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,

+ Q

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=

11,1

(t)6ql +

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+

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(t)

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([13*],

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111)

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s

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II

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dv

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v (t),

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mlft=p+f.

(3.8)

\

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=

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=

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+

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+ v (t),

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=

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(t).

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m = m (t) = m

o

+m

l

(t).

cr

«

•.

rAe

m

o

=

const

-

He1I3MeHHaH

(~

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(~~

r

J

)

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.

I1TaK,

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m

l

(t)

;;>

0 - rrrc. 3.3.

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nopHp;Ka:

peaRTlIBHaH

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l

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l

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(8

-

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dml/dt; v -

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a6COJIIOTHOH

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m;

8 -

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2

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t +

dt

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ee

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p -

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(2.4).

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a1)

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I

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ill

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a C OCHMH £ H

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(3.10)

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no

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paa

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(2)

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no

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no

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a

z

a

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a

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.),

0

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all all

28

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)

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1

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'

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29

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BLI'IHCJIHIOTCH

no

cPopMYJIaM:

W

r

=

~'-

np2,

W<\I

=

2':~

+r1ji.

/'0

CJIeAOBaTeJIbHO,

HMeeM

ABa

AHcPcPe-

pe

HIJ;HaJIbHLIX

ypaBHeHIIH,

RamAoe

lIS

ROTOpLIX

eCTb

ypaBHeHHe

BToporo

no-

pHAKa

OTHOCIITeJIbHO

r H

'IJ':

m

(~

-

r~2)

, p +

Ir,

}

(3.16)

.m

(2~~

+

r-iji)

=

1<\1,

PHC.

3.5.

rAe

Ir

=

arm,

1<\1

=

a<\lm,

a a

r

II

a<\l

CYTb

npoeRIJ;HH

BeRTopa

OTHOCH-

TeJIbHOH

CKOpOCTII

OTAeJIHIO~HCH

'IaCTHIJ;LI

Ha

HanpaBJIeHHe

paAHyca

H

nOnepe'IHOe

HanpaBJIeHHe

CooTBeTcTBeHHo.

PaSAeJIHB

06a

ypaBHeHHH

(3.16)

Ha

m

(t)

H

nOJIaraH,

'ITO

CHJIa

p

eCTb

CIIJIa

BceMHpHoro

THrOTeHHH

([22*

1,

CTp.

176),

OROH'IaTeJIbHO

nOJIYYIHM:

..

. v m

oh2

r - r

'Y

+-

,2

= a

r

_

m '

)

" (3.17)

r1ji

+

2,:

~

=

a<\l

~

.

m

J

3AecL

v =

Vo

M,

Vo

-

nOCTOHHHaH

BceMHpHoro

THrOTeHIIH.

IIYCTb

npH

OTCyTCTBIIH

ynpaBJIHIOm;eH

peaKTIIBHoH

CIIJILI

TO'IKa

m

MomeT

ABHraTbCH

no

RpyroBoH

TpaeRToplIH

ro

paAHyca

r =

const>

0,

TaR

'ITO

to = 0, r

o

= 0.

TorAa

lIS

ypaBHeHHH

(3.17), B

o

KOTOpLIX

npa-

BLIe

'IaCTH

nOJIOmeHLI

paBHLIMII

HyJIIO,

onpeAeJIHIOTCH

SHa'IeHHH

oc-

TaJIbHLIX

cPasOBLIX

RooPAHHaT,

cooTBeTcTByIOm;Hx

ABHmeHHIO

lIO

KpH-

BOH

ro:

,po

=

at,

'ljJo

=

<X,

<X

=

-V

vlrg.

3anHrneM

ypaBHeHIm

(3.17) B

HOpMaJIbHOH

cPopMe

(2.1):

tII

=

Y2,

1

th = - ++

YIY:

+

Ul,

I

Y

1

(3.18)

t13

=

Y4,

. 2

Y4112

+'

1 J

fj4 = -

--

-

U2'

'!II

YI

,-.

1

_'

..

,

,.

C

O"~

c>

:

..

:::.\

".-:..~/

30

llOC'l'AHOBHA

SA,ll;AQH OE

YIIPABJIEHHH

[1'JI. 1

3,n;eCb OOOSHaTfeHOI

Yl

=

r,

Y2

=

;,

Ys

=

'i',

Y4

= = arin/m,

*,

Ul

~

= allin/m.

TIpe,n;nOJIOmHM

Tenepb,

TfWO

B

MOMeHT

BKJIIOTfeHHH

ynpaBJIHIOID;HX

Bos,n;eHcTBlriI

<!lasoBhIe

KOOp,n;HHaThI

Hame1'O

OO'I>eK1'a

MaJIO

O1'JIHTfaIOTCH

01'

HX

SHaTfeHHH

Ha

BhIopaHHOH

KpyroBoH

op6HTe

r

o

,

a

BeJIHTfHHa

yn-

paBJIHIOID;eH

peaKTHllHOH

CHJIhI

f

opaBHIITeJIbHO

HeBeJIHKa,

TaK

TfTO B

npon;ecce

Bce1'O

ynpaBJIeHHH

TOTfKa

OCTaeTCH

B ,n;OCTaTOTfHO

MaJIOH

OK-

pecTHOCTH

yKasaHHoro

KpyroBoro

,n;BHmeHHH.

ECJIH

BBeCTH

OTKJIO-

HeHHH

Xl

=

Yl

- r

o

, X

2

=

Y2'

X

s

=

Ys

-

at,

X

4

=

Y4

- a

<!lasoBhIX

KOOp)l;HHaT

01'

HX

SHaTfeHHH

Ha

KpyroBoH

op6HTe

1),

TO ,n;JIH

HHX

HS

(3.18)

nOJIYTfaIOTCH

ypaBHeHHHl

Xl =

Q;2,

1

X2

=

11

(Xl,

X2,

Xs!

X4)

+

Ul!

(3.19)

Xs

=

X4,

X4

=

Y2

(Xh

X2,

xs,

X4)

+

1s

(Xl,

X2,

Xs!

Q;4)

.

U2,

r)l;e

V

X2

(X4

-+-

a.)

1

rl=-IXl+ro)2-(XI+rO)(X4+a)!

12=,-2

xl.<j?rO

~1s=xl+ro'

MbI

npe,n;nOJIa1'aeM,

TfTO

sa

Bce

BpeMH

,Il;BHmeHHH

Ha

paccMaTpH-

BaeMOM

HHTepBaJIe

[ta,

t/3]

KOOp)l;HHaThI

Xl

(t), X

g

(t), X

s

(t) H X

4

(t)

OCTaIOTCH ,n;OCTaTOTfHO

MaJIhIMH.

Tor,n;a

MOmHO

COCTaBHTb

ypaBHeHHH

JIHHeHHOrO

npHOmm<eHHH

B

OKpeCTHOCTH

KpHBOH

r

O

)l;JIH

CHCTeMhI

(3.18)

HJIH,

TfTO TO

iI<e

car-fOe,

B

OKpeCTHOCTH

Xl

=

0,

X

2

=

0,

X

s

=0,

X

4

= °)l;JIH

CHCTeMbI

(3.19).

3TH

ypaBHeHHH

HMeIOT

BH)l;

Xl =

X2f

X2

=

a21

x

l +

a24

x

4 +

Ul,

(3.20)

Xs

=

XII,

:V4

=

a42Q;2

+

~U2'

3

(

art

)

3-

2 ( Orl ) 2 ( 8r2 )

..

,n;eCb

a21

=

axJ.

0 =

~,

a211

=

ax;-

0 = roa,

a42

=

aX2

0 =

2a.

1 •

= -

-,

~

=

(Ys)o

=

-.

TIPH

aTOM

U<peAnOJIaraeTCH,

qTO

U = °

ro

roo

npH

Xl

=

0,

X

2

= 0, X

s

=

0,

X

4

-

0.

(CHMBOJI

(

)0

OSHaTfaeT,

qTO

BblpaiKeHHe,

CTOHm;ee

B

cK06KaX,

illlIqHCJIHeTCH

HpH

OTKJIOHeHHHX

Xl

=

0,

X

2

=

0,

X

s

=

0'-X

4

= 0.) .

3)l;eCb,

BnpOqeM,

CJIe)l;yeT

C)l;eJIaTb

em;e

O)l;HO

nOHCHeHHe.

)J;BHiI<e-

HHe

Xl

(t) = ... = X

4

(t)

=

OnpH

U = °

1f,eycmoU'lU60

no

Koop,n;HHa-

Te

X

s

(t).

31'0

OSHaTfaeT,

TfTO

npH

OTCyTCTBH:Fl

ynpaBJIHIOm;HX

CHJI

yKa-

1) BeJLlIqlIIThI

Xi

(t)

OITHChIBaIOT

OTRJIOnelllie TOql(H m (t)

01'

6JIlI3ROii H

neii

cDo60~non

Tl?q:KIJ

m.,

ADJiJmy~eiiclI

no

HpyroBoii

op6nTO

1'0,

§ ill .

11Pj;IMEPbl

JHIHEt1H

hrJ(

Y

flJ'JIBJ!Hi,MhIX

eIIe-rroM

3'1

saHHaH

KOOp)l;HHaTa

C

TeTfeHHeM

BpeMeHH

)l;onOJIbHO

OhICTpO

nospac-

TaeT

no

aOCOJIIOTHOH

neJUrTfHHe

H

'He

OCTaerCH,

BOOOm;e

rOBopH,

orpa-

IHI'IeHHOH

npH

t

-+

00,

l\ame

eCJIH

Bce'

HaTfaJIbHbre

OTKJIOHeHHH

Xi

(t

a

)

(i

=

1,

...

,4)

MaJIhI.

TI03TOMY

nOJIb30Ba1'bCH

ypaBHeHHHMlf

(3:20)

MbI

MomeM

JIlfmb

npu

TOM

YC,If,oeuu,

'ITO

)l;errCTBYIOm;He

CHJIbI

U

I

(t),

,u

2

(t)

o6ecne'/ueaIOTn

MaJIOCTb

O1'RJIOHeHHH

,Xi

(t)

B

TeTfeHJIe

Bcero

paCCMaTpHBaeMoro

OTpeSI,a

BpeMeHH

fta,

t{3],

JI

31'0

yCJIOBHe

)l;OJIrRHO

npOBepHTbCH

no

XO)l;Y

paOOTbI

C

ypaBHeHJIHMlf

(3.20).

B § 2

OTMeqaJIOCb,

qTO

O)l;JIH H TOT i-He OO'I>eHT

MomeT

HH01')l;a

JIMeTb

HeCHOJIbIW

<!lasOB.bIX

BeHTopOB

paSJIHqHOH

pa3MepHOCTJI.

,lJ;aHH.bIH

npHMep

,n;OCTaBJIHeT

HaM

HaH

pas

TaIWH

CJIyTfaH.

,lJ;eHcTBHTeJIbHO,

BMeCTO

<!lasoBo1'O

neHTopa

Y = {Yl'

Y2'

Ys,

Y4}

Monum

BSHTb

<!lasOB.bIH

BeHTop

MeHbIIIeH

pasMepHOCTJI,

a

HMeHHo,

npHHHTb

sa

<!lasOB.bIe

IWOp-

,n;HHaT.bI

BeJIHqUHDI

r (t), t (t), X

(t)

=

'ljJr

2

,

TaH

RaH

COBoHynHOCTb

HX

y,n;oBJIeTBOpHeT

BceM

yCJIOBHHM

onpe,n;eJIeHHH

2.1.

BeJIUqHHa

X

HB-

JIHeTCH

OOOOm;eHH.bIM

HMnyJIbCOM,

cooTBeTcTByIOm;HM

ROop,n;uHa'l'e

Ys

=

'Ij.J,

HOTopaH

6y,n;eT

Il;MKJIUTfeCHOH

npH

OTCyTCTBHU

peaHTuBHoH

CJIJI.bI.

YTfUT.bIBaH,

'fro

W<jJ

= 2

t,p

+

rip

= ;

:t

(r

2

'IjJ)

U

'IjJ

=

x/r2,

nOJIyTfaeM

JIS CUCTeM.bI (3.17)

,n;H<!l<!lepeHIl;Ua~bHbIe

ypaBHeHuH,

onM-

cblBaIOJIJ;Me

HSMeHeHue

<!las'oBbIX

HOop,n;HHa'l'

r,

t,

X DO

BpeMeHlI:

..

v X

2

m'

a

r---+_+

r

3

r

-

'- r

2

m

J

. m

(3.21)

X =

ra<jJ-.

• m

110JIa1'aH

neJIHTfUH.bI a

r

H

a<jJ

nOC'l'OHHHhIMU

BO

BCe

BpeMH

,n;BumeHHH,

T.

e.

CTfUTaH,

TfTO

BbIOPOC

MaCC.bI Tn

l

npOHSBo,n;HTCH

opHeHTHpOBaHHO

OTHOCHTeJIbHO

CHCTeM.bI

HOop,n;HHaT,

CBHsaHHOH

C

'l'OqHaMM

Tn

H

M,

sanHIIIeM

cHcTeMy

(3.21) B

HOpMaJIbHOH

<!lopMe

(2.1):

l

i =

Z2,

J

. v

z;

Z2 = -

-2-

+ 3 +aru,

(3.22)

zl zl

is

=

zla<jJu.

3,n;eCb

0003HaTfeHo:

Zl

= r,

Z2

=

t,

Zs =

X,

u = in/Tn.

3Ha

t

IeHHH

<!lasOB.bIX

HOOp,n;HHa'l'

Zl'

Z2'

Zs

Ha

Hpy1'oBOH

opouTe

pa,n;Hyca

r

6YJWT

Tenepb

cJIe,n;yrom;MMM:

o

Zl

= ro,

Z2

= 0,

Zs

=

Xo

=

Yvro.

OOOOm;eHH.bIH

HMnyJIbC

Xo

onperr,e.JIHe'l'CH

us

nepBoro

ypaDueHHH

CH-

CTeM.bI (3.21)

npu

7-

0

= 0,

7h

=

O.

33

32

(:.>

IS>

~~:J

l~·;

ITOCTAHOBRA

3AAA~II

OB

YITPABnEHIIII

[rJI.

1

§ I.]

ITOCTAHOBKA

3AAA~H

on

YITPABnEHIIH

IIocTylIaH

TaR

me,

RaK

JI B IIpeAIiI,n;yIIJ;eM

npIIMepe,

MO}KHO

COCTa-

(2)

BTOpOH

cnoco6

COCTaBJIeHIIH

ypaBHeHIIH

JIIIHeHHOro

npII6JIII-

BHTb

ypaBHeHHH

JIJIHeHHOro

npII6JIHmeHHH

B

OKpeCTHOCTH

KpHBOH

ro.

meHHH

3aKJIIOlJaeTCH

B CJIe,n;YIOIIJ;eM.

8TJI

ypaBHeHHH

,n;JIH

,n;aHHoro

npIIMepa

3anHCIilBaIOTCH

TalC

Xl

=

X2'

j

. v 2

YWo

.

X2

= -""""""3

xi

+

--3-X3

+

aru,

(3.23)

TO

X3

=

rOa<jlu.

3,n;eCb

Xl

=

Zl

- r

o

, x

2

= Z2 -

0,

x

3

=

Z3

-?Go

eCTb

OTKJIOHe-

HHH

cPa30HIilx

Koop,n;JIHaT

01'

IIX

3HalJeHJIH

Ha

KpyroBoH

Op6JITe

roo

ECJIII

BIilIIOJIHHIOTCH

yCJIOBIIH,

06eCnelJHBaIOIIJ;JIe

6JIJI30CTb

pac-

CMaTpIIBaeMoro

ynpaBJIJleMOrO

,n;BIImeHJIH

K,n;B'IImeHIIIO

IIO

KpyroBoH

TpaeKTopIIII

r

o

,

TO IICCJIe,n;OBaHHe

JIIIHeHHOH

CIICTeMIiI

(3.23)

,n;aeT

IIOJIe3HYIO

IIHcPopMaIJ;IIIO 0

,n;BIImeHIIII

B

CIIJIy

IIOJIHOH,

HeJIIIHeHHOH

CIICTeMIiI

,n;HcPcPepeHI~IIaJIbHIilX

ypaBHeHIIH

(3.22).

Bo

Bcex

paCCMOTpeHHIilX

IIpIIMepax

:,meMeHTIiI

MaTpI:ll:~

A

II

B

CYTb

nOCTOHHHIile

BeJIIIlJIIHIiI,

T.

e.

CIICTeMbl

JIIIHeHHOro

IIpII6JIIImeHIIH

HB-

JIHIOTCH

CIICTeMaMH

JIIIHeHHIilX

,n;IIcPcPepeHIwaJIbHbIX

ypaBHeHIIH

C

no-

CTOHHHbIMII IW<lcPcPIIIJ;IIeHTaMII.

O,n;HaKo

HeTpy,n;HO

YKa3aTb

cIITyaIJ;IIIO,

Kor,n;a

<lJIeMeHTIiI

MaTpIIIJ;

A

II

B 6y,n;YT cPYHKIJ;lIHMII

BpeMeHII.

TaK,

eCJIlI

B

IIpIIMepe

5°

B

KalJeCTBe

,n;B1I1HeHIIH, B

OKPeCTHOCTII

KOToporo

JIIIHeapII3yeTcH

CIICTeMa

(3.18),

B3Hl'b

,n;BIImeHIIe

He

IIO

OK-

pymHocTII,

a

ITO

<lJIJIIIncy,

TO

MaTpIIIJ;a

A

6y,n;eT

HMeTb

<lJIeMeHTbI,

ne-

plIo,n;IIlJeCKII

MeHHIOIIJ;lIeCH

co

BpeMeHeM.

ECJIII

me

ClJIITaTb,

'ITO

BIiI-

6poc

MaCCIiI IIpOII3BO,n;IITCH IIO

onpe,n;eJIeHHOH

IIpOrpaMMe

B

MeHHlO-

IIJ;IIXCH

HalIpaBJIeHHHX

OTHOCIITeJIbHO

CIICTeMIiI

Koop,n;IIHaT,

CBH3aHHOH

C

06'beKTOM,

TO a

r

II

a<jl

6y,n;yT

3a,n;aHHbIMII

cPYHKIJ;IIHMII

BpeMeHlI,

a

CJIe-

,n;OBaTeJIbHO,

II

KOMITOHeHTIiI MaTpJIIJ;:b:r B

6y,n;yT

TaKme

ITepeMeHHIilMII

BeJIIIlJIIHaMII

.

B

3aKJIIOlJeHIIe

3aMeTHM

CJIe,n;YIOIIJ;ee.

ECJIII

B

IIpOIJ;ecce

ylIpaBJIe-

HIIH

CJICTeMa

cOBepmaeT

,n;BJImeHHe

B

oKpeCTHocTII

HeKOToporo

IIO-

JIOmeHHH

paBHoBecIIH

IIJIJI

B6JI1I3II

H8KoToporo

3a,n;aHHoro

,n;BIImeHlIH,

TO

npJI

COCTaBJIeHIIH

ypaBHeHIIH

JIIIHeHHOro

IIpII6JIIImeHHH

y~06HO

nOJIb30BaTbCH

CJIe,n;YIOIIJ;IIMII ,n;ByMH

npHeMaMII.

(1)

B

BIilpameHIIHx

,n;JIH

KlIHeTHlJeCKOH

II

IIOTeHIJ;lIaJIbHOH

<lHeprIIH,

B

BIilpameHIIHx

,n;JIH

<lJIeMeHTapHoH

pa60TIiI

YlIpaBJIHIOIIJ;lIX

II

BHemHIIX

B03MYIIJ;aIOIIJ;IIx

CIIJI,

a

TaKme

B

BIilpameHIIHx

,n;JIH cPYHKIJ;IIH,

onIICIiI-

BaIOIIJ;lIX ,n;IICCIInaIJ;IIIO

II

rJIpocKOnIIlJeCKIIe

<lcPcPeKTIiI, BIilnIICIilBaIOTCH

JIlImb

lJJIeHIiI,

IIMeIOIIJ;IIe

HaIIHII3mHH

nopH,n;oK

IIO

COBOKylIHOCTII

ne-

peMeHHbIx

qi'

if.

i JI u.

3aTeM

IIO <lTIIM

BeJIIIlJIIHaM

B COOTBeTCTBHH C

06-

IIJ;IIMII

IIpaBIIJIaMII

MexaHIIKII

COCTaBJIHIOTCH

ypaBHeHIIH

,n;BIImeHIIH.

TaKlIM

nyTeM

6Ii1JIII COCTaBJIeHIiI

ypaBHeHIIH

IIepBOrO

IIp1I6JIIIrKeHIIH

(3.15)

B

IIpIIMepe

4°.

CualJaJIa

COCTaBJIHIOTCH"

IIOJIHIile

HeJIHHeHHIile

ypaBHeHIIH

,n;BIIme-

HHH

paccMaTpHBaeMoH

ynpaBJIHeMOH

CIICTeMIiI.

IIocJIe

<lTOrO cPYHKIJ;HII,

onpe,n;eJIHIOIIJ;IIe

<lTJI

ypaBHeHHH,

pa3JIaraIOTCH

B pH,n;Ii1

no

BeJIIIlJJIHaM

Aqf, Llqf,

Llqi

H

U,

onlICIilBaIOIIJ;JIM

OTKJIOHeHIIe

ynpaBJIHeMOrO

,n;BH-

meHIIH

qi

(t)

01'

3a,n;aHHoro

q~

(t). B

IIOJIyqeHHIilx

pH,n;ax

coxpaHHIOTCH

JIHmb

lJJIeHIil

He

BIilIDe

nepBoro

nopH,n;Ka

MaJIOCTII

no

<lTIIM

BeJIlIlJIIHaM.

RO<lcPcPIIIJ;HeiJ:TLl

nplI

Llqi'

Llq

il

Aq

i

II

U B

ypaBHeHIIHX

IIepBOrO

npJI-

6JIJImeHIIH

onpep;eJIHIOTCH

B

TaKOM

CJIylJae

npOII3Bo,n;HIilMJI

COOTBeT-

CTByIOIIJ;IIX cPYHKIJ;JIH

no

qi' iIi'

(Ii

JI

U,

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([15*],

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2

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q(k)

=

{q(~)}

(k = 1,... ,

s)

Ha-

aLIBalOTCH

.l/,UHeUHO

Heaa8UCUMbl.MU,

Kor)1.a

JIlO6aH

HX

JIHHeHHaH

KOM-

6HHan;HH

q =

llq(l)

+

...

+

lsq(S)

)1.aeT

BeKTOp

q, OTJIHTIHLIH OT

HyJIe-

Boro,

eCJIH

TOJIbKO

Cpe)1.H

'IHCeJI

lie

eCTb

OTJIH'IHLIe

OT

HyJIH.>"lIsBecT-

HO ([21 *],

CTp.

276),

'ITO

BeKTopLI

zk

(t)

(k

= 1,..., n)

JIHHeHHO

Heaa-

BHCHMLI

IlpH

Bcex

t

Ha

OTpe3Ka

[ta,

t~l

Tor)1.a

1'1

TOJIbKO

Tor)1.a,

Kor)1.a

OHH

HeaaBHCHMLI

no

KpaHHeH

Mepe

npH

O)1.HOM

aHa'IeHHH

t =

1:'

1'18

aToro

OTpeaKa.

C'IHTaH,

'ITO

Kam)1.LIH

Ha

BeKTopOB

z(f;)

(t)

HBJIHeTCH

BeKTopOM-CTOJI6n;OM,

COCTaBHM

MaTpHn;y

zP)

(t)

...

zi

n

)

(t)

Z (t) =

z~)

(t)

z~n)

(t)

IIOCKOJIbKY

npH

Bcex

paCCMaTpHBaeMLIX

SHa'IeHHHX

t

BeKTopLI

z<1e)

(t)

(k

=

1'00"

n)

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TO

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Z (t)

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1'1,

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t

1'13

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[t

oc

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t~]

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Z-l

(t).

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X

[t,

toJ

= Z

(t)

·Z-l

(to)'

(5.3)

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to)

=

{xi}

(t,

to)}

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HBJIHIOTCH

pemeHHHMH

ypaBHeHHH

(5.2),

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OHH

cyTb

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I\HH

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Z(kj

(t)

(k

= 1'00,' n).

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MaTpHD;a

X

It,

t

p

)

ija;JLI.BlleTY~

¢lI

H

{)Q.JIf,cHma.l£1:IH9!j,

MCl-rpHn;eH

CJlCreMbJ

(5.2).

38

~

<,

·~

..

,

:.

C'

, .

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' "

./

C

[rJI,

2

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JIIUIEJIlHbIX

CMCTEM

BaJI\Ho,

'ITO

npM

t =

to

OHa

06pam,aeTCH

B

ep;MHM'IHyIO

MaTpHIJ;y

E.

II03TOMY

qaCTH,oe

pemeHHe

z (t,

to,

ZO)

CHCTeMhI

(5.2),

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m,ee

yCJIOBHIO

z

(to)

=

ZO

=

{z~},

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paBeHCTBOM

z (t,

to,

ZO)

= X

[t,

to]

zoo

(5.4)

B

nop;p06HOH

3anMCM

paBeHcTBo

(5.4)

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BHP;

Xll(t;

to)

.

..

X

ln

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to)

'(Zl

:(t)]

=

[J~]~

X

nl

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to)

.

..

X

nn

(t,

to)

zn(t)

- n

h

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(t,

to)

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i=1

n

h

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to)

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ECJIM

B

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(5.2)

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aij

MaTpHIJ;hI

A

cyTI,

nOCTOHH-

HhIe

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TO

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aHaJIHTH'IeCKHH

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pemeHHH

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CJIep;OBaTeJIbHO,

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H

HBHhIH

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MaTpHIJ;hI

X It,

toJ.

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z(l»

(t)

B

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no

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8HJIepa

B

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z(l»

(t) =

pUc)

(t) i'le

t

,

rp;e

p(1,)

(t)

-

BeKTop-<pyHKIJ;MM,

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nOCTOHH-

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01'

t,

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xapaKTepHCTH'IeCKOrO

ypaB-

HeHHH

all - A

...

al

n

=0.

(5.5)

I

A-AEI_

ani

...

ann

- A

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pemeHHH

z(lc)

(t)

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pi

k

)

(t»

3aBH-

CMT 01' CBOHCTB

KopHeH

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ypaBHeHHH

(5.5). B

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eCJIH

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Ai

npOCThIe,

TO

BeKTOphI

p(k)

(t)

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nOCTOHHHhIMH

_._

."

_

..

" ·

·.t.

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:

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····j·!·a'±iiiti

§ 5]

<I:>OPMyJIA

lwIDn

39

297), [25*]

(CTp.

57-64).

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npep;nOJIaraTb,

'ITO

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B

06-

m;HX

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C

3THMH

BonpocaMIL

ECJIH

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aij

(t)

CYTb

<PYHKIJ;HH

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t, TO

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JIH

MOIKHO

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MeTOp;

nOCTpoeHHH

B

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<popMe

pemeHHH

z (t)

ypaBHeHHH

(5.2)

npH

n;>

2. B

3THX~i

CJIy'Ia-

HX,

KaK

npaBHJIO,

3JIeMeHThI

xij

(t,

to)

<pYHp;aMeHTaJIbHOH~MaTp.n

IJ;hI X [t,

to]

npHxop;HTCH

onpep;eJIHTb

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cnoc06aMH.

B

p;aJIbHeHmeM

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npep;nOJIaraeTCH,

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MaT-

pHIJ;a

X

[t,

to]

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6hITb

TaK

lIJIH

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HaHp;eHa.

II03TOMY

MaT-

pHIJ;y

X

[t,

to]

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C'IlITaTb

1I3BecTHoH

P;JIH

Bcex

Tex

3Ha'IeHlIH

t

H

to,

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B

paccyIKp;eHlIHx.

(IIplI

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nyTeM

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X

[t,

toJ.)

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X

(t)=

X (t,

to,

X

O

)

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(5.1),

KOTopoe

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pHeT

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yCJIOBHIO

X

(to)

= x

o

,

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<pOpMyJIOH

Ko-

mH

([12*],

CTp.

80-82):

t

X (t) = X [t,

to]

XO

+

~

X [t, 't'].

[B

('t')

U

('t')

+w

('t')]

d't'. (5.6)

t,

ihy

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MhI

3p;eCb

8bL8of)umb

He

6yp;eM,

a

JIlImb

np08epUM

ee

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PaBeHcTBo

(5.6)

npOBepHeTCH

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npa-

BOH

'IaCTH

ero

B

ypaBHeHHe

(5.1). B

caMOMp;eJIe

I),

j;

(t) = X

[t,

to]

X

O

+X [t,

t]

[B

(t) U(t) +w (t)] +

t

+

~

X

[t,

't']

[B

('t')

U

('t')

+w ('t')]d't'.

to

Ho

X

[t,

t]

= E H X

[t,

't] = A

(t)

X ft,

't],

TaK

KaK

CTOJI6IJ;hI

xU) (t, 't)

MaTplIIJ;hI

X [t, 't]

nplI

KaIKp;OM

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3Ha'IeHlIH

't

HBJIHIOTCHpemeHlIHMlI

ypaBHeHMH

(5.2).

II03TOMY

t

j;

(t) = A (t)

{X

[t,

to]

XO

+

~

X

[t,

't']

[B

('t')

u

('t')

+w ('t)]

d't'

} +

t

,

+B (t) u(t) +w (t)

=-

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

1)

3Aecb

BhIqHCJIJIIOTCJI

npoH3BoAHhIe

OT

MaTpm~.

HarrOMHJilM,qTO

nOA

rrpoH3-

BeKTOpaMH

p(le)

,

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H3

ypaBHeHHH

BOAHOii

OT

MaTpll~hI

MhI

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H8

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Bcex

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HX

rrpoH8BoAHhIMH.

,r:(JIJI

orrepa~HH

)wepepepeH~llpOBaHHJI

(A -

AleE)p(le)

= 0 (k = 1,... , n).

MaTpH~

COXpaHJIIOTCJI, C

He8HaQUTeJIbHhIMU

H8MeHeHHJIMH,

npaBHJIa·

AHepcPepeH-

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cPYHK~Hii

'([3*],

CTp.

100).

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Toro,

MhI

AHcPcPepeH-

t

~pyeM

HHTerpaJI

B

rrpaBoii.qaCTH

(5.6)

no

napaMeTpy

t.

MCIWMaJI

rrpOH3BOAHaJI

IIOCTpoeHHe

pemeHHH

Z (/')

(t)

B

06m,eM

CJIy'Iae

KpaTHhIX

KopHeH

CKJIaAhIBaeTCJI

H8

rrOAhIHTerpaJIbHOrO

BblpameHHJI

rrpH

1:'

= t H

H8

IIHTerpaJIa

OT

Ak

M

npaBHJIa

P;JIH

BhI'IHCJIeHHH

KOMnOHeHT

nOJIHHOMOB

p(k)

(t)

onH-

qacTHOii

rrpoH8BoAHOii

rro

t,

B8JITOii

OT

nOAbIHTerpaJIbHOrO

BhIpameHHJI

([23*],

CfJ:HhI,

HanpHMep,

B

KHHrax

I19*]

(CTp.

69-74),

[21*]

(CTp.

288-

T.

2,

CTp.

670).

Красовский Н.Н. Теория управления движением

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