;eM 0603Ha'laTh cI>asoBbIlt

BeKTOp

ynpaBJIHeMOrO

06'1>eRTa

CHMBOJIOM

x (t), a

cI>asoBbIH

BeKTOp

Ha6-

JIIOp;aeMOro

06'1>eKTa

060sHaWM

'1epes s (t). :UTaK, paCCMOTplIM

p;Be

CHCTeMbI:

x= A

(t)x

+B

(t)u

+w (t)

(36.1)

H

s = P (t)

s,

(36.2)

Y = G (t)

s.

(36.3)

3p;eCh, RaR 06hIqHO, x H S - n-MepHbIe

cI>asoBbIe

BeKTOpbI; U -

r-

J4epHbIH BeRTop ynpaBJIeHHH; y 03HaqaeT m-MepHbIH BeKTop-eHraaJIj

0,:

~~.,,~

..

§

36J

COOTHOIDEHlIE

P:BO:f1CTBEHHOCTH

299

A (t), B (t), P (t), G (t) - HenpepbIBHbIe

MaTpH~bI

COOTBeTCTBYIOm;HX

HSMepeHlIH.

o n

pep;

e

JI

e

HH

e

36.1.

CucmeMbL

(36.1)

U

(36.2),

(36.3)

na-

8bLealOmCR

COnpRJlCennbWU,

ec.fl,U

8bLnO.fl,nRlOmCR

coomnOutenuR

P (t) = -

A'

(t),

G

(t)

=

B'

(t).

(36.4)

QeJIeC006paSHOCTb

TaKoro

onpep;eJIeHl'IH 6YAeT

HCHa

HS

AaJIhHeH-

mero:

CHa'laJIa

MbI

nOKai-ReM,

'ITO

saAa'la

13.1

06

ynpaBJIeHl'IH H sap;a'la

31.1

0

Ha6JIIOp;eHl'I1I,

nOCTaBJIeHHbIe

COOTBeTCTBeHHO

P;JIH

conpHmeHHbIX

CHCTeM

(36.1)

H

(36.2),

(36.3),-9RBHBaJIeHTHbI,

a saTeM 6YAeT yCTaHoB-

JIeHO

TaRme H COBnap;eHlIe npaBHJI MIIHl'IMaRCa, YRasaHHbIX B

§§

17

H

33

AJIH

9TlIX

saAa'l.

,D;JIH

ynpom;eHIIH BbIImaAOR MbI npOBep;eM

pac-

cymp;eHHH

06

9RBlIBaJIeHTHOCTlI 3ap;a'l

13.1

II

31.1,

nOJIaraH, 'ITO

ynpa-

BJIeHlIe U H ClIrHaJI y

HBJIHIOTCH

CRaJIHpHbIMlI. B

COOTBeTCTBlIlI

C

9THM

B ypaBHeHlIHX

(36.1)

II

(36.3)

cI>YHR~HII

B (t) = b (t)

II

G (t) = g' (t)

HBJIHIOTCH

Torp;a n-MepHbIMII BeRTopaMH (nepBbIH -

CTOJI6~OM,

BTO-

paH -

CTpOROH).

11TaR, cpaBHlIM Sap;a'lH

13.1

H

31.1.

MbI

yme

SHaeM,

qTO

3ap;a'la

13.1

06 ynpaBJIeHHH

CHCTeMoH

(36.1)

liS

TO'lRlI X (ta) = x

a

BTOqRY

X

(t~)

=

x~

CBOP;HTCH

K ypaBHeHIIHM

(15.1)

P;JIH

IICROMOH

cI>YHR~lIH

U (T). 8TH ypaBHeHIIH B BeRTopHOH

cI>opMe

IIMeIOT

BHP;

t{3

~

X

[t~,

TJ

b (T) U (T)

dT

=

c,

(36.5)

t

a

rp;e

t

C =

x{3

- X

[t{3,

tCl.]

x

a

-

~

X

[t~,

T]

w ('t)

dr,

(36.6)

t

a

npllqeM

X [t,

npH

U

(t)

=

O.

to] -

cI>YHAaMeHTaJIbHaH

MaTpH~a

CHCTeMbI

C

p;pyroH

CTOpOHbI,

'1T06bI HaHTH

onepa~IIIO

(36.1)

t

qJ[t,

y('t)]

=

~

y(T)V(t,

T)dT,

.

t-{j.

paspemaIOm;yIO sap;a'ly

31.1

0

Ha6JIIOp;eHHH

BeJIlI'IIIHbI

'YJ

(t) =

p's

(t)

no

ClIrHaJIY y (T), Hap;JIemllT onpep;eJIIITh

HeH3BeCTHYIO

cI>YHR~IIIO

v (t, T), ROTopaH

yp;OBJIeTBOpHeT

yCJIOBlIIO

t

~

{y("t)lp's(t)}v(t,

T)d"t=p's(t),

t-{j.

RaROB

6bI

HH

6b1JI

BeRTOp s (t). CHrHaJI y (T) CBHsaH

s (t)

COOTHOmeHl'IeM

y (T) =

g'

(T) S

[T,

t]

s (t), rp;e

(36.7)

C BeRTopOM

S [t, t

0

1 -

Красовский Н.Н. Теория управления движением

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