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)l.OnOJIHRTeJIbHblH nOTOK

qo

(t)"paBeu

-qQ

(t) =q-qJ

(t).

(3)

r

~>

'384

VnPABJlEHI1E

BOJiHOBbIM.lnpOUECCAJM1

LIJHl CJIyll8H. KOr.na

q1

(l)ltMeeT

Blt.n~

H306pameHHblH

RapHc.

6.7.2,

rpa<p

HK

.nonOJIHI1TeJIbHOrO

nOTOKa

qo

(t)

HMeeT

BH.n.

H306pameHHbIH

Ha'pHc.

6.7.3. TaKI1M

06pa30M.

Hallo

C03,llaTlillonoJiHHTeJIbHbIH

no-

TOK

qo

rn;

BeJIlilIliH8 KOTopor6 B KamnbIH

MOMeHT

BpeMeHH t paBHa'

pa3HOCTH

tpenHero

3HaqeHHH nOTOKa q H MrHOBeHHoro

3HaqeHHYl'

I

~

..

!t

I'

,:

TP,~g;R~

-'

PHC.

6.7.4.

)

~

nOTOKa

ql

(t), KOTOpbIH .naeT

OCHOBHOH

UHJIHH.np.

PeaJIH3aUHIO:Horo

yCJIOBHH,

T.

e.

yCJIOBHH

(3). MomHO

ocy~ecTBHTb3a

CqeT

aBTOMa-

,THlIeC}(OH

tJIe.nH~eH

CHCTeMbI,

~bIIIOJIHeHHOH,

HanpffMep;

no,

txeMe,

IIf306p8)KeHHoH

Ha

pHC.

6.7A: ' ,

§ 8. Mo.neJlHpOB8HHe

CHCTeMbI

ynpaBJlSleMoro rarneHHH nYJlbcaQHH

.

ra3a

B TpyoonpoBo.nax (nepeMeHHble

eMKOCTb

H conpOTHBJleHJfe)

PaccMoTpHM

cxeMy,

H306pameHHyIO

Ha

pHC.

6.8. L

npouecc

H3-

MeHeHHH

aaBJIeHHH P B CeqeHHH I HarHeTaTeJIbHOrO

Tpy60rip

oBo

.na

3aBHCHT

OT

xapaKTepa

li3MeHeHHH

eMKOCTH

V

Ha.n

nOpUIHeM n pe-

rYJIHTopa.

<Plt3HlIecKH

HCHO

Ii

nocJIellYIOlQee Mo.neJIlipOBaHlie TaKHX

npouec-

COB

Ha

ABM

(aHaJIOrOBQH BblllHCJIHTeJIbHOH

MaUJHHe)

nOKa3bIBaeT,

liTO MomHO HaHTIi

TaROH

3aKOH

.n'BH>KeHHH

nopUJHH n

peryJIHT6pa.

liTO nyJIbCaU'HH .naBJIeHHW B CeqeHJHt I

Tpy6onpoBona

6y.nYTCBeJl.eHbJ

.no

CKOJIb

yro.nHO MaJIOrO ypOBHH.

MeTOJl.bI

opraHH3aUHIi

TaKoroJJ.BHmeHHH nopUJHH n (a

cJieJl.OBa-

TeJIbHO,

Ii

li3MeHeHHH 06beMa V Han nOpUIHeM) H Mo.neJIHpOBaHHH

paCCMOTpeHbl .naJIee B

3TOH

r

JIaBe,

.,

AHaJIOrHlIHaH

cHTya~IHH

HMeeT

MeCTO

H B

TOM

CJIyllae, Kor.na pe-

ryJJHpylOr1r,HM opraHOM, npe.nHa3HaQeHHblM llJIH rarneHHH

nyJibcaUHH,

cJIymwt

He'

nepeMeliHaH

eMKocTb, a nepeMeHHoe

COIlpbT~IBJIeHHe.

<PH~

,~uqeCl)H

nepeMeHHoe

COUpOTH!3.neHl'le

MomeT

npe.1lCTaBJI1m;o

COOOH

,,"'"

.'"

-I

-C""

',',;/

MO.nEJ1HPOBMII<fE

CHCTEMbI

rAI1lBHI<f51

nYJlbCAUHA:

385

,!Ii!1a<pparMy ynpaBJISleMOrO

nepeMeHHoro

celleHIHl,

3aCJIOHKy

If

.np.

I!O-BH.nHMOMY,

3HepreTHlIecKH

B 60JIbWHHCTBe CJIyllaeB Bblro.nHee

_,

B'

KalleCTBe

peryJIupylOw.ero

op-raHa

Bce-TaKH npHMeHHTb nepeMeH-

H-YIO

eMKOCTb.

I

';'

PaccMoTpHM

ytlaCTOK

Tpy6onpOllO.na

C BCTpoeHHhIMUHJIHH.npOM,

,8

KOTOpOM

MO>KeT

nepeMew.aTbCH

nopweHb

n

(peryJIHTOp

.naBJIeHiHl).

:1

3.neCb

P

c-

.naBJIeHHe B UH-

JIHH)I.pe

peryJIHTOpa

(ClIHTaeM

I

~'ero

COBna.nalOw.HM C .naBJIeHH-

I

eM

B celleHHH I

Tpy6onpoBO-

I

.na), V

-o6beM

Ha.n

nopWHeM.

"

,.lI.JI51

Mo.neJIHpOBaHH51

y,n06Hee

I

i

nepeHTtI

K

MaCCOB,bIM

pacxo-

j

.naM. I10:)TOMy

nycTb·

G

li

,

~n

.

G

m-

MaCCOBble

pacxo.nbl

ra3a

B

CelleHH51X

II

H I I I

Tpy6o-

npoBo.n

a

•

PRC.

6.8.1.

By)I.eM C'lHTaTb, liTO

npo-

II.eCCbl

B

U~JIHH.npe

npoxo.n51T

npHnOCT051HHOH

TeMnepaType.

Tor.na

cnpaBe.nJIHBO

ypaBHeHHe

co·

, I

CT051HH51

PV=gRTu.

M

,

(1)

r.ne g""",;,ycKopeHHe CHJIbl

TH>KeCTH,

R-ra30Bafl:

nOCT051HHa51,

Tu.-

TeMnepaTypa

B II.HJIHH.npe peryJI51TOpa, M -

Macca

ra3a

B II.HJIHH.npe

perYJIHTopa.

I1pO)I.H<p<PepeHII.HpOBaB

o6e'lIacTH

ypaBHeHH51 (1)

no

BpeMeHH t,

nOJIyllHM

dP. dV

dM

VdT+P

dt

=gRTu.(ft.

(2)

113

(2)

BH.nHO,

liTO

eCJIH

npHH51Tb 3aKOH

H3MeHeHH51

06beMa

B

BH.ne·

dV

__

1_

[

dM(t)

dt.-P(t)

gRTu.-ar-+~(P(t)-'-P*)J,

(3)

r)I.e

~-npOH3BOJIbHOe

nOJIOR<HTeJIbHOe 'lHCJIO,

P*-HeKOTopoe

(R<e-

JIaeMoe).naBJIeHHe,

TO

Ha

OCHOBaHHH

(2), (3)

)I.JI51)I.aBJIeHH51

P (t)

B II.HnHH.npe perYJI51TOpa cnpaBe.nJIHBO

ypaBHeHHe

.

~

V

(t)

d:/f)

+P

(t)

=P*.

(4)

npH

V (t) > 0

(c

YlleToM

orp~HHlIeHHOC1'H

nOH

<pyHKII.HU)

yp'aB-

H~HHe

(4) onHCblBaeT aCHMnTOTHtleCKH yCTOHlIHByIO cHcTeMy, TaK

'lTO

P

(t)

---+

P*.

Bbl6opOM

)I.OCTaTOtIHO

60JIbWOrOK03<p<pHUHeHTa

~

3TO

CTpeMJIeHHe MomHO C)I.eJIaTb

)I.OCTaTO'lHO

6bICTpbIM. <PH3HlIeCKH

13

A.

r.

5yl'KOBCKliA

"

YnPABJJ~HHE"--~O~~OB

bIM[1P01'1ECCAMH'

386

3TO srCHO:

ypaBHeHHi

(4)-3T0

ypaBHeHHe«HHepII,HOHlfOrO»

, , 1

c,

nepeMeHHOH

«nocTOHHHoib>

BpeMeHH

't=<f

V

(t~.,

I1TaK,

3aKOH

ynpaBJIeHHH

(4)

n03BOJIHeT

no.zmep>KHBaTb

,!{aBJIeHHe'

IfJlU

no

dV

([[

'iIeHHH

SBlma:

BceA

CKO-

(5)

B O,!{HOM

H3

celIeHHH

Tpy60rrpOBO){a,

a

CJIe~OBaTeJIbHO,

H

BO

Tpy6e

npaBee

perYJIHTOpa

Ha

>KeJIaeMOM

3HalIeHHH

p•.

D:JIH

cTa6HJIH3aII,HH

Cpe){HerO

nOJIO>KeHHH

nopillHH

B

~HJIHH){pe

perYJIHTOpa

He06xo){HMO

Bbl6HpaTb

p.

'paBHblM

Cpe){HeMy

){aBJIeHHIO'

PCP

B

celIeHHH

I

Tpy6hI.

PeaJIH3aII,HH

3Toro

3aKOHa

perYJIHpOB8HHH'

CBH3aHa

C

He06xo){HMOCTbIOBbIIIHCJIemi1f

npOH3BO){HOH

dM/dt,

pOCTH

H3MeHeHHH

MaCChI

ra3'a

B

II,HJIHH){pe

perYJIHTopa.

3Ty

npOH3-

BO){HyIO

MO>KHO

BhIlIHCJIHTb,

HCnOJIb3yH

COOTHOIIIeHHe (CM.

pHC.

6.8.1)

dM

di

=

GII-G

m

,

r){e

pa3HOCTb

B

npaBoil:

IIaCTH

<pOpMyJIbI

,(5),

KaK·

6y){eT

nOKa3aHO

HH>Ke,

MO>KHO

BbIIIHCJIHTb

(npH6JIH>KeHHo)

IIepe3

){aBJIeHHe

ra3a

B

ce-

lIeHHHX

I,

IV

H V

Tpy60npOBO){a.

'

HapH){y

C

3aKOHOM

(4)

paccMoTpHM

TaK>Ke){pyroH

(<<yJIyIIIIIeH-

HbIH»)

3aKOH

H3MeHeHHH

o6beMa

V,

a

HMeHHO:

dV

1 [

dM'

-,

dP]

dt=P(/)

gR~ndi+{3(P(t)-P*)+(V-V(t»dT'

(6)

3){eCb

V

-HeKoTopblH

<pHKcHpoBaHHbIH

06beM.

HeTpy){HO

BH){eTb,

<iTO

3aKoHy

(6)

cooTBeTCTByeT

cJIe){YIOru.ee

ypaBHeHHe,

onHcbIBaIOru.ee

npoII,ecc

H3MeHeHHH

){aBJIeHHH

ra3a

B,

II,HJIHH){pe

perYJIHTopa:

~

V

d:t)

+P (t) =

P*.

(7)

3){eCb,

B

OTJIHIIHe

aT

(4), V=

canst,

TaK

liTO

(7)

OTpa>KaeT

HHep-

ll.HOHHOe

3BeHO

C

HeH3MeHHOH

nocToHliHOH

BpeMeHH

't,

=13

]-

V.

IlpH

3TOM

P(t)=P(O)/J-:

t+P.(I_e-:

t

).

(8)

Bbi()OPOM

K09<p<P

H

II,HeHTa

~

MO>KHO

nOJIYIIHTb

KaKOH

yrO){HO

6hICTPbIH

npOII,ecc

yCTaHOBJIeHHH

){aBJIeHHH

P (t)

Ha

ypOBHEl

P*.

IlpOH3BO){HyIO

dM/dt,

KaK

H»

(6), MO>KHO

BbIpa3HTb

qepea

){BB-

JIeHHe

ra3a

P B

II,HJIHH){pe

H

){aBJIeHHe

ra3a

P

IV,

P

v

B

CeqeHHlIx

IV

'H

V

Tpy60npOBO){8.

KaK

6y){eT

HCHO

H3

){aJIbHeHiliero,

t

J!!. '

2S

TP

S'

tit

=gRTnt;l

[P

I

v('t)-2P('t)+Pv('t)]d't,

(9)

o

'·MO,Z1,E.riHP08AHMi;:

nEPEMEiIHor:I EMKoCnl

C?

387

d2M - 2·S

Tp

,

dt

2

=gRT

n

--;v-[P

IV

-2P+P

v

],

(9a)

'r){e

STP'~nJIoru.a){b

CeqeHHH

Tpy60npoBo.u.a,

~l-paCCTOHHHe

aT

ce-

I ){O

celIeHHH

IV

H

V.

'OTCIO){a

3aKOH

peryJIHpOBaHHH

(6)

MO>KHO

npe){CTaBHTb

B

BH){e

" t

1,"

2S

Tp

r ',

= P

(t)

gRT

n

t;l

~

[PIV

t

r

)-2P

('t) + Pv('t)]

d't+

+~

[P

('t)-P*]

+

[if

- V (t)]

~

.

(~O)

YnpaBJIeHHe

no

ypaBHeHHIO

(10)

6y){eM

Ha3bIBaTb

ynpaOJleliUeM

Memooy

UOeaJlbl-tOeo

08U:JICeHUft.

§

9.

~O){eJIHpOBaHHe

nepeMeHHoH

eMKOCTH

.

!

JiI3MeHeHHe

TaKa

i (t) B

KOH){eHCaTOpe

nepeMeHHOH

eMKOCTHC

(t)

onUCbIBaeTCH,

KaK

H3BeCTHO,

ypaBHeHHeM

i (t) =

~

[c

(t) U

(t)],

(1)

r){e

u

(t)-HanpH>KeHHe

Ha

KOH){eHCaTOpe

(pHC.

6.9.1).

I1){eH

MO){e-

JIHpOBaHHH

nepeMeHHoH

eMKOCTH C (t)

COCTOHT B TOM,

qT06bI

B

KOH-

'Hi)

rll}

){eHCaTOpe

HeKOTopOH

npOH3BOJIb-

HOH nOCTOHHHOH eMKOCTH

Co

nOk

){ep>KHBaTb

TOK i

(t),B

TOIIHO-

u(t)

Co

uo(t)

I

to

CTH

paBHblH

~

[c

(t) u

(t)],

3a

CqeT

~t--=:o

---i(t)

H3MeHeHHH

nOTeHUHaJIa

uo(l)

Ha

eMKOCTH

Co,

T.

e.

B

cxeMe

PHC.

6.9.1.

PHC,

69.2.

pHC.

6.9.2.

OlIeBH){HO,

tn06bI

nOk

){ep>KHBaTb3ToT

TOK i (t), ){OJI>KHO

BbmOJIHHTbCH

paBeHCTBO

.

('

t")

= d

Ie

(t)

u (t)l =

deo

[u

(t)

-Uo

(t)J

(2)

t dt

dt'

LlJIH

BbInOJIHeHHH

3Toro

paBeHCTBa

){OCTaTOqHO

BbInOJIHHTb

yCJIOBHe

C

(t)

U (t)

=:=C

o

[u

(t)-u

o

(t)].

(3)

Pa3peiliaH

3TO

IpaBeHCTBO

OTHOCHTeJIbHO U

o

(t),

Mbl

H

HaH){er;t

Tpe6yeMoe

H3MeHeHI1e

HanpH>KeHI1H

U

o

(t) B

cxeMe

C

nocToAHHOH

13*

389

388

eMKOCTbIO

Co

(pRC.

6.9.2),

Mo,nemlpYlOmee

nepeMeHHylO

eMKOCTb C(t):'

, ,

VnPABJlEHRE

BOJlHOBbIMC'~OUECCAMR

~~.:.

U

o

(t)

=-U

(t)-c(t)

U

(t).

Co

OCTaeTCH TOJIbKO

peam130BaTb

B 3JIeKTpOHHOH

Mo,neJIHpyromeA

yCTaHoBKe

q>OpMyJIy (4).

CxeMa

STon

MOJJ.eJIH

H300paJKeHa'

Ha

pHC.

6.9.3.

~

am

~

-art}r

I I

C

q

kftJ

Illi

IfHO)/(llmeObHOe

11

2

38eHO

R

s

PRC.

6.9.3.

"..

npOBepHM,

lITO

cxeMa

pHC.

6.9.3 JJ.encTBHTeJIbHO

06ecnelIHBaeT

TOK i

(t)

d

[e

(~/

(t)j.

113

cxeMbI

pHC.

6:9.3 HMeeM

u

(t)]

. _

d[u(t)-uo(t»_

d

[

U(t)-uo(t)+e(t)c;;-

_de(t)u(t)

t (t) -

Co

dt

-

Co

dt -

dt

'

qTO H TpeOOBaJIOCb

nOKa3aTb.

ConpOTHBJIeHHe

R

1

JJ.OJIJKHO

BbIOHpaTbC5I B03MOJKHO

60JIbillHM,

.lJTOObI

He

nOJJ.rpyJKaTb HCTOlJHHK

Hanp5lJKeHH5I

u (t),

KpOMe

Toro,

R

4

/R

z

=I,

RaIRz=lOOjc

o

'

L

T

lr

L

T

ll(tjr

~E

I

;c

m

Puc. 6.9.4.

PRC.

6.9.5.

TaK

KaK

eMKOCTb

Co

51BJIHCTCH

BeJIHlJHHOA

npOH3BOJIbHOA,

TO

ee

HaJJ.o

BblopaTb

onTHtdaJIbHO:

He

OlJeHb

150JIbrnoii,

qTOObI

Harpy3Ka

He

ObIJIa

BeJIHKa,

H

He

OlleHb,

MaJIeHbKOH,

lITOObI

He

ObIJI BeJIHK

K03<p<P

HI

J,HeHT Ile

o

'

.-

....

;

C

~

'"

:,i.j.

SJlOK

VCPE,nHEHHS!

I1TaK,

cxeMa

pHC.

6.904

npe.n.cTaBJIHeT

coooii

OJIOK

nepeMeHHoH

-,

eMKOCTH

(fl

E).

BXC?JJ.HbIMR

BeJIHlJHHaMH 31'OrO OJIOKa 5IBJI5IIOTCH u (t)

'H~C(t).

CxeMa,nO.D.KJIlOlJeHH5I OJIOKa

nepeMeHHoH

eMKOCTH K Mo.n.eJIH TPYOO-

npoBo.n.a

nOKa3aHa

HIl

pHC.

6.9.5. L

T

H

cT-napaMeTpbI

Mo.n.eJIH TPYOO-

,npoBo.n.a

(HH.n.yKTHBHoCTb H

eMKocTb).

-::-

§ 10. BJIOK ycpe.n.HeHHH

)lJISI

nOJIYlJeHH5I cpeJJ.Hero ..n.aBJIeHHH P*,

HrpalOmero

POJIb

3a-

.n.JI5I

CHCTeMbI

ynpaBJIeHH5I,

Heooxo.n.HMO BbIlJHCJI5ITb BeJIHlJHHY

o

1

ST

'

P*

=

T~

P

(1,

or)

dT,

(1)

o

r.n.e

T

o

-H8HOOJIbillHA

nepHO.n. KO,lIeOaHHH, P

(1,

t)-,llaBJIeHHe

B TOlJKe

x = 1 B MOMeHT

BpeMeHH

t.

TaK

KaK

OnepaIJ,HH (1) B aHaJIOrOBOrr

MaillHHe

peaJIHsyeTcSI

CJIOJKHO,

TO

JKeJIaeMOe P*

MOJKHO

nOJIYlJaTb

Henpe·pbIBHd,

T.

e.

KaK

<PYHKIJ,HIO

BpeMeHH

t,

nponycKaH

P (t,

;)

lJepeSanepHo.n.HlJeCKOe

3BeHO C nepe.n.aTOlJHOH <pYHKIJ,Heii

W (p) =

(T

+1)-1,

p

KOTopoe

OlJeHb npOCTO

peaJIH3yeTc5I

B Mo.n.eJIH•

E.n.HHCTBeHHoe,

lJTO

HyJKHO,-

3TO

no.n.oOpaTb

COOTBeTCTBYlOmylO

nocToHHHylO

BpeMeHH

T

3Toro

SBeHa.

aHa

onpe.n.eJISIeTCH 3a.n.aHHeM

nepHo.n.a

ycpeJJ;HeHHH

To,

COBna.n.alOmero C

nepHo.n.oM

HHsilleH

rap-

MOHHKH B pa3JIOJKeHHH CJIOJKHOrO nepH'o.n.HlJeCKOrO KOJIeOaHHH P

(1,

t)

B

p5l.n.

no

cHHycaM.

,UeiicTBHTeJIbHO,

JKeJIaeMOe

ycpe,nlieHHoe

.n.aBJIe-

HHe

p*

(t),

KaK

<PYHKIJ,H5I

t, nOJIYlJaeTCSI

H3

perneHHSI CJIe.n.YlOmero

JJ.Hq>epepeHIJ,HaJIbHOrO

ypaBHeHH$l:

T

dP;t(t)

+ P*

(t)

= P

(1,

t). (2)

PelIleHHe

3Toro

'ypaBHeHHH

(npH

HyJIeBbIX

HalJaJIbHbIX

yCJIOBH5IX,

a

HX

.n.eHcTBHTeJIbHO

MO>KHO

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)

k

71

=

n-

, A

k =

100MG

aM

t

2DS2

p ,

81

.-p

M

p

Mt STP

1_

k

}

(9)

82=~

111'

0,

I

Mp, STP

k =

_._'

Mt""A1 '

83

Mo, U

Mal

c'

k =

--

M

t

's--'-l •

1

81 Mp.

Tp

u

no

aHaJIOrHtIHbIM

epOpMyJIaM paCCtIHTblBaIOTCH K03<PlP

H

a.

HU

'ftI

MOAeJIeH

APyrHx

ytIaCTKOB

Tpy6~.

PHe.

6.12.5.

Бутковский А.Г. Методы управления системами с распределенными параметрами

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