Бутковский А.Г. Методы управления системами с распределенными параметрами

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lITO lp H e 5IBJI5IIOTC5I

CJIYllaHHbIMH,

paBHOMepHO

pacnpe;ll.e-

JIeHHblMH

BeJIHlIHHaMH

co

SHalleHH5IMH B

.l\HanaSOHelp

E

[0,465;

0,548],

HE

[1,0;

1,2].

3TO

cooTBeTcTByeT

18%-HoMY

H

20%-HOMY

QrHOCH-

TeJIbHblM OTKJIOHeHH5IM BeJIHq.HH lp H e

OT

MHHHMaJIbHblX

3HalIeHHH

•

CootBeTcTBYIOII~ee

MaKCHMaJIbHOe OTKJIOHeHHe

rJIy6HHblueMeHTauuH

~TJmax

=

0,2.

BblllHCJIHM,

KaKoe

pacnpe.l\eJIeHH6

Bep05ITHOCTeH 6yJ.{eT

HMeTb BeJIHIIHHa

~TJ

B

3THX

YCJIOBH5IX . .llJISI BblllHCJIeHHH BOCnOJIb-

syeMC5ITa6JIHuaMH

CJIYllaHHblx

BeJIHlIHH C

paBHoMepHblM

pacnpe.l\e-

JIeHHeM B

oTpesKe

[0,

1].

CJIYllaHHble

BeJIHIIHHbl lp H e

clIHTaeM

HesaBHCHMbIMH.

nJIOTHOCTb

pacnpeJIeJIeHH5I

CJIyqaHHOH BeJIHIIHHbl

~TJ,.

nOJIyqeHHa5I

B

pesyJIbTaTe

npOBe.l\eHHSI

200

3KcnepHMeHToB

no

MeTO.l\y

MOHTe-KapJIo,

npe.l\CTaBJIeHa

Ha

pHC.

8.3.1.

Pacnpe.l\eJIeHHe

BepOSIT-

HOCTeH BeJIHqHHbl

~TJ

MaJIO

noxome

Ha

paBHoMepHoe

pacnpe.l\eJIeHife

HCXO,ll;HbIX

norpeillHocTeH.

Bepo5ITHoCTb

CJIyqaSI,

KOr.l\a

~TJ

6Y.l\eT

npH.Ha.l\JIe}KaTb

oTpesKy

[0,10;

0,125];

pacnOJIO}KeHHoMy

B Cepe.l\HHe

HHTepBaJIa,

npHMepHo

B 3

pasa

60JIbille

Bep05ITHOCTH nOna.l\aHHSI

Ha

oTpesKH

[0;

0,125],

[0,175;

0,200].

nYCTb

Tenepb

CJIYlla~Hble

BeJIjH-

qHHbl

lp H e

HeSaBijCHMbl

Ii

pacnpe.l\eJIeHbl

HOpMaJIbHO C

MaTeMa-

THqeCKHMH 0}KH.l\aHH5IMH

Mql

=

0,5065,

Me

=

1,1,

a

oTpesKH

[0,465;0,558]

H

[1;

0;

1,2]

SIBJI5IIOTCSI

95%-HbIMH

.l\OBepHTeJIbHbIMH

HHTepBaJIaMH

3THX

pacnpe.l\eJIeHHH.

nOJIYlleHHa5I

B

3THX

YCJIOBH5IX

nJIOTHOCTb

pacnpe.l\eJIeHH5I

BepoSITHocTeH

.l\JI5I

~TJ

npHBe.l\eHa··

Ha

pHC.

8.3.2.

3KcnepHMeHTbl

6blJIH

npOBe.l\eHbl

C nOMOLUblO

Ta6JIln~·

CJIyqaHHblX

qliceJI,

HMelOllI,HX

HOpMHpoBaHHoe

HOpMaJIbHOe

pa<:npe-

.l\eJIeHlfe.

OueHKa

95

%

-Horo

.l\OBepHTeJIbHOrO

HHTepBaJia

.l\JI5I

BeJIH':

lrHHbI

~TJ,nOJIYlleHHa5I

Ha

OCHOBaHHH

pesyJIbTaTOB

100

3KcnepH-

MeHTOB,

cooTBeTcTByeT

oTpesKy

[0,05;

0,20].

npHMep

nOKaSblBaeT,

qTO paH.l\OMHSaUH5I

norpeillHOCTeH

nOSBOJI5IeT

cymecTBeHHO

COKpaTHTb

pas6poc

rJIy6HHbl

ueMeHTaUHW

OT

Sa.l\aHHOrO 3HatleHHSI.

Bce

CKa3aHHoe

HMeeT OTHOIlleHHe K

JII06blM

.l\H<P<PYSHOHHbIM

HJIH

TenJIOBblM

npoueccaM,

MaTeMaTHllecKoH

MO.l\eJIblO KOTOpblX

SIBJI5IeTC5I

O.l\MepHOe

ypaBHeHHe

.l\H<P<PYSHH,

paccMaTpHBaeMoe

.l\JI5I

nOJIy6ecKo~

HellHoro

npocTpaHcTBa

..

B 3aKJIIOqeHHe

saMeTHM,

lITO

.l\JI5I

npOBe.l\eHH5I

oueHOK

pacope"·

.l\eJIeHHH Bep05ITHOCTeH

BeJIlfqHH.

no

KOTOpblM Mbl

oueHHBaeM

KaqeCTBO

.1

H cTa6HJIbHOCTb

npouecca

(B

HallleM

CJIyqaeB

KalleCTBe

TaKOH BeJIH':

qHHbl

6blJIa

npHHSITa

rJIy6HHa

ueMeHTaUJfH).

He

Tpe6yeTC5I

rpOMOS.l\-

KHX

BbIIIHCJIeHHH, eCJIH HCrrOJIb3YlOTC5I JIHHeHHbIe MO.l\eJIH

npouecca

ueMeHTaUHH.

nOCKOJIbKY

qHCJIeHHble

3KcnepHMeHTbl

npOBO.l\5ITC5I

. B 3TOM

CJIyqae

no

SIBHbIM

<popMyJIaM.

npHBe.l\eHHblM

B

[186],

06'beM

BbiqHCJIeHHH

.l\OnycKaeT

npOBe.l\eHHe

HX

BpylIHylO

HJIH. C nOMOLUblO

MaJIblX

UBM,

nprtcnocot$JIeHHbIX

.l\JI5I

HH1KeHepHblx

paclIeToB.

'.

~.

1

;.

468

YnPABJiEH~E

TEnJIO,IUi<%><%>Y3HOHC;~H

npoUEceA:MH

[r

JI.

S"

§

4.

Y

ftpaBJfeHHe

HarpeBoM

MaccHBHofO

Tena

C

yqeTOM

TepMO"HanpHlKeHHH

.

npH

HarpeBe

MaCCIiBHbI}{

TeJI

B03HliKaIOT BHyTpeHHlie

TeMnepa~

TypHble

HanpsDKeHIiH, KOTopble

MoryT

OrpaHlilJHBaTb

CKOpOCTIl

Ha-

rpe'Ba, oCb6eHHo B HaqaJIbHOH

lIH3KoTeMnepaTypHoH

CTa.n;lili.

Dpouecc

HarpeBa

.u.OJI)f{eH

6bITb npoBe.u.eH TaK, lJT06bI TepMOHanpSDKeHIiSI

He

rrpeJ3hllUaJIH MaKCHMaJIbHO .u.onyCTIiMbiX 3HalJeHHH

(c

TOqKIi 3peHiiSi

nOSlBJIeHHSI

pa3JIHlJHbIX MIiJ{po.u.eepeKTOB, a

TaK.>Ke

B03MO.>KHOCTH

pa3-

pymeHIiH

TeJIa). B qaCTHOCTH,

nplipemeHli1i

3a.u.aq

OnTIiMaJIbHOrO

no

6bICTpo.u.eHCTBIiIO

HarpeBa

MaCCIiBHbIX

TeJI

npli.n;eTCSI

YlJHTbIBaTb

'He

Tb'JIbKb OrpaHHlJeHHH Ha ynpaBJIHIOlll,ee B03.u.eHCTBlie

(TeMnepaTYPy

rpeIOw.eH

Cpe.u.

bI

),

HO

Ii.

OrpaHHlJeHHSI Ha epa30BbleKoop.u.HHaTbI

(Tep-

. MOHanpSl.>KeHHSI). ECTeCTBeHHO,

lJTO

pemeHlie

3a.u.alJi:!

OnTHMaJIbHOrO

no

6b1CTpo.u.eikTBHIO

HarpeBa

MaccHBHbI'o TeJIa C

YlJeTOM

OrpaHHtIeHHH

,

Ha'

TepMOHanpSl.>KeHHH ropa3.u.0

CJIO.>KHee,'

lJeM'

6e3 ytIeTa 9THX

orpa:-

'HlilJeHIiH

[49;

II,

110].

HH.>Ke

paccMoTpliM

O.u.HH

nplieM,

KOTOpbIH B

.n;aHHOM

cJIytIae

n03BOJIHeT OrpaHHlJeHHSI H8 epa30BbIe, Koop.u.HHaTbI 3aMeHHTb

orpa-

,

HHlJeHHSlMH

Ha ynpaBJISIIOlll,ee B03.u.eHcTBlie,

lJTO

B

CBOIO

OlJepe.n;b

cyw.ecTBeHHo

ynp0lll,aeT

BhI60p .u.OnyCTIiMOH

CKOpOCTIi

HarpeBa,

B

lJaCTHOCTIi

npH

peilleHli1i

3a.u.alJ

OllTliMaJIbHOrO ynpaBJIeHIiSi. ,

PaccMoTpHM

npouecc

HarpeBa

nJIaCTIiHbI

(6aJIKIi)

TOJIIll,HHOH

2S,

p,acnpe.u.eJIeHHe

TeMnepaTypbI

Q(x,

t)

BHYTPIi

KOTOpOH

onHCbiBaeTCSI

ypaBHeHHeM TennonpoBo.u.HOCTIi

Q=aQ",

O<t~T,

-S<x<S,

(1)

co

CJIe.u.yIOIll,HMH

HalJaJIbHbIMH

Ii

rpaHlilJHbIMIi

YCJIOBHSlMH:

Q (x,

O)=Qo,

(2)

~-a

aQ

I

=(J,

[u

l

(t)-Q

(S, t)],

(3)

x

x~s

-A.

~~

Ix:

-s

=

(J,

[u

2

(t)

- Q(:-.S, t)],

(4)

rAe

u

1

H u

2

-TeMnepaTypa

rp,eIOlll,eH

Cpe.u.bI,

a,

A.,

(J,-COOTBeTCT-

BeHHO

TeMnepaTyponpOBo.n;HOCTb

11

TenJIOnpOBo.n;HOCTb MaTepliaJIa

Ii

K09epepHUHeHT

TeoJIo06MeHa.

Pacqpe.zteJIeHlie

(J (x,

t)

TeMnepaTypHbIX HanpH.>KeHIiH B nJIaCTHHe

onliCbIBaeTCH COOTHomeHlieM

(CM.

[115])

0'

==

~

1~E

8 Q (x,

t)

+

2S

(11_

8

) 5

S

~EQ

(x,

t)

dx+

-s

s

f

2S&3/i:"6)

S

~'ExQ

(x,

t)

dx, (5)

,

-s

.

7·f

,,,-,,

(:',,:

;

C.~·

'S1

~(...

, ynPABJIEHH'E C Y'lETOM TEPMoHAnp'Sl)KEHHR

469,

, '

'rAe

~-Kb3<pepHuHeBT

JIHHefmoro

TeMoepaTyp-HOrO pacIllHp'eHHSI"

E:-MOlJ"yJIb

ynpyrocTH,

8_KO'9epepHuHeHT

TIyaccoHa,

K = 1

COOTBeT-'

cTByeT cBo6olJ"HOH nJlaCTHHe, K = °cOoTBeTcTByeT 6ecKOHetIHOH,

nJlac-,

'

TIiHe,

eCJlH

3aKpenJI~HHe

ee

KpaeB

npenSlTcTByeT

H3rH6y.

TIPH

HarpeBe

nocTOSlHHbIM TenJIOBbIM nOTOKOM, T. e. KorlJ"a

iJ'Q/ax

=const

(npli

x

==

± S), B peryJlSlpHOM

pe.>KHMe

pacnpelJ"eJIeHHe

TeMnepaTypbI

no

TOJIIll,HHe nJlaCTHHbI

MO.>KHO

onHcbIBaTbypaBHeliHeM

napa60JIbi

',.0

Q(x,

t)

=c

(t)

+c'

(X+C")2,.o

(6J

rlJ"e

c(t)_JlHHeflHaHepYHKUHH

BpeMeHH,t,

c'

H

C"-KoHCTaHTbI.

'ECJIH

~

==

const,

E =

const

H

TeMnepaTypHOe

pacnpelJ"eJIeHlie

onlicbIBaetcH

ypaBHeHHeM (6), MaKcliMaJIbHble pacTHrFlBaIOIll,He (Jmax

Ii

C.>KHMaI<>Ill,He

(Jmln

TeMnepaTypHble

HanpH.>KeHH

H

B COOTBeTcTBHIi C (5) onpe,/le-

JIHIOTCH

epopMy

JIOH

~E

S

aQ

\ '

(7)

(Jlllax

min

=

1-8

'3

ax

x:s

f (11),

f,/le

f

(11)-epYHKuIiH

01'

K09epepHulieHTa HecHMMeTplittHOCTH

BarpeBa

11

= (S +c")/2S.

3lJ"eCb

npHHHTO, ,tITO TenJlo06MeH C

OlJ"HOH

CTOpOHbI

nJIaCTHHbI

(x

= S)

HBJIHeTCH

«BelJ"yIll,HM», a C

~pyroH

CTOpOHbI

nJI1i-

CTi:!HbI

(x

=_

S)

OH

ytIH1'bIBaeTcH

tIepe3

K09epepHuHeHT HecHMMeTpHtI-

HOCTH

HarpeBa

11.

TaK,

npH

CIiMMeTpHtIHOM

HarpeBe

11

= 0,5, a

npH

OlJ"HOCTOpoHHeM

HarpeBe,

C TenJIOH30JIHpOBaHHo

H

.n;pyroH

CTOpOHOH

nJIaCTHHbl, 11;= 1,0. '

<l>yHKUHSI

f

(11)

onpelJ"eJIHeTCH cJIelJ"YIOIll,IiM

06pa30M.

,UJlH nJlaCTHHbI:

1

3

(11

-1)

+

-,:L

,

11

< 1 }

lJ"JIH'

(J

11

~

1

max'

2

3--

J.L

'

f(I1)=

2

3--

11<°,5

}

J.L

'

1 '

11

~

0,5

lJ"JIH

(Jmln;

-,:L

,-3,

6aJIKH:

lJ"JlH

cB060lJ"HOH

1 '

4'

lJ"JlSl

(Jmax'

f(I1)=

{

-2~

.u.JIH

(Jmln:

TaK

KaKnpH

HarpeBe

Hali60Jlee

onaCHbI paCTSlrHBaIOI1lHe

HaIill'll·'

)KeHIiSl,

BBelJ"eM

orpaHHtIeHHe:

(Jmax

~

(J~ax,

t8)

r,/le

(J~ax

_ npe,/leJlbHO ,/lOnycTHMOe pacTHfHBaIOUI.ee HanpH.>KeHHe.

,

"

:~

\

.

'~:'

"

\

470:

TEriJIO.llH4>4>V':HiOHQH

npoQECCAMH

l'>-TeMnepaTypa,

rrpH

KOrOpoHMaTepHaJI

HMeeT,lJ.OCTaTo'lHylO

r,lJ.e

XO,lJ.HMO

Q

(S,

t).

MeHH

'IeHHe

r,lJ.e

MaJIbHOe

•.JIeHHIO

',-D.n;e

nJIaCTH'IHOCTb

,lJ.JIH

rroraIIIeHHH

TepMOHanpH.>KeHHH,

MO.>KHO

orrHTb('

'J'aKH

Y'lHTbIBaTb

TOJIbKO

OrpaHH'IeHHe

(10).

MOMeHT

BpeMeHH

t

a

,

KOr,lJ.a

iorpaHH'IeHHe

(9)

Tep5IeT

CHJIy,

MO.>KeT

ObITb

orrpe,lJ.eJIeHH3

BbIpa2i<eHH5I

VnPABJIEf:l~E

crJI,8:

,

Y'lI:fTbIBa~

OrpaHHQeal1e

(8),

MO.>KHQ

onpe.n;eJIIfTbnoepopMYJIe

(n,

MaKCHMaJIbHO

.n;onycTHMoe

3Ha'leHHe

TenJIOBoro

nOToKa,

T.

e.'

max

aQ

lax

(npH

x = S),

I:f

B

CBOIO

O'lepe.n;b

no

rpaHHIU{OMy

yCJIO'

BHIO

(3)-orpaHH'IeHHe

Ha

TElMnepaTypy

rpelOl1.{eH Cpe.n:bI:

C

l

u (t) = A

(t)

= Q (S,

t)

+

as'

(/4)

,:

(9)

r.n;e'

Cm=3A(1-8)0'~ax/~E-K03epepH~HeHT,

3aBHCHl1.{HH TOJIbKO

OT

MaTepHaJIa

HarpeBaeMoro

TeJIa.

IlpH

U

I

(t)

= A

(t)

HMeeT

MeCTO MaKCHMaJIbHO .n;orrycTHMbIH

(rro-

CTOHHHbIH) TerrJIOBOH rrOTOK

Ha

rroBepxHOCTb

x =S;

T.

e.

BbIIlOJIHHeTCH

yCJIOBHe

peryJIHpHOro

pe.>KHMa

HarpeBa,

npH

KOTOpOM cnpaBe.n;JIHBO

BbIpa.>KeHHe

(6). Cne.n;oBaTeJIbHO, B

peryJIHpHOM

pe.>KHMe

HarpeBa

MO.>KHo

'Iepe3

BHeIIIHHH

TerrJIOOOMeH Cy.n;HTb 0

TeMrrepaTYPHbIX

iIa-

rrpH.>KeHHHx B

nJIaCTHHe

H

B6aJIKe.

'.

PaccMoTpHM

Terrepb

rrpaKTH'IeCKoe

rrpHMeHeHHe

'BbIpa.>KeHHH (9),

rrpH

COOJIIO.n;eHHH

KOToporo

BbIIlOJIHHIOTCH

OrpaHH'IeHHH

Ha

TeMne-

paTypHbIe

HarrpH.>KeHHH.

IlycTb

Ha

TeMnepaTypy

rpelOl1.{eH

cpe;itbI

HaJIO.>KeHO,

KpoMe

Toro,

orpaHHlIeHHe

UI(t)~A2=const,

O~t~T.

(10)

JJ.,JIH

rrpoCTOTbI

C'IHTaeM,

'ITO

Ha'laJIbHaH

TeMrrepaTypa

nJIaCTHHbI

Q (x, 0) =

const

= Q

0"

H

TepMOHanpH.>KeHHH

OTCyTCTByIOT,

T.

e.

a (x, 0) =

0.

B

Ha'laJIbHOH

CTa.n;HH

HarpeBa,

Iwr.n;a

paCTHrHBalOl1.{He

TepMOHarrpH.>KeHHH

He

.n:OCTHrJIH el1.{e MaKCHMaJIbHO

.n;onycTHMoH

BeJIH'IHHbI,

HeoOxo.n;HMO

Y'lHTblBaTb

TOJIbKO

OrpaHH'IeHHe

(10).

Ha'lH-

HaH

C MOMeHTa t

10

Kor.n;a

am

ax

(tI)

=a~ax,

HeooXO.n;HMO,

KpOMe

orpa-

HH'IeHHH

(10),

Y'lHTbIBaTb

el1.{e

OrpaHH'IeHI:fe

(9).

MOMeHT

BpeMeHH

ti.

MO.>KeT

ObITb onpe.n;eJIeH

H3.

BbIpa~eHHH

Q

(S,.

ti)

= Q

o

+

LlQ

m8Xl

(11)

r.n;e

LlQmax-MaKCHMaJIbHO

.n;orrYCTHMbIH neperra.n;

TeMrrepaTypbI

rro

TOJIl1.{HHe rrJIaCTHHbI C

TO'lKH

3peHHH

.n;orrycTHMbIX

TepMOHanpH.>KeHHH,

onpe.n;eJIHeMbIH

H3

epOPMYJIbI

0"

-

~E

.1Qmax'

(/4)

max

-

1-8'

3

p,2'

(12)

CTporo

rOBOpH,

epOPMYJIa (12) cnpaBe.n;JIHBa

rrpH

peryJIHpHOM

pe.>KHMe

HarpeBa.

B MOMeHT

BpeMeHH

t

,

orrpe.n:eJIeHHbIH H3 (11) H

1

12

I

)',

:zteHCTBHTeJIbHO

HMeeT

MeCTO

O'max

(II):%;

q~u,

To'

e.B

MOMeHT

tt,

HMeeTCH

3arrac

no

TepMOHanpH.>KeHHHM.

ECJIH

min Q (x, t)

~

(\

-s,x,s

CiPA~JJE~I1E

C YQETOM TEPMOHAnp5!)I(EHl1fl

.4(1

Q (S, t

2

)_l'>+LlQmax'

(13)

Cne,lJ.OBaTeJIbHO,

,lJ.JI5I

BbIIlOJIHeHHH

OrpaHH'IeHHH

Ha

BHyTpeHHHe

TepMOHanp5I.>KeHH5I (8)

rrpH

HCrrOJIb30BaHHH

COOTHOIIIeHH5I (9)

Heoo-

HerrpepbIBHo

H3MepHTb

TeMrrepaTypy

rroBepxHocTH

rrJIaCTHHbI

ITo

BbIpa.>KeHH5IM

(11)

H

(13)

orrpe,lJ.eJI5IIOTCH MOMeHTbI

Bpe-

11

H t

2

,

Me.>K,lJ.Y

KOTOPbIMH

.lI.0JI.>KHO

ObITb BbIIlOJIHeHO

orpaHH-

(9), HCrrOJIb3YIOm;ee TaK.>Ke

TeKym;ee

3Ha'leHHe

TeMrrepaTypbl

rroBepxHocTH

Q(S,

t),

tI~t~t2'

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nOBepxHocTH:

Q

(S,

t)~

Qt.

JJ.JISI

pelIIeHHSI

CHCTeMbI

ypaBHeHHH

(3)-(6)

C,lteJIaeM

nepeMeHHhlX

no

~opMYJIaM

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(x,

t)

+

Q.

*+

~*_l~c

(x

-8),]

u(t)=v(t)+Q*,

..

x=l(t)y,.

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~

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npOH3BOJIbHOH,

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JIHlIIb

TOJIbKO

KycOqHO-HenpepbIB-

l* 3a.ztaH- "

CBH-

n

C

(12)

(13)

SaMeHy

HMeTb

],

HOH,

~YHKUHH

v(t)

pelIIeHHe

CHCTeMbI

(3)-(6)

B

KBa3HCTaUHOHap-

HOM

npHOJIH)KeHHH

C

yqeTOM

HOBbIX

nepeMeHHblx

(13)

oy,lteT

.BH,lt

q (y,

cp)

= J

. =

i;

An(y,

l)[(Qo-Q*)e-~~(l)lp+

S

lJ.~(l)e-~~(l)

(q>-,)

W(S)d

s

n=J

0

r,lte

A l _ 2

[Si~:(l~lr

COS/Ln(l)]

sinf/Ln(l)(I-y)J

n (y, ) - ,

/Ln

(I) - sin

/Ln

(I) cos

/Ln

(l)

,

lJ.n·-

nOJIO)KHTeJIbHhle

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KOPHH

TpaHCueH,lteHTHoro

ypaB-

neHHSI

~

=Bi

(l)

=-lJ.

ctg

lJ.

(:Bi

-KpHTepHA

5.HO)

,

6e3pa3MepHoe

BpeMSI cI>YPbe

t

S

ad.

cP

= fl

(.)]2

o

Il,

"HaKoHeu.,

.

Qc-Q*

"

v(<p)+~

W(cp)

= I

I

+Bi

(I)

B

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nepeMeHHblx

KOHeqHOe

pacnpe,lteJIeHHe

TeMnepaTypbI

B

3aTBep-

v .

aT

o

0

.zteBlIIeH

qaCTH

CJIHTKa

npH

CP.

CPo

=

(l*)2

y,lteT

HyJIeBblM,

T.

e.

q(y,

CPil)=O,

O~y~l.

(15)

•

~

.~,,:'

'''I.

~-;.

~

p";-.,

(

..

~:';

§

5)

CJIHTOK C )l(H,lu<oR CEP,ltUEBHHOR

.41.7

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B

KBa,ltpaTHbIX

CJ{OOKax

~OPMYJIbI

(l4),OQQ--

3HaqHTb

qepe3

X

n

(Ip),

TO

JIerKO

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'ITO

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x"

(Ij):i.

=

1,

2~

...

, y,ltOBJIeTBOpSIeT

06bIKHoBeHHoMy

,ltH~~epeHu.HaJIbHoMy';

ypaBHeHHID

nepBoro

nOpSI,ltKa

X

n

(cp)

=

-lJ.~

(l) X

n

(l) +

lJ.~

(l)

W

(cp)

(16)

Ha'laJIhHhlM

yCJIOBHeM

Xn(O)=

Qo-Q*

=-v.

(17)

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n

(cp),

KaK

6bIJIO

nOKa3~HO

B §

17

rJIaBbI

3, HMeIDT

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OT

BpeMeHH

K03~~HUHeHTa

npH

npOCTpaHCTBeH-

HbTX

rapMoHHKax

pa3JIO)KeHH5ITeMnepaTypbI

B CJIHTKe B

rapMOI;!H-

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C

npHeMJIeMOH

)l:JI5I

npaK-

THKH

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~YHKUH5IMH

Xl

(cp)

H

X

2

(Ip).

.

n03TOMY

C

yqeTOM

HOBblX

nepeMeHHblx

(13)

ypaBHeHHe

(1)

npH-

MeT

BH,lt

2

X

s

(Ip)

=

~

CnX

n

(cp)

+K

(18)'

n=1

npH

HaqaJIbHhlX

YCJIOBH5IX

.,'\..

X

3

(0)

=lo-l*,

(19)

r,lte

C = 2 [sin

/A-n

(l)

-

/Ln

(l)

cos

/Ln

(l)]

~l

n

/Ln

(I) - sin

/Ln

(I) cos

/Ln

(l)

a'

K

.

g..!:

Q*-

Qc

f'

a

S-l*

.

I

,~!

I

YpaBHeHHe

(8) C

yqeTOM

(13) H (16)

nepenHIIleTcSI

B CJIe,ltyIDlII,eM

I

BH,lte:

, 2

w

aT

dd

=koliU(Ip)-l'

[w(cp)+Q*]

+k

l

2

~

An

(O)X

n

(cp)

(20)

<p

n~1

C

HaqaJIbHhlM

yCJIOBHeM

Qc-Q*

V

(0)-

B'

(I)

w(O) =

II

(21)

I+

Bi

(l)

Tenepb

"3a,ltaqa

OUTHMaJIbHOrO

ynpaBJIeHH5I

no

ObICTpO,lteHCT-

BHID MO)KeT

6bITb

C~OPMYJIHpOBaHa

CJIe,ltyIOlII,HM

06pa30M.

Tpe6yeTcH

HaHTH

ynpaBJIeHHe

U

(Ip),

nepeBO,ltSllU,ee

cHcTeMy

(16), (18), (20)

H3

.

,~.

,..,

Бутковский А.Г. Методы управления системами с распределенными параметрами

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