Красовский Н.Н. Теория управления движением

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60

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M

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06

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(9.3).

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cMcTeMy

(9.3)

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paapemMTh,

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6M

HM

6MJIM

lJHCJIa

Ci

(i

=

1'00"

71,).

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caMblM

yCTRHaBJIMBaeTCH

cym,ecTBoBa-

HMe

ynpaBJIeHHH

UO

(t) B

BMAe

(9.2) M

onpeAeJIHeTCH

ero

BeJIMlJMHa.

HOKameM

Tenepb,

'ITO

pa3HOCTb

1jJ

(t) = U (t) -

UO

(t),

MemAY

JIIO-

6bIM

pemeHMeM

u (t)

aaAaqM

4.1

II

HaHAeHHbIM

ynpaBJIeHHeM

UO

(t)

opTOrOHaJIbHaK

!f>YHKD;HH

UO

(t).

¥paBHeHHH

(8.5)

YAOBJIeTBOpHIOTCH

npH

U = U (t)

II

npM

U =

UO

(t).

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cnpaBeAJIHBM

paBeHCTBa

~ ~

.

~

h(i)

[til'

'1']

[u

('1')

-

UO

('1')]

d'1'

= \ h(i)

[t/3'

'1']

'I\J

('1')

d'1'

=

O.

(9.5)

tat

roc

¥MHomaH

paBeHCTBa

(9.5)

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Ha

l~,

CKJIaAMBaH

nOJIyqeHHhle

COOTHomeHHH

M

yqMTMBaH

paBeHCTBO

(9.2),

nOJIyqHM:

t/3

~

uO('1')1jJ('1')d'1'=O, (9.6)

t

oc

'lTO

M

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UO

(t) M'I\J (t).

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BMAeTb,

qTO

H,

Ha060poT,

BCHKaH

!f>YHKD;MH U

(t)

=

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(t) +

1]

(t),

rAe

UO

(t) -

ynpaBJIeHHe

(9.2), a

1],

(t) -

npoHa:'

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K

h(i),·

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4.1.

AJIH

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B

aTOM,

AO-

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npeAhTAym,He

paccYfKAeHlfH

B

06paTHoM

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KaK

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nocTpoHTb

6eCqllCJleHHoe

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~MH

1]

(t),

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K h(i),

aaKJlIOlJaeM"

qTO

pemeHHe

aaAaqH

4.1

He

HBJlHeTCH

eAIIHCTBeUHbIM.

Tenepb

AJIH

nOJlHOrO

nOATBepmAeHMH

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uO"

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no.

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U (t)

=1=

UO(t),

TorAa

t/3

~

'1>2

('1')

d'1'

>

O.

(9.7)

tat

¥'IDTMBaH,

'ITO

u(t)

=

UO

(t)

+'I\J

(t),

HMeeM:

t/3

tjJ

U

11

2

=)at

[u

(l')]2dl' =

~at

IUD

(l') +

'I>

(l') Pdl' =

tjJ

t/3

tjJ

=

~

[uO('1')]2d'1'+ \

["'(l')]2d'1'+2

~

uO('1')"'('1')d'1'. (9.8)

tlJ

rlJ

t.

C

',',

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9]

PEWEHME

3A,n;A'IM

61

IIOCJleAHee

CJlaraeMOe

B

npaBoH

qaCTM

paBeHCTBa

(9.8)

BCJleACTBMe

(9.6)

eCTh

Hynh,

BTopoe

CJlaraeMOe

BCJleACTBMe

(9.7)

llOJlOm:rreJlbI:IO.

CJleAOBaTeJlhHO,

Ma (9.8)

BhlBOAMTCH

HepaBeHCTBO

II

u"

2 >

II

UO

II

2 lIJ1M

II

u

II

>"

UO

II·

TaKMM

06paaoM,

cnpaBeAJlMBOCTb

rMnOTe3bl

nOJlHOCTbIO

npoBepeHa,

CJleAOBaTeJlhHO,

BepHa

TeopeMa.

Teo

p

eM

a 9.1.

Ec.f/,U

UMnY.f/,bCUb~e

nepexooub~e

(/iyun/1uu 06'benma

h(l)

[t/3'

t l,

...

, h(n)

[t/3'

t]

.f/,uueUno

neaaeucU:Mb~,

mo aaoalla 4.1 06

ynpa8.l/,enuu

UMeem

6ecllucMnnoe

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peuunuu.

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(t),

,R,8.I/,,R,lOUfeec,R,

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rt~,

t]

(i

= 1,

..

.

,71,),

m.

e.

ynpa8.l/,enue

. 0

euoa (9.2),

coe

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li

onpeOe.f/,,R,lOmC,R,

ua

a.f/,ce6paullecnou

cu-

cme:Mb~

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neoonopoonbLx ypaenenuu. .

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ecex

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(t)

aaoallU (4.1) y.npaeMnue

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(t)

UMeem

nau.MenbutylO

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noPMY

II

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n (/iynn/1uu

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nu

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UO

(t)

H3

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{(Xij}

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(9.3).

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(9.3) B

MaTpH'IHOH

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I

Q.lo

= c,

I

rAe

lO

-

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{l~,

...

,

l~}.

CJIeAOBaT8J1bHO,

-

to

=

Q-l.C.

(9.9)

t

'PaBeHCTBO

(9.2) C

YlleTOM

(9.9)

MomHO

3anHcaTb

B

!f>opMe

I

UO

(t)=

h'

[til,

t]·lo

=

h'

[til,

t]

Q-l.C,

rAe

h [til,

t]

-

BeKTop

{h(l) [til, t l,

...

, h(n) [til, t l}.

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aaMeHHR

BeKTop

c

ero

BMpameHMeM

(8.4),

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(t)

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-

h'

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t]

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tat]

x

at

_

~

X

[t/3.

l']

W

('1')

dl'}'

l1a

paBeHCTBa

(9.10)

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Oln

npaeeb~X

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aaoa'lu x

at

-6U,R,

w

(t)

.f/,uneiino,

tat

(9.10)

uaHAeHHoe

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(t)

CB

OHCTB OM: ynpa8.l/,euue

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(t)

u

xfJ

u om

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3

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em;e

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TaJIbHyro

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X

It,

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t{3

UO

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(t)

.

X'

[t{3,

t]

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[t{3,

tocl

x<x

-

~

X

[t{3,

1'1

W

(1')

d1'}.

t<x

(9.11)

§ 10.

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cxeMa,

MOAeJlHpYIOm:aJl

yrrpaBJIeHHe

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CHalJ:aJIa

em;e

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UO

(t).

IIpeffip;e

Bcero,

HanOMHMM

CJIep;yrom;ee

paBeHCTBO:

X

It,

tol = X

It,

t.,l.

X-I

[to,

tal,

ROTopoe

nOJIYlJ:aeTCJI

liS

06m;ero

paBeHCTBa

(5.3) B lJ:aCTHOM

CJIyqae

Z (t) = X

It,

t<Xl.

3aMeltHH

Tenepb

t

Ha

t{3,

a

to

Ha

t,

nOJIylJ:HM:

X

[t{3,

t]

= X

[tp.,

t<X]

•

X-I

It,

t<xl.

YlJ:HThIBaH

nOCJIep;Hee

paBeHCl'BO,

BblBO,IJ;l'lM

HS

(9.11),

"ITO

UO

(t) =

b'

(t) rX-l [t, t<Xll'so,

(10.t)

rp;e

n-MepHbIH

nOCl'OHHHbIH

Bel\l'0p

SO

= {SI'

...

, sn}

onpep;eJIeH

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H0

III

e

HlIeM

§ 10]

CXEMA,

MO,ll;EJIHPYIOmMI

YTIPABJIEHHE

63

MaTpHIl;a

X

It,

t",]

yp;OBJIeTBOpHeT

ypaBHeHHIO

X

[t,t,.]

= A (t) X

It,

t",l

no

onpep;eJIeHHIO

ee

RaR

<pYHp;aMeHTaJIbHOH

MaTpMIl;bI.

IIOp;CTaBJIHH

X

HS

nOCJIep;HerO

paBeHCTBa

B

(to.4),

HaHp;eM:

AX

ft,

till

X-I

It,

tal +X

It,

tal

X-I

It,

tocl

= 0,

UO

(t)

= h' (t) s

(t)

(10.6)

BeRTopa

b'(t)

Ha

BeRTOp

s (t),

HBJIHIOm;HHCH

peIIIeUlIeM

'ypaBHeHHH

t{3

.

(10.5).

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lJ:TO

npH

t = t

Clt

cnpaBep;mIBO

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SO

=

X'

[t{3,

t<X]

Q-l

{X,9 - X

[t{3,

t",l

x<x

- \ X

[t{3,

1'1

W

(1')

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i

s(tClt)

=

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t",

t{3

I

r

,1l;aJIee,

MaTpHIl;a

S (t) =

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1

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CJlegyrorrre-

=

X'

[t{3,

t<X]

Q-l

[x{3

--

X

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tocl

x<x

-

:)

X

[t{3,

1']

W

(1')

d,;

J-

(10.7)

My

MaTpHlJ:HoMy

ypaBHeHHW:

t

oc

1

S (t) = -

A'S

(t).

(10.3)

.'

1

IbaR,

MbI

yCTaHOBHJIIl.

CJIep;yIOm;HH

peayJIbTaT.

,

B

r.aMOM

p;eJIe,

no

onpep;eJIeHHro

MarpHIl;bI

X

-·l

ft,

t",

1

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0

pa-

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ero

no

t H

nOJIbSYHCb

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([3* l,

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372)

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01'

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MaTpHIl;,

nOJIylJ:HM:

X

It,

tlJ]X-l

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t<ltl

+ X

It,

t.IX-

1

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=

O.

(10.4)

Teo

p

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a 10.1.

YnpaMenue

UO

(t),

1f,omopoe

peutaem

8aOaIJ,y

4.1 u

u.ueem

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amOM

nau.uenbutylO

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[uOl,1£8M/,em-

CI£

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(t).

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(10.5)

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10.1,

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(t)

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64

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6bI

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peanbHbIe

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cxeMy,

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(t), 6y,n:eM

Ha3bI-

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cucmeMOU

ynpa6.fleUU!L.

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ynporu;eHHH,

l.JTO

A - nOCTORHHaR MaTpHJJ;a.

RaK

BM,n:HO

Ha

paBeHCTBa

(10.6),

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onpe,n:eneHHH

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(t)

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JJ;HH

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aHaTb

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onHCbIBanOCb

61>1

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Mo,n:eHb ,n:oHmHa BNl.JHCnHTb aHal.JeHHH

nepeMeH-

Horo

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s

(t)

OT

MOMeHTa ta,n:o MOMeHTa

tfJ.

Pu60Ta

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e.

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(t). B Kal.JeCTBe

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B MO-

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t

a

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npMHHTh

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s

(til()

=

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BeK-

Topa

SO

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.

tf!

z (tfl) = X [tfl' t",]

x'"

+

~

X

[t

fl

,

'1:]

w

('I:)

dr.

(10.8)

t",

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X'

[tfJ,

tal

Q-l

Cl.JMTaeM

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aapaHee.)

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(tfJ)

TaKme

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Ha

Mo,n:enM. ::ha Mo,n:enb

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6bITb

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MeHM t

a

<

'I:

<

tfJ

ypaBHeHMeM

~~

=

Az

+w

(-r)

(10.9)

C

Hal.JanbHNM

ycnoBMeM

Z (t

a

) = x"'. O,n:HaKO OKaaNBaeTCH

menaTenb-

HbIM

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Bl.Il.JIICHeHHH

KaK

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6NCTpee

(aa

BpeMH

8

(tfJ

-

ta»

H

TaK,

l.JTo6N

MX OCYlJJ;eCTBHeHHe Hal.JIIHaHOCb

,n:(l

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ynpaBneHHH,

a

KOaqaHOCb

B MOMeHT

trl.'

::horo

MomHO

,n:OCTHl.Jb,

ecnH

B

ypaBHeHUH

(10.9)

BNnOHHHTb'

aaMeHy

BpeMeHH

t = t

a

-

t - t )

-

8(tfJ

-

'1:)

H nOHOJl<lITb y

(t)

= z

(

tfl +

~

.

Tor,n:a

tPaao-

Boe

COCTORHMe CMCTeMN,

BbIpa6aTbIBaIOru;eH

BenWIMHy

y

(t

a

)

= z (tfJ),

6y,n:eT

onHCNBaTbCH

ypaBHeHHeM

8

~;

=

Ay

+v

(t),

(10.10)

r,n:e 8 -

,n:OCTaTOl.JHO

Manoe

nOHomHTeHbHoe

l.JIICHO,

t - t )

v (l) = w

(

tfl +

~

.

(cc-'C.

~I

§ 10]

C~EMA.

MOJ~EJJ[TPYIOrn:Af{

YUPABJIEHyl8

65

Mo,n:enb,

onHCblBaeMaH

ypaBHeHHeM

(10.10), 6y,n:eT

pa60TUTb

B 1/8

paa

6NCTpee,

'-leM MCXO,n:HblH

06'heKT

(8.1).

(ManOCTb

8

06eCnel.fHBaeT

6bl-

CTpo,n:eHCTBlIe

aToH

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CTBHH

C «POPMYJIOH

ROllIH

(7.7).

<PyH,D;aMeHTaJIbHaJI

MaTpH~a

CHCTeMbI

(12.1)

HMeeT

BH,D;

X[t,~J=II~

t~~II·

CJIe,D;OBaTeJIbHO,

HMnyJIbCHaJI

nepeXO,D;HaJI

MaTpH~a

H a,D;eCb

TaKOBa:

H It,

~J

=

[

hW[t

(2)'

~J]

= X It,

~J

. b =

111

t-~II[O]

=

h It,

~J

0 1 v

=

[V(t:~)],

H

,D;BHmeHHe

cHcTeMbI

(12.1)

onHcMBaeTCJI

paBeHCTBaMH:

t

Xl

(t) =

~

h(l) ft,

~J

u

(~)

d~,

o

I

X

2

(t)

= j

h(2)

ft,

~J

u

(~)

d~.

o

B

'IaCTHOCTH,

npH

t =

tfJ

HMeeM:

tf>

XdtfJ) =

.\

h(l)

[til,

~J

u

(~)

d~,

(12.3)

o

tf3

X

2

(tf3)

= J

h(2)

ftil,

~J

u

(~)

d~.

. (12.4)

o

Tenepb

BOCnOJIb3yeMCJI

HeHOTOpblMH

nOHJITHJIMH,

CBJIaaHHI>IMH

C

.rtUlteuJtbZ.M,U onepalfUJt.M,U B

npOCTpaHCTBax

«PYHK~H.

(qHTaTeJIIO,

He

BJIa,D;eIOIqeMY

3THM

MaTepHaJIOM,

MM

yme

nopeKOMeH,D;OBaJIH

BO

BBe,D;eHHH

K ,D;aHHOH

'IaCTH

n03HaKOMHTbCJI

npe,D;BapHTeJIbHO

C

§§

54-60

Ha

llpH-

JIOmeHHJI

HJIH

'IHTaTb

aTH

naparpa«pbI

no

XO,D;y

Hay'IeHHJI

CJIe,D;YIO~ero

HHme

OCHOBHoro

TeKCTa.

B

COOTBeTCTBHH

co

CKaaaHHI>IM

B

,D;aJIbHeH-

llIeM

6y.n;eM

JIHllIb

HanOMHHaTI:.

OCHOBHMe

He06xo,D;HMMe

«paKTMHa

Teo-

pHH

JIHHeHHbIx

onepa~HH,

He

paa6HpaJI

HX

nO,D;p06HO,

HO

yKaaMBaJI

Kam,D;bIH

paa

cOOTBeTcTBYIO~He

paa,D;eJIbI

B

llpHJIOmeHHH.)

PaCCMOTpHM

«PYHK~HH

h(l)

[tfJ,

~J

H

h(2)

[tfJ,

-rJ

OT

aprYMeHTa

-r

(0

<

-r

<

tfJ)

KaK

3JIeMeHTM

h

(~)

(/iyltJl,lfUOlta.rtblt020

npocmpaltcmea

X[o(;~~j,

I'M

lt0p,Ma

p [h

(-r)1

onpe)J;eJIHeTCJI

paBeHcTBoM

(CM.

§ 57)

Ig

,

'I,

Pllt(1')J=b

h

2

(1')d-r]'.

o

..

_...-:....:..:...:..~.:...-.:-.L_;

__

..

..:

.:~.

':'.':

..

':

.'.

'.;"

~

.

..:

-~'~,.:_~~.:

",,

,:'.'

~

':~":

C

j 12]

~lllHKEHI1JI

YllPABJlHEMbIX

CIlCTEM

II

JIUliEvlHbIg

OllEPAL{HII

71

HanOMHHM,

'ITO

HopMa

p [h

(-r)J

xapaKTepHayeT

«y)J;aJIeHHe»

«PYHKIIHH

h

(-r)

OT

«pyHKIIHH

h

(-r)

=

O.

l1aBecTHo,

'ITO

KamAaJI

«pyHKIIHJI

U

(1'),

,D;JIJI

ROTOpOH

HHTerpaJl

1f3

~

u2(~)d~<

00,

(12.5)

o

nopOm,D;aeT

B

npOCTpaHCTBe

x[J~ill]

.rtUlteUltylO

onepalfUIO

lil

qJu[h(1')J = j

h(-r)u(-r)d~.

(12.6)

o

llpH

3TOM

ltop.Ma

p·[qJuJ

onepa~HH

qJu

[h

(-r)J,

xapaKTepH3yIOIqaH,

rpy60

rOBopH,

«MO~HOCTb»

3TOH

onepaIIHH,

B CBOIO

O'Iepe.n;b

BblpamaeT-

CH

paBeHCTBOM

(CM.

§§

58,

59)

p.

[qJuJ

=

1f3

1/,

[j u

2

('r)

d-rJ

.

(12.7)

()

TaK

KaK

e03,MO:JIC1We

ynpaBJIeHHe

U

(-r),

O'IeB

H,D;HO,

y)J;OBJIeTBOpHeT

yCJIOBHIO

(12.5),

TO

Ha

OCHOBaHHH

paBeHCTB

(12.3), (12.4)

npHXO)J;HM

K BbIBO)J;Y,

'ITO

aHa'IeHHH

Xl

(tfJ),

X2

(tfj)

Koop)J;HHaT

«p8.aOBOrO

BeKTopa

X

(t)

B

MOMeHT

t = tfj

MomHO

paCCMaTpHBaTb

KaR

peayJIbTaT

JIHHeH-

HOH

onepaIIHH

qJu

[hJ

(12.6),

IIopom)J;eHHoH

«pyHKIIHeH

U

(-r)

H

BbInOJI-

HeHHOH

Ha)J;

«PYHK~HriMH

h(ll

(-r)

=

h(l)

[tfj,

-r

J = v (tfj -

-r),

h(2l('r)

=

h(2)

[tfj,

-r

J =

v.

BOJIee

Toro,

MM

BHAHM

TaKme,

'ITO

BbI6paHHaJI

HaMH

Ha

«pHaH'IeCKHX

coo6pameHHH

HHTeHCHBHOCTb

x

[uJ

(12.2) B

)J;aH-

HOM

CJIy'Iae

KaK

pa3

HMeeT

CMMCJI

HOpMM

p*

[qJuJ

(12.7)

onepaIIHH

qJu

[h

(-r)J.

QT06bI

nOJIy'IHTb

3TO

COBna)J;eHHe,

MbI

H

paCCMaTpHBaJIH

3)J;eCb

«pyHKIIHH

h(l)

('r)

H

h(2)(-r)

KaK

3JIeMeHTbI

npOCTpaHCTBa

Xt~;I']'

TaKHM

06paaoM,

MomHO

aanHcaTb:

..-

If)

~

h(l)

(~)

u

(~)

dt

=

qJu

[h(l)

(1')

J,

o

tii

~

h(2)

(-r)

u

(-r)

d~

=

qJu

[h(2)

(,;)] ,

o

T.

e.

Xl

(tfJ)

=

qJu

[h(l) (,;»)' X

2

(tfJ)

=

qJu

[h(2)

(,;»)' r)J;e

qJu

-

JIHHen-

HaH

onepaIIHJI

Ha)J;

«PYHKIIHHMH

Ha

X{~:IFJ'

'

72

'':,·:·:·

..

,.

"

C

••

J"

IIOCTAHOBHA

3AJJ:A'IH

013

OIITMMAJlbHOM

YIlPABJTEHIIM

[rJI.

"

IIepeHp;eM

Tenepb

H

OOJIee

OOm;eMY

CJIY'IaIO.

IIycTI>

nOBep;eHMe

ynpaBJIHeMOH

CMCTeMbI

OnMCblBaeTCH

ypaBHeHMeM

1;

= A

(t)x

+B

(t)

u

+w

(t).

(12.8)

RaR

H B

paccMoTpeHHoM

BbIme

'IaCTHOM

npHMepe,

3anHmeM

3a-

ROH

p;BHiReHIIH

CHCTeMbI

(12.8)

no

<popMyJIe

RomH

(7.7):

t t

Xi

(t)

=

Xli)'

(t,

t

a

) X

(ta)

+

~

Xli)'

(t,

1:')

tv

(1:')

d1:'

+

~

h[i)'

[t,

1:']

U

(1:')

d1:'

t

a

t

a

(i=1,

...

,n).

(12.9)

3p;eCb

X[il'

(t,

.)

=

{Xij

(t,

.);

j = 1,

...

,

n}

- i-H

cmp01'i,a

<pyH-

p;aMeHTaJIbHOH

MaTpHIJ;bI

X

(t,

.)

CMCTeMbI

(5.2);

h[i]'

[t,

.]

=

{hiAt,.);

j = 1, ... ,

r}

- i-H

cmp01'i,a

HMnyJIhCHOH

nepexop;HoH

MaTpHIJ;bI

H

[t,

.]

CHCTeMbI

(12.5)

(CM.

§ 7).

IIpH

t =

t~

H3

(12.9)

CJIep;yeT:

tf!

Xi

(t(3)

=

X[i]'

(t(3,

t

a

) X

(ta)

+

~

xli)'

(t(3,

1:')

tv

(1:')

d1:'

+

t

a

tf!

+

~

h[i]'

[t{'l,

1:']

U

(1:')

d1:'

(i = 1,

...

, n).

(12.10)

t

a

BeRTOpHbIe

<PYHRIJ;HH

h[l]

(.)

=

h[I][t~,

.l,

...

,

h[n]

(.)

=

h[n]

[t~,

.]

OT

aprYMeHTa

•

oyp;eM

C'IMTaTb

Terrepb

9JIeMeHTaMH

npOCTpaHCTBa

.6Jiht

a

,t(3]

HHTerpHpyeMbIx

BeRTOp-<pyHRIJ;MH

h

(.)={h

j

(.)}

(j~1,

...,

r;

t

a

<•<

t~)

C

HOpMOH

(CM.

§ 57)

t(3

,.

P [h

(1:')]

,---

{~

[~

h~

(1:')J

d1:'t

• (12.11)

t

a

3=1

HanOMHMM,

'ITO

CHMBOJI

:1a[ta,

td

H30paH

Hm.fH

P;JIH

0003Ha'IeHMH

RaROrO-JIHOO

HOpMHpOBaHHOrO

U:pOCTpaHCTBa

<PYHRIJ;HH

(CM.

§ 57).

STOT

CHMBOJI

MbI

oyp;eM

ynoTpeOJIHTb

BCHRHH

pa3,

Rorp;a

HopMa

pac-

CMaTpHBaeMoro

npOCTpaHCTBa

He

<pHRcHpyeTcH

HJIH

Rorp;a

P;JIH

pac-

CMaTpHBaeMoro

npOCTpaHCTBa

C

BbIopaHHOH

HOpMOH

HaMH

He

orOBO-

peHO

CTaHp;apTHoe

0003Ha'IeHHe

(RaR

B

p;aHHOM

CJIY'Iae).

lI3BecTHo

(CM.

§§

59, 60),

'ITO

RaiRp;aH

BeRTop-<pyHRIJ;HH

u(.)

=

{Ul

(.),

.••

, U

r

(.)},

P;JIH

ROTOpOH

IIHTerpaJI

tl3

r

.

~

r~

U;(1:')]d1:'<,oo,_

(12.12)

t

a

t=1

....

'.

',',;,.'.:

..

~~'.:...:.........~::...:.:

.:".

".

:'.

',:

.....

',,;'

':'.:'.:.

',:

..

:_---'----.~._,~~_

...

:~:~,_,.:.:.·T~~

c

§ 12]

JJ:BIIIREHHff

YITPABJIHEMbIX

CHCTEM

H

JIHHE:f[HbIE

OITEPAIJ;HII

73

nopOiRp;aeT

B

paCCMaTpHBaeMOM

npOCTpaHCTBe

fla[ta'

t(3]

JIHHeHHyIO

onepaIJ;HIO

t{3

'Pu

[h

(or)]

=

~

h'

(1:')

U

(or)

dor

.

(12.13)

t

a

IIpH

9TOM

HopMa

OnepaIJ;HH

'P

u

[h]

BblpaJImeTCH

paBeHCTBOM

Jl/'

P*['Pu]

=p*[u(or)]

=.

1(3

j

[,~

uf(1:')]dT:

2.

(12.14)

1

la

t-1

HanOMHHM

em;e,

'ITO

BCe

Te

<PYHRIJ;HH U

(.),

ROTOpbIe

rrOpOiRp;aIOT

BCeB03MOiRHbIe

OnepaIJ;HH

'Pu

[h] B

RaROM-JIHOO

npOCTpaHCTBe

fla[ta,tfj]'

COCTaBJIffIOT

B CBOIO

O'Iepep;b

HeROTOpOe

npOCTpaHCTBO

fla~t«,

til]'

B

9TOM

npOCTpaHCTBe

fla*

,Ha3bIBaeMOM

COnpffiReHHbIM

R

HCXOp;HOMy

npOCTpaHcTBy

fla,

HOpMa

<PYHRIJ;HH U

(.)

Onpep;eJIHeTCff

paBeHCTBOM

p*

[U

(.)]

= p* ['Pul.

lIHa'Ie

rOBOpH,

HOpMa

<PYHRIJ;IIII U

(.)

B

fla*

npHHMMaeTCH

paBHOH

HopMeOnepaIJ;MII

'P

u

[h

(.)l,

nOpOiRp;eHHOH

9TOH

<pyHRIJ;HeH.

B

p;aHHOM

CJIy'Iae

H3

(12.11)

II

(12.14)

CJIep;yeT,

'ITO

HOp-

Ma

p* [U] B

COnpffiReHHOM

npOCTpaHCTBe

fla*

COBnap;aeT

C

HOpMOH

P [h] B HCXOP;HOM

npOCTpaHCTBe

fla.

lI3

(12.13)

3aRJIIO'IaeM,

'ITO

nOCJIep;HlIe

CJIaraeMbIe

B

paBeHcTBax

(12.10)

MOiRHO

paCCMaTpHBaTb

RaR

pe3YJIbTaT

JIHHeHHOH

OnepaIJ;HII

'Pu

[h

(.)l,

nopOJI<p;eHHoH

<pYHRIJ;lIeH

U

(or)

M

BbInOJIHeHHOH

Hap;

9JIe-

MeHTaMH

h[l]

(.),

..•

,

h[n]

(.)

1I3

npOCTpaHCTBa

fla[ta,tlll'

T.

e.

1{3

~.

h£il'

(T:)

U

(T:)

dT:

=

'Pl.,

[h[i]

(T:)]

(i = 1, . ,

.,

n).

lot

RpOMe

Toro,

eCJIH

IIHTeHCIIBHOCTb

%

[u]

BbIopaTb

B

BIIp;e

1(3

r

)1/

2

%(U)=.~

[.~U~(T:)]dT:

1

t

a

.-1

TO

1I3

(12.14)

BbITeRaeT,

'IT09Ta

IIHTeHCIIBHOCTb

%

[U]

MOiReT

ObITb

HCTOJIROBaHa

RaR

HopMa

p*

['Pu]

onepaIJ;MH

'P

u

[h] (12.13),

T.

e •

p*

['PiL]

= p* [U] = %

[ul.

Hplmep;eM

Tenepb

em;e

OP;MH,

HeCROJIbRO

OOJIee

CJIOiRHbIH

rrplIMep.

PaCCMOTpMM

CHOBa

CIICTeMY

(12.8),

rp;e

oyp;eM

nOJIaraTl>,

'ITO

,~,-._::

.....

~.

74

"':::,

',:",'>

'_v

c/

C

[rJl.

4

IlOCTAHOBKA

aA,L\Al.lM

or;

OllTIIMAJIhHOM

YIIPABJIEHIIlI

u -

CKaJlJIpHaJI

cPYHKIJ;IIJI. B 3TOM

cJlYlJae

paBeHcTBo

(12.10)

rrplIHlI-

MaeT

BlIP;

x

1/3

et

Xi

(t{3) =

xli]'

(tf3,

t

et

)

+

~

Xli]'

(t/3 ,

't')

w

('t')

d't'

+

let

1/3

+'

hl

i

)

[t/3, 't']u('t')d't'

(i=1,

...

,n),

(12.15)

J

let

rp;e

hli)

(.)

= hl

il

[til,

.]

=

hi

[til,

.]

-

TenepI>

ena.l£Jtp1UJl,e

cPYHK-

D,IIII

1)

(i

= 1,

...

, n;

til.

<

1"

<

til)'

B

OTJlIIlJIIe

01'

rrpep;bIP;YIIl,ero

cJlYlJaJI

oyp;eM

ClJIITaTb,

lJTO cPYHRD,IIII hl

i

)

(1")

JIBJlJIIOTCll

3p;eCb

3JIe-

MeHTaMII

cPYHRD,IIOHaJIbHOrO

rrpOCTpaHCTBa

2[1a.

t,s]'

rp;e

nopita

onpep;e-

JIeHa

paBeHcTBoM

1{3

P [h

('t')]

= \ Ih

(1")

I

d1".

(12.16)

fa

113BecTHO

(CM.

§§

57, 59), lJTO

Ramp;aJI

orpaHlI'-IeHHall

II

IIHTerpIIpy-

eMall

cPYHRIJ;IIJI u

(1")

rrOpOllip;aeT

B

npOCTpaHCTBO

2[t(1,t{1]

JIIIHeHHYIO

OnCpalWIO

t{3

CPu

[h ('t')] =

.l

It

('t')

u

('t')

d't',

III.

rrpIIlJeM

HOpMa

3TOH

OnepaD,IIII

Onpep;eJIJIeTCJI

paBeHCTBOM

p*

[CPu]

=

p*

[U] =

sup

IU

(1")

\.

(12.17)

Ramp;oe

603Jt,0:JICnOe

ynpaBJIeHIIe

U

(1")

JIBJIlleTCll

II

orpaHlIlJeHHOH

II

IIHTerpIIpyeMoH

cPYHRD,IIeH.

I103TOMY

II3

paBeHCTBa

(12.17)

MbI

CHOBa

3aRJIIOlJaeM,

lJTO

Te

CJlaraeMble

B

BblpameHIIJIX

(12.15) P;JIll

:l:i

(til),

KO-

Topble

3aBIICJIT

01' U

(.),

MOiI{110

TpaRTOBaTb

RaR

pe3YJIbTaT

JIIIHeHHOH

OrrepaIJ;IIII

CPu

[h]

Hap;

cPYHRD,IIJIMR

hl

i

)

(1").

I1pII

3TOM,

eCJIR

BbIOpaTb

RHTeHCIIBHOCTb

x

[U

] =

sup

Iu

(1")1

(til. <

1"

< til), TO

Ra

(12.17)

CJIe-

P;yeT,

lJTO

OnJITb

RaR

paa

p*

[U] = x [U].

l:ITaR,

B

rrpIIBep;eHHbIX

BMilie

rrpRMepax

MbI

rrORaaaJIII,

lJTO

B

BbIpa-

meHRJIX

P;JIJI

XI

(til)

CJIaraeMbIe

1{3

j

h[i]'

[t{3,

't']

u ('t')d't', (12.18)

1(1

3aBIICJIII(IIe

01' U

(1"),

MomHO

TpaRTOBaTb

RaR

aHalJeHRJI

JIRHeHHOH

one-

1)

B AaJIbBeii:meM B Tex CJIY'IalIX, KorAa

cmp01>u

h[i]

[til'

-r]

MaTpl~bI

11

[tp'

-r]

6YAYT

C1>a.M/,pa,I1.U

If

IwrAa

aTO

Hy}KHO

6YAeT rrOA'IepKHyTb,

6YAeM

0603IIaQaTb

HX

CIIMBOJIOM

h(i)

[til'

-r].

§ 121.IJ:BRH\EHII1I

YIIPABJIHEMbIX

CIICTEM

II

JIHHEfiHbIE

Ol1EPAll,IIII

75

paD,IIII

CPu

[h

(1")]

(12.13),

nopollip;eHHoH

cPYHRIJ;lIeH U

(1")

II

BbInOJIHeH-

HOH

Hap;

BeRTopaMII

h[l]

[til,

1"],

...

,

h[n]

[til,

1").

.

I1pII

3TOM B

ililIpOROM

RJlaCCe

CJlyqaeB,

ClJIITaJI

cPYHRD,RII

h[i]

(1")

=

h[i]

[til,

1"]

3JIeMeHTaMII

nop;Xop;JIIIl,ero

cPYHRD,IIOHaJlbHOrO

npOCT-

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Красовский Н.Н. Теория управления движением

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