Красовский Н.Н. Теория управления движением

tf

'f'

>,'.""

fiW-···

c

..........

·

",,\.·=<rror.r-'j')'r'

','':"'

..

:,'

,','."

•• ,::,

....

i;:;.:;r:.:;..-.:.;,., :

.•...

,-

't"':.'ijf:<:';"';'};;;,

..

~'gtIJ

n

~:'_

•..

:.)

LrJl.

10

iiI'POiUil

3A,Ii;A'lA

()

BC1'PE'iE ,Ii;nHtREHltl1:

340

u = U

O

(42.15)

npll

JII060M

BOSMomHOM

BLI60pe

v =

iJ

(t)

llJIll

v =

= v [t, Y (t), z (t),

/.L

(t), v (t)]

06eCne'lllBaeT

BpeMH

,n;o

BCTpe'lll

He

60Jlbme,

'1eM

TO,

npll'leM

npll

aKCTpeMaJlhHOM

ynpaBJleHllll

(42.16)

v =

VO

[t,

Y (t), z (t),

/.L

(t), v

(t)]

=

v~

[t],

onpe,n;eJreHHOM

no

npaBllJIY

(42.6)

BCTpe'la

ocym;ecTBllTcH

TO'lHO

B

MO-

MeHT

t =

11,0

=

ta.

+

TO.

CJle,n;yeT

no,n;'1eprmYTb,

'ITO

STOT BLIBO,n;

co,n;epmllT

TaKme

yTBepm,n;eHlle

0

TOM,

'ITO

TO'lKa

{x

(t),

~

(t)}

~O

caMoro

MOMeHTa

BCTpe'lll

He

nOKll,n;aeT

06J1aCTb

r.

81'01'

BamHLIB

<p

aKT

,

KOTOpLIB

TaKme

CJle,n;yeT

llS

coxpaHeHllH

06J1aCTll

G(2)

BHYTPll

G(l),

MLI

MOJl'laJlllBO

llMeJlll

B

Bll,n;y

B

npe,n;LI,n;ym;llx

paccym,n;eHllHx.

TaKllM

"

j

06pasoM

nOJIyqaeM,

'ITO

Tv.-,v'

=

max

T

'Un,v'

Tenepb

OCTaJlOCb

npo-

'I

v

"

,I

BepllTb,

'ITO

npll

U =

uo,

,n;eBCTBllTeJlbHO,

BbIIIOJlHHeTCH

YCJlOBlle

MlI-

'i

,

HllMyMa

no

u.

IIpe,n;noJIomllM

,n;JIH

aTOB

rr;eJIll,

'ITO

MLI

BLI6paJlll

Ka-

KO~-Hll6y,n;b

,n;onycTllMoe

ynpaBJIeHlle

U =

u"

[t,

y,

z,

/.L,

v].

IIpll

aTOM

MLI

,n;OJlmHLI

y'lecTb

B

COOTBeTCTBllll

C

nrpOBOB

nOCTaHOBKOB

sa-

,n;a'lll,

'ITO

npeCJle,n;OBaTl3,Jlb

He,

rapaHTnpOBaH

01'

BLI60pa

npeCJIe,n;ye-

MLIM

napTHepOM

ynpaBJleHllH

(42.17)

v =

v"=

~ua

11

Ho

Tor,n;a

B

nporr;ecce

,n;BllmeHllH

npll

t

>-

to

Bce

BpeMH

6y,n;eT

BLInOJl~

~:

HHTbCH

paBeHCTBO

v

(t)//.L

(t)

= v

(t{)//.L

(to)

=

const,

KOTopoe

nOJIy-

".

'1aeTCH

llHTerpllpOBaHlleM

ypaBHeHllB

~

= - u

2

/2/.L

= - [u

a

]2/2/.L'

V=

-v

2

/2v = -

[u

a

]2v/2/.L

(CM.

(42.3»

liIs

SToro

paBeHcTBa

BLIBO-

,n;llTCH,

'ITO

ynpaBJIeHlle

Va

TaKme

,n;OnYCTllMO

II

KpOMe

TOrO,

'ITO

Ha

OTpeSKe

[to,

to

+

TO]

6y,n;eT X

[U

- v] <

~

(to)

=

/.L

(to)

- v

(to),

a STO

OSHU'IaeT

COrJlaCHO

npe,n;LI,n;ym;eMy,

'ITO

BCTpe'la

y

(t)

II

z (t)

npn

t <

to

+

TO

HeBOSMO>KHU.

li'rar{,

,n;eBcTBllTeJlbHO,

ynpaBJleHlIe

~,

(42.15) n (42.16)

06eCne'lnBaeT

llcrWMLIB

MllHIIMaKC.

MLIpas06paJlll

cIITyarr;IIIO,

Kor,n;a

IIcxo,n;HLIe

,n;aHHLIe y

(to),

z

(to),

/.L

(to),

v

(to)

TaKOBLI,

'ITO

TO'lKa

{x

(to),

~

(to)}

JIemlIT

B

06J1aCTII

r.

He

pas6llpaH

no,n;p06HO

npo'rIIBOnOJIOmHyIO

cnTyarr;IIIO,

OTMeTIIM

JIllmb,

'ITO

Tor,n;a

nplI

JII060M

u =

U",

max

T"

= {(Xl}

II

sa,n;a'la

40.1, CJle,n;OBaTeJIbHO,

pemeHIIH

He

v u

,v

IIMeeT.

q

T06LI

y6e,n;llTbCH

B

CKasaHHOM,

,n;OCTaTO'lHO

CHOBa

BLI6paTb

yn-

paBJleHlle

v B BII,n;e (42.17).

TaKIIM

06pasoM,

nOJlY'lae~

cJle,n;yrom;IIB

BLIBO,n;.

Teo

p e M a 42.1. B

c,n,y<tae

o8nomunnbLX

o6'benmoe

(42.1)

U (42.2)

aa8a<ta

40.1.

UMeem

peUJ,enue

moe8a U

mO.ltbnO

moe8a, noe8a

l-ta'ta.ILbHbLe

yc,n,oeUR-

x

(to)

= Y

(to)

- z

(to)

U

~

(to)

=

/.L

(to)

- v

(to)

co8ep;JJcamcR-

e

o6.ILacmu

r.

OnmUMa.ILbJ-aJLMU

ynpaeJtenUR-MU,

paapeUJ,alO-

UfUMU

moe8a

aa8a<ty

40.1,

R-8.lLR-lOmCR-

ancmpeMMbHbLe

ynpaeJtenuR-

u

o

, v

o

.

/J.lLR-

ebL"tUC.lLO/,UR-

amux

ynpa8.ILenuu

na8.ILe:JJCum

e

na:JlC8bLU

me-

,

·:

..

··

'

...

~',

:.

"

c

I

§

4.2]

3AAA

4:A

b i3CTPE

qE

O)J;lw'rnnlIbIX

OB'tEHTOB

341

nyUfuu

MOMenm

epeMeJ-/,U

t =

ta.

paapeUJ,amb

aa8a'ly 42.1 a

npe8e.ILbnO:A/,

6bLcmpo8eucmeuu U

nO.ltaeamb

1

)

11

v

u(t]=uo=~(t)T'

v[t]

=vo=w~(t)I'

(42.18)

liITaK,

eCJIII

npeCJIe,n;OBaTeJIb

meJIaeT

nplI,n;epiKllBaTbCH

OnTIIMaJlb-

Horo

ynpaBJIeHllH

U =

UO

B

COOTBeTCTBlIll

C

yKasailHbIM

npaBllJIOM,

TO

nporr;e,n;ypa

BLIlJllCJIeHllH

TeKym;lIx

SHa'leHllB

ynpaBJleHIIJI

UO

[t] CBO-

,n;llTCH K

HenpepLIBHOB

KoppeKTnpOBKe

BeJlll'lllHLI

TO

II

<pYHKrr;lllI

w~

('t) B

COOTBeTCTBllll

C

llSMeHHlOm;llMllCJI

no

Xo,n;y

nporr;ecca

YCJlO-

BIIHMll

sa,n;a'llI

42.1, r,n;e

ta.

= t. A STO B CBorO

O'lepe,n;b

COrJIaCHO

Ma-

TeplIaJIy

lIS

§

27

npllBo,n;IIT

K

HenpepLIBHoB

KoppeKTlIpOBKe

HalIMeHb-

mero

KOpHH

TO

ypaBHeHIIJI

(27.5),

KOTopoe

BMeCTe

C

YCJIOBlIeM

MaKCII-

MyMa

(17.1)

onpe,n;eJlHeT

ynpaBJleHlIe

w~

('t).

IIpOIIJIJIIOCTplIpyeM

Ha

npocTeBmeM

npllMepe

nporr;e,n;ypy

BLI'IIIC-

JIeHIIH

ynpaBJleHIIH

U =

UO

II

V = vO,

BLITeKaIOm;yIO

lIS

TeopeMLI

42.1.

iIPH~IeP

42.1.

IIycTb

rrpeCJIeAYIOID;HM

H

rrpeCJIeAyeMMM

061>eKTaMll

JIBJIJI-

IOTCJI COOTBeTCTBeHHO

MaTepHaJIbHIiIe

TO'IKll

m

l

H m

2

,

AB:lUHYID;HeCJI

rro

ropllaOH-

TaJIbHOB

OCR

~

rroA

AeBcTBlleM

HeKoTophIX

yrrpaBJIJIIOID;IlX

CHJI

it

II

v,

CTeCHeHHbIX

OrpaJm1lBffiIJIMII

00

~

u

2

(r)

dr

<

f12

(t),

~

v

2

('I:')

d'l:'

<v

2

(t).

(42.19)

t

TplJ6yeTCJI,

'ITo6hI

B

MOMeHT

BCTpe'lll

COBilaJIll

He

TOJIbKO

KooPAllHaTbI

TO'leK

~(l)

II

~(2),

HO

RllX

CKOpOCTll

~(l)

R

~(2).

PaCCMOTpllM

ABlllReffiIe

B

AByMepHoM

¢aao-

BOM

rrpOCTpaHCTBe

q =

{ql'

q2}

=

g,

~}.

TorAa,

060aHa'laJI

Yl

=

~(l),

Y2

=

~'(l),

Zl

=

~

(2l,Z2

=

~(2)

II

IIOJIaraJI,

He

YMeHbmHB

06ID;HOCTll,

m

l

= m

2

= 1,

aaIIll-

meMypaBHeHRJI

)l;BRlReHHJI

o6~eKToB

B

HOpMaJIbHOH

¢opMe

ill

=

Y2,

Y2

= u,

(42.20)

.21

=

Z2,

2:2

= V •

(42.21)

IIOCKOJIbKY

o6~eKThI

(42.20) H (42.21)

0AHOTllIIHhI,

TO

MbI

MOlneM

corrOCTaBHTb

llCXOAHOii

rrp06JIeMe

40.1 AJIJI

CllCTeM

(42.20)

II

(42.21)

aai-\a

qy

42.1 0

IIpeAeJIb-

HOM

6hICTpOAeiicTBHH

AJIJI

CllCTeMhI

Xl

= x

2

'

x

2

=

11',

(42.22)

rp;e

Xi =

Yi

- Zi

(i

= 1,2),

11'

= u -

v.

lITaK,

Tpe6yeTCJI

HaBTH

OrrTllMaJIbHOe

yilpaBJIeHHe

Il'~

('1:'),

CTeCHeHHoe

YC-

JIOBlleM

00

~

11'2

('I:')

d'l:'

<

~2

(t)

(~(t)

=

f1

(t),

- v

(t»

(42.23)

--------

t

1)

3aMeTHM,

'ITO

B

CTaU;IlOHapHOM

CHY'lae,

KorAa

A

II

B

nocToHHHN,

ynpaB-

JIeHllJI

uO

II

VO

(42.18)

nOJIyqaIOTCJI

He

aaBllCJIID;HMll

JIBHO

OT

t.

, (

~~~.

',"

."

.'

."

..

,.

",

,.,

;,..

.;.....

','

..

',

',',' :-:.,,'.<:.;.:-:::::-

..

::.:.:

..

:

..

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

·::-:::::-,~·:·'";;h:;;.jtii

C?

:342

nrpOBAff

3AnAqA

0

BCTPEQE

nBHmEHM~

trJI.

io

J:I

rrepeBOAH~ee

oth,eKT (42.22)

J:l3

COCTOHlillH

X

(t)

=

{Xl

(t), X

a

(t)}

B

rrOJIOme-

HJ:le

X

(t

+

T)

= 0

aa

HaJ:lMeHbmee

BOaMOmHOe

BpeMH

TO

= min

T.

Pemall

:ny

w

aaAa'ly

cnoco6oM, J:laY'leHHblM B

§§

27, 29,

nOCJIe

<lJIeMeHTapHblx

BblKJIaAOK

nOJIy'lRM:

2

w~

(t) = -

[TOP

[3

X

I

(t) +2Toxa (t),l.

(42.24)

fAe

TO

=

TO

(Xl'

Xa,

~)

HBJIHeTCH

HaUMeHbIIIHM

IIOJIOmJ:lTeJIbHblM

KopHeM

ypaB-

HeHlfH

~a

(t)T3 - 4

x~

(t)

12

-

12x

I

(t)x

a

(t)T -

12

xi

(t) =

O.

' (42.25)

I

TaKJ:lM

o6paaoM,

UCKOMble

yIIpaBJIeHJ:lH uO

J:I

v

O

,

KaK

<lTO

CJIeAyeT

ua

(42.18),

j

J:IMeIOT

B HameM

rrpJ:lMepe

BRA

2(1

[TOP

ztO

fYI,

Y2,

ZI,

Z2,

(1,

v] = -

«(1-

v) [3

(YI-

ZI),

+

2TO

(Y2

-

Z2)l.

(42.26)

2v

[TOP

VO

[YI,

Y2,

ZI,

Z2,jA.,

v]

=-

«(1-v)

[3(YI-ZI)+2TO(Y2-

Z

2)]'

(42.27)

I

Ha

PJ:lc.

42.1 rrpeACTaBJIeHbl

peaJIJ:laa~J:lR

ABJ:lmeHIDI

06'beKTOB

(42.20)

II

(42.21)

AJIH

Ha'laJIbHblX

YCJIODJ:lB

i

I

.........

:.r

..

····;······

I

01······ 7

Tl!.~~

to

=

0,

YI (0) =

-1,

Ya

(0) = 0,

ZI

(0) =

Za

(0) = 0,

fl.

(0) =

1,

v (0) =

1/

3

~

(~2.28)

B CJIy'lae, KOfAa o6a rrapTHepa rrpHAep-

mHBaIOTCH

cTpaTerHH

(42.26),

(42.27).

I,

~

lia

peayJIbTaTOB

HaCTOHII~ero

rraparpacP

a

CJIeAyeT,

'ITO

eCJIH

rrpeCJIeAYIO~HH

Bcer-

I:'

Aa

6YAeT

Bbl6J:1paTb

ynpaBJIeHIle U =

UO

~

!

(42.26),

TO,

KaK 6bl

lill

Bbl6IlpaJI Aorry-

PIlC.

42.1.

CTIlMoe

ynpaBJIeHHe rrpeCJIeAyeMbm,

BCTpeqa

o6HaaTeJIbHO

COCTOIlTCH

He

rroame,

."

'leM 'lepea

TO

(to)

(B

HameM rrpHMepe AJIlI Ha'laJIbHblX

yCJIOBIDI

(42.28)

TO

= 3).

Ha

pRC.

42.2Il

42.3

Hao6pameHbl ABHmeHllH

06'beKTOB,

KorAa U =

UO

(42.27), a

v (t)

==

0

(pHC.

42.2)

II

v (t) =

=;

-10

sin

10

t

(pHC.

42.3).

T'

T

B

<lTHX

CJIyqaHX BCTpe'la rrpoHc-

z,

-I

01......

XOAHT

B HeKoTopbIe

MOMeHTbI

t'

t =

to

+ T,

rrpeAIDeCTBYIO~Re

(

MOMeHTy

BpeMeHJ:I

t = t

o

+

TO

(to)' T

=t.76

3Aecb IlHTepecHo

CpaBHJ:lTb

, ,

OrrTHMaJIbHble

ABHmelillH 06'beK_ T

EZ,SO

Ta

(42.20),

KOTopbIe

ocy~ecTBH-

d-T-T'-!J,4b

JIJ:lCb

B

npo~ecce

rrpeCJIeAOBaHJ:lH,

-I

C TeMii

OrrTHMaJIbHbIMH

ABHme-

HllHMH, KOTopbIe

rroJIytIHJ1J:1Cb

PIlC.

42.2.

6bI,

ecmi

6bl 3apa1tee

6btAO

U3-

eecm1tO,

'ITO 06'beKT

(42.21)

B

rrpo~ecce

npeCJIeAOBaHllH

He

ABJ:lHeTClI

C

MeCTa

IlJIJ:I

6YAeT

yrrpaBJIlITbCH

no

aaKony

v (t) =

-10

sin 10 t (KaK

<lTO

H

ocy~ecT

BHJIOCb

Ha

caMOM

AeJIe).

8TH

rrOCJIeAHHe

ABHmeHIDI 060aHatIeHbI Ha

PJ:lc.

42.2

H

42.3

rryHl{THpHbIMii JIHHIllIMR.

0,',

\'.'.'

:/

I

I.

§ 42]

3A,lJ;AQA 0

BCTPEQE

O,lJ;HOTMrrHbIX

OB'bERTOB

343

B

paCCMaTpIlBaeMOM

rrpHMepe cTpaTerJ:lU

(42.26)

J1

(42.27)

MomHO

6bIJ10

JIerKO

HaHTJ:I,

HCXOAH

cpaay

lia

rrpaBll'J1a

<lKCTpeMaJIbHOrO

npJ1~eJIJ:lBaHJ:llI.

0("\"

72~~

c(l)

f/,=t,

-I

PJ:lC.

42.3.

Pac.

42.4.

06JIaCTH

AOCTHmHMOCTH

C(J)

H d

2

)

COOTBeTCTBeHHO

AJIH

06'beKTOB

(42.20) a

(1t2.21)

aAeCb

CYTb

06JIaCTJ:I

IT

npocTpaHcTDe q =

{ql'

Q2}'

OrpaHJ:ltIeHHbIe

<lJIJ1lUlCaMJ:I:

12 12

[TO]3

[ql-Yd

t

)]2-

[TO]2

[QI--YI(t)]

[q2+yz(t)]

+

[1~0]

[Q~

+

q2Y2

(i) +

Y~

\i)]

=

(12

(t)',

(42.29)

12 12

[TOP [qi - ZI (t)]2 -

[TOP

[ql

-

Zl

(i)]

[q2

+

Z2

(i)]

+

4

+

[TO]

[q~

+

Q2

z

2 (t) +

Z~

(t)]

= v

2

(t). (42.30)

8TJ:I

06JIaCTH

Hao6palReHbI

Ha

p~IC.

42.4

AJIfI HcxoAHoro

ITOJIOmeHHlI

06'beKTOB

(42.28).

TaK

KaK

<lJIJIHnCbl

(42.29)

II

(42.30)

rroAo6HbI,

TO

onpeAeJIeHHe

TOtIKR

qO,

B

KOTOpOB

OHll

KacaIOTClI, H ynpaBJIeHlIH,

npH~eJIJ:lBaIO~RX

ABJ:lmeHJ:llI

(42.20)

J:I

(42.21) B :lTy TO'lKy,

He

npeACTaBJIlIeT

oco6oro

TpYAa.

TeopeMa

42.1

rroKa3bIBaeT,

'ITO

yrrpaBJIeHMH

UO

M

UO

(42.18)

o6ec-

rrelfMBaIOT

min

max

T, a OTCIOp;a

CJIep;yeT,

'ITO

yrrpaBJIeHMe

u = UO.

u v

p;eHCTBMTeJIbHO,

HBJIHeTCH

B

orrpep;eJIeHHOM

CMIJCJIe

OrrTMMaJIbHIJM

P;JIH

rrpeCJIep;OBaTeJIH.

ECJIH

6IJ

yrrpaBJIeHHH

UO H

UO

COCTaBJIHJIH

cep;JIo:ayIO

TOlfKY

p;aHHoH

MrpIJ,

T.

e.

o6ecrrelfMBaJIM

6IJ

M

~ta1£CUMUH.

BeJIHlfHHIJ

TU,v.

TO

3TO

03HalfaJIO

6IJ,

'ITO

M U =

UO

(42.18)

eCTb

yrrpaBJIeHHe.

orrTH-

MaJIbHOe

C

TOlfKM

3peHMH

rrpeCJIep;yeMoro.

R

COiKaJIeHHIO,

B

CJIylfae

orpaHMlfeHHH

(40,6)

Borrpoc

TaK

yp;06HO

He

pellIueTcH.

He

paa6HpaH

~MTya~MIO

B

o6JII;eM

CJIy-lfae,

OrpaHHQH~CHrrpOCTIJM

rrpHMepoM,

• •

••

..

• ! •

.'

• . •

••

• •

....

•

"1i"f"2Wit

,~....:.~~_c;.=_.;..._._

0'

<"

[rJl.

10

HrpOBMI

SAAAqA

0

BCTPEqE

ABHlREHHf[

;344

llpUMep

42.2.

PaccMoTpUM

CUCTeMhI (42.1) U (42.2),

orrHChIBaeMhIe

yrrpillR-

HeHHHMU

rrepBoro

rropHp;Ra

(42.31)

iJ

= u,

z

= v,

v -

CI,aJIHphI.

B COOTBeTCTBHlI C (42.3)

HaMeHeHUH

BeJIU'UIH

rAe

y,

Z,

u,

00

p.2

(t)

~

u

2

(or)

dor,

v

2

(t) =

~

v

2

(or)

dor

t

OllIIChlBalOTCJI

ypaBHeHUJIMl[

2

u2

v

(42.32)

~~

= -

2fAo

' v = - 2V .

CHCTeMa

(42.4) CBOP;IlTCJI B

p;aHHOM

cJIyqae

R

ypaBHeHuIO

(42.33)

:i:

= w

(x

= y -

Z,

W = U - v).

ECJlu

Mhl

peJllHM

Terrepb

cooTBeTcTBYIO~YIO

aap;aqy

(42.1) 0

rrpep;eJlbHOM

6hICTpop;eRCTBUU,

TO

nOJlyqUM,

'ITO

x (t) x

2

(t)

(42.34)

w~

(t2

~=

-

TO'

TO

=

~2

(t)'

~

(t) =

P.

(t) - v (t)

II,

CJIep;OBaTeJlbnO,

!l-V

(42.35)

u=uO[lI,

z,

p.,

v]=-p.

Y-Z

'

p.-v

(42.36)

v = V

O

[y

Z

II,

v] = - v

---

.

) 't"'"

y-z

IIoRanwM,

'ITO

napa

yrrpaBJIeHuii:

(42.35) H (42.36)

He

06eCOe'llIBaeT

cep;JlOBYIO

TO'lKy,

rp;e

max

min

l'

=

min

max1'=

T°.

IIycTb

to

<

O.

IIocTapaeMcH

v

'U.

U v

\

oop;o6paTh

na'IaJlhHhIe

YCJlOBBJ1 x (to) = y (to) - Z (to) >0 B s(to) =

p.

(to)-

_ v

(to)

U

yopaBJleHue

u (t)

(to

<:;;;

t <0)

TaR,

'lT06bl

Bce

BpeMH

OpB

t

;;;;.

to

BP;OJIb

p;BHiKeHHH

x (t) (42.33),

rp;e

1l0JlOmeHO

u = u (t) H V = v

O

(42.36),

BhI-

1\.,

1l0JlHHJIOCb

YCJloBue

"

{x

(t),

S

(t)}

Ef,

T°

(x

(t), S

(t»

= 1 (42.37)

U

'lT06b1BCTpetIa

y (t) = z (t)

ocy~ecTBUJIaCb

rrpH

t =

O.

)J.JlH

BhIllOJlIleHHlI

yc-

JlOBHH

(42.37) BCJlep;CTBUe (42.34) p;OCTaTO'lHO,

'lT06hI

BhIllOJlHHJlOCb

YCJloBue

S (t) =

Il.

(t) - v (t) = x (t) >

O.

(42.38)

'YnpaBJleHUe

v = V

O

(42.36)

6yp;eT

Torp;a

yP;oBJleTBOpJITb

paBeHcTBy

(42.39)

v =

VO

=

-v

(t).

lIa

(42.32) U (42.39),

BhI6upaH

v (0) =

1,

llOJlY'lHM:

-

.!..

t'

v(t)=e

2 (42.40)

Torp;a

113

(42.33), (42.38), (42.39) H(42.40)

CJIep;yeT,

'ITO

p;OJIn,HO

6hITb:

-

1:...t

(42.41)

p.\t)=e

2

+x(t),

_1:...t

2 • )

(~2.42)

u (t) = - e +

x(t

•

I

r~

'\<....:..:---

§ 42]

3AAAqA

0

BCTPEQE

0AHOTHrrHbIX

OB'bEK'rOB

I

345

Teoepb

4JYIlK~HH

x (t) OOpep;eJIHeTCJI

D3

p;lt4J4Jepelll.\IofaJIbHOrO

ypaBHemuI,

IWTopoe

OOJIyqaeTcH

OO,l:J;CTaHoBKoii: (42.41), (42.42) B

oepfloe

ypaBHeHF.le

(42.32).

!

8Top;H4J4JepeH~HaJlbHOe

ypaBHeHHe

I'lMeeT

BlIP;

[

-2:-

t

JI/

2

x=-x-

xe 2 + x

2

• (42.43)

'YpaBHeHHe,

(42.43)

rrpH

yCJIOBHH

x

(0)

= 0

MOffiHO

paapemUTb

B

4JopMe

pJIp;a

t

2

x (t) = T +

<p

(t),

<p

(t) =

ctata

+

ctIt

I

+

..

','

(42,44)

OpU'leM

P;JIJI

4JYIlK~UU

<p

(t)

OOJIyqaeTCJI

p;H4J4JepeH~HaJIl>Hoe

ypaBHemle

cP=

f

[t,

<p]

C

rOJIOMop4JHoii:

opaBoii:

'laCTblO

B OKpeCTHOCTH

TOtIRlt

t = 0, x =

O.

OTclOp;a

COrJIaCHO

TeopeMe

Room

([12*],

CTp.

55) CJIep;yeT CXOP;HMOCTb

pJIp;a

(42.44)

opn

p;OCTaTO'lHO

MaJIhIX

3HaqeHHJIX

t.

RpoMe

Toro,

llpU

MaJIhIX

t

4JYIlK~nJI

x (t) (42.44)

1l0JIOffiHTeJIbHa.

IITaR,

llpU

p;OCTaTO'lHO

MaJIhIX

3Ha'leHUHX

t (to

<:;;;

t <

0)

1l0cTpoeHo

yllpaB-

JIeHHe

u (t) (42.42)

_'!"t

t .

u (t) = - e 2 +

2"

+

cp

Talwe,

'ITO

XOTJI

llpU

Bcex

t

113

UHTepBaJIa

[to, 0)

yllpaBJIeHHe

v =

vO

[y (t),

z(t), Il.(t), v(t)]

BhI6HpaeTCH

Bce

BpeMJI

B COOTBeTCTBJiU C

npamWOM

(42.18),

llpH-

'leM

ORa3hIBaeTCJI

Bce

BpeMH

T°

[x

(t), S

(t)]

==

1,

op;HaRO

BCT})etIa

p;BuffieHuH

y (t) H z (t)

ocy~eCTBJIHeTCH

B p;eHCTBUTeJIbHOCTH

llpU

t =

O.

CJlep;OBaTeJIbHO,

B

p;aHHOM

rrpHMepe

BpeMH

p;o

BCTpetIlI

l'

OplI

V =

VO

OKaaJllBaeTCH

MeHbllIe,

'leM

T°

[x

(to),

~

(to)],

eCJIlI

I

to

I <

1.

IIpHBe,n;eHHIiIH

rrpHMep

,n;OKa3IiIBaeT

BI>ICKaSaHHoe

BIiIIIIe

YTBepm,n;e-

HHe

0

TOM,

'ITO,

BOOOm;e

rOBOpH,

BeJIH'IHHa

TO

[x

(to),

~

(to)]

He

HB-

JIHeTCH

MaRCHMHHOM

BpeMeHH

,n;o

BCTpe'IH

y (t) H z (t), a

rrapa

yrrpaB-

JIemlH

UO

H V

o

(42.18)

He

HBJIHeTCH

ce,n;JIOBOH

TO'IROH

COOTBeTcTByIO-

m;eH

Hrpl>I,

,n;ame

eCJIH

OrpaHH'IHBaTbCH

yrrpanJIeIIHHMH

U H v,

He

BIiIBO,n;Hm;HMH

,n;BHmeHIIH

(42.1), (42.2), (42.3)

HS

06JIaCTH

r.

I1MeHHo,

rrpHMep

rrOKaSIiIBaeT

BOSMOiI\HOCTb

TaKHX

OTKJIOHeI-IHH

yrrpaBJIeHHH

U

01'

uO,

rrpH

ROTOpI>IX,

XOTH

,n;BHmeHHe

(42.2)

Hoy,n;eT

Bce

BpeMH

rrpH

t

>-

to

yrrpaBJIHTbCH

rro

rrpaBHJIY

(42.18),

BCTpe'Ia

rrpoHsoH,n;eT

paHbIIIe,

'IeM

B

MOMeHT

t =

to

+

TO

[y(t

o

),

z(to),

f1(t

o

),

v(to)] =

min

max

TtL,v'

u v

3aMeTHM,

o,n;HaKO,

'ITO

OTCyTcTBHe

ce,n;JIOBOH

TO'IKH

CBHsaHO

B

paso-

opaHHOM

TOJIbRO

qTO

npHMepe

cym;ecTBeHHO

C

HIITerpaJIbHI>IM

xapaK-

TepOM

OrpaHH'IeHHH

(40.6). B

aTOM

MomHO

ObIJIO

ObI

yoe,n;HTbCH,

paso"'

opaB

CJIy'IaH,

Kor,n;a

OrpaHH'IeHHH

(40.3) n (40.4)

HMeIOT

BH,n;

II

U

('r)

11-<

f1

H

II

v (or)

II

-<

v,

rrpH'IeM

,n;OJIlRHbI

COBrra,n;aTb

Bce

tPa30BbIe

Koop,n;HHa-

TbI.

Tor,n;a

MM

yBH,n;eJIH

ObI,

'ITO

rrpaBHJIO

aI'CTpeMaJIbHOrO

rrpHU;emI-

BaHHH

OOeCrre'IHBaeT

He

TOJIbRO

M-UNUJlwnc

T,

HO

H paBHI>II1:

eMy

Man-

CUM-UN,

rrpH'IeM,

CJIe,n;OBaTeJIbHO,

BliIrrOJIHHeTCH

yCJIOBHe

(40.12)

ce-

oAoeou

mOtl.nu.

O,n;HaRo

s,n;eCb

cepbeSHIiIM

OOCTOHTeJIbCTBOM,

3~lTPY,n;-

346

. "

.\

.

:.'

~

'. '

:.':

:',:

;.:-~

..

: :

'7-.:

.::.'-:

•.

.''''.

'J'!

.'.-'

::,-,,--:

.

,'or;

•..

~

......

"

o

tr.il.

10

t1I'POllMi:

SA,t(A'lA

0

:eC'I'PEqE

,r(BHml!1Hlnt

HHIO:r.u;HM

HCCJleAOBaHHe

np06J1eMbI,

HBJlHeTCH

BonpOc 0

KJlaCCe

Tex

ynpaBJleHHH

u (t) H v (t) (HJlH u

[tl

H v

[tl,

HJlH,

HaKOHe~,

U

=U[t,

y,

z,

11,

V],

V = V [t, y, z,

11,

vl),

KOTOp:bIe

AonycTHM:bI

AJIH

KamAOro

Ha

napTHepOB. B nOATBepmAeHHe CKaaaHHoro

paCCMOTpHM

cJleAYro-

:r.u;HH

npHMep.

/71

o /

••••••

Z

1/

1"=1,0

')1-0.9

.............

/ I I

»-

1'=2

oI

..

' }

1°_2

t

oI

/,1

,

::::::r=~b'g

..

,

1"=2

'Y

-I

-/r-

1/

PIIC.

42.6.

PM.

42.5.

llplIMep

42.3.

IIycTb

,n;BlIlReWUI npecJIe,n;yIOm;ero

II

rrpeCJIe,n;yeMoro 06'beR-

TOB,

RaR

II

B rrpHMepe

42.1,

onpe,n;eJIJlIOTCJI

,n;mp¢epeWJ;HaJIbBLIMII ypaBHeHIIJlMII

(42.20)

II

(42.21),

HO

B OTJIWIlIe

OT

/t-!,;O

y~/aO

.'/.....

t···

'.

:

\Z,

/ '

.''''

Or

\../

~'p#'

(42.19)

orpaHII'leHIIJI Ha ynpaBJIJlIOm;l1e

B

03,n;eHcTBIIJI

IIMeIOT

BII,n;

i u

('t')

1<

f.t,

I v

('t')

\ <

V,

(42.45)

r,n;e

f.t

11

V -

rrOJIOlRRTeJIbHhIe

rrOCTOJlH-

!.

HbIe.

11

nycTb

CHOBa

B

MOMeHT

BCTpe'lR

Tpe6yeTcJI

COBna,n;eHRe

ROop,n;IIHaT

TO'leR

£(l)

II

£(2)

R

IIX

CRopOCTeii

~(l)

Ii

~(2).

ECJIII rrOCTpOIITb

,n;JIJI

paCCMaTpRBaeMblX

06'beRTOB

COOTBeTCTBYIOm;lIe

06JIaCTR

,n;OCTRlRIIMOCTR

C y'leTOM

(42.45),

TO

pemeRRe

aTOH

3a·,n;a'lR,

BbITeRaIOm;ee

R3

npaBIIJIa aRCTpeMaJIbHOrO npWJ;eJIRBa-

HIIJI,

TaROBO:

aRCTpeMaJIbHbIe

ynpaB-

.

JIeHRJI

U = Uo

II

V =

Vo

orrpep,eJIJI-

IOTCJI

3aKOHOM

uo=

f.t,

vo=

V,

eCJIII

TO'lKa

{Xl

(t), X

z

(tn,

Xi

(t) =

Yi

(t) -

_ Zi (t)

JIelRRT

HRlRe KpHBOH

\.

Xz!

xz!

= 0,

(42.46)

Xl+

2(IJo-V)

PIIC.

42.7.

Ii

u

o

=

-I!,

VO

=

-v,

eCJIII

TO'lKa

{Xl

(t),

Xz

(t)}

JIe;KRT BbIIDe RpllBOH

(42.46)

Ha

rrJIOCKOCTII

Xl'

Xz

. 3aMeTRM, 'lTO B HameM CJIy'lae

yK~3aHHbIe

aRCTpe-

MaJIbHhIe yrrpaBJIeHRJI 6y,n;yT

Ii

orrTRMaJIbHhIMII B

CMbICJIe

3a,n;a'l1l

40.1

R, 60JIee

TO-

1'0,

ORR

06ecne'lRBaIOT TaR;Ke yCJIOBlIe

(40.12)

ce,n;JIOBoii:

TO'lKH.

rIa

plIc.42.5,

42.6

I

·0"-

I

§

42]

SA,ll;AQA 0

BCTPEqE

O,ll;HOTHIlHbIX

OB'bEKTOB

347

R

42.7

R306palReHhI HeRoTopbIe peaJIlI3a:n;1I1I

,n;BRlReHRH

06'beRTOB

B cIITyaiVIJIX,

aHaJIOrR'lHbIX paccMoTpeHHbIM BbIIDe, B npRMepe 42.1

(CM.

pRC.

42.1,

42.2

II

42.3)

.(aTlI peaJIlI3a:n;RlI rrOJIy'leHbI C

nOMOm;bIO

anaJIOrOBbIX Bbl'lIICJIlITeJIbHbIX yCT-

pOHCTB).

BalRHOH oc06eHHocTbIO

JlBJIJleTCJI

3,n;eCb

B03HIIKHOBeHlIe

CKOJl,b3J1,UfUX

pelRllMOB

(CM.

plIC.

42.8

II

42.9,

iI306palRaIOm;lIe

UO[t]

B CJIy'laJlX, Kor,n;a

rrpe'

CJIe,n;yeMbIH

rrapTHep

OTRJIOHJleTcJI

OT

OnTllMaJIbHOH CTpaTerllH).

PIIC.

42.8.

PIIC.

42.9.

liTaK,

paao6paB npHMep 42.3,

M:bI

BHAHM,

'ITO

B

AaHHOM

cJly'Iae

nOCTaHOBKa

aaAa'IH 0 npeCJleAOBaHHH

no

npHH~Hny

06paTHOH

CBHaH

AOJlmHa AonycKaTb

peaJIHaa~HH

BoaAeHCTBHH

u

[tl

H V

[tl

e:r.u;e

60Jlee

06:r.u;eH

npHpOAbI, 'IeM BBeAeHHbIe

HaMH

paHee AonycKaeMble ynpaBJIe-

HHH

u (t) H v (t).

8TO

06CTOHTeJlbCTBO

Tpe6yeT cepbeaHoro MaTeMaTH-

'IeCKOrO aHaJlHsa.

IT

P

II

M e

'l

a H

II

e

42.1.

IIpaB,n;orro,n;06HbIe paccylR,n;eHIIJI 1I3

§§

41,

42

M;OlRHO

CTporo !Jl0pMaJIJil30BaTb. OITlIpaJICb

lIa

nOCJIe,n;onaTeJIbUOCTb {t,.}

(k

= 0.

1,

...

;

tk+l-t"

=

6>0)

CMeHbI

3Ua'leUHH u

[t]

C

rrOCJIe,n;Ylom;IIM

rrpe,n;eJIbHblM nepexo,n;OM

npll

6

.....

0.

8TO

ocym;eCTBJIfIeTCJI TaK. Orrpep,eJIHM Bemr'lHH[,[

it

[t,,] peKyppeUTHO:

it[to]-MoMeHT

rrorJIOm;eUHJI

,n;JIJI

y[to]

..

z[t

o

],

I![t

o

],

v[to],

rrpRy[t

k

+

1

],

z[tk+l],

f.t[tk+d,

v

[t"+l],

it

[td.rrOJIaraeM

it

[t"+l]

=

min

(it

ltd,

it~+1)'

r,n;e

it?+1

-

MOMeHT

no-

rllOm;eHlIJI

,n;JIJI

y

[tk+d,

z [th·+d,

IJo

[t

li

+).],

v [t,,+1]' I1yCTb

,n;aJIee

qO

ltd

JIp

O

[t,,]-

TO'lKlI

JI3

06JIaCTeH GCl>

[tk]'

Y

[tli],

IJo

[tli],

{jo

[th:l

JI

G

C

2}

[t/,],

z [t/,], V [t,,],

{jo

[td.

,n;rrJI

KOTOpbIX ,n;ocTIIraeTCJI

max

p

min

q

II

p - q " rrpH p

113

GCZ}

R q

1I3

GCl>.

8KCTpeMaJIbHOe yrrpaBJIeHIIe

UO

orrpe,n;eJIJIeTCJI

Tor,n;a

paBeHCTBOM

UO

[t]

=

uO

[y

[td.

z

[tli],

f.t

ltd,

v

[tli],

it

[tli]]

(42.47)

(t"

~

t <t

k

+

l

)

ll3

yCJIOBRJI

rrpHll,eJlIIBaHlIJI B MpMeHT t = tk

,n;BHlHeHRJI

(40.1)

B T.O'lKy

qO

[tk]

(K

MOMeHTy

t =

it

[t"J). B c,JIy'lae

o,n;HOTHrrHbIX

06'beKTOB yrrpaBJIeHRe UO

(42.47)

348

"

,,'"

....

.

,,'

.••••.••

'

...

, .... • .•. ;

...

~;.;

...

; "

",",

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r

...

,

\~:~;:

:

[rn.

to

HrpOBAli 3AAA"IA 0 BCTPE'IE ABHmEHHfi

06eCne'llIBaeT

MlIHlIMaKC

lu'

=

minu

lu

=

>(}

[to]

-

to.

rAe

lu

=

sup

[lim

sup

(SUp

T~,v)]'

(42.48)

E>O &-+0 v

3Aech

to

-+

T~

v -

MOMeHT

t,

KorAa

BnepBble BblnOJIHlIeTClI nepaBeHCTBO

II

y

(t)

_ i

(t)

II

<

e:

B

cny'lae

OrpaHlI'leHlIB:

Ha MrHOBeHHble

aHa'leHlIlI

CUJI

(Ha-

npRMep.

II

u

II

<

11,

IIv

II

<

v,

11

>

v)

yTBepmAeHue

coxpaHlIeT

CUJIy U AJIlI

aa-

Aa'lU

0 BCTpe'le

no

'laCTU

KoopAlIHaT.

B 06m;eM

cJIy'lae

HeoAHoTunHblx 06'LeK-

O

TOB

nplI

OrpaHlI'leHlIlIX

BlIAa

1\

u

11<

11.

II

v

II

< v

ynpaBJIeHlIe

u

(42.47)

TaKme

06eCne'lliBaOT C6JIIilHOHlIO B

CMblCJIe

(42.48) He

noame

'leM

B

MOMOHT

t =

>(}

[to].

eCJIlI

l.'OJIhKO

lIMeeT

MeCTO

Heoco6oe

nOrJIbm;eHlIe 06JIaCTeii:

G(2)

06JIaCTlIMlI G<l'

(CM.

CHOCKy

ua

CTp. 333)

..

CJIeAyeT

Ao6aBliTh,'lTO

B

cJIy'lae

HOojJ;HOTlInHblx 06'beKTOB (40.1), (40.2) U

nplI

OrpaIllI'lOHRlIX

p*[u('t');

t < 't'] <

11,

p"'[v('t');

t<'t']<

v

MOMeHT

norJIOm;emIlI

>(}O[t]

MQll.{lIO

OnpOAeJIlIThKaK

UaJilMeUhml!ln KOpOUh

ypaBHeHlIlI

min

{I1P[B'('t')

¥'[>(} , 't']I;

t<

't' <

>(}]

-

vp[Q'('t')Z'[>(},'t']l;

t < 't'

<;

>(}]-+

Ul

U~l

+

(¥[il',

tJy[t]

-

Z[>(},

t]z[tJ)'I}

= 0, (42.49)

rlJ:e

Y l!I

Z-CooTBeTcTB6HHO

<pyuAaMeHTaJIhHhIe

MaTpllIlbl

AJIlI

CUCTeM

(40.1) l!I

(40.2),

.a

p-HopMa,

K KOTOp.oii

conplImeHa

aajJ;aHHalI

HopMa

p'.

¥puBHeHHe

(42.49). BblBOAllTClI

IIa

onpeAeJIeHMlI BeJIlI'lHHbl

>(}o

KaK

CJIeACTBlle COOTHOilleHIIn

BllAl

(17.13), (17.17), onpeAeJIlIIOm;llX 06JIaCTII AOCTl!Iml!lMOCTll

C(l)

II

d

2

).

3aMoniM

em;e, 'lTO

HaIIllI

paccymAeHIIlI

rrepeHoclITclI

6ea TpYAa

II

Ha CJIy-

'ran

lIrpoBblx

aaAa'l,

rAe

ocym;ecTBJIlIeTCJl MaKCl!IMIIU (MIIHlIMaKC) pacCTOlIHllJl

Mell.{AY TOTfKaMII

{y

(il')}m

U {z

(>(})}m

B <pIIKClIpOBaHHblB:

MOMeHT

BpeMeHll t =

>(}.

3TH aaAa'll!l

HMeIOT

60JIee

peryJIlIpHblH

xapaKTep,

'leM

paCCMOTpeHHble

Bblille.

l1x

aHaJIll3

MomHO peKOMOHAOBaTh 'll!lTaTeJIIO B Ka'leCTBe

ynpamHeHHlI.

B 3aKmO'IeHHe

HaCTOHm;eR

rJIaBbI He06xop;HMO

OTMeTHTb,

'ITO

Bam-

HbIM

HHcTpyMeHToM

npH

pemeHHH HrpOBbIX aap;a'l B TeopHH ynpaBJIHe-

MbIX

CHCTeM

H,

B 'laCTHOCTH, B aap;a'lax 0 npeCJIep;OBaHHH

HBJIHeTCH

MeTOp;

p;HHaMH'leCKOrO nporpaMMHpOBaHHH

[28a,

341].

Op;HaKo

06-

cymp;eHHe

yKaaaHHoro

MeTop;a

BbIXOP;HT

aa paMKH cop;epmaHHH p;all-

HOR

KHHrH.

0>

r.f1,

a (J a 11

JUIHEIIHbIE

CIICTEMbI

C

OCOEEHHOCTlIMM

§

43.

CncTeMa C

nOCJIep;eHCTBHeM,

peaJIHayrom;aa Haomop;aeMyro BeJIH'IHHy

BbIme,

B rJIaBaX 8

1I

9, MbI H3y'laJIH aap;a'lY 0 Ha6mop;eHlUI JIHHeR-

BOR

p;IIHaMH'leCKOR

CHCTeMbI,

onpep;eJIHH

HCKOMbIR

4>aaoBbIR

BeKTOp

x

(t)

Ha

OCHOBaHIIH

IIH4>opMaIJ;IIII 0 npeiJbLcmopuu Ha6mop;aeMoR BeJIH-

qHHbI

y

(t)

(31.6).

Tenepb

MbI yCTaHoBIIM

CBHab

Mei-Kp;ypemeHHeM

aa-

p;aqH 0 Ha6mop;eHlUI C HeKOTopbIMII Bo.npocaMH

ua

Teopuu

JIUHeRHbIX

p;II4>4>epeHIJ;IIaJIbHbIX

ypaBHeHHR C 3ana30lileaJ-tUeJlt apZYMeJ-tma.

06-

cyiRp;eHIIe

aTOH

CBHaH

H

COCTaBJIHeT

OCHOBHoe

COp;epJKaHIIe

HaCTOH-

m;ero

II

nOCJIep;yIOm;ero naparpa4>oB.

TeM

caMbIM, C

OP;HOR

CTOpOHbI,

MbI npOHJIJIIOCTpIIpyeM

em;e

paa pemeHIIe aap;a'lU 0 Ha6JIIOp;eHHH,

a C

p;pyroR

CTOpOHbI,

p;ap;IIM

HOByIO

IIHTepnpeTaIJ;HIO

4>aKToB,

H3BeCT-

HbIX B TeopIIH ypaBHeHIIR C aanaap;bIBaHIIHMII.

IIpII

aTOM

HaM nOTpe-

6YIOTCH

OTp;eJIbHbIe

CBep;eHHH

Ha

TeopHH

CHCTeM

C

nOCJIep;eRCTBHeM

(B

06'beMe, HanpHMep, KHHrH

[26*]).

BnpO'leM,

KaK

H B

p;pyrnx

paaA~~

JIax p;aHHoR MOHorpacPHH, Bce

OCHOBHbIe

HCnOJIbayeMbIe

nOHHTHH

6y":

.p;yT KpaTKO

nOHCHHTbCH

no

xop;y

p;eJIa.

IhaK,

paCCMOTpHM

JIHHeRHyIO p;UHaMII'lecKyIO cHcTeMy

dx

(t)

=

Ax

(t),

(43.1)

lit

npep;nOJIaraH, 'ITO peayJIbTaTOM IIaMepeHIIR

ee

p;BHmeHHH

HBJIHeTCH

BeJIII'lIIHa

Y (t) =

g'

x (t).

(43.2)

3p;eCb

A -

nOCTOHHHaH

(n X n)-MaTpIIIJ;a, g -

nOCTOHHHbIR

n-Mep-

HbIR BeKTOp.

,IJ;JIH

ynpom;eHHH

MbI

OrpaHII'lHBaeMCH CJIyqaeM, Kor;n;a

Y

(t)

- CKaJIHp. Byp;eM paCCMaTpIIBaTb aap;a'lY

34.1

0 Ha6JIIOp;eHIIH

BeJIIIqIIHbI

1]

(t)

=

p'

x

(t)

(43.3)

no

Hp;eaJIbHoMy CIIrHaJIy

Y (t +

'1')

. g'x

(t

+

'1')

(-i}

<

'1'

< 0).

(43.4)

RaK

YKaabIBaJIOCb B §

34

(CTp. 289), p;JIH

Toro

'lT06bI

pemIITb

aTy

aap;a'ly,

nOJIeaHO

aa;n;aTbCH

KaKHM-JIH60 yCJIOBIIeM

OnTHMaJIbl;IOCTH,

UMeIOm;UM

CMbICJI

OrpaHH'leHHH Ha

HOpMy

paapemaIOm;eR

onepau;lm,

(2;,

350

JIHHE:A:HbIE

CHCTEMbI

C

OCOBEHHOCTJIMH

[rJI.

11

cp.

[y

(t+r)

\

fJ

(t)]=fJ

(t).

M:bI

6yp;eM

HCKaTb

Onepa~HIO

cp

[y

(t

+

'1')]

B

BHp;e

°

cp[y(t

+'1')]

=

~

b\(t+'t)dV('t),

(43.5)

-if

rp;e V

('1')

-

WYHK~HJI

C OrpaHH'IeHH:bIM

H3MeHeHHeM

Ha

OTpeaKe

[

-'it,

0],

H BbI6epeM B Ka'IeCTBe

yCJIOBHJI

OnTHMaJIbHOCTH

Tpe60BaHHe

p'

[cp]

=

var

[V

('1')]~

=

min.

(43.6)

,1J;JIJI

nOCTpOeHHJI

OIITHMaJIbHOH

paapemaIOm;eH

onepa~HH

cpo

[y (t +

'1')]

BOCnOJIbayeMcJI

npaBHJIOM MHHHMaKca

33.1;

COrJIaCHO

aToMy

npa-

BHJIy npemp;e Bcero

Hap;JIemHT

HaHTH

MHHHMaJIbH:bIH

CHrHaJI

{yO

(t +

T)

I

fJ

(t)

=

1}

Ha

YCJIOBHJI

(33.6),

T.

e.

Ha

yCJIOBHJI

(CM.

06

aTOM

nop;p06Hee B §

33,

CTp.

284)

pC

=

max

I

{yO

(t + '1')'

fJ

(t) =

1}'

=

-{)~-:~o

=

min

maxl{y(t+'t)lfJ(t)=1}I.

{43.7)

y

-~~T~O

TIpep;nOJIOmIIM, KaK

II

B

p;pyrIIx

aHaJIOrII'IHbIX CJIy'IaJlX, paa06paH-

H:bIX

B:bIme,

'ITO

CIIrHaJI

{yO

(t

+

T)

I

fJ

(t)} = 1

TeM

IIJIII

IIH:bIM

cno-

C060M

HaHp;eH.

HaM

IIaBeCTHO,

lITO

IICXOp;HaJI

aap;a'Ia 0 Ha6JIIOp;eHIIII

IIMeeT pemeHIIe Torp;a

II

TOJIbKO

Torp;a, Korp;a

pO

>

O.

TIOJIQmIIM

B

(43.3)

p'

=

g'A.

Torp;a

06J1aaTeJIbHO

pO>

O.

B

caMOM

p;eJIe,

IIMeeM:

y

(t

+

T)

=

g'x

(t

+

T)

=

g'

X

[t

+T,

t]

x (t), (43.8)

rp;e X

[t

+ T,

t]

-

WYHp;aMeHTaJIbHaJl

MaTpII~a

CIICTeM:bI

(43.1),

06pam;aIOm;aJlcJI

npII

T = 0 B ep;IIHII'IHyIO

MaTpII~Y.

,1J;JIJI

B:bInOJIHeHIIJI

'II,

yCJIOBIIJI

pO

> 0

p;OCTaTO'IHO,

'IT06:bI B:bIpameHIIe (43.8)

He

06pam;aJIOCb

TOJKp;eCTBeHHO

B HyJIb

npII

-'it

< T < 0 HII

npII

KaKOM

aHa'IeHIIII

BeKTopa

x (t),

CTeCHeHHOM

yCJIOBIIeM

.

fJ

(t) =

p'

x (t) =

g'

Ax

(t) =

1,

(43.9)

,1J;IIwwepeH~IIpYJl

npaBYIO 'IaCTb

paBeHCTBa

(43.8)

no

T, Y'IIIT:bIBaJl

paBeHCTBO

dX

[t

+ T,

t]ldT

=

AX

[t

+ T,

t]

H

COOTHOmeHHe

(43.9),

nOJIy'IIIM

npH

T = 0

(:-r:

[g'X

[t

+

1',

t]

x

(t)]

)T=O

=

g'

Ax

(t) = 1.

(43.10)

113

(43.10)

CJIep;yeT,

'ITO

Tomp;eCTBO

y (t +

T)

=0

npII

yCJIOBIIH

(43.9)

HeBoaMomHO.

8TO

H

p;OKaa:bIBaeT

HepaBeHCTBO

pO>

O.

IfTaK,

paapemHMOCTb aap;a'IH 0

Ha6JIIOp;eHHH

npH

p'

=

g'

A p;oKaaaHa 1).

. 1)

IIoJIe3Ho

3aMeT.II:Tb,

'ITO

3TOT <!JaRT MOlI\HO 6bIJIO BbIBeCT.II:

.II:

H3

06mHX

Co-

06palI\eHHH,

01IlipalICb

Ha

TO

06CTOJITeJIbCTBO,

'ITO

WYHR~lilI

if

(t)

eCT!>

<!JYHK~liJI

ilHllJIHTH'leCRaJI.

..

" . .

..

'

..

;.:

:.;..

:

:c.

'::';...

'.'

..

'

>:.

;'::::'.

'::':'>::'.:':'

'-:·;:>·.;,;i.;'

r

'~:

§ 43]

CHCTEMA

C

nOCJIE,!J;El1:CTBIIEM

351

Tenepb

MomHO

onpep;eJIHTb

IICKOMyIO

onepa~HIO

cpo

[y

(t

+

T)

J,

OnHpaJlCb Ha

yCJIOBHe

MaKcIIMyMa

(33.16):

°

q;O

[{

yO

(t

+

'1')

,

fJ

(t) =

1}]

=

~.

{yO

(t

+

'1')

I

fJ

(t) =

1}

dVo

(1')

=

-&

°

=

max

~

{yO

(t +

'1')

IfJ·(t) = 1}

dV

(1')

(43.11)

v

_~

npH

var

IV

(T)]~~

=

lI

p

o.

C TaKoH aap;a'IeH

M:bI

yme

BCTpe'IaJ!HCb B

§

23,

pemaJl aap;a'Iy 06 ynpaBJIeHHII C

MHHHMaJIbHbIM

IIMnyJIbCOM

yn-

paBJIJlIOm;eH

CIIJI:bI.

BamHo

aaMeTHTb,

'ITO

WYHK~IIH

~(T)

= I

{yO

(t

+

T)

I

fJ

(t)

=

1}

I

p;OCTIIraeT

CBoero HaH60JIbmerO aHa'IeHHH JIHmb B

KOHe'IHOM'IIICJIe

m

HaOJIHpOBaHH:bIX

3Ha'IeHIIH T =

--:-

Tj

(Tj>

0;.

j =

1,

...

,

m)

Ha

oTpeaKa

-'it

< T <

O.

B

caMOM

p;emj, IIHa'Ie

WYHK-

~HJI

yO

(T) p;OJIjfma

6:bIJIa

6:bI

6:bITb

nOCTOHHHOH

BeJIH'IHHOH

(cpaBHH

C paccymp;eHHHMH Ha

CTp.

185),

a

aTO

IICKJIIO'IeHO

yCJIOBIIeM

(43.10).

TIOBTOPJlH

Tenepb paccymp;eHIIH

Ha

§

23

Ha

CTp.

186,

npIIxop;HM K Bbl-

BOP;y,

'ITO

WYHK~IIH

VO

(T)

HBJIJleTCH

KycO'IHO-nOCTOHHHOH

C paapbI-

Ball-HI:

Henpep:bIBHOCTH

B TO'IKaX T = -

Tj

H,

CJIep;OBaTeJIbHO,

m

fJ

(t) =

cpO

[y

(t +

1')]

=

L;

ajY

(t

-

'tj).

(43.12)

j=1

3p;eCb

'Iepea aj 0603Ha'IeH:bI CKa'IKH

WYHK~IIII

VO(t)

B TO'IKax T = -

Tj.

TIpop;IIwwepeH~IIpyeM

Tenepb

paBeHCTBO

(43.2)

no

t. Y'IIIT:bIBaH

npH

aTOM

COOTHOmeHHH

(43.1),

(43.3) H

(43.12),

npHxop;IIM K

BbIBOP;y,

'ITO

Ha6JIIOp;aeMaJl

BeJIH'lHHa y (t)

yP;OBJIeTBOpHeT

CJIep;yIOm;eMy

ypaB-

HeHIHO

C

aanaap;:bIBaHl'IJIMH

Tj apryMeHTa

t:

m

.

"',

Y (t) =

L.J

ajY (t -

'tj)'

(43.13)

j=1

TaKHM 06pa30M,

lW1WBa

6b~

HU

6bl1la

BeKmop-(/jyHKq,UR,

x (t), R,B-

,l/,R,lOUfaR,CR,

peUleHUeM

ypaBHeHUR,

(43.1),

Ha6,1/,108aeMaR,

BMU'l(,UHa

y (t)

(43.2)

R,B,I/,R,emcR,

peUleHueM

ypaBHeHUR,

(43.13).

06paTHoe aaKJIIO'IeHIIe, Bo06m;e rOBopH,

HeBepHO.

He

P;JIH

BCH-

Koro

pemeHHH y

(t)

ypaBHeHIIH (43.13)

MomHO

nop;06paTb

BeKTOp

x (t),

yp;OBJIeTBOpHIOIU;IIH

ypaBHeHIIIO (43.1) H

TaKOH,

'ITO

WYHK~IIII

X

(t)

II

Y

(t)

6YP;YT

yp;OBJIeTBOpHTb

COOTHomeHIIIO

(43.2).

8TO

eCTeCT-

BeHHO,

TaK

KaK

paBeHCTBO

(43.2)

II

ypaBHeHHe

(43.1)

onpep;eJIHIOT

co-

rJIaCHO

(43.8) KOHe<tHOMepHOe

ceMeHCTBO

WYHK~IIH

y (t). Op;HaKo

co-

rJIaCHO

TeopIIII

p;IIwwepeH~HaJIbH:bIX

ypaBHeHIIH C aanaap;bIBaHHHMII

apryMeHTa

([26*],

CTp.

9) pemeHIIJI y

(t)

ypaBHeHIIH (43.13)

npII

t

;;>

to

nopomp;aIOTCH

BceBOaMOmH:bIMII

Henpep:bIBHbIMH

Ha'l(,a,l/,bHb~MU

352

--

cP"-'XF

n

'':'

.

'x;'

.

......

~

..

',',

"

•..

,.

'

..

'

..

>"

..

-

'.~;'

.

",".".,

•.

'.',.',

':'"','

',"

•.

:

•.

:.',";,.

~".:»:

:.:

.....

.;.:.'~

...

:::'::.~:':

.::;

:~'fz:;-;~~;

•

r~

\~~"

[rJI.

11

JIIIHE1iUbIE

CIICTEMbI

C

OCOBEHHOCTHMII

HpIIBhIMII

Y

(to

+

-e)

(-\t

<:

-e

<:

0)

(IIJIII

BCeBOSMOmHIiIMII

AII<P<P

e

-

peHIJ;IIpyeMhIMII

<PYHKIJ;JIHMII

!J

(to

+ 't)

II

T.

A.)

II,

CJIeAOBaTeJIbHO"

ilTII

pemeHlIH

B

06IIJ;eM

cJIy'Iae

COCTaBJIHIOT

6eCK,OHe'/f,HOMepnoe

ceMeHcTBo

<pyHRIJ;IIH.

IITaK,

B

aTOM

naparpa<pe

Mlil

nOKaSaJIII

BO

BCHKOM

CJIY'!ae,

'ITO,

HaROBIiI

61i1

.HII

61i1JIII

CIICTeMa

(43.1)

II

COOTHOmeHIIe

(43.2),

MomHO

nOCTpOIITb

ypaBHeHIIe

C

nOCJIeAeHCTBIIeM

(43.13),

cpeAII

pemeHIIH

HO-

Toporo

6YAyT

cOAepmaTbCH

Bce

<PYHRIJ;IIII y (t),

onpeAeJIeHHhle

paBeH-

CTBOM

(43.2)

II

ypaBHeHIIeM

(43.1). .

PaccMoTpIIM

npocToH

IIJIJIIOCTPIIPYIOIIJ;IIH

npIIMep.

II

PIIl~leP

43.1. IIyCTb

CIlCTeMa

(43.1)

IlMeeT

BlIP;

(43.14)

:t

l

=

-x

2

,

X

2

=

Xl

II

B

KaJI{ghlii:

MOMeHT

BpeMeHIl t

B03MmKHO

ll3MepeHIle

II

3arrOMllHaHRe Ha oTpe3Ke

3t

t -

2'"

<"t'

< t

KOOpp;IlHaThI

x

2

(t).

EygeM

pemaTb

3aga'ly

0

Ha6mogemm

BeJIIl-

qllHhI

'1

(t)

(43.3),

KOTOpaJI

3gecb,

O'leBHp;HO,

cOBrragaeT C KoopgllHaToll

Xl

(t) Cll-

CTeMhI

(43.14).)J;JIJI

rrOCTpOeHllJI

pa3pemalOm;ell orrepan;IlII

cpO

[y

(t

+

"t')]

(43.12)

He06xop;HMO HallTH 'lllCJIa

aj

H

"t'j.

C

STOll

n;eJIblO

onpegeJIIiM lI3

yCJIOBRJI

(43.7)

MHHIlMaJIbHbIll

ClIrHaJI

{yO

(t +

"t')

I

'1

(t) = 1}.

«:DYHgaMeHTaJIbHaJI

MaTpHn;a

CII-

CTeMhI

(43.14)

llMeeT Bllg

cos

"t'

-

sin

"t'11

X

[t

+"t',

t]

=.

'.

SIll"t' cos"t'

II

II, CJIegoBaTeJIbHO, ClirHaJI

y (t +

"t')(43.4)

paBeH

y (t +

"t')

=

Xl

(t)

sin"t'

+ X

z

(t)

cos

1'.

(43.15)

YCJIOBlIe

(43.7)

rrplIHIlMaeT Bllg:

HaMTH

pO

=

min

max

I

sin

l'

+

Xz

cos

l'

I·

x:::

't'

TaK

RaK

-n/2

<

"t'

< 0,

TO

lfOTPYP;HO

co06pa3111'I>,

'lTO

pO

= 1

II

MllHflMYM

goc-

TflraOTCH, HarrpllMep, UpII X

z

= 0,

II

TOflW,

CJIegoBaTOJIbHO,

I

:1

{yO

(t +

1')

I

'1

(t) = 1} =

sin

1'.

«:DyHKn;IlJI

~

(1')

= I sin1' I

gocTllraeT

HaIl60JIbmerO SHa'lemIJI

pO

= 1 B TO'lI,e

"t'

=

-1'1

=

-n/2.

)J;aJIee,

ll3

CBollcTBa

MaKcIIMYMa

(43.11)

rrOJIY'laeM

cpasy,

'lTO

01

= 1. IITaR,

llCKOMaJI

orrepalWH

cpO

[y

(t

+ 1')] (43.12)orrHChIBaeTCH paBcH-

CTBOM

"

Idt)

= -

Xz

(t

-

~).

YpaBHeHlle

(43.13)

IIMeeT,

CJIeAOBaTeJIbHO,

BliP;

d~~t)

=_y

(t

_

~)

(43.16)

;.:~:

ii.

p.

<:

.....

'

§

41.]

RAHOHH"IECHHE

KOOP.D:HHATbI

CHCTEM

C IIOCJIE.D:EJlrCJBHEM

353

§ 44.

KaHOmilJeCRHe

Roop.a;HuaTbI

CHCTe,,!

c

rrocJIe.a;eiicTBHell'I

H sa;:J;alJa 0

Ua6JIID.a;eHHH

06CYP;'HM

CBH3b

MelliAY

ypaBHeHIIeM

(43.13) H

CHCTeMOH

ypaBHeHHH

(43.1), (43.2)

60JIee

nOAP06HO.

,IJ;JIH

Toro

'IT061i1

pa3JIH'IaTb

o6Ufee

peme-

HHe

ypaBHeHHH

(43.13) H

ceMeHCTBO

<PYHRIJ;HH y (t),

onpeAeJIHeMOe

CIICTeMOH

(43.1), (43.2),

BBeAeM

nepeMeHHyIO

z

(t)

H

6YAeM

paCCMaT-

pHBaTb

ypaBHeHHe

m

dz

It)

~

--

= a·z

(t-t:.)

(M.1)

dt

] ] •

j=1

Y

CJIOBHMCa

0 CJIep;yIOI:u;eM

060SHa'leHHH.

MHomecTBo

'IaCTHhlX

peme-

HHH

z (t)

ypaBHeHHH

(44.1),

rropomAeHHoe

BceB03MomHhIMH

Henpe-

pIilBHO-AH<p<pepeHIJ;HpyeMIiIMH

Ha'laJIbHhIMH

KpHBIiIMH

Z ('t)

(-

\t

<:

't

~

<:

0),

6YAeM

0603Ha'laTb

CHMBOJIOM

{z (t)}.

IIoAMHomocTBo

aTHX

pe-

meHHH

z (t),

KQTOpOe

COBnaAaeT

C

ceMeHCTBOM

<PYHRIJ;HH y (t),

onpe-

AeJIHeMhIX

paBOHCTBaMH

(43.2), (43.1),

6YAeM

0603Ha'laTb

CHMBOJIOM

{y (t)}.

lIs

TeopHH

AH<p<pepeHIJ;HaJIbHlilX

ypaBHeHlIH

C OTKJIOHHIOI:u;HMCH

apryMeHToM

H3BeCTHO

[237],

'ITO

B

JII060M

ABHmOHlIH

z

(t)

HS

ceMeHCT-

Ba

{z

(t)}

MOmHO

BIiIAeJIHTb

K,aHOHU'leCK,Ue

KOOPAHHaTIiI

qi

(t)

(i

=

1,

2,

...

),

HBJIHIOIIJ;HecH

npOeRIJ;HHMH

6eCKOHe'lHOMepHoro

BeK-

_

Topa

q (t).

8TH

RaHOHlI'IeCKIIe

ROOPAHHaTIiI

BhIAeJIHIOTCH

rrpH

nOMOI:u;H

JIHHeHHIiIX

OnepaIJ;HH

'¢ [z

(t

+ 't)

J,

KOToplile

npOH3BoAHTCH

HaA

OT-

peSRaMH

Z

(t

+ 't)

(-

\t

<:

't

~

0)

RpHBOH

z = z (t).

OnepaIJ;HII

'¢i [z

(t

+'t) J

HMeIOT

BHA

m 0

qi(t)='¢i[Z(t+-e)J=BiZ(t)+~

~

z(t+-e)Bij(-e)d.

(44.2)

j=1-"t

j

(i

= 1,

...

, n),

r.a;e

nOCTOHHHlile

Bi H

HerrpephIBHhIO

<PYHRIJ;HII

Bij

('t)

BhlpamaIOTCH

HS-

BeCTHIiIM

o6pa30M

'Iepe3

pemeHlIH

Z *(t)

ypaBHeHlIH

C

orrepemaIOI:u;HM

apryMeHToM

dz*

(t)

= _

dt

conpHmeHHoro

R

ypaBHeHlIIO

3)l;eCb

HBHIiIH

BHA

BOJIHlJ.HH

Bi

H

m

~

ajz·

(t +

-ej),

j=1

(44.1).

MI>I

He,

6YAeM

npHBOAHTb

<PYHRIJ;HH

8ij

('t),

OTCIiIJIaH

'IIITaTeJIH

R

crreIJ;HaJIbHlilM

HCTO'lHHKaM

[237, 3231.

BalliHo

JIHmb

OTMeTHTb,

'ITO

aTH

BeJIH'IHHhI

HeTpYAHO

BhI'IHCJIHTb,

eCJIH

H3BeCTHIiI

ROpHlI

12

H,

H.

HpacoBCImfi

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

.•

'""'

..

"'-'-.

L'

._'.

.,

".'

..

,

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

''-,'"

·.;;:";l':"';;

'

:'::

:-

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

" '

Lj

354

JIHHEfiHbIE

CHCTEMbI

C

OCOBEHHOCTffMH

[rJI.

11

xapaRTepHCTHlJeCROrO

ypaBHeHHff

m

~

ajeXp(-A't'j)-A=O,

(44.3)

;=1

OTBelJaIOm;erO ypaBHeHHIO

(44.1).

KaHOHHlJeCRHe Roop,n;HHaTbI

qi

(t)

y,n;OBJIeTBOpHIOT

CI:ICTeMe

OOMRHOBeHHMX

JIHHeHHMX ,n;H$!JlepeHIQlaJIb-

HMX ypaBHeHHH C rrOCTOffHHbIMH ROs$$HIJ;HeHTaMI:I, MaTpHIJ;a

ROTOpOH

HMeeT iROp,n;aHoBy

$OpMy

([3*], CTp.

127).

CTpyRTypa

STOll:

MaTpI:IIJ;M

aaBHCHT

CHOBa

0'1'

RopHeH A

ypaBHeHlIff

(44.3).

ECJIH MM BMoepeM

RaRI:Ie-JII:IOo

n

RaHOHI:I1JeCRI:IX

ROOPAI:IHaT

qi

(,n;JIH

orrpe,n;eJIeHHOCTH

i =

1,

.•.

, n), OrpaHI:I1JI:IBmI:ICb

,n;JIff

rrpOCTOTbI

CJIylJaeM, Ror,n;a Bce

COOTBeTcTByIOm;I:Ie

ROpHI:I

Ai

rrpOCTble,

TO

rrOJIylJI:IM

CI:ICTeMy

ypaBHeIDIH

qt

(t) =

Atgi

(t)

(i

= 1, ... , n).

(44.4)

KaiR,n;oii TaROH

CHCTeMe

Roop,n;HHaT

qi

(t),

y,n;OBJIeTBOpHIOIIJ;HX

ypaB-

HeHHffM

(44.4),

oy,n;eT

OTBe'IaTb

If,Olle'llloJUep1l0e

ceMeHCTBO

get)}

4lYHR-

IJ;HH

~

(t), ffBJIffIOIIJ;I:IXCff pemeHHeM ypaBHeHHH

(44.1)

H

I.fMeIOIIJ;HX

cor-

JIaCHO

(44.4)

BH,n;

n n

S(t) =

~

Siqi (t) =

~

li exp (Ait),

(44.5)

i=1

r,n;e

Si - HeRoTopMe tPHRcHpOBaHHbIe HeHyJIeBble rrocTOHHHMe,

Ii-

UPOH3BOJIbHMe UOCTOffHHMe.

BepHeMCff R

HCXO,n;HOH

CHCTeMe

(43.1), (43.2),

ROTopaH

uo-

po,n;HJIa

ypaBHeHHe

(44.1).

IIpHMeM

Teuepb,

'ITO 3Ta

CHCTeMa

BUOJIHe

HaOJIIo,n;aeMa.

IIyCTb,

RpoMe

Toro,

OIIffTb

,n;JIH

yrrpOIIJ;eHHH,

IWpHH

Wi

(i

=

1,

.•.

, n) xapaRTepHCTH'IeCROrO ypaBHeHHff

det

(A

-

<oE)

= °

CHCTeMbI

(43.1)

upOCTMe.

lIaBecTHo

([12*J,

CTp.

82),

'ITO cHcTeMy

(43.1)

MOiRHO

upHBeCTH

HeOCOOMM

JIHHeHHMM

upeoopaaoBaIDIeM

W =

Tx

(44.6)

K RaHOHH'IeCROH

$opMe

W(t) =

<OiWi

(t)

(i

=

1,

...

, n).

(44.7)

COOTHOmeHHe

(43.2)

rrpHffiIMaeT Tor,n;a

BH,n;:

n

Y (t) = g'T-1

w

(t) =

~

~iwdt),

(44.8)

i=1

r,n;e

'Iepea

St

0008H3'IeHa

i-ff

ROMUOHeHT8.

BeRTopa-cTpoRH

g'T-t,

rrpH-

",-:-,

.":.,,:......

:

C

§ 44] RAHOHH"IECHHE HOOP,[(1iHAThI

CHCTEM

C 110CJIE,IJ;E1WTBl1EM355

lJeM rrOCTOllHHbHi

Si

(i

=

1,

:

••

, n)

y,n;OBJIeTBOpHIOT

HepaBeHCTBaM

Si

=F

0

(i

= 1,

..

" n), (44.9)

TaR RaR B

aTOM

H

TOJIbKO

B

aTOM

CJIylJae BMUOJIHffIOTCff yCJIOBHff nOJI-

HOH

HaOJIIO,n;aeMOCTH

CHCT8MM

(43.1)

rro

BeJII:I1JHHe

y (t)

(43.2).

(CHcTe-

Ma

(44.7)

MOiReT HMeTb ROMrrJIeRCHMe Roa$$HIJ;HeHTM.

BCIO,n;y

BMme,

o,n;HaRO,

UpH

paccMoTpeHHH aa,n;a'I 0

HaOJIIO,n;eHHI:I

Bce paCCMaTpHBae-

MMe BeJIH'IHHbI rrpe,n;uOJIaraJIHCb ,n;eHcTBHTeJIbHMMH. qHTaTeJIb,

ROTO-

poro

CMyTHT TaROH rrepexo,n;, MOiReT uepeBeCTH cHcTeMy

(44.7)

B

,n;eH-

CTBHTeJIbHYIO

$OpMy.

MbI He

,n;eJIaeM

3Toro,

TaRHaR

aanHCb rrOJIy'IH-

JIaCb

OM

OOJIee

rpOMoa,n;RoH.)

lIa

COOTHomeHHH

(44.7),

(44.8)

H

(44.9)

BMTeRaeT, 'ITO

ceMeHCTBO

{y (t)}

COCTOHT

H3

$YHRIJ;HH Y

(t)

BH,n;a

n

Y (t) =

~

li

exp

(<Oit),

(44.10)

i=1

r,n;e

Ii

- npOH3BOJIbHble nOCTOffHHMe.

OTCIO,n;a

H H3

paccMoTpeHHblx

BMme

CBOHCTB

DemeHHH z (t)

MOiRHO

c,n;eJIaTJ.>

BbIBO,n;,

'ITO

cpe,n;H

RaHO-

IDI'IeCRHX Roop,n;HHaT

qi

(t)

(i

=

1,

...

, n)

ypaBHeHHff

(44.1)

MOiRHO

BMopa'fb

n TaRHX Roop,n;HHaT

qk

(,n;JIff

onpe,n;eJIeHHOCTH

nepeHyMepyeM

HX TaR: k =

1,

...

, n), ROTOpbIe

Ha

$YHRIJ;HffX Y (t) H3

{y

(t)}

Oy,n;yT

COBna,n;aTb

C Koop,n;HHaTaMH W

k

HCXO,n;HOH

CHCTeMbI

(43.1),

(43.2).

B03Bpam;affcb

Tenepb

K rrepeMeHHbIM

Xi

(t), npHxo,n;HM R

CJIe,n;yIOIIJ;eMy

BbIBO,n;y.

llycmb

ypaellellue c

nOC.ll£OeUCmBueJU

(44.1)

nocmp0e1lo

O.M/'

BnO.lu/,e

lla6AlOoaeJUou

cucmeJUbL

(43.1),

(43.2)

malf"

If,alf,

amo onucallO B §

43.

Toeoa

JUO:JIC1l0

y1f,aaamb

n AUlleUllbLX onepa4uu Buoa

in 0

Idz(t+'t'»)=akz(t)+~

~

z(t+'t')akj('t')d't'

(44.11)

j=1-"j

(k=1,

...

,n),

1f,OmOpbLe

onpeoeMlOm

nepeJUel-lllbLe

£k

(t)

= I

k

[z

(t +

't')J

(k

= 1,

...

,

n),

(44.12)

YOOBAemBOpRlOUflle

cucmeJUe

(43.1),

npu'leJU

l-la

(/iYl-l1f,4URX

z (t) = Y (t)

U8

{y

(t)}

BeAU'lUl-lbL

£k

(t)

COBnaoalOm

c 1f,oopouuama.Mu

(/iaaoBoeo

Be1f,mOpa

X (t)

CUCmeJUbL

(43.1).

Onepa4uu

I

k

Iz

(t

+ 't)l

RBMlOmcR

AUlleUl-lblJUU

If,0JU6ul-la4

URJUu

onepa4uu,pi

[z (t + 't)l

(44.2):

n

I

k

[z (t +

't'»)

=

~

d,d'li'i

[z (t +

't'»),

(44.13)

i=1.

eoe

d

ki

-

FlAeJUellm

k-u

cmpo1f,U

U

i-eO

cmoA64a

JUampu4bL

T-\

o6pam-

1l0U

1f,

OOl-lOU

ua

JUampu4T,

npU800RUfux

cucmeJUl/

(43.1)

1f,

1f,al-lOl-lU'lec-

1;oii

¢oJMte

(44.7),

It"

~~-"--~

".

~

'..

",.,

"

..

;,:.,:-;,

.:

: :,.-..

.'.-:

,',' ;'

..

:<::::

::.\<. ';:':.,':

:.:;:

:.::

: :7l7-i

c

JUIHEfrHbIE

CHCTEMbI

C

OCOBEHHOCTflMH

[I'JI.

11

356

IilTaK,

onepan;MH

(44.11)

MomHO paCCMaTpHBaTb KaK TaKI!e JIHHeH-

Hble onepan;MH, KOTopbIe

,n;JIH

BnOJIHe

HaOJIIO,n;aeMoH

cMcTeMbI·(43.1),

(43.2)

BOCCTaHaBJIHBaIOT

KOOp,n;HHaTbI

Xi

(t)

no

CHI'HaJIy y (t + 't)

(-

,'}

< 't

-<

0)

(43.4).

ITo

l}lopMe

OHM

OTJIH'IaIOTCH

OT

onepan;MH

<P

[y

(t + 't)]

(43.5),

paCCMOTpeHHbIX B §

43,

TaK KaK

TaM

MCKaJIMCb

onepan;MH,

MMeIOIIJ;Me

lWUJJ.t,elibUlYlO

li0pMy

(43.6).

O,n;HaKo

,n;JIH

CHCTeMbI

(43.1),

(43.2)

MomHO

CTpOMTb

onepan;HH (Pi [y (t + 't)],

BOCCTaHaBJIH-

BaIOIIJ;IIe

Koop,n;HHaTI>I

Xi

(t)

no

CHI'HaJIy y

(t

+ 't)

(43.4)

M

MMeIOIIJ;Me

l}lOpMy

o

Xi

(t) =

<Pi

[y (t +

,;)]

=

Vi

(0) Y(t) +

~

Y(t

+,;)

Vi

(,;)d,;, (44.14)

-~

aHaJIOrM'IHyIO l}lopMe

(44.11),

He

OOpaIIJ;aHCb

npH

9TOM

K ypaBHeHIIIO

(44.1),

!Ia

OCHOBaHIIH

KOToporo

ObIJIII nOCTpoeHbI orrepan;IIH t

k

[z]'

One-

pa~IIIItk

[z]

(44.11)

MomHO paCCMaTpIIBaTbnpII

9TOM

KaK pacnpocmpa-

lieHUe

onepan;IIH

<Pi

[y]

C JIIIHeHHOro MHomeCTBa

{y

(t)} Ha

MHomeCTBO

{z

(t)}

1).

Onepan;HII

<Pi

[y]

BH,n;a

(44.14)

MomHO

nOCTpOHTb,

eCJIH

pe-

llIaTI> 3a,n;a'Iy

34.1

0

HaOJIIO,n;eHIIII

CHCTeMbI

(43.1),

(43.2)

npII

yCJIOBIIII

p*

[<Pi

[y]]

=

min,

r,n;e

HopMa

P*

[<Pi]

onepan;HH

<Pi

[y]

(44.14)

onpe,n;eJIHeTCH

(CM.

§§

58-

60

H

TaoJIHn;y12.1)

paBeHCTBOM

o 1

p.

[<pd

=

{or

(0)

+

~

vr

(,;)

d,;

p--.

(44.15)

-~

TaKIIM

nyTeM

,n;JIH

3a,n;a'III 0

HaOJIIOp;eHMH

CIICTeMbI

(43.1),

(43.2)

Mom-

t

.;

'1

~~

HO

nOCTpOMTh

onepan;IIH

<P~

[y]

BII,n;a

(44.14),

IIMeIOlIJ;He

HalIMeHhllIIIe

~;

d

B03MomHbIe HOpMbI

p*

[<p?]

(44.15).

ITOCKOJIhKY

onepan;HII t

k

[z]

~.

(44.11)

HBJIHIOTCH

pacnpOCTpaHeHMeM onepan;HH

<Pk

[y]

(44.14)

C ce-

MeHCTBa

l}lYHKn;HH {y (t)}

Ha

ceMeHCTBO

l}lYHKn;IIH {z (t)}

II

HopMa

P*

[<pn

JiBJIHeTCH

MIIHHMaJIhHO B03MomHOH,

TO

,n;OJImHO

BbIllOJIHHThCH

HepaBeHCTBO

p.

[<p7]

-<

p

[fd·

(44.16)

ITpIIBep;eM

IIJIJIIOCTPMPYIOIIJ;IIH

npIIMep.

1)

STO

BhIpameHlle

CJIe)l;yeT

rrOJmMaTb

Tall.

IIyCTb

B HeKoTopoM rrpocTpaHcTBe

j9

BhI)l;eJIeHO

HeKoTopoe rrO)l;rrpocTpaHcTBo

j9m,

B

KOTOpOM

orrpe)l;eJIeHa

JIHHeH-

HaH

orrepan;HH

<p(m)

[xl.

fOBOpHT, 'ITO orrepan;HH

<P

pacnpocmpaH,JI,em

<p(m)

Ha

j9,

eCJIll

OHa

orrpe)l;eJIeHa Ha Bcex 8JIeMeHTax

x·

H3

j9

H 3Ha'leJmH

ee

Ha

:meMeHTax

x

FI3

j9m

COBrra)l;aIOT

CO

3Ha'leHllJIMll

<p(m)

[xl.

B CBJI3ll C 3TllM B03HllKaeT Borrpoc,

Bcer)J;a

JIll

MOiKHO

pacrrpOCTpUHHTb

3ap;aHHylO

orrepan;HIO

<p(m)

[xl

Ha

Bce

rrpo-

CTpaHCTBO

j9.

Oco6eliHO UIiTepeCeH cJIyqaii, IWr)l;a

Pm

[<p(m)l

= p

[<pl.

STa 3a)l;aqa

na31.dBaeTCJI

np06JIeMOii 0 pacrrpOCTpaHeHllll JIlmeiiHOH orrepan;llH. OHa

rrOJIO-

,lKHreJIbHO

peu;raeTCll

TeOpe140J!:

X~JJa-Banfl.xa

(CM.

l6*],

CTp.

74; l14*],

CTp.

173).

~

C·""

>.

'-!.-

•.

~

§

44]

RAHOHH'IECRHE

ROOP.JJ:HHAThI

CHCTEM

C

IIOCJIE.JJ:EfrCTBHEM

357

IIpDMep

44.1. BhIIIIe, B § 43,

MhI

nOCTpOHJIH

ypaBHeHHe C rrocJIe)l;eiiCTBlIeM

(43.13),

cpe,n;H

pemeHJIii KOToporo

CO)l;epmaTCJI

Bce

cPYHKn;JIH

y (t), onpe)l;eJIJI-

eMbIe

paBeHCTBaMJI

(43.2), (43.1).

CJICTeMa

(43.1),

(43.2)

BnOJIHe

Ha6JIIO)l;aeMa.

HaH,il;eM

Terrepb JIJIHeuHylO

orrepan;HIO

11

[z

1

(44.11),

BbI)l;eJIHIOIIJ;YIO

KOOp)l;llHaTy

Xl

(t)

lI3

peirreJmJI z (t +

T)

ypaBHeHlIJI (43.16). XapaKTepRCmqecKoe ypaBHeHlIe

CHCTeMhI

(43.14)

HMeeT

)l;Ba

KOpHJI:

WI

=

i,

W2 =

-i,

KOTophIe, KaK H

,n;OJImHO

6hITb,

JIBJIJIIOTCJI

TaKme KOpHJIMII xapaKTepIICTMqeCKOrO ypaBHeHHJI

/..,

+

+exp

(-

~

/..,)

= 0,

COOTBeTCTBYIOIIJ;erOCllCTeMe

(43.16). Onepan;llJI'l/Ji

[z]

(44.2),

BhI,il;eJIJIIOIIJ;He

Te

KaHOHlIqeCKHe

KOOp)l;HHaThI

ql'

q2'

KOTophIe

OTBeqalOT

KOpHJIM

~

= i,

/"'2

= - i,

eCJIH

3TH

orrepan;IIM HaUTII

no

rrpaBHJIaM, orrHcaHHhIM B

[234],

[323],

rrpHJmMalOT

BM

o

ql

(t) =

1\JI

[z

(t +

T)]

= z (t) + i

~

ei'l:

z

(t

+ T) dT.

l'

2

o

q2

(t) =

1J'z

[z

(t +1')] = z (t) - i

~

e-i'l:

z (t +

T)

dT.

l'

2

TaK

KaK

Xl

= n

2

~

4

{(

~

+

i)

ql +

(~

-

i)

q2}

,

TO

HCKOMaJI

onepan;JIJI

11

[z (t + T)] (44.11)

orrpep;eJIHTCJI

paBeHCTBOM

o

4n

\

4n

sin

l'

- 8 cos T

II [z (t + 1')] = n2 + 4 z (t) +

.l

n

2

+4

z(t+1')d1'.

(44.17)

"

2

IIocTpoHM TeIiepb onepan;HIO

<p~

[y (t + 1')] Bllp;a (44.14),

pemalOIIJ;YIO

3a-

p;a'ly 34.1 0

Ha6JIIOp;eHHH

KOOp)l;HHaThI

Xl

(t)

CHCTeMbI

(43.14).

TaK

KaK corJIac-

HO

!1'a6JIHn;e

12.1 rrpH BhI60pe

HOpMhI

p'

[<Pi]

;(44.15)

JIMeeM

p

[h

(T)]

={

h

2

(0) +

o } 1

+

~

h

2

(1')

d1'

2,

TO

yCJIOBHe

(33.16),

H3

KOToporo

orrpep;eJIHeTCH

MRHlIMaJIbH.blii:

...-8

CHrHaJI

{yO

(t

+

1')

I

'YI

(t)

=

1},

rrpHHlIMaeT

3p;eCb

BJIP;:

HaUTH

o 1 1

po=m~~{x;+

~

(SinT+x2COST)2d1'P-

=~:n{x;(1+

~)~X2+

Z}2

" .

.2

STOT

MJIHHMYM

pO

p;OCTlUaeTCJI

npM X

2

=

2/(4

+:Tt)

FI

pUBen

pO

=

Yen

(4

+

n)

-4)/4

(4

+ n).

O'J'!lI9l1a

If.

Ha

paBeHcTBa (43.15)

rr9JI}'q~eM,

'ITO

MijIDlMa,JU,HhIU

CHrllaJI{VO

(t

+1')

I

:!I

,I'

0',

--'------

...

_--"".

','

",- -

.'

: ..,'-

.....

;.,

..

;,

...

"

.....

'.'.':'

.

....

,

"',,

;:

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

.,.,>::

";~".

.

..

, ',':::,":' : :

o • • • ,

6

'

'.:

.••••1

358

JIHHE1lHbIE

CHCTEMbJ C

OCOBEHHOCTHMH

[rJI,

11

I

Xl

(t) =

1}

eCTT,

«PYHK:qHH

{yO

tt

+

't')

I

'1

(t) =

1}

=

sin

't'

+

.

4 +

2

11:

cos't'.

(44.18)

YCJIOBHe

MaKCHMYMa

(33.25)

aanmrreTCH

TenepL B

BHAe-

2

°

m:x

{

4 +

11:

VI

(0) +

~

(sin't'+4:1I:

cos't')

vd't')

d't'}=1

(44.19)

7t

2

° 1

npH

{v~(O)+

~

vi

('t')

d't'P""

=

Y4(1I:+4)/(1I:(1I:+4)-4)

7t

2

lIa

cOOTHOmeHHH

(44.19)

neTpYAHO

aaMeTHTL,

'ITO

«PYHK:qHH

v~

('t'),

onpeAeJIHIO~

~aH

HCROMYIO

OIITHMaJILHYIO

onepa:qHIO<p~

[y

(t

+

't')]

(44.14),

rrponopn;HOHaJILHa

MHHlIMaJILHOMY

ClIrHaJIY

(44.18) H

MHOlRHTeJIL

nponopn;HOHaJILHOCTH

paBeH

(1/

p

O)2.

TaKHM

o6paaoM,

orrepan;HH

<P~

[y

(t

+

't')]

orrpeAeJIHeTCH

paBeHcTBoM

°

11:+4

{2

<pdy(t

+ 't')] =

11:(11:+4)-4

11:+4

y(t)-+-

°

+

~

(sin

't'

-+-

11:

~

4 cos 't') Y

(t

+ 't')

d't'}

.

(44.20

7t

2

CpaBHHDaH

HOpMLI

onepa:qRH

<p~

(44.20) H (44.17),

BRAHM,

'ITO,

AeHCTDIITeJILHO

1 1

2

p*

[<pO]

= { 4

(4

+

11:)

}2-

<

p*

[II]

=

{411:

'(11:

+

411:

+

12)}2

1

11:(4+11:)-4

11:2+4

§ 45.

3aAa'la

00

ycnQIWeHHH

CHCTellf

C

IIOCJIeAeuCTBHeM

B

HaCTOHm;eM

naparpaipe

MbI

npOp;OJDRHM

paCCMOTpeHlie

JIHHeHHbIX

CHCTeM C

nOCJIep;eHCTBHeM,

op;HaKO

Tenepb

06CYP;HM P;JIH

HliX

OP;HY

aap;ally

06

ynpaBJIeHHH.

IbaK,

nYCTb

HeKOTopaH

ynpaBJIHeMaH

CHCTe-

Ma

onHCbIBaeTCH

ypaBHemIeM

1;

(t) =

Ax

(t) +

Gx

(t -

t'})

+

Bu,

(45.1)

rp;e

A H G -

KBap;paTHbIe

n-MepHbIe nOCTOHHIThIe

MaTpHn;bI,

nOCTOHH-

HaH

MaTplIn;a

B,

KaK

06I>IlJHO,

lIMeeT

IIOpHP;OK

m X n.

3anaap;bIBalme

t'}

TaKrKe

npep;nOJIaraeM

nOCTOHHHbIM.

IIpH

BbI6paHHoM

ynpaBJIemm

u

(t)

(t

>t

ex

)

ABHmeHHe

ClICTeMbI

(45.1)

onpep;eJIHeTCH

ero

npe8bLCmo-

pueu,

m.·

e.

lta'la.t£bUOU

~PU80U

x

(t)

=

XO

(t)

(t«.

-

t'}

-<

t

-<

t«.)

(CM. §

4~).

RaK

H B

CJIyllae

06bIKHoBeBHhIx

cHcTeM,

,rpm

CHCTeMU

:~~:

:

:.;:~.::g;;:.\

(!>

"--.:.:'--

~

45J

3A.D:A

qA

013

YCIIOHOEHlfH

CHCT1!1M

C IfOCJIE,n:Et:J:CTBHEM

359

(45.1)

MomllO

nOCTaBHTb

aaAallY

0

npHBeAeIDIlI

ee

B COCTOHHHe

paBIIO-

BaCHH

x (t) =

O.

IIpH

aTOM P;JIH

nOJIHOrO

ycnoKoeIDIH

06bIImOBeHHOH

JIlIHeHBOH

ClICTeMbI

-

j;

(t)

=

Ax

(t) +

Bu

(45.2)

p;OCTaTOllIIO

6bIJIO,

KaK

MbI

BHAeJIH,

npHBecTH

ee

K

lIeKoTopoMy

MOM

elI-

Ty

BpeMeIDI

ts

B

COCTOHlIlIe

x

(t(3)

= 0 H

aaTeM

C6POCHTb

ynpaBJIeHlie

(T.

e.

nOJIOmHTb,

liTO'

u (t) =0

npH

t >

t(3).

B

peayJII>TaTe

npH

Bcex

t ;>

t(3

CHCTeMa

(45.2)

OCTaBaJIaCb

B COCTOHHliH

nOKOH

x (t) =

O.

)l;JIR

CHCTeMbI (45.1) AeJIO,

Op;lIaKO,

06CTOlIT

CJIOmHee.

)l;eHcTBlITeJII>HO,

eCJIlI

QbI

AJIH

aTOH

CHCTeMbI

MbI

HMeJIH

x

(t(3)

= 0

II

u

(t)

= 0

npH

t >

t(3,

TO

oTcrop;a

BCJIep;CTBHe

HaJIlI'IHH

B

ypaBHeBHH

(45.1)

CJIaraeMoro

Gx

(t -

t'})

em;e

HliKaK

6bI

He

BbITeKaJIO,

liTO

x (t) = 0

npH

t ;>

t{3.

B

CBHaH

C

YKaaaHBbIM

06CTOHTeJIbCTBOM

aaAalla

00

ycnoKoeIDIII

CHCTe-

MbI

(45.1)

ipopMyJIHpyeTcH

CJIeAyrom;HM

oopaaoM.

3 a A a

II

a 45.1

IIycmb

3a8aub~

M,O.MeUmb~

8pelrf,enU

ta.

H

tf;

>

ta.

u

U38ecmua npe8bU:mopUR,

XO

(t)

(tex

-

t'}

-<

t

-<

ta.)

8euJICeuuR,

x (t).

Tpe6yemcR,

uaumu

ynpa6.ll.euue u

(t)

(ta.

-<

t

-<

t(3

+t'}),

~omopoe

o6ecne'l,U8aem pa8eucnwo

x (t) = 0

npII

t(3

-<

t

-<

tf]

+t'}. (45.3)

MbI

paccMoTpHM

JIHml>

IIpocTeHmyro

CHTyan;liIO,

KorAa

BeKTOp

x (t)

HBJIHeTCH

p;ByMepHbIM,

a

ynpaBJIeHlie

u (t)

eCTb

CKaJIHp.

BbI6epeM

t«.

=

0,

t(3

=

t'}

+

e,

rAe

e >

O.

IIpHMeM

em;e,

liTO

MaTpIIn;a

G

Heoco-

6aH,

a

BeKTopbI

b = B H

Gb

JIHHeHHO

HeaaBHCHMbI

1).

Torp;a

JIlIHeHBO

HeaaBHCHMI>IMH

6YP;yT

H

BeKTopbI

b, G-1b.

IIocJIep;HHe

p;Ba

BeKTopa

npHMeM

aa

6aaHCHI>Ie

B

p;ByMepHoM

npOCTpaHCTBe

{x}

H

npe06paayeM

ypaBHeHlie

(45.1) K

BOBMM

ipaaoBbIM

Koopp;HBaTaM,

nOJIaraH,

liTO

B

IIOBOH

ClICTeMe

KOOPAHHaT

BeKTopbI

b H

G-lb

HMeroT

BHA

(45.4)

b

=[

~].

C-

1

b = [

~

] .

B

p;aJII>HeHmeM

MbI

6yp;eM

C'IIITaTb,

'ITO

yKaaaHHoe

npe06paaOBaHHe

BbIIIOJI;HeIIO,

op;HaKO

B

n;eJIHX

COKpam;eHHH

nHCbMa

coxpalIHM

P;JIH

BOBbIX

KOOPAHlIaT

H

HOBbIX

napaMeTpOB

CHCTeMbI (45.1)

CTapbIe

060a-

BalleHHH.

ClIcTeMa

(45.1)

TorAa

no

ipopMe

aanHOH

He

HaMeHHTCH,

HO

MllTpHn;a

G

6yp;eT

yme

HMeTb

Cnen;HaJIblIbIH

BHA

c =

II

gll

1

II

gH

0 '

1)

AHaJIRa

APyrHx

CJIy'laeB

rrpeAOCTaBJI1IeM

'lRTaTeJIlO.

i

Красовский Н.Н. Теория управления движением

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