Красовский Н.Н. Теория управления движением

1110

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1.41

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e

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, A

n

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bU)

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,

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.

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n

j-

1

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),

a

aTO

rrpOTHBO-

pe'lHT

rrpep;rrOJIOmeHIIro

0

TOM,

'ITO

cpep;H

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(19.4)

eCTb

mj

JIlIHeHHO

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llTaK,

p;eHCTBlITeJIbHO,

BTOPOH

CJIy'laH

HeBoa-

MOmmI.

CJIep;OBaT8JIbHO,

Hallie

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0

JIHHeHHOH

HeaaBHCHMO-

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nepBbIX

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BeKTopOB

b

(j)

Ab(J)

A

m

j-

1

b(j)

, ,

..

.,

Ha

cOBoKyrrHocTH

(19.4)

p;OKaaaHO.

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sB

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s

>-

In

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MaTpH~bl

K

m

(19.3),

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H

p;oI<aabIBaeT

crrpaBep;JIHBOCTb

JIeMMbI.

l1pep;nOJIOIHHM

Tenepb,

'lTO

paHf

MaTpH~blK

(19.2)

paBeH

In

(rn

-<

n),

H

060aHa'lHM

KaKHe-HlI6yp;b

rn

JIJlHeHHO

HeaaBHCHMblX

BeK-

TopoB:noH

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K

CHMBOJIaMH

k[l]

=

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k[2]

=

{k

C

:]}

,

..•

...

, k[m] =

{k[7]}.

060aHaQUM

'lepea

X

m

BeKTopHoe

rrop;rrpocTpaHcTBo,

rropomp;eHlIOe

aTHMH

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B n-MepHOM

npocTpaHcTBe

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n'

KaK

MbI

yBHp;HM

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rrop;npOCTpaHCTBO

X

m

HrpaeT

OCHOBHyro

POJIb

npu

HaylJeHHH

CBOHCTBa

ynpaBJIJIeMOCTH

CHCTeMbl

(19.1).

lIay-

'lHM

noaToMy

HeKOTopble

BamHble

P;JIH

Hac

CBOHCTBa

uaHHoro

nop;rrpo-

CTpaHCTBa.

JI

eM

M a 19.2.

IIoonpocmpaucrneo

X

m

,

nopo:JICoeuuoe ee1f,mopa.Mu

k[I],

k[2],

...

,

k[m],

o6paayem

uueapuaumuoe

noonpocmpaucmeo

cu-

cmeMb~

(19.1),

m.

e.

tjJa.1oeb~u

eell,mop x (t)

amou

cucmeMbL

ocmaem-

CR, e X

m

,

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e Ua'la.ILbUMU

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ou

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e

amOM

noonpo-

cmpaucmee.

lIHa'le

rOBopH,

JIro60e

p;BH}1<eHHe

{Xi

(t)}

CHCTeMbI

(19.1) C

Ha'laJIb-

HbIMH

yCJIOBHHMII

{Xi

(t,,)},

yp;OBJIeTBOpJIrollJ;HMH

paBeHcTBy

m

Xi

(t,,) =

'"

A.

0

k[j]

(i=1,

...

,n),

(19.5)

. LJ J t

j=1

rrpH

Bcex

t

HMeeT

BHP;

m

_ ''V

[j]

.

Xi

(t) - LJ

A.j

(t) k

i

(L =

1,

...

, n). (19.6)

j=1

A 0 K a a

aTe

JI b C

l'

B O.

IIpoBepKy

yTBepmp;eHHJI

JIeMMbl

P;O-

CTaTO'lHO

npoBeCTH

JIHllib

P;JIH

In

< n,

TaK

KaK

rrpH

m = n

rrop;rrpOCT-

paHCTBO

:Jt'm

rrpoCTo

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co

BceM

rrpOCTpaHcTBoM.Ai'

n'

KOTOpOMy

rrpHHap;JIemHT

BeKTop

X (t).

3arrHllieM

aaKOH

p;BHmeHHH

CHCTeMIil

(19.1)

no

tPopMyJIe

KOlliH

(5.6):

I

x

(t)

= X

[t,

["l

x (t,,) +

~

X

[t,

1']

Bu

(1')

d1'.

(19.7)

I"

lIaBecTHo

([3*],

CTp

373),

'lTO

tPYHp;aMeHTaJIbHaH

MaTpH~a

X

[t,

t,":!

CHCTeMbI

(19.1),

HMerollJ;eH

rrocToJlHHyro

MaTpH~y

A,

MomeT

6bITb

rrpe,o;-

CTaBJleHa

B

BHp;e

MaTpMHoro

pJlp;a

(CM.

TaKme

CTp.

39)

(t

t)

S

X[t,toc]=eA(t-I,,)

=E+(t-t,,)A+

...

+

~loc

A

8

+

...

,

(19.8)

CXop;JIIqerocJI

rrpH

BCex

t.

IIYCTb

BeKTopbl

k(Si)

-

CTOJI6qbI

MaTpH~bI

ASB,

TaK

'ITO

ASB

=

{k(S1)!l

..

,

k(S,,)}.

Ha

OCHOBaHHH

JIeMMbI

19.1

HMeeM:

k(\)

=

v~Si)k[l]

+ ...

+v~i)k[m].

(19.9)

Y'IJITbIBaJl

(19.5), (19.8), (19.9),

3anHUIeM

paneHCTBO

(19.7) B

CJIep;yrollJ;eM

BHp;e:

moo

8

100

"

x (t) =

~

[~

(t

~/,,)

A

8

A.1J

k[j]

+

~

[~

.(t

~

1")S

~

k(Si)u

i

(1')Jd1'

=

3=1

8=0

I"

s=o

'=1

1n

co s

=

~

{~

(~~Itoc)

ASA.~

+

3=1.

s=O

I

00

r m

(t -

1")8

.~

vjSi)U.i

(1')J

dt}

k[n =

~

A.j

(t)

kCj]

.

+

~

[~

loc

8=0

'=1

.

3=1

110CJIep;Hee

COOTHOIIIeHJIe

p;OKaabIBaeT

JIeMMy.

Terrepb

MbI

BbInOJIHlIM

OP;HO

JIHH(lHHOe

Heoc060e

npe06paaoBamre

X =

Ty,

(19.10)

KOTopoe

B

CBJlaU

C

HHBapHaHTHocTbro

rrop;rrpOCTpaHCTBa

X

m

06JIer'lIIT

HCCJIep;OBaHue

CBOHCTBa

ynpaBJIJleMOCTH

CHCTeMbI

(19.1).

MaTpH~y

rrpe06paaOBaHJIJI

T =

{du}

orrpep;emIM

CJIep;yrollJ;HM

06pa30M:

B

Ka'leCT-

Be

rrepBblX

m

CTOJI6~OB

d(i)

=

{d

u

}

(i

= 1, ,

m)

aToH

MaTpH~bI

B03bMeM

JIHHeHHO

HeaaBUCUMbIe

BeKTopbI

k[I],

,

kern]

-

CTOJI6~bI

MaTpu~bI

K (19.2), a B

Ka'leCTBe

OCTaJIbHbIX

(n - m)

CTOJI6~OB

d(i)

(i

= m +1,

...

, n)

B03bMeM

KaKUe-JIH60

BeKToplil

d~)

(j

=

In

+

1,

...

,n),

rrOp;'lHmJHHble

JIUllib

TOMy

yCJIOBUro,

'1T06bI

Bce

BeKTopbI

d(i)

(i = 1,

...

, n)

OKaaaJIHCb

JIUHeHHO

HeaaBJICHMbIMH.

ECJIH

paHr

MaTpU~bI

K

'paBeH

m < n, TO

TaKOH

BbI60p

BeKTopOB

d(i)

MomeT

6bITb

Cp;eJIaH

6eC'IUCJIeHHbIM

MHomeCTBOM

crroc060B,

eCJIH

m = n, TO

MaTpH~a

T

ep;HHCTBeHHa.

B

HOBDIX

rrepeMeHHbIX

CHCTeMa

(19.1)

aarrHIIIeTCJI

TaK:

tJ

=

Py

+ Qu,

(19.11)

rp;e

P =

T-IAT,

Q =

T-IB.

."0,

(

:',

:.\

<.:.'J

{li2

CBo1!rcTBA

OTlTJIMAJIbHOI'O

YITT'AB.TIEHMR.

[1'11.

6

ClIpaBe~JnlBO

CJlO~YIOm;Oe

YTBepm~eHl'Ie.

31

e M M a 19.3.

MamrulfbL

P U Q

UMelOm

6uiJ

=

\I-~+~~II

'

Q=

~--1-L--ll

'

P

die

PI

-

(m

X

m)-Mampulfa,

P

2

-

m X (n - m)-.Mampuqa, P

a

-

- (n - m) X (n - m)-JJtampuqa, Ql -

(m

X r)-JJtampUlfa.

II

pu

9mOM

pane

.MampuqbL·

N = {Ql, PI

Qv

...

,

P~-lQd

pa6en

n'L.

)J;

0 R a 3 a

l'

e

JI

h C

l'

B

O.

)J;ORameM CHa'laJla

nepBoe

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HHO

JlOMMhI. 3aMOTHM,

npom~o

BCOI'O,

'ITO B HOBhIX rropeMOHHbIX

Yi

rHnOpnJl

OCI,

OCTh

Y +1 = 0,

...

, Y = 0

(19.12)

m

n

OCTb

He 'ITO lIHOO, RaR

rro,I:~np.OCTpaHCTJ3

0

:J£'

m'

B caMOM

p;OJIO,

no

BbIOOPY

npeoopa30BaHlIJI

(19.10)

P;JlJIBORTOpa Y C Roopp;HHaTaMH {Yl'

...

,

Ym,

0,

...

, o} HMeeM HaR

pa3

.

x =

k[llYl

+

...

+

k[m

l

ym

,

rp;e

k[il-

oa3

HCHbiO

BeRTopbI X

m

. CorJlaCHO JleMMe

19.2

nop;npOCTpaH-

CTBO

:J£'m,

a,

CJle,n;OBaTeJlhHO, H rHnepnJlOCROCTb

(19.12),

eCTh

HHBapHaH-

THoe nop;npOCTpaHCTBO

CHCTeMhI

(19.11).

II03TOMY,

eCJlH

Bce

Yi

(to)=O,

U>m+1),

TO

npH

t>

to

p;BHmeHHe Y

(t)

p;OJlmHO YP;OBJleTBOpJITb

pa-

BeHCTBY Y (t) = °

P;JlJI

Bcex

i > m +1. ECJlH, op;HaRo,

BonpeRlI

i

JleMMe

19.3,

npHHJITb,

'ITO

JlHOO

Pij

=1=

0,

JlHOO

qis

=1=

°

npH

HeHO-

TOpbIX

i > m +

1;

j < m +

1;

s =

1,

...

, r,

TO

MomHO

ObIJIO

ObI

nop;oopaTh

TaRoe

ynpaBJleHHe

u (t) H nOCTpOHTb TaRoe lIaCTHOe

peme-

HHe

{y1

(t)}

(i

=

1,

...

, n)

ypaBIIeHMJI

(19.11)

C HalIaJlbHbIM yCJlOBM-

eM

y1

(ta.)

= °

(i

= m +

1,

...

, n),

P;JlJI

HOToporo

B HeHOTopbIH MO-

MeliT

t

=1=

tao

BbIIIOJlIIJIJlOCb

ObI

HepaBeHCTBO

y~

(t)

=1=

°

npM

S > m.

IIoJlYlIeHHoo

npoTHBOpO'llIO p;OHa3bIBueT cnpaBep;JllIBOCTb

nepBoro

yT-

Bepa,~eHl1JI

JleMMbI.

,l(oRaa,oM

Teneph

BTopoe

yTBepiI,p;eHlIe

JleMMbI. )J;JlJI

3'1'01'0

pac··

CMOTplIM MaTpHQY

K(l)

_

{Q

PQ·

pn-l

Q}.-

'.

-

.,

,

..

-,

-

= {T-1B,

T-lATT-1B,

...

,

(T-lAT)n-lT-1B}

=

T-lK.

RaH

cJlop;yeT 1I3 P;ORa3aHHoro

yn,e

nepBoro

yTBepmp;eHHJI

HaCTOJIm;OH

"

:":,':~)

C

YIlPABJIJiEMOC'l'b

Cl1CTEM C

IIOC'l'OJUIHhIMl1

llAPAME'l'PAMH

143

§ 19l

JleMMLI,

MaTplIQa

K(l) lIMoeT BlIP;

X(l)

=

II

Ql'

PlQt.

...

, pr;'-lQh

...

,

p~-lQl

II.

t·

0, 0,

..•

,

0,

•..

, °

I

TaR

RaR T _

HeocooaH

MaTplIQa,

TO

paHr

MaTplIQbI

K(l) p;OJlmeH COB-

nap;aTb c

paHrOM

MaTplIQbI

K ([3*], CTp.

60)

lI,

CJle,n;oBaTeJlbHO,

OH

,n;OJI-

meH

ObITh

paBeH

m. B

CMJlY

JleMMLI

19.1

m JlHHeHHO HeaaBHCHMbIX

CTOJlOQOB

3TOH

MaTpHQbI co,n;epmaTcH B MaTpHQe

K~2)

_

(Q

PO

p

m

-

l

Q) = I

Qt.

P1Qt. , Pr;'-lQl\\

- , "

..

"

II

0,

, ° .

CJlep;OBaTeJlbHO,

paHr

MaTpHQbI N

TaRme

paBeH

m, TaR RaR

nepBbIe

m

CTpOR

MaTpHQbI COBnap;aIOT

CO

CTpORaMH MaTpHQbI

N,

a 3J1eMeHTbI

OCTaJlbHIJX

(n-m)

CTpOR

CyTh

HyJlH.

JleMMa

nOJlHOCTb10 p;oRaaaHa.

JITaR,

MbI BHP;lIM,

lITOnpeOopa30BaHlIeM

(19.10)clIcTeMa

(19.1)

nplI-

BOP;HTCH

R p;ByM ClICTeM;aM:

•

(1)

(I)

+

(2)

+

Yi = Pit

1/1

-+

...

+

Pim1!""

Pi,

m+1

Ym+1

+

qH)

Ul

+ ... +

ql~)UI'

+ (2)

. . . Pin

Yn

+

(i

= 1,

...

, m), (19.13)

• (3) + (3)

(i

= m +1,

...

,

n),

(19.14)

1/i

= Pi,

lIltl1!mtl

+ ... PinYn

nplIlJeM

ynpaBJleHlIO

U

(t)

He 01,a3blBaOT BJIHHHlIH

na

IlOBep;eHHe

ne-

peMeHHbIX

Yi

(t)

nplI

i > m.

IIepeHp;eM

Tenepb

R

oocymp;eHlIIO

slJllJleHTlIBHbIX YCJlOBHH

ynpaBJlJI-

eMOCTlI

CHCTeMbI

(19.1).

COrJlaCHO TeopeMe

17.1,

,n;JlH

1'01'0 lITOOLI 3ap;a-

lIa

4.1

00

ynpaBJleHlIH

HMeJla

pemeHlIe,

Heooxop;lIMO

II

p;OCTaTOlIHO,

lITO-

·Ob1

BbIIIOJlHHJlOCb YCJloBHe

n n

(19.15)

po

=

min

p

[2j

lih[il

(t)

] >0

npH

2J

liCi

=

1,

. I

i=l

{=l

r,n;e

BeRTOp C

onpep;eJlea

paBeacTBoM

(15.2),

T. e. B HarneM

cJlyqae

c=x~-X[t~,tatl~lIt.

(19.16)

n

f

B1>lpameHHe

h(t)

=

2J

l{Mi

l

(t)

B

(19.15)

COCTaBJleHO

1I3

He-

'1=1

npepLIBHLIX

!f>YHRQHH

h[il(t)

(toc

<:

't

<:

t~).

II03TOMY

HopMa

p [h ('t)]

j

H,

CJlep;OBaTeJlbHO, BeJllIlJHHa

po

MomeT

oopaTlITbCJI B

HyJlb

JIlImb

B T0M

CJlYlIae,

eCJlH

cym;ecTByIOT TaRHe

1I1ICJIa

l~

(i

=

1,

...

, n),

P;JlJI

ROTO-

pbtX

cnpaBep;JlHBO COOTHomeHlIe

11.

.11.

~

l~h[il

(t)=

°

IIplI

~

l~Oi

= 1

i=l

r·'.··

r·.

~:"

~>

144

CBOnCTBA

OIITMMAJIbHOrO

YIIPABJIEH1111·

[rJI.

6

§ 19]

YIIPABJIllEMOCTb

ClfCTEM

C

IIOCTOllHHbIMM

IIAPAMETPAMH

145

HJIM,

B

MaTpIf'IHOH

cPopMe,

=(lO)'H[t/3,'t] = (l°)' X

[t/3,

't]

B.

BhI'IHCJIMM

IIPOH3BO}J;HhIe

cPYHKll;HH

(l0)'

H

[t/3,

.r) =0

npH

(l0)'

e =

1.

(19.17)

~

('t)

npH't

=

t/3.

llMeeM:

t

(MhI

He

orOBapHBaeM

a}J;eCb

KOHKpeTHhIH

BhI60p

HOpMhI

p.

,Il;JIll

onpe-

(

dB~

) = (lo)'

ABB.

(19.19)

}J;eJIeHHOCTH

'IHTaTeJIb

MOiKeT

rrOJIaraTb,

HarrpHMep,

'ITO

BeJIlI'IHHa

p

d't ..

=tll

onpe}J;eJIeHa

paBeHCTBOM

(18.1).)

HaM

nOTpe6yeTcll

Tenepb

em;e

O}J;HO

BcnOMOraTeJIbHOe

rrpe}J;JIo~

meHHe.

JI

eM

M a

19.4.

TOJ/Caecm80

(19.17)

8bmOAuRemcR

moeaa u

mOAbJf,O

moeaa,

K,oeaa

8eK,mOp

e

(19.16)

He

npuuaa.n,eJ/Cum

noanpocmpancm8Y X

m

.

(HarrOMRHM,

'ITO

BeKTOp

C M.bI yCJIOBHJIHC.b

C'IHTaT.b

HeHYJIeBhIM.)

,II;

0 K a 3

aTe

JI b C

l'

B

O.

,Il;oKaiKeM

CHa'laJIa

He06xo}J;HMOCTb

yT-

BepiK}J;eRHH

JIeMMhI.

,Il;JIH

8Toro

rrpe}J;nOJIOiKHM

01'

npOTHBHoro,

'ITO

HMeeT

MeCTO

paBeHCTBO

(19.17)

H

HeHyJIeBOH

BeKTOp

c

(19.16)

npH-

Ha}J;JIeiKHTX

m

·

IIpo}J;HcPcPepeHIwpoBaB

paBeHCTBO

(19.17)

no

't

(12

-

1)

pa3

M

y'lHThIBall,

'ITO

H

[t/3,

't]

= X

[t/3,

't]

B

(CM.

§

6)

M

dX-I

[t/3,

't]/d't =

-X-I

[t/3,

't]

A

(CM.

§

10),

nOJI¥'IHM

cHcTeMy

TOiK}J;eCTB:

(lO)'

X

[til,

't]

B =0,

(l0)' X [til'

't]

AB

- 0,

(l0)' X

[til.

't]

A

n

-

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3Ta

CIICTeMa

He

HBJIHeTCH

onO/ll-te ynpaB/lJlCMoii.,

II60

B

IIpOTIIBHOM

CJIyqae

BeRTopIiI

b

U

) ,

AbU),

:

..

. . . , A

n-

1

b(i) ,

RaR

MbI

SHaeM,

6hlJIII

6bI

JIIIHeHHO

HeSaBI1CHMhl

II

He

cY.I.IJ.eCTBOBaJIO

6hl

BeKTopa

p,

ROTOpbIH

YAOBJIeTBOpHeT

yCJIOBIIHM

(19.23).

IIPII

3TOM Mhl BIIAIIM

TaRiHe,

11'1'0

IIHBaplIaHTHIilMII

ROOPAIIHa-

TaMH

6YAYT

Te

II

TOJIbRO

Te

KooPAIIHaTbl

Yi

=

p'

x,

KOTOpIilC

OllpeAeJIe-

HIiI

BeRTopaMII

p,

0pTOrOHaJIbHIilMH

II:

noonpocmpancmey

ynpae/lReMO-

cmu

CUCmeMb~

(19.1).

§ 20.

YnpaBJIHeMOCTb

HeCTaIJ;HOHapHbIX

CHCTeM

Tellepb

MIiI

AaAIIM

AOCTaTOlIHoe

yCJIOBHe

1I0JIHOH

'I

AJIH

HeCTaIJ;IIOHapHoH

CHCTeMbI

t!

\'

j;

= A (t) x

-+

B (t) u.

OrpaHHTIIIMCH

SAeCb

CJIylIaeM,

RorAa

3JIeMeHTbI

yllpaBJIHeMOCTH

(20.1)

MaTpJilIJ;IiI

A(t)

II

B(t)

JilMeIOT

HellpephlBHhIe

IIpOJilaBOAHIile

'BIIJIOTb

AO

(n -

1)-1'0

110-

pHAKa

110

KpaHHeH

Mepe

B OKpeCTHOCTJiI

HeRoTopoH

TOlIRJiI t = t.

lIa

OTpeSKa

[ta,

till

(B

TOlIRax

t = t

a

JilJIH t = til

pellb

HAeT,

eCTeCTBeHHO,

JIHmb

0

IIpaBbIX

IIJIU

JIeBIilX

IIpouaBOAHbIX

cooTBeTcTBeHHo).

BYAeM

paCCMa'rpJilBaTb

MaTpUIJ;IiI L)e (t),

orrpeAeJIeHHIile

B

OKpeCT-

HOCTH

TOlIRH

t. CJIeAYIOm;UMH

peKyppeHTHbIMU

COOTHomeHUHMU:

dL/;_1

L

1

(t) = B (t), L

k

(t) =

AL

h

-

1

-

-d

t

- (k = 2,

...

, n).

CnpaBeAJIUBa

CJIeAyI<1m;aH

TeopeMa.

Teo

p e M a

20.1.

IIycmb

lta

ompea1'ie

[ta,

till

MOJ/CltO

y1'iaaamb

mo'l{,-

1'iy

t =

t.,

e 1'iomopoii. pal-la MampulfbL

K

(t)

=

{L

1

(t),

.

..

, L

1I

(t)}

(20.2)

paeeH.

n.

Toaoa

cucmeMa

(20.1)

onO/lH.e

ynpaB/lReMa l-la

Ompe81'ie

[tilt,

till.

§

:dO]

YIIP

AB.JIHEMOCTb

HECT

AL(UOHAPHblX

CHCTEM

149

A 0 R a

saT

C JI b C

'1'

B

o.

HaM

CJICAyeT

AOIWSaTb,

lITO IIPJil

yCJIO-

BHJIX

TeopeMhl

CllpaBeAJIHBO

HepaBeHCTBO

pO

=

min

p

[B'

('t)X'

[til,

'tlll

>0

npu

l'c

=

1,

(20.3)

I

T.

e.

AORaaaTh,

lITO

He

MomeT

6hlTb

TomAeCTBa

(l0)' X

[tll,tl

B

('I')

=0.

(20.4)

II

peAIIOJIOmIIM,

HallpOTUB,

lITO

cym;eCTBYIOT

1IlICJIa

l~,

AJIH

ROTOpIilX

3'1'0

TomAeCTBO

BIilrrOJIHHeTCH.

TomAecTBo

(20.4)

IIpoAJilcPcPepeHIJ;HpyeM

110

BpeMeHH.

TorAa

IIpH

'I'

= t.

6YAeM

UMeTb:

.~

g'

(tJLd

t

.)

=

0,

...

,

g'

(tJ

L

n

(tJ =

0.

(20.5)

3AeCh

HeHyJIeBOH

n-MepHIilH

Bel,TOp

g (t*)

OllpeAeJIeH

paBeHCTBOM

g'

(tJ

= (l°)' X

[til,

tJ.

IIYCTb

k(j)

(t)

(j =

1,

...

, nr) -

n-MepHIilH

BeKTOp,

KOMllOHeH-

TaMH

KOToporo

HBJIHIOTCH

3JIeMeHTbI

j-H

ROJIOHKII

MaTplIIJ;bI

K(t)

(20.2).

TaK

KaK

IIplI

t =

t.

paRr

MaTplIIJ;bl

K

(t)

paBeH

n,

TO

cpeAII

BeKTopOB

kU>

(tJ

IIMeeTCH

n

JIIIHeHHO

HeaaBIICIIMhlX

k[lJ(tJ,

...

•••

,

k[n]

(tJ. B TO

iHe

BpeMH

lIS

CooTHollIeHIIH(20.

5)

CJIeAyeT,

11'1'0

HeHy-

JIeBOH

n-~epHbIH

BeKTOp

g

(tJ

opTOrOHaJIeH

RO

BceM

BeKTopaM

k

U

)

(tJ

H,

B

lIaCTHOCTII,

OH

opTOrOHaJIeH

R n

JIlIHeHHO

HeaaBHCIIMIilM

BeRTopaM

k[l]

(tJ,

...

,

k[n]

(tJ,

11'1'0

HeBoaMomHO.

IIOJIylIeHHoe

rrpOTIIBopelIlIe

II

AORaaIilBaeT

TeopeMY.

IIpeAJIOmeHHbIe

yCJIOBIIH

yllpaBJIHeMOCTII

AJIH

HeCTaIJ;IIOHapHOH

JIIIHeHHOH

CIICTeMbI,

RaR

MIiI

BIIAIIM,

IIMeIOT

60JIee

CJIOmHbIH

BIIA,

lIeM

AJIH CHCTeMbI C 1I0CTOHHHbIMII

rrapaMeTpaMII.

RpOMe

'1'01'0,

3TU

yCJIOBIIH

BKJIIOlIaIOT

rrpOIIaBOAHIile

0'1' cPYHKD;IIH

aij

(t)

II

bij

(t),

lITO

He

BcerAa

YAo6HO.

CpeAII

HeCTaD;IIOHapHhlx

CIICTeM,

OAHaKO,

oc060e

MeCTO

aaHII-

MaIOT

JIIIHeHHbIe

CIICTeMIJ

C

rrepIIOAlIlIeCKIIMu

MaTpIID;aMII

A (t)

II

B

(t).

B 3TOM

CJIylIae

MomHO

yKasaTb

AOCTaTOlIHble

yCJIOBHJI

yllpaBJIHeMOCTII,

KOTopIile

BbIpamaIOTcH

AOBOJIbHO

IIpOCTO

lIepea

MaTpIID;y

X

[ro,

0

l,

rAe

ro

-

lIepIIoA

cPYHKD;HH

aij(t)

II

bij

(t).

flpII

3TOM MIiI

orpaHIIlIIIMcH

CJIylIaeM,

KorAa

3JIeMeHTbl

aij

(t)

II

bij

(t)

MaTpIID;

A(t)

II

B(t)

HBJIHIOT-

CH

aHaJIIITIIlIeCKIIMII

cPYHKD;IIHMII

BpeMeHII.

CrrpaneAJIIIB

CJIeAYIOm;IIH

peayJIbTaT.

Teo

p

eM

a

20.2.

IIycmb

f)/leMeHmbL

aij

(t) U

bij

(t)

Mampulf

A

(t)

U B (t) 0 ypael-leH.UU

(20.1)

RO/lRlOmCR

al-la/lumU1f,ec1'iU~£u

U nepuoou1f,e-

C1>UMU

ifiylt1>lfURMU opeMeHU t c nepuoooM

ro.

Toaoa

O/lR

moao,

'l{,m06b~

cu-

cmeMa

(20.1)

6bL/la enO/llte ynpaO/lReMoii.

lta

ompeaKe

[ta,

till,

oocmamo'l{,-

HO,

'l{,m06bL

paH.B

MampulfbL

, X

n

1

K

(ro)

=

{B

(t

a

),

X

[ro,

Ol

B

(ta),

.

..

-

[ro,

Ol

B

(t

a

)}

(20.6)

6bL/l

paeeH

n.

"'·"'·

C

.......

_:;~

~~~;

C

CRoVrCTRAOITTHMAJIMIOrO

YTIPAD.JJEHHH

[l'JI. G

150

,IJ;

0 H a 3

aTe

JI b C

'1'

B O.

ITpep;nOJIOmUM

0'1'

npOTUBHoro,

'ITO

paHr

MaTpUIV:'I

K

(00)

(20.6)

paBeH

n, HO

CUCTeMa

(20.1)

He

HBJIHeTCH

BnOJIHe

ynpaBJIHeMOH

Ha

OTpe3Ke

[tex,

t~].

RaH

OTMelIeHO

B.bIIDe,

<lTO

B03MomHO

JIUIDb

npu

yCJIOBUH,

'ITO

cym;ecTByeT

HeHyJIeBOH

BeKTOp

lO,

yP;OBJIeTBOpHIOm;UH

paBeHcTBy

(20.4),

T.

e.

g'

(,;) =

(lO)'

H

[t~,

,;] = (l°)'X [t.s, ,;] B

(,;)

=-

0

npu

(l°)'

c = 1,

t

ex

-<,;

-<

t~.

(20.7)

3JIeMeHT.bI

«pYHp;aMeHTaJIbHOH

MaTpHD;.bI

X [til, ,;]

HBJIHIOTCH

aHaJIUTH-

qeCKHMH

«PYHKD;HHMII

,;

([12*

l,

CTp.

55),

nO<lTOMY

HOMnOHeHT.bI

BeHTopa

g'

(,;),

CTOHm;ero

B

JIeBQj!

llaCTH

Tomp;eCTBa

(20.7),

HBJIHIOTCH

aHaJIIITII-

'IeCHHMH

no

T.

Ho

Torp;a

BeHTOp

g'

(-r.),

Tomp;eCTBeHHO

paBH.bIH

HyJIIO

npH

Bcex

,;

H3

OTpe3Ha

[tex,

t.sl,

6yp;eT

HyJIeBbIMB006m;e

npH

Bcex

-

00

<

,;

<

00

H,

B

'laCTHOCTH,

npH

T = t

ex

, t

ex

-

00,

t

ex

-

2<.U,

.••

BOCnOJIb3yeMcH

Tenepb

TeM,

lITO

«pYHp;aMeHTaJIbHaH

MaTpHD;a

X

[t~,

T]

nepHOp;HlJ:eCHOH

CHCTeM.bI (20.1)

yp;OBJIeTBOpHeT

yCJIOBHIO

([19* ],

CTp.

152)

X [t.s, t

ex

.::.-

kw] = X

[t.s,

t

ex

] Xl, [w, 0]

(20.8)

(k -

D;eJIoe

'lHCJIO).

ITOJIaraH

B (20.7) T = t

oc

-

kw

(k

=

0,1,

...

, n - 1) H

yqHT.bIBaH

(20.8) H COOTHOIDeH:Ue B

(tex

+

koo)

= B

(t

ex

),

nOJIyqHM

cHcTeMy

pa-

BeHCTB:

(l0)' X

[tp,

t

ex

] B

(tex)

= 0,

(l0)' X [t!J'

tad

X [w,

0]

B

(toc)

= 0,

(20.9)

it

i

l

(l0)' X [t!J' t

ex

] Xn-1[w;

0]

B(tex) =

O.

TaK

HaI{

MaTpHD;a

X

[t~,

t

oc

] -

Heoc06aH,

TO

BeKTOp

q'

=

(l0)' X

[t~,

t

ex

]

HBJIHeTCH

HeHyJIeB.bIM.

Op;HaHO

eCJIH

paHr

MaTpHlI.bI

K

(00)

(20.6)

paBeH

n,

TO,

OlIeBHp;HO,

HeJIb3H

nop;06paTb

HeHyJIeBOH

BeKTOp

q',

HOTOp.bIH

yp;OBJIeTBOpHJI

6.bI

yCJIOBHHM

(20.9).

ITOJIyqeHHoe

npOTHBopelIHe

H

p;0Ka3.bIBaeT

TeopeMy

20.2.

ITop;lIepHHeM

Tenepb,

'ITO

yCJIOBHe

nOJIilOH

ynpaBJIHeMOCTH,

c«pop-

MyJIHpOBaHHoe

B

TeopeMe

20.2 (20.1),

HBJIHeTCH

J/,UUJ,b f}ocmamot{HblM,

HO

OHO

He

HBJIHeTClI

Heo6xop;HM.bIM

yCJIOBHeM

ynpaBJIHeMOCTH.

IToHa~eM

<lTO

Ha

npHMepe.

lIpDMep

20.1.

PaccMoTpllM

CIICTeMY

ypaBHeHHD:

Xl

= X

2

, X

2

= -

Xl

+

u.

iha

CRCTeMa BrrOJIHe

yrrpaBJIHeMa

Ha

JII060M

OTpeSlI:e

[t

oc

'

tjlj

H,

B qaCTUOCTII,

ua

oTpeaKe

[0,

2n],

TaK

K8K

S,lI,eCb

A =

11_

~ ~

II

' b =

[~J

§

20]

o'IIPABJUIEMOCTb

HECTAIJ;I10H

APH

J,IX

CHCTEM

151

II

BeKTopbl

b =

(0,

1),

Ab

=

(1,0)

JIHHeHHO

ueBaBHCHMbIe,

T. e.

BWIOJIHeHbI

YC-

JIOBlUI

TeOpeMl>I

20.1.

O,ll,HaKO yCJIOBRII

TeopeMbI

20.2

Ha

OTpeSKe

[0,2nj

He

BbI-

rrOJIUHIOTCH;

TaK

RaK

cos

211:

sin

2n

II

X

[211:,

0] = _

sin

211:

cos

211:

= E

II

R

MaTp~a

K

(2:rt),

CJIe,ll,OBaTeJIbHO,

T8ROBa:

K

(2:1:)

=

(~~),

T. e.

RMe8T

paur,

paBm.r:D:

e,ll,lI~e.

B SaRJIIOqeHHe OTMeTRM

e~e,

qTO rrpe,ll,rrOJIOmeHHe

06

aHaJIHTHqHOCTH :me-

MeHTOB

MaTpII~

aij

(t)

II

bij

(t)

MaTpII~

A (t) R B (t)

IIBJIlIeTCH

cy~ecTBeHHbIM.

HarrpHMep,

paccMoTplilM

yrrpaBJIHeMYIO

C,H-

CTeMY j'(t),

,',_._

x = -

~

(t)

[E

+

'I/J

(t)

Gj-l

Gx+

bu. !

---~..,-

, ,

7lZL

(20.10)

3,l1,eCb

x -

n-MepHIilli:

BeKTOp;

HHqHalI

MaTplilI~a;

G

--.:

(n

X

E - e,ll,H-

n)-MaTplil-

,

,~

~a

C rrOCTOIIHHblMH

II

,lI,OCTaTOqHO

MaJIblMll

0 ! Z t

aJIeMeHTaMll;

b -

n-MepHblH

rrOCTOIIH-

HblH

BeRTOp;

u-

CKaJIlIpIIalI

4lYHK~IlII,

PRC.

20.1.

orrHCblBaIO~alI

yrrpaBJIIIIO~ee

BOS,lI,eHCTBRe;

'I/J(t)

- rrepIIO,!J,lilqeCRalI

CRaJIlIpHalI

iPYHR~HII

C rrepHO,ll,OM W =

1,

rpalPHl\

ROTOpoii

liso6pameH

Ha

plilC.

20.1.

Ey,ll,eM

rrpe,ll,IIOJIaraTb,

qTO

BeKTopbl

b,

(E + G)-lb, ... ,

(E

+

Gf(n-l)

b JIlilHeHHO HesaBMCIIMbJ.

)l;JIlI

ClilCTeMbl

(20.10)

MMeeM:

X [t,Oj = [E +

'I/J

(t)

Gj-l,

H

[1,

t]

=

[E

+

Gj-l

(b

+

'I/J

(t) Gb). (20.11)

Ms

(20.11)

BblTeRaeT,

qTO CrrpaBe,ll,JIlilBO Tom,ll,eCTBO

g'

(t) = (to)' H [1,

tj

==

0, (0

~

t <1),

R8ROB 6b1

Hlil

'6blJI

BeRTop

lO,

0pTOrOHaJIbHhlH

R

BeRTop8M

[E

+

Gj-lb

lil

[E

+Gj-lGb. OTCIO,lI,a CJIe,ll,yeT, qTO CllCTeMa

(20.10)

He

IIBJIReTCII

BIIOJIHe

yIIpaB-

JIlIeMoii:.

IIoRameM,

O,ll,HaRO, qTO

yCJIOBlle,

lPMrypllpYIO~ee

B

TeopeMe

20.2,

BJ)l-

IIOJIHHeTCII.

)l;eHcTBHTeJIbHO,

B

paccMaTpliBaeMoM

CJIyqae

MaTpR~a

K

(w)l~(20.6)

HMeeT BB)];

K

(w)

= {b,

(E

+

G)":l

b,

...

,

(E

+

G)-(,t-

l

)

b}.

PaRr

STOB

MaTpM~bl

rro

IIpe,ll,IIOJIOmeHllIO

paBeH

n,

lil,

CJIe,ll,OBaTeJIbHO,

eCJIli

6bl

TeopeMa

20.2

6blJIa

BepHa

Il,ll,JIII

HeaHaJIHTllqeCRHX

aij

(t)

lil

bij

(t),

TO

rro

aTOH

TeopeMe

CRCTeMa

(20.10)

6blJIa

6bl

BrrOJIHe

yrrpaBJIlIeMoH.

O,ll,HaRo

Mbl

TOJIbRO

qTO yCTaHOBlilJIlil, qTO

SToro

HeT.

MTaR,

,lI,eHCTBlilTeJIbHO, upe,ll,IIOJIOmeHlile

06

aHa-

JIlilTRqHOCTlil

lPYHK~lilH

aij

(t) n

bij

(t)

cy~ecTBeHHO

,lI,JIR CIIpaBe,ll,JIRBOCTR

Teope-

MIll

20.2.

i

B

3aHJIIO'IeHHe

<lTOrO

rraparpa«pa

BBep;eM

em;e

OP;HO

nOHHTHe,

KOTO-

poe

nOTpe6yeTcH

HHme.

RaK

M.bI

'BHp;eJIH,

P;JIH

rrOJIHOH

yrrpaBJIHeMOCTH

JIHHeHHOH

CHCTeM.bI

Tpe6yeTcH,

'ITo6bI

!PYHKD;MH

Mil

[t~,

,;]

6bIJIH

Ha

OTpe3He

t

ex

-<

T

-<

til

JIHHeHHO

He3UBMCHM.bIMH,

T.

e.

Tpe6yeTcH,

'ITo6.bI

!

F

C~:\

,~.~.)

I

152

CBO~CTBA

OIITHMAnbHOrO

YIIPABnEHHH

[rJI.

8

nrooaR

nHHenHaH

KOMOHHa~HH

n n

~

lih[i]

[til, t']

npll

~lt>O

i=l

He

oopam;anaCb

B

Hynb

Tom)J;eCTBeHHO

Ha

eceJlt

OTpe3Ke

[tat,

t,sl.

O)J;HaKO

HHor)J;a

menaTenbHO,

'ITOOLI

4>YHK~HH

h£il

[t,s,

-d

oona)J;anH

oonee

CHnbHLIM

CBonCTBOM:

lIMeHHO,

lIHor)J;a

BamHO,

'ITOOLI

BeKTop-4>YHK~HR

(16.2)

He

oopam;anaCb

Tom)J;eCTBeHHO

B

Hynb

HU

Ha

1'>a1'>OM

UHmepea.lle,

nemam;eM

BHyTpll

OTpe3Ka

[t

lX

, t,sl,

lI,

oonee

Toro,

BamHO,

'ITOOLI

::lTa

4>YHK~HH

Morna.

oopam;aTbCH

B

Hynb

nHmb

B

OT)J;enbHLIX,

H30nllpo-

BaHHLIX

TO'IKaX

1".

B

TaKHX

cny'IaHX

cHcTeMy

OY)J;eM

Ha3LIBaTb

Heoco-

6emwu

(Ha

OTpe3Ke

[t

lX

,

t,sl).

HeTpY)J;HO

npOBepHTb,

'ITO

CHCTeMa

(19.1) C

nOCTOHHHLIMH

IIapaMeTpaMH

aij

II

bij

HeocooeHHaH

Tor)J;a

II

TonbKO

Tor)J;a,

Kor)J;a

OHa

enO.llHe

ynpa8.llJl.eJlta.

EcnH

Koa4>4>H~HeHTLI

aij

(t)

II

b

ij

(t)

CyTb

nepeMeHHLIe

-

nonomeHHe

cnomHee.

O)J;HaKO,

BO

BCHKOM

cny'Iae,

H

TaKaH

ynpaBnHeMaH

CHCTeMa

OY)J;eT

Heoco6elUWU,

ecmi

B

1'>aJICfJou

mO't1'>e

t =

t.

H3

OTpe3Ka

[t

lX

,

t,s]

BLInonHHeTCH

ycnoBHe,

YKa3aHHoe

B

TeopeMe

20.1.

HaKoHe~,

MbIoY)J;eM

Hcnonb30BaTb

em;e

O)J;HH

TepMHH:

CKameM,

'ITO

ynpaBnHeMaH

CHCTeMa

Heoco6eHHaJl.

no

eoafJeucm8U1o

Uj,

ecnH

nHHenHaH

KOMOHHa~HH

n n

~

lih[i]

[til,

t]j

npH

~

It >0,

i=l

COCTaBJleHHaH

H3

HMnynbcHLIx

nepeXO)J;HbIX

4>YHK:U;Hn

CHCTeMLI

no

aToMy

B03)J;enCTBHro,

MomeT

oopam;aTbCH

B

Hynb

nHmb

B

OT)J;enbHLIX

H30nHpOBaHHbIX

TO'IKaX

t

H3

[t1X'

t,s]

.

CHcTeMa

(19.1) C

nOCTOHHHbIMH

napaMeTpaMH

HeocooeHHaH

no

Uj

Tor)J;a

H

TonbKO

Tor)J;a,

Kor)J;a

BeKTO-

pLI

b

U

),

AbU),

...

, A

n-!b<i)nHHenHo

He3aBHCHMLI.

3)J;ecb

bU)-

j-n

cTonoe~

MaTpH~LI

B.

~OCTaTO'IHLIM

ycnoBHeM

HeocoOeHHocmu

HeCTa-

~HoHapHon

CHCTeMLI

(20.1)

no

B03)J;encTBHHM

Uj

HBmreTCH

mmen-

HaH

He3aBHCHMOCTb

BeKTopOB

lU)

(t)(l) ,

...

,

lU)

(t)(n)

npH

Bcex

t

113

OTpe3Ka

[t

lX

,

till. 3)J;eCb

lU)

(t)(l) =

b(i)

(t),

...

,

lU)

(t)(i+1)

=

dlHl/dt

- A (t)

l~{j.

l.!wraTonb,

pa300paBIIIHn

MaTepHan

)J;aH-

Horo

H

IIpep;bI)J;ynwro

naparpaq,oB,

MomeT

)J;OKa3aTb

caMocToHTenbHo

Bce

BbICKa3aHHbIe

TonbKO

'ITO

yTBepm)J;eHHH,

'ITO

MLI

H

peKOMeH)J;yeM

npop;enaTb

B

Kli'IeCTBe

ynpanmeHHH.

§

21.

YnpaBJIeHlIe

MDHHMaJIbHOD:

CUJIOD:

B

aTOM

naparpaq,e

MbI

H3Y'IHM

O)J;HH

xapaKTepHbID:

H

'IaCTO

BCTpe-

'Iarom;HD:CH

THn

3a)J;a'I

00

onTHManbHOM

ynpaBneHHH.

IIMeHHo,

pac-

CMOTPlIJVI

3a)J;a'Iy

13.1, r)J;e

HHTeHCHBHOCTb

%

[u]

onpe.ueneHa

paBeH-

CTBOM

%

[u]

=

max

y* [u (t')].

(21.1)

tlX~~.,;;tp

§21J

ynPABnEHHE

MMHIfMAnhHO~

CM,lIO:fI

153

f

3)J;eCb y*

[u]

-

KaKaH-HHOY)J;b

ltOpMa

BeKTopa

u.

TpeoyeTcH,

cne-

)J;OBaTeJlbHO,

nOCTpOHTb

ynpaBneHHe

UO

(1"),

nepeBO)J;Hm;ee

cHcTeMy

(13.1)

H3

COCTOHHHH

x

(t

lX

)

= xex B

COCTOHHHe

x (til) = X

P

If

):PIH

KO-

Toporo

BenH'lHHa

%[u]

(21.1)

npHHHMaeT

HaHMeHbmee

B03MomHoe

3

Hall

eHHe,

B

3a)J;a'laX

ynpaBJleHMH

4>YHK~MH

U

(1")

3a'lacTyro

HMeeT

CMLIcn

OOOOm;eHHOM

CHilbI,

a

HHTeHCHBHOCTb

%

[u]

(21.1)

o~eHHBaeT

Tor)J;a

HaHoonbmee

3Ha'leHHe

aTOM

,CHnbl.

IIoaToMy

H3y'laeMLIe.

B

.

)J;aHHoM

naparpa4>e

3a)J;a'lH

MbI

Ha30BeM

3a)J;a'laMH

00

ynpaBneHHH

MUHUMa.llbltOU

CU.Ilou.

Hall

HeM

Ifccne)J;OBamIe

co

cny'IaR

CKanJipHOrO

B03)J;eMCTBHH U

(1").

IIHTeHCHBHOCTb

%[u]

(21.1)

HMeeT

a)J;eCb

BIoIP;

%

[u,]

=

max

Iu (t')

I.

(21.2)

tlX~'t~t(3

HaM

H3BeCTHO

(CM.

§ 12),

'lTO

BenHTIHHa

%

[u]

(21.2)

MomeT

OLITb

HCTonKOBaHa

KaK

HopMa

p*

[u]

4>YHK~HH

U

('1:)

B

npOCTpaHCTBe

.At {u(1")}. B

COOTBeTCTBHH

C

oom;en

MeTo)J;HKoM

pemeHHH

aap;a'IH

13.1

HopMe

p*

[u]

= %

[u]

(21.2)

conOCTaBnHeTCH

HopMa

p [h

(1")],

4>Mry-

pHpyrom;aH

B

npaBHne

MHHHMaKca.

MLI

yme

3HaeM

H3

§ 12,

'ITO

t(3

P[h('t')]=~lh('t')ldt'

(21.3)

t·

ex

(CM.,

B

'laCTHOCTH,

CnpaBO'lHYro

TaonH~y

12.1).

TaKHM

00pa30M,

q,YHK~HH

h

(1")

MbI

paCCMaTpHBaeM

KaK

3neMeHTbI

npOCTpaHCTBa

2{h

('1:);

t

lX

-<

1"

-<

t,s}

HHTerpHpyeMbIx

4>YHK~Hn,

a

)J;onycKaeMLIe

ynpaBneHHH

U

(1")

-

KaK

aneMeHTbI

npOCTpaHCTBa

.At

{u

(1");

t

ex

-<

1"

-<

t,s}

orpaHH'leHHbIX

HHTerpHpyeMLIx

4>YHK~Hn~

IIpemp;e

Bcero,

cne)J;yH

npaBHny

17.1,

Hap;nemHT

onpe)J;enllTb

MH-

HHManbHyro

4>YHK~Hro

hO

(1").

B

paCCMaTpHBaeMOM

cnyqae

CKanHpHO-

ro

Bel3)J;encTBHH

U (t)

q,YHK~HH

hO

(1")

eCTb

hO

(1")

=

b'

(1")

S

fT,

tll]lO,

r)J;e

BeKTOp

lO

= {l°l,

...

,l~}

npe)J;CTaBnHeT

pemeHHe

cnep;yrom;en

3a-

)J;a'lll

Ha

ycnoBHLID:

aKCTpeMYM:

HanTH

til

pO=min~

Ib'(t')S[t',tll]lld't'

npH

l'c = 1.

(21.4)

I

t/ll,

IIocne)J;H1'm

3a)J;a'la

MomeT

TaK:

B

npOCTpaHCTBe

{ll"'"

1.Jl'

(lb'" ., in)

ObITb

reOMeTpH'leCKH

HHTepnpeTHpoBaHa

In}

cpe)J;H

nOBepXHOCTen

til

=

~

I

b'(t')

S [t', til]

lldt'

= P (21.5)

ta.

CBOtfC'fBA

On'rHMAnLHOrO

SnPABJIEHHJl

[fJl.

6

1.54

TpeoyeTcH

HaHTH

TY,

KOTOpaH KaCaeTCH

nJIOCI<OCTH

l'c

=

1.

TO'IKa

KaCaHfIJI

{I~,

... ,

l~}

OnpeAeJIIIT

IICKOMhIH BeKTOp

lO.

OnpeAeJIeIDIe

1JHCeJI

IO

i

(i

=

1,

...

, n)

113

(21.4)

eCTb

caMoe

TpYAHoe

MeCTO B

npo-

IJ;ecce

nOCTpoeIDIH

ynpaBJIeIDIH

UO

(t).

0'1'0

001>HCHHeTCH

TeM,

'ITO

nOBepxHocTII

(21.5)

yme

He

HBJIHIOTCH 3AeCh CTOJIb npOCThIMH

no

q,op-

Me

II

CTOJIb

rJIaAKHMII,

KaK

aJIJIlInCOHAhI IIJIH aJIJIlInTlI1JeCKlIe IJ;HJIHHlJ;-

phI

<I>

(tl"'"

In) =

const

(18.4),

HrpaIOIIJ;He aHaJIOflI1JHYIO POJIb B §

18.

OAHaKo

3aAa1Ja

(21.4)

xopomo

pemaeTCH

'IHCJIeHHO.

IIpHMep

TaKoro

pemeIDIH

nOKa3aH

B §

24.

HanoMHIIM,

1JTO

3aAa1JY

(21.4)

AJIH YAOO-

CTBa BMqHCJIeHHH IJ;eJIeCOOOpa3Ho

HHorAa

aaMeHHTh

3aAa'IeH

00

on-

peAeJIeHHH

,n.cmpe.MaJlbUOU

q,YHKIJ;HH

h

a

(r) = b

'

(or) S [';,

til]

k

a

= b' (,;) sa

(,;),

rAe

BeKTop

k

a

=

sO

(tfj)

HaxoAHTcH

H3

yCJIomm:

t{3

VO

=

max

k'c

= k

a

'

c

npH

~

Ib'

(-r)

S

['t',

t(3)

k Id't' = 1

(21.6)

k

till.

(CM.

npaBHJIO

17.2).

11pH

aTOM

'IlICJIa

pO,

v

ll

H

BeKTophI

lO,

k

a

CBH3aHhI

COOTHomeHHHMlI

pO

V

O

=

1,

k

a

= V

O

lO.

il,

IITaK,

oYAeM

C'IllTaTh,

1JTO

BeKTOp

lO

TeM

lIJIlI

lIHhIM

CTIOCOOOM

HaH-

,,,

'"'

AeH

H,

CJIeAOBaTeJIhHO, H3BeCTHhI

MlIHlIMaJIbHaH

q,YHKIJ;lIH

1)

hO

(,;)

II

1JlICJIO

pO.

IIpeAnoJIOmllM,

1JTO

'IHCJIO

pO

OTJIlI1JHO

0'1'

HyJIH.KaK

MhI

I!'

yme

3HaeM,

npll

aTOM

H TOJIhKO

npll

aTOM

yCJIOBlIlI

IIcxOAHaH

3aAa'Ia

:111

13.11IMeeT

pemeIDIe.

COfJIaCHO

npaBHJIY

17.1

MhI

AOJImHhI

Teneph

orrpeAeJIIITb

IICKOMoe

OrrTlIMaJIbHOe

yrrpaBJIeHHe

UO

(,;),

orrIIpaHch

Ha

yCJIOBMe MaKCIIMYMa

(17.1).

110CJIeAHee

rrpMIDIMaeT

3AeCh

BIIA

~

-

~

hO

(,;)

UO

(,;)

d't'

=

max

~

hO

(,;)

U

(1:)

d-r

= 1

(21.

7)

till.

to(

U

npM

max

Iu

(,;)

I =

1/po

MJIM, MHa1Je,

rrpM

Iu

(,;)

1<

1/

p

o.

,T

MaKcMMyM

lIHTerpaJIa

B

npaBoH

'IaCTM

(21.

7)

oYAeT,

O'IeBMAHO,

AOCTHraThCH B

TOM

CJIyqae,

KorAa

B

KamAhIH

MOMeHT

BpeMeIDI

't',

no-

AMHTerpaJI:bHaH

q,YHKIJ;HH

hO

(,;) u

(,;)

OYAeT MaKCHMaJIhHoH. 3aAa1Ja,

CJIeAOBaTeJIbHO, CBOAHTCH K IIOCTpoeHMIO <PYHKIJ;HII

UO

(,;),

aoco-

JIIOTHaH BeJIH1JlIHa KOTOpOB:

He

npeBocxoAHT

1/

p

O

II

KOTopaH

ooecne-

1)

Bce

p;aJ1bHeHmlie paccylRp;eHliH B paBHoH

CTerreHH

OTHOCHTCH

KaK K

MU,

Hu"ua,f/,bHOU

lPYHK~HII

hO

('t),

TaK

II

K a,;cmpe..ua,f/,bHOU

lPYHK~IIII

h

a

('t). Op;HaKo

P;J1H

orrpep;eJIeHHOCTII

6yp;eM pa6oTaTb,

I<aK

rrpaBHJIO, C

lPYHK~HeH

h

O

('t).

0,,\

~~'_:~

155

§

21]

YIIPABJIEHHE

MlIHHMAJIhHOil

CHJIOil

'IHBaeT

MaKcHMyM

rrpOM3BeAeIDIH

hO

(t)

UO

(,;)

, T.

e.

hO,('t')

UO

('t')

=maxhO(-r)u('t')

npH

lu(-r)\<1/pO

(ta<-r<t(3)'

(21.8)

tt

EYAeM

npeAnOJIaraTb

B

AaJIhHeHmeM,

1JTO

q,YHKIJ;HH

hO

('r)

oOpaIIJ;aeTCH

B

HyJIb

JIRmh

B

KOHe1JHOM

1JRCJIe

R30JIIIpOBaHHhIX

3Ha'IeHHH

't' =

';j'

TaKMe

q,YHKIJ;RR

hO

(,;)

oYAeM

Ha3hIBaTh

ueoco6blMu.

1I3

peSYJIbTaTOB

§

20

CJIeAyeT,

1JTO

yKaaaHHoe

yCJIOBRe BhIIIOJIHHeTCH, eCJIII CRCTeMa

(20.1)

HBJIHeTCH ueoco6euuoil.

IITaK,

nYCTb

<PYHKIJ;HlI

hO

(,;)

Heocooall.

TorAa

HCHO,

'ITO

pemeHlIe

aaAa1JlI

(21.8)

p;OCTaBJIlIeTClI

BhIpameHIIeM

1

UO

(-r) =

-0

sgnhO

(1:),

(21.9)

p

npII1JeM <PYHKIJ;IIH

UO

(,;)

orrpeAeJIeHa

BCIOAY,

KpoMe

IWHe1JHOrO

1JRCJIa

R30JIHpOBaHHhIx

3Ua1JeIDIH

,;

=

't'j,

rAe

nepeMeHHaH

hO

(,;) oOpaIIJ;aeT-

CH

B

HYJIh.

B MOMeHThI

BpeMeHII

,;

=

';j3Ha1JeIDIH

UO

('t'j)

CRJIbI

UO

(1:)

HBJIHIOTCH

uecY/1fecm6eUUbl.MU,

II

MX

MomHO,

BOOOIIJ;e

ronopH,

BhIoHpaTb

npOH3BOJIhHbIMR

lIa

oTpe3Ka

[-1/po,

1/p°l.

OAHaKo,

npII

aTOM

MbI

AOJImHbI RCXOP;HTb

IIS

HameH

AorOBopeHHOCTH 0

HenpepbIBHOCTII

<PYHK-

IJ;IIII

UO

(,;)

crrpaBa,

TaK

'ITO

B KOHe'IHOM

C1JeTe

noaAeHCTBIIe

UO

(,;)

liI

B T01JKaX

,;

=

';j

OKa3bIBaeTCH

onpeAeJIeHHhIM

OAHOaHa'IHO.

IITaR,

MbI

BIIAIIM,

1JTO

eCJIlI

MIIHIIMaJIhHaH

q,YHKIJ;IIH

hO

(,;) HBJIa--

eTCH

HeOCOOOH,

TO

yCJIOBIIe MaRCIIMYMa

(21.7)

((21.8»

onpeAeJIHeT

e8uu-

cm6eUUbl.M

oopaaoM

on'rIIMaJIbHOe

ynpaBJIHIOIIJ;ee Bos,u;eHcTBliIe

UO

(,;),

IIMeIOIIJ;ee

BHA

(21.9).

YnpaBJIeHlIe

UO

(,;)

BIIAa

(21.9)

l1pIIHHTO

Ha-

3bIBaTb

pe.l£eUUbl.M

no

aHaJIOrIIII

C

Ha3BaHIIeM

TeXHII1JeCKMX yCTpOHCTB,

ROTopbIe

MoryT

Haxo,u;liITbCH B P;BYX COCTOHHIIHX.

3aMeTIIM

eIIJ;e

CJIeAYIOIIJ;ee.

ECJIII

YAaeTCH rrOCTpOIITb y,u;OOHhIH

aJIrOpIITM

P;JIH

pemeHHH

3ap;a'III

(21.4),

'1'0

HeTpYAHO

yKaaaTb

BbI1JIIC-

JllI'feJIbBYIO

cxeMY,

MOAeJIIIpyIOIIJ;YIO

yrrpaBJIeIDIe

(21.9).

TaKaH

cxe-

Ma

CTpOIITCH TaR

me,

RaR

aTO

CAeJIaHO B §

10

IlPM

pemeHIIII

3a-

Aa1JII

00

ynpaBJIeIDIH

C

MIIIDIMaJIbHOH

:meprIIeH.

OnHmeM

3,u;eCb

KpaTKo

paooTy

CIICTeMhI

yrrpaBJIeHIIH,

BhIpaoaTbIBaIOIIJ;eH neJIII1JH-

BY

(21.9).

11YCTh K MOMeHTy t =

trx.

B

BYI

pemeHa

aaAa1Ja

(21.4)

II

HaHp;eHhI,

CJIeAOBaTeJIbHO,

BeJIII'IIIHbI

SO

(tell)

= S

[trx.,

tfj]

lO,

pO

>

O.

3Ha'IeHIIe

sO

(till.)

B

TOT

me

MOMeHT

BpeMeHH

t =

trx.

-nocTynaeT

B

MoAeJIb,

<pa30Boe

COCTOHHHe

ROTOpOH orrHChIBaeTCH

conpHmeHHOH

cIIcTeMoH

(17.3).

C

aToro

MOMeHTa MOp;eJIb

Ha'IIIHaeT

BbIpaoaTbIBaTh

BeRTOp

SO

(t).

3Ha-

1JeHHH

rrepeMeHHoH

SO(t)

II

BeKTOp-q,YHKIJ;HH b (t) rrop;aIOTCH B

BYII

TaR,

1JTQ

Ha

BbIxo,u;e

aToro

yCTpoHcnia,

Ha1JIIHaH C MOMeHTa t =

trx.,

BhIpaoaThIBaeTCH

MIIHIIMaJIhHaH

q,YHKIJ;HH

hO

(T)

= b

'

(or)

SO

(,;).

110-

C'l'OHHBaH

pO

H

rrepeMeHHaH

hO

(,;)

nocTynaIOT

Ha

BXO,u;

peJIeHHOrO

3Be-

aa PS.

Ha

BhIXOA6

ROToporo

npH

t~

< t < tfj

BhIpaoaThIBaeTcH

156

c·

CBOflCTBA

Ol1TUMAJlbHOrO

~'l1PABJIEHnll

[rJI.

6

BeJIll'll1Ha

(21.9)

ynpaBJIHIOm;erO

BOSlJ,eHCTBHH

UO

(t).

CUrRaJI

UO(t)

npeo6pasyeTcH

B

YCUJIUTeJIbHOM

yCTpoHcTBe

H nOlJ,aeTCH

Ha

o6~eKT

ynpaBJIeHUH

B

«popMe

peaJIbHO

lJ,eHcTByIOm;Hx

CUJI.

IIp

H M e

'I

a H H e

21.1.

06CYP;IIM Tenepb

KpaTKo

Te

cIITyaIJ;IIH, KOTophIe

BOSMomHhI

npR

HapymeHllH

YCllOBHff 0 Heoc06eHHocTH MHHIIMallbHoH

~YHKn;HH

hO

(T).

Ecml

CHCTeMa

(20.1)

cTau;uoHapHaff (T. e. MaTpuu;a A

Ii

BeKTop b -

no-

CTOffHHble),

TO

lIS

pesyllbTaToB

§ 19 cllep;yeT

(CM.

CTp.

144),

'ITO sap;a'la

13.1

IIMeeT

pemeHlIe

Torp;a

II

TOllbKO Torp;a, Korp;a BeKTOp c

(15.2)

llemRT

B

no8npocmpancm6e

:JC

m

ynpaBllffeMOCTR

CRCTeMhI

(19.1)

(CM.

CTp. 146).

Ho

Torp;a HaM

p;OCTaTOqHO

OO?Jef(m

x(t)

i=Ax+bu+w

r-------

I

:

II

IsQ(tj

I

x4'

~

BY/

s=-A's

I'

.....

,

I

L

I

~

PRC. 21.1-.

paCCMOTpeTb

mlmh

BnOllHe

ynpaBllffeMyIO

qaCTh

::lTOH

CRCTeMhI,

RrHopHpyff

oc-

TallhHbIe Koopp;RHaThI. Op;HaKo 3Ta

nop;CHCTeMa

(19.21)

ffBllffeTCff Heoc06eHHoH.

II03TOMY eCllH sap;aqa HMeeT

pemeHlle,

TO

~YHKn;Hff

hO

(T)

06ffsaTellbHo

eCTb

Heoc06aff. Cllep;oBaTellbHo, B cTan;HoHapHoM

cllyqae

sap;aqa

13.1

llH60

Bo06rn;e

He

HMeeT

pemeHHff,.llH60

lIMeeT e8uncm6ennoe pemeHHe, onpep;ellffeMoe paBeHcTBoM

(21.9).

B

cllyqae

me nepeMeHHhIx napaMeTpoB ClICTeMhI MomeT CllyqRThCff, 'ITO

XOTff

hO

(T)

=1=

0 Ha

6ceM

oTpesKe [t",

tfillI

pO>

0,

HO

hO

(T)

==

0 Ha RaKoM-n1l60

oTp;ellbHoM

npoMemyTKe

tty, t

s

],

npHHap;JIemarn;eM

oTpesKy

[t

'

tfll.

(IIOJIeSHO

oc

saMeTlITb, op;HaKO, 'ITO

3TO

BOSMomHO

nHlIIb

Torp;a, Korp;a A

(t),

b

(t)

- HeaHa-

nRTllqeCKlle

~YHKn;1I11

BpeMeHH t.)

IIYCTb,

HanpllMep,

TaKoH

npoMemyToK

tty, tsl

ep;HBcTBeHHbIH.

Torp;a

P;JIff

Bcex

T,

He

npHHap;llemarn;HX eMy, ynpaBJIeHHe

UO

(T)

CJIep;yeT

onpep;eJIffTb

COrJIaCHO

(21.

9).

,IJ;nff

MOMeHTOB

me T

liS

npoMemyTKa

tty' tsl COOTHomeHIDI (21. 7),

(21.8)

yme.He:pa60TaIOT. B

3TOM

cJIyqae rrpoH3Bep;eHlile

hO

(T)

UO

(T) paBHO HyJIIO Ha

n;enOM

IIHTepBaJIe tty,

tsllI

yTBepmp;aTb,

'ITO

ynpaB-

JIeHRe

UO

(t) ffBllffeTCff peJteunbtM U e8unCm6ennbtM,

IICXOP;ff

TOnbKO

HS

yCJIOBIDI

Mal,cIIMyMa

(21.8),

Mbl

yme

He

MomeM. OKaSbIBaeTCff,

op;HaKO,

'ITO H B

caMOM

06rn;eM

cllyqae

ynpaBneHHe

uO

(T) Bcerp;a

MOJImO

BhI6paTb peneHHhIM, KaK

3TO

UOKasaHO B crreIJ;n:aJIbHhIX

pa60Tax

(CM., HanpHMep,

[365a,

6]),

K

KOTOpbIM

MhI

H

OTChIJIaeM

'lHTaTeJIff.

qTO

me

KacaeTCff Bbl'lHCJIeHHff

UO

(t) D KOHKpeTHhIx

cllY'laffX,

lwr,!{a

hO

(T)

==

0 Ha

OTp;eJIbHbIX

HHTepB:.J.nax BpeMeHH,

TO

sp;eCb

MomHO BOCnOJIlr

·.····.···

.

'.

. ,

. ..

c

§ 21]

VIlPABJIEHHE

MHHHMAJIbHO:A:

CHJIO:A:

157

SOBaTbCff

npep;eJIbHbIM nepexop;OM

OT

Heoc06eHHoro CJIy'laff,

nO,lJ,06HO

TOMy,

Ral,

3TO

onRcaHO HRme B

p;pyrllx

aHaJIOrHqHbIX cHTyaIJ;RffX

(CM.

CTp.

199).

IIpJIBelJ,eM

Tenepb

npJIMep

MlIHJIM1I3an;JIH

BeJIlI'IHHbI

(21.

2).

llpDMep

21.1. PaccMoTpIIM

cllyqaH

p;BHmeHllff MaTepllaJIbHOH TOqKIil

m,

onllcaHHbIH B

n.

5°

§ 3,

II

06CYP;HM

aap;a'ly 0 nepeBop;e

3TOH

MaTepHaJIbHOH TOq-

KII

Ha

KpyroByIO

Op6HTy. B KaqeCTBe ynpaBJIffIOrn;erO B03p;eHcTBllff 6yp;eM

pac-

CMaTpHBaTb CIIJIY, B03HHKaIOrn;yIO

BCJIep;CTBlle

'BhI6pacbIBaHliff peaKTHBHOH

Mac-

ChI

C nOCToffHHoii:

no

Mop;yJIIO

OTHOCHTeJIbHoii:

CKOpOCTbIO

B HanpaBJIeHllH,

nep-

neHp;HKyJIJIpHOM K pa)l;Hycy-BeKTopy TO'lKH. YpaBHeHllJI p;BHmeHHff B mmeHHOM

rrpH6JIHmeHIIH

HMeIOT

BlIP;

(3.23):

Xl = X

2

,

1:

2

= -

Xl

+

X3'

X

3

=

U,

(21.10)

rp;e

Xi

CYTb

OTKJIOHeHfffI

~aaOBblX

Hoopp;lfHaT TO'IHff

OT

lIX

SHa'leHHlI

Ha

HPY-

rOBon: op6HTe. '(EP;IIHff'lHble

3HaqeHffffH03tPtPlI~[1eHTOB

B

(23.10)

IIollY'leHbI

aa

C'leT BbI60pa cooTBeTcTBYlOrn;I'lX

MacIIITa6oB.)

By.n;eM

pelIIaTb

aap;a'Iy 0 UOCTpO-

emmynpaBneHllff

uO

('r), nepeBOp;IIrn;ero cllcTeMY

(21.10)

liS

COCTOfIHlflI x

(0)

=

{XlO,

X20,

X30}

B

COCTOIIHlle

x

(T)

=

{O,

0,

O}

aa BpeMfI

T.

MUHIIMHaaU;1I1I Be·

llH'llIHhI

x

[uJ

(21.2)

OaBaqaeT meJIaHlIe ocym:ecTBIITb yHaaaHHhIll: uepeBOp; 6ea

60nhlIIHx

p;o.nOllHHTeJIbHbIX

ueperpyaoR.

Cif!oPMYlllIpOBaHHalI aap;a'Ia IIMeeT

pe-

UIeUlIe

uO

(T)

UplI

JII060M

aHa'leHilll

x

(0),

TaR HaR

CHCTeMa

(21.10)

eno.n.H.e

yn

pae.M1.e.M.a.

,IJ;eil:cTBI'lTellbHO, BeHTophI

b=

[~l

Ab=

f!l

A'bnJ

llHHeMHO

HeaaB.IlCIilMhI,

If,

CJIep;OBaTellbBO, BhIrrOllElIIIOTCII yCJIOBliff TeopeMbI

19.1.

Ho

Torp;a

paccMaTpllBaeMalI

CllCTeMa

(21.10)

fIBJIffeTCff TaKme

II

Heoco6eIIHon:

(CM.

§

20,

CTp.

152)

H,

CJIep;OBaTeJIhHO,

peilieHlle

uO('t') eJlIlHCTBeHHoe

II

orrpep;e-

lllIeTCII paBeHcTBoM

(21.9)

By.n:eM HCRaTb

pemeHlle

u

O

('t'),

oUIIpalIcb

Ha

all:cm-

peMa.n.bnylO

if!YHRIJ;HIO

hG(T) H

HCXOP;If,

CJIep;OBaTeJIbHO,

Ha

yCJIOBllff

(21.6).

([lyH-

.n:aMeHTaJIbHaff MaTpll:u;a 8

[T,

Tl

CIICTeMbI

0....-

51

= 82,

82 =

81,

5a

= -

82,

COnpJIJI{6HHOM

R

CII:CTeMe

(21.10)

(rrpH u

==

0),

HMeeT

BHP;

cos

(T

- T)

sin

(T

-

't')

0

8

['I:',

T]

=

II

-

sin

(T

- T)

cos

(T

-

'1:')

0

1 - cos

(T

-

'1:')

-

sin

(T

-

T)

1

OTCIOp;a

BhITeRaeT, 'ITO

~YHKn;HJI

h

('t')

= b'

('t')

8['1:',

Tlk,

~HrYPHpYIOrn;aJI

B

(21.6),

B

p;aHHOM

clly1Jlae paBHa

h (T) =

1£1

[1

-

cos

(T

-

T)l

+ k

2

sin

(T

-

T)

+ k

3

•

(21.11)

3ap;a'la

(21.6)

MomeT 6bITb TerrepL c4opMYJIHpoBaHa TaR: Baii:TK

'110

=

IIIax

(klCI

-+

1~2C2

+-

kacs)

(21.12)

I.

-

rrpH

T

~

Ik

l

[1

-

cos

(T

- 't')]

-+

k

2

sin

(T

-

't')

+kal

d'l:'

= 1,

°

158

0,:'"

~:'>

CBOllC'l'BA

OIlTI1MAJlhHOrOYIlPABJIEHl11I

Lrn. 6

rAe

KOMnOHeHTbI

ci

BeKTopa

C

OnpeAeJIHIOTCH

paBeHcTBaMIl:

CI

=

(xso

- XIO)

cos

T

~

X20

sin

T - XSO,!

C2

= -

(xso

- XIO)

sin

T - X20

cos

T,

(21.13)

Cs=

-

X30.

PaCCMOTpliIM

caMblii

npocToH

CJIy'laH,

Korp,a

T = 2n.

BBeAeM

060BHa'leHHJi:

Pr=-kr,

P2

=

-k

2

,

~s

= k

I

+

1c

2

, }

(21.14)

CI

=

Cs-

CI,

C2 = - C2,

Cs

=

Cs.

Tor,n;a

aa,n;a'Ia

(21.12) CBOAIlTCH K CJIe,n;yIOm;eMy:

HaHTIiI

V

O

=

max

(CIPI

-t

C2P2

+

CaPs)

(21.15)

p

2"

..

nplil

'l'

(PI.

P2.

Ps) =

~

I V

P~

+

P~

sin

(1'

+~)

+-

Psi

d1'

= 1, r,n;e

tg

~

= PI/PS'

o

BLI'IIICJIHH

IIHTerpaJI

B (21.15),

nOJIyqaeM:

1 =

'l'

{PI, P2.

PS}

=

"If

. IPsi

..

f------.

4 Y

P~

+

P~

-

P~

+

41

ps I

arCSIn

V -

npn

I

Ps

I< Y

P,i

+

P;;,

=

~+~

,

(

2n I

Psi

IIpFI

Ips',

~

-V

pi

+

p~.

(21.16)

YpaBHeHlfe

(21.16)

onpeAeJIHeT

B

TpexMepHoM

npOCTpaHCTBe

{PI'

Pa,

p.;}

He-

IWTOpyIO

nOBepXHOCTb

@[.

B

TO

me

BpeMH

paBeHCTBO

elPI +C

2

Pa+ C

a

Pa

= v

nplil

paaJIll'IHLIX

v

aaAaeT

B

aTOM

me

npOCTpaHCTBe

ceMell-

CTBO

UJIOCKOcTeHfJJ.

TaKllM

06-

paBOM,

aa,n;a'la

(21.15) CBOAllTCJI

K

orrpeAeJIeHIilIO

TOll

rrJIOCKOCTH

O

'C

I

PI

+C

a

Pa

+C

a

Pa

= v

,

(21.17)

p,

KOTopaH

KacaeTCH

nOBepxHocTH

(21.16)

li

AJIH.

KOTopoll

vO>

0.

TO'IKa

I<acaHHH

{g~,

pg,

p~}

aToll

IIJIOCKOCTH C

rrOBepXHOCTbIO

(21.16) J_H

onpeAeJIHT

k

IiIcKOMblii

/lz

, a

{a

a

a}

TX

BeKTOp

I P - Pl'

P2'

,Pa'

ua

PRe.

21.2.

~

(21.16)

HenocpeACTBeHHO

BLITe·

KaeT,

'ITO

nOBepXHOCTb

@l,

orr-

peAeJIHeMaH

aTHM

paBeHCTBOM,

npeACTaBJIHeT

c060i

rrOBepXHOCTb

Bpam;eHIiIH

BOKpyr

OCH

Pa,

cHMMeTpH'IHyIO

OTHOCHTeJIbHO rrJIOCKOCTIiI

Pa

0.

IIoaToMY

=0

AJIH

cymAeHHH

0

xapaKTepe

nOBepxHocTH

(j

AocTaTO'IHO

paccMoTpeTh

ee

ce-

'loHUe,

Hanpmlep,

nJIOCIWCTbIO

Pa

=-

0.

Ha

pHC.

21.2

Ra06pameHa

aTa

no-

n,

~L

§ 21]

YUPABJlEHllE

1,HIHI1MAJIhHOll

CHJloll

159

BepXHOCTb.

Ma

3Toro

plilcyHKa

HeTpYAHO

aaMeTRTb,

'ITO

CCJIll

BeKTOp

C =

{C

I

,

C

a

,

CaL

,OnpCAeJIHCMLIU

Ha'IaJIb,HLIM

COCTOHHHeM x (0)

CIlCTeMLI

(21.10),

HanpaBJIeH

no

OCli

Pa

('1'.

e.

CI

= 0,

(;a

= 0), TO

nJIOCKOCTb

fJJ (21.17)

KacaeTCH

nOBepxHocTII

(21.16)

no

Kpyry,

06paBOBaHHOMY

Bpam;eHlleM

OTpeaKa

AB

BOKpyr

OCII Pa'

Pe-

IDeHlie

aaAa'lli

(21.15) AOCTaBJIHeT B

TaKOM

CJIy'Iae,

O'IeBliAHO,

JII060H

BeKTop

l'

=

{p~,

p~,

p~},

KOHeD;

KOToporo

JIemllT

Ha

aTOM

Kpyre,

T.

C.

BeKTOp

pa =

=

{p~,

p~,

2~

sgn

cs}

,

rAe

(p~)a

+

(p~)a

~

(

2~

r.

IIplI

aTOM

'IRCJIO

VO

= 1/pO =

= ICal/2rc,

llTorAa

liB

CooTHoIDeHRH

(21.11), (21.14) CJIe,n;yeT,

'ITO

aKCmpe:.ltaltbnaS£

WYHKD;IIH

ha

(1')

p8BHa

. 1

h

a

('t")

=

p~

cos

l'

+-

Pg

S11l1'

+-

2n:

sgn

Ca.

OrrTllMaJIbHOe

yrrpaBJIeHHe

UO

(1')

orrpe,n;eJIHeM

Terrepb

COrJIaCHO

(21. 9), r,n;e CJIe-

AyeT

JIHIDb

aaMeHHTb

k

O

(1')

Ha

h

a

(1').

Y'IIITLIBaH

npll

aTOM

HepaBeHCTBO

(p~)2

+

(p~)2~

(1/2n:)2,

OKOH'IaTeJILHO

nOJIy'IaeM:

1

UO(1')

=

2n:

Ca·

(21.18)

IIYC'J!L

TenepL

BeKTop

C =

{c

I

,

ca,C

a

}

He

HanpaBJIeH

rro

OCII

Pa

(CM.

pIIC.

21.2).

118

PllCYHK8

BIIAHO,

'ITO

B

aTOM

ciIy'lae

nJIOCKOCTb fJJ (21.17)

KacaeTCH

nOBepxHocTII

@l

B

ep;RHcTBeHHoH

TO'IKe

N C

KooPARHaTaMIiI

p~,

p~.

l1a

pIiIC.

21.2

nOJIyqaeM,

'ITO

V

O

= C

1

P~

+

~2P~

+

csp~,

a

liB

paBeHCTB

(21.11), (21.14)

HaxoAHM

a.KcTpeMaJIbHYIO

WYHKD;1iIIO

kG

(1'):

h

a

(1') =

P~

cos

l'

+-

p~

sin

l'

+

p~.

(21.19)

Tor,n;a

comaCHO

(21.!:l)

OnTlIMaJIbHOe

ynpaBJIeHHe

paBHO

U

O

=

VO

sgn

(p~

cos

l'

+

P~

sin

l'

-t

p~).

(21.20)

PaCCMOTplIM

Tenepb

Ba,n;a'IY l1.JIH'

HeKoTopLIX

KOHKpeTHLIX

Ha'IaJIbHLIX

YCJIOBIiIH.

IIyCTb

B

MOMeHT

t = °

CKOpOCTb

,n;Bllmym;eHcH

TO'IKll

rrepneHp;HKyJIHpHa

K

pa,n;Hycy-BeKTopy,

a

KOJIH'IeCTBO

,n;BHmeHIiIH

TO'lKH

B aTOT

MOMeHT

paBHO

ee

KOJIll'lecTBy

p;BH}KeHIiIH

Ha

aap;aHHoH

KpyroBoH

op6HTe.

TaKHM

06paaoM"

nycTb

X

1

0 =

1,

X20

= 0, X

3

0 = 0.

Ma

paBeHCTB

(21.13), (21.14)

JiaxOP;IIM,

'ITO

C1

=

1,

C2

"'"

0,

C3

=-=

0.

IIJIOCKOCTb

/fJ

(21.17)

KaCaeTCH

B

aTOM

CJIy'Iae

nOBepXHOOTII

Ifl

(2-1.16)

B

TO'IKe

D

(CM.

pIIC.

21.2) C

Koopp;HHaTaMH

P~

= 1/4,

p~

-=

0,

P~

= 0,

H,

CJIe,n;OBaTeJIbHO, 'IIiICJIO

VO

= 1/4.

Ms

(21.20)

rroJIy'Iaell,

'I'1'0

OrrTIiIMaJIbHoeJ

yrrpaBJIeHHe

UO(~

paBHO

UO

(tl

= 4

1

sgn

cos

or.

(21.21)

IIop;

TaKHM

Boap;eUCTBHeM

TO'lKa

CHa'laJIa

pa8rOHHeTCH

B

Te'IeHHe

BpeMeHH

°

~

t <

<n/2; B

MOMeHT

1'1

=rc/2

CJIep;yeT

nepeKJIIO'IeHlle

ynpaBJIeHHH,

H

npH

rc

/2

~

1'<3n/2

1'0'IKa

TOp!ri:OBHTCH; B

MOMeHT

1'2

= 3rc/2 CJIep;yeT

HOBoe

nepeKJIIO'IeHHe,

TO'lna

CHOBA

paBrOHHeTCH

ll,

HaKOHen;,

npE:

1'a

=

2rc

TO'IKa

no

naCaTeJIbHOH

npHxop;HT

Ha

aap;aHHYIO

KpyroByIO

Op6HTy;

B aTOT

MOMeHT

ynpaBJIeHHe

c6paCLIBaeT-

CH

(CM. pIiIC. 21.3). B

peayJIbTaTe

TO'IKa

OK33hIBaeTCH

Ha

npyroBoH

op6HTe

C

TeM

Красовский Н.Н. Теория управления движением

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