Красовский Н.Н. Теория управления движением

40

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5]

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1

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-

or)

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1')

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~

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42

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A

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2

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1:)

0

7[1-

I

x~

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-

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x~

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cos

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0

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sin

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(t

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1:)

I xO

X 3

+S

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1

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j

(tn,

to

W (t) =

{W

k

(tn

CYTb

cPYHKIJ;HH

60.1tee

o6u{-e'U

npupof)u,

qeM

onpe-

.n;OJIeHHhIe

BbIme

BeJIHqHHbI

(6.3), (6.4). ,IJ;JIH

::\Toro

CJIe.n;yeTJIHillb

YC-

CJIe.n;oBaTeJIbHO,

B

3TOM

CJIyqae

CHMBOJI

dU

(t)

B (6.7)

MomHO

JIOBIITbCH

nOHHMaTb

HHTerpaJIbI

B

npaBoH

lJaCTH

(6.5) B CMhICJIe

U1f,-

TpaKTOBaTb

KaK

06bIlJHbIH

.n;IIcPcPepeHIJ;HaJI.

OTMeTHM

TaKme,

qTO

cnpa-

!

me3pal108

Cmullmbeca.

TeopHH

HHTerpaJIa

CTnJITbeCa

II3JIaraeTCH

Be.n;JIHBO CJIe.n;YIOm;ee

BamHoe

HepaBeHCTBO:

no.n;po6HO

B

I{ypcax

~maJIH3a

([23*1,

T.

3,

CTp.

89-122)

II

B

Kypcax

t

w

no

TeopHH

cPYHKIJ;HH

.n;eHcTBHTeJIbHOrO

rrepeMeHHoro.

CmaToe,

HO

I~

h(l)dU(t)I<suplh(t)l·var[U(t)];~.

(6.9)

BnOJIHe

.n;OCTaTOqHOe .n;JIH

HameH

IJ;eJIH

H3JIOmeHHe

TeopIIII

HHTerpaJIa

to

t

CTHJITbeCa

qHTaTeJIb

MomeT

HaHTH

B

KHHre

(5*]

(CTp.

157).

;!l,JIlf

y.n;o6cTBa

qTeHHH

npHBe.n;eM

3.n;eCb

onpe.n;eJIeHHe

H

OTMeTHM

COBceM

B

KalJeCTBe

npHMepa

paCCMOTpHM

em;'e TOT

CJIyqaH,

Kor.n;a

h (t) = y

KpaTKO

OCHOBHbIe

oco6eHHoCTH

::\Toro

HHTerpaJIa.

H U (t) = 0

rrpH

to

< t <

it,

U

(t)

= 1

npH

'6'

< t <

two

Tor.n;a

MHTerpaJI

CTHJITbeCa

rryTeM

Henocpe.n;CTBeHHoro

BbIqHCJIeHHH

HHTerpaJIbHOH

CYMMbI

nOJIylJHM:

t",

t

~

h (t) dU (t)

(6.7)

t,

~

h

(r)dU

('t') = 0

rrpH

t <

'6',

~h(1:)dU('r)=y

npH

t>it.

t

rJ

t,

i

46

c

p.

/

.

c:-,····

-'.'

~)

'.,:;.-

CBOYl:CTBA

JUIHEYl:HbIX

CHCTEM

[rJJ.

2

PaCCMOTpIIM

TaKme

HeCKOJIbKO

60JIee

06n:r;IIH

CJIy'IaH,

KOrp;a

<PYHK-

D;IIJI h (t)

HerrpepbIBHa

Ha

[to,

t,,,),

a

<PYHKn;IIJI

U(t)

HerrpepbIBHa

II

p;II<p<pepeHn;IIpyeMa

BCIOP;y,

KpOMe

TO'IKII

t =

iJ,

rp;e

OHa

IIMeeT

pa3-

pbIB,

HO

rrpII

aTOM

HerrpepbIBHa

C.lLe8a.

BbI'IIICJIJIJI

rrpep;eJI

IIHTerpaJIb-

HOH

CYMMbI

Sm,

HeTpyp;HO

rrpOBepHTb

paBeHCTBa

t t

~

h

(T)

dU

(T)

=

~

h

(T)

U

(T)

dT

rrpII

t <;

iJ,

t,

to

t

(6.10)

~

h

(T)

dU

(T)

= h

(iJ)

[U

(iJ

+

0)-

U (iJ)] +

t,

t

+

~

h

(T)

if

(T)

dT

UpII

t > f)'.

t.,

BepHeMCJI

I{

paBeHCTBaM

(6.5).

IIOCKOJIbKY

<PYHKD;IIII

hij

[t,

T]

HerrpepbIBHbI,

npaBbIe

'IaCTII

aTIIX

paBeHCTB

HMeIOT

CMbIC.TI

rrpH

.TII06bIX

<PYHKD;IIJIX

U

j

(T)

II

WI>

(T),

IIMeIon:r;IIX

HapaCCMaTpIIBaeMOM

OTpe3Ke

[to,

tOl]

OrpaHII'IeHHOe

II3-

MeHeHIIe

val'

[Uj],

val'

[W"J.

IIOaTOMy

MO}lmO

rrpIIHJITb,

'ITO

II

B

TaKOM

60Jlee

o6m;eM

cJIy'Iae

paBeHCTBa

(6.5)

(IIJIII

paBeHCTBO

(6.6»

3ap;aIOT

p;BHiKeHIIe

x

(t)

cIIcTeMbI,

yrrpaBJIJIeMOH

BeKTopOM

dU

(t)

=

{dUj(t)}

H

rrop;BepiKeHHoH

B03Myn:r;eHIIJIM

dW(t)

=

{dWdt)}.

MomHo

rrpOBepIITb,

'ITO

BeKTOp

x

(t)

6JIarop;apJI

CBOHCTBaM

<pYHp;a-

MeHTaJIbHOH

MaTpHD;bI

X [t,

to]

H

CBOHCTBaM

HHTerpaJIa

CTH.TITbeCa

H

Torp;a

yP;OBJIeTBOpJIeT

yC.TIOBHJIM,

orrpep;e.TIRIon:r;IIM

<pa30BbIH

BeK-

TOp

x

(t)

(CM.

orrpep;eJIeHHe

2.1).

3aMeTIIM

en:r;e,

'ITO

rrpaBbIe

'IaCTH

B

BeKTopHOM

paBeHCTBe

(6.6) C.TIep;yeT

TpaKTOBaTb

KaK

HHTerpaJIbI

CTHJITbeCa

0'1'

MaTpH'IHbIX

<PYHKD;HH H H X

no

BeKTOPHbIM

<PYHKD;IIJIM

U (t) H W (t).

IIpH

aTOM

rrop;

IIHTerpa.TIOM

CTII.TITbeCa

tOl

~

H (t)

dU

(t) = q

(H

= {hij}, i = 1,

...

,

n;

j = 1,

...

,

r)

t,

6yp;eM

rrOHIIMaTb

BeKTOp

q C

KOMrrOHeHTaMH

r tOl

qi =

~

hij

(t)

dU

j

(t)

(i=1,

...

,n).·

3=1

t,

,1];JIJI '1'01'0

'IT06bI

rrOp;'IepKHyTb,

'ITO

P;JIJI

paCCMaTpHBaeMOH

JIlmeH-

HOH

CHCTeMbI

MomHO

p;orrycKaTb

yrrpaB.TIeHHH

dU

j

(t)

(H

B03Myn:r;eHIIJI

dWI>

(t»,

nopomp;aeMbIe

<PYHKD;HJIMH C

OrpaHII'IeHHbIM

II3MeHeHHeM,

II

H36emaTb

npH

aTOM

rpOM03p;KOH

3anHCH

(6.5),

6yp;eM

B

COOTBeT-

§ 6]

OBOBmEHHbIE

YPABHEHHJI

)];BHtREHHJI

47

CTByIOn:r;IIX

C.TIy'IaJIX

orrIICbIBaTb

cIIcTeMy

He

ypaBHeHHeM

(5.1), a

ypaBHemfeM

dx

= A (t) xdt +B (t)dU + dW. (6.11)

II

pH

aTOM

3anHCb

(6.11) JIB.TIJIeTCJI

.TIHillb

CU~t80A,u'{eC1i,u.M

yKa3a-

HHeM

Ha

TO,

'ITO

p;BIImeHIIe

CHCTeMbI

onHCbIBaeTCJI

paBeHCTBaMH

(6.5),

COCTaBJIeHHbIMH

no

MaTpIID;aM

A H B B

COOTBeTCTBHII

C <POPMYJIOH

ROillII,

H B

aux

paBeHCTBax

U

j

(t),

WI>

(t)

cyTb

<PYHKD;IIII

C

OrpaHII'IeH-

HbIM

H3MeHeHIIeM.

B

TaKHX

cJIy'IaJIX6yp;eM

rOBopIITb,

'ITO

B03MOiKHbI

o606Ufel-ll-lbLe

yrrpaB.TIeHHJI

dU

(t) H

B03Myn:r;eHHJI

dW

(t).

06CYP;HM

xapaKTep

p;BIImeHHH

ynpaB.TIJIeMbIX

cHcTeM,

onHCbIBae-

MbIX

ypaBHeHHJIMII

B

<popMe

(6.11). Cp;e.TIaeM aTO

Ha

npHMepe

ypaBHe-

HHJI

dx = A (t)

.1J·dt

+ b (t)

dU

(t),

(6.12)

rp;e

U (t) -

CKaJIJIpHaJI

<PYHKD;IIJI

II,

C.TIep;OBaTe.TIbHO, b

(t)

-

n-MepHbIH

BeKTOp.

PaccMoTpHM

ynpaB.TIeHIIe

'IaCTHOro

BHp;a.

IIYCTb

<PYHKD;HJI U

(t)

HerrpepbIBHa

rrpH

to

<;

t <;

iJ

II

rrpII

iJ

< t <;

tOl'

II,

KpOMe

'1'01'0,

npH-

MeM,

'ITO

BHyTpH

YKa3aHHbIX

IIHTepBa.TIOB

OHa

HenpepbIBHO

p;II<p<pepeH-

D;HpyeMa.

IIPH

t =

f}

rrpIIMeM

U

(iJ

+

0)

- U

(iJ)

= 1.

Torp;a

P;.TIJI

t E [to,

iJ)

H.TIII P;JIJI t E

(iJ,

tOl]

HMeeM:

n

xd

t

+

~t)

-

xdt)

=

~

[Xij (t +

M,

to)

-

xij

(t, to)]

x~

+

j=l

t+t.t

(6.13)

+

~

h(i)

[t

+

M,

T]

dU

(,t)

+

t

+

~

{h(i)

[t

+

~t,

Tj

-

h(i)

It,

T]}

dU

(T)

t,

n ,

(h(i)

[t,

T]

=

~

Xij (t,

T)

b

j

(T),

i = 1,

...

,

n;

M

>0).

;=1

Pa3p;e.TIIIB

paBeHCTBO

(6.13)

Ha

~t

11:

nepeXop;JI

K

npep;e.TIy

npH

~t

-+-0,

rrO.TIy'IHM

C

y'IeTOM

CBOHCTB

<pYHp;aMeHTa.TIbHbIX

MaTpHD;,

KaK

H

BbIille

.

Ha

CTp.

39,

paBeHCTBO

dx .

dt

= A (t) x +b (t) u,

rp;e

u (t) = U (t).

TaKHM

06pa30M,

Ha

HHTepBa.TIe

p;H<p<pepeHD;HpyeMocTII

U

(.)

p;BHiKe-

HHe

x (t)

orrHCbIBUeTCJI

CHOBa

p;H<p<pepeHD;Ha.TIbBbIM

ypaBHeBIIeM

(5.1).

.,.:~ ~

..

~.'

..

::

..

,

'.',"

'.

..:

,::.':.;,

':::~'zit+W"iM

.,,','

"...

::'..

.

...

.........

'.

,":

:~~,.~.:;.

C

'~

...

'.,

..

'.

;)

48

CBOfiCTBA

JIHHEfiHbIX

C11CTEM

[rJI.

~

TIyCTb,

O,lwaRO,

t =

'it.

Tor.u;a,

HCXO.u;H

orrHTb

11a

paBeHcTBa

(6.13)

H

omlpaHCb

Ha

CBOHCTBO

(6.10)

HHTerpaJIa

CTIiJITbeca,

rrOJIY'lHM:

'Xi

('it

+

0)

-

Xi

(tt) = b

i

('it).

CJIe.u;OBaTeJIbHO,

rrpH

rrpoxom.u;eHHH

qepea

TOqRY

t = tt

.u;BHmym;aH-

CH

TOqKa

X (t)·

COBepIDaeT

CRaqOR,

rrpHqeM

BeJIH'IHHbI

Xi

(t)rrpeTep-

rreBaIOT

paapbIBbI

LlXi

= b

i

(tt).

AHaJIOrHqHOe

paccym.u;eHHe

MomHO

rrpHBeCTH

H

.u;JIH

r-MepHoro

BeRTopHoro

yrrpaBJIeHHH

dU

(t) H

.u;JIH

B03Mym;eHHH

dW

(t).

B COOTBeTCTBHH

CO

CRa3aHHbIM

.u;BHmeHHe

X (t),

aa.u;aBaeMoe

pa-

BeHCTBaMH

(6.5),

ORaabIBaeTCHpaapbIBHbIM,

eCJIH «PYHKD;HH U

j

(t)

H

WI,

(t)

TeprrHT

paapbIBLI

B

RaRHX-JIHOO

TOqRaX

t =

'it.

MeXaHHqeC-

Roe

HCTOJIROBaHHe

~noro

OOCTOHTeJIbCTBa oy.u;eT .u;aHO B CJIe.u;YIOm;eM

rraparpa«pe

rrpH

paCCMOTpeHHH

nOHHTHH

HMnyJIbCHOH

uepexo.u;HoH

«PYHRD;HH

JIImeHHOH

CHCTeMbI.

CeHqaC

yCJIOBHMCH

JIHIDb

0 CJIe.u;yIO-

m;eM.

ECJIH

He

c.u;eJIaHO

HHRaRHX

orOBopOR,

TO

«PYHRD;HH Uj (t),

W

k

(t)

npe.u;nOJIaraIOTCH

RyCOqHO-HerrpephIBHbIMH

H RYCOqHO-.u;H«P-

«pepeHn;HpyeMbIMH

Ha

OTpe3Re

to

<, t <,

to>,

a B

TOqRaX

paapbIBa,

ec-

JIH

OHH

eCTb,

,aTH «PYHRD;HH

rrpe.u;nOJIaraIOTCH

HenpepbIBHbIMH

CAeoa.

Tor.u;a

.u;BHmeHHe

X

(t)

(6.6)

oRa3bIBaeTcH

TaRme

1£enpepU01£bZM

CAeoa.

§ 7.

IlMnyJIb6HaJl

nepeXO)l;HaH

4>yHRD;iiH

OO'l>eKTa

BBe.u;eM

BamHoe

.u;JIH

TeopHH

JIHHeHHbIX

yrrpaBJIHeMbIX

ClICTeM

no-

HHTHe

HMnyJIbCHOH

nepeXO,lJ;HOH «pyHKD;HH.

TIYCTb

ynpaBJIHeMhIH

OO'l>eRT

onHCbIBaeTCH

BeRTopHhIM

AH«P«Pe-

peHD;HaJIbHhIM

ypaBHeHlIeM

1;

= A (t)x + b (t)1'),

(7.1)

r.u;e,

RaR

OObIqHO, X - n-MepHbIH

BeRTOp,

A(t)

-

HenpepbIBHaH

MaTpllD;a

pa3MepHOCTII

n X n,

1')

-

CRaJIHpHaH

nepeMeHHaH,

b

(t)

-

HenpephIBHhIH

n-MepHbIH

BeRTOp.

0003HaqllM

ClIMBOJIOM

1')£

(t) «PYHR-

D;lIIO,

onpe.u;eJIeHHYIO

paBeHCTBaMlI

1')£

(t) = 0

rrpll

t <0 1I t

'>

e,

} (7.2)

'YJ£(t)=1/e

npHO<,t<e,

r.u;e e -

HeROTopoe

nOJIOmllTeJIbHOe

qHCJIO

(pllC.

7.1).

PaccMoTpHM

ypaBHeHlIe

(7.1)

npH

1')

=

1')£

(t

-

'it)

('it

> 0).

81'0

paBHOClIJIbHO

npe.u;nOJIOrKeHlIIO,

qTO

B03.u;eHcTBlIe

1')

ORa3hIBaeT

BJIlIH-

HHe

Ha

OO'l>eRT

JIlIIDb

B

TeqeHlIe

BpeMeHlI'it

<,

t <

'it

+

e.

TIPll

aTOM

He3aBlIClIMO

01'

'it

1I e

lIMnYJIbC

p

()Toro

B03.u;eHCTBlIH

paBeH

e.u;lIHlID;e:

ll+£

p

[rid

=

~

1')£

(t -

'it)

dt = 1.

':',

,

f

'

.•

::.'~,)

C

I

49

l'

§

7]

lIMIIYJIbCHAlI

IIEPEXO,IJ;HAII

lI>YHln~HlI

OB'bEKTA

j'

Ilpe.u;noJIomHM

TaRme,

qTO

.u;o

MOMeHTa

BpeMeHH

t =

it

ClICTeMa

\

(7.1)

Haxo.u;lIJIaCb

B

COCTOlIHlIH

paBHoBeclIH

X =

0,

H

0603HaqllM

ABJ(-

I

meHlIe

X (t),

nopomp;eHHoe

B03AeHcTBueM

1')

= 'le (t - tt), CHMBOJIOM

t

x<e) (t, tt). .

f

COrJlaCHO

«pOpMyJl6

ROIDU

(5.6)

lIMeeM:

x(e)

(t,

it)

= 0

npll

t <tt,

t

rrpll

'it

<,"t <

tt

+

e,

x(£)

(t,

tt) =

~ ~

X

[t,

or] b (or)

dor

(7.3)

II

ll+£

x(£)

(t,

'it)

= e

1

.)

\'

X

[t,

or] b (or)

dor

npll

t

'>

it

+

e.

&

IloTpeoyeM

Tenepb,

qTOObI

e

cTpeMlIJIOCb

R

HyJIIO.

TorAa

BeRTop-«pyHR-

'le{t}

I

~.

h(i:/I/eJ

t

tJ.

e

t 0

o

Pac.

7.2.

Pile. 7.1.

D;lIH

x<£)

(t,

'it)

(7.3)

npll

Ram.u;OM t oy.u;eT

CXOAlITbCH

R

HeRoTopoH

BeR-

TOp-«pyHRD;lIH

h

[t,

'it],

npllqeM

h

[t,

'it]

= 0

npll

t<'it,

} (7.4)

h

It,

'it]

= X

[t,

'I't]

b

(\)0)

l1pll

t >

\)0.

}\OMnOHeHTbI

h(i)

[t,

\)0]

«PYHRD;lIl1 h

[t,

\)0]

aMeIOT,

CJIe.u;OBaTeJIioHO,

BlI.u;

(pllC.

7.2)

l1pll

t <,

\)0,

h(i)

[t,

\)oj

= 0

n

h t [t,

0]

=

LJ

Xij

(t,

\)0)

b

j

(\)0)

IIpll

t

>\)0,

(')

'"

j=l

rp;e

xii

(t,

'it)

_

aJIeMeHTbI

«pYHp;aMeHTaJIbHOH

MaTpllD;LI

X

[t,

'it]

OP;HOpO,ll,HOH CHCTeMhI .i = A

z,

cooTBeTcTBYl(Ill~eH

ClICTeMe

(7.1).

. C

p;pyrQH

CTOpOHhI,

COrJIaCHO

paBeHC'IBy

(5.4)

BeRTop-«pYHRD;HH

h

[t,

\)0]

onllCbIBaeT

p;BlImeHlIe

OO'l>eRTa (5.2),

ROTOpbIH

AO

MoMeHTa

•

II

'............. ;

..

...:.

..

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

"

..

:

..

c·

':',':

',:;';

C

50

CBOJ/fCTBA

JIHHEfIHb1X

CHCTEM

[rJI.

2

§

7]

nMn3'JIbCHM1

IJEPEXOAHMI

<l>,,'HKI~nH

OB'bEHTA

51

to

-,-

1'l'

HaXO,o;HJICH B COCTOHHHH paBHOBeCHH Z = 0, a B MOMeHT

to

= 1'l'+ 0

cpasy

OKa3aJICH B COCTOHHHH Z

(to)

= b

(to)

= b

(1'l')

H ,o;aJIb-

me

Ha'laJI

,o;BHraTbCH

Ii

COOTBeTCTBHH C

O,o;HOpO,o;HJiIM

ypaBHeHHeM

(5.2).

BbISBaTb

TaKoe

,o;BHmeHHe x

(t)

= h [t, {}] Y CHCTeMbI (7.1)

C nOMOm;bIO Bos,o;encTBHH

1']

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-

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6,

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t

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t>1'l'.

to

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[1']£]

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B COOTBeTCTBHH C 3THM

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x (i) =

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-

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a

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t

r t

X (t) = X

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7.1.

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r),

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6

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3):

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(t)

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Tpe60BaHHHM

3ap;alJH

4.1.

CJIep;OBaTeJIbHO,

MomHO

omlIp;aTb,

lJTO B

06m;eM

CJIylJae

3ap;alJa

4.1

HMeeT

He

eAHHCTBeHHoe

pe-

meHHe.

B

3TOM

rraparpa<pe

rrpIIBop;HTCH

3BpHCTHlJeCKHe

co06pameHHH,

yKa-

3bIBaIOm;lIe

rrYTb

P;JIH

orrpep;eJIeHlIH

op;Horo

H3

pemeHHH

U

(t)

ypaB-

HeHlIH

(8.5). C

aTOH

D;eJIbIO

corrOCTaBIIM

ypaBHeHHHM

(8.5)

ypaBHeHHH,

aHaJIOrlIlJHbIe

IIM

B

H3BeCTHOM

CMbICJIe,

HO

COCTaBJIeHHbIe

He

P;JIH

HC-

KOMOH

<PYHKD;HII U (T:)

(ta.

<

T:<

t(3),

a

AJIH

HeKOToporo

HeH3BeCTHoro

m-MepHoro

BeKTopa

v =

{VI'"''

v"J.

,D;JIH

rrOJIylJeHHH

3THX

ypaBHe-

HHH

pa306beM

OTpe30K

[ta.,

t(31

Ha

m

lJaCTeH

TOlJKaMlI

ta.

=

T:o

<

...

<

< T:j <

'tj+1

<

...

< 1:

m

=

t(3.

CocTaBlIM

BeKTopbI

g(l)

=

{gel)

gel)}

g(n) - {g(n) g(n)}

rITe

g(i)=h(i)

[to

....

]

th.

1 , . •

OJ

,

...

, -

1.'"

0'

m '

f..J"

j p'

foIJ

Jm

(j=1~

...

,m;

i=1,

...

,n),

dt:j='t'j-t:j_l.

MHTerpaJIbI

~

m

~

cyMMaMH

2}

g~i)Vj'

h(i)

[t{3,.

t:]

u

(t:)

dt:

3aMeHHM

rp;e,

CJJ.ep;oBaTeJIb-

ta.

j=l

HO,

BeJIH'IHHbI

Vj

3aMeHHIOT

3HaqeHHH

u ('t'j).

f[OCKOJIbKY

npH

60JJb-

mHX

m

HHTerpaJJbI

Ifl

~

h(i)[t

ll

, t:] u(t:)

dt:

ta.

npH6JIIImeHHO

rrpep;CTaBJIJIIOTCH

cyMMaMH

m

~

h(i)

[tp,

t:j]

dt:jU

(t:o)

=

2}

gJi)v;,

;=1 ;=1

MOmBO

npep;rrOJIaraTb,

'ITO

CB

OHCTBa

COCTaBJIeJ;lHbIX

HaMH

BCrrOMora-

TeJIbHbIX

ypaBHeHHll

6YAyT

HMeTb

MHOrO

06rn;ero

CO

CBOHCTBaMH

,"."

__

._._._,

..

,

.:

...

,.

....

.....

_~~,-,-"""",,-,--,-~,-,-",-,-

• .:...'

~.~""

0..-'-"_'

""'.

~-,---,,,.:...,c,-,-,

C'·.

·····

..

'

(·~':.'~i

c

"

.:<:

..

:.

..;',.,.,

......

:':"4'

56

IIPEJJ:BAPIlTEJIhHOE

PEmEHIlE

SAJJ:Aqll

0]3

YIIPABJIEHIlIl

[rJl.

3

IICXOP;HbIX

ypaBHeHIIH

(8.5).

IIo<lToMY

llaYlleHlIe

BcrrOMOraTeJIhHl>IX

ypaBHeHIIH

AOJImHO

nop;CKaaaTh

nYTh

AJIH

pemeHIlH

ypaBHeHIIH

(8.5)

II

P;JIH

HCCJIeAOBRHlIH

nOJlyqeHHbIx

pemeHIIH.

MTaK,

paCCMOTpHM

BcnOMOraTeJIhHl>le

ypaBHeHlIH

m m

II

g}1)V

j

= C

l

,

••

'.'

II

gt)V

i

=

C

'

n

i=l

i~l

B

p;aJIhHeHmeM

P;JIH

HarJIHp;HOCTII

npllMeM,

liTO

n =

HeHIIH

(8.6)

aanllmyTcH

TalC

g~l}Vl

+ +

g~)Vm

=

Cl,

}

gi2)Vl

+ +

g~)Vm

=

C2'

ECJIIl

m-:MepHbIe

BeKTopbI

gel)

II

g(2)

JIIlHeHHO

aaBIICIIMbI,

T.

e.

eCJIII

cym;eCTBYIOT

1IlICJIa

II

12'

P;JIH

KOTOphIX

llg(l)

+

12g(2)

=

0

nplI

l~

+

l~

=f=

0,'

TO

MomHO

nop;oopaTh

([3*

1,

CTp.

50-52)

TaKlIe

1IlICJIa

C

l

Il

C

2

,

'ITO

ypaBHeHIIH

(8.7) OYP;YT

HepaapemlIMbl

(Hap;o

JIlImh

P;JIH <lTOH :u;eJIIl

BaHTh

Ci

TaKIIMIl,

lITOObI

IlC

l

+

12

c

2

=f=

0).

IIo<lToMY

oyp;eM

npep;IIOJIaraTI:"

liTO

BeKTopbl

g(i)

,/f,uneuno

neaaeucu-

MbL

II,

CJIep;OBaTeJIhHO,

B

'1aCTHOCTII,

HII

Op;IIH

lIa

<lTIIX

BeKTopOB

He

paBeH

HyJIIO.

PaBeHcTBa

(8.7)

p;onycKaIOT

nOJIesHyIO

reoMeTplIlIecKyIO

HHTep-

npeTa:U;IIIO.

JIeBbIe

lIaCTII

paBeHCTB

(8.7)

IIMeIOT

CMbICJI

CKaJIHpHblX

npmtaBeAeHIIH

BeKTopOB

g(i)

Ha

BeKTOp

v,

T.

e.

ypaBHeHIIH

(8.6)

Mom-

HO

aanlIcaTh

TaK:

g(i)'

v =

Ci

(i

= 1,2). (8.8)

BeJIH'IHHbI

ai

=

(g(i)'

v)/

~

g(i)

II,

rAe

~

g(i)

II

-

MOp;yJIb

BeKTOpa

g(i),

HBJIHIOTCH

npoeK:U;IIHMIl

IlCKOMOro

BeKTopa

v

Ha

HanpaBJIeHIIH

llaBeCT-

HbIX

BeKTopoB

g(i).

CJIep;OBaTeJIhHO,

ypaBHeHlIH

(8.8)

MomHO

aamlCaTb

B

BIIp;e

IIpg(i)

v =

ai

(i

= 1,2).

TaKIIM

oopaaoM,

sap;alJa

COCTOIIT

B

CJIep;YIOm;eM:

TpeoyeTcH

HaHTIl

BeI{TOp

v,

npoeK:U;IIIIKoToporo

Ha

sap;aHHble

BeKTopbl

gel)

H

g(2)

P;OJIIRHbI

ObITb

paBHhl

sap;aHHbIM

qlICJIaM

a

l

If

a

2

(CM.

pllC.

8.1,

rp;e

P;JIH

HarJIHp;-

HOCT!.'I

BbIopaHo

m = 3,

T.

e.

BeII:T0phl

g(l)

II

g(2)

npep;nOJIaraIOTCH

OObI'lHbIMIf

TpeXMepHbIMH

OqeBHp;HO,

YKasaHHbIM

v,

KOHe:u;

KOToporo

JIeIRIlT

lJeHHeM

nJIOCKOCTeH

:Ttl H

g(2)

H

npOXOp;Hm;HX

lIepes

pbl

yCJIOBHHM

aap;R'lH

He

BJIeTBOpHIOm;HX

yCJIOBHHM

iHam;HH

B

nJIOCKOCTII

BeKTopOB

gel)

Il

g(2).

8TOT

BeKTOp

VO

HBJIHeTCH

BeKTopaMH).

yCJIOBHHM

yp;OBJIeTBOpHeT

JIIOOOH

BeKTOp

Ha

npHMoH

I -

I,

HBJIHIOm;eHCH

nepece-

:Tt

2

,

nepneHp;HKyJIHpHbIX

K

BeKTopaM

gel)

H

TOllKH

Ql

Il

Q2'

HHKaKHe

p;pyrHe

BeKTO-

yp;OBJIeTBOpHIOT.

Cpep;H

BeKTopOB

v,

yp;o-

a

ap;a'llI

,

MOIIUIO

Bbl)J;eJIHTb

BeKTOp

VO,

JIe~

(8.6)

2.

Torp;a

ypaB-

(8.7)

§

8J

8BPIlC'l'IFIECHHE

COO]3PAlI,EHIIfI

57

JIHHeHHOH

KOMoHHa:U;HeH

BeKTopOB

gel)

Il

g(2),

T.

e.

VO

=

g(l)ll

+

g(2)12'

(8.9)

,IJ;JIH

BbI'lllCJIeHlIH

1IlICeJI

II

H

12

nOACTaBIIM

BbIpaIReHHe

(8.9) P;JIH

VO

B

ypaBHeHHe

(8.8).

IIoJIyqHM

CHCTeMyp;Byx

JIHHeHHbIX

ypaBHeHHH

P;JIH

p;ByX

HeHSBeCTHbIX

II

Il

12:

<xnll

+

<x

12

1

2

= c

l

,

~lll

+

~212

= C

2

,

rp;e

<Xii

=

g(i)'g(i).

Onpep;eJIHTeJIh

;noH

CHCTeMbJ

He

paBeH

HyJIIO,

eCJIIl

BeKTopbI

gel)

Il

g(2)

JIIlHeHHO

HeaaBHCIlMbI(qTO

MbI

II

npep;nOJIOIRIIJIIl

Bblme),

HOO

ll

12

!

<X

= <Xll<X22 -

<X12<X21

=

<X21<X22

l

~

II

g(l}

11

2

•

II

g(2)

11

2

-II

g(l)

11

2

.

II

g(2)

11

2

Cos

2

'I\'

=

II

g(l}

11

2

•

II

g(2)

11

2

sin

2

'1\',

rp;e

'"

-

yroJI

Memp;y

BeKTopaMIl

gel)

H

g(2).

.

PaapemaH

nOJIyqeHayIO

cHcTeMy

ypaBHeHHH,

onpep;eJIHM

IlCKO-

MbIH

BeKTOp

Vo.

8TH

paccyIRp;eHlIH,

OTHOCHm;HeCH

K

onpep;eJIeHHIO

BeKTopa

VO,

OqeBHp;HO,

cnpaBep;JIHBbI

He

TOJIhKO

B

paCCMOTpeHHOM

HarJIHp;-

HOM

CJIyllae

n = 2, m = 3,

HO

II

npH

JIIOObIX

n H m

:>

n.

lIa

pHC.

8.1

CJIep;yeT,

liTO

BeKTOp

VO

OOJIap;aeT

OP;HHM

aaMellaTeJIhHbIM

CBOHCTBOM.

lIMeHHo,

KaKOB

Ohl

HlI

ObIJI

p;pyroH

BeKTOp

v,

yp;OBJIeTBO-

pHIOm;HH

yCJIOBHHM

aap;allH,

cnpa-

Bep;JIHBO

HepaBeHcTBo

~

VO

~

<

II

v

II·

(8.10)

B

caMOM

p;eJIe,

COrJIaCHO

pHC.

8.1

HMeeM

v =

pO

+p.

Ho

BeKTOp

p

op-

TOrOHaJIeH

K

BeKTopy

vo.

CJIep;OBa-

TeJIhHO,

B

npHMoyrOJIhHOM

TpeyrOJIh-

HHKe,

COCTaBJIeHHOM

BeKTopaMIl

VO,

v, p,

BeKTOp

v

HBJIHeTCH

rHnoTeHyaoH,

OTKYP;R H

CJIep;yeT

cnpaBep;-

JIIIBOCTh

HepaBeHCTBa

(8.10).

JIraK,

MbI

nOJIyqHJIIlp;OCTaTOllHO

HarJIRp;HOe

npep;CTaBJIeHlIe

0

CBOH-

CTBax

pemeHlIH

BcnOMOraTeJIhHoH·

aap;allH.

PaCCMOTpeHHaR

reOMeTpH-

'IecKaR

IlHTepnpeTa:U;HR

nop;CKa3bIBaeT

HaM

onpep;eJIeHHhle

npe,ll;nOJIO-

IReHHR

0

CBOHCTBax

pemeHlIH

u (t)

IlCXO,Il;Hoi

npOOJIeMbJ.

8TH

CBOH-

CTBa

MbI

BbIBep;eM

B CJIep;yIOr:u;eM

naparpa4Je.

I

II

r;;-

,Ji

z

I

.J

~

fill

Pile. 8.1.

,';,

n

C:I

58

UPE)l;BAPHTEJIbHOE

PEIDEHUE

3A)l;AqM

OB

YllPABJIEHUJI

ern.

:l

§ 9.

PemeHHe

3a~a'lH

BOCllOJIb3yeMcH

3p;eCb

1l0HHTHHMl1,

IWTopb~e

P;JIH

<PYHK~l1H,

3a-

p;aHHbIX

HB.

oTpe3Ke

t

..

< 't <

t~,

COOTBeTcTByIOT

CKaJIHpHoMy

llPO-

113Bep;eHl1IO, MOP;YJIIO

11

CBOHCTBy

opTOrOHaJIbHOCTl1

KOHe'lHOMepHbIX

. m

BeKTopOB.

CKaJIHpHoMy

llp0113Bef(eHl1IO

p;ByX

BeKTopOB

a·b =

~

a,b,

i=l

COllOCTaBJIHeTCH

cr;,a.M£pnoe

npoua8e8enue

(<X,

~)

p;ByX

<PYHK~HH

<X

(-t)

11

~

('t)

(t~

< 't <

t~),

Ollpep;eJIHeMOe

paBeHCTBOM

tfl

(<X,

~)

=

~

<X

(-r)

~

(-r)

a-r.

i~

BeJIl1'll1Ha

ifl

'/

2

P

[<X]

=

[~

<X

(-r)

a-rJ

"

t

oX

COOTBeTcTByIOm;aH

llOHHTl1IO

MOp;yJIH

m-MepHoro

BeKTopa

<X,

Ha3hI-

BaeTCH

nopMOU

1)

<PYHK~l1l1

<X

('t). ,IJ;Be

<PYHK~l1l1

<X

('t)

11

~

('t)

(to:

<'t <

<

t~)

Ha3bIBaIOTCH

opTOrOHaJIbHhIMH,

eCJIl1

HX

CKaJIHpHOe

llpOH3Be-

p;eHHe

(<X,

~)

paBHO

HyJIIO,

T.

e.

eCJIH

BhIllOJIHHeTCH

paBeHCTBO

ifl

~

0:

(-r)

~

(-r)

a-r = 0.

to:

He

onpep;eJIHH

3p;eCb

Ha1160JIee

ml1pOKHH

KJIaCC

<PYHK~l1H

<X

('t),

~

('t),

P;JIH

KOTOp:bIX

~eJIeC006pa3Ho

paCCMOTpeHl1e

YKa3aHHhIX

nOHHTl1H,

OrpaHH'll1MCH

B

3TOH

rJIaBe

KycO'lHO-HellpephIBHbIMl1

<PYHK~l1HMH.

06cymp;eHl1e

BcnOMOraTeJIbHOH

3

ap;a'll1

,

npl1Bep;eHHoe

B § 8,

nop;-

CKa3bIBaeT

CJIep;YIOm;yIO

rl1nOTe3Y

0

pemeHllHX

3ap;a'lH

4.1.

r H

nOT

e 3

a.

IIycmb

rjiynr;,/tuu

h(I)[t~,

'tl, ... , h(n)

[t~,

't] (8.2)

.l/,uneuno

neaa8UCUMbL,

m.

e.

h

(-r)

= h(l} [tfl,

-r]lI

+... +

h(n)

[tFl,

't']ln

=f=

° (9.1)

npu

to:

< 't <

tFl,

li+

... +

l~

=F

0.

Toe8a aa8alia 4.1

UMeem

peute-

nue u (t). Peutenue aa8aliu 4.1 ne

.f/,8.I/,umc.f/,

e8unCm8ennbLM.

Cpe8u

peutenuu u (t)

MOJICnO

8bL6pamb

ynpa8.l/,enue

UO

(t),

.f/,8.1/,.f/,lOUfeeCR

.l/,uneu-

nou

r;,oM6ulia/tueu rjiynr;,/tuu h(i), m.

e.

UO

(t) =

h(ll

[t

Fl

,

t]l~

+... +

h(n)

(tFl'

t]l~.

(9.2)

Ynpa8.l/,enue

UO

(t)

UMeem

naUMenbutylO

e8r;,.I/,u808Y-

IiOpMy P

[UO

(t)]

1)

llT06hI

oTml'IRTb

BeJIHQHHY

p[a]

01'

~pynl:X

BH~OB

HOpMhI,

KOTophIe

BCTpe-

TllTCll

HHme,

6y~eM

HR3hIBRTb

ee

HHor~R

ee"ir,uooeoi!.

(o606II:\eHHoii)

HOp",tOU.

,.'.

."

;.

.'

: : : :.

.,:':"

-::.

"",',:,

,,',.':,

,.'",,..<

.

.',

·:

..

··,:::·,:···'.:'.~~,,'::':,"t',/,'./~.'4·

,

,:,~~.~.

·c·

..

59

PEIDEHl1E

3A):J,A

"IU

§ 9]

cpe8u ecex peuteliuu u (t) aa8aliu 4.1.

Ec.l/,u

u (t) -

.l/,lo6oe

peutenue

aa8aliu, mo paaliocmb'i' (t)

==

u (t) - U

O

(t)

opmoeon~bHa

r;,

ynpa8.l/,enUlo

. 0 0

U

O

(t).

fJuc.l/,a

lI,""

In,

onpe8eAAlOUfue

peuteliue

UO

(t),

y8ooMm80p.f/,lOm

cucme~te

.l/,UlieUlibLX

ypaonenuu.

IIpoBepl1M

cnpaBep;JIHBOCTb

3TOH

rHnOTe3bI.

Byp;eM

l1CKaTb

ynpa-

BJIeHHe

UO

(t)

B

<popMe

(9.2).

IIo):J,cTaBl1B

BhIpameHHe

(9.2) B?lfeCTO u ('t)

B

ypaBHeHllH

(8.5) H

BhInOJIHHB

HHTerpHpOBaHl1e,

nOJIy'll1M

cHcTeMy

u

lO

ypaBHeHHH

P;JIH

onpep;eJIeHHH

BeJIH'll1H

1,·

..

,

n:

•

<Xlll~

+ ... +

<Xlnl~

=

CI,

I

. . . . . . . . . . . . • (9.3)

<XnIl~

+ ... +

<Xnnl~

=

en,

rp;e

K03<p<pH~l1eHThI

<Xij

onpep;eJIeHhI

paBeHCTBaMl1

ifl

<Xi}

= \ h(i)

[t

,

-r]

. h(j) [tfl,

t]

a-r

(i,

j = 1,

...

,

n).

(9.4)

Fl

to:

IIoKameM,

'ITO

onpep;eJIl1TeJIb

!J.

{<Xij}

Cl1CTeMbI

(9.3)

He

paBeH

HyJIIO.

,IJ;eHcTBHTeJIhHO,

113

(9.1)

CJIep;yeT

HepaBeHcTBo

tf!

tFl

~

h

2

(-r)

a-r =

~

{h(l)

[tfl'

-r]lI

+... +

h(n)

[t

Fl

,

-r]ln}2

a-r

=

t«

t~

n

IIpH

~

l7=F

0.

_

~

<Xilli

l

) >°

i=1

.i,

j=I

CJIep;OBaTeJIbHO,

npH

Haml1X

npep;llOJIOmeHl1HX

KBap;paTH'lHaH

<popMa

n

<D(lx,

.

..

,In)=

~

<Xi)li

l

)

i,

j=I

OT

nepeMeHHhIx

l i

HBJIHeTCH

onpep;eJIeHHO

llOJIOml1TeJIbHOH,

T.

e.

OHa

MomeT

npHHl1MaTb

JIHmb

llOJIOmHTeJIbHbIe

3Ha'leHl1H,

Korp;ali

+

...

.,.

+

l~

=F

0.

1I3BecTHo,

op;HaKO,

'ITO

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Красовский Н.Н. Теория управления движением

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