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(23)

<!>OpMyJIy

(23)

MOiKHO

paccMaTpHBaTb

HJIH

KaK

orrepaTopHylO,

HJIlI

B

H306paiKeHHHx,

T.

e.

-

u(p)

u (p) = R

(p)

•

(24)

113(23) H

(24)

BH.l(HO,

1JTO

u

(t)

YiKe

He

6 y.l(eT,

Bo06~e

rOBopH,

¢HHHTHOii

<pyHKUHeii.

PaccMoTpHM

rrpocToH

rrpHMep,

HJIJIIOCTPHPYIO~Hii

nOT

cJIyqaii.

l1ycTb

3a.l(aHa CHCTeMa

jj

=

it,

y (0) =

I,

.Ii

(0) =

1.

(25)

2

3.l(eCb d (p) = p

,

M I (p) = P H HMeeTCH

06~Hii

KopeHb

PI

=

O.

B CTapOM CMbICJIe CHCTeMa

HeyrrpaBJI$IeMa.

3.l(eCb HeJIb3H

cpa3Y

BOCrrOJIb30BaTbcH

<popMyJIoii

(19).

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yrrpaBJIeHHe

v

(t)

= u(t),

TO

CHCTeMa

rro

OTHOllIeHHIO K 3TOMY

yrrpaBJIeHHIO

CTaHeT

yrrpaBJIHeMoii.

JIerKo

BH.l(eTb,

1JTO

<PHHHTHoe

yrrpaBJIeHHe

v

(t)

= - 6

(t)

-6

(t).

(26)

OTclO,lta

t

u (t) = Jv

(T)

dT=-

6

(t)-]

(t),

(27)

o

f.l(e

1

(t)-e.l(HHH1JHaH

<PYHKUH5I.

YrrpaBJIeHHe

(27)

06eCrre1JHBaeT

npHBe.l(eHHe

KOOp.l(HHaTbi y

(t)

B

HyJIb

3a

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1

...

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.'

p-,

/

~~>

/

np5lMOa

M~TO..q

~.8~

yrrpaBJIeHHe

u

(t),

KaK

BH)l,HO

H3

<POpMyJIbI

(27),

He

51BJISleTCH

.«pHHHTHOH

<pyHKUHeH.

He

Tpe6y51

«pHHHTHOCTH u

(t),

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He

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BHH~

MBHH5I HB

KOpHH

d (p) H M", (p) H

3arrHcaTb

it

(p) B BH,lte .

U(P)--~~~),

yo-(Uo,·

..

,

Un-I)'

(28)

rAe

L (p,

yo)

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OT

P CTerreHH n

-I,

aaBHcHw.HH

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HhlX

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yo.

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yO)

rrOJIytlaeTcH

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npe06pa30BaHI1Sl

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K JIeBOH

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(10).

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TIepeH.l(eM

Terrepb

K

paccMoTpeHHIO

pacrrpe,lteJIeHHbIX

CHCTeM.

3.l(eCb

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orrpe.l(eJIeHHH

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B TOM,

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B

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it

(Z),

y)l,OBJIeTBopHIOw.eH HHTeprrOmlI.J.HOHHOH

np06JI~Me

:,'

(CM. (32) §

10

nOH

rJIaBbI):'

-

." ...

~

Ii

(Zk)

=

~

(Zk)

=

~k'

k=O,

±l,

±2,

...

,

(29)

B3HTb

«PYHKI.J.HIO

~

(z),

T.

e.

rrOJIOiKHTb

.

it

(z)

=

~_(z).

(3b)

.

,

I

J

ECJIH

rrpH

3TOM

~

(z)

Cp.a3y

OKa3bIBaeTCH

I.J.eJIOH

aHaJIHTHtleCKOH

'.

,j

«pyHKI.J.HeH,

TO

OHa

YiKe

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SlBJISleTCSl

pellIeHHeM

3a)l,atlH

«pH'

HHTHoro

yrrpaBJIeHHH,

TaK

KaK

U(z) =

~

(z)

aBTOMaTH1JeCKH

pellIaer

'I

HHTeprrOJISlI.J.HOfIHYIO

rrp06JIeMy

(29).

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MHOrHX

CJIytlaHX,

KaK

MbI

3TO

YBH)l,HM )l,aJIee,

~

(z)

cpa3Y

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<pyHKUHeH,

a B

Tex

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He

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ee

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0

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l'

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+

l'

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i

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C KBa.l(paTOM

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e.

rrpH

z=ro.

j

.

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~

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l'

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paBHoe

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rrpH

t <

O.

Y)l,OBJIeTBOpHTb-

j

3TO

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He

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TaKHM

06pa30M,

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~

(z)

HJIH

~

(z) +

l'

(z) d

(z)

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,I

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2)

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3)

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rrpe06pa30BaHHe

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(z

= ro),

paBHoe

HyJIIO"

rrpH

t <0 (eCJIH Ha1JaJIbHble yCJIOBHSl 3a)l,aHbI

rrpH

t =

0);

4)

HMeTb 3a)l,aHHylO CTerreHb (eCTeCTBeHHO,

He

MeHbWYIOMHHH"

MaJIbHO

B03MOiKHOH);

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10

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~29b

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yC}JIEHM5I

[M.3

5)

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C

KBa,!{paToM

Ha

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(npH

Z=ffi)

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HU~eM

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pilCTHHa

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OCH

He

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n lJeJIHtIHHbI Iffil.

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pSl,!{npHMepOB.

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3a,!{atIy(1)-(7~

§

11

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00

ycnoKoeHHH

KOJIeoJIIomeAcSI

CTpyHbI.

113

<pop-

;MyJIbI (11)

.9Toro

}Ke.

n,aparpa<pa,

saMeHSISI k

HIl

Z,

nOJIytIHM

11

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(z)

=

co-

Jsin zx [IQo (x) +:

Qdx~J

dx.

(a

1)

JIerKo

BH,!{eTb, tITO'

TaK

onpe,!{eJIeHHaSl<pYHKnHSI Ii

(z)

B

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sa,!{atIe,

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pernaeT

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sa,!{atIy

:(11)

§

11

3TOH r JIaBbI,

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neJIoii

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<pyHKnHeH

CTeneHH n

H,

no

KpaiiHeii

Mepe,

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Ha

,!{eiicTBHTeJIbHOH

OCH

npH

Z =

ffi.

JIerKo

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ynpaBJIeHHe

,!{JISI

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. sa,!{aqH

Ha

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t e

(-

n,

n):

11

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sin

COX]

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QO(x)P-l

lffi~

dx-

o

I

11

n

[sin

COX]

- I 5Q

J1 J1

-

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dX=-2

o(x)[u(t+X)-u(t-X)jdx_

o •

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11

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(t-x)]

dx=

o

_Q'<O~Q'H)+;

U

Q,(Xldx-J'

Q,(x)dx

l

(32)

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<pyHKnHH

Qo

(x) H

Qdx)

s,!{eCb CtIHTaIOTCSI

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1·

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(t)

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sa

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3),

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(t)

npH

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O.

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s,!{eCh

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co

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(t)

C)(BHHyTCSI H

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JKe

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(OCMbICJIHTb) (j>HHHTHoe

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(32).

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no

(j>opMyJIe

(32)

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ObI OblTb HCnOJIbSOBaHO

KaK

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eCJIH

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t

=-n

~HaJIH,

lITO B MOMeHT

BpeMeHH

t . 0

HaCHcTeMy

no,!{eHcTByIOTBosMyIII,eHHSI

B BH,!{e

H~qaJIbHbIX

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0)=

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0)

=

Qdx).

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JleHHe

(32)

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c npe,!{BH,!{eHHeM

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CtIHTaTb,

tITO

ynpallJIeHHe

THna

(32)

HatIHHaeTCSI B MOMeHT

BpeMeHH

t =

O.

3TOMy

cooTBeTcTByeT

C,!{BHr Ha- n

no

t

BnpaBo,

T.

e.

'.,'

····

, "

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:-~'

c

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OP

51Morf METO)].

291

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t

:::;;

2n

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Q,(I-.l-

..

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I

+.)

+;

[T

Q,(Xldx-

-r"

Q,(x)

dx

]-

(33)

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B 3TOM

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MbI ,!{OJI>KHbI npe)l.BH,!{eTb COCTOSlHHe CHCTeMbI

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)].eiiCTIJHH

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(B

BH)l.e TOJIbKO

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yCJIOBHii)B

MO·

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BpeMeHH

t=

n,

T.

e.

MbI )]'OJI)i{HbI SHaTb Q (x, n) H Q

(x,n),

<I>yHKnHH Q

(x,

n)

HQ

(x,

n)

MO>KHO

nOJIytIHTb

KaKperneHHe

CHCTeMbI

(1)-(7)

§

11

3TOii;

rJIaBbI

npH

HyJIeBbIX

rpaHHtIHbIX

yCJIO;'

BHSlX,

T.e.

KOr)l.a U (t)

O.

Tor,!{a

B CHJIy

(33)

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ynpaBJIeHHe

MOJKeT ObfTb

sanHcaHO

B HeCKOJIbKO HeOObIqHOM BH,,!{e:

U

(t)=

Q

(/-n,

n)-n

Q

(-I+n,

n)

+;

[if

Q(x,

n)

dx-

-r

11

Q(X,

n)

dX].

(34)

o 0 .

MOJKHO

npoBepHTb,

tITO eCJIH MbI BbItIHCJIHM Q (x, n) H Q(x,

n);

H

nO)l.CTaBHM B

(34),

a

saTeM

npOBe)l.eM HeOOXO)l.HMbIe

npeoopaSOBaHUSI,

TO

MbI npH,!{eM

K(j>opMyJIe

(37)

§11

:)Toii

fJIaBbI.

KOHetIHO,

nOJIy-

tIeHHOe

ynpaBJIeHHe

THnll

(33)

MOJKHO

OTHeCTH K

KJIaccy

(j>HSHtIeCKH

HepeaJIHsyeMbIx

CHCTeM,

TaK

KaK

ynpaBJIeHHe

,!{OJIJKHO

HatIHHaTbCSI

sa

KOHetIHOe BpeMSI,

paBHoe

n,

)1.0

tlatIaJIa)l.eiiCTBHH

BosMYIII,eHH~

OT

HaqaJIbHbIX

yCJIOBHii,

HatIHHaIOmHX

CBoe ,!{eHcTBHe B MOMeHT t=O,

T.

e.

nOCJle

HalfaJla

ynpaBJIeHHH.YnpaBJIeaHe

(34)

yJKe

<pHSHqecKf!

peaJIHsyeMo,

O,!{HaKO Hy>KHO

yMeTb

npe)l.CKasaTb:

(BqItIHCJIHTb)

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Ha

BpeMSI n

Bnepe)].

OT

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BosMymeHHii

B BH,!{e

HatIaJIbHbIX

yCJ:IOBHH Q (x, 0) = Q 0(x), Q(x,

0)

~--

Ql (x).

4TOObI

nOJIytIHTb

Bce

MHOJKeCTBO

(j>HHHTHbIxynpaBJIeHHii

npH

T = 2n, Ha)l.O

K

it

(z)

B (j>opMyJIe

(31)

)].ooaBHTb

tIJIeH '\'

(z)

sin

nz

(T.

e.

-)l.OOaBIITb

tIJIeH, cooTBeTcTByIOlIJ.HH HyJlb-(j>HHHTHOMy

ynpaBJIeHHIO).

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crBeHHO

B

(34)

HyJKHO )l.OOaBHTb

tIJIeH,

BbIpa>KaeMblii

no

<popMyJIe

(38)

§

11

3TOH

r

JIaBbI.

. .

PaccMoTpHM

Tenepb

sa)].atIy

(1)-(5)

§

15

3TOH

rJIaBbl

,!{JISI

it

(jk~)

--,-

-{

5Q0 (x)

sin

kx

dx,

ypaBHeHHH

TenJIOnpOBO)l.HOCTH.

Do

(j>opMyJIe

(16)

roro

naparpa<pa

HMeeM

:Jt

k=O,

I,

...

(35)

o

DOJIar::ISl

ik9=z,

HOJIytHfM,

qTO

k=Vjz.

OTCIO)l.a, no,!{cTaBJI5UI

3TO

3HatIeHHe

k B

(35),

nOJIytIHM,

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KatIeCTBe<pyHKnHH

U

(z)

10·

~:),.

3~eCb

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292

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JI. 3 METOlJ,bl

<I>~HHTHOrO

yQBJlEH~51

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B3~Tb

n .

U

(z)

==

-

,~.J

Q0

(X)

sin x

Vii

dx. (36)

r

/Z

0

..

~ha

<»YHKIlH~

u

(z)

eCTb

<»YHKIlH~

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CTeneHH,

H

BpO~e

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u

(t)

==

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(ro)]

oy~eT

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l'

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n V jz,

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u(z)

==

-

/ii

S

11

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(x)

sin

x

VIZ

dx +

l'

(z)

sin

n Vjz (37)

o

M01KHO

C~eJIaTb

TaKoH,

qTOObI cylll,ecTBOBaJIO u

(t)

=

F-l

[it

(ro»),

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n~TH

TpeooBaHHSlM,

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00

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(§

13

3TOH

rJlaBbI).

H3

ypaBHeHHSI

(28) §

13

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qTO

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it

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==

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S

I

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J 0

(~k}IX)

dx-

o

1

- J

1

(~k)

S

Qo

(x) J0

(~k

VX)

dx.

(38)

o '

KaK

JIerKO

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3~eCb

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saMeHa

{-

~k

==

Z

(CJIe~OBa

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==

2z)

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z,

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Бутковский А.Г. Методы управления системами с распределенными параметрами

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