Красовский Н.Н. Теория управления движением

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10.

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u (t)

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tjlYH~HeH,

a CHCTeMa (23.1) -

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ynpaB-

JIeHHH

BR,n;a u (t) = 6 (t - t

lX

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u (t)

B BH,n;e

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u (t) =

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t

lX

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2

6(1)

(t-

t

lX

) + ...

+P

6(n-l)

(t - t

IX

),

(23.76)

n

r,n;e CHMBOJIbI

6(i)

(t - t

IX

)

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4.1

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u(t) BH,n;a (23.76),

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tl'l n

C =

~

X

[t{j,

tJ

b

L~

P 6(i-l)

(t

- t

IX

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dt,

(23.77)

i

t

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(15.2).

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qaCTb

paBeHCTBa

(23.77)

rro

HaBeCTHblM

rrpaBHJIaM

pa60TbI

C

0606m;eHHblMH

tjlYH~RHMH,

nOJIyqHM

Ha

(23.77)

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q =

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b +

P2

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...

+

Pn

An-

1

b (q = X-l[t{j,

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C),

(23.78)

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BblBe,n;eHHblM B § 19.

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qTO

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(23.76)

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(23.76), OqeBH,n;HO,

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lX

, t

oc

+

8],

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u (t),

meJIaTeJIbHO

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a

)

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HarrpHMep,

1

6 (t - t

a

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= -

npH

t

oc

-<

t <t

oc

+

8,

6 (t - t

a

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= 0

npH

t;>

t

a

+

8,

13

W 1 <

13

6(1)

(t

- t ) =

6 (t - t

lX

) = 48

2

rrpH

t

oc

-<

t t

oc

+

2"

'

oc

1

8

= - 48

2

rrpH

t

a

+-

2'

-<

t <t

lX

+-

8,

6(1) (t - t

oc

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rrpH

t;>

t

lX

+

8,

...

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TaK

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1)

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(CM.,

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[6*, 26*1.

CM.

TaK1Ke

§ 47,

CTp. 387).

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6

202

§ 24.

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pemeH~e

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00

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13.1.

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17.1,

TO

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n

min

p

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1:']

l]

= min p

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(1:')]

=

pO

(24.1)

t=I

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n

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B

Tex

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:11,'1

B §

18-21,

3ap;a'ly

(24.1), (24.2)

y,ll;aBaJIOCb

pemlITb

lIa

9JIeMeHTap-

HLIX

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mere

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II

paaJIlI'IHbIe

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rro

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[8*

J.

3,l1;eCh

me

MLI

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(B

rrplIH:qHrrel),

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K

HameH

aa,ll;a'Ie

(24.1),

(24.2).

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rrpe,n;rroJIaraTh,

'ITO

CIIC1'eMa

(13.1),

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KOTOpOH

rrOCTaB-

JIeHa

aap;a'la

13.1

06

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yrrpaBJIeHIIII,

HBJIHeTCH

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yrrpaBJIHeMOHHa

aa,ll;aHHOM

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t(1.

-<:

't

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tf'>

(CM.

§§

19,

20)

II,

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<pyHK:qHII

h[i]

('t),

<plIrYPlIpYIO~lIe

B (24.1),

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0

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rrp06JIeMe,

aHaJIOrH1J:HOH

aa,ll;a'Ie

(24.1), (24.2), HO C

MeHbmlIM,

1J:eM

n,

KOJIlI'IeCTBOM

<PYHK:qIIH

h[j]

('t).

IIOCJIe,ll;HlIe

<PYHK:qHII

h[j]

yme

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HeaaBIICIIMLIMH(CM.

§ 16,

CTp.

108).

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C',

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2

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PEIIJEHHE

:IAJ(A

'HI

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K

paCCMOTpeHlIIO

yrroMHHyToro

1J:lICJIeHHOrO

MeTo,n;a

pemeHIIH

aa,ll;a'lH

(24.1), (24.2).

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n-MepHoe

npo-

CTpaHCTBO

rrepeMeHHbIX

lI,

...

,l,...

YpaBHeHlIe

(24.2)

orrpe,ll;eJIHeT

n

rrpo-

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(:n:)

(CM.

plIC.

24.1,

rAe

1I.JIH

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n = 3).

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aa,n;a'lH

(24.1),

(24.2)

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orrpe,ll;eJIeHHH

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(:n:)

TaKOH

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lO={l~,

...

,

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BKOTOpOH

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(24.1).

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(CM.,

HarrplIMep,

§§

23,

21),

HO 91'0

He

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rrpHH-

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B

pac-

cym,ll;eHIIH.)

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n

J

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(24.3)

, B

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l.

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ee

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(CM.

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§ 13).

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,

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= 'I'}.

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C

rlIrreprrJIOCKOCTbIO

(n),

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0;

r

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204

CBoHCTDA

OIJTHMA.TlhHOfO

nlPABJIEHlUI

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aTO

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li

yme

He

cBffilaHhI

HHKaKHMH

COOTHorneHHHMH.

BBe)l;eM

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~

rrapaMeTp

tt,

TpaKTyH

ero

KaK

BpeMH,

OTC'IHThIBaeMOe

npH

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TO'IKH

{li} B)l;OJIh

~

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l(l)

K

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lO.

CJIe)l;OBaTeJIhHO,

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KpHBOH

~

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li

HBJIHIOTCH

q,yHKU;HHMH

0'1'

tt,

T.

e.

li = li (tt).

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yCJIOBHe

HaH-

cKopeirnefo

y6hIBaHHH

q,yHKU;HH

(0

[ll"'"

In-d

B)l;OJIh

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~

Tpe-

6yeT,

'IT06hI

BeKTOp

CKOpOCTH

dl/dtt

= {dli/dtt}

TO'IKH

1 (tt), )l;Bpr-

mym;eicH

rro~,

6hIJI

Bce

BpeMH

KOJIJIHHeapeH

BeKTopy

grad

(0

=

{fJ(J)/fJl

i

}

(i

=

1,

...

, n

-1),

HO

HarrpaBJIeH

B

npoTHBOnOJIOmHyIO

CTOpOHy.

i:ITo

yCJIOBHe

)l;aeT

)l;Hq,q,epeHU;HaJIhHOe

ypaBHeHHe

dL

i

ow

[L

l

•...•

Ln_d

(24.4)

d'l}- = - 8 oLi

f)l;e

8 > 0 -

Koa<pqlHU;HeHT

rrponopU;HOHaJIhHOCTH,

orrpe)l;eJIHIOm;HH

CKOpOCTh

)l;BHmeHHH

rro

KpHBOi

~'

8'1'0'1'

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co06pameHHi

y)l;06CTBa

C'IeTa

H B

rrpou;ecce

B'hI'IHCJIeHHi

ero

MomHO

MeHHTh.

qHCJIeHHOe

HHTerpHpOBaHHe

ypaBHeHHH

(24.4),

OTrrpaBJIH-

HCh

0'1'

TO'IKH

l(l) H

COCTaBJIHeT

B

)l;aHHOM

CJIy'Iae

MeTO)l;

HaHcKopei-

i

rnero

crrycKa

)l;JIH

np06JIeMhI

(24.1),

(24.2).

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'IaCTHOCTH,

eCJIH

pernaTh

aTO

ypaBHeHHe

MeTO)l;OM

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d tt, TO )l;JIH

BhI-

"',

'IHCJIeHHH

rrOCJIe)l;OBaTeJIhHOCTH

TO'IeK

1

(j)

= 1 (tt

j

) (j =

1,2,

...

),

"""

\

"Il

CXO)l;Hm;HXCH

K

TO'IKe

lO,

nOJIy'IHM

peKyppeHTHhIe

COOTHorneHHH

l

(i+1)

=

l(j)

_

[OW

[L

l

•

•••

, L

n

_

l

]]

A.Q.

t t 8

"L

(

')

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(24.5)

II,I!

vi

1=1

J

~I

OrrHcaHHhIi

MeTO)l;

MhI

paa6epeM

Terreph

Ha

CJIe)l;yIOm;HX

KOHKpeTHhIX

rrpHMepaX.

PaCCMOTpHM

aa)l;a'IY

0

rrpHBe)l;eHHH

B

MepH)l;HaH

fHpOCKOrrH'IeCKOrO

KOMrraCa

C

rH)l;paBJIH'IeCKHM

ycrroKOHTeJIeM

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KOJIe6aHHH.

YpaBHeHHH

rrpen;eCCHOHHOfO

)l;BHmeHlIH

aTOfO

fHpOKOMrraCa

~l

= al2

X

2 +

a13Xa

+

u,

)

X2

=

a21

x

lt

(24.6L

Xa

=

aa2

X

2

+

aaaXa,

BaHThI

H3

KHHrH

[1936],

f)l;e

'IHTaTeJIh

H

MomeT

HaiTH

nOHCHeHHe

Me-

XaHH'IeCIWfO

CMhICJIa

BeJIH'IHH

Xl'

X

2

, X

a

•

Roaq,q,HU;HeHThI

ypaBHeHHi

(24.6)

CYTh

nOCTOHHHhIe

BeJIH'IHHhI,

onpe)l;eJIeHHhIM

06pa30M

BhIpama-

IOm;HeCH

'Iepe3

napaMeTphl

CHCTeMhI.

8TH

napaMeTphI

fHpOKOMnaca

TaKOBhI,

'ITO

xapaKTepHCTH'IeCKOe

ypaBHeHHe

CHCTeMhI

'

(24.6)

····

......

'.

c.

..

~,'

205

qlIC,TIEHHOE

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3A,D;AQlI

OR

YIJPAB.TIEHlIH:

§

21.]

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U =0

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Bem;eCTBeHHhIi

KopeHh

"'1

> 0 H

)l;Ba

KOM-

nJleKCHhIX

conpHmeHHhIX

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"'2.a

= a ±

i~

(a

<

0).

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fllpOKOMnaca

B

Mepll)l;lIaH,

T.

e.

npllBe)l;eHlIe

ero

B

COCTOHHlIe

Xi

=

0,

nOCJIe

nycKa

B

XO,IJ;

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nyTeM

npllJlO-

meHlIH

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ny-

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a

24.1.

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3aKOH

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aToi

ClI-

JlhI

lICXO)l;H

Ha

Tpe60BaHlIH,

'IT06hI

K

MOMeHTy

BpeMeHlI

t/3

= T

rll-

l

pOKOMnac,

orrHchIBaeMhIH

ypaBHeHHHMlI

(24.6),

6hIJI

npllBe)l;eH

113

Ha'IaJIhHOrO

COCTOHHHH

X (t

a

)

= X

(0)

=

X"-

B

MeplI)l;lIaH,

T.

e.

B

COCTOHHHe

X (T) =

x(3

=

O.

Y)l;06HhIH

3<p<peKTHBHhIH

crroc06

perne-

HHH

3TOH

aa)l;a'IH

}J;aH

B

U;HTHpOBaHHOH

KHHre

[19361,

npHlJeM

TaM

lICKaJIOCh

ynpaBJIeHlIe

u (t)

rro

BoaMomHOCTH

rrpOCTOH

CTpyKTyphI

H

He

CTeCHeHHoe

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orrTHMaJIhHOCTH.

3)l;eCh

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cpaBHeHlIH

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aa)l;a'IY

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Tex

me

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H B

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[1936],

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UO

(t), )l;JIH

KOToporo

MaKCHMaJIhHOe

aHa'IeHlIe

MO)l;yJIH

I

UO

(t) I

Ha

0'1'-

peaKe

[0,

T]

eCTh

BeJIlI'IHHa,

HaHMeHhrnaH

lIa

B03MomHhIX,

T.

e.

(24.7)

max

(I

UO

('t')

I,

0

-<

't'

-<

T)

= min.

't

U

OIITHMaJIhHOe

yrrpaBJIeHHe

UO

(t),

pemaIOm;ee

aa)l;a'IY

(24.6),

(24.7),

RaK

rroKaaaHO

B §

21,

lIMeeT

BH)l;

U

O

(t) =

~sgnB'S

[t/3'

't']

lO

=

~sgn

(h

n

h(i)

(t)

In

=

p P

i=l

=

;0

sgn

(l~h(l)

(t) +

l~h(2)

(t) +

l~h(a)

(t»,

(24.8)

r)l;e

BeJIH'IlIHhI

p

°

II

lO

.i

HBJIHIOTCH

perneHlIeM

aa)l;a<IH

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-

3KC-

TpeMyM

T a

(24.9)

min

~

Ih lih(i)

('t')

I

d't'

=

min

r

[llt

l2'

la]

=

pO

I °

i=l

I

rrpH

llC

+

l2C2

+

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=

1.

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Ci

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l

IOTCH

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rro

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MaTpHu;e

CliO--

TeMhI

(24.6).)

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Haxom)l;eHHH

'IHCeJI

l~,

rrpll

KOTOphIX

)l;OCTHraeTcH

MlIHHMyM

(24.9),

rrpHMeHHM

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BhIrne

rrpou;e)l;YPY

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crrycKa

rro

BeJlll'IlIHaM

li'

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113

BhIpameHHH

(24.9)

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l2'

la]

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la'

8'1'0

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C)l;eJIaTb,

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a

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206

r

,,::'

~,:.,

[rJl,

G

cnOnCTBA

OTITHMAnhHOrO

YTIPAnnEHHR.

(CM. (24.12)).

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<pyHR:qInO

T

ro

[lb

l21

=

~

III

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('r)-

:~

h(3)

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o

+l2

(h(2)

(-r)

_

Cz

hIS)

(-r))

+

hIS)

(1')

I

d-r.

(24.10)

Cs Cs

T

II

pOH3BO,IJ;HaH

8£/8

Z

OT

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£ =

~

I!

(.-r,

l) I

dT

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o

CH,

RaR

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ITO

<pOpMyJIO

o~

T

\

[0/

(1',

I)

-]

ar=

J

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sgn!(T,l).

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°

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8ro/8Z

j

(i

= 1,2)

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S,IJ;eCb

BH,IJ;

T

~

= \ [hIll

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-

~

h(3)

(-r)J

san

[ll

(h(l)

(T)

-

~

h(3)

(T))

+

'.

all

J

cs"

Cs

o

..

+

l2

(h(Z)

(-r)

-

::

h(3)

(T))

+

h(3~s('r)J

d-r,

....

11'

~ll

'\;1

T

~

=

\'

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(-r)

_

Cz

hIS)

(-r)-J

sgn

[ll

(h(l)

(-r)

-

~

h(3)

(-r))

+

alz J

cs

Cs

iii

o

~Ii

+

lz

(h(Z)

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_~

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+

h(3)

(1')Jd-r,

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H,

CJIe,IJ;OBaTeJIbHO,

,IJ;H<p<pepeHJJ;HaJIbHlile

ypaBHeHHH

(24.4), OITHClil-

BaIOIlJ,He

H3MeHeHHe

BeJIH'IHH

li

({)o)

(i

= 1,2) B,IJ;OJIb

RpHBOii:

~,

npHHHMaIOT

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<pOpMy:

T

~

=

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_ B [hIll ('r) -

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sgn

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(h(l)

(-r)

-

Cs

~

h(a)

(-r))

+

Cs

o

+lz

(h(Z)

(-r)

_

~

h(3)

(-r))

+

h(3:

(1')J

d-r,

Cs

s 1(24.11)

T

~~

= _ B

~

[h(Z)

(-r)

-

*h(3)

(-r)J

Sgn[ll

(h(l)

(-r)

-

*h(3)

(-r))

+

o

+

lz

(h(Z)

(t)

-

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(-r))

+ his)

(1')J

dT;.

Cs

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r

'

....

:.:

...

C

207

tIMCJIEBHOE

PEIDEHI1E

3A,IJ;A

qJ![ OB YITPABJIEHI1l1

§ 2

/

,J

llpUMep 24.1. CucTeMa (24.11) HHTerpIIpOBaJIaCb

Ha

8B~M

MeTOAOM

RQ-

He'lRhIX pasHoCTeil:,

T.,

e.

AeJIO

CBOAIIJIOCb

R

nOCJIeAOBaTeJIbHoMY

Bbl'lUCJIeHlIlO

'BeJIH'lIIH

{l~),

l~)}

(i

=

1,2,

...

)

H3

cooTHomeHlIil:

BHAa

(24.5),

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B

COOTBeTCTBIUl

C (24.11). Bbl'lHCJIeHIUl

BeJIIICb

AO

Tex

nop,

nORa,

He

npeRpa~aJIOCb

nOHlIlReHlIe 3Ha'leHlIH w

[ll'

lz]

B npeAeJIax Bhl6paHHoii:

TO'lHOCTH.

TIPH

8TOM

B

np0D;ecce Bhl'lHCJIeHlIil: IIHTerpaJIhI B npaBbIX 'laCTHX (24.11)

TaRlRe

3aMeHHJIHCb

IIX npII6JIIIlReHHhlMII BhlpalReHHHMlI 'lepes

cooTBeTcTBylO~He

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MhI.

3aAa'la pemaJIaCb

npII

CJIeAylO~Hx

AaHHhlx:

hIll

(t)

=

ct(l)

eA,t

+

(.1(1)

eell

COS

(.It

_

(.I(Z)

e

ILt

sin

(.It·

n

t'n

t'

t'11

t'

•

h(Z)

(t) =

ct~i)

eA,t

+I3W

eelt

cos

I3

t

-13~i)

eelt

sin

I3

t

;

hiS)

(t) =

ct~{

eA,t

+ I3W

eell

cos

I3t

-I3W

erJ.t

sin

I3t;

ct

(l)

=

-0

4438'

I3U)

= 1,4140;

13~~

= - 0,0572;

11

,t

13~~)

= 0,0207;

13~~

= 0,0559;

ct~~)

=

-0,0207;

(24.12)

l3~i)

= - 0,0503;

l3~i)

= ::-0,0304;

ctn)

= 0,0503;

1..1

= - 0,8824·10-11;

ct

= - 0,3088·10-11;

13

= 0,9481.10-11; T = 1800;

xdO)

= 0,3;

X2

(0) =

Xs

(0) = 0,004.

Xl

(T)

=

X2

(T)

=

Xs

(T)

=

O;Cl

= - 0,0287;

Cz

= 0,0139;

Cs

= - 0,0101.

8TH

AaHHhle, l,aR

YlRe

OTMe'laJIOCb,

nOJIyqalOTCH

n

COOTneTCTBlIH

C

lIX

BhlpalRe-

HlIHMH

(8.4),

(15.2),

eCJIII

BOCnOJIb30BaTbCH

'laCJIOBhIMlI 3Ha'leHlIHMlI

AJIH

napa-

MeTpoB

aij

(24.6)

II

AJIH

BeJIH'lIIH T

II

xf,

x~,

YRa3aHHhlx B

RHlIre

[1936].

HHTerpllpOBaHlIe ypaBHeHllii: (24.11)

6h1JIO

BhlnOJIHeHO

Ha

D;lltPPOBoil:

Bbl'lll-

CJIllTeJIbHoii: MamllHe

~'YpaJI

Ib,

acxoAH

ll3

Ha'laJIbHOrO

yCJIOBHH

l(l) =

{OJ

-145

-298,

5643} C marOM

~'6'

= 18.

TIOJIY'lHBmaHCH

npll

aTOM

nOCJIeAOBaTeJIbHOCTb

l

(j)

1I

W

[l(j)]

npllBeJIa R BeJIlI'lllHaM '

I~

=

.lim

I;i)

= 0,8397;

pO=

lim

w[l(j)]

=2147,5177;

]-"00

;-+00

Ig

=

.lim

I~i)

=

-140,2842;

I~

=

.lim

I~)

= - 294,4653.

]

..-.00

]

..-.00

CJIemlBaTeJIbHO, MlIHlIMaJIbHaH

WYHKD;lIH

hO

(1')

B

AaHHOM

cJIy'lae

HMeeT

BlIA

h

O

{t)

= _ 12,2804e-

O

,8ll24'l

o

-'

t

+13,0951e-

O

,3088'IO-'

t

cos

0,9481·10-1lt-

_1,0618e-

O

,30B8'1O-·

t

sin

0,9481.

lO-

S

t, (24.13)

a

OnTHMaJIbHOO

ynpaBJIeHlle u

O

(t)

Ha

OCHOnaHnH

(24.8)

tt

O

(t)

= 0,4657.10-11 sgn

(_

12,2804e-O,88Z4-]O~'t

+13,0951e-

o

,S088'10-·t

X

cos 0,9481.10-1lt _1,0618e-O.3088.10-·t

sin

O,9481.10-

8

t. (24.14)

'

I\.'

\

0,002

..

".'

..: ..

.;.

,;,,- ';-'. ',".'

..

i!

I

();

r

I

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....:_~

208

CBOnCTBA

onTUMAnhHoro

vnPABnEHUH

[r.l.

6

§

24)

qlWnEHHoE

PEillEHUE

3A,r:r;A~U

OB

VIIPABnEHUH

209

'. "..

',;.;:L·

.-

rpa<IJ1lli BToro

OIITHMaJIbHOrO

ylIpaBJieHHR

IIpHBe~eH

Ha

pHC.

24.2,

r~e

TaKffie

IIpe~CTaBJIeHbI

cooTBeTcTByIOJIVle

KpHBbIe

xl

(t), X

2

(t) H X8 (t).

O~HOBpeMeHHO

I /, /

~;>\

•

o ,

M~

\ r

lIMO

t

/MM

u"(0--0,468

'/O-J

UO(t)-_0,488-/0-3

-0,002,

i'

Pnc,

24.2.

\.

'.

lIlll

•

~

~~;

~Ii

1\1

1

uft)

=0,0522

'/0-

3

900

!2IJ0

1/800

t

300

u(t)=-8,0487·/0-

J

u(t)~-0,540t-!O-3

-0,002

Puc.

24.3.

Ha

pHC.

24.3 H306pameHbI

rpa<IJ1lli

YlIpaBJIeHlIJI u (t) H rpa<IJIlliH

<lJYHK~Hil:

Xl

(t),

%2 (t),

%3

(t), OTBe'laIOlIwe

pemeHUIO

TaKOR

me

aa~a'lH

06

ylIpaBJIeHHH,

~aHHOMY

B KHHre· [1936]. IIoJIeauo

cpaBHHTb

ynpaBJIlilHHe

-

0,6401.10-

3

IIpH 0

~

T <600,

It

(T) = {

0,0522.10-

3

IIpIl

600

~

T <1200, (24.15)

- 0,0437 ·10-3

IIpIl

1200

~

T < 1800,

Han,l.WHHoe

B KHHre

[1936]

H

HMeIOIIW8

HOpMy

max

I u (t) I =

0,6401.10-

3

C orrTH-

t

MaJIbHbIM

yrrpaBJIeHHeM (24.14).

Rax

H

CJIe~OBaJlO

omH~aTb,

MbI

BH~I1M,

'ITO

max

I

uo

(t) I =

0,4657.10-

3

<

max

I u (t) \ = 0,6401.10-3.

t t

O~HaKo

paaHH~a

Me~y

HopMon

OrrTBMaJIbHOrO

yrrpaBJIeHHJI

Uo

(t) H HopMon

yrrpaBJIeHHR

u (t),

HaR~eHHoro

B KHHre [1936]6oJIee

rrpOCTbIM

crroco60M,

He

CTOJIb

BeJIHKa

(CM.

pHC.

24.2, 24.3).;

OOCy.n;UM

Terrepb

oOJIee

CJIOllmyIO cHTyaU;HIO. PaccMoTpUM MaTe-

.l

pUaJIbHyIO

TO~KY

MaCCbI

m,

.n;BHmym;YIOCH

B u;eHTpaJIbHOM

rrOJIe

CUJI

THroTeHHH, c03.n;aBaeMoM

.n;pyron

T01JKOn

MaCCbI

M

(M~

m), rrop;

.n;encTBHeM

ynpaBJIHIOm;en

peaKTHBHon

CHJIbI, B03HHKaIOm;en rrpH BbIO-

pacbIBaHHH peaKTHBHoH

MaCCbI

rro KaKoMy-JIUOO HarrpaBJIeHlfIO, JIe-

mam;eMY B

rrJIOCKOCTU,

r.n;e

.n;BumeTcH

T01JKa

m

(CM.

rr.5°

§ 3).

IIycTb,

KaK

H BbIme, B

npHMepe

23.1, TpeoyeTcH rrepeBeCTH

aTy

T01JKY

Ha

aa.n;aHHyro

KpyroByIO

OpOUTy

I'"

pa.n;Hyca

roo

O.n;HaKo

Terre

ph

MbI

I,

HeCKOJIbKO

yCJIOmHUM a

a.n;a1JY

, npe.n;rronomUB, rrpem.n;e

Bcero,

1JTO

Ha-

rrpaBJIeHUe, rro KOTOpOMy BbIopaCbIBaeTCH peaKTUBHaH

Macca,

Terrepb

.,'

He

HBJIHeTCH

IPHKCHpOBaHHbIM

(KaK

aTO

ObIJIO

B rrpUMepe 23.1),

HO

MomeT

HaMeHHTbCH

B rrpou;ecce yrrpaBJIeHHH. Tor.n;a B ypaBHeHUHX

.n;BHmeHHH

(3.23), K

KOTOpbIM

MbI

CHOBa

.n;OJImHbI

3.n;eCb

oopaTHTbcH,

BeJIU1JUHbI

ar

U

a<l>

CJIe.n;yeT

C1JHTaTb

rrepeMeHHbIMH. B

COOTBeTCTBUH

C

3THM

BMeCTO

o.n;Horo CKaJIHpHOrO yrrpaBJIeHUH U, KOTopoe

IPHrypupyeT

B

ypaBHe-

.1'

HUHX (3.23),

BBe.n;eM

Terrepb BeKTopHoe yrrpaBJIHIOm;ee Boa.n;encTBue

{Ul'

u

2

} C KOMrroHeHTaMH U

l

=

aru,

U

2

' =

a<l>u,

IITaK,

rrpHMeM,

1JTO

.n;BHmeHHe

T01JKU

m B

JIUHeHHOM

rrpHOJIHmeHUH orrHCbIBaeTCH

ypaB-

HeHHHMU:

:ih =

X2,

]

. v 2

Vvro

X2 =

--Xl

+---

X3

+Ub

(24.16)

rg

1;3 = r

u

u,2.

B03Bpam;aHCb K

rrOCTaHOBKe:

;Ja.n;a1JU,

tlOTpeoyeM

em;e

p;OrrOJIHU-

,1

TeJIbHO,

1JTOObI

.n;oOaB01JHbIe

yCKopeHUH, B03HUKaIOm;He 3a

C1JeT

yrrpaB-

JIHIOm;UX

yCUJIHn, He rrpeBocxo.n;HJIH Harrepe.n;

3a.n;aHHhIX

rrpe.n;eJIOB,

rrpU1JeM

coxpaHUM

nOmeJIaHUe 0 HaHMeHbmeM pacxo.n;e peaKTHBHOH

MaCChI.

IIpooJIeMa

rrepeBo.n;a

T01JKU

C

UCXO.n;HOH

OPOUTbI

Ha

OPOHTY

fa,

c!fJopMyJIUpOBaHHaH BhIme, TpaHCIPOpMHpyeTcH Terrepb B

CJIe.n;yIO-

m;yIO

3a.n;a1JY·

3 a.n; a q a 24.2.

IIycmb

8eu:JICel-lue

:MamepUa.l£bl-lOU

mo'f.r;,u

m

onUCbUJaemCSL

cucme:MOU

.l£Ul-leUl-lblX

8wj)(pepel-llfUMbl-llitX ypael-lel-l

uu

(24.16).

Tpe6yemcSL

l-laumu

ynpae.l£el-lue

U = {u

l

,

u

2

},

r;,omopoe

6bl

nepeeo8u.I£o

Cucme:My

(24.16)

U3

l-la'f.a.l£blW30

COCmOSLl-lUSL

X

(0)

=

XCI

e

l'i.Ol-le'f.l-lOe

COCrnOSLl-lUe

x

(T)

= 0

3a

epe:MSL

T U

mar;,oe,

"lm06bl

npu

iJmO:M

c:

(~>

C::

[rJI.

6

CBOnCTBA

OllTHMAnbHOrO YllPABnEHHH

210

UMnY//,bC

CU//,b£

6b£//,

,M,Uliu~ta//'eH

T

~

(Ui

(t)

+

11.;

(t»'/'d1: = min,

(24.17)

o .

a

6MUv.Ulia

ynpa6JtJ£lOUWU

CUJtb£

lie

npe6ocxo8UJta

6b£

Hanepe8

3a8aliHOZO

3lia"le1iUJ£

f.t,

m.

e.

(24.18)

lUi

(t) +

11.;

(t)]'/' <

l.l.

npu

6cex

t.

3a,n;alIa

24.2 OTJIHlIaeTCH

01'

3ap;alIH,

H3YlIeHHo

H

HaMH

B § 23,

TeM, lITO co,n;epmHT p;OnOJIHHTeJIbHOe

orpaHHlIeHHe

(24.18) 1). ,l];JlH

ee

pemeHHH

paCCMOTpHM

CHalIaJIa

BcnOMoraTeJIbHYIO

3a,n;alIY 13.1, rp;e

HHTeHcHBHoCTb

BbloepeM

B BHp;e

"'A

[u]

=

max

[max

(Ui

(1.')

+

u~

(t»'f"

"t

T

'J..

~(Ui(';)

+

11.;

(t»'/'dt].

;'·'1

o

;".:

3p;eCb

'J..

-

nOKa

em;e HeOnpep;eJIeHHOe nOCTOHHHoe nOJIOmUTeJIbHOe

~Il

lIHCJIO,

cnocoo

nop;oopa

KOToporo

oyp;eT

nOHCHeH

HHme.

MHTeHCHB-

"'I

HOCTb

"'A

[11.]

TaKoro

pop;a

yme

BCTpelIaJIaCb

Bblme,

B § 4, rp;e H OblJI

~l

nOHCHeH

ee

CMblCJI.

BcnoMOraTeJlbHyIO

3ap;alIy

MomHO

pemaTb

cnoco-

OOM,

onHcaHHblM

B ::nOH

MOHorpa<pHH.

COrJIaCHO TaojIHIJ;e 12.1 (CM.

§ 12)

BeJIHlIHHa

p*

[11.]

=

"'A

[11.]

(24.19) HMeeT CMblCJI HOpMbl

p*

[11.]

BeKTop-<pYHKIJ;HH

11.

(,;) =

{U1

(,;),

11.

2

(,;)}.

;:ha

HopMa

COBnap;aeT C

HOpMOH COOTBeTCTByrom;eH

mIHeHHOH

onepaIJ;HH

. .

T

CPu

[h

(t)]

=

~

h'

(t)

U

(t)dt,

o

BblnOJIHHeMOH B

npocTpaHcTBe

HHTerpHpyeMblx

BeKTop-<pyHKIJ;HH

h

(,;)

=

{hI

(,;), h

2

('t)} C HOpMOH

p[h(t)]

=

sup

~[hi(t)+

h;(t)]'/'dt,

)

~ ~

I

(24.20)

o(Ll)=min(+,T).

r

1)

B

OTJUi'IHe

OT

§

23

ap;ecb

He

Tpe6yeTCH

Ilepexop;

K

o6061I\eHHhlM

yllpaBJIe-

HHHM;

TaK

I,aK

P;OIlOJIHHTeJIbHOe

orpaHH1IeHHe

(24.18)

06eClle1JHBaeT

coxpaHeHHe

OIlTHMaJIbHOrO

ynpaBJIeHHH

B

KJIaCCe

06hl'IHbIX

p;orrycKaeMhlx

ynpaBJIemui:

u

(t),

KOTophle

BCJIep;CTBHe

aToro

OrpaHU1JeHHH

He

MoryT

Bhlpomp;aThCH

B

HMIJYJIbCHhle

Boap;eaCTBHH,

KaK

aTO

6hlJIO

B § 23.

(24.19)

§ 21.]

qUC.JJRHIJOE

PEJIJEHHE

3A,n;A'In

OR

,'nPAJJ.lIEHHn

211

3p;eCb

CHMBOJI

Ll

03HalIaeT

cucTeMy

npoMemyTKoB

H3

OTpe3Ka

[0,

T],

H,

CJIe,n;OBaTeJIbHO, B (24.20)

MaKcHMyM

BbllIHCJIHeTCH

no

BceM

B03":

MomHblM

CHCTeMaM

TaKHX

OTpe3KOB C

3ap;aHHoH

cyMMapHoH

P;JIHHOH

(HJIH, TO'IHee

rOBopH,

J.tepou

(J

(Ll».

TaKHM

00pa30M,

HMeeM CJIep;yrom;yro

BcnOMOraTeJIbHyro

3ap;alIy:

cpep;H

MHomeCTBa

HHTerpHpye¥blx,

OrpaHH'IeHHhIX

Ha

[0,

T]

<PYHK-

IJ;HH

11.

(t) =

{U1

(t),

11.

2

(t)}

HaHTH

TaKyro

<pyHKIJ;Hro

11."

(t),

KOTopaH

X

oc

OOeCne'IHJIa Obl nepeBOp; CHCTeMbl (24.16) H3 nOJIOmeHUH x

(0)

=

B nOJIOiReHHe x (T) = 0

3a

BpeMH

r H

KOTopaH

If

MeJia

Obl

rrpH

aTOM

HaHMeHbmyro

BOaMOiRHYIO

HOpMy

p*

[U"!

=

%"

[u"!

(24.19).

3a-

MeTHM, rrperKp;e

Bcero,

'ITO p;aHHaH

aap;a'Ia,

p;eHCTBHTeJIbHO,

paapemH-

Ma,

TaI(

I(aR

CHc'reMa (24,16) HBJWeTCH BrrOJIHe

yrrpaBJIHeMOH,

B

'leM

'IHTaTeJIb

MOiReT

JIerKO

yoep;IfTbcH,

npOBepIfB

YCJIOBIfH

yrrpaBJWe-

MOCTH

(CM. § 19).

MTaK,

aap;aBillIfCb

HeKOTopblM

'IUCJIOM

'J..

>0,

00-

paTHMCH

P;JIH

pemeHIfH

cooTBeTcTByrom;eH

3ap;alIH

K

npaBIfJIy

17.3

•.

B eOOTBeTCTBHIf C

aTIfM

rrpaBHJIOM Hap;JIemHT

pemHTb

3ap;aqy

(17.10),

(17.11).

0003HaQIfM

CHMBOJIOM hA('t).

M,U1iUMMbHyT0

<PYHKIJ;HIO,

n

II

(t)

= B's" (,;) =

B'S

It,

t

oc

!

SA

(toc)

=

B'S

It,

toc!l"

=

~

ltl

il

(t),

i=l

OTBeQaIOm;yIO paCCMaTpHBaeMOH 3ap;aQe.

HaM

Hap;JIemHT, CJIep;OBa-

TeJlbHO,

HaHTH

BeKTop

l"

=

{l~

,... ,

l~},

P;JIH

KOToporo

BeKTop-<pYHR-

IJ;HH

hA('t)

HMeeT

HaHMeHbmyro

B03Moamyro

HOpMy

(24.20), T.

e.

Hap;-

JIemIfT

pemIfTb

CJIep;yrom;yIO

3ap;aQy

Ha

YCJIOBHhIH

aKcTpeMyM:

2 3

min

max

~

[~

(~ldf/l

(t)

)2T'

d,; =

I

~

Ll.

k=l

i=l

2 3

'/

=

~

[~C~

l\!f,il(t)YJ

'dt=pA

(24.21)

Ll."

1.=1

2=1

npH

YCJIOBIfH

3

~

l\Ci = 1,

o(LlA)=o(Ll)=min(+,

T),

(24.22)

i=l

r,n;e

f~il

('t) = {f[i]('t)} -

CTOJIO~bl

MaTpHIJ;bI

B'S

['t, t

oc

],

Ci =

= -

xf.

,l];JIH

pemeHHH

aap;aQH OTLICKaHHH

MHHHMyMa

(24.21), (24.22)

(npu

BhIopaHHoM

<pHRcHpoBaHHoM

'J..)

npIfMeHHM

npoIJ;ep;ypy

onpep;e-

JIeHHH

lA

If

d"

MeTop;oM'

HaHcRopeHmero

cnycKa

no

BeJlMQHHaM li'

CocTaBHM

<PYHKIJ;HIO

. - 2 3 .

'/

r [l1,

l2'

ls! =

max

H

~

(~

ltf~2]

(t)

rJ

'dt

(24.23)

t.

Ll.

k=l

i=l

~_.-

- --<..-,.... ••

-'

•.

_,-._'~~'

-'

..

~.~-

'..

..~.

:_":_"

.•

-.'.

~.~~~_._

.~

c>

[rJI.

r,

CBofiCTBA OTITHMAnhHOrO

ynPABnEHUff

212

H HCKJIIOlJ:HM

H3

Hee

nepeMeHHyIO

la'

nOJIb3YHCb

COOTHOIneHHeM

(24. 22), n

OJIy'lHM

ePYHKn;HIO

ffi

Ill!

l2

1

ffi

[lb

l21

=

mIx

~

[±

li

(t~.iJ

(1:)

-

::

tf

3

]

(1:»)

+

~

t~a]

(1:)r r

d1:

=

~

«=1

i=1

'

2 2

[a]

,

=

~

[2}

L

2}

li

(tL

i

]

('t')

-

~:

t£3]

(1:»)

+

fie

C

(~)

rr

d1:.

(24.24)

~

(I)

k=1

i=1

8 ' '

Tenepb

HaM

CJIeAyeT,

HCXOAH

H3

KaKorO-HH6YAb

HaqaJIbHOrO

yCJIO-

BHH

{l~)}

(i

=

1,2),

cnycTHTbcH

B

CHJIy

ypaBHeHHii

(24.4)

no

Kpn-

BOH

~

AO

HCKOMOii

TOqKH

{l~},

rAe

ffi

[l~,

l~l

=

min.

'YpaBHeHHH

(24.4)

HMemT

B

AaHHOM

CJIyqae

BHA

dli

= _ e

l-

~

(tli]

(1:)

-

2..

tL3]

(1:»)

X

dD

J

~

(I)

1.=1

X

(±

[±

lj(tf/

J

(1:)-

:~

t~3](1:»)+

fh3~8(~)

rf'/'Jd1:. (24.25)

1.=1

j=1

.,

.~

qacTHbIe

npOH3BOAHbIe

B

npaBOH

qaCTH

(24.25)

nOJIyqaIOTCH

AHePeP

e

-

peHn;HpoB'aHHeM

no

II

KaK

nOAbIHTerpaJIbHOrO

BbIpameHHH

B (24.24),

~!

TaK

H

Bcero

HHTerpaJIa

no

nepeMeHHoH

06JIaCTH

HHTerpHpoBaHHH

II (l).

OAHaKo

BCJIeACTBHe

Toro,

'ITO

npH

II = II (l)

npaBaH

qaCTb

(24.24)

HMeeT

MaKCHMj7M,

OKa3bIBaeTCH,

'ITO

nOCJIeAHee

AHePcPepeHlJ;HpOBaHHe

HMeeT

pe3YJIbTaTOM

HyJIb.

3AeCh,

BnpOqeM,

B03HHKaeT

AOnOJIHHTeJIbHaH

TPYAHOCTb

cnycKa,

KOTopaH

COCTOHT B

TOM,

'ITO

CHCTeMa

OTpe3KOB

II (l) B

npaBoii

qaCTH

(24.25)

3aBHCHT

01'

3HaqeHHH

II

('I'l').

CJIeA0BaTeJIbHO,

eCJIH

MbI

6YAeM

pernaTb

ypaBHeHHH

(24.25),

HanpHMep,

MeTOAoM

KOHeqHbIX

pa3HocTeH,

T.

e.

6YAeM

HCnOJIb30BaTb

peKyppeHTHbIe

COOTHorneHHH

(24.5),

KOTO-

pbIe

3AeCb

npHHHMaIOT

BHA

l1

j

+

J

)

=

l~j)

_

p.Ll'l'l'

~

l

[2}

2

(tf

i

]

(1:)

-

::

t1

3

]

(1:»)

X

~(I)

},=1

X

(±

[±

li

(tL

i

]

(1:)

-

;:

t£3

1

(1:»)

+

fLa]c~'t)rf"'J

d't',

(24.26)

1.=1

i=1

TO

HaM

npHAeTCH

Ha

Kam)l;OM

mare

nepeCqHTLIBaTb

cHcTeMy

OTpeBKOB

II

(l(j»,

HCXOAH

HB

yCJIOBHH

(' 2 a

2J'I

(24.27)

j

[2}

(~

lilli)

(1:»)

•

d1:

=

max.

~

(I)

«=1

i=1

~

,:,"

.,,:.

c·

"

.

~

2/.]

LIHC.lIEHHOE

PEJIlEHHE

3A,n:A'nr

OB

Y'TIPABJIElBUI

213

IIoHcHHM

O,n:HH

npocTeiirnHH

cnoc06

npH6JIH'l{eHHOrO

BbIqHCJIeHHH

CHCTeMbI

OTpe3Koa

II

(l(j»

,

JIerKO

peaJIH3yeMbIii

Ha

n;HePpOBOH

BbI-

qHCJIHTeJIbHOH

MarnHHe.

OrpaHHqHMCH

npH

aTOM

CJIyqaeM,

KorAa

a (ll) =

1/').

<

T,

TaK

KaK

B

CJIyqae

II = [0,

Tl

B

npon;e,n:ype

cnycKa

no

li

He

B,CTpeqaeTCH

HHKaKHX

oc06eHHOCTeHno

cpaBHeHHIO

C

npe-

,n:bIAYID;HM

npHMepOM,

H60

TorAa

CHCTeMa

oTpe3KoB

II

(l(j) = [0,

TJ

He

MeHHeTCH.

IfTaK,

nycTb

a (ll (l(j») =

1/')..

B03bMeM

ePYHKl~HIO

2 2

f[3](~)J

'I

w

('t',

l) =

{~

[~

li

(t~i]

(1:)-

::

tL3]

(1:»)

+

~_

2}'.

(24.28)

1.=1

i=1

HCHO,

'ITO

riPH

KamAoM

l(t>

CHCTe1YIa

Ll

(l(j) COCTOHT

H3

OTpe3KOB

Ha

[0,

Tl,

r,n:e

ePYHKn;IIH

W

(L,

l)

npHHHMae1'

3HaqeHHH

He

MeHbrnHe,

qeM

BHe

CHCTeMbI

II (l(j». OTCIO,n:a

CJIeAyeT

TaKOH

cnoc06

OnpeAeJIe-

HHH

aTOH

CHCTeMbI

Ll

(l(j) ,

OTBeqaIOID;HH

KaKoMy-JIH60

Ha60py

qHCeJI

W)

(i

= 162).

Pa306beM

OTpe30K

[0,

Tl

Ha

m

paBHbIX

H

IleCbMa

Ma-

JIbIX

qaCTeH

TOqKaMH

L i =

ill't

(i = 0, 1,..., m).

3aMeHHM

HHTerpaJI

(24.2~)

cooTBeTcTByIOID;eii

HHTerpaJIbHOH

CyMMOH

H

MaKcHMyM

HH-

TerpaJIa

(24.24)

3aMeHHM

TaKme

MaKcHMyMOM

nOH

HHTerpaJIbHOH

CyMMbI.

IIOCJIeAHHH

MaKcHMyM

,n:OCTHraeTCH

Ha

CHCTeMe

OTpe3KOB

[1:

i

q

',:

1:

i

q

+

Ll1:1

(q = 1,

...

,

p),

]

cr

(d)]

1 (24.29)

p=E

[

----x:r'

a(Ll)=T'

3Aecb

CHMBOJI

E[a(Il)/IlLl

03HaqaeT

n;eJIyIO

qaCTb

qHCJIa

a

(ll)/ll't.

CyMMapHaH

AJIHHa

aTHX

OTpe3KOB,

pacnOJIOmeHHbIX

B

06JIaCTH

HaH-

60JIbrnHX

3HaqeHHH

cPYHKn;HH W ('t, l) (24.28),

paBHHeTCH

npH6JIHmeH-

HO

BeJIHqHHe

1/')..

BbIqHCJIHM

noaToMy

3HaqeHHH

ePYHKn;HH W ('ti, l) =

=

WI

B

TOqKaX

AeJIeHHH

'ti H

paCnOJIOmHM

nOJIyqeHHbIe

qHCJIa

Wi

B

nopH,n:Ke

y6bIBaHHH

WI,:>

WI.:>

...

:>

Wi

.

TorAa

npH6JIHmeHHO

, m '

HCKOMaH

cHcTeMa,

Ha

KOTOpOH

AocTHraeTcH

MaKcHMyM

(24.24),

6YAeT

CJIaraTbCH

H3

OTpe3KOB

['tq,Lq+ll'tl

(q=1,

...

,p).

(24.30)

3HaqeHHe

ePYHKn;HH

ffi

[lf

j

),

l~j)]

onpep;eJIHTCH

Tenepb

npH6JIHmeH-

HbIM

paBeHCTBOM

P

ffi

[lm

lUl],.-.. M

~

w·

(24.31)

1,

2

~

."'-I

'q'

q=1

a

ypaBHeHHH

(24.26)

BanHrnYTc~

B

BHAe

p 2

~

{~

[fLiJ

(~iq)

-

~:

fLal

(~iq)

]}

l(i+1) =

l(j)

_ e

q";'1

h=1

Ll1:Ll'l'l'

(24.32)

" Wi

,

q

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

.:

.....

;:

..

:

.':

'.:

..

,;,j

r.

\::

. . J

---..:-~-

214

CBO:fIC,'1'BA

OIITHMA.JJhHOrO

YJIPABJIEHHJI

[rJl.

6

nplIqeM

TOqHOCTb

BbIqlICJIeHlIH,

eCTeCTBeHHO,

nOBbImaeTCJI

C

yBeJIlI

qeHlIeM

qlICJIa

m.

IfTaK,

Tenepbp;JIJI

O~pep;eJIeHlIJI

qlICeJI

l~

II

ClICTeMbI

OTpeSKOB

Ll

A'

lICXOP;JI

lIS

KaKlIX-JIlIOO

3HaqeHlIH

l~l),

Hap;JIemlIT

HaHTlI

nOCJIep;OBa-

TeJIbHOCTlI

l(j)

II

Ll

(l(j».

Torp;a

l~

= lim lii)

nplI

j

-+

00

.

IIpH

aTOM,

K

COmaJIeHHro,

nOCJIe

Kamp;oro

mara

BbIqHCJIeHHH

no

<popMyJIaM

(24.32)

nplIXOp;lITCJI

CHOBa

BbIOlIpaTb

ClICTeMY

OTpe3KOB

Ll

(j),

rp;e

<PYHKD;lIJI w

(t,

l(j»

nplIHlIMaeT

HalIOOJIbmee

3HaqeHlIe.

.

HaHp;JI

qHCJIU

lA

H

COOTBeTcTByroID;Yro

cHcTeMy

Ll

A,

MbI

MomeM

on-

pep;eJIHTb

H

MHHHMaJIbHYIO

<PYHKD;HIO h

A

(.)

=

l~/[l)(.)

+

l~

1[2](.)+

1 .

+ls

1[3]

(.)

H

HOpMy

OnTHMaJIbHOrO

ynpaBJIem'IJI

u

A

(.),

paBHyIO

x[

u

l

J = 1/

pA.

IIocJIe

nora

Haxop;HM

HHe,

lICXOP;JI

H3

YCJIOBlIJI

MaKclIMyMa

caMO

OnTHMaJIbHOe

T

~

h

l'

(T) u

A

(T)

dT

o

=

T

m:x

~

0

hA'

(T) U

(T)

dT

~

\

nplI

~

I

r;

1

II

u

(T)

II<""T'

p

').,

T 1

II

U

(T)

II

dt

<

""T

.

•.

1 P

o

ynpaBJIe-

(24.33)

HeTpYP;HO

cooopa3lITb,

qTO

ynpaBJIeHlIe

Ul(T)

,

yp;OBJIeTBOpJIIOID;ee

YCJIOBlIJIM

(24.33),

oyp;eT

OTJIlIqHbIM

0'1'

HyJIJI

JIlImb

Ha

ClICTeMe

0'1'-

pe3KOB

Lll'

nplIqeM

Ha

aTlIX

OTpe3Kax

<PYHKD;lIJI u

l

(.)

yp;OBJIeTBOpJIeT

yCJIOBHIO

II

Ul(T)

II

= 1/

pl

H,

KpOMe

Toro,

BbInOJIHJIeTCJI

COOTHome-

HHe

ul(-r)

=~(-r)hl

(-r)

(CM.

aHaJIOrlIqHbIHCJIyqaH

B

§§

21,23).

CJIep;o-

BaTeJIbHO,

Tenepb

MbI

MomeM

sanHcaTb,

qTO

UA(T)

= h

A

(-r)/

pl

\I

h

l

(-r)/1

.

npH

-r

H3

CHCTeMbI

OTpe3KOB

Ll

A H u

l

(-r)

= 0

npH

T

BHe

Ll

A'

TaKHM

oopaSOM,

3ap;aB;lJICb

BeJIHqHHOH

')."

MbI

MomeM

pemHTb

BcnOMOraTeJIbHYIO

sap;aqy:

HaHTH

<PYHKD;HIO u

A

(-r),

paSpemaIOIn;YIO

sap;aqy

0

npHBep;eHHH

ClICTeMbI

(24.16) B

COCTOJIHlIe

x (T) = 0 H

HMeIOIn;YI{)

npH

aTOM

HaHMem,myIo

HOpMy

Xl

lull

= 1/pl.

ll.JIJI

Toro

qTGObI

HaHTH

Tenepb

pemeHHe

HCXOP;HOH

Sap;aqH,

Heooxop;HMO

nop;06-

paTb

').,

TaK,

qTOObI

yp;OBJIeTBOpJIJIOCb

YCJIOBlIe

max

II

u

l

(T)

II

= A

1

=

~t.

~

P

JIerKo

cOOOpaSlITb,

qTO

c

pOCTOM

').,

BeJIHqHHa

max"

ul(-r)11

paCTeT,

~

.

npHqeM

MomHO

npOBepHTb,

qTO

max

II

u

l

(-r)lI

eCTb

HenpepLIBHaJI

~

l

<PYHKD;HJI

').,

II

BLInOJIHJIeTCJI

COOTHomeHHe

lim

max

II

u (t)

\I

=

00

nplI

t .

').,

-+

00.

OTcIOp;a

BbITeKaeT

CJIep;yroIn;ee

SaKJIroqeHHe:

Hcxop;Han

sap;aqa

lIMeeT

pemeHHe,

eCJIlI

CYIn;eCTByeT

TaKoe

SHaqeHlIe

').,1

> 0

r

(~:.:,;

120

'lHC.'1:EHHOE

PEmEHHE

3A,IJ;A'lU

OB

ynPABJIEHIUI

215

napaMeTpa

')."

P;JIJI

KOToporo

BbInOJIHJIeTCJI

yCJIOBHe

1

A

max"

u

•

('t')" =

-1-

<

/1.

~

P 1

B

TaKOM

CJIyqae

pemeHHe

3ap;aqlI

CBOP;HTCJI K

nOCJIep;OBaTeJIbHO-

My

pemeHHIO

onucaHHbIX

TOJIbKO

qTO

BcnOMOraTeJIbHbIX

3ap;aq,

rp;e

BeJIHqlIHbI

').,

nop;OlIpaIOTCJITaK,

qTOObI

BeJIlIqlIHa

max

IIUA(-r)11

=

1/pA

CXOp;lIJIaCb

K

/1.

B

qaCTHOCTH,

npHOJIUmeHHOe

onpep;eJIeHHe

HCKOMoro

SHaqeHlIJI

').,

=

').,0

MomeT

ObITb

CBep;eHO,

HanplIMep,

K

no-

CJIe;L\OBaTeJIbHOMY

p;eJIeHHIO

oTpe3Ka

[AI'

').,21,

rp;e

1/

pAl

<

/1

<1/

pA!.

IIpll:Mep

24.2.

OrrlIcaHHhIM rryTeM

3a,n;aqa

6hIJIa pelIleHa rra MalIlHHe «YpaJI

11»,

rrpHqeM

6hIJIH

BhI6paHhI

CJIe,n;YIOlI\He

,n;aHHble:

ro

=10,

v = 1000, T =

11,/2,

I.t

= 0,34. (QlICJIa 6hIJIlI BhI6paHhI TaRUMlI,

I1CXO,n;H

JIlIIIIb

113

coo6pameHHH y,n;o6-

CTBa

CqeTa.) ClIcTeMa (24.16) rrplIHlIMaeT

Tor,n;a

BlI,n;

XI

=

X2,

I

~2

= -

Xl

+ 0,2

x

s

+

UJ,

(24.34)

Xs =

10U2.

•

)l;orrycTI1M,

qTO

B HaqaJIbIllJii

MOMeHT

t = °

ClICTeMa

rraxol:vmaCb B

COCTOHHHlI

x(1,

=

{O,

2;

0;

0,

1}.

Tor,n;a BeRTop C = - {x(1,}=

{-O,

2;

0;

-0,

1}.

CHcTeMa,

COrrpH}ReHHaH

R

ClICTeMe

(24.34), lIMeeT

BH,n;

81

=

8

2,

SI

= -

SI,

8S=

-0,282.

cDyH,n;aMeHTaJIbrraH

MaTplIn;a

ee

TaROBa:

cos t

sint

0

S

[t,

0]=

-

sint

cos t °

0, 2

(1

- cos

t)

-

0,2

sin

t 1

YqHThIBaH,

qTO

MaTpnn;a

B'

B

l~aHHOM

cJIyqae

UMeeT

BU,n;

B'

=

II

~

~ ~o

II

'

rrOJIyQHM

~YHRn;lIll:

[

I]

1

[IJ]

[

sm

.

'I'

]

1

(T)-

1

-

[

IpJ

-

2-2cOST

/[2]]

[ cos T ]

[I

[SI]

[0]

/[21

(T)

= 1 =

,j[sJ(T)=

1 = .

[

1

[2

J

-

2

sin

T f[S}

10

_ 2

·2

IIPlI

YRaaaHHblX

lICXO,n;HbIX

,n;aHHhIX

B pe3YJIbTaTe

BhIqHCJIeHUH

Ha

9U:BM

ORaabIBaeTCH,

QTO

HymHoe 3HaqeHlIe rrapaMeTpa

')"

=

'),,0

eCTb

'),,0

~

2.5465,

.t

=4

l;e

2

I

//

I

//

.

1/

r .

'

---'

---.----

.-.

;,_._._._.-

-I--~

~~~~~...c.-'"~~-'-'--"--'-'-"""-~~'~"':"'~'","",'"".'~'.

'':;.'

...•..

',",'

.,<

...

~,.;.:

;',,"

.:

..

:

....

':

..

:::.

·'.:'<':7-:.fM'0~;

216

CBORCTBA

OrrTRMAITbHOrO

yrrPABITEHRH

[rJI.

6

l~

=

-5;

l~

=

-3,3;

l~

=0.

llpli

aTOM

pO

= 2,9738

H,

CJIeAOBaTeJIbHO,

1/po =

= 0,3364

;::;;

It = 0,34.

ClicTeMa

oTpe3RoB

Llo

=

LlAo

nOJIytIHTCH

TaRoif:

[0; 0,1885], [0,9739; 1,5708], a

MHmIMaJIbHaH:

cPYHKD;HH

h,0

('t')

=

h,'A

o

('t')

=

{h~,

h~}=

{5

sin't' - 3,3

COST;

6,6 sin't'+ 10 cos 't'-10}.

Ha

pHC.

24.4

npliBeAeH

rpacPliR

cPYHKD;HHW

('t',

I).

TaRHM

oopaaoJ(j

w .

(f'

I)

On~HMaJIbHOe

ynpaBJIemIe

UO

(t)

AJIH

UCXOA-

01

0,1.9

W.

HOH

3aAatIH

onpeAeJiHeTCH

BhIpamemIJlMH:

uO

(t) _ 5

sin

t - 3,3 cos t n

1 -

2,973811

hO

(t)

11

pH

t

U3

CHCTeMhI

OTpe3KOB

.1.0,

uO(t)=

6,6sint+10cost-1O

n H t

H3

2

2,

9738

11

h

O

(t)

II

p

f'

CHCTeMhI

oTpe3KoB

.1.°,

Puc.

~4.4.

u~

(t)

==

0

(k

= 1,2)

npH

t

BHe

Llo.

I

3TliM

Ii

3aBepmaeTcH

pememIe

3aAaQU

24.2.

rpacPHKu

ynpaBJIeHIISI

UO

II

cooTBeTcTByIOlIJ;He

OnTUMaJIbHhIe

TpaeKTopli1i

npH-

t

BeAeHhI

Ha

pRC.

24.5-24.7.

I

PaccMoTpRM

Tenepb

CHOBa

aap;alIY

(oocymp;aBmyrocH

yme

HaMRB

§§

21,

23)

0

nepeopoce

MaTepRaITbHOii

TOlIKR

MaCCbI m B

HalIaJIO

KO-

OPP;RHWf, HO

Terrepb

npR

HeCKOJIb-

~

,

KO

RHbIX

orpaHRlIeHRHX,

HemeJIR

u'

Te,

KOTopbIe

ObIJIR

BbIopaHbI

pa-

Hee.

IIMeHHo,

nOTpeoyeM

R

ap;eCb,

lITO

ObI

rrpR

p;BRmeHRR

TOlIKR

P;O-

oaBOlIHbIe

yCKopeHRH,

BoaHRKaro-

I

UO(tJ

m;ne

aa

C'IeT

ynpaBJIHrom;RX

peaK-

11

2

TRBHIJX

CRJI,

He

rrpeBOCXOp;RJIR

aap;aHHbIX

npep;eJIOB

R

pacxop;

BbI-

0H

_·L--T--/t

opaCbIBaeMoii

MaCCbI

m1

ObIJI

MR-

0,19

0..'17

\ Z

HRMaJIeH.

::hy

aap;alIy

MbI

paCCMOT-

pHM,

onRpaHCb

Ha

Te

cooopame-

HRH,

KOTOpIJe

ObIJIR

npRBep;eHbI

B § 21, HO B

KalIeCTBe

HHTeHCRB-

HOCTR x [ul

BbIoepeM

BeJIRlIRHY

(24.19).

TaKRM

oopaaoM,

pemaH

aap;alIY,

RCXOP;H

Ra

npaBRITa

17.1,

'r-'u,o(

t.'

npRxop;RM

K

Heooxop;RMOCTR

on-

PHC.

24.5.

.

pep;eJIRTb

MRHRMaJIbHyro

<PYHK-

~Rro

h

A

('t),

KOTopaH

Haxop;RTcH

ap;ecb

Ra

cJIep;yroII(ero

COOTHomeHRH

min

max

\ ([II

(1-

-r)

+

I2]2

+[l3

(1-

1:)+

I41

2

)'hd-r

=

p'A

(24.35)

.1 4 i .

§ 24)

TUWJIEHHOE

PEIDEHIIE

3A,I(ATJR

OD

YIIPABJIEHIIH

217

npR

yCJIOBRH

ll+

I~g

+l4g=1,

cqa)

=

min[+,

1J.

(24.36)

OopaTHM

em;e

paa

BHHMaHHe

lIHTaTeJIH

Ha

TO,

lITO

HHTerpHpOBa-

HHe

ap;eCb Bep;eTCH

He

no

BceMy

oTpeaKy

[0,

1],

a

no

CH-

x(t)

CTeMe

HHTepBaJIOB,

cyMMapHaH

0,

If

p;JIHHa

KOTOpbIX

(J

(Ll)

paBHa

min

[1/A,

11.

IIop;ooHo

TO.MY

KaK

aTO ObIJIO Cp;eJIaHO B §

21,

MomHO

nOKaaaTb,

'ITO

MHHH-

MaJIbHaH

<PYHK~HH

h'A

HMeeT

TaKoe

me

CTpoeHHe,

KaK

H B

aap;alIe,

paCCMOTpeHHbIM

B §

21,

T.

e.

Ha

nJIOCKOCTH (t,

h~)

rpa-

<pHK

<PYHK~HH

h~

('t)

HBJIHeTCH

aHTHcHMMeTpHlIHbIM

OTHOCH-

TeJIbHO

npHMoii

1: = -

1/

2

, a

<PYHK~HH

~

('t)

nOCTOHHHa

H

o'J['

. X

J

j

-0,1

nOJIOmHTeJIbHa.

CJIep;yeT

aaMe-

-0,2

THTb

npH

aTOM,

lITO

Korp;a

A

PRC.

24.6.

BaHTO

Ha

oTpeaKa

[0,

1],

CH-

CTeMa

HHTepBaJIOB

COBnap;aeT

co

BceM

oTpesKoM

[0,

11

H

MbI

npOCTO

nOJIylIaeM

TO

me

pemeHHe

sap;alIH,

lITO

H B §

21.

IIpH

A > 1

j;

Puc. 24.7.

ClICTeMa

HHTepBaJIOB

aA

oyp;eT

COOTBeTCTBeHHO

cJIep;yrom;eii:

a

A

=

{[

O~

2~

J,

[1-

2~

, 1

J}

.

...

.

'.'

"

.'

".,.

-."

,," • • ..

t,

,.;..

..

;-"".1:':=

i

c,

~.~~:.>

CBortCTBA

OII'rHMAJIhHOrO

YlIPADJIEI-TITfl

(rn.

6

218

TorAa,

RaR

H B § 21,

orrpeAeJIeHHe

MHHHMaJIbHOH

q>yHRIWH

h"A

('t)

CBOAHTCH

CHOBa

R

3aAaqe

0

BblqHCJIeHHH

eAHHCTBeHHoro

qHCJIa

l~,

ROTopoe

HaxOAHTCH

Terrepb

H3

yCJIOBHH

'/,"A

min

[~

ll~(+-~r

+C

/1

rr

a

,,+

I,

o

+

~

1

[In+-~r

+ e

/1

rrd~J

=p"A.

1-1/2"A

YrrpaBJIeHHe

u"A

('t)

oYAeT

HMeTb

BHA

I

u"A

(~)

= 1

[l~

(1-

1')

+

l~

]

rrpu

~

H3

A"A,

p"A

II

h"A

('r)11

l;

(1

_

~)

+

l~

I

u"A(~)

= 0

rrpH

'1'

BHe

A"A'

I

!

B

TOM

CJIyqae,

RorAa

MbI

rrpHAaeM

OCHOBHoe

3HaqeHHe

MHHHMH3aD;H!

pacxOAa

peaRTHBHOH

MaCCbI,

BeJIHqHHa

AOJDRHa

ObITb

OOJIbmOH

(rrpH

A

~

00

rrpHXOAHM

R

HM-

l'T

J

rryJIbCHoMy

yrrpaBJIeHHIO,

pac-

/'

..........

,

'r

CMOTpeHHoMy

B § 23). B

TOM

/

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Красовский Н.Н. Теория управления движением

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