Красовский Н.Н. Теория управления движением

',','

:

....

~:

;.'.

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::.

','

.':

..

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;:='w'hfai'Ii-J

c

ITPOBnEMA

ITPE~EnbnOrO

BMCTPO~EfiCTBHH

lrn.

7

240

COOTBeTcTByIOm;aH

tPYHR~RH

(23.12)

PT

=

min

sup

r [B'

(-r)

S [-r, ta;]l]

npR

l'xa;

=-1

(27.15)

I "

eCTb, Boo6m;e rOBOpH, .'IRmb Hey6blBaIOm;aH

tPYHR~RH

T.

MbI

He

6y~eM

aa,n;epmRBaTbCH Ha aHanRae yCJIOBRii, 06ecnelIRBaIO-

m;RX

HenpepbIBHOCTb

tPYHR~RR

PT.

OTMeTRM

JIRmb,

1JTO

3Ta

Henpe-

PbIBHOCTb

npOBepJIeTCH 6ea

oc06oro

Tpy,n;a

npR

ycnoBRHX W (t) = 0,

xp,

= °

,n;JIH

mRpORoro

Rpyra

BeJIR1JRH

x [ul, HanpRMep, B

CJIY-

1Jae,

OTBelIaIOm;eM

OrpaHR1JeHRIO

Ha 3HeprRIO ynpaBJIeHRH

RJIR

Ha

MaRcRMyM ynpaBnHIOm;eii

CRJIbI,

eCJIR

CRCTeMa

(26.1)

BnOJIHe

ynpaB-

nHeMa.

CBoiicTBa

cTporoii

MOHOTOHHOCTR

R HenpepbIBHOCTR

tPYHR~HR

PT

ORaabIBaIOTCH

BamHbIMH ,n;JIHpemeHHH

aa,n;a1JH

0 rtpe,n;enbHoM 6bICTpO-

,n;eiicTBRR

no

cne,n;YIOm;eii npH1JRHe.

IIycTb

CYI~ecTByeT

XOTH

6bI

O,n;HO

1JRCJIO

T

1

,

,n;nH

ROToporo

cpe,n;R

,n;onycRaeMbIX

ynpaBneHRii

Haii,n;eTcH

ynpaBJIeHRe

u(l) (t),

;rrepeBo,n;Hm;ee

cRcTeMy

(26.1)

Ra

nOJIOmeHRH

xa;

1

B nOJIOmeHRe

xp,

aa BpeMH 4

) -

ta;

= T1

npR

yCJIOBRR

P*

[U(I)]

=

.,.

= x

[u(l)]

-<

/1.

Tor,n;a

6y,n;eT

cnpaBe,n;JIRBO

HepaBeHCTBO

PT,

).

11/1.

Ec-

1'41;

JIR

tPYHR~RH

PT

HBJIHeTCH

HenpepbIBHoii

R

eCJIR

cnpaBe,n;JIRBO

COOTHO-

"'Ill

IIIeHRe

1)

lim

PT

= °

npR

T

-+0,

TO

MomHO

YTBepm,n;aTb,

1JTO

cYUfecm-

eyem

HaRMeHbmee nOJIOmHTeJIbHOe

1JRCJIO

TO,

npR

ROTOpOM

BbIIIOJIHHeTC.H

paBeHCTBO

PT.

=

11/1.

ECJIH R TOMy

me

tPYHR~RH

PT

CTporo

MOHOTOHHa,

'"

ro

TaRoe

1JRCJIO

TO

6y,n;eT

e,n;RHCTBeHHbIM,

T.

e.

npR

c,n;eJIaHHbIX

BbIme

....

i:

npe,n;nOJIOmeHRHX, 06ecnelIRBaIOm;RX

cTporyIO

MOHOTOHHOCTb

Henpe-

PbIBHOii

tPYHR~RR

PT, ypaBHeHHe

(27.5)

RMeeT

e,n;RHcTBeHHbIii nOJIOmR-

~:

..

TeJIbHbIR RopeHb

T =

TO,

ROTOpbIii R onpe,n;eJIHeT

OnTRMaJIbHOe

BpeMH

nepexo,n;Horo

npo~ecca.

RpoMe

Toro,

onRpaHCb Ha TeopeMbI 0 HeHBHbIX

tPYHR~RHX,

HeTpy,n;HO

npOBepRTb,

1JTO

npR

yCJIOBRH

cTporoii

MOHOTOH-

HOCTR

nepeMeHHoii

PT

BeJIR1JRHa

TO

- RopeHb ypaBHeHRH

(27.5)

-

aa-

BRCRT

Henpepbl67-£O

OT

Roop,n;RHaT

xf

H

OT

rrapaMeTpOB

CRCTeMbI.

Oc06eHHo nOJIeaHbI YRaaaHHbIe cBoiicTBa

npR

pemeHRR

aa,n;a1JR

26.1

B Tex

CJIy1JaHX,

Ror,n;a

aa,n;a1Ja

06

OnTRMaJIbHOM

ynpaBJIeHRR

npR

yCJIOBHH

MRHHMyMa cooTBeTcTBYIOm;eii

RHTeHCRBHOCTR

He

pemaeTCH

B aaMRHyToii tPopMe.

IIycTb,

HanpRMep,

Tpe6yeTcH HaRTR

OnTRMaJIbHOe

no

6bICTpo,u;eii-

CTBRIO

ynpaBJIeHRe

npR

OrpaHR1JeHRH Ha

RHTeHCRBHOCTb

(21.1)

BR,n;a

x[u]=supmax(luj(-r)I>-<I-I-

(j=1,

...

,r,1').ta;).

(27.16)

" j

1)

8TO

COOTHOmelme

OIIJIT:b-TaKlI

B:bIITOJIHJIeTCJI C OqeBIl,l:J;HOCT:bIO B

CJIyqae

OrpaHrrqeHllJI

Ha

aHeprllIO

lIJIlI-Ha

MaKclIMyM

ynpaBJIJIIOill;ero

B03p;eHCTUliJI

II

MomeT

He

llMeT:b

MeCTa

nplI

orpalmqelmll

Ha

lilMIIYJI:bC

ynpaBJIJIIO~eH

CllJI:bI. B IIOCJIeJJ;-

HeM

CJIyqae

MomeT

OKa3aT:bCJl

TO

=

O.

····,

,

>'

c

§ 27]

PEmEHHE

3A~A

lIH

0

BbICTPO~EnCTBHH

241

YpaBHeHRe

(27.5)

,n;JIH

onpe,n;eJIeHRH

TO

B

COOTBeTCTBRR

C peaYJIbTa-

TaMR

Ra

§ 21 npRHRMaeT

a,n;eCb

BR,n;

(CM.

(21.27» •

....

ta;+T

r n

.....

1

tJi

PT=

~

C2}/.2}

l~(T)hij(ta;+T,-r)I)d-r=

(27.17)

..

r:-

It"'

tIX.

)=11=1

r,n;e

hij

(tp"

1')

-

3JIeMeH'FbI

RMnyJIbCHOii nepexo,n;Hoii

MaTpR~bI

H

[tp,,1']

=

S'

(,;,

tp,]B

(r),

a

Z~

(T) -

1JRCna,

,n;aIOm;Re

pemeHRe

aa,n;a1JR

ta;+T

r n

min

[

~

(~/

~

likij

(ta;

+

T,

-r)

/)

dT ] =

PT

(27.18)

tIX.

j=1

i=1

n

npR

yCJIOBRR

~

liei (T) = 1.

.

i=1

RaR

HaM

yme

RaBeCTHO,

aa,n;a1Ja

(27.18)

B 06m;eM

cJIy1Jae

He

pe-

maeTCH, R

COmaJIeHHEO,

B aaMRHyToR tPopMe. II03TOMY, onpe,n;eJIHH

TO

Ra

ypaBHeHRH

(27.18)

RaRRM-HR6y,n;b

1JRCJIeHHbIM

cnoc06oM,

MbI BbI-

Hym,n;eHbI Ha

Ram,n;OM

mare

lIRcneHHO (HanpRMep,

MeTo,n;OM

HaRCRO-

peiimero

cnycRa,

CM.

no

aTOMy

nOBO,n;y

§

24)

Haxo,n;HTb

HOBbIe

aHa1Je-

HRH

BeJIR1JRH

zf

(T). ECJIH yCTalIOBJIeHO,

1JTO

tPYHR~HH

PT

HenpepbIB-

Ha R

06JIa,n;aeT

CBoiicTBOM

cTporoii

MOHOTOHHOCTR,

TO

,n;nH

OTbICRRHRH

TO

MomHO

npe,n;JIOmHTb

CJIe,n;YIOm;yIO

cxeMy

C1JeTa,

peaJIRayIOm;yIO

npocTeiimRii

npo~ecc

nOCJIe,n;OBaTeJIbHOrO

npR6JIRmeHRH.

BbI6RpaH

Ra

paayMHbIx

co06pameHRii

HeROTopoe

1JRCJIO

T

1

,

onpe,n;eJIRM MRHR-

MaJIbHYIO

tPYHRIJ;RIO

h~,

(1');

eCJIR

ORameTCH, lITO P

[h~,

('r)]

=

PT.

>

11/1,

TO

BO'lbMeM T

2

,

paBHoe T

1

/2, a

eCJIR

BbIIIOJIHHeTCH HepaBeHCTBO

npOTRBOnOJIOmHOr9

·CMbICJIa

PT,

<

11/1,

TO

CJIe,n;yeT

nonomRTb

T

2

=.

=

2T

l'

IhaR,

UYCTb T2

onpe,n;eJIRJIOCb.

ECJIH

,n;JIH

HalIaJIbHOrO

npn-

6JIHmeHRH T

1

6bIJIO

PT,

>

11/1

R

CHOBa

ORaaaJIOCb

PT,

>

11/1,

TO

no-

JIaraeM onHTb T

3

= T

2

/2, R TaR

nocTynaeM

,n;o

Tex

nop,

nORa

He

npR-

,n;eM

R HepaBeHcTBy

PT.

)

<

11/1.

TeM

caMbIM

RCROMoe

lIRCJIO

TO

ORaabI-

BaeTCH

aaXBa1JeHHhIM

B BRJIRy T

j

<

TO

< T

j

-

1

.

,IJ;aJIee

nOHHTHbIM

06paaoM

COCTaBJIHeM

nOCJIe,n;OBaTeJIbHOCTb

cmRMaIOm;HXCH OTpeaROB,

co,n;epmam;RX

TO.

ECJIR

me

6blJI0

PT,

<

11/1

R

CHOBa

nOJIyTIRJIR

PT,

<

.

<

11/1,

TO

B:bI6RpaeM T

3

=

2T

2

,

R TaR nocTynaeM,n;o Tex

nop,

nORa

He

•

nOJIyTIRTCH

PTj

>11/1. Tor,n;a

TO

CHOBa

nona,n;aeT B BRJIRy T

j

-

1

<

TO

< T

j

R

T.

,n;.

YRaaaHHbIii

npo~ecc

npo,n;omRRM,n;o Tex

nop,

nORa

He

6y,n;eT

BbInOJIHeHO

C aa,n;aHHoii

T01JHOCTbIO

paBeHCTBO

P

[h~

('t)] =

. m

=

11/1.

B RTore noctpoRTCH

nOCJIe,n;OBaTeJIbHOCTb

TIRCeJI

T T

2

,

••.

1

,

l.

. .. , T

m

,···,

CXO,n;Hm;aHCH

K

TO.

8Ta

CXO,n;RMOCTb

,n;ORaabIBaeTCH

6ea

~~~-~~"'.""'~-~-'~--":"

.....

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

.....

.:

_.

..:

.....

_,_

,.:.::....:_L.:.:~._'

_'~'

~,~,'_.~,."

",

,',',-.::,'

':,'

....

..:...:~~,~~~

'

.....

:

::.'.:.,

,.:.,

::::".

::.

'",

',"

..

,'.;';".

"::':~::.:

c:

l'

,

,,=-,~

>

242

llPOBJIEMA

IIPE,LI)!lJIbHOrO

BbICTPO,L(EJ./lCTBlIlI

(rJl,

7

TpY,ll,a,

eCJUI

TOJIbKO

Sa,ll,atIa,

,ll,eiiCTBIfTeJIbHO,

If

MeeT

perneHIfe,

T.

e.

eCJUI

lim

PT

>

1/fJo

npll

l'

.-.

00.

llocJle

Toro

KaK

T°

onpe,ll,eJleHO,

caMo

OnTIfMaJlbHOe

no

6bICTpO,ll,eiicTBluO

ynpaBJlemm

B

cJly'Iae

(27.16)

sa-

.

lllIrneTCH

COrJlaCHO

(21.32)

tPopMyJloii

u?(t) .

fJosgnh~o,i

(t) (t

a

<t<t

a

+To,·i=1,oo.,r).

(27.19)

llOHHTHO,

'ITO

,ll,JlH

HaXOm,ll,eHIIH

HymHoro

HaM

ROpHH

ypaBHeHIIH

(27.5),

nOMIIMO

onlIcaHHoii

HalInpocTeiirneii

cxeMbI

C'IeTa,

MoryT

6bI~1>

nplIMeHeHbI

II

,ll,pyrlIe

BbI'IIICJlIITeJlbHbIe

npo:u;e,ll,ypbI,

9tPtPeRTlIBHo

pernaIDIIJ;We

clTY

Sa,ll,atIY.

O,ll,Ha

lIS

HIIX

TaROBa.

,l];onycTIIM,

HanplI-

Mep,

'ITO

Ha

j-M

mare

npo:u;ecca

nOCJle,ll,OBaTeJlbHbIX

nplI6J1lIme-

HlIii

rrplI

l'

= T

j

peaJIIISOBaJIOCb

HepaBeHCTBO

P

[h~.

,

(1i),

t

a

<

11

<t

a

+

+ Til =

PT'

<

1/

fJo·

ECJlII

cJle;rfYIDIIJ;ee

rrplI6J1lImeHlIe

T

j

+1 rro,ll,-

J

6lIpaTb

lIS

YCJJOBIIH

p[h~.(1i),

ta<11<ta+Tj+tl=1/!Jo,ToTj+b

6Y,ll,Y'I1I

60JIbrne

T

i

,

He

rrpeBJsoii,ll,eT,

O,ll,HaIW, IICROMOH

BeJJlI'IlIHbI

T°,

rrOCROJIbRy

P[h~'I(1i),

t

a

<11<

ta+Tj+tl

=PT'I<P[h~,(11),

J+ J+

,

1

4!1!

t

a

<1i

< t

a

+ T

j

+11

=

[t'

~j

~I

MMeH

,ll,JJR:

HaH,ll,eHHOrO

T

jn

MIIHIfMaJJbHYID

tPYHR:U;IIID

h~.

(1i),

-

J+1

i

..

rrO,ll,6epeM

1'j+2

CHOBa

lIS

ypaBHeHlIfI

p

[h~j+1

(1i),

t

a

<

1i

<t

a

+Tj+

2

J =

=

1/!Jo

II

T.

,ll,.

TaRIIM

06pasoM,

Ha'IIIHafl

C

l'

= Tj,

rrOJIy'IaeTCH

11:

1

rrOCJJe,ll,OBaTeJlbHOCTb

T

i

,

T

j

+

1

,

.•.

, T

m

, .

..

,

MOl-Wm01l1lo

cxo,ll,HIIJ;aH-

CH R

T°. 'Y,ll,06HO

TaRme

pernaTb

ypaBHeHlIe

(27.5)' MeTO,ll,OM XOP,ll,.

ROHe'IHO,

nplI

perneHlI1I

ROHRpeTHbIX

saj];a'I

0

rrpe,ll,eJlbHOM

6bICT-

pO,ll,eiiCTBHII

MoryT

BCTpeTIITbCH

nplIMepbI,

ROr,ll,a

Ha,ll,06HOCTb

B

'IIIC-

JleHHOM

perneHlI1I

sa,ll,a'IH

OTrra,ll,aeT.

Ho,

R

COmaJleHIIID,

TaRlIe

cJlyqalI

HBJlHIDTCH

IICRJlID'IIITeJlbHbIMII

H,

RaR

rrpaBIIJlO,

rrpHXO,ll,IITCH

pernaTb

sa,ll,a'IY

'IHCJleHHO.

3aMeTIIM,

HaROHe:u;,

'ITO

BeCb

C'IeT

MomHo

BecTII,

'"

I

,orrlIpaHCb

He

Ha

MIIHIIMaJlbHYID

tPYHK:U;IIID

h~

('r),

HO

IIcnOJlbSYH

9RCT-

peMaJlbHyIO

tPYHK:U;IIIO h

T

(1:")

II

pernaH,

CJle,ll,OBaTeJlbHO,

He

3a,ll,a'IY

(17.4)

HJIII

(17.10), (17.11), HO

sa,ll,a'IY

(17.5)

IIJlII

(17.12).

llplIBe,ll,eM

Terrepb

,ll,Ba

npOCTblX

IIJlJlIDCTplIpyIDIIJ;HX

rrplIMepa.

IIpB~Iep

27.2.

II

yCTb

ABliJJIWnHe

MaTepHaJIbHO:H

TOqRH (CM.

II.

1

0

§

3)

OIIHCbI-

BaeTCJI

AHcPcPepenqHaJIbHhIMH

ypaBneHHJIMH

:t

- x }

.1 -

2'

(27.20)

X

2

=

-Xl

+

U,

ROTopbIe

IIOJIyqaIOTCJI

H3

Cl!:CTeMbI (3.5)

rrpH

eg = 1,

'V

= 0, m = 1, a (t)

==c

1.

Tpe6yeTcR

rrepeBecTH

cHcTeMy

(27.20)

lI3

COCTOJIHHJI X

(ta)=

X

(0)

=

{2,'1}

B

rrOJIOlReHHe x (til)

=x

(T)

:=

{O,

O}

3a

nallMeHbmee

B03MOlRHOe

BpeMH

TO

rrpH

yc-

JlOB~IJI,

'lTO

I u (t)1 <1.

EYAeM

peman

STy

3a,llaqy

MeTO,llOM rrOCJTeAOBaTem.-

lJhIX rrpH6JTJ1iHenHR,

orrupaRCb

B

rrpoqecce

cqeTa

na

rrpaBHJJO 17.4.

CHcTeMa,

C':--

(;;.

§

21)

PElItEIlIIl!:

3A,tl;A

ttIf

0 Sb1CTl'O):(E1tCTBIfII

243

coupRmeHHaJI

RcnCTeMe

(27.20),

HMeeT

BlIA

$1

=

-$2'

8

2

r:::::=

$1'

H,

CJIeAOBaTeJIbHO,

cPYHAaMeHTaJIbHaH

MaTpnqa

S ['t',

tocl

llTO:H rrOCJIeAHe:H CHCTe-

MbI

TaROBa:

COS

(ta

- 't') -

sin

(t

a

-

't'))

S ['t', t J =

(

.

at

sin

(ta

- 't')

cos

(ta

- 't')

TaR

RaR

B (27.20)

B('t')

=b

=

[~J,

TO,

0603HaqaR

$ (ta) = k H

yqHTbIBaJI,

qTO

t = 0,

rrpHXOAHM

R

BhIBOAY:

rrpH

oc

RalRAOM

tf3

= T

3aAaqa

(17.12),

BbITeRaIDIqaR

H3

npaBHJIa

17.4,

rrpeBpaIqaeTcH

3AeCb B

3aAaqy:

Ha:HTH

a+ k

aJ

'V

0

__

1

max

[k

l

x

1

2x2

=

T-

PT

(27.21)

k

UpH

yCJIOBHH

T

P

[B'

('t') s ('r)J = P [b'S ['t',

tal

kJ= j I

kd(I)

('t') +k

2

f(2)

('t') I

dT

=

1,

(27.22)

°

rAe

/1)('t')

=

-sin

't',

P)('t')

=

cos

't'.

(27.23)

.D:JIH

narJIJIAHOCTH

6Y,lleM

pemaTb

3aAaqy

(27.21), (27.22)

AJIa

Ra}UAoro

cP

HR

-

cHpoBaHHoro

T

rpacPnqeCRH.

tIHCJIa

k~

(T),

pemaIOIque

STY

3aAallY,

OIIpeAeJIII-

IOTCJI

TorAa

RaR

ROOPAHHaThI

Ha

IJJIOCROCTH

{kl'

k

2

}

TOqRH

I,acaHHR

rrpRMO:H

klX~

+ k

2

X:

=

'V~

H RpHBO:H (27.22),

ypaBHeHHe

ROTOpOH,

yqHThIBaJI

(27.23), MOlRHO

3aUHcaTb

B

60JIee

npocToM

BRAe:

T-e

(lei

+

Ie;)

II,.

~

I

sin

'I't

I

cl'l't

=

1,

(27.24)

-E

rAe

'I't=

't'

-

e,

tg

e = /k

•

k

2

l

BHA

RpHBbIX

(27.24) H

cooTBeTcTByIDIqHe

IIpJIMbIe

UpH

pa3JIH'lHhIX

3Haqe-

HHRX

Tj,

BCTpeqaIDIqHxcR

no

XOAY

CqeTa,

R306palReHhI

na

pHC.

27.2.

BbI6Hpall

BRaqeCTBe

HaqaJIbHOrO

npH6JIHlRemIJI

T

I

= 2It,

oUpeAeJIlleM

HCROMYIO

BeJIHqHHY

TO

no

cxeMe,

nOAo6HO:H

OAHO:H

H3

onHcaHHhIx

BhIme

cxeM

no-

CJIe,llOBaTeJIbHhIX npH6JIlIJI:\eHH:H. B

HToreHaxoAHM

IIocJIeAOBaTeJIbHOCTH

T = 2It,

T,}.

= It, T

2

=3,5,

T4 = 3,75 H

'V~,=

0,56,

'V~2

= 1,13,

'V~3

= 1,07,

'V~,

=

1

1,05,

ECJIR

Hac

yCTpaHBaeT

nocJIeAnee

npH6JIHlRenHe

R

pemeHHID

3aAaqH,

TO

Momno

IIOJIOlRHTb

TO

= T

4

= 3,75.

ROOp,llHHaThI

TOqRH

uacaHHII

k~

=

k~

(3,75),

244

..

'.

';'.'

...•.

,"

..

-.'"

'-'0°.,',

..

:'~::-

.:',

..•.•

::

',:.:'::

•

.;..

C

"""

C,""'"

',,:.;

~:

:':

:.

~

-.:....:_.v'

····,'·.':;ti~:-:'··

..

h

[rJI. 7

rtpotnBMA hPBAEnbHOrO tMCTPoABftcTBHa

k~

=

k~

(3,75)

OKaahIBaIOTca

rrplI

aTOM

TaKlIM1I:

k~

= 0,45,

k;

= 0,16. B

TaKOM

cJlY'Iae,

rrplIIDIMaa

h

r

,

(1.')

=

k~

(1'4)

/1)

(1.')

+

k~

(1'4)

jC2)(1.')

aa

aKCTpeMaJlb-

HYIO

mYHK~

h

TO

(1.')

,

Haxop;lIM

rrp1I6JIWKeHHOe

OrrTlIMaJIbHOe

rro

6bICTpop;eiicTBlIIO

k

2

\

k,xt+k2xt

=v!

k,

Z".zg,lLO

.Zj

/1

0

=/

I

o

l/=-

PRC.

27.3.

yrrpaBJleHlIe

U

(t) = 1,05 sgn

hTo

(t) = 1,05 sgn [0,45

sint

- 0,16 cos

t].

flepexop;Hbm

rrpo:u;ecc

B

ClICTeMe

(27.20)

rrpR

u =

uO

(t),

CMOp;eJlRpOBaHHbIii

Ha

ABM,

rrpep;CTaBJIeH

Ha

pRC.

27.3.

PlIe. 27.2.

§ 27)

PEIllEHllE

3AM

TIll

0

BbICTPOAE1iCTBHll

245

IIpB~[ep

27.3.

PaccMoTplIM

aa,u;a'Iy

0

H3.RCKOpenmeM

rrpRBe,u;eIDIlI

rlIpOI\OM-

rraca B

aa,u;aHHoe

COCToaHlIe

rrplI

YCJlOBlIlI

OrpaIDI'IeHHOCTlI

BeJllI'IRHbI

yrrpaBJlaIO-

II\en

CRJlbI.

PaHee, B § 24,

6bIJla

pemeHa

aa,u;a'Ia

24.1 0

rrpRBe,u;eHRR

aToro

'He

rRpOCKOrrR'IeCKOrO

KOMIJaCa

B

MepR,u;RaH

MHIDIMaJlbHOll

ClIJlOn

aa

BbI6paHHoe

aapaHee

BpeMa

1'=1800

eel'.

TaM

6bIJlO

Han,u;eHO,

B

'IaCTHOCTR,

'ITO

max Iu

(1.')

1=

'I'

=

0,47.10-

3

•

Terrepb

MbI

orrpe,u;eJlRM

MIlIDIMaJlbHOe

BpeMa,

aa

KOTopoe

MomHO

rrepeBecTR

rRpOKOMIJaC

B

aa,u;aHHoe

COCToaHHe

rrplI

YCJlOBRlI

sup

I U

(1.')

I

~

/-L,

'I'

rrplI'leM

,u;Jla

cpaBHeHRa

C

rrpRMepoM

Ra

RHRrli

[1936]

(CTp.

21)

RHTepecHo

rro-

JlOmllTb

/-L

=

0,64.10-

3

•

BbIrrRmeM

eII\e

paa

ypaBHeH:lUT,

cooTBeTcTBYIOII\ne

rrpe:u;ecclIoHHoMy

,u;BII1He-

HRIO

rlipOK

OMIJaca:

:t = 1,53921 x + 1,53921

.062

x

-+

u

1 41,1368 2 41,1368 ' 3 ,

X2

= 41,1368.10-0

Xl,

:1:

3

=

-1,5

·10-

3

X2

-

1,5.10-

3

X3.

flyCTb

aa,u;aHO

Ha'IaJIbHOe

COCToaHlie

xIX.

= {0,3; 0,004; 0,004},

tIX.

= °n

orpa~

HlI'leIDIe

I u (t) I

~

0,6401.10-

3

•

Tpe6yeTca

rrepeBeCTlI

rnpoRoMnac B

rrOJIome-

i3

IDle

x =

{O;

0;

O}

aa

HalIMeHbmee

BoaMomHoe

BpeMa

T°. )l;Jla

,u;aHHoro

npHMepa

ypaBBeIDIe

(27.17)

rrpnMeT

BH,u;

T

PT

=

~

Il~h(l)

(T,

or)

+l~h(2)

(T,

'1:)

+

l~h(3)

(T,

'1:)

I

d1.'

=

10

3

/0,6401 = 1562,5,

° , (27.25)

rAe

h(i)

(1',

'1:)

(i

= 1,2,3)

orrpe,u;eJleBbI

cP0pMYJlaMH,

ROTopbIe

CJIe,u;yIOT

na

WOPMYJI

(24.12) rrplI t =

l'

-

'1::

h(l)

(T,

'1:)

=

lXi~

eAt

(T-'l')

+

~i~)

eIX.

(T-'l')

cos

~

(T

-

or)

-

~i~

eIX.

(T-'l')

sin

~

(T

-

'1:),

h(2)

(T,

'1:)

=

IX~~

eAt

(T-'l')

+

~~~)

eIX.

(T-'l')

cos

~

(T

-

'1:)

-

~~i)

eIX.

(T-'l')

sin

~

(T

-

'1:),

U

M

3

l

(T,

'1:)

=

IX~~)

eAt

(T-'l')

-+

~~~

e

(T-'l')cos

~

(T

-

'1:)

-~

~i)

eIX.

(T-'l')

sin

~

(T

-

1.').

'YpaBBeIDIe

(27.25)

pemaJlOCb

Ba

8I.J,BM

cnoco6oM

nOCJle,u;OBaTeJlbHbIX

rrpR-

6J1nmeIDIn,

rrpn

aTOM

Ha

Ram,u;OM

mare

pemeHRe

COOTBeTCTBYIOII\en

aa,u;a'IR

(27.18)

onpe,u;eJIaJlOCb

MeTo,u;OM

HaHcRopenmero cnycRa. B peayJIbTaTe

rrpH

1'1

= 1800,

1'2

= 1300,

1'3

= 1250,

1'4

= 1240

nOJly'IHJIOCb,

'ITO

PT.

= 2147,5,

PT,

= 1651,

PT.

= 1583,

PT,

= 1562.

Tor,u;a

MomBO

rrOJlOmHTb

T°

= 1'4' flpR

l'

=

1'4

OKaaa-

JlOCb,

'ITO

l~

(1'4) = - 1,050,

l~

(1'4)

=-198,6,

l~

(1'4)

=

-498,5.

CJle,u;OBaTeJIbBO,

h~o

(1.')

=

l~

(TO)

h(l)

(TO,

'1:)

-+

l~

(TO)

h(2)

(TO,

t') +

l~

(TO)

h(3) (TO,

'1:)

==

= _ 20,49. eAt (TO-'l')

-+

19,48.eIX.

(To-'l')

cos

~

(TO

-

'1:)

+

-+-

4,11.eIX. (TO-'l')

sin

~

(TO

-

'1:),

TO

= 1240.

".;

.:..:

,'

;;

::;:.<.,

,.:::

..

:.:<

..

':

:

".

,

...

.

....

~~'-'~~

(~.

c.:

240

rrpOBnEMA

rrPE~~nhHOrO

BNCTPOAE~CT,BHa

[rJI.'7'

§ 27]

PEillEHIiIE

3AP;A

qH

0

BblC'f

POP;ErtCTBI111

247

.j

3Ta

tPYHIUJ,Hll

B

COOTBeTCTBHH

C (27.19) onpep;eJIlleT

HCll:OMOe

OnTllMaJlbHOe

yn-

paBJIeHHe: .

(

...,-

0,6401.10-

8

np.li

°

~

t <725,

u (t) =

~

+0,6401.10-3 np.li

725

~

t <

1066,

l-

0,6401.10-

8

npn

1066

~

t

~

1240.

IIpoll,ecc

HaacKope.umero

nepeBo):J;a

rRpoKoMnaca

B Mep.lip;.IiaH,

np

oc

'IlIT

aHHlilli

Ha

8l\BM, npeACTaBJIeH rpatPHKaMR

Xl

(t),

X

2

(t), X

3

(t) Ha

pRC.

27.4.

x

IbaR,

B

cJIyqaax,

ROnt;a

ypaBHeHHe

(27.5)

HMeeT

ep;HHCTBeHHhIH

p;eH-

CTBHTeJIbHblH

RopeHb,

3a-

p;aqa

0

rrpep;eJIbHOM

OhI-

cTpop;eHcTBHH

Bcerp;a

MO-

iKeT

OhITb

pemeHa

p;OCTa-

TOqHO

a~~eRTHBHO

oe3

OOJIbmHX

rrOMex

B'

qHC~

JIeHHOH

rrpOIt;ep;ype.

llpH

OTCyTCTBHH

iKe

MOHOTOH-

7

HOCTH

<PYHRIt;HH

PT

B03-:-

HHRaIOT

p;OrrOJIHHTeJIbHhIe

"i

uO(t)~tJ,540/Io-J

I i

oI

300

\1in

I

/.g~fJ

n~'

Tpyp;HOCTH,

CBH3aHHbIe

C

I

-0,01.

orrpep;eJIeHHeM

HaHMeHb-

"",,

U°(t)=

-0,

5401'10-

1

mero

rrOJIOiKHTeJIbHOrO

u0(t)~-o,fj40/

·/o-J

RopHa

ypaBHeHHH

(27.5)"

'"

,.'1

8T~

TPYP;HOCTH

MoryT

cy-

,

PRC. 27.4.

I

..

,'

IIt;eCTBeHHO

YBeJIHqHTb

OO'beM

BhIqHCJIHTeJIbHOH

paOOThI

H

MoryT

TaRiKe

rroTpeooBaTb

01'

BhI'IHCJIHTeJIH

Orrpep;8JIeH-

Horo

HCRyccTBa.

OTMeTHM

Terrepb

eIIt;e

CJIep;YIOIIt;ee.

OCHoBHoe

BHHMaHHe

B

aTOM

rraparpa~e

MhI

yp;eJIHJIH

oocyiKp;eHHIO

rrpOIt;ep;yphI

OThICRaHHH

onTH-

MaJIbHOrO

no

OhICTpop;eHCTBHIO

ynpaBJIeHHH.

llpH

aTOM

He

aRIt;eHTH-

pOBaJIOCb

BHHMaHHe

Ha

Bonpocax

0

cyIIt;eCTBOBaHHH

pemeHHH

H

Ha

p;pyrHx

nop;OOHbIX

npOOJIeMax.

Op;HaRO

H

TaRHe

BonpOChI

pa3pema-

IOTCH

3p;eCb

oe3

Tpyp;a,

eCJIH

orrHTb

,He

BOCnOJIb30BaTbCH

pe3yJIbTaTa-

MH

H3

npep;hIp;yIIt;HX

naparpa~OB.

TaRHM

rryTeM

MOiKHO

nC)JIyqHTb

OOJIbmOe

qHCJIO

pa3JIHqHblX

TeopeM

0

cyIIt;eCTBOBaHHH

pemeHHH

3a-

p;aqH

npep;eJIbHOrO

OhICTpop;eHCTBHH,

0

HerrpephIBHOH

3aBHCHMOCTH

ero

OT

HaqaJIbHhIX

yCJIOBHii,

01'

rrapaMeTpOB

yrrpaBJIHeMOH

CHCTeMhI

H

T.

p;.

qHTaTeJIb,

pa300paBmHRcH

B

npep;bIp;yIIt;eM

MaTepHaJIe,

MOiKeT

C

ycrrexoM

Cp;eJIaTb

aTO

caMOCTOHTeJIbHO.

llpHBep;eM

p;nH

npHMepa

JIHmb

OP;HH

nop;ooHhIH

pe3YJIbTaT.

PaCCMOTpHM

ynpaBJIHeMyIO

cHcTeMy

1;

=

Ax

+ bu,

(27.26)

P;JIH

ROTOpOR

nOCTaBJIeHa

3ap;aqa

00

OnTHMaJIbHOM

ObICTpop;eiicTBHH

X

OC

H3

TOqKH

X

(ta)

= B

TOqRY

X

(tf3)

= 0

npH

yCJIOBHHX.

Iu (t)

1<1,

(27.27)

tf3

~

Iu (t) I

dt

< 1. (27.28)

t

oc

Byp;eM

nOJIaraTb

MaTpHIt;y

A H

BeRTOp

b

nOCTOHHHbIMH.

COTJIaCHO

06IIt;eR

npoIt;ep;ype

pemeHHH,

onHcaHHOR

B

aTOM

naparpa~e,

p;aHHOR

3ap;aqe

Hap;JIeiKHT

conOCTaBHTb

3ap;aqy

13.1

00

OnTHMaJIbHOM

ynpaB-

JIeHHH

x

(toc)

-+

x

(t~)

npH

yCJIOBHH

MHHHMyMa

HHTeHCHBHOCTH

tf3

x[uJ

=

max[s~plu(t)l,

~

lu(t)ldt]

(27.29)

t

oc

H

pacCMOTpeTb

BeJIHqHHY

PT

= 1htT

(T

=

t{ol

- t

a

),

ROTopaH

B

p;aH-

HOM

CJIyqae

COrJIaCHO

MaTepHaJIy

H3

§ 24

onpep;eJIHeTCH

paBeHCTBOM

PT

=

~in

m~ax

[~

Ib's

('r)

I

dt,

a

[L\J

=

min

(1,

T)J,

(27.30)

rp;e

s (t) -

p;BHmeHHe

conpHiKeHHoH

CHCTeMhI

oS

= -

A's,

(27.31)

CTeCHeHHoe

RpaeBblM

yCJIOBHeM

s'

(ta)

x

a

=

-1.

(27.32)

CnpaBep;JIHBO

yTBepiKp;eHHe.

II

ycmb

f)J1,!t

aaf)aw-w80

l-la'lWJ1,bU080

YCJ1,08U!t

x

(toc)

= x

a

8MnOJ1,u!t-

emc!t

COOm1-tOlUeHue

I

lim

[min

max

(\

Ib's

(t)

I

dt

)J

>1

(s'

(0)

X

OC

= - 1), (27.33)

T~oo

s

~

i

m08f)a

aaf)a'J.a

06

onmUJo£aJ1,b1-lOM

6b~cmpof)eucm8uu

UMeem

pelUeJ-tue

UTa

(t) = U °(t),

1'>OmOpOe

!tM!temC!t

(/iYJ-tK/fUeU,

npUUUMa10Ufeu

J1,UlUb

mpu

alia'J.eUU!t: 1,

-1

U 0,

npU'J.eM

aUa'llel-/,u!t

± 1 npUUUMa1Omc!t

ua

MHOJICeCm8e

@

mO'J.e1'>

t, Mepa

1'>Omop080

ecmb

min

(1,

TO).

B

caMOM

p;eJIe,

eCJIH

yCJIOBHe

(27.33)

BbInOJIHerrO,

TO

<lT0,

BO

BCH-

ROM

CJIyqae,

03HaQaeT,

qTO

npH

p;OCTaTOqHO

OOJIbmOM

T =

t(3

- t

oc

P;JIH

p;aHHoro

BaqaJIbHOrO

yCJIOBHH

x

(tot).

= x

a

HMeeM

PT

>0

H,

.CJIep;OBaTeJIbHO,

Tor)l;a

pa3pemHMa

3ap;aqa

13.1

00

ynpaBJIeHHH

.

a

.

OTcIOp;a

CJIep;yeT,

'ITO

neRTOp

x

npHHap;JIeiKHT.

I{

noiJnpocmpaHcmey

ynpaM!teJoWCmu

.7{11l

CHCTeMhI

(27.26)

(CM.

CTp.

146).

Ho

B

TaROM

.",

..

::":.

.~.

:::.:.~;.<;~~

.',"

~-'~.

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

........

:~,-'

~.~:~::.,

<::>:

-:."

,:,"

,:.::

',,'

.~

....

<~:'.'

..

;,:

..

···.~~;;i:·.:.

248

rrpOBllEMA

rrPEAEllbHOrO

BMCTPOAEfiCTBHR

[rJI.

7

Myqae

MhI

MOmeM

rrpHBeCTH

ypaBHeHHe

(27.26) R

KaHOHHqeCROMY

BH~y

(19.13), (19.14) H

pemaTb

sa~aqy

0

npe~ellbHOM

OhIcTpo~eHcT

BHH

llHmb

~mI

8nO.lLl-le

ynpa8.MI.e.MOU

qaCTH

(19.3)

CHCTeMhI,

uel-lOpupyJl.

rrpOqHe

Roop~HHaThI,

TI:lR

RaR

BClle~CTBHe

npHHa~llemHOCTH

x

a

R X

m

Bce

<lTH

Roop~HHaThIcorllaCHO

MaTep:u:ally

§ 19

OY~YT

Tom~eCTBeHHhI

MH

HymIMH

rrpH

Bcex

t:>

t

a

,

RaR

OhI

HH

BhIoHpallocb

yrrpaBJIeHHe

U (t).

JITaR,

He

HapymaH

OOm;HOCTH,

MhI

MomeM

CqHTaTb

paCCMaTpH-

BaeMyIO

cHcTeMy

BrrOJIHe

ynpaBJIHeMoH.

Ho

~JIH

TaROH

CHCTeMhI

Be-

JIfiqHHa

PT,

orrpe~eJIeHHaH

COOTHomeHHeM

(27.30),

eCTb

HenpephIB-

HaH,nOJIOmHTeJIbHaH

H MOHOTOHHO

HeYOl>IBaIOm;aH

<pyHR:qHH

npH

T > 0,

y~OBJIeTBOpHIOm;aH

yCJIOBHIO

lim

PT

= 0

rrpH

T

--+

O.OT-

CIO~a

H

HS

(27.33)

CJIe~yeT,

'ITO

ypaBHeHHe

(27.5),

T.

e.

s~eCb

ypaB-

HeHHe

PT

= 1,

HMeeT

HaHMeHbmHH

rroJIOilUlTeJIbHhIH

RopeHb

T =

TO,

KOTOphIH

H

HBJIHeTCH

OnTHMaJIbHhIM

BpeMeHeM

TO

rrne~eJIbHOrO

OhICT-

po~eHcTBHH.

CBoHcTBa

OnTHMaJIbHOrO

ynpaBJIeHHH

UO

(t) =

UTO

(t)

CJIe~YIOT

Tenepb

cpasy

HS

MaTepHaJIa

§

24

(CM.

CTp.

214).

ECJIH

em;e

rrpe~rrOJIOmHTb,

'ITO

MHHHMaJIbHaH

<PYHR:qIlH

h~o

('t) =

b'

s~o

('t)

OT-

JIHqHa

OT

Tom~ecTBeHHoii

nOCTOHHHOH,

TO

cpasy

me

npH~eM

K

BhI-

~,

BO~y,

'ITO

UO

(t)

eCTb

RycOqHO-nOCTOHHHaH

<pyHK:qHH,

a eCJIH

npll

nOM

(!l

BocnOllbSOBaTbCJI

TeM,

'ITO

CHCTeMa

OTpeSKOB

L1

0

Ha

[t

a

, t

a

+ TO],

1/:

ROTopaH

y~OBJIeTBOpHeT

yCJIOBHIO

~I

,t'

max

[~

I

b's~o

('t) I

d-rJ

=

II

'~

IN"

=~

Ib's~o(-r)ld't=1,

Cl

[L1]

= min (1,

TO),

(27.34)

llO

e~IlHCTBeHHa,

TO

cpasy

yoem~aeMCH

B

e~HHCTBeHHOCTH

OrrTHMaJIbHOrO

no

ObICTpo~eHCTBHIO

yrrpaBlleHHH

UO

(t),

RaK

<lTO

CJIe~yeT

HS

cBoiicTBa

MaRcllMyMa

(17.1),

ROTOpOMy

npH

~aHHOM

S~il

(t)

~OJImHO

y~OBJIeTBO

pHTb

Ram~oe

OnTHMaJIbHOe

ynpaBJIeHHe

UT'

(t).

ECJIIl

OTKasaTbCH

OT

BhICKasaHHhIX

TOJIbKO

'ITO

anpHopHhIX

npe~nOJIQjI\eHIlH

0

CBOHCTBax

MHHHMaJIbHOrO

~BllmeHHH

s~o

(t),

TO

~JIH

pemeHHH

Bonpoca

0

e~IlH

CTBeHHOCTH

OilTllMaJIbHOrO

ynpaBJIeHHH

UO

(t)

(B

cJIyqae

TO

> 1)

HymeH

oOJIee

~eTaJIbHhIii

aHaJIIlS,

KOTOphIH

MomHO

npe~JIOmllTb

qllTaTellIO

B

RaqeCTBe

yrrpamHeHHH.

(TOqHO

TaR

me

HymeH

~onOJI

HHTeJIbHhIH

aHaJIHS

cym;ecTBoBaHlIH

pemeHHH

B

CJIyqae,

Ror~a

lim

PT

= 1

rrpH

T

--+

00

.)

B

CJIyqae

TO

-<

1

e~HHCTBeHHOCTb

peJIeH-

Horo

OnTHMaJIbHOrO

yrrpaBJIeHHH

UO

(t)

CJIe~yeT

Bcer~a

cpasy

HS

CBOHCTBa

MaRcHMyMa

(1:7.1).

HaRoHe:q,

eCJIIl

paccMoTpeTb

~JIH

CHCTeMhI (27.26)

Ty

me

sa~aqy

o

rrpe~eJIbHOM

OhIcTpo~eHcTBHH,

HO

JIllmb

IIpll

OrpaHHqeHIlH

(27.27),

TO

cpasy

IIOJIyqHM

YTBepm~eHHe.

c

§ 27]

PEIllEHHE

3A~A

'IH

0

BbICTPOAEi1CTBHH

249

Teo

p e M a 27.2. 3afJa"ia

UMeem

pewel-lue

UO

(t)

moefJa

U

mO.lLblW

moefJa,

'/'i,oefJa

T

lim

[min

(JI

b's

(-r)

I

d-r)J

> 1

(s'(t,,)x"=-1).

(27.35)

T~oo

s

t

a

OnmUMa.ll.bl-lOe

ynpa8.11.el-lUe

UO

(t)

efJUlicm8ewiO U

Jl.8.II.Jl.emCfl.pe.ll.eu-

l-lOU

rPYl-llflfUeU,

npUl-lUMalOUfeu

fJ8a

3lta"iel-lUJI.

+ 1.

3~ecb

aHaJIHS

cym;eCTBeHHO

ynpom;eH

no

cpaBHeHHIO

C

npe~hIp;y

m;HM

CJIyqaeM,

nOCROJIbKY

cpasy

BH~HO,

'ITO

BeJIH'IHHa

PT

HBJIHeTCH

CTporo

MOHOTOHHoii

BospaCTaIOm;eii

<pyHK:qHeH

OT

T, H

cpasy

BHAHO,

'ITO

MHHHMaJIbHOe

~BHmeHHe

s~o

(t)

onpe~eJIHeT

OnTHMaJIbHOe

ynpaB-

JIeHHe

UTo

(t)

HS

CBOHCTBa

MaKcllMyMa

(17 .1)

e~HHCTBeHHhIM

oopasoM.

3aMeTHM

em;e,

'ITO

B

CJIyqae

nO.ll.l-lOU

ynpa8.ll.Jl.eJlwcmu

CHCTeMhI

(27.26)

~JIH

BhIrrOJIHeHHH

yCJIOBHH

(27.35)

npH

.ll.1060M

XliX

HeOOXO~IlMO

H

~OC

TaTOqHO,

OqeBH~HO,

qTOOhI

HH

O~IlH

ROpeHbf.ti

(i

= 1,... ,

n)

xapaRTe-

pllCTHqeCROrO

ypaBHeHHH

det

(-A'

-

f.tE)

= 0 CHCTeMhI (27.31)

He

IlMeJI

OTprr:qaTeJIbHOii

~eiiCTBHTeJIbHOH

qaCTH.

A

~JIH

<lTOrO

Heooxo-

~HMO

H p;OCTaTOqHO,

qTOOhI

Bce

ROpHH

'Ai

ypaBHeHIlH

det

(A

-

'AE)

=

= 0

HMeJIH

HerrOJIOmHTeJIbHYIO~eiicTBHTeJIbHYIO

qaCTb,

TaK

KaR

HS-

BeCTHO,

'ITO

'Ai = -

f.ti

(i

=

1,

... ,n).

JITaR,

MhI

paSOOpaJIH

cnocoo

pemeHHH

Sa~aqH

26.1,

OCHOBaHHhIH

Ha

npaBHJIe

MHHIlMaRca.

Bo

BBe~eHHIl

yme

OhIJIO

YRasaHO,

'ITO

BeCbMa

OOm;He

npHSHaRH

OrrTHMaJIbHOCTH

~aeT

npultlfun

JlW'/'i,CU.MYMa

JI.

C.

IIoHTpHrIlHa.

IIOCMoTpllMTerrepb

~JIH

cpaBHeHIlH,

RaR

omICaHHoe

HaMH

pemeHHe

CBHsaHO

C

pemeHHeM

Toii

me

Sa~aqH

Ha

OCHOBe

npHH-

:qHrra

MaRcHMyMa.

IIpH

nOM

OrpaHHqHMCH

oocym~eHHeM

~aHHoro

Bonpoca

Ha

MaTepHaJIe

ROHRpeTHoii

Sa~aqH.

I1MeHHo,

npe~rrOJIOmHM,

'ITO

sa~aHa

CHCTeMa

(27.26)

co

CRaJIHpHhIM

yrrpaBJIeHlleM

U

(t)

11:

'ITO

~JIH

,noH

CHCTeMhI

TpeoyeTcH

pemHTb

sa~aqyo

OhIcTpo~eiicTBHH

26.1

npH

yCJIOBHH

(27.27).

OopaTIlMcH

BHaqaJIe

K

npHH:qHrry

MaRcHMyMa

JI.

C.

IIoHTpHrHHa.

IIycTb,¢

(t)

OSHaqaeT

pemeHHe

conpHmeHHoii

CHCTeMbI

'¢

= - A',¢,

(27.36)

COBrra;o;aIOm;ei

C (27 .31).

Tor~a

~JIH

C<pOpMyJIHpOBaHHoii

HaMH

sa~a

qH

npHH:qHn

MaRcHMyMa

qHTaeTCH

CJIeAYIOm;IlM

oopasoM:

~JIH

Toro

qTOOhI

yrrpaBJIeHHe

UO

(t) OhIJIO

OrrTHMaJIbHhIM,

Heooxo~HMO

cym;ecT-

BOBaHHe

TaKOH

HeHyJIeBOii

HenpephIBHoii

BeKTop-<pyHR:qHH

'¢O

(t),

y~OBJIeTBOpHIOm;eii

(27.36),

'ITO

~JIH

Bcex

t,

tao

-<

t

-<

tg

=

tao

+

TO

<pyHK:qHH

;;e

['¢

(t), x (t), t,

ul

= '¢'

(t)

[Ax

+

bul

(27.37)

".':

',.;..' "'::.'.'

.,.:,:.:,:.-,

.>

.......

,',

...

'

....

::.

.

..

',.'.J:,~<

..'

,>::'·.:r"'~·i·":··.:':·'::·:::_'~~·~"~~~':_:'~':;';

':~:~':',~~;;Z~

c~

250

TJPOD.ITEMA

ITPEl\EJIhHOrO

DhICTPo,n:EnCTBlUI

(rJl.

7

nepeMeHHoro U

AOCTHraeT

B

TOlIRe

u

~-=

UO

(t)

MaRcHMyMa

no

BceM

3HalIeHHRM

U,

AJIR

ROTOpbIX

IU I<

It

=

1,

npHtleM

max

:;e

[,¢O,

x,

t,

u]

=:;e

[,¢O,

x,

t,

UO]

'=

1.

(27.38)

u

3aMelIaR Tenepb,

lITO

CJIaraeMOe

'¢'

(t)

Ax

0'1'

U

He

3aBHCHT,

HaxoAHM,

lITO

tPYHR~HH

,¢O

(t) H

UO

(t)

OOR3aHbI

YAOBJIeTBOpRTb

COOTHomeHHIO

I'ljJ°

(t)]'

buO

(t) =

:;e.

[,¢O,

uO,

t]

=

max:;e.

['ljJ0,

u,

t]

=

max

['¢O' (t) bu(t)] (27.39)

u u

~PH

IU I<

11.

= 1.

YnpaBJIeHHe

UO

(t) CJIeAyeT HaXOAHTb

no

BeRTopy,¢o (t)

COrJIaCHO

(27.39).

IIpH

3TOM,

eCJIH

BeRTOp

'¢O

(t)

H3BeCTeH,

TO

HaxomAeHHe

tPYHR-

~HH

UO

(t)

He

COCTaBJIReT

TpYAa

]I

npeACTaBJIHeT

COOOM

npo~eAYPY,

aHaJIOrHlIHYIO HaxOlHAeHino BeJIHlIHHbI

UO

(t)

no

tPYHR~HH

hO

(t)

H3

COOTHOilleHHH

(21.8).

TaR,

'eCJIH

BeJIHlIHHa

,¢O'(t)

b

He

ooparu;aeTCH B

TomAeCTBeHHbIM

HyJIb HHrAe Ha

[toc,

toc

+

TO],

TO

UO(t)

eAHHCTBeHHO

H onpeAeJIReTCR

paBeHCTBOM

UO

(t) =

11.

sgn

['¢O'

(t)b].

II

OAlIepRHeM,

(')lwaRo,

lITO

yCJIOBHR

(27.38)

HJIH

(27.39)

npH~Hna

MaRcHMyMa

He

1

:

1

.,

II

YRa3bIBaIOT

RBHO

Ha

cnocoo

BbIoopa tPyHlm;HH

'ljJ0

(t) cpeAH Bcex

pe-

Ii

illeHHM ypaBHeHHR

(27.36),

T.

e.

He

YRa3bIBaeTCR

cnocoo

BbIQHCJIeHHH

1

RpaeBoro

yCJIOBHR

,¢O

(toc)

HJIH

,¢O

(t(3).

KpOMe

'1'01'0,

He

YRa3bIBaIOTCR

.'

YC.lIOBI1R

AJIR

onpeAeJIeHHR

tIHCJIa

tg

=

toc

+

TO.

IIOCJIeAHHe

OOCTOR-

TeJIbCTBa

MoryT

BbI3BaTb aaTpYAHeHHR

npH

nOCTpoeHHH BbIlIHCJIHTeJIb-

HOM

cxeMbI

AJIR

ROHRpeTHoro onpeAeJIeHHR ynpaBJIeHHH

UO

(~.

OopaTHMCH Tenepb R TeopeMaM

27.1,

27.2

H npaBHJIY MHHHMaI{Ca.

IIpHBJIeRaR

COOTHOilleHHR

(17.1),

(17.4)

H y'lRTbIBaR, 'lTO s (t)

eCTb

peilleHHe

CHCTeMbI

(27.31),

cOBnaAaIOru;eM

C

(27.36),

HaxoAHM corJIac-

HO

TeopeMe

27.1,

'lTO

t(3

°

~

°

\

b's~o('t)uo('r)a-r=max

\

b's~u(-r)u(-r)a-r=1

"

toc

npH

IU ('t) I<

11.

= 1.

II

(21.8»,

lITO

b's~o

(t)

UO

(t) =

u(~)

.1

toc

IIOCJIeAHee

COOTHOilleHHe

OaHa'laeT

(CM

(21.7)

max

(b's~o

(t) u)

npH

Iu I<

fA.

= L

(27.40)

u

ConOCTaBJIRH

(27.39)

C

(27.40),

BHAHM,

'lTO

3TH

yCJIOBHR

npOCTO

COB-

rraAaIOT

"

eCJIH

nOJIOmHTb

,¢O

(t) =

s~o

(t). 0AHaRO npaBHJIO

MHHHMaRCa

COAepmHT

AOnOJIHIITeJIbHyIO

HHtPopMa~HIO

0 BeRTope

s~o

(t),

OOJIer-

tIaIOru;yIO

ero

HaxomAeHHe. IIMeHHo,

H3

npaBHJIa

17.1

(HJIH

17.3)

H3-

BeCTHO,

'lTO

3'1'0'1'

BeRTOp

s~o

(t)

eCTb

ABHmeHHe

(27.31),

06JIaAaIO

l

uee

G:':.

C,~,

§

28]

ynPABJIEHHE

ITO

IIPHH~HITY

OBPATHOJ1:

CBll3H

251

CJIeAYIOru;HM

cne~HtPHlIeCRHM

CBOMCTBOM:

to

(3

I

~

I

b's~o(-r)

I

a-r

= msin

~

Ib'S

(-r)

I

a-r

= 1

i

t« t«

I

npH

s'

(tg)

c (T) =

l'

(HJIH

npH

s'

(t«)

X

OC

= - 1 B npaBHJIe

17.3),

I

H

HBJIHIOIIIeeCH

B

3TOM

CMbICJIe

MUl/,U.Ma.ltb]-{b~M

ABHmeHHeM yRaaaHHoM

I

CHCTeMbI

(27.31).

KpoMe

'1'01'0,

nOCJIeAHee

ypaBHeHHe

AOCTaBJIHeT

HH-

I

tPopMa~HIO

AJIH

BbIlIHCJIeHHR

TO.

OTMeTHM

TaRme,

lITO

yCJIOBHR

(17.1);

(17.4)npaBHJIa

MHHHMaRca wopMyJIHPYIOTCH

kaK

'l£eo6xoaUMb~e

H

aOCmamO'll£bW

AJIR

OnTHMaJIbHOCTH

11°

(t).,

B

TO

BpeMH

RaR

npH~Hn

I

MaRcHMyMa

(27.38)

tPOPMYJIHpyeTcH B Raq:ecTBel£eo6xoaUMoao

yCJIO-

BRR.

CneAyeT, BnpolIeM, nOAlIepRHyTb, 'lTO B

JIHHeMHOM

CJIylIae,

RaR

MblllHAHM,

3TJiI

HeOOXOAHMbIe

yCJIOBHR,

AaBaeMbIe

npH~HnOM

MaR-

I

cHMyMa,

HBJIHIOTCH

H

AOCT,aTO'lHhIMH.

HaROHe~,

TeopeMa 27.1

BRJlIO-

traeT B ceoH eIIIe yTBepmAeHHe 0

cY/lfecmeoeal£UU

peIlIeHHH

UO

(t H

ero

o~eHRY·.

3aMelIaHHH, aHaJIOrH'lHhIe

TOJIbRO

lITO

BhICRaaaHHhIM,

MoryT

OhITb

CJJ;eJIaHhI

He TOJIblW

'ITO

OTHomenHIO' R

aa,n:alIe

0

ObICT.pOAeMCTBHH,

HO

H liD.

·OTHomeHintfKo

BceMy,:'l\'pyry

JiHHeMHhIX

3aAa'l

ynpaBJIeHHR,

paCCMaTpHBaeMhIX B

HaCTOHII1;eM

RHHI'e.

IIpH

3TOM,

OAHaRO,

CJIeAyeT

I

lIMeT!? B BHP;y, 'lTO B OTJIH'lHe

0'1'

MeTOAOB,

onHCRHHhIX B

AaHHOM

MO-

HorpatPHH,

npH~Hn

MaRcHMyMa JI. C. IIOHTpHrHHa

eCTh

OOIIIHM

Ii

npH~Hn

OnTHMaJIbHOCTH,

OXBaThIBaIOIIIHM

lIpeaBhITJaMHO

illHPORHM

Rpyr

pa3HooopaaHhIX aaAalI RaR

AJIH

JIHHeMHhIX, TaR H

AJIH

HeJIH-

HeMHhIX

CHCTeM.

§

28.

YnpaBJIeHHe no

npHHqHny

oopaTHoH

CBUBH

I

B -rex' BaAalIax

00

ynpaBJIeHHH, ROTophIe paCCMaTpHBaJIHCb B

upeAhIAYIIIHX

naparpatPax,

ynpaBJIRIOIIIHe BoaAeMcTBHH U pa3hICRHBa-

iHcb

B!tPopMe

tPYIIR~HM

U (t)

0'1'

BpeMeHH t,

npH'leM

Ha'laJIbHOe H

RO-

He'lHOe

yCJIOBHHI

x

(toc)

=X

OC

H x

(tf])

=

x(3

npeAnOJIaraJIHCb aaAaH-

Hi.IMH aapaHee H cTporotPHRcHpoBaHHbIMH.

IIO,ll;OOHbIe

3aAa'lH

HOCRT

iIasBaHHe

rrpOOJIeM

npOapaMMl£OaO

ynpaBJIeHHH.

3'1'0'1'

TepMHH

nOA'lep-

KHBaeT

Ty

M;hICJIb,

lITO

B paCCMaTpHBaeMhIX CJIylIaHx

ABHJI{eHHe

x (t)

oC,YIIIeCTBJIHeTCH

no

nporpaMMe,

AHRTyeMoM BhIopaHHhIM aapaHee

BHOJiHeonpeAeJIeHHhIM ynpaBJIeHHeM

U (t).

3aAa'lH

nporpaMMHoro

yn:paBJIeHHlI

COCTaBJIRIOT

BamIJhIM

Rpyr

npOOJIeM B TeopHH 'ynpaBJIH-

e¥~xrip.o:g~<?COB~

O)J;HaRO

BO

MHorHX CJIy'laRX

YlIp,~IiJI~fIHe

no'

npor-

p8.:MM:eu··

=U

.(t) .

He

..

n.OAXOAH'f,

ORaShIB

aHCb

HeYAOBJI8.TBO

pHTeJIbHI>IM

B"'peaJIbn:r:.rX

YCJlOBHHXipaOOThI

CHCTeMhI.

OCHOBHOH

HeAoCTaTOR

3'1'01'0

ci:rijcb6a~effCrBJfii:

cocrour

B rOM,

'{TO

OR

He

Y'lI{'rIimaeT

p;onO.JIJIH'l'eJI~-

-----~--'"._~.-

..

~.

__

.

-~~.~.

~.~.~.

'-'.~'-'~.

~'-'-'~'.'

.:

...

',.

".:

'.,'

.....

~~

:;",:,",:

..

,:,:.:'.,

~~:';'~~'~'~'~~?:~3~

C"

~~.~~

"

ITPOBnEMA

ITPEAEnbHOrO

BMCTPOAEnCTBMH

L

rJI.

7

252

HLIe

06CTOHTeJIbCTBa,

KOTOpLIe

MoryT

B03HHKHYTb

rro

xop;y

rrpon;ecca.

TaK,

eCJIH B

CHCTeMe,

pa60TaIOm;eH

rro

3apaHee

orrpep;eJIeHHOH

rrpor-

paMMe

u

(t),

B03HHKHyT

B

HeKOTopbIH

MOMeHT

t =

t.

Herrpep;BHp;eHHLIe

B03Mym;eHHH

~X

(t.)

<pa30Boro

BeKTopa

x (t),

TO

eCTeCTBeHHO,

qTO

3ap;aHHaH,

HeH3MeHHaH

rrpH

t>-

t.rrporpaMMa

u

(t)

rrOBep;eT

06'I>eKT,

HaqHHaH

C

MOMeHTa

t =

t.,

B COCTOHHHe X (tfJ),

OTJIJIqHOe,

Bo06m;e

rOBopH

(H,

MOllieT

6LITb,

BeCbMa

cym;ecTBeHHO),

01'

3ap;aHHoro

rro-

JIOllieHHH

xfJ.

B

rrop;06HLIXCJIyqaHx

B03p;eHCTBHe

u P;OJIlliHO rrOaTOM.Y

<pOpMHpOBaTbCH

C

yqeToM

p;OrrOJIHHTeJIbHOH

HH<popMan;HH,

rroCTyrraIO-

m;eH

B

OpraH

yrrpaBJIeHHH

rro

xop;y

rrpon;ecca.

8TOMy

yCJIOBHIO

01'-

BeqaeT

yrrpaBJIeHHerronpun/funy

o6po,mnou

C8.f/.8U,

KOTOpLIH

HBJIH-

eTCH

KpaeyrOJIbHLIM

KaMHeM

B

TeopHH

peryJIHpOBaHHH.

CYTb

:3TOrO

rrpHHn;Hrra

COCTOHT B

CJIep;yIOm;eM:

6

r;,o,:JICf}b~u

.MO.Menm

t 808f}eucm8ue u

onpef}e.tf,.f/.emc.f/.

no,

ocn08o,nuu unrfiop.Mo,quu 0

mer;,YUfe.M

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8 amom

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1,

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I

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u = U

[xI,

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n

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1)

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n

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[XI,

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n

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HO B

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U = U

[Xl

[tl,,,,,X

n

[tJ]

(HJIH

U

[t,

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COOTBeT-

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e.

u

[t]

= u

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n

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= u

[Xl

[t],

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n

[tJ].

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qTO

B

p;aHHoH

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[t]

1)

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YlI

4

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HeHIiH

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co,n;epJRaT

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t

Ii

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He

IiSMeHJlIOTCJl

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B

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cTan;HoHapHoH

sa,n;aqH

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paccMaTpHBaeMaJl

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aTOM

lIaparpalfle

sa,n;aqa

28.1, ,n;aJRe eCJIH B

ypaBHeHIiJIX

(28.1)

MaTpIin;hI

A H B JlBJIJlIOTCJI

IIOCTO-

JIHHhIMH

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sa,n;aqa

29.1,

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B § 29, 6y,n;eT

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eMH

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a;q:bl{hIX

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C

1I0CTO~HHhIMH

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.

~.

I.:

.-

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I

§

28)

YITPABnEHME

ITO

ITPHH~HITYOBPATHOnCBH3H

253

I

0603HaqaeT

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pa3

TO

p;BHlliemie

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U

[t).

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em;e,

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B

BHp;e

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cxeMLI,

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pHC.

28.1.

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rro

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60JIee

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qeM

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p;ByX

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ill

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.r(t)

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•

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(t)

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28.1.

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B

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= u

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peryJIHTOpe

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xop;y

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01'

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[t]

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He

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01'

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[t).

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H

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u

[t).

•

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u [t,

X]

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B

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[28a,

136a,

213r].

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H

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C

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Cp;em.I.Tb :31'0

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rro

TOH

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0

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COC-

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C

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3 a

p;

a q a

28.1.

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X

(tfJ)

=

xfJ,

8

1;

,.

9fno.My

.MO.Menmy

t

ua

K,.ILo,cco,

80a.MO:JICnb~X.

IIycmb

8o,f}o,nb~

.MO.Menm

8pe.Me1iU

t =

tfJ

u coc-

lwmopoe

1ieo6xof}uJrtO

npu8ecmu cucme.My

= A

(t)x

+B (t) u

(28.1)

=

tfJ

ynpo,8.ILe1iUe.M

u =

UO

[t,

x],

8b~6po,n1ibUt

MO.Menm t =

trr.

nG,'l{,G,.lLG,

npo/feccG,

u ucxof)nQC

~_._----=--,---,-':"'_'

.,.,

,.'"

',

••••.

' •

;:.:',,'

.....

,,:::.

:,:':":',

"

••.

'

•.

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.:.

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~~'-'-"

254

COCmOR.liUe

e,

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to

<

trL

<

t~,

-

00

<

xi

<

00

(i

= 1,...,

n).

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liO

It,

x)

mpe6yemcR. ebz6pamb maK,

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npu

Jl,1060J.t

lia'l{aJl,bliOM

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rL

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liepaeelicmeo

~

rL

I

[t

a

,

xc",

Uo)

-:-

~

II

UO

[t)

11

2

dt

<I

[toc,

x ,

u]

=

~

II

u (t)

11

2

dt,

(28.2)

trL

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6bL

liU

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UJI,U

U =

= U

[t]

= u

[t,

x (t)J,

npueooR.utee

cucmeMy (28.1) e flyjJcfloe

coc-

mOR.flUe

x

(t{3)

.:.....

x~.

'

IlpR

<lTOM

no~

B03MomHbIMR

ynpaIilleHRHMRu

(t)

RU

[t,

x)

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HHMaIOTCH

3~eCb

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nO~CTaHOBKe

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pbIX B

(28.1)

nOJIyqaIOTcH

AR<p<pepeirn;RaJIbHbIe

ypaBHeHRH,

YAO-

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CYID;eCTBOBaHRH

pemeHHii:(cM.

cTp.18),

B AaHHoii: 3aAa'le

MomHO

orpaHH'UITbCH, HanplIMep,

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II

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'ITO

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II

nplIMeM B AaJIbHeH-

','

meM AJIHOnpeAeJIeHHOCTII.

~I

I,

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3aAa'lII;'

BYAeM npeAnOJIaraTb,

t"

'ITO

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(28.1)

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ynpaBJIHeMa Ha JII060M

oTpe3Ket

rL

<

t

<;;

t{3

(CM.

§

20).

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'ITO

MOMeHT

t = t

a

II

3Ha'le-

i

",

HlIe

x (t

a

)

= x

a

KaK-HH6YAbBbI6paHbI, .H. 3a<pIIKclIpyeM

IIX

Ha

Bpe-

1'''

I

MH.

TOrAa, OnlIpaHCb Ha MaTepRaJI

1I3

§

18,

MbI

CMO}KeM

pemlITb 3a-

Aa'ly

13.1

nplI

yCJIOBlI1I

MIIHHMyMa

IIHTeHCIIBHOCTlI

x [U) (18.1).

t:

TaKyIO

3aAa'ly

MomHO

Ha3BaTb npozpalrtJtfloii saAa'leii:, coomeem-

;

:!

cmeylOuteii saAa'le

28.1.

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IISBeCTHO,

'ITO

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U

Cr}

(ta <

'T

<

t(3),

paspemaIOID;aH

<lTy

nOCJIe~HIOIO

saAa'ly,

IIMeeT

BRA

u

('T)

= M

('T,

ta,

t(3)

x

a

+ N

('T,

ta'

t~)

x~,

(28.3)

rAe

MaTplID;bI M

II

N HenocpeAcTBeHHo

onpeAeJIHIOTCH

lIS

<pOpMyJI

(18.10)

II

(15.2).

d>YHKD;IIIO

(28.3)

YAo6HO

AJIH

AaJIbHeii:mero o6osHa-

'1lITb

CIIMBOJIOM

Ut

a,x

«('T), a

COOTBeTcTByIOID;ee

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npo-

rpaMMHoe ABlImeHlIe x (t)

--,--

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x

ta

,x

a

(t). B

'1aCTHOCTII,

nplI

'T

=

trL

lIMeeM:

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(la)

= M

(la,

ta,

t(3)

x

a

+ N

(la,

t

a

,t(3)

x~.

(28.4)

a,X

TaK

KaK BeJIlI'IlIHbI

t~

II

x~

no

yCJIOBIIIO

sa~aHbI

II

nOCTOHHHbI,

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npaBYIO

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paBeHCTBa

(28.4)

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<PYHK-

~IIIO

en

t =

t«.

II

X = x

a

,

T.

e.

Ut

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(?8.5)

.

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.'

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.

·"

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C

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7

rL

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npOUaeMbH.:bl.M,U

C?

§ 2ll]

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no

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2!)S

rue

B

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C

(18.10),

(15.2)

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P =

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(t)

= - D

:t]

.2J

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Jk

[t]

X

ks

[t~,

t]

h

Ji

[t

p

,

t],

J,k=l

n

. 1

~

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=

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[t]

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jk

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...

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pas

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28.1.

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(28.6)

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t]

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II

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x]

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Ae

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(28.5),

(28.6)

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UO

(28.7)

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no

CBOHM

apryMeHTaM H

HBJIHeTCH,

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x ' x

a

(t)

npH

t

a

< t <

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la

eCTI>

pemeHHe x

U]

ypaBHeHHH

.f (t) =

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+

BuO

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xl

=

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+B

(px

+ q), (28.8)

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[t]

=

UO

[t,

Xl

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(t)]

= U

l

.a

(t).

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a,x

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t

a,X

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y

< t <

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a (ly) = y (t

y

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=

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y

, X

Y

].

(

'J

D)

U

t

a'x

U

t

y,X

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.(.1

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u

la.

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tao

x

a

(t) H

BClle~CT

Bue

(28.9)

IIMeeM:

1;

ta' x

a

(ty)

=

AXI~.

x'"

(t

y

) +

Bu

ta

,

x'"

(l,)

=

Ax

t

",.

x

a

(ly) +

Bu

[ty,

X la'

xa

(ty)],

-------~._~

..

'

.'

....

_

...

. .

.

,',

.

':."

"::.

:,':::

::.':

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..

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·.:.::.::·;·.·~.::·.::~·'.?~·a:

c:

256

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(rJI.

7

a 3'1'0 B

CHJIy

npOHaBOJIbHOCTH

t

y

H

p;OKaabIBaeT,

'ITO

<PYHKIJ;HH X

t

a,x

.a

(t)

yP;OBJIeTBOpHeT

ypaBHeHHIO

(28.8),

npH'IeM,

p;eB:CTBHTeJIbHO,

u

t

.a

(t) =

a,X

= U (t, x a

(t)].

,IJ;aJIee,

CorJIaCHO

HaBeCTHOB:

TeopeMe.

(19,21]

t

a,X

Ha

TeopHH

p;H<p<pepeHIJ;HaJIbHbIX

ypaBHeHHB:

aaKJIIO'IaeM,

'ITO

(28.8)

npH

JIIOOOM

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yCJIOBHH

X (t

a

) = x

a

HMeeT

eoultcnweltltoe

pemeHHe

X

(tl,

HOO

npaBaH

'IaCTb

ero

yp;OBJIeTBOpHeT

yCJIOBHHM

JIHn-

mHIJ;a

no

nepeMeHHbIM

Xi'

II03TOMY

OnTHMaJIbHOe

nporpaMMHoe

P;BII-

meHHe

x a

(t)

eCTb,

p;eB:CTBHTeJIbHO,

ep;HHCTBeHHoepemeHIIe

t

a'x

X It]

ypaBHeHHH

(28.8),

OTBe'IaIOIIJ;ee

Ha'IaJIbHbIM

yCJIOBHHM

X

(t

a

)

= x

a

,

'ITO

II

TpeOOBaJIOCb

npOBepHTb.

TaKHM

oopaaOM,

OnTHMaJIbHOCTb

ynpaBJleHHH

UO

= r It, xl

(28.7)

p;o1<aaaHa.

ITop;Bep;eM

HTOr.

HaMH

YCTaHOBJleHO

CJIep;yIOIIJ;ee

npaBHJlO.

l(

Mf,

moeo

'WW6b~

pel.UUmb

aaoal(,y

28.1 06

onmUMa.t£bltOM

ynpa-

6.1teltUU

UO

It, xl

cucmeMOU

(28.1) no

npUltupny

o6pamltou

C6Rau,

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ltaumu

onmUMaJ/,bltOe

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t

.a (t)

OJ/,R

coom-

a'X

6emcm6YJOUf£u

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aaoal(,U

npu

J/,J06bLX

80IJ:AWJlCltUX

t

a

U

x

a

(npu

(18.1». Toeoa

'iii:

UO

It,

x]

= Ut,x (t).

(28.10)

I!I!

t."

8'1'0

yTBepmp;eHlIe,

p;OKaaaHHoe

TOJlbKO

'ITO

B

CJIY'!ae

MHHHMHaa-

IJ;HH

BeJIH'IHHbI

I (28.2)

Ha

p;BHmeHHHX

JlIIHeB:HOB:

CHCTeMbI

(28.1),

"i

B

p;eB:CTBHTeJlbHOCTH

coxpaHHeT

CBOIO

CHJIy

P;JlH

BeCbMa

mHpoKoro

'i::

KJIaCCa

aap;a'I.

B

caMOM

p;eJIe,

<paKTH'IeCKH

B

paccymp;eHHHx

MbI

HC-

nOJlbaOBaJIH

JIHmb

CJIep;yIOIIJ;He

OOIIJ;He

<paKTbI:

(1)

oonycmuMocmb

ynpaBJleHHH

UO

= r It, xl, (2)

npHHIJ;Hn

onmUMaJ/,bltOCmU,

(3)

eOUlt-

Cm6eltltOCmb

pemeHHB:

x It]

ypaBHeHlIJI

(28.8).

KpoMe

Toro,'

AJlJI

BOaMO}HHOCTH

rrOCTpOeHlIJI

<PYHKIJ;HHUo

It, xl = r It, xl

YAOOHbIM

OOCTOJITeJIbCTBOM

JIBHJIaCb

ep;HHCTBeHHOCTb

nporpaMMHoro

ynpaBJIe-

aHJI

U

a,x

a (t), HO

nOCJIep;Hee

OOCTOJITeJIbCTBO

He

JIBJIJIeTCJI

CTOJlb

Bam-

t

HbIM.

GAHaKO

npaKTH'IeCKaJI

IJ;eHHOCTb

C!fJOPMYJIHpOBaHHOrO

BbIme

npaBHJIa

OrpaHH'IeHa

TeM

OOCTOJITeJIbCTBOM,

'ITO

B

CJIy'IaHX

BeJIH'IHH

I,

OTJIH'IHbIX

OT (28.2),

Korp;a,

CJleAOBaTeJIbHO,

B

COOTBeTcTByIOIIJ;HX

rrporpaMMHliIX

aap;a'Iax

H

BeJIH'IHHbI

'X

OTJIH'IaIOTCJI

OT

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'X

(U] (18.1),

He

YAaeTCJI

BbIpaaHTb

TaK

me

npOCTO

JIBHyIO

aaBHCHMOCTb

U

t

xa (t

a

)

OT t

a

,

x

a

•

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He

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H B

3THX

CJIy'IaJIX,

eCJIH

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a,

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U

t

a (t

a

)

MOmeT

ObITb

ocyIIJ;eCTBJIeH

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ObI-

a,x

CTpO

no

cpaBHeHHIO

C

peaJIbHbIM

p;BHmeHHeM

x (tl,'

yKaaaHHoe

npa-

BHJIO

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OKaaaTbCJI

rrOJIeaHbIM.

MbI

eIIJ;e

BepHeMCJI

H

HeMy

HHme,

B

rJIaBe

10

npH

Hay'IeHHH

aap;a'IH

0

rrpeCJIeAOBaHHH.,

c,,·

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~

29]

ClIHTEa

OTI'l'MMAnbHblX

CHCTEM

IIpH:\Iep 28.1. PaCCHOTpUM n Ha'IeCTne

HJImOCTpaJ~IHl

aana'ly

1I:L

1.

,LJ,JIH

nOR

:Jal\aqH

nporpaMMHoe

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lit

llMeTb BllP;

U

.a.(t)=

t

a,X

CJIep;OBaTeJIbHO,

npHHqnny

06paTHoH

'a,

6

(t(a

- t

)3

[-

2x~

-

x~

(til

- tal]

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a

-I-

(tr~

taP

[x~

+

~

x~

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6

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(-

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-

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-I-

a

257

U3 rrpUMepa

x a (t} 6yp,eT

+

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tlX,)2

[x~

+

~

x~

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a

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g.

_

OllTHMaJIbIIOe

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uO

(t,

xl.

CpopMHpyeMoe

no

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U MHHHMI13HpyroID,ee nemrqMHY

tf!

J =

~

[u~

(t) +

u~

(t)]

rit,

ta,

CJIe,Il,yeT

3,I1,eCb

onpe,Il,eJIJITb

no

cPoPMYJIe

2 -

xl+3(I(a-t)X2.

J

uO[t,x]=Ut,X{t)=-(1

~t)2

[

2 .

g(I{3-t)2

.

(a

X3+;r(t{3-t)XI-

6

MTaR, yMeH

pemaTb

<lcP<peRnlDHo

11

6blCTpO COOl'BeTcTBylOID,yro

rrporpaMMnyro

sap;aqy 13.1

rrplI

yCJIOBHH

MHHHMyMa

HHTenCHBROCT.H

%

(ul

(18.1), Mbl

MonWM

·uaHT.H

pememre

UO

[t,

xl

(28.10)

II

,Il,JIH

HCXO,Il,HOH

3a,Il,a"lH 28.1.

§ 29.

CHHTea

OnTHMaJIhlIblX

CHCTeM

npeAeJIhHOrO

ObIcTpoAeucTBHH

3ap;a'Ia

0

npOepaMJMlOM

ynpaBJIeHHlI

U (t),

KOTopoe

oOeCne'IMBaeT

HaHCKOpeB:mee

BoaMO>KHoe

npHBeAeHHe

ClICTeMbI

(26.1) B

aap;aHHOO

COCTOJIHHe

x =

x{3,

ObIJIa

yme

lIBY'!eHa

B § 27.

3p;ecb

MbJ:

OOCYP;HM

aHaJIorlI'IHyro

rrpOOJleMy,

npep;nOJlaraJI,

Op;HaI<D,

Tenepb,

'ITO

yrrpa-

BJIeHHe

U

<popMHpyeTCJI

no

npult4uny

o6pamltou

C8Rau.

TaKyIO

npo-

OJIeMy

HaabIBaIOT

eIIJ;e aap;a-qeB: 0

CUltmeae

OrrTHMaJlbHOB:

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cTByIOIIJ;eB:

CHCTeMbI.

B

COOTBeTCTBHH

C

MaTepHaJIOM

Ha

§ 26

paCCMaT-

pllBaeMaJI

HaMH

aap;a"la

<pOpMyJlI1pyeTCJI

CJleAYIOIIJ;HM

oopaaoM.

3 a

p;

a

'I

a 29.1.

IIycmb

aaoalio

COCmORltUe

x =

·x;q,

6

Jf,omopoe

lte06XOOUMO

npueecmu cucmeMy (28.1), U

oe060peno

oepaltU1leltUe

lta

pecypcu

80aMO:JfCltb~X

ynpaeJ/,eltuii

'X

(u

('r), .. ;>

tl

~

It

(t):

(29.1)

9 H. H. HpaconclHI1i

,'-,...

'''.,'

•

e'

::'"

',','

:;',

.-

::.::::.:

••.•

-::

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I,

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~:

•

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::

•

.,.:

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1 f

~

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G

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258

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(rn.

7

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8

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ynpaBJl,eJ-lllU U

[t,

X,

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on-

7nU.7'rW.I/,bnOe

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UO

[t,

X,

/1],

J;;ornopoe

US

.1/,/-06020

803MO:JICn020

na'la.l/,bn020

cocmoa,nua,

ta, x

a

06eCne'lU8aem

naUCJ;;OpeUIUee

npu8ef)enue

cuerne,ubL 8 nO.l/,OJlCenue X =

xi3

(no

cpa8nenUlO

c

.I/,/-06blM

f)PY2UM 803,u0-

;)lCnbl.M

ynpaBJl,enue,u

U (t)

U.I/,U

U [t, X,

/1],

cmeCnennbl.M YC.l/,08ue,u (29.1».

MI>I

He

OYAeM

rrOflCHflTb

rrOCTaHoBKy

3TOH

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H

orrpeAeJIflTb

CMbICJI

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B

HeH

TepMHHOB,

TaK

KaK

AJIH

lIHTaTeJIH,

rro-

3HaKOMHBIlIerOCH

C

MaTepHaJIOM

H3

§§

26, 27, 28,

Bce

31'0

AOJImHO

ObITb

HCHI>IM.

OrOBopHM

JIHIlIh

OAHO

HOBoe

OOCTOHTeJIbCTBO,

rrOHBHB-

IlIeeCH

3AeCh.

B

lIHCJIO

aprYMeHTOB

4>YHKn;im

U

BOIlIJIa

Tenepb

H

Be-

JIJlqHHa

/1.

81'0

BI>I3BaHO

TeM,

lITO

YKa3aHHaH

BeJIlPIHHa

/1,

On;eHHBaIO-

lIJ,aH

pecypcl>I

yrrpaBJIeHHH,

KOTOpl>Ie

MomHO

HCnOJIh30BaTh,

HalIHHaH

C

HeKOToporo

MOMeHTa

BpeMeHH

t = ta,

MomeT

3aBHceTb

01' t

a

H

OI>IT1>,

CJIeAOBaTeJIhHO,

BeJIHllHHOH

nepeMeHHOH,BJIHHIOlIJ,eH

Ha

BbI-

OOp

ynpaB.iIeHHH

U.

IhaK,

HepaBeHCTBO

(29.1) B

AaHHOM

CJIylIae

Ha-

AO

rrOHHMaTb

CJIeAyIOlIJ,IIM

00pa30M.

ITYCTb

HaM

H3BeCTHO,

lITO

B

CHJIy

CJIOmHBIlIHXCH

OOCTOHTeJIbCTB

B

HeKOTOpl>IH

MOMeHT

t = t

a

B

npou;ec-

ce

yrrpaBJIeHHH

peaJIH30BaJIOCh

HeKOTopoe

3HalIeHHe

/1;=

/1

(ta).

TorAa

MomHO

HCnOJIb30Ba1'h

JIHIlIb

TaKHe

ynpaBJIeHHH

U,

AJIH

KOTO-

pl>IX

cnpaBeAJIHBO

yCJIOBHe

x

[u

('t), 't;:> tal

<:

/1

(ta), (29.2)

H

31'0

yCJIOBHe

AOJImHO

BWnOJIHHTbCH

AJIH

peaJIH3an;HH

U

[t],

onpe-

AeJIeHHbIX

BI>IOpaHHbIM

3aKOHOM

ynpaBJIeHHH

U [t,

X,

/1],

BO

BCHKOM

CJIYlIae,

AO

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t =

1'r,

rrpeAIlIecTByIOlIJ,erO

MOMeHTy

t =

t~,

KorAa

X

(tfJ)

= Xfl.

(HanOMHHM,

lITO

MOMeHT

t =

t.3

no

CMI>I-

CJIy

3aAalIH

29.1

3apaHee

HeH3BeCTeH

H

HBJIHeTCH

KaK

pa3

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ITpu

3TOM,

eCTeCTBeHHO,

AOJImeH

ObITb

TaKme

H3BeCTeH

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KO-

TOPbIH

onpeAeJIHeT

H3MeHeHHe

BeJIHlJ:HHI>I

/1

(t) C

TelIeHHeM

BpeMeHR

t.

OOl>IlIHO

31'01'

3aKOH

onpeAeJIHe1'CH,

HCXOAH

H3

yCJIOBHH

pacXOAOBa-

HHH

pecypCOB

ynpaBJIeHHH.

HanpHMep,

eCJIH

pelIh

HAeT

00

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HHH

3a

ClIeT

OTopaCbIBaHHH

peaKTRBHOH

MaCCbI,

TO,

KaK

MI>I

3HaeM

(CM.

§ 23),

orpaHHlIeHHe

(29.2)

MomHO

3anHCI>IBaTb

B

BRAe

t;i

~

IdU

(t)

I

<:

/1

(ta),

teL

rAe

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It

(teL)

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peaKTRBHoH

MaCCI>I,

KO-

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npR

t >

tao

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lITO,

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t{3

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MbI

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B

BRAe

(CM.

CTp.

233)

00

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I

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I

<:

It (ta), (29.3)

t

a

rAe

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00

rro)J;"IepKHBaeT

KaK

pa3,

lITO

MOMeHT

t{'"

Korp;a

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"

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:',

':,~::

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§

29]

CIIHTE3

OllTHMAJIbHbIX

CHCTEM

259

lIHTCH

rrpOn;ecc

yrrpaBJIeHHH,

HaM

He

H3BeCTeH,

H

MbI

Ha

Bce

6y-

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BpeMH

t

;:>

t

a

OOJIaAaeM

pecypCOM

yrrpaBJIeHIIH,

paBHbIM

It

(t

a

).

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H3MeHeHHe

It

(t)

3a

BpeMH

t

a

<t

<:

t

y

<

t{3

orrpeAeJIHeTCH

ec-

TeCTBeHHO,

HCXOAH

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paBeHCTBa

t

y

It (t

y

)

= It (ta) -

~

I

dU['!]

I,

(29.4)

teL

rAe

au

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no

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CJIe,llyeT,

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/1

B

YCJIOBHflX

(29.1)

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eTCH

nOCTOHHHOH

(HanpHMep,

eCJIH

orpaHRlleHHe

(29.1)

HMeeT

BHA

II

U (t)

II

<:

It)· B

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It

B

3anRCR

U [t,

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29.1 0

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coxpaHHeT

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onHcaH

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npep;l>Ip;ylIJ,eM

na

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28.1,

HOO,

KaK

yme

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31'01'

nYTh

BOOOlIJ,e

HMeeT'

CMbICJI

P;JIH

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npOOJIeM

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no

npnHn;nnyoopaTHOH

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CJIe-

p;OBaTeJIhHO,

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29.1

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paccMoTpeTh

co-

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3ap;alIY

26.1 0

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Tex

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t

,

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a

H

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=

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29.1

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Te

me

(H

elIJ,e

ycyrYOJIeHHIiIe

3p;ech)

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yme

OTMelIa-

JIHCh

B § 27.

ITomaJIYH,

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npOCTo

3THM

nyTeM

peIlIaeTCH

3a-

p;alIa

29.1

CHOBa

B

CJIylIae

orpaHIIlIeHHH

00

~

IJU

(t)11

2

dT<:lt

2

(t). (29.6)

t

81'0

CJIyqaH

MI>I H

BI>IOepeM

n03ToMY

B

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IIJIJIIOCTpaU;IIH

00-

cymp;aeMoro

MeTOAa.

IIpH

3TOM

B

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co

CKa3aHHI>IM

BI>IIlIe

H3MeHeHIIe

BeJIHlIIIHIiI

It (t) C

II3MeHeHHeM

t

onpeAeJIHeTCH

paBeHCTBOM

t

y

~2(ty)-It2(t<x)

=-

~

IJu(1')11

2

dr.

t~

9·

Красовский Н.Н. Теория управления движением

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