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pellIHTb

B KJIaCCe

0000-

rqeHHblx

'yrrpaBJIeHHH

u (t). ,UeHCTBHTeJlbHO,

rrO)lXo.n;SI~HM

BblOOPOM

u(z)

MOR<HO

C)l.eJIaTb q

(z)

= 0, T.

e.

Koop.n;HHaTa

q (t) MrHOBeHHO

nepeH.lI.eT B

HyJIeBOe

COCTOHHHe. 113 (8) BH.n;HO, qTO

TaKoe

U(z)

orr

pe.n;eJIHeTCH C

1I0MO~bIO

rrpHpaBHHBaHHH

HyJIIO

qHCJIHTeml

.n;pOOH

(8):

it

(z)

+x (0)

---b'%

(z)

e-

JZT =

O.

\12)

B

TaKoil:

aanHcHoea

KaKoro

Obi

TO

HHOblJIO

OrpaHHqeHHH

06rqHO"

CTH

MO)RHO

CtIHTaTb,

tITO q

(t)

= 0

rrpH

t < 0 H X

(t)-epHHHTHaH

epyHKl{HH

Ha

HOCHTeJIe

[-

T,

OJ.

TaKR<e

Oy.neM

CtIHTaTb,

KaK

,Mbl

Bcer.n;a

!nO

.n;eJIaJIH

paHee,

tITO u

(t)

= 0

rrpH

t <0 H t >

T.

Tenepb

MOR<HO

nOCTaBHTb

aa.n;atIy

epHHHTHoro

ynpaBJIeHHH.

He-

06XO.n;HMO HaMTH

ynpaBJIeHHe

u (t), 0

~

t

~

T,

KOTopoe

nepeBo.n;HJIO·

6bl

CHCTeMy (3)

H3HaqaJIbHOro

COCTOHHHJI (4) B COCTOHHHe rrOKOH

aa

KOHelIHOe

BpeMH

T,

T.

e.

Heooxo.n;HMO BbInOJIHHTb yCJIOBHH:

q (I) = 0

npH

t

~

T,

(5.)

u(t)=O

npH

t~T.

(5')

B

npHHl{Hrre

rrOCJIe.n;oBaTeJIbHOCTb

pemeHHH

STOil: 3a.n;aqH

TaKaH

)Re,

KaK

H rrOCJIe.n;OBaTeJIbHOCTb

pemeHHJI

aa.n;atI

epHHHTHoro

yrrpaB-

neHHH

.n;JIJI

CHCTeM

oe3

3arra3.n;bIBaHHH.

TIpHMeHHH

K

ypaBHeHHIO

(3)

C'

HaqaJIbHblM

yCJIOBHeM (4)

npeoo-

paaoBaHHe

«I>ypbe,

nOJIyqHM

jroq

(ro)-x

(0)

+aq

(ro)

+bx

(ro)

e-/

WT

+bq

(ro)

e-/

WT

= u

(ro).

(6)

OTcIO.n;a

nerKO

Haxo.n;HM,

tIro

q(ro) =

U(ro)+X(O)-bX(ro)e-/

WT

jro+a+

be-/

WT

(7)

B COOTBeTCTBHH C

TeopeMoil:

BHHepa-D3JIH,

.n;JIH

Toro

tITOObl

q

(I)

ObIJIO epHHHTHOii: epyHKII.Heil:

Ha

[0,

T], Heooxo.n;HMO, tITOObl

,epyHKUHH

q(w), orrpe.n;eJIHeM3H

paBeHCTBoM

(7),

MOrJIa

oblTbpacnpo-

CTpaHeHaHa

BCIO

rrJIOCKOCTh

KOMnJIeRCHoro

nepeMeHHOrO

RaR

II.eJIaff

8-

i

'I;·

"

'I

~!t,\

METO,nbl

4>HHHTHOroyB~JlEHH51

228

OrclO,/t'l

u(z)

==-

x (0)

+bx

(z)

e-iz't.

(13)

3aMeHH51 B

3TOM

paBeHCTBe Z

Ha

ro

H BbInOJIH51H OOpaTHOe

npe-

oopa30BaHHe

ypbe B

0606w.eHHbIX

q>yHKI{H51X, nOJIytIHM, tITO HCKO-

Moe

q>HHHTHoe

ynpaBJIeHHe

paBHO

u(t)

= - x (0)

c5

(t)

+bx

(i

-T),

O~i~T=T.

(14)

T3KI-JM

oopa30M,

BpeMH T <}>HHHTHoro

ynpaBJIeHHH

,n3}l{e B KJI3CCe

oOoow.eHHbIX

(y,napHbIX

HJIHHMnYJIbCHbIX)

ynpaBJI51IOw.HX B03,neH-

CTBHH 3,/teCb He MO}l{eT ObITb C,/teJIaHO MeHbIIIe BpeMeHH 3ana3,/tbIBa-

HH51

T,

T.

e.

minT=T.

3,necb

MbI CTaJIKHBaeMc51. C

TaKoH

ocooeHHOCTbIO,

xapaKTepHOH

.

AJI'51

CHCTeM

C

3aIIa3,nbIBaHHeM,Kor,na

yIIpaBJlHeMa51 Koop,nHHaTa

MrHOBeHHO

IIepelliJIa

B

Tpe6yeMoe

COCTOHHHe

q= °H TaM OCTaeTCH;

HO

3TO

IIOJlO}l{eHHe B TetIeHHe BpeMeHH 0

~

i

~

T HepaBHOBeCHO.

JlJIH

TOro

tITOObI

nOMep}l{HBatb

Koop,nim3ty

q B

3TOM

IIOJlO}l{eHHH

q = 0, Ha,no

eW.e

IIo,naBaTb

yIIpaBJI51IOw.ee B03,neHCTBHe u

(i),

oIIpe,/te--

JI51eMOe

q>bpMyJIOH (14), B

retIeHHe

oTpe3Ka

BpeMeHH [0,

T].

TIocJIe

3Toro

BpeMeHH,

IIpH

i

->

T,

MO}l{HO

npeKpanlTb

IIo,naBaTb

ynpaBJIHIO-

w.ee B03)I.eHCTBHe, T.

e.C)I.eJIaTb

u

(t)

= °

IIpH

t > T, T.

e.

CHCTeMa

,/teHcTBHTeJIbHO oy)I.eT HaXO)I.HTbC51.B

paBHoBecHoM

COCT051HHH,

xa-

paKTepH3yeMoM

paBeHcTBaMH (5) H (5').

'MeTO)I.HKa perneHH51 3a)I.atI q>HHHTHoro yIIpaBJleHH51, paCCMOTpeH-

BaH'

HaMH Ha ,/taHHOM tIaCTHOM

IIpHMepe,

B

IIpHHUHIIe

MO}l{eT ObITb

.nerKO

oooow.eHaHa

CJIytIaH,

KOr)I.a

ynpaBJlHeMbIH

OO'beKT OIIHCbI-

Bae1'C51

oow.eH

JlHHeHHOH

cHcteMoH06bIKl:lOBeHHbIX

)I.Hq>q>epeHl./,HaJIb-

Hbix

ypaBHeHHH

C

nOC1'051HHbIMH

K03<p<}>HUl1eIitaMH H C IIOCT051HHhIMH

SaIIa3)I.blBaHHHMH

(He

3aBHC51Ill.I1MH 01'

BpeMeHH),

KOTOpbIX MO}l{eT

ObITb B )I.aHHOH

CHC1'eMe

HeCKOJIbKO.

§ 10. cf)HlfHTHoe yrtpaBJIeHHe pacnpe)I.eJIeHHblMH CHCTeMaMH

HatIHHa51 C

3Toro

II3parpaq>a,

B

31'OH:

rJIaBe

MbI oy)I.eM

paccM8T-

pHBa1'b 3a)I.atIy q>HHH1'Horo

yIIpaBJIeHHH

)I.JIH

ynpaBJI51eMbIX

oo"eKTOB

c paCIIpe)I.eJIeHHbIMH

napaMe1'paMH.

JlJIH

OOJIbrneH OIIpe)I.eJIeHHOCTH

,H

IIpOC1'OTbI' H3JIO}l{eHH51 6y)I.eM

paccMaTpHI3aTb

O)I.HOMePI!bIe

npo-

crpaHc1'BeHHo

paCIIpe)I.eJIeHHble OO'beK1'bI, ,/tBH}l{eHHe K01'OpblX

onH-

o CbIBae1'CH )I.Hq>q>epeHII.HaJIbHbIMH

ypaBHeHHHMH

B tIaCTHblX

IIPOH3BOk

HbIX

BTOporo

IIOpH)I.Ka.

OooOw.eHHe

Bcex

paccy}l{)I.eHHH

Ha'

cJIytIaii

60JIee

BbICOKOH

pa3MepHocTH

IIpocTpaHc1'Ba

H oOJIee

BbICOKoro

IiOp5l)I.Ka

ypaBHeHHH

)I.OBOJIbHO

OlleBH)I.HO.

HeKoTOpble

IIpHMepbI

TaKHX

p6'beK1'OB (}l{eC1'KaH OaJIKa,

yIIpyra51

)I.ByMepHaH IIJIaC1'HHa) Oy)I.y1'

paccMo1'peHbI B IIOCJIe,/tYIOIUHx

naparpaq>ax.

~

"

'C::''':,

':.:~,:

229,

':..:.

PAcnPE,nEJlEHHbIECHCTEMbI

;,'JiraK,

paccMoTPHM

ynpaBJI51eMbIH oObeK1',

COCTOHHHe

KOToporo

OIIHCbIBaeTC51

q>yHKuHeH Q(x,

i),

onpe)I.eJIeHHOB

npH

i

~O

11

xE

[xo,

Xl]'

nyCTb

IIpH

t > 0

Il

X

o

< X <

Xl

q>yH}{UH51 Q

(X,

i)

nOA-

tlHH51eTC51

ypaBHeHI1IO

~

aQ

+bQ

+cQ

= A(x)

Q"

+B

(x)

Q'

+C

(x) Q +

+f(x,

i,

u(i),

v(x),

w(x.

t».

(1)

rAe

a, b,

c-nbct051HHbIe

K03q>q>HuHeH1'bI, U (f), v (x), w(x"

t)

'""'-

pacnpe)I.eJIeHHoe ynpaBJI51lOm,ee

B03.!leik1'BHe.

K03q>q>HUHeH1'hI A; B

H C B

TaKora

po,/ta

33)I.atIaX,

pemaeMl:>I

X

Me1'O)I.aMHq>HHH1'HOr()

ynpaB-

JIeHHH,

t.mryT

33,BHCeTh

OT

npocTpaHcTBeHHoH

nepeMeHHoH x.

npH-

Mepbl

pemeHH51 Tai<HX 3a)1atI MbI 1'aK}l{e

paccMoTPHM

)1aJIee

..

nYCTb

3a.naHbI

KpaeBbIe

yCJIOBH51

(2)

aoQ' (x

o

'

i)

+~oQ

(x

o

'

i)

= U

o

(f),

a

l

Q'

(Xl'

t)

+

~l

Q

(Xl'

i)

= u

l

(i)

(3)

H HatIaJIbHbIe

yCJIOBH51

(~

Q (x, 0) =

Qo

(x),

(5)

Q(x,

0)=

Ql

(x).

3.n

ao,

~o,

a!'

~l-noc1'oHHHble

tIHCJIa, Uo

(i)

H u

l

(i)

-ynpaB-

eCb

JlHIOw.He

(rpaHHtIHble)

B03.nei\cTBH51.

3a.natIa

<!>HHHTHoro

ynpaBJIeHHH

Tenepb

MO}l{eT ObITb

noctaBJIeHa

B cJIe.nYIOw.

eM

BH)I.e.

HaH1'H

1'aKHe

ynpaBJI51IOW.

He

q>yHKUHH

U

(i),

v (x), ,w(x,

i),

u

l

(i),

U

2

(f),

tlTOObl

B MOMeRT BpeMeRH

i = T ynpaBJI5JeMblH OObeRT, oriHchIBaeMbIH

ypaa-

HeltHHMH

(1)-(5),

npHmeJI

B

COCT051HHe

paBHoBecH51. T.

e.,

H"hIMH

CJIOBaMH,

tltoohI

ObIJIH Bblrto.iI.HeltlsI

YC;1IOBH51

(6)

Q(x,.

T)

==0.

(7)

Q

(x,T)

=0.

nepeH.n

Tenepb

K

onHcaHHIO

MeTo,na peIIIeHH51, Cq>OpMyJIHpO-

eM

BaRHoH

3a.natIH <!>HHHTHoro

ynpaBJIeHHH.

TIpe}l{.n

e

Bcero

IIpHMeHMM

npeoopa30BaHlie

ypbe

no

BpeMeHHoH

nepeMeHHoH

i K

OCHOBHOMY

ypaBHeHlfIO

(1).

Tor.na

C yqeTOM HatIaJIbHbIX yCJIOBHH (4),

(5).1ierKo

BH.neTb,tlTO

IIOCJIe l1pHMeHeHH51

Iipeoopa30BaHHH

ypbe K

06eHM

tIaCT51M

ypaBHeHH51 (1), IIOCJIe HeCJIO}l{HbIX

aJireOpaHtIeCKHX

npeo6pa-

30BaHHH,

nOJIytIHM cJIe.nyIOw.

ee

ypaBHeHHe:

d

2

Q(x

ro)

dQ

(x

ro)

•

i'i

A(x)

dx§

+B (x)

ai

' +

(aro

2

-l

bro

-

c

-

c

(x»

Iol

(x.

ro)

=

=f(x,

ro)-(ajU'

+

b)

Qo(X)-Ql(X),

X

o

~X~Xi'

(8)

"

..

C

:~:"

,

.,'

,""J

.:......:..;/

,j

230

METO~br

I1HI1THOrO

YIIPABJIEHI151

flpHMemlfl

npe06paSOBaHHfl

<l>YPbe

K 06eHM lIaCTflM

(2)

H(3),

BblpamalOIIJ,HX rpaHHIIHble yCJIOBHfl,

HaH)I,eM

-

dQ

- _

_

~o

dx

(X

o

'

ro)

+-

~o

Q

(X

o

,

ro)

= U

o

(ro),

dQ

_ - _

a

l

dX

(Xl'

ro)

+-

~IQ

(Xl'

ro)

= U

t

(ro). (10)

B

ypaBHeHHRx(8)-(10)

<PYHKUHfl

Q

(X,

ro)

eCTb

npe06paSOBaHHe

<l>YPbe

no

apryMeHTy

t<PyHKUHHQ

(x, t), TaKHM 06pasoM, He06-

XO,ll,'HMO

pelllHTb KpaeBylO sa)l,ally (9)" (10)

OTHOCHTeJIbHO

<PYHKUIfIf

Q(x,

ro)

)l,JIfl 06bIKHoBeHHoro

)l,1f<p<pepeHUHaJIbHOrO

ypaBHeHlffl

(8).

3)1,eCb

roflBJIReTCRnapaMeTpoM. "

nYCTb

hI

(X,

ro)1f

h

2

(x,

ro)

eCTb

<pyH)I,aMeHTaJIbHOe

pellleHHe

OOblKHOBeHHoro

)l,1f<p<pepeHUlfaJIbHOrO

ypaBHeHlffl

no

x,

a

ro

paCCMaT-

plfBaeTCfl KaK napaMeTp. Tor)l,a, KaK,

IfSBeCTHO,

pellleHlfe ypaBHe-

'HHR,

(8)

MomHO

aanlfcaTb B

CJIe)l,ylOIIJ,eM

BIf)l,e: _ ' '

..;..

.'

.

·.1

rl'"

Q

(X,

ro)

=

clhdx,

ro)

+-c

2

h;(x;

ro)

+-

F

(X,

ro), (11)

f)l,e

F(x,

ro)

= L

2

h

(X,

ro)

S

;c

Wk (hI

(S.

ro);

h2

(S.

ro»

X

k

k = 1 ;Co W (hI

(~,

ro).

h

2

(S.

ro»

x[l(s,

ro)-(ajto+-b)Qo(1;)-Qd~)]d~,

(11a)

hI

(x.

ro)

h

2

(x,

ro)

I

W

{hI

(X,

ro), h

2

(X,

ro»

=dh

l

(x.

ro)

dh

2

(x.

ro)

(12)

dx

I

.If Wk

(hI

(x, ro), h2 (X, ro» -onpe)l,eJII1TeJIb, IIOJIYllaeMbIH

1f3

onpe)l,e-

-JIHTeJIfl

W (hI

(X,

ro), h

2

(X,

ro» B pesyJIbTaTe

3aMeHbI

k-ro cToJI6ua

Ha

cToJI6eu

(0,1),

k=

1,2.

BeJIHIIHHbI H C B (11)

)l,OJImHbI

6b1Tb

c

I

2

onpe)l,eJIeHbI TaK, IIT06bI

BblIIOJIHRJIHCb

rpaHlfllHble

YCJIOBlffl

(9),

(10).

'BbIlIlfCJIlfM

Ha

OCHOBaHHIf

(11)

dQ

(x.

ro)

= C

dh

l

(x,

ro)

+-

c

dh2

(x,

ro)

+-

dF

(x.

ro)

(13)

dx

I

dx

2

dx

dx'

r)l,e _

'dF

F(x

o

,

ro)

=

0,

dX

(Xii,

ro)

=

O.

(14)

nO)l,cTaBJIflfl (13) If (11) B (9)

If

(10)

COOTBeTCTBeHHO

IIpH

x=x

H X = X

j

If ylllfTbIBafl,

tITO

cnpaBe)l,JIlfBO

(14),

nOJIyllHM

o

- - -

dh

l

( )

+-

'

dh

2

( )

+-

A h ( )

cIaO

dx

-'

xo,

ro

c

2

a

O

dX

Xu,

ro

CWo

I

Xo,

ro

+-

c2~oh2

(X

o

,

ro)

= U

o

(ro), (15)

dh

l

(

X

to

)"

dh

2

)'

dF

( )

h'

cla

1dx

ro

+c.a.£iX(xl,ro

+-a

1dX

~l'ro

+-ct:J

IJ

1

t(xo,ro)+-

',f

+-

c'~lh.

(x

o

,

ro)

+-

~J

(Xl'

ro)

=

Udro).

(16)

"\,-~

-.

(:;;

~)'

\

PACIIPE)I,EJIEHHbIE

CI1CTEMbI

-23-1

J~pynnHpYH

tIJIeHbI

nplf

HeH3BecTHbIxc

l

H

C.

If

IIepeHocH cB06o;ltHbie

}IJIeHbI B npaBYIO qaCTb, nOJIyqlfM clfcTeMY aJire6paHtIeCKHX JIHHeii-

H&lX ypaBHeHHH

OTHOCHTeJIbHO

C

l

H

c.:

'

('(aoh~

(x

o

,

ro)

+-~Ohl

(x

o

'

ro»Ci

+-

(aoh; (x

o

'

ro)

+-P.h

2

(x

o

' ro» c.

=,u

o

(ro),

(17)

(aIh~

(Xl'

ro)

+Plh

l

(Xi' ro» C

l

+ (alh;

(Xt>

ro)

+-

P

l

h

2

(Xi' ro»

C.

=0:

-Ui

(ro)-aiF'

(Xi,

ro)-PIF(Xi'

ro). (18)

OTCIO,A8

, d

2

(ro)

01 (ro) =

6,1

(ro)

A

(ro)

0.

(ro) = .i(ro) ,

(19)

,

r,lle

~

(ro)

=1

CXoh~

(xo.

'ro)'+

Pohdxo,

ro)

aoh~

(xo.

ro)+

~oh.

(xo,

ro)

I,

(2(»

,I%]h

l

(Xi.

ro)+Plhdxo,

ro)

I%]h.(Xl>

O»+~lh'(Xl>

ro)

~dCJ)

= I

~o

(ro)

lXoh~

(Xo,

ro)+

~oh.

(Xo.ro)

I,

(21)

Ul

(ro)-alF'

(Xl>

ro)-PIF

(Xl.

ro)

alh. (Xl'

ro)+

P

l

h

2

(Xl>

ro)

~2

(ro) = I

aoh;

(xo,

ro)+

Pohl

(xo.

ro)

~o

(ro),

/.

(22)

I%]hdxl'

ro)+Plhdxl.

(0)

udro)-alF'

(Xl>

ro)-~lF(Xi,

0»

nO,llCTaBJIHH nOJiytIeHHbIe

ne;>

<popMyJIaM

(19)

3HatIeHHfl

cdro)

H

c.

(ro)

B (11), nOJIyqHM

- ( _

ddro)

1idx,

ro)+

d.

(ro)

h2 (x,

(0)

+'F

()

(23')

Q

x,

ro)

- , d

(ro)

x,

ro

• ,

Tenepb

HCIIOJIb3yeM TeopeMy

BHHepa-ll3JIH

HJIH

ee0600IIJ.eHHe,

,llaHHOe

IIlBapueM.

B CHJIy.,310H TeopeMbI

,llJIH

TOro,

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2n 5dx

T =

2T

= tz

=i:

2 A (x) .

(24r)

o

B lIaCTHOCTlf, eCJIH K03<p<PHUHeHT A (x)

He

3aaHCHT

OT

x H,

KaK

B

HaweM

npHMepe,

A = I,

TO

MHHHMaJlbHoe BpeM5I

ycnoKoeHHH

- -...-.-

"j'

,,',

,:

i·

.~.:,

; fHnEPBOJ,Il<lllECKI1E

CI<lCTEMbl

,237

C

<c.

TaKliM

06pa30M,

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ynpaBAeHUe

(nyAb-ynpa:

BA

e-i1.u;e)

..

.z(J1H

BOJl.HOBOrO

ypaBHeHHH

(CTpyHbl)

6y)l.eT HMeTb

BH.n

U0(t)

=<'>

(t)-5(t-2n),

(26)

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ueJlhle

cepHH

caMoracHlUHxCH

ynpaBJIeHHH

nyTeM

"".\.';<'

. YMHO)KeHHH

OCHOBHOH

HyJlb-WyHKUHH

j

sin

nz

Ha

npOH3BOJll HyIO

ueJIyIO

<PYHKUHIO

HyJIeBOH CTeneHH y (z), ECJIH,

HanpHMep,

nOJIO)KHfb

.y(z)=z~

(ueJIaH

clJyHKuHHHyJIeBOH

CTeneHH),

TO

3TO

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COOT-

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TOMy,

lIro

JlI06ble

npbH3BO)l.Hhle

OT

<PYHKUHl1

(26)

no'

Bpe

1

MeHH

TaK)Ke6y.nyT

HyJlh-ynpaBJIeHHHMH,

T.

e.

u~

(t)

=

-4-

[6(nl

(t)

-

6(n)

(t

-

2n»,

(27)

, '

H JIio6aH JIHHeHHaH KOM6HHaItHH clJyHKUHH BH.na

u~

(t)

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6y~eT

Hynh-ynpaBJIeHHeM.

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BH)I.

HyJlb-ynpaBJleHHH

tdO*HO

nOJlYllHTh

nyTeM

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HliH

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sin

Uln

Ha

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no

t

opHrHHaJia

clJyHKUHH i

sin

Uln.

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06pa30M,

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ypaBHeHHH

6y.neT HMeTb

BH)I.

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n

BnpaB~)

o _ { 1/2

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0 ~ t

~

2n, 8

U-

1

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t <0, t > 2n.

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(t)

3TO

HyJlb-ynpaBJIeHHe

MO)KHO

3anHcaTh

B BH)l.e

1

(29)

U!.1

(t)

=2'

[1

(t)-

1

(t-2n)].

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ynpaBJIeHHH

C npOH3BOJlbHhlMH

ij:;lllaJIhHhlMH YCJlOBHHMH HaM 6y)l.eT y.n061'l0 B3HTh B KalleCTBe

HyJlb-

\"

'

..

ynpaBJleHHH

clJyHKUHlO

L1

(z)

= i

sin

nz

(30)

IIJlH

L1

(Ul) = i

sin

nUl.

(31) ,

OTCIO)l.a

L1'

(Zk)

=

L1'

(k)

:;=

in

cos

nZk

=

in

cos

kn

=

in

(-1)k.

(32)

'-

1"!

.1

1

I1TaK,

no

clJ0pMYJIe

JIarpau)Ka

AJIH onpe.neJIeHHH clJHHHTHoro

ynpaB-

JIeHHH u (t)

Ha)l.O

ItaliTH

06paTHoe

npe06pa30BaHHe

<l>ypbe

)l.JIH

cJle·

- nYIOxueH clJyHKUHH:

'"

. (33)

U(Ul)=

I:

~kn«i)~nk~~_l)k+Y(Ul)SinnUl,

,"\

k=-'"

'"

-

T=2

~

dx=2n.

(25)

r)l.e

~k

onpe)l.eJl5lIOTC5I ct>opMyJloA (22) H '\'

(z)

- npOH3BOJIbH8H H

o

Бутковский А.Г. Методы управления системами с распределенными параметрами

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