Красовский Н.Н. Теория управления движением

<·

'.'

_.~>.

C

[rHo 6

CBO:A:CTBA

OITTHMAJIbHOrO

YITPABJIEHHfl

180

yCJIOBI1e,

!{OTOpOe

06eCne'lMBaeT

JIMHeHHYIO

HesaBHCMMOCTb

epYHIUI

M

H

h(i)

[t/3,

t]

ua

JII060M

HHTepBaJIe

('t,

~)

MS

OTpeaKa

[t

oc

,

t/3J.

COCTaBMM

neKTOp

t/3

c =

~

h

[t(:l,

--r]

UO

(-r)

d-r.

(22.40)

t

...

CnpaBeAJIIIBO

YTBepiKAeHlIe:

OnmlM£a.l/,b"Jf,oe

ynpa8Muue

UO

(t)

RB-

.MtemcR,

pe.tteuuou

!jJYUri,l1ueu,

ri,omopaR,

YOO8.ltem80pR,em

YC.tt08UlO

UO(t)

=

I.l.

sgn b' (t)

SO(t),

(22.41)

eae

SO(t)

-

MUUUMa.ttbuoe

a8UJICeUUe

cucmeMbL

8 = -

A'(t)s,

(22.42)

cmecueltuoe

YC.tt08ueM

c'S°(t/3)

= 1

(22.43)

U

paapeutalOUfee

npu

amOM

aaoa'tY

ua

MUlWMyM

t/3

~

Ib' (t) s (t) Idt = min.

(22.44)

t

4

HaMeTIIM

.nJIaH

AOKa3aTeJIbcTBa,

AeTaJIbHOe

npoBeAeHlIe

KOToporo

npeAocTaBIIM

qIITaTeJIIO.

IIpeAnOJIOiKlIM

OT

npoTIIBHoro,

'lTO

HeKOTO-

poe

OllTlIMaJIbHOe

ynpaBJIeHlIe

UO(t)

C

max

II

UO

(t)

11

=

f.A.

yCJIOBIIIO

. t

(22.41)

He

YAOBJIeTBOpfleT,

nplI'leM

eCTeCTBeHHO

npeAnOJIaraeTCfl,

'lTO

paBeHcTBo

(22.41)

He

BbIllOJIHfleTCfl

no

cym;ecTBy,

T.

e.

OHO

He

Bbl-

nOJIHfleTCfl

Ha

MHOiKeCTBe

SHa'leHlIH

t

lIS

[loc,

t/3]

HeayJIeBOH

MePbI.

TorAa

lIS

MaTeplIaJIa

npeAblAym;lIx

naparpaepoB

BblBOAIITCfl,

'lTO

cy-

m;ecTByeT

ynpaBJIeHlIe

U = v (t),

YAOBJIeTBOpflIOm;ee

paBeHcTBY

t/3

c=

~-h[t(:l,-r]v(-r)d-r,

(22.45)

t

oc

npRQeM

maxllv

(t)

11<

I.l.-ke

(22.46)

t

II

max

Iv (t) -

UO

(t) 1<

e,

(22.47)

t

rAe

8 -

HanepeA

Bbl6paHHoe

npOlI3BOJIbHO

MaJIOe

nOJIOiKlITeJIbHOe

'lMCJIO.

(B

Ka'leCTBe

v (t)

AOCTaTO'lHO

Bbl6paTb

ynpaBJIeHlIe

v(t) =

=

(1

-

8)

UO(t)

+8

VO(t),

rAe

VO(t)

-

epYHK~lIfl,

YAOBJIeTBOpflIOm;afl

·

·.::

' :

c

181

YITPABJIEHIIE

C

MIIHMMAJIbHbIM

HMrIYJIbCOM

§

23]

yCJIOBlIIO

(22.45)

II

MMeIOm;afl

nplI

8TOM

HUIIMeHblITylo

nosMomayIO

HOpMy

p*

[v]

=

max

II

v (t)

II.

HeTpYAHo

y6eAMTbcll,

TITO

npM

HalIT8M

t

npeAnOJIOiKemiM

p*

[val

= v <

f.A..)

B

clmy

ypaBHeHMfl

(22.39)

MSMe-

HeHlIe

()X(I)(t)

ABMmeHlIfl

x(t),'

BblSBaHHoe

MSMeHeHMeM

ynpaBJIeHlIfl

6u(t) = v (t) -

u,o(t),

onpeAejIfleTcfl

n

JIIIHeHHOM

npM6JIHmeHlIlI

paBeHcTBoM

t/3

6X(I)

(t(:l)

=

~

h[t(:l,

-r]

(v ('t) -

UO

(-r»

d-r,

t

oc

npM'leM

npaBafl

'laCTb

8Toro

paBeHcTBa

BCJIeACTBMe

(22.40) M (22.45)

paBHa

HyJIIO.

OTcIOAa

C

y'leTOM

(22.47) HblBO).1;HTCfl,

1.1TO

paCCMaTpMBae-

Moe

BapblIpOBaHlIe

ynpaBJIeHlIll

U (t)

npHBO).\HT

UU

caMOM

L(eJIe

JIMlIIb

K

1I3MeHeHlIIO

()X

(t/3),

KOTopoe

IIMeeT

BTOPOH

nop'flAOK

MaJIOCTH

OT-

HOCHTeJIbHO

8.

IIocJIe

8Toro,

c.rreAYfl

cxeMe

paccyjUAeHMH,

nplIBeAeH-

HOH

B 8TOM

naparpaepe

BblllIe

(CM.

CTp.

173),

MOjUHO

nOCTPOIITb

noc-

JIeA0J:laTeJIbHOCTb

epYHK~IIH

v (t) =

VI

(t) , v

2

(t), ... ,

CXOAflm;YIOCll

K

ynpaBJIeHlIIO

u. (t),

KOTopoe

npHBOAUT

ABMiKeHMe

x (t) (22.37) B

3aAaHayIO

TO'lKy

x

(t/3)

=

x/3.

IIpH

8TOM

BCJIeACTBHe

(22.46) H (22.47)

nplI

MaJIOM

8 > 0

6YAeT

max

II

u. (t)

II

<

f.A..

TeM

caMblM

npMXoAHM

B

t

npoTIIBOpe'lMe

C

npeAnOJIOmeHlIeM

06

onTMMaJIbHOCTII

ynpaBJIeHlIlI

U

O

(t).

IIoJIy'leHHoe

npoTIIBOpe'llIe

AOKaSbIBaeT

HalITe

YTBepiKAeHlIe.

3aMeTIIM,

"ITO

aHaJIorlITJHbIe

paccymAeHlIfl

npoXOAllT

II

B CJIyTJae

BeK-

TopHoro

ynpaBJIeHlIlI

u (t), a

TaKiKe

B

CJIyTJaflX

APyrlIx

HOpM

P*

[uJ.

§ 23.

YnpaBJleHUe

npu

yCJIOBUH

MHHuMyMa

HMnyJlbca

ynpaBJlJIIOm;erO

.Bo3AeiicTBIIH

.

B 8TOM

naparpaepe

Mbl

pUCCMOTplIM

saAUl.J.Y

06

onTMMaJIbHOM

yn-

paBJleHIIII,

BKJlIOTJaIOm;ylO

YCJIOBlIe

MIIHlIMyMa

MHTeHCMBHOCTII

x

[dU]

ynpaBJlHIOm;ero

Bos.n;eHcTBHH,

B

cJly'lae,

Korp;a

ypaBueHHH

,n;BHiKeHlIH

sa,n;aHl>l B

o606m;eHHOH

epopMe

(6.11),

T.

e.

dx (t) = A (t)x (t)dt +

B(t)dU

+

dW

(t). (23.1)

HanoMHlIM

TJIITaTeJIIO

(CM.

§ 6,

CTp.

47),

'lTO

sanMCb

(23.1)

lIBJllI-

eTCH

JlHlITb

CIIMBOJIH'leCKHM

yKasaHlIeM

Ha

TO,

"ITO

.n;BMjHeI-lHe x (t)

ynpaBJllIeMOrO

06~eKTa

onMCblBaeTCH

paneHCTnOM

t t

x (t) = X

[t,

t

oc

] X

OC

+

~

H

It,

-r]

dU

+

~

X

It,

-r]

dW.

(23.2)

t

oc

t

oc

x

oc

(t

~

t <

t(:l;

x (t

oc

) = ; x

(t(:l)

=

x(:l).

oc

-------

182

c

CBO:fiCTBA

OnTUMAJIbHOrO

ynPABJIEHUH

[rJI. n

IIOHCHUM

CHa'IaJIa

rrYTb,

KOTOpbIH

rrpHBop;HT

Hac

K

3ap;a'IaM,

H3Y-

'IaeM.bIM

B

aTOM

rraparpaepe.

C

aTOH

~eJIbIO

paCCMOTpUM

06.bI'IHyIO

yn-

paBJIHeMyIO

cHcTeMy

j)

(t) = A (t) x (t) + B (t) U + w (t) (23.3)

H

npep;rrOJIOmHM,

'ITO

BeJIH'IHHa

U

HMeeT

CM.bICJI CUJI.bI.

IIYCTb

HaM

Tpe-

6yeTcH

pemUTh

P;JIH

CUCTeMbI

(23.3)

aap;a'Iy

06

OrrTHMaJIbHOM

yrrpaB-

.ITeHHH

rrpH

yCJIOBHH

MUHHMyMa

nO.llU080 u.Mny.llbca

CJIy-qae

B

Ka'IeCTBe

HHTeHCHBHOCTH

yrrpaBJIeHHH

x

B.bI6paTb

BeJIH'IHHy

tfl

x[u]

=

~

In

(-r)ld-r,

ta.

eCJIH

U -

CKaJIHp,

HJIH

BeJIH'IHHy

tfl

x[n]

=

~y·[u(-r)]d-r,

ta.

CUJI.bI

u.

B

aTOM

[ul

CJIep;OBaJIO 6.bI

(23.4)

(23.5)

eCJIu

U -

BeKTOp.

3p;ecb

CHMBOJI y*

[u]

CHOBa

03Ha'IaeT

KaKyIO-HH-

6YP;h

HOpMy

BeKTopa

U.

Op;HaKo

Ha

rrpocTbIX

rrpHMepax

MomHO

y6ep;HTbcH

B

TOM,

'ITO

TaKaH

3ap;a'Ia

He

6yp;eT

HMeTb

pemeHUH

He

TOJIbKO

B

KJIaCCe

B03MOmH.bIX

yrrpaBJIeHHH,

HO

H B

·60JIee

mHpoKoM

RJIaCCe

p;orrycKaeM.bIX

yrrpaBJIe-

HHH.

I1MeHHo,

OKa3.bIBaeTCH,

'ITO

yrrpaBJIHIOm;He

CHJI.bI U (t),

rrepeBO-

p;Hm;He

CHcTeMy

(23.3)

H3

COCTOHHHH

xa.

B

COCTOHHHe

x

J3

H

HMeIOm;He

HaHMeHbIITUH

rrOJIH.bIH

HMrryJIbC,

cop;epmaT

B

Ka'IeCTBe

CJIaraeM.bIX

MrHOBeHH.bIe

yp;apH.bIe

B03P;eHcTBHH,

OrrHC.bIBaeM.bIe

()-epYHK~HHMH

Bpe-

MeHH.

YKa3aHHoe

06CTOHTeJIbCTBO

H

B.bIHymp;aeT

Hac

rrepeHTH

0'1'

CHCTeMLI

(23.3) K

0606m;eHHoH

CuCTeMe

(23.1) H

paCCMOTpeTb

P;JIH

rrOCJIep;HeH

aap;a'Iy

4.2.

IIPH

aTOM

HHTeHCHBHoCTH

x

rid

(23.4), (23.5)

aaMeHHIOTCH

COOTBeTCTB8HHO

CJIe,uYIOm;HMH

HHTeHCHBHOCTHMH

x

[dU]

.o606u/euuoao

ynpa8.lleTwf£

dU

(CM.

IIpHJIOmeHHe,

§ 60):

tfl

x [dU] =

~

I

dU

I,

(23.6)

ta.

tfl

x[dU]=

~y·[dU].

(23.7)

ta.

YqHTbIBaH

p;aHHyIO

<pHaHqecKYIO

HHTeprrpeTa~HIO,

MhI

6yp;eM

Ha-

aLIBaTb

HHTeHCIiIBHOCT!;f

(23.6), (23.7) nO.llUblMU uMny.llbcaMu

yrrpaBJIH-

IOm;ero

Boap;eHcTBHH

dUo

CKa3aIHIOe

ap;eCb

W

06'tfICHrrer

III_HMeHeHliIe

TepMHH<t

«HMUYJIbC

yrrpaBJIHIOm;eru

Boap;eHCTBHH»

B

3arJIaBHH

HaCTOH-

m;ero

rraparpaepa.

I

",(

l'

CD

2:l

J

YllPABJIEHHE

C

MHfHIMAJlbHbIM

HMI1YJIb

COM

183

Cym,eCTBeHHO

OTMeTUTb,

~IT()

Bee

paCCMaTpUBaeMblO

II

::JTOM

napar-

paepe

aap;a'IH

MhI

MomeM

T'aJOI,e

OObe,uHIUITb

U

no

epop]\-[aJIbHoMY

MD.-

TeMaTU'IeCKoMy

npu3HaKy.

I1MeHHo,

B

COOTBeTCTBHU

C

M3TepUaJIOM

§

L4

aaKJIIO'IaeM,

'ITO

BeJIHqHHa

x

[dU],

o~eIlHBaIOm;aH

pecYPcbI,

aaT-

pa'IHBaeMble

Ha

yrrpaBJIeHHe

Ha

OTpeaKe

BpeMeHH

tu.

<:

t <

tJ3,

MomeT

TpaKToBaTbcH

RaK

HopMa

p*

[~u]

x

[dU]

= p* [UJ = p*

[~u

1 (23.8)

JIHHeHHOH

orrepa~HH

til

~u

[h

(-r)]

=

~

h'

(-r)

dU

(-r),

ta.

rropomp;aeMoH

epYHK~HeH

U

(-r)

B

rrpOCTpaHCTBe

fliJ

{h

(-r);

ta.

<:

-r

<:tJ3}

HerrpepLIBH.bIX

r-MepH.bIX

BeKTop-epYHK~UH

h

(-r)

= {hI

(-r),

... ,h

r

(-r))

C

rrop;XOp;Hm;eH

MeTpHKOH'

.

p[h(-r)]

=

max

y[h(-r)].

(23.9)

ta.<-r.,,;;tll

IIpHMep.bI

TaKoro

B.bI60pa

HOPMbI

p [h ('t)] P;JIH

Hau60JIee

'laCTO

BCTpe-

qaIOm;liIXCH

B

rrpHJIOmeHHHX

HHTeHcHBHocTeH

x

[dU]

YKaaaHbI

B

Ta6:"

mI~e

14.1,

KOTOpOH

MLI

H

6yp;eM

rrOJIbaOBaTbCH.

CKaaaHHoe

rr03BOJIHeT

cepopMymlpOBaTb

HCXOp;HyIO

rrp06JIeMY

4.2

KaK

aap;a'Iy

14.1.

BLIme,

B § 15,

rroKa3aHo,

'ITO

aTa

3ap;a-qa

CBOP;HTCH

K

rrp06JIeMe

MOMeHTOB.

IIPH

aTOM,

Op;HaI{O,

npep;rrOJIaraJIOCb,

'ITO

CHC-

T.8Ma (23.1)

rrop;BepmeHa

CKaJIHpHoMy

B03p;eHcTBHIO

dU

H

HHTeHCHB-

HOCTb

aToro

Boap;eHcTBHH

x

[dU]

HM.8eT

BHP; (23.6).

AHaJlOI'H'IH.bIe

paccy)Hp;eHHH

MomHO

rrpOBeCTH

H B

CJIy'Iae

BeI{TOpHOro

yrrpaBJIHIO-

m;ero

B03p;eHcTBHHdU C

HHTeHCHBHOCTbIO

(23.7).

MbI

Ha

HHX,

op;HaKO,

OCTaHaBJIHBaTbCH

He

6yp;eM,

TaK

KaK

HHKaHHX

rrpHH~UnHaJIhHO

HO-

B.bIX

MOMeHTOB

no

cpaUHeHHIO

CO

CKaJIHpHl:'IMrCJIY'IaeM

dU

ap;eCb

He

BoaHHKaeT.

YCJIOBHH

paapemHMOCTH

rrp06JIeMLI

MOMeHTOB,

K

KOTOpOH

CBO,7;l;HT-

CH,

CJIep;OBaTeJIbHO,

aap;aqa

14.1,

BbIHCHeHbI

B § 16.

OnHpaHcb

Ha

aTH

yCJIOBHH,

B § 17

MLI

CepopMyJIHpOBaJIH

rrpaBHJIO

MHHHMaKca

P;JIH

pe-

meHHH

aap;aqH

13.1.

8THM

rrpaBHJIOM,

HeCHOJIbKO

CJIep;yeT

rrOJIb30BaTbCH

H

rrpH

pemeHHH

aap;a-qH

BHp;oHaMeHeHHe

COCTOJiIT

JIHmb

B

3aMeHe

YCJIOBHH

H (17.6)

COOTBeTCTBeHHO

yCJIOBHJIMH

1)

t(J

tf)

~

hO'

(-rtdUo =

max

~

hO'

(T:)dU =

""

U

t'a.

t

oc

BHp;OH3MeHeHH.bIM,

14.1.

YKaaaHHoe

MaKcHMyMa

(17.1)

1 (23.10)

1)

B

,n;aJ1hHeil:meM,

pa60TaH C o606ru;eHHItIMH yrrpaBJJCHHHMH, MhI

6y,n;eM

CCbI-

JIaThCH

Ira

rrpaBHJ1a

17.1,

17.2;o,n;HaKo

qHTaTeJ1h

,n;OJImeH

rrOMHHTh,

qTO

npHaToM

YCJ10BHH

MaKcHMyMa

(17.1),

(17.6)

saMeHHIOTCJI

Ha

YCJ10BHH

BH,n;a

(23.10),

(23.11).

(eM. TaKme § 47, CTp.

387.)

-I

c::'

()

184

CBOl1:C'l'BA

OlrrMMAJIbHOrO

YllPABJIEHHJI

[rJI,

6

rrpH

X [dU} =

1/pO

HJIH

tf3

O

~

h

'

(-r)

dUO

=

max

~

hO'

(-r)

dU =

,,0

(23,11)

, t U t

oc

rrpH

x [dU} =

,,0.

OrrlIpaJICb

Ha

rrpaBlIJIO

MlIHHMaKca

17.1,

rrplIcTyrrHM

Terrepb

K

pemeHHIO

aa,IJ;a'lH

14.1.

IIepBbIii

:narr

pemeHHJI

COCTOlIT B orrpe,IJ;eJIe,.

HHH

MunUMa,//,blwa

<PYHKD;HH

hO

(t)

(16.15),

rropom,IJ;aeMoii

MHJhIMaJII>-

HbIM ,IJ;BHi-KeHHeM

SO

(t)

COrrpJImeHHOii

CHC'l'eMbI

(17.3):

!to

('t) = B'('r) X

X

SO

('t).

~

Ra1\

II

rrpem,IJ;e, aTO ,IJ;BHmeHHe

SO

'('1:)

MbI

mD;eM B

BlI,IJ;e

1)

SO

('1:)

= S

['1:,

tfl}·lO, s' (tfl)' c = lO'. c = 1. 3a,IJ;a'la COCTOlIT, CJIe-

,IJ;oBaTeJII>HO,

B orrpe,IJ;eJIeHHlI rrOCTOJIHHOI'O

BeKTopa

lO,

,IJ;JIJI

KOTopoI'o

<PYHKD;HJI

hO

(1:)

=

B'

(1:)

S

!'t.,

tfll·lo

JIBJIJIeTCJI

pemeHHeM

aa,IJ;a'llI

(16.8).:IIocJIe,IJ;HJIJI

<pOpMyJIlIpyeTCJI

a,IJ;ec(CJIe,IJ;YIOm;lIM

oopaaOM:

HaiiTlI

pO=min

max

y[B'(-r)S[-r,

tf3}.l] =

max

y'[B'(1:)sO(-r)) (23.12)

I

tQt~1:~tf3

toc~1:~tf3

-

rrpH,

l'e

=

1.

RaK

H B ,IJ;pyI'Hx

CJIy'laJIX,

B,IJ;eCI>

MomHO

HCXO,IJ;HTI>

lIa

rrpaBHJIa

17.2,

paOOTa1t C a"cmpeMa,//,bnoa

<py.:mD;ueii

h

o

(1:)

=

B'

(T)

S

[T,

tflJ

k

O

=

B'

(T)

sa

(T),

orrpe,IJ;eJIJIlI

ee

Terrepl>

lIa

yCJIOBHJI

vO=maxk"D=ko'·c

,

(23.13)

Ii

rrpH

max

y

[B'

(-r)

S

[-r,

tf3J

.kJ = 1.

toc~1:~tf3

3a,IJ;a'lll

(23.12), (23.13),

KaK

H

aHaJIOI'H

aa,IJ;a'l

(16.8),

(10.17),

CyTh

TPY,IJ;HhIe rrpOOJleMbI,

I\OTopbIe,

K COmaJIeHHIO,

He.

CBO,IJ;HTCH

K

KJIaCClI'leCKlIM

BapHaD;lIOHHbIM

aa,IJ;a'laM. O,IJ;HaKO B HaCTOHm;ee

BpeMII

Bce-TaKH

HaBeCTHbI

HeKoTopbIe

a<p<peKTHBHhle

rrpHeMI.I

HX

'lHCJIeHHOI'O

pemeHHH,

CBllaaHBbIB

r,

MeTOII,aMH

JIlIHenHOI'O

H

BI.myKJIOI'O

rrpOI'paM-

MHpOBaHHH

(CM.,

HarrpHMep,

paoOTY

[8*l).

3aMeTHM

em;e,

'lTO

aa,IJ;a'ly(23.12)

MomHO

paCCMaTplIBaTI>

TaKme

KaK

rrpOOJIeMy

ua

meopuu

npu6,//,u:HCenuJl,

r}iyH"LfUU.

IIOHCHlIM

aTO.

IIYCTb

,IJ;JIJI

rrpocToTbI

U

(T)

- CKaJIJIpHaJI <PYHKD;lIH. Tor,IJ;a,

rrOJIaraJI

.n;JIJI

orrpe,IJ;eJIeHHOCTH C

n

=1=

0

II

BbIpamaH

In

'lepea

It

(i

<n),

CBe,IJ;eM

(23.12)

K aa,IJ;a'le

Ha

aOCOJIIOTH~

MlIHlIMYM

n-l

pO

=

min

max

I

~

lig(i)

(-r)

- g

(-r)

I,

I

toc~1:~tf3

i=1

1)

llJIH,

Hor~a

<lTO

y~o6Hee

lIO rrpaBlIJIy 17.3 B

BH~e

SO

('t')

= S

[1:',

tocll

o

, s'

(toc)

XOC

=

10'

XOC

=

-1.

(C:'-

(~;)

§ 23)

YIIPABJIEHHE

C

MHHMMAJIbHbIM

HMIIYJIbCOM

185

r,IJ;e

<PYHKD;HH

g(i)

(T)

H g ('t)

BbIpamaIOTcH

HaBeCTHbIM

oopaaoM

'lepea

<PYHKD;HH

h(i)

(T),

a'lHCJIa

It

(i

= 1, ... ,n)

yme

He

CBHaaHbI

HHKaImMH

COOTHOmeHHJIMH.

Ho

rrOCJIe,IJ;HHH

aa,IJ;a'la

eCTI:.

He

lITO

HHoe,

KaK

aa,IJ;a-

n-l

'la

0

BbIoope]

JIHHeiiHOii

KOMOHHaD;IiH h

(T)

=

2J

It

g(i)

('t),

COC-

i=1

TaBJIeHHOii

HB aa,IJ;aHHbIX <pyHKD;H:ii

g(i)

('t) H rrpHOJIHmaIOm;e:ii

Hau-

,//,y'l(,UJ,UM 06paaoM (B

MeTpHKe

rrpOCTpaHCTBa

'£)

,IJ;pYI'YIO aa,IJ;aHHyIO

<PYHKD;HIO

g

(T).

TaKHM

oopaaOM,

B

(23.12)

MbI,

,IJ;e:iicTBHTeJIbHO,

HMe-

eM

THrrH'lHYIO

rrpOOJIeMY

HalIJIy'lmeI'O

rrpHOJIHmeHHH

<PYHKD;lI:ii B

rrpo-

CTpaHCTBe

'£.

R COmaJIeHHIO, BbIJICHeHlIe

aToro

OOCTOHTeJII.CTBa

He

O'leHb

rrOMoraeT

,IJ;eJIy,

TaK

KaK,

HeCMOTpH

Ha

rJIyooKyIO

MaTeMaTH'leC-

KyIO

paapaOOTKY

OOm;HX BorrpOCOB,

TeopHH

rrpHOJIIUKeHlIH

<PYHKD;H:ii

He

lIBOOHJIyeT BbI'lHCJIHTeJII.HbIMH

aJIrOpHTMaMH

,IJ;JIH

a<p<peKTHBHoro

pemeHHH

KOHKpeTHbIX

Ba,IJ;a'l,

rrO,IJ;OOHbIX

aa,IJ;a'le

(23.12).

Bo

BCHKOM

CJIy'lae,

oY,IJ;eM

,IJ;aJIee

rrpe,IJ;rrOJIaraTb,

'lTO

MHHHMaJIbHYIO

<PYHKD;HIO

h

O

(T)

(HJIH

aKCTpeMaJIbHyIO

<PYHKD;lIIO

hO

('1:»

y,IJ;aJIOCI. TeM HJIH

HHbIM

crrocoooM

Ha:iiTH,

a BMeCTe C TeM CTaJIO HaBeCTHI.IM H 'lHCJIO

pO.

IIpe,IJ;rrOJIaraeM,

KaK

BCer,IJ;a,

'lTO

aa,IJ;a'la

HMeeT

pemeHHe,

T.

e.

pO

>

O.

,IJ;JIH

'1'01'0 'lTOObI

rrOCJIe,IJ;Hee

yCJIOBHe BI.IllOJIHHJIOCI.

rrpH

JIIO-

ObIX

XOC

H

x/3,

HeOOXO,IJ;HMO

H ,IJ;OCTaTO'lHO, 'lTOObI

H'

[t/3'

TJ

·l

=

B'

(T)

S

IT,

t(3J

·l

=f=.

0

rrpH

l

=f=

O.

A

aTO

yCJIOBHe B

CBOIO

O'lepe,IJ;b

aKBHBaJIeHTHO YC.iIOBHIO rrOJIHo:ii yrrpaBJIJIeMOCTH, ,IJ;aHHoMy

Teope-

MO:ii

19.1.

TaKHM

oopaaoM

yCTaHaBJIHBaeTCJI, KCTaTH,

'lTO

HCrrOJIb30Ba-

HHe

OOOOm;eHHbIX

yrrpaBJIeHHii

dU(t)

He

pacmHpHeT

CBO:iiCTB

yrrpaBJIHe-

MOCTH CHCTeMbI (CM.

aaMe'laHHe

Ha

CTp.

145).

MbI

rrO,IJ;OmJiH

TerrepI.

KO

BTOpOMy

aTarry

pemeHHH

aa,IJ;a'lH, KOI',IJ;a,

HMeJI MHHHMaJIbHYIO

<PYHKD;HIO

hO

(T),

HaM

Ha,IJ;JIemHT orrpe,IJ;eJIJ'ITI.

orr-

THMaJII.HOe

yrrpaBJIeHHe

dUO

(T),

orrHpaHCb

Ha

yCJIOBHe

MaKcHMyMa

(23.10).

I

PaccMoTpHM

CHa'laJIa

cJIytIaii

CKaJIHpHOI'O

yrrpaBJIHIOm;erO

BOa,IJ;e:ii-

CTBHH

dU.

I1HTeHcHBHoCTb

yrrpaBJIeHHH

'X

[dU}

orrpe,IJ;eJIHeTCH B aTOM

CJIy'lae

paBeHCTBOM

(23.6)

H

rroaToMy

(CM. TaOJIHD;y

14.1)

p

[h

(-r)}

=

max

Ih

(-r)

I·

(23.14)

tlt~1:~tf3

By.n;eM rrpe,IJ;ll()JIaraTb,

'lTO

<PYHKD;HH

~.,<T)

= Ih

O

(T)

I,IJ;OCTHraeT

CBoero

HaHOOJIl>meI'O

aHa'leHHH

pO

(23.12)

JIHmb

B

r.OHe'lJ,HOM

'lHCJIe

TO'leK

1:}

(j

=

1,

•.

"

m)

Ha

rrpoMemyTKa

t

oc

< T < tfl.

HarrpHMep,

B CJIY-

'lae

rrOCTOHHHbIX

rrapaMeTpOB

CHCTeMbI (A =

const;

B = b =

const

B

(23.1»

yKaaaHHoe

yCJIOBMe BbIrrOJIHHeTCJI, eCJIH

BeKTopbI

b,

Ab,

...

. , " A

n-

1

b

mme:iiHo

HeaaBHCHMbI

II

MaTpHD;a

A

HeocooaH.

B

caMOM

,IJ;eJIe,

eCJIH ObI y

<PYHKD;HH

~

(T)

= I

hO

('t) I

Ha

OTpeaKe

[[oc, t(3tObIJIO

oecKo-

He'lHOe

'lHCJIO

MaKcHMyMOB,

TO

rrpOHaBO,IJ;HaJI

dhO

('t)/d1:

HMeJIa

ObI

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aTOM

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HyJIe:ii.

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hO

(1:)

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<PYHKIWll

C

:>·

0

:'·

,',":"

',.;

..

.

'.

)

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186

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[rJI.

6

aHaJIlITlI'IeCKaH,

II

rronoMy

B TaKOM

CJIy'lae

OHa 6I:JJIa 6I:J

rro-

CTOHHHOH. Op;HaKO

rrpH

yKaaaHHI:JX

HaMlI

YCJIOBlIHX k

O

('r)

6I:JTb

rrOCTOHHHOH He

MomeT.

,D;eHcTBlITeJIbHO, eCJIlI rrpep;rrOJIOmlITb, 'ITO

k

O

('r)

= b'so

('r)

=

const,

TO

BI:JrrOJIHHJIHCb 6I:J COOTHomeHlIH

dh

O

Ci')

/

d-r

= -

{Ab}'

SO

(-r)

_ 0,

..

"

dnh

o

/

din

=

(_1)n

{Anb}'

SO('I:)

,

lIa

KOTOpI:JX B ClIJIY JIlIHeHHOH HeaaBlIClIMOCTlI BeKTopOB b,

Ab,

...

,

An-1b

BI:JTeKaJIO

6M

paBeHCTBO

SO

(or)

= 0,

KOTopoe

HeBoaMomHO BCJIeli-

CTBlIe

KpaeBoro

yCJIOBHH

c'SO

(tfJ)

= 1.

TIoKameM

Terrepb,

'ITO 4>YHKIJ;lIH

UO

(or), orrpep;eJIHIOm;aH lICKOMoe

orrTlIMaJIbHOe

yrrpaBJIeHHe

dUO

('r),

HBJIHeTCH

Kyco'lHo-rrocToHHHOH

C

paapI:JBaMH

HerrpepMBHOCTlI

TOJIbKO B

TO'lKaX

T:j

(j

= 1,

...

, m).

,D;JIH

9Toro

p;OCTaTO'lHO rrOJ{aaaTb, 'ITO rrOJIHOe

lIaMeHeHlIe

val'

[UO]

4>YHJ{-

IJ;lIH

UO

('I:)

Ha

JII060M

OTpeaJ{e

[t

1

,

t

2

],

He

cop;epmam;eM

TO'leJ{ 'l:j> paBHO

Hymo.

TIOJ{ameM

8'1'0.

B caMOM p;eJIe, eCJIlI OTpeaOJ{

[t

1

, t

2

]

B;e

cop;epmlIT

TO'lel{

T:

j,

TO

max

Ih

O

('I:) 1<

wax

Ik

O

('I:)

I·

(23.15)

t,~-r:~t.

ta~-r:~t(3

C

p;pyroH

CTOpOHlJ,

OIIlIpaHCb

Ha

yCJIOBHe MaJ{CHMyMa (23.10),

M:bI

MomeM

aarrlIcaTb

HepaBeHCTBO

max

Ih

O

('r)

I·

val'

[U°lt~

= p

rho

(-r)]

.

p*

[UO

(t)]

=

ta~-r:~t(3

t(3

j h

O

('t)dUO ('t)-<,

max

IhO('I:)I·

val'

[U°J;'

+

t

toc~-r:~t,

'"

+

max

Iho('r)l·val'[UO]~:+

max

IhO('I:)I·val'[[,T()]/~.

(23.16)

t,~-r:~t.

t,~-r:~t(3

')

O'leBlIp;HO,

BCJIep;CTBHe paBeHCTBa

~

o~

.

~

val'

[U°ltoc

=

val'

[UOk" +

val'

[U

lt,

+

val'

[U°lt.

COOTHomeHlIH (23.15), (23.16)

MoryT

op;HOBpeMeHHO BI:JrrOJIHHTbCH

JIlImb

rrplI

yCJIOBIiflI

val'

[[70 (-r)P'

t,

=

O.

(ho

H p;OI{aaI:JBaeT

Hame

yTBepmp;eHHe.

.

IIycTb

Terrepb

yrrpaBJIHIOm;ee Boap;eHcTBHe

dU

- BeKTOp H HHTeH-

CHBHOCTb x

[dU]

9Toro

Boap;eHcTBlIH orrpep;eJIHeTCH

paBeHcTBoM

(23.7).

RaJ{

H

rrpemp;e,

MM

6yp;eM

rrpep;rroJIaraTb,

'ITO BeKTop-4>yHIrn;lIH

~

('r)

=

l'

[kO

(T:)]

Ha (23.12) p;OCTHraeT

CBoero

HaH60JIbmerO

aHa'le-

HlIH

pO

(23.12)

JIHmb

B KOHe'IHOM '1HCJIe TO'leK

T:j

(j

= 1,

...

,

m.).

§

23J

YIIPABJIEHHE

C

MHHHMAJIbHbIM

HMIIYJIbCOM

187

Torp;a

BeKTop-4>YHKIJ;lIH UO

('I:),

orrpep;eJIHIOm;aH lICKOMoe

OrrTlIMaJIb-

Hoe

yrrpaBJIeHlIe

dUO

(t), HBJIHeTCH

Kyco'lHo-rrocToHHHOH

4>YHKIJ;lIeH

C

pa3pMBaMlI

HerrpepMBHOCTlI

TOJIbKO B

TO'lKaX

'tj

(j

= 1,

...

, m).

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6yp;eM sp;eCb OCTaHaBJIHBaTbCH

Ha

p;oKaSaTeJIbCTBe

nora

yT-

Bepmp;eHlIH,

TaK

KaK

OHO

CKJIap;bIBaeTCH

lIa

paccymp;eHHH,

COBepmeH-

HO

aHaJIOrlI'IHlJX

TeM,

KOTopbIe

MbI

rrplIBeJIH

BMme

B

CJIy'lae

CKaJIHp-

Horo

Boap;eHCTBlIH dU.

TaKlIM

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MI:J

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K BbIBOP;y, 'ITO lICKOMaH 4>YHKIJ;lIH

UO

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HMeeT BlIP;

q(O)

= 0

rrplI

t",

-<,

-r

-<,

'tl,

q(l)

rrplI

'tl <

-r

-<,

't2,

yo

(-r)

= : :

(23.17)

. .

/

q(m)

rrplI

-r

m

<

-r

-<,

t(3,

eCJIlI

-r

m

=/=

t(3, l'IJIl'I Bl'IP;

(

q(O)

= 0

rrpH

t",

-<,

-r1,

q(l)

rrpH

'tl <

-r

-<,

't2,

j

Uo

('t) = :

(23.18)

q(m-l)

rrpH

-r

m

-

1

<

-r

<'t

m

,

l

q(m)

rrpH

'l: = t(3,

m

ecJtlI

-r

m

=

t(3

(rrpep;rrOJIaraeTCH,

'ITO 'tj >-rj-l)'

060aHa'IHM

p(j)

=

q(j)

-

q(j-l)

(j

= 1,

...

,

m).

(23.19)

YCJIOBlIe

MaKclIMyMa

(23.10)

TpaHc4>opMlIpyeTcH

Terrepb

B YCJIOBlIe

.ho'

('tj)

p(j)

=

max

h

O

'

(-rj)

p

rrpH

y*

[p]

= y*

[p(j)]

(23.20)

p

(j

= 1,

...

,

m.),

rrpH'IeM

y*

[p(l)]

+... +y*

[p(m)]

=

11

pO.

(23.21)

RpOMe

Toro,

BeKTopbI

p(j)

(j

= 1, ,

..

,

m)

p;OJImHbI yp;OBJIeTBOpHTb

ypaBHeHlIHM

h[iJ'

[t(3,

'tl] .

p(l)+

•

••

+

h[W

[t(3,

'em] •

p(m)

=

Ci

(i = 1,

..

.

,n),

(23.22)

B KOTopbIe

rrpeBpam;aeTcH

sp;eCb ClICTeMa (15.6). B '1aCTHOCTH, eCJIH

dU - CKaJIHp,

TO

YCJIOBlIe (23.20)

OSHa'laeT,

'ITO

sgn

pU)

=

sgnhO(T:j).

ECJIlI

me

yrrpaBJIHIOm;ee Bosp;eHcTBlIe

dU

-

BeKTop

H

1'*

[dU]

=

"""

II

dU

II,

TO

YCJIOBlIe (23.20)

OaHa'laeT

l

'ITO BeKTOpbI p(j) H k

O

('rt)

O~IifHa~OBO

HarrpaBJIe~w,

188

···.··

..

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c

[rJI.

6

CBOnCTBA

OITTHMAllbHOrO

YITPABllEHHH

MCITOJIh3yJI

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b-<PYHRI\IIlI

(CM.

§ 7),

MhI

MomeM

TaRme

CRa-

3aTh,

'ITO

B03AeB:CTBlIe

Ha

cHcTeMy

(23.1)

0606m;eHHoro

yrrpaBJIeHIIH

dUo

(t),

orrpeAeJIJIeMOrO

<pYHRIJ;lIeH

UO

('r)

(23.17), (23.18),

aRBII-

BaJIeHTHO

B03AeB:CTBIIIO

Ha

CIICTeMY

(23.3)

IIMrrYJIbCHOrO

yrrpaBJIeHIIJI

m

U (t),

paBHoro

U (t) =

2}

p(;i)

6 (t -

tj).

3Aecb

rrOCTOJIHHhIe

BeRTopbI

;=1

p(:i)

(23.19)

xapaRTepll3yroT,

CJIeAOBaTeJIbHO,

BeJIH'IIIHy

MrHOBeHHhIX

IIMrrYJIbCOB

II

06JIaAaIOT

CBOB:CTBOM .

sgnpV)

=

sgn

h2

(tj)

(k

=1,

...

,

r;

l=

1,

...

,m).

(23.23)

MTaR,

MhI

MomeM

a.arrllcaTb

OrrTHMaJIbHOe

yrrpaBJIJIIOm;ee

B03AeB:-

CTBlIe

dUo

B

BIIAe

In

dUO

(t) =

2}pW

6 (t -

tj)

dt.

(23.24)

j=l

3aMeTIIM,

'ITO

rrpll

t{3

=

'tm

BeJIH'IIIHy

6 (t -

'tm)

B (23.24)

CJIeAyeT

rrOHIIMaTb

B CMhICJIe

6-

(t -

'tm)

(CM.

§ 7,

CTp.

50).

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HamllX

paccymAeHIIB:

MhI

06'LeAlIHlIM

B

CJIeAYIOm;yro

TeopeMY·

Teo

p e M a 23.1. PaccJ.wmpult£ aaoa"ly 14.1.

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MunUMU-

aupyemcit

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x [dU] euoa (23.7).

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-

mo

ua

oeu:JICenuu

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cucme:/ttb£

(17.3), yooMemeopitlO14UX If,pae-

eOMy

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so'

(t{3)

·c

= 1,

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om";'.fl,onenuit

om

nY.fl,it

nopO:JICOaeMOU

UM

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('t)

=

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noao e

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30ecb

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pO

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om ny.fl,it,

mo

3aOa'la

14.1

06

onmUMa.fl,bnOM

yripaMenuu

cucmeltwu (23.1)

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pemenue,

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pO

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.,;,a.,;,

paa MunUMMbnoe ana"lenue

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x

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1/

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o.

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(t)

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noc.fl,eOOeame,n,b-

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onUCb£eaeMblX

b-(/jYlt";'/fUitMU

pO)

6 (t

-.'tj)

(j

= 1, ... ,m) u

npU.fl,aaaeMblX

11-

ynpae.fl,iteMOU

cucmeMe

e

me

MOMenmbl

epeMenu

'tj

(j = 1, ...

,m),

/'i,oeoa

om";'.fl,onenue

om

nY.fl,it

I'

rho

('t)]

Mwi,lUltMbUOU

(/jY11./'i,lfUU

kO

('t)

oocmueaem

ceoeeo

liau6Mb-

meeo

alW'lellUit pO. B 'lacmllocmu,

ec.fl,U

y*[dU]

=lldUII,

mo

I'

rhO]

=

II

kO

('t)

II

U lianpaeMNue

eer;,mopa

pU) coenaoaem C uanpaMenueM

eer;,mopa

kO

(Lj)'

c:::~!':

~.-'

§ 23]

VITPABJIEHHE

C

MIIHHMAllbHhIM

IIMITVJIhCOM

189

Ha

OCHOBaHlI1I

aTOB:

TeopeMhI

MhI

MomeM

yRaaaTh

CJIeAyrom;yIO

rrpOIJ;eAYPY

pemeHIIJI

3aAa'IlI

06

OrrTIIMaJIhHOM

yrrpaBJIeHHH

CIICTe-

MOB: (23.1).

,n;JIJI '1'01'0

'IT06hI

pemllTh

yRaaaHHyIO

3aAa'IY,

CJIeAyeT;

1)

rro

aaAaHHOB:

HHTeHCIIBHOCTII

x [dUJ (23.7)

BhI6paTb

MeTpllRy

p [k('t)] (23.9)

TaR,

'IT06hI

BeJIlI'lllHa

x [dUJ

HMeJIa

CMhICJI

HOpMhI

p*

[UJ (23.8);

2)

HaB:TII,

OITlIpaJICb

Ha

YCJIOBIIJI

(23.12),

MIIHHMaJIbHOe

ABlImeHlIe

SO

(t)

COrrpJIjReHHOR

CIICTeMhI

(17.3),

orrpeAeJIIIB

TeM

caMhIM

MHHII-

MaJIbHyro

<PYHRIJ;IIIO kO

('t)

=

B'

('r)

SO

('r)

H

MaRcHMyM

pO

(23.12)

ee

OTRJIOHeHIIJI

'l' [kO

('r)

J 0'1'

HyJIJI;

3)

orrpeAeJIIITb

MOMeHThI

BpeMeHII

t =

't

j

(j

= 1,... , m), B

ROTophIe

6'1'0

OTRJIOHeHlIe

I'

[kO

('t)J

AOCTlIraeT

CBoero

Hall60JIbmerO

aHa'Ie-

HlIJI

pO;

4)

AJIJI

orrpeAeJIeHIIJI

OIITIIMaJIbHOrO

yrrpaBJIeHHJI

dUO

(23.24) BOC-

rrOJIhaOBaTbCJI

yCJIOBHJIMII

MaRcHMyMa

(23.20), (23.21);

ecmr

aToro

ORameTCJI

HeAOCTaTO'IHO,

TO

Hap;JIemllT

COCTaBIITb

CIICTeMY

ypaBHe-

HIIB:

(23.22)

II

ITpllBJIe'Ib

eeR

orrpeAeJIeHIIIO

BeJIH'IIIH

pW.

IIYHRThI

2)·

-

4)

COOTBeTcTByIOT

rrpaBHJIaM

MHHIIMaRCa

17.1,

17.3,

orrllpaIOm;IIMCJI

Ha

MUUUMa.fl,bnylO

<PYHRIJ;IIIO kO

('t).

AHaJIOrll'I-

HhIM

06pa30M

OHII

CTpOJITCJI,

eCJIII

3a

OCHOBy

BhI6HpaeTCJI

ar;,cm-

peMMbnait

<PYHRIJ;IIJI k

a

(r),

<PlIrYPllpyrom;aJI

B

rrpaBIIJIaX

17.2, 17.4.

B

'IaCTHOCTII,

rrYHRThI

2) H

3)

<pOpMyJIlIpyIOTCJI

TorAa

CJIeAYIOm;IIM

06pa30M;

2)

HaRTH,

OrrllpaJICh

Ha

yeJIOBlIe

(23.13),

aRCTpeMaJIbHOe

ABHmeHlIe

sa

(t)

COrrpJIjReHHOB:

CIICTeMhI(

17.3),

orrpeAeJIIIB

TeM

caMhIM

aRCTpe-

MaJIbHyro

<PYHRIJ;IIIO k

a

('t)

=

B'

('t)

sa

('t)

II

'IIICJIO

VO

(23.13),

orrpe-

AeJIJIIOm;ee

HOpMy

OrrTIIMaJIbHOrO

yrrpaBJIeHIIJI

dUO;

x

[dUO]

=

p*

[uoJ

=

vO;

3)

orrpeAeJIIITb

MOMeHThI

BpeMeHH

t =

't

j

(j

=

1,

...

, m), B

ROTO-

phIe

OTRJIOHeHlIe

'l' [k

a

('r)

J

6RCTpeMaJIbHOB:

<PYHRIJ;HII k

a

('t)

0'1'

HyJIJI

AOCTHraeT

CBoero

Hall60JIhmerO

3Ha'IeHIIJI,

paBHoro

eAHHIIIJ;e.

TeopeMa

23.1,

Ha

OCHOBe

ROTOpOH

MhI

1I3JIOmllJIlI

rrpOIJ;eAYPY

pe-

meHHJI

3aAa'IlI

06

OrrTIIMaJIhHOM

yrrpaBJIeHlI1I

CIICTeMOB: (23.1),

yCTa-

HOBJIeHa

HaMil

B

rrpeArrOJIOmeHIIII,

'ITO

OTRJIOHeHlIe

I'

[kOJ

0'1'

HyJIJI

MIIHIIMaJIbHOB:

tPYHRIJ;1I11

kO

('t)AocTlIraeT

CBoero

Hall60JIbmerO

3Ha-

'IeHllJI

pO

JIlImh

B

OTAeJIbHhIe

1I30JIlIpOBaHHhIe

MOMeHThI

BpeMeHII

't

J

(j = 1,... , m). B

CJIy'IaJIX,

RorAa

aTO

YCJIOBlIe

He

BhIrrOJIHJIeTCJI

(Harr-

pllMep,

RorAa

kO

('t)

_

const),

AJIJI

orrpeAeJIeHIIJI

yITpaBJIeHHJI

dUO

Heo6'xoAIIMo

rrpllBJIeRaTh

AOrrOJIHHTeJIbHhIe

coo6pameHHJI.

He

06-

cymAaJI

nOT

Borrpoc

rroAP06HO,

OTMeTHM

JIlImb,

'ITO

H B

TaRIIX

CJIy-

'IaJIX

yrrpaBJJeHlIe

dUO

(t)

CJIeAyeT

HCI<aTI> B

BHAe

rrOCJJeAOBaTeJIbHOCTH

HMIIyJIbCOB.

IIpH

aTOM

MOMeHThI

BpeMeHIi

t =

Lj

CJIeAyeT

BhI6HpaTb

TorAa

li3

yCJIOBHJI,

'lT06hI

paHr

MaTpHIJ;hI

F

{kf/

J

('t;)}

(i

= 1, ..., n;

k = 1,... , r; j =

1,

...

,m)

CHCTeMhI

(23;22)

6hIJI

paBeH

paHry

ee

190

r

\~~.-;:

CBOaCTBA

OrrTHMAnbHOrO

YrrPABnEHH~

[rJI.

6

pacmHpeHHOHMaTpHn;LI (F,

C)

H

qTOOLI

pemeHHe

3TOH

CHcTeMLI'y.n;OBJIeT-

BOpHJIO

OrpaHHqeHHIO

(23.21).

BOJIee

Toro,

Ha rrpHMepax,

paCCMaT-

pHBaeMLIX HHme, MomHO yoe.n;HTbcH,

qTO

qHCJIO

m

MOMeHTOB

Bpe-

MeHH

t = Tj

y,n;aeTCH

BLIopaTb

He

rrpeBocxo.n;HIIJ;HM

nopH,n;Ka n

CHC-

TeMLI

(23.1).

MLI

He

oy.n;eM

OCTaHaBJIHBaTbCH

Ha

no,n;pOOHOM

ot5m;eM

pa300pe

B03HHKaIOm;eH

3.n;eCb

cHTyan;HH,

OTCLIJIaH

HHTepeCYIOm;erOCH

qHTaTeJIH K

Cnen;HaJIbHOH

JIHTepaType [1100,.n;;

123a;

3890].

OT-

MeTHM

JIHmb,

qTO

onpe.n;eJIeHHe

MOMeHTOB

Tj

MomeT

OLITb,

HarrpHMep,

ocym;eCTBJIeHO

nyTeM

MaJIOH

.n;eqJOpMan;HHrrapaMeTpoB

CHCTeMLI,

HaqaJIbHLIX

yCJIOBHH

HJIH

BH,n;a

HHTeHCHBHOCTH

%

[dU]

C

nOCJIe,n;yIO-

m;HM

npe.n;eJIbHLIM nepexo,n;OM

(CM.

HHme

KOHen;

3Toro rraparpacf>a,

a TaKme

KOHen;

§

24).

ITepeii,n;eM K KOHKpeTHLIM rrpHMepaM.

llpBMep

23.1. PaccMoTpBM 3aAa'Iy, nOCTaBJIeHlIYIO B § 14 (CTp. 90), 0

nepe-

BOAe

MaTepHaJIbHOH TO'lRH m Ha

KpyroByIO

Op6HTy,

oXBaTLIBaIO~YIO

IJ;eHTp

Macc

M.

BYAeM npeAnOJIaraTb H 3AeCb, 'lTO peaKTlIBHaH Macca BLI6pacLIBaeTcH

C nocToHHHOH

no

MOAYJIIO

OTHOCHTeJIbHOH

CKOpOCTbIO

B UanpaBJIeHHH,

nepneH-

AHKyJIHpHOM

It

paAlI.ycy-BeKTopy TO'lKH. YpaBHeHlI.H ABlI.meHHH TO'lRB

MoryT

6LITb

TorAa

3anHcaHLI B

BRAe

(CM.

(14.13), (21.10»

II',

dXI=X2dt,

I

Ih

;

dX2.= -

Xl

dt+

Xs

dt,

(23.25)

l

~ll

;

dxs=

dUo

B Ka'IeCTBe

HHTeHClI.BHOCTH

" [dU],

OIJ;eHHBaIO~eH

pecypcLI,

3aTpa'lHBaeMHe

Ha ynpaBJIeHHe Ha oTpeaKe BpeMeHH

0':;;;;;

t

.:;;;;;

.T, MLI BLI6epeM BeJIB'lHHY (23.6)

nOJIHOrO

HMUYJIbCa

ynpaBJIHIO~eH

CRJILI

dUo

B § 14 nOKaaaHO, 'ITO MHHHMH3a-

IJ;.l<IH

HMUyJIbCa (23.6) MomeT 6LITb

lI.CTOJIKOBaHa

KaK

HCnOJIHeHHe

Tpe60Bamm

0

Hall.MeHbmeM

pacxoAe

peaKTHBHoH

MaCCLI

ml

(0). B

COOTBeTCTBHH

C

o6~eH

npo-

~,

IJ;eAYpon pemeHHH,

npemAe

Bcero,

HaAJIemaJIo6LI onpeAeJIHTb

no

3aAaHHoH HH-

TeHCHBHOCTH

" [d

Uj

(23.6) HOpMy P [h ('r) j (23.9).

Ho

MLI

yme

3HaeM, 'lTO

p [h

(1')

j onpeAeJIHeTCH B

aTOM

CJIyqae paBeHcTBoM (23.14). CJIeAyH npaBHJIY

17.2, HanAeM 3KCTpeMaJIbHYIO

lPYHltIJ;RIO

h

a

(1'), onRpaHcb Ha

yCJIOBRH

(23.13)·

<I>YHKIJ;HH

h

(1')

=

B'

(1')

S [1', tf3j.k,

lPHrypRpYIO~aH

B (23.13), BLI'lHCJIeHa

ym

e

D

§

21

II

OnpeAeJIHeTCH

lPOpMyJIOH

(21.11).

3anHmeM

ee

B

DHAe

h

(1')

= k

l

-$-

k

s

+'-

-V

k~

-+

k~

•

sin

'(T

~

l'

-

~),

(23.26)

rAe

tg

~

= k

l

/k

2

•

3a,n;a'la (23.26) MomeT 6LITb Tenepb

CW,OPMYJIHpOBaHa

TaK: BaHTH

,,0

=

max

(clk

l

",

Csk

2

+

caks)

(23.27)

k

npH

max

Ik

l

-t-

k

s

+-

-V

k~

+-

k~.

sin

(T

.....

't'

....

~ll':;;;;;

1, rp;e

KOMUOHeHTLI

ci

BeKTO-

O~"<~T

pa

C onpeAeJIeHbl paBeHcTBoM (21.13).

,IJ;JIH

HarJIHAHOCTH

paCCMOTpHM

CJly'IaH,

KorAa

T = 4n.

TorAa

HeTpYAHO

coo6paaRTb, 'ITO 3ap;a'Ia (23.27)

CDOARTCH

K

CJIeAYIO~eMY:

HaHTH:

V

O

=

max

[CIXI

+-

C2X2

+-

csXs]

(23.28)

l\

~,

r·

(~~,:)

§

23J

YUPABJI8HHE

C

MHHHMAJI.hHhIM

HMIIYJIl>COM

191

npu

I

Xs

I+

-V

X~

+

X~

= 1.

(2:1.29)

~lleCb

BBe~eHLI

0603Ha'leHI1H:

Xl

=

leI,

X2

=

le

2

,

Xa

=

leI

+Irs;

;;;,

=

:CSO

- X.lO,

C2

==

-

Xzo,

Ca

=--

X~O.

PaBeHcTBo (23.29) onpeAeJIHeT D rrpocTpaHcTBe

{Xl'

X2'

Xs} lIOBepXHOCTb

@.,

COC-

TaBJIeHHYIO

H3

KOHYCOB

(pRC. 23.1).

reo-

MeTpH'leCKH aaAa'la (23.28), (23.29) oaHa-

'IaeT, 'ITO cpeAR

Bcex

nJIOCKOCTen

fP, Ka-

.

caIO~HXCR

rrOBepXHOCTR

(ijj,

CJIeAyeT

Hanni

Ty,

KOTopaH opTOrOHaJIbHa K BeKTopy

c;

T01JRa

KacallRH {Xr,

X~,

X~}

3TOM

IIJlOCKO_

CTH

C

nOB'epXHOCTbIO

@f

.II

orrpeAemlT HCKO'

MbTM

BeHTOp

X

a

=

{X~,

X~,

X~}·

lia

pHC.

23.1

DHAHO,

'ITO

eCJIH

BeKTOp C

opTOrOHaJleH 60KOBon

ITOBepXHOCTJ!:

@,

TO

ITJIOCKOCTb

HacaeTCR HOHH'IeCKOn rrOBepx-

HOCTH

ITO

06paaYIO~eM

H,

CJleAOBaTeJIbHO,

aaAa'Ia (23.28), (23.29)

HMeeT

6eC'IHCJleH-

Hoe

MHomeCTBO

pemeHHH

Xa.

ECJIH

me

BeKTOp

C

He

opTOrOHaJleH K

ITOBepXHO-

CTR

@.,

TO

ITJIOCKOCTb

KacaeTCR @ JIHmb B

OAHon

TO'lKe H,

CJleAOBaTeJlbHO,

aaAa'Ia

(23.28), (23.29)

HMeeT

eAHHCTBeHHoe

pe-

meHHe

Xa.

P.lIC. 23.1.

TaKHM o6paaoM, BIrAHM, 'ITO pemeHHe X

a

3TOH

aaAa'IH

AOCTaBJIHeT

BeKTOp

X

a

=

{;;;'FJ!ci

+c:'

c2/

Vc~

+

c:'

OJ,

t23.30)

eCJlH

yrOJI

1jJ

(CM.

P.IIC.

23.1) TaKOB:

:It - _ :It

1jJ

=

2"

-

arccos

/

cs/l/lcl/

=

2"

-

arccos

I

xs

o

I :It

"Ii·

<T'

r

{x30

-

XI

o

)2

+ x:

o

+

x;o

.IIJlH

BeKTOp

(~3.31)

a -

X =

{O,

0,

sgn

cs},

(~3.32)

eCJlR

:It I

X80

I :It

1jJ

=

2"

-

arccos

"Ii

;> T . (23.33)

r

{xso

-

XlOj2

+

x~o

+

x;o

'

IIpe.n;IToJIomHM CHa'IaJla, 'ITO

BLIITOJIHHeTCH

yCJIOBHe

(23.31).

TorAa

H3

(23.26),(23.28),

(23.30) ITOJIy'laeM, 'lTO 3KCTpeMaJlbHaR

lPYHKIJ;HH

ha

('r)

olIpe-

AeJIReTCH

paBeHCTBaM.II

a

h

(1')

= -

sin

(1'

+

~)

=

V(xso

-

XIO)2

+

x:

.hO(T), (23.34)

o

(.g~=~qO

(0':;;;;;

~

<

~:It).

~t'~o

••

_~_.'

• I w

....

· ,

-'

P'

'"

,

.....

L '

c'

[rJl.

·6

CBoficTBA

OITTMMAllbHOrO

YITPABllEHHfl

192

CJIe.o;OBaTeJIhHO, OTRJIOHeIDIe

OT

HyJIH

'Y

rhO

('t')]

= I h

O

('t') I

.MUHu.uaJl,l>/·/,Ou

WyH-

I<D;HH

h

O

('t')

.o;OCTHraeT

CBoero

MaRcHMyMaj

1

(23.35)

pO

=

---;:====~;==

V

(XSO

- XI0)2

~

X;o

B MOMeHTM BpeMeIDI

. n

(XI0

-

XSo)

(23.36)

't';=-2'

('2j-1)-

~O

~=arctg

.

X20

U= 1

•...

,4).

eCJIIi

X20

=F

0, HJIH B MOMeH'l'bI BpeMeHH

't'j = n

(j

-

1)

(j

= 1,..., 5),j (23.37)

eCJIH

X 0

=0.

"YCJIOBHH

MancHMYMa (23.20), (23.21)

s.o;eCh

He.o;ocTaTO'lHO

ll:J1JI

2

nOJIHora- onpe.o;eJIeIDIJI

Bcex

BeJIH'lHH

p(j).

II

03TOMY

COCTaBHM

cHcTeMy

ypaBHcHHH

(23.22). B CJIy'lae (23.31)

npH

X

2

0

=F

0 3Ta

CHCTeMa

J-IMeeT

BHll:

(1

-

sin

~O)

p(l)

+

(1

+

sin

~02

p(Z)

+

(1

-

sin

~01

p(S)

+ 1

+

(1

+

sin~)

p(4)'=

-

XIO.

(2:1.38)

_

cos

SO

pel)

+-

cos

SOp(Z) -

cos

SOp(S) +

cos

~p(4)=-xzo,

p(l)

+

p(Z)

+

p(3)

+

p(4)

=

-X

3

0'

H

pesyJIhTaT

ee

paspemeIDIJI

npH

OrpamI'leHHH

(23.21)

I

pel)

I+I

p(Z)

I+ I

p(S)

I+I

p(4)

1=

V(-X-SO---XI-0)-2-+-

X

-;-0

CJI

eAyICJIJ,EH:

p(ll,

p(Z),

p(S),

p(4)_

npOHSBOJlhHhle

lJECJJa,

C1ECECHHhle

yCJJ

CBI-fJIM

I

p(l)

I+I

p(3)

I=

~

{V(XSO

-

XIO)2

+-

X;o

-

XSO

sgn

X20} ,

sgn

p(l)

=

sgn

pes)

= -

sgn

X9

-0'

~

(23.39)

I

p(Z)

l-.t I

p(4)

I=

~

{V'(X80

-

xlO)2

+

x;o

+

XSO

sgn

X20} ,

sgn

p(Z)

=

sgn

p(4)

=

sgn

XZO,

IIpn

X20

= 0

CHCTeMa

(23.22), cooTBeTcTByICJIJ,aJI CJIy'laIC (23.31). HMeeT BEll:

O.

p(l)

+-

2p(2)

+-

O.

p(3)

+-

2p(4)

+

O.

p(5)

= -

XI0'j

O.

p(l)

+

O.

p(Z.) +

O.

pes)

+-

o.

p(4)

+-

O.

p(5)

=

0,

(23.4.0)

pel)

+

p(Z)

+

p(S)

+-

p(4)

~

p(5)

= -

XSO,

H

pesyJIhTaT

ee

paspemeIDIJI

npH

OrpaHH'leHHH (23.21)

I

pel)

I+ ...

+-

I

p(o)

I=

V(X30

-

XIO)2

-I-

X~o

§

2.~]

r

(".'.::,~

..

···

VITPABllEHME

C

MMHMMAllbHbIM

HMITVllbCOM

193

CJIeAYICJIJ,UH:

pel)

,.,.,p(6)

- JIIC6Me

1

p(Z)

+

p(4)

= -

2"

XIO,

'IRCJIa,

CTeCHeHHMe

yCJIOBHHMH

sgn

p(Z)

=

sgn

p(4)

=

sgn

(X30 -

XI0).

1

pel)

+

pes)

+

p(o)

= - XS

O

+

2"

XI

O

'

(23.41)

sgn

pel)

=

sgn.p(S)

=

sgn

plO)

= -

sgn

(XSO - XIO)'

TaKHM

o6paSOM,

eCJIH

Ha'laJIhHble

YCJIOBliIJI

xio

(i

= 1, 2,

3)

yP;OBJIeTBOpHICT

COOTHomeHHIC (23.31).

TO

OnTHMaJIhHOe yrrpaBJIJIICJIJ,ee

BOSp;eHCTBHe

dUO

(23.24)

MOiKeT 6MTh npep;CTaBJIeHO B BHp;e:

(npH

X

2

0

=F

0)

dUO(t)={p(I)6(t-~

-I-so)+p(Z)6(t-;

n-l-s

o

)

+

+p(S)6(t~

~

n+so)

+p(4)

6

(t-;

n+so)}at.

(23.42)

rp;e

p(i)

(i

= 1

....

,4) yP;OBJIeTBOpJIICT (23.39);

(rrpH

X

2

0 =

0)

dUO

(t) =

{pel)

(\ (t) +

p(Z)

(\

(t

_

n)

+

p(3)

6(t

-2n)

+

p(4)

6

(t

- 3n) +

+

p(li)

6-

(t

- 4n)} dt, (23.43)

rAe

p(i)

(i

= 1,... ,5) yp;OBJIeTBOpJIICT (23.41). .I:Is (23.39), (23.41) CJIep;yeT. 'lTO

'lHCJIO HMnyJIhCOB

p(i)

6 (t - 't'i) B (23.42), (23.43) MOiKeT 6MTh

BM6paHo

(H

He-

op;HosHa'lHoI) He npeBOCXOp;HIII,HM nopJIJJ;Ka

CHCTeMM.

.

IIpell:rrOJIOiKlIM

Teneph,

'ITO Ha'laJIhHMe yCJIOBHJI yp;OBJIeTBOpHICT

COOTHO-

meHHIO (23.33).

Torp;a

HS

(23.26), (23.28), (23.32)

nOJIy'laeM,

'lTO MHHHMaJIhHaJI

!PYHKD;HJI

h

O

('t') TOiKp;eCTBeHHa

rrOCTOJIHHOH:

1

hO('t')

=-

-,

(23.44)

X80

lI,

CJIep;OBaTeJIhHO. OTKJIOHeIDIe Ih

O

('t') I ee

OT

HyJIJI

eCTh

TaniKe

BeJIH'lHHa nOCTO-

JIHH8H,

paBHaH

Ih

O

('t') I = 1/ Ix

so

I=

pO.

(23 45)

OrpaHH'lHMCH

ll:JIH

npocToTM

CJIyqaeM,

norp;a

BeRTOp C

(CM.

pHC. 23.1)

HanpaB-

JIeH

no

OCH

)(;3'

T.

e. X

2

0 = 0, XI0 = X80. H

npOBepHM,

'lTO OnTHMaJIhHOe

ynpaB-

JIHIOJIJ,ee

Bosp;eii:cTBHe

dUO

MOiKeT 6MTh sp;ech

TaniKe

nocTpoeHo

B BHll:e (23.24).

"YnpaBJIHICJIJ,ee

BOSll:eHCTBHe

dUO

HJIJ,eM

B Bnp;e

dUO

=

{p(l)

6

(t

-

't'1)

+

p(Z)

()

(t

-

't'tn

dt.

(23.46)

rll:e 't'1,

't'2

-

HeRo'fopMe

MOMeHTM

BpeMeIDI

HS

npoMeiKyTKa

0 < t < 4n.

CHcTeMa

ypaBHeIDIH

(23.22) MOiKeT 6MTh npe.o;cTaBJIeHa B

BHp;e

pel)

cos

't'1

+

p(Z)

cos

't'2

=

0'1

(23.47) .

p(l)

sin

't'1

+-

p(Z)

sin

't'2

.

0,

pel)

+-

p(Z)

= _ X80.

IIonpo6yeM

BM6p8Th,

HanpllMep,

't'1

=

n,

't'2

= 4n.

PaspemaJI

Torp;a cHcTeMy

ypaBHeHHH

(23.47)

npH

OrpaHJ-I'leIDIH (23.21)

I

p(l)

I+I

p(Z)

I = I

zao

I,

7 H. H. KpacoBcKult

194

p

....

(.'-'::

[rn.

6

CBOnCTBA

OITTHMAnLHOrO

YITPABnEHllff

HaxOAllM,

'ITO

1

p(l)

=

p(2)

= - '2 Xso·

TaKHM

06paaoM,

OIITHMaJIbHOe

ynpaBJIHIOII\ee

BoaAeucTBHe

dUO

(t)

MOlReT

6bITb

BbI6paHo

paBHbIM

Xso

dUo (t) =

-""2

{ll (t -

n)

+

1I-

(t - 4n)} dt.

llpHBeAeHHbIU

npHMep

nOKaSbIBaeT,

'ITO

paCCMaTpHBaeMaH

B 3TOM

napar-

pa¢e

saAa'la

06

OnTHMaJIbHOM

ynpaBJIeHHH

HMeeT,

Bo0611\e

rOBopH,

He

eAHHCT-

BeHHoe

pemeHHe.

IIpHMep

23.2.

PaccMoTpliM

saAa'ly

06

ynpaBJIeSHH

MaTepUaJIbHOii,

TO'lKOii,

p;BHlRYII\eiicfI'

R

BepTJ1RaJIbHOii

nJIOCKOCTJI

nOA

AeiicTBHeM

peaRTHBHoii

CHJIbI H

CHJIbI

THlReCTH.

3Ta

saAa'la

YiRe

Ilay'laJIaCb

B

S§

18 H 21.

060611\eHHbIe

ypaBHemfH

p;BHlReHHH

TO'lRH

MoryT

6bITb

npeACTaBJIeHbI

B

BHAe

dX1

=

X2

dt

,

dX2=

dU

1

, }

(23.48)

dxs

=

x(dt,

dX4

=

dU

2

-

gdt.

3AeCb

BeJIH'IHHbI

dU

H

dU

BBOP;JlTCH

TO'lHO

TaR

me,

l(aK

H B

npuMepe

14.1.

MbI

l

2

6YAeM

MHHHMHBHpoBaTb

BeJIH'IllHy

1

(23.49)

~

[dU] =

~

VdU~

+

dU~.

o

"

31'0

OBHa'iaeT

(CM.

§ 14),

'ITO

MbI

XOTHM

OCYII\eCTBllTb

nepeBoA

TO'lRH

Ha

COCTOH-

HHH

X (0) =

{-1,

0,0

O}

B COCTOJIHHe x (1) =

{O,

O,O,O}

C

MHHHMaJIb-

HbIM

paCXOAOM

aanaca

peaKTUBHOH

MaccbI

m1

(0).

IIpelRAe

Bcero,

nOJIbBYHCb

Ta6-

JIH~eH:

14.1,

onpeAeJIHeM

HOpMY

P [h

(or)]

(23.9)

no

aaAaHHoii

llRTeRCHBROCTH

x [dU] (23.49):

P

[h

('t')] =

max

[h~

(or)

+h

2

(or}]'/

••

0.,;;

..

";;1 2

1'1

O

EYAeM

HCXOADTb

DB

npaBllna

17.1.

MHHHMaJIbRylO

¢YHK~HIO

h

('1:')

6YAeM

onpeAeJIflTb

HB

yCJIOBHH

(23.12).

<IlYHl(~HH

h

(or)

=

B'

('1:')

S

(or)

=

B'

(or)

S X

X

[or,

t

].l

H

nocTOHHHbIH:

BeKTOp

C

HB

(23.13)

BbI'lHCJIeHbI

YlRe

B § 18

(CM.

CTp.

13

187). B

COOTBeTCTBllH

C

3TllMH

AaHHbIMH

saAa'la

(23.12)

c4JopMyJlHpyeTCH

TaK:

RauTll

(23.50)

pO

=

min

max

{[l1

(1

-

or)

+-

i

2

]2

+

[Is

(1-

or)

+-14P}'/2

I 0";;"";;1

UpH

h +

~

Is

+-

g14

=

1-

HeTpYAHo

aaMeTllTb,

'ITO

npll

JIlO60M

BbI60pe

1

RaH60JIbmllM

aHa'leHlleM

T

WYHK~Hll

f

('1:',

i)

=

{[i1

(1

-

or)

+ 1

2

]2

+

[1

3

(1

-

or)

+ l(]2}'/'

6YAe

OARO

liB

ee

SRa'leRHH:

f (0, i) =

[(11

+ 1

2

)2

+

(13

+ 1

4

)2]'/2,

•+ 2

'I.

!(1,l)=(lt

It)'

p.":

t:',:.:

t 23]

YITPABJIE'HHE C

MHHHMAJIbHblM

HMllYJIbCOM

195

3a4JuKcllpyeM

aRa'leHUe

4JYRK~UH

A.

(l)

= f (1,1):

A.

(1)

= f (1,1) =

11,

H

pemHM

aaAa'ly:

HaUTH

[23.51)

~

=

min

f (0,

l)

I

g

rrpH

f

(1,l)

=

11,

i

1

+

Zi

s +

gl4

= 1.

)J;JlH

pemeIDUI

3aAaqR

(23.51)

BOCnOJIb3yeMcH

MeTOAOM

HeonpeAeJIeRHLIX

MROII{llTeJIeH

JIarpaHma

(CM.,

HanpHMep,

§ 18). B

peayJIbTaTO

rrOJIy'laeM,

'ITO

pemeHHe

AOCTaBJlJIIOT '1HCJIa

'~

0_-=

g11

4

l

l1

-

g'+

4 '

'

4

-

-yg2+-4

0

__

2 0

o

2g

0

19=~g14'

1

3

=

g2

+-

4 -

2l

4

•

(23.52)

TIoAcTaBJlJleM

BeJIH'lHRbI

(23.52) B

Bhlpa-

lReRlle

(23.51);

HaiiAeM

rI>YHK~lllO

~=~(11):

2

'I

~

(Tj)

=

111-

-y

g2

+4

\.

(23.53)

TIOCTpOHM

rpa4JHI(H

4JYHK~llii

S = S

(l'J)

PHC.

23.2.

H S =

l'J

(pHC.

23.2).

TaK

l(aK

'1RCJIO

pO

(23.12)

paBRo

MHHHMYMy

HaH60JIbmeH

113

BeJIH'I~lR

f (0,

i)

11

f

(1

,i),

TO

Ha

pHC.

23.2

aaKJlIO'IaeM,

'ITO

31'01'

MHHHMYM

AOCTHraeTCJI

B

AaHROM

cJly'lae

npH

yCJlOmlH,

KorAa

f (0,

i)

= f

(1,

i),

T.

e.

IIpH

l'J

=

11°

= 1/ V

g2

+

4.

CJleAOBaTeJlbRO,

1

pO

= l;

(Tj

0

2=

110

=

..

r'

(23.54)

.

f.

g2+4

qHcJla

I?

(i

= 1,

...

,4) .

OI(OH'IaTeJIbRO

onpeAeJIJlIOTCJI

paBeHcTBaMH

(CIL

(23.52»:

4

2

K

l~

=

g2

+

4'

19

= -

g2

+4

19=O,

l~

=

g2

+4

(23,.552

H,

CJJeAOBaTeJlbHO,

MIUlHMaJlbRaJl

!PYHl(l\HH

hO

('1:')

paBRa

h~

(or)]

[-

g'

~

4

or

+-

g2

~

4]

hO

(or)

= = .

(23.56)

[

hg(or)

_g_

g2+4

MbI

YlRe

OTMeTBJIH

BlilIIIe,

'ITO

OTKJlOReHHe

y

rhO

(or)]

"'"

{[h~

(or)]2

+

[hg

(or»)2}'/.

3TOH: WYHKIJ;HH 01'

SYJlJI

AocTHraeT

Ra1l60JlbmerO

aRa'lCHHH

B

MO-

MeRT

BpeMeHH

t = °H t = 1.

Tenepb

AJIJI

onpeAeJleRHJI

OilTHMaJIbROrO

ynpaBJIe-

HHJI

dUO

(23.24)

CJleAyer

HaullI

lHlKTOPbI

p(1)

JI

p(2),

JICXo,qJl

Ha

YCJlOBIIH

(23.20),

(23.2

1

). . .

7'

197

.

,-,

.•••

••

co,

••

'

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

;

..

:~.

'-~'

•••

',

.,.;.

~.~

>';',

~'_~.~i~'

c;

c·

196

CBO~CTBA

OrrTHMAnbHOrO

yrrPABnEHHff

[rJI.

6

~

23]

YIIPABJIEHHE

C

MHHHMAnbHbIM

HMIIYnbCOM

BeRTopbI

hO(O)

If

hO

(1)

paBHbI

COOTBeTCTBeHHO

BeOOXO,LIHMO

orrpe,LIeJIRTL

Cpe,LIR

!pYBKIJ;Hii

h

("r)

BIf,LIa

(1

-

1")

11J

h)

[

(23.61)

(1"

=

.(1--r)12'

ho(o)=[g';'],

hO(1)=[-g;~4].

npBqeM,

RaK

H

npelli,LIe,

p [h (-r)] =

max

[hf

(-r)

+

h:

(,r)j'/2.

3a,LIa'Ia

(23.12)

g2

+4

g2+

4

"t

)

npHHHMaeT

a,LIeCL BR,LI:

HauTR

pO

=

min

max

[(1

-

T)2

Ii

+

(1

_

T)2

l~

]'/2

BeRTopbI

p(l)

11

p(2)

KOJIJIlfHeapHbI

h

O

(0) H h

O

(1);

rr03ToMY,

yqUThlBaJI

(23.21),

I

"t

T.

e.

B ,LIaHHOM

cJIy'Iae

rrpR

C111

+C

3

1

3

= 1.

COBepmeHHO

JICHO,

'ITO

!PYHRIJ;RJI

~

(T)

=

[(1-T)2

1~

+

+(1~1")1~]'!2

rrpR

JIIOObIX 1

1

,

1

3

,LIocTRraeT

CBoero

HaROOJIhmerO

aHa'IeHlIJI

rrplI

II

p(l)

~

+

II

p(2)

II

=

Vg

2

+4 ;

(23.57)

-r

= 0 H 3TO

aHaqeHHe

paBHO

(1~

+

l~)

'f,.

OTCIO,LIa

cpaay

CJIe,LIyeT,

'ITO

Tpy,LIHO

cOOOpaaHTL,

'ITO

~

~.

1

1°

---

1°

-

po=

..

1"-

1 =

ei

+

e~

, 3 -

ei

+

e;

.

II

ei

+

e~

p(l)

= [ il

p(2)

=

[-;

1

l

R !PYHKIJ;RJI

hO

(T)

RMeeT

BR,LI

I

C1

TaRlfM

oopaaoM,

RCROMoe

OrrT1IMaJILHOe

yrrpaBJIeHlfe

dVo (23.24)

RMeeT

BlfA

hO

(1")

=

ei

:

~~

(1

-

-r)

J'

(23.58)

dVO(t) = [ i

l~(t)dt+[~1l~-(t-1)dt

2

(1-

'-

c +e

2

T)

l 3

, .

I

R

CBO,LIH~CJI,

CJIe,LIOBaTeJILHO, R ,LIByM

RMnyJILCaM.

TaRRM

oopaaOM,

HCROMoe

OnTHMaJILHOe

ynpaBJIJIIO~ee

BOa,LIeUCTBRe dVo (t)

,

Iii

MbI

HayqlfJIIf

CeHqaC

aa,LIaqy

0

rrepeopoce

B

BepTRKaJIhHOH

llJIOCROCTIf

npe,LICTaBJIJIeT

cooo.o

e,LIRHCTBeHHbIu

RMrryJILC

B

Ha'IaJILHbIU

MOMeHT

,LIBRllieHHlI

."

.,'

TJIllieJIOH

MateplfaJILHOH

TO'u{If,

rrpe,LIIIOJIaraJI

aaAaHHbIMR

ee

IIOJIOllieHRJI

R

CROpO-

CTR B

HaqaJILHhIH

If

ROHeqnbIH

MOMeHTbI

,LIBHllieHHJI.

PaccMoTplfM

TerrepL

rro,LIoo-

HYIO

llie

aa,LIaqy,

CqlfTaJI,

O,LIHaRO, aa,LIaHHbIMH

JIHmb

HaqaJILHOe

If

ROHeqHoe

rro-

JIOllieHHJI

TO'IRIf

H

ee

Ha'IaJILHyIO

CROpOCTL

R

He

OrpaHlfqlfBaJI

ee

CROpOCTL

B

RO-

He'IHbIH

MOMeHT

,LIBRllieHHJI

(CM.

§ 13).

IITaR

paccMoTpHM

aa,LIa'IY

00

yrrpaBJIeHRR

dvo(t)

=

MaTepRaJIbHOH

TOqROH,

,LIBHlliy~e.oCJI

rro,LI ,LIeHCTBHeM

peaRTlfBHo.o

CHJIbI

If

ClfJILI

TJIllieCTIf

COrJIaCHO

ypaBHeHHJIM

(23.48)

Ha

COCTOJIHHJI x

(tJ

=

.{x~},

t(X

= 0 B

"II

COCTOJIHHe x

(t{3)

=

{O,

xdt(3)' 0,

X4

(t{3)}'

t{3

=

1,

r,LIe

Ha

BeJIHqlIHLI

x

2

(1),

x

4

(1)

He

HaJIOllieHO

HRKaRlIX

OrpaHlI'IeHlf.o.

8TO

,LIBRllieHRe

TpeoyeTCJI

ocy~ecTBHTJ,

'ITO,

ROHe'IHO,

HeTpy,l(Ho

ObIJIO

cooopaaRTL

cpaay

H3

!pHaH'IeCRRX

OOCTOJITeJILCTll.

C

MRHlfMaJILHbIM

paCxO,LIOM

aanaca

peaRTHBHOH

MaCCbI

In.1

(0).

II03TOMY

B

Raqe-

CTBe

RHTeHCRBHOCTlI

yrrpaBJIeHlfJI

MbI

CHOBa

BbIOHpaeM

BeJIlI'IlIHY

(23.49).

3a,LIa'Ia,

Paa6epeM

em;e

OP;HH

cJIyqaii,

Korp;a

MHHHMaJIbHaJI

<PYHKD;HJI

hO

(-I:)

CJIe,LIOBaTeJILHO, COCTORT B

onpe,LIeJIeHlfR

BOa,LIeHCTBlIJI dVo,

y,LIOBJIeTBOpJIIO~erO

yCJIOBRJIM

(CM. § 1): .

1

~

(1

-

1"1

dV~

=

1-

x~

=

e1,

°

1

\' g

.)

(1-1")dVg=T

-

x:

=

e2.

<>

x

[d

VO]

=

min

x [d

V].

u

i_xIX.

2

o(t)dt

1(1-

X~)2

+(g/2 -

X~)2

g/2 -

xIX.

4

o(t)dt

1(1.-

X~)2

+(g/2 -

X~)2

OKaaItIBaeTCJI

TOmp;eCTBeHHO

rrOCTOJIHHOii

1'1,

CJlep;OBaTeJIbHO,

Korp;a

3Ta

<PYHKD;HJI

He

p;OCTaBJIJIeT

HH<popMaD;HM

0

Tex

MOMeHTax

BpeMeHH

t =

T:j,

B

ROTopIiIe

p;OJImHIiI

rrpHJIaraTbCJI

yrrpaBJIJIIOm;He

HMrrYJIbCHIiIe

Boap;eiicTBHJI.

Ha

CJIep;yIOm;eM

HHme

rrpHMepe

6yp;eT

rrOKaaaHO,

KaK

3TH

MOMeHTIiI

MomHO

orrpep;eJIHTb,

HCrrOJIbayJI

MaJIyIO

p;e<popMaD;HIO

rrapaMeTpOB

CHCTeMIiI

(CM.

BIiIme

CTp.

190).

IIpH

3TOM

MIiI

OrpaHH'IHMClI

JIHmb

paa6opoM

rrpocTeiimeii

KOHKpeTHoii

aap;a'lH

•.

PaaBHTHe

.orrHCIiI-

BaeMIiIX

p;aJIee

paccymp;eHHii

H

HX

06ocHoBaHHe

B

06m;eM

CJIyqae

Mom-

HO

nopeKOMeHp;OBaTb

HHTepeCYIOm;eMYCJI

qHTaTeJIIO

B

KaqeCTBe

nOJIe3-

(23.59)

(23.60) .

Horo

caMOCTOJITeJIbHOrO

HCCJIep;OBaHHJI.

11a

peayJILTaTOB

§§

15-17

BbITeRaeT,

'ITO

rrpR

pemeHlIlI

aa,LIa'IR (23.59), (23.60)

IIpuMep

23.3.

PaccMoTpRM

MaTepRaJILHyIO

TOqRY

Ha

npJIMOH

;.

IIycTL

MbI

MOllieM

B

OCHOBHbIX

'IaCTJIX

CJIe,LIOBaTL

yRaaaHHou

BacTp.

184

npOIJ;eAype.

,lTy

TO'IRy,

Maccy

ROTOpOD: m

rrpRMeM

paBHon:

e,LIHHRIJ;e,

TpeoyeTCJI

nepe·

OcoO/'lijJj:OCTb

aJlech

COC1'OJI1

JlUmL

B 1'014,

'ITO

MUIIHUaJIbBYIO

WYRKIJ;lIIO.ho

(t'}

BeCTJl

us

Ha'IaJJh~0J'9

1;I{>J10J1(!lI,lRH

~

(0)

""

~Q:

=

-1,

~

(0)=

~(X

=

'Y

(v>

1)

B

198

p.

~~'~:"

[rJI.

6

CBOflCTBA

OITTHMAllbHOrO

YllPABllEHHff

COCTOflHne

nOKOfl

S(1) =

S(.l

= 0,

~

(1)=

~(.l

= 0

npn

yCJIOBMJi

MIfHHMyMa

HM-

nyJIbcaynpaBJIflRlIJJ;eH

CHJIhI u (t), T.

e.

npH

yCJIOBHH

1 1

I

l'

(23.62)

.)

IU

(1')

I

d1'

=

~

I

dU

(1')

I=

min.

o 0

::ha

'aaAa'la,

KOHe'lHO,

pemaeTCfl

cpaay

Ha

<pllall'leCKHX

C006palllemlH,

OT-

Kyp,a

BhIBOp,HTCfl,'lTO

OnTHMaJIhHOe

ynpaBJIeHHe

UO(t)

MQ;KHO

Bhl6paTb

B <pOpMe

ep:HHCTBeHBOrO

HMnyJIbCHOrO

BoaAeHCTBHfl

uO (t) = -

vf>

(t

_

~

) , (23.63)

UOTopoe

npllJIaraeTCfl,

CJIep,OBaTeJIbBO, B MOMeHT t =

1'1

= 1/v.

Mhl,

Op:BaUO,

npOCJIeAliM

aAeCb

peryJIflpBhlJli

(<Hay'lBhlJli»

nYTb

pememw

aTOH

aap,a'lU

C TeM,

'lT06bI

npOJImnOCTpHpOBaTb

IIa

HeM

MeTOA

Ae<popMaUlIJI

CHCTeMh1.

COCTaBHM

ypaBHeHIDI

ABJIlIIeHHfl

B

BOpMaJIbBOH

<popMe (23.1).

fIOJIarafl

S =

Xl'

~

= X

2

,

nOJIy'lJIM:

dX1

=

X2

dt,

dX

2

= dU. (23.64)

CHcTeMa,

conpflllleHBaJl

U CHCTeMe (23.64),

HMeeT

BHA

(23.65)

81

= 0,

82

= -

81,

II

.ee

<pYBAaMeHTaJIbHafl

Ma1'pHua

S [t, tol

TaUOBa:

'Iii

(23.66)

S [t,

to]

= ( 1 0 ) •

:>

- (t -

to)

1

"

B

AaHHOM

CJIy'lae

B=

(

~

) ,

ta.

= 0, t(.l = 1,

noaToMy

~RBHMaJIhHYRl

<pyHmi;uRl

hO

(1')

B COOTBeTCTBIIH C

Ma-

~h

TepHaJIOM

Ha

CTp.

184~189

Hap:JIelllHT

HcuaTb

B

BRAe

(23.67)

hO (1') =

B'S

[1', t ] •

10

=

(0

1)

( 1

0)

(

I~)

=

I~

-1~1',

0

oc

-1'

1 1

• 2

rp,e

BeKTOp

1°

=

(l~,

l~)

flBJIfleTCfl

pemeHHeM

aaAa'lR

(23.68)

max

11

2

-

11

1'1

=

min

(0 <

l'

<1)

't

I

npH

yCJIOBHH

I'Xa.

= (iI, 1

2

) (

-:

) =

vl2

-

11

= - 1. (23.69)

HeTpYAHo

npoBepHTb,

'lTO

nOCJIeAHRlRl

aap,a'ly

(23.68), (23,69)

paapemaRlT

'lI1Clla

l~

=

0,

l~

=

-lIv.

CJIeAOBaTeJIbHO,

MHHnMaJIbHaJl

<pyHI'I\HI1

hO

(1')

=

-V-

1

=

const

····

:',

,

.~~

..

c

§ 23]

YIlPABJIEHHE

C

MnHHMAJIbHbIM

l1MrrYJIbCOM

199

..

ouaahlBaeTCfl

TOlllp:eCTBeHHO

paBHoH

nocToflBHOMY

'lHCJIy,

H AeHCTBHTeJIbHO, B

AaHHOM

cJIy'lae

aTa

<PYImUHfl

Be

AOCTaBJIfleT

lIBpopMaUHU

0

MOMeHTax

t = 1';,

Korp:a

AOJIlIIHhI

npHJIUraTbCfl

lIMUyJlbcHhle

noap,eHCTBHfl.

COCTaBHM

BcnOMOraTeJIbnyRl

aaAa'ly.

ConOCTaBHM

CHCTeMe (23.64)

BOByro

CHCTeMY

dX1

=

X2dt,

dX2

=

-e

Xl

dt +

dU,

(23.70)

rAe

nOCTOflHHaJl

e>

0 - MaJIhlJli

napaMeTp.

fIpH

8 --+ 0

ypaBHeHHJI

(23.70)

nepeXOAflT

B

Hcxop,Hhle

ypaBBeBHfl

(23.64).

PaccMoTpHM

CBOBa

aaAa'ly

06

ynpaB-

JIeHHH

npH

yCJIOBHH (23.62),

BO

Tenepb

Yllle

AJIJI

BcnOMOraTeJIhHOH

CHCTeMhI

(23.70),

nOJIarafl,

OAHauo,

CBOBa X

OC

=

(-1

,v),

xf.l

= (0,0).

CHcTeMa

(23.70)

YAOBJIeTBOpfleT

yCJIOBHJlM,

yuaaaHHhlM

Ba

CTp.

185

-186.

fIoaToMY

MHHHMaJIb-

BaJl

<PYImD;Hfl

h~

(1'), OTBe'laIOIJJ;afl HOBOH

aaAa'le

06

ynpaBJIeHHH

cHcTeMoH

(23.70),

06flaaTeJIhBO

6YAeT

neoco6ou

11

6YAeT,

CJIeAOBaTeJIbBO, AOCTaBJIJlTL

HB-

.

<popMaUHRl 0

MOMeHTax

t = 1'(i), B

uOTophle

AOJIlIIBhl

npllJIaraThCfl

lIMUYJIbCBhle

BoaAeHCTBHJI.

(HanoMHHM,

'lTO

t =

1'(j)

- aTO Te

TO'lUU,

B

UOTOphlX

<PYBU1VIfl

II

h~(t)

II

AOCTHraeT

CBoero

Ban60JIhmerO

aHa'leBIHI.)

Bhl'lnCJIHJI

aTn

aHa'leHHH

1'(j)

11

paccMaTpnBaJl

nx

nOBeneHHe

npH

e --+ 0,

MhI

11

orrpeAelIllM

MOMeHTbl

r (£) 0

1'j

=

1m

1';

up

I!

8

->

•

(23.71)

B

KOTophle

AOJImHhl

npHJIaraTbCH

HMrryJIbCBhle

ynpaBJIeBl1fl

B IICXOP:BOH

aaAa'le

06

ynpaBJIeHHH

cHcTeMoH

(23.64).

(CTporoe

060cBoBaHHe

Toro

npaBAonoA06Boro

yTBepmAeBHfl,

'lTO

BeJIH'lllHhl

(23.71), AeHCTBIITe.ITbBO, B 06IJJ;eM

cJIy'lae

AOCTaB-

JIJlRlT

nymBble

aBa'leHHJI

t = 1';,

Mhl

OCTaBJIJleM B

cTopoBe

H

OrpaHH'lHMCfl

BlI}Ke

JIlIIDb

cooTBeTcTByRlIJJ;HMH

BhlliJlaAKaMH,

onpeAeJIJlRlIJJ;HMH 'lHCJlO

1'1'

yme

uaBeCT-

Boe

BaM

ap,eCh B

'laCTBOM

CJIy'lae

Ha

<pllaH'leCUHX

C006pa}KeHHH.)

MTaK,

Ba'IHeM

pemaTh

aaAa'ly

06

ynpaBJIeBII11

cHcTeMoH

(23.70),

OnHpaJlCb

Ba

TeopeMy

23.1.

CHcTeMa,

conpflmeHHafl

I,

CHCTeMe (23.70),

UMeeT

BHA

8

1

=8

$2'

$2

=

-8

1

,

(23.72)

H

ee

<pYHAaMeBTaJIbBaJl

MaTpllI\a

S£

[t, tol

TaKOBa:

cos

e (t -

to)

e

sin

e (t -

to)

II.

S [t,

to]

= 1

£

-eSin8(l-to)

cose(t-t

o

)

II

CJIep,OBaTeJIhBO,

.MHHlIMaJIbBYRl

<PYHUI\lIRl

h~

(1')

Hap,JIemHT

aAeCL .l1CIW'fL B

BHAe

h

O

(1')

=

B'S

[1'

0]

1(£)

= I(E)

cos

e1'

-

1(£)

-

1

sin

81'

(23.73)

£

£'

2

Ie'

rAe

BeKTop

1(£)

= (I\E),

l~»

flBJIfleTCfl

pemeHHeM

aap,a'lH

p£

=

max

112

cos

e1'

-

11

..!..

sin

e1'

I=

min

(0

<

l'

< 1)

(23.74)

't

8 I

npH

yCJIOBHH (23.69).

MomHo

co,o6paaHTb,

'ITO

B

AaUBOM

CJIy'lae

UpH

MaJII.lX

8B8'leHIDIX

8

rpa<pHK

MHHHMaJIbHOH

<pyImIWH

h~

(1')

(23.73)

6YAeT

IIMeTh

BlIl1.

~~06pameHHhlH

Ha

puc.

23.3,

npH'leM

1

1\£)

=

I~

+

II.

(e) =

11

(e),

I~E)

=

Ig

+

12

(e) = - v

-I-

/2(e),

Красовский Н.Н. Теория управления движением

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