Красовский Н.Н. Теория управления движением

o

~~.,

t60

CBO:aCTBA

OnTllMAJILHOrO

ynPABJIEHlUI

[rJI.

6

me

ROJLIrqeCTBOM

ABHmeHHH, ROTopoe

OHa

HMeJIa

B Ha'IaJIbHhlH

MOMeHT

t = 0.

OTMeTHM,

'ITO B

AaHHOM

CJIyqae BpeMH pasrOHa H BpeMJI TopMomeHHH

ORaSaJIHCb

0AHHaROBbIMH.

IIycTb

Tenepb TO'IKa 06JIaAaeT B Ha'IaJIbm.m

MOMeHT

BpeMeHH

60JIb-

illllM

ROJIH'IeCTBOM

ABllmeHHH, 'IeM Ha

saAaHHOH

RpyroBoH

op6HTe. IIpHMeM,

HanpHMep, XIO =

0,052,

Xao

=

0,

Xao = -

0,145.

TorAa

liS

(21.13),

(21.14)

nOJIy-

~

qaeM, 'ITO

c~

=

0,197,

Ca

=

0,

c~

=

0,145.

B

aTOM

cJIyqae,

nJIOCROCTb

tfi

(21.17)

RacaeTCH nOBepxHocTIi

(ffJ

(21.16)

B

HeRoTopoH

TO'IRe

G

nJIOCROCTIi

Pa

=

---L~

= 0. !\oopp;IIHaThI

aToH

TO'IKII JIer·

//-

/

''-,

RO

HaXOAHTCJI

rpa4lH'IeCRH -

no

AO-

CTaTO'IHO

TO'IHO

BhI'Iep'IeHHoMy ce-

qeHHID

nOBepxHocTH

(ffJ.

8TO

II

6hIJIO

BhIIIOJIHeHO,

IIplI'IeM

IIOJIy'IlIJIOCb

P~

= 0,216;

Pg

=

0;

\

,

P~

= 0,119.

I

CJIeAOBaTeJIbHO,

'IHCJIO

VO

=

0,06.

,

COrJIaCHO

(21.20)

OnpeAeJIHeM

Te-

'%

nepb

IICROMoe

OnTIiMaJIbHoe

ynpaB-

JIeHHe:

~J!;z

/.

/

UO(t)

=

0,06

sgn(0,216

cos:r+

0,119).

/

SAecb,

TaR

me

RaR

II

BhIille, TO'IKa

/

-,0..",

---'

~_.

:_::.:~_

._//

BHa'IaJIe

paSrOHJIeTCH,

saTeM

TOpMO-

t=O

-----

.;;::;C

SIITCH

H, HaKoHen;,

CHOBa

paSrOHHeT-

CH.

OAHilKO

BpeMH

pasroHa

11

TOp-

MomeHHH nOJIy'IaeTCH

yme

paSJIH'I-

m/~

PHC.

21.3.

HhIM.

ITepeHAeM

Tenepb

K

CJIylIaID

r-MepHoro

BeKTopHoro

ynpaBJIeHllH

u (t).

CHOBa

B

COOTBeTCTBIUI

C OOm;eH

MeTOAllKOH

no

HopMe

p*

(u]

=

= % [U]

HaxOAHM

HS

TaOJIHIJ;bI 12.1

HOpMy

P [h ('t)],

cPHl'ypllpyIDm;yID

B

npaBHJIe

MHHHMaKca

17.1. -

IfMeeM:

t{3

P

[h

('r)] =

~

y (h

(-r)]

d-r, (21.22)

ta.

npHl:J:eM y [h (,;)] -

::lTO

RaK

pas

Ta

HopMa

B

r-MepHOM

npOCTpaHCTBe

BeKTopOB

h,

AJIH

KOTOpOH

HopMa

y* [U]

HBJIHeTCH

conpHiReHHoH.

B

lIaCTHOCTH,

eCJIH

%

[u]

=

max

y*

(u

(-r)]

=

max

Ilu

(-r)II,

(21.23)

ta.<;;;",<;;;t{3'

t

at

<;;;",<;;;t{3

TO

t{3

P [h

(-ri]

=

~

Ilh

('1:)

IId-r;

(21.24)

ta.

§ 21]

ynPABJIEHI1E

MHHIIMAJIbHO:a

ClifJIo:a

c'

161

eCJIH

me

%[u]

=

max

I

ui

(-r)

I,

(21.25)

t

at

<;;;",<;;;t{3

i

TO

t{3

r

p(h('C)]

--:-

~~

Ihd-r)ld'C.

(21.26)

t«

i=l

If

B

BeKTopHOM

CJIylIae

npemAe

Bcero

HaAJIemHT

orrpeAeJIUTbMHHII-

MaJIbHyID

4>YHKIJ;HID

hO

(-r)

=

B'

(-r)

S

(-r,

til]

lO,

rAe

n-BeKTop

lO

HB-

JIHeTCH

pemeHHeM

3aAatIH

t{3 t{3

pO

=

~

y (h

O

('C)]

d'C

=

min

~

y

[B'

(r)

S

[-r,

t{3]l]

d'C

(21.27)

tat I tat

npH

l'e

= 1.

RaK

H B

CRaJIHpHOM

CJIylIae,

<)Ty

saAalIY

AJIH

YAoocTBa

BbIlIHCJIeHlIH

IIHorAa

IJ;eJIecooopa3Ho

saMeHHTb

npOOJIeMOH

00

onpeAeJIeHHII

h

a

(,;).

IIpll

3TOM B (21.6)

BMeCTO

BeJIHtIIIHbI

I

h'

('1:)

S

L-r,

t{3lkl

CJIeAyeT

nH-

caTb

BeJIHlIlIHY

r fB'

(-r)

S

[-r,

til]

kL

TaR

RaR

3'1'0'1'

3Tan

pemeHIIH

3aAallH

He

HMeeT

RaKHX-JIIIOO

cym;ecTBeHHbIX

OTJIlIlIlIH

no

cpaBHeHHIO

CO

CKaJIHpHbIM

CJIyllaeM,

TO

Ha

ero

oocymAeHlI1I

OCTaHaBJIIIBaTbCH

He

OYAeM,

nOJIaraH

AaJIee

cPYHKIJ;IIID

hO

(.)

(h

a

(.»

II

lIHCJIO

pO

1I3BeCT-

HbIMlI.

IIp

eAnOJI

omHM,

lITO

1IlICJIO

pO

OTJIHtIHO

OT

HyJIH,

H,

CJIeAOBa-

TeJIbHO,

3aAaqa

13.1

HMeeT

pemeHHe.

MbI

AOJImHbI

Tenepb

onpeAeJIIITb

IICKOMOe

OnTHMaJIbHOe

ynpaBJIe-

HHe

UO(t),

onllpaHCb

Ha

YCJIOBlIe

MaKCHMYMa

(17.1),

ROTopoe

npllHII-

MaeT

SAeCb

BHA

t{3

~

h

O

'

(-r)

UO

(-r)

d'C

=

max

~

h

O

'

(-r)

u

(-r)

d-r

(21.28)

~

u

~

npll

max

y*

[u

(.)]

=

1/

po

IIJIH,

IIHalIe,

npll

y*

[u

(-r)]

<

1/

p

o.

Pac-

0:

cymAaH

TaK

me,

KaK

II

BbIme

-

npll

nepeXOAe

0'1'

(21.7) K (21.8),

Ha-

XOAIIM,

liTO

nOAbIHTerpaJIbHOe

BbIpameHHe

B (21.28)

AOJImHO

ObITb

MaKCIIMaJIbHbIM

B

KamAbIH

MOMeHT

BpeMeHII

';,

T.

e.

hO'

('t)

UO

(.)

=

max

hOI

(.)

U

(.)

(2~

.29)

u

npll

y*

[u

(.)]

<

1/

p

o.

IIycTb,

B

lIaCTHOCTH,

y*

[u]

=

II

u(,;)

II·

(21.30)

IIpeAnOJIOmHM

TorAa,

lITO

BeJIHlJHHa

II

hO

(,;)

II

oopam;aeTcH

B

HyJlb

.JIHmb

B

KOHellHOM

lIlICJIe

1I30JIHpOnaHHblX

TOlIeK

• =

.j'

B 3TOM

6 H. H. HpacoBcRHli

'"

"

'.'

C

,--'-

162

CBOYICTBA

OIITJ1MAJILHOrO

YIIPABJIEHMH

[I'JI. 6

CJIyqae

tPYHI\IJ;lHO

hO(T:)

,

RaR

II

BMme,

6Y,1J;eM

Ha3MBaTb

l-leoco6ou.

TaR

RaR

tPYHR~IIH

hO

(T:)

neoco6aR,

TO

yCJIOBIIe

(21.29)

03HaqaeT,

'ITO

BeRTopM

hO

(T:)

HU

o

(T:)

B

Ran\p;bIH

MOMeHT

BpeMeHH

T:=!==

T:j

,1J;OJImHbI6bITb

ROJI-

JIMHeapHbI

M

HopMa

r*

[u]

=

II

UO

(T:)

II

BeRTopa

UO

(T:)

IIpH

Ram,1J;OM

T:

=1=

Tj

p;OJImHa

6bITb

MaRCMMaJIbHOH,

T.

e.

paBHoH

1/

p

o.

TaKMM

06-

pa30M,

B

CJIy'lae

(21.30)

OIITHMaJIbHOe

ynpaBJIeHMe

UO

\T)

HBJIHeTCH

efJul-lCmeel-ll-lblM

H

paBHMM

1

hO(T)

(21.31 )

UO

('t')

= po

Ilh

O

(T)II

B

JII060H

MOMeHT

BpeMeHH

T:

=f=

T:j.

B

MOMeHTM

T:

=

T:j

3Ha'leHHH

UO

(T:j)

BOa,1J;eHC'fBHH

UO

('r)

CHOBa

HBJIHIOTCH

l-lecYUfecmeel-ll-lbl.MU,

HO

MM

,1J;oJIm-

,HbI

HX

BbI6paTb

BO

BCHROM

CJIyqae

TaR,

'1T06bI

tPYHRIJ;HM Uj

(T:)

HBJIH-

JIHCb

HenpepbIBHbIMM

cnpaBa.

PaCCMOTpHM

Tenepb

CJIy'laH,

Ror,1J;a

r*

[u]

=

max

lUi

('t')

\.

(21.32)

i

llpe,1J;noJIO}RMM

a,1J;eCb,

'ITO

yme

Ramp;aH

tPYHR~MH

hOi

(T:)

(i

= 1,oo.,n)

06pam;aeTcH

B

HyJIb

JIMmb

B

ROHe'tHOM

'IHCJIe

M30JIHpOBaHHMx

MO-

MeHTOB

T:

=

T:~).

(81'0

yCJIOBMe

BbIIIOJIHHeTCH,

eCJIM

CMCTeMa'

(20.1)

l-leoc06ert/-laR

no

JW:JfCfJoMY

e08fJeucmeU10

Ui

(T:)

(CM.

§ 20).) Tor,1J;a

H3

•

yCJIOBHH

(21.29)

aaRJIIO'IaeM,

'ITO

MCROMoe

OnTMMaJIbHOe

ynpaBJIeHHe

UO

(T:)

CHOBa

HBJIHeTCH

e,1J;HHCTBeHHMM, M Ram,1J;aH

ero

ROMnOHeHTa

UO

i

(T:)

onpe,1J;eJIHeTCH

paBeHCTB

OM

° 1 °

(21.33)

Ui

('t')

= ---0 sgn

hi

('t')

p

B

JII060H

MOMeHT

BpeMeHH

T:

=f=

T:~i).

CyMMHPYH

peaYJIbTaTbI

HamMX

paccym,1J;eHMH,

MM

MomeM

ctPop-

MyJIMpOBaTb

CJIep;YIOm;yIO

TeopeMy.

Teo

p

eM

a 21.1. PaccMompUM 8afJa'lY 13.1,

afJe

MUI-lUJrt,U8Upy-

emcR

Ul-lmel-lCUel-lOCmb

x

[u]

eufJa (21.1).

llycmb

sO

(t)

- mo

U8

fJeu-

:JlCel-luu

conp£:JfCel-ll-lou

CUCme.MM

(17.3),

cmeCl-lel-ll-lblX

ICpaeebl.M

YC.l£08Ue.M

c'so

(tr»

=

1,

fJ.I£R

IComopoao

nopo:JfCfJae.MaR

UM

rjiyI-lIC/{UR

hO

(T:)

=

=:

B'

(T:)

SO

(T:)

yfJoe.l£emBOpRem

YC.I£oeulO

(21.27).

Ec.l£u

'lUC.I£O

pO

Om.l£U'lI-lO

om

l-lyAA,

mo 8afJa'la 13.1

060nmUMa.I£bl-lOM

ynpa8.l£el-luu

UMeem

peUlel-lUe,

npu

amOM

ee.l£U'lUl-la

pO

U

onpefJeAAem

ICaIC

pa8

MUI-lUMyM

x

[UO]

ee.l£U'lUl-lbl

OnmUMa.l£bl-lOU

ynpa8.l£RlOUfeu

CU,/£bl

11,0 (t),

pael-lblU

1/

p

o.

lfcICo.MaR

cU.I£a

UO

(t)

l-laxofJumcR

U8

YC.I£oeUR

(21.29).

B 'lacmI-lOcmu,

ec.I£U

cucme.Ma

l-leOC06el-ll-laR

no

ICa:JfCfJoMY

e08fJeucmeuIO

Uj, mo 8afJa'la

UMeem

efJul-lcmeel-ll-loe

peUlel-lueuo (t), npU'leM

ee.l£U'lUl-la

r*

[UO

(T:)]

e

}W;JlCfJbIU

.MO.MeI-lm

epe.MeI-lu

.MaICCUMa.l£bl-la.

B

aaRJIIO'leHHe

naparpatPa

npHBe,1J;eM

em;e

O,1J;HH

ROHRpeTHbIH

npM¥ep.

c"

C·:)

§

21J

YIIPABJIEHME

MMHMMAJIbHOn

CMJIOn

163

IIpHMep 21.2.

PaccMoTpHM

.n;BItlKeHHe

TJDI\eJloii

MaTepllaJIbHOii

TO'IKll

B

BepTJilKaJIbHOii

rrJIOCKOCTll,

OIIllCaHHoe

B

rr.

3° § 3.

Hama

aa.n;aqa aaKJIIOqaeTCJI B

TOM,

qTo6bI

rrOCTpOItTb

yrrpaBJIJIIO~ee

Boa.n;eiicTBlle

UO

(t),

06eCrreqItBaIO~ee

rre-

pe6poc

TO'IKlllla

COCTOJIHllJI

xa.

= X (0) =

{-1,0

O,O}

B COCTOJIHMe

xll=x(1)=

=

{O,

O,O,O}

aa

BpeMJI

til -

t(J.

= 1 - 0 = 1

IIpIt

yCJIOBMH

MMHllMyMa

HHTeH-

CMBHOCTM

x[uJ

=

max

[U~(T)

+

u~

(T)I'(,·

(21.34)

...

IIocTaBJIeHHaJI

aa.n;aqa MMeeT

pemeHMe

uO

(t),

TaK

KaH

CMCTeMa (3.10) BrrOJIHe

yrrpaBJIHeMa.

OHa

JIBJIJIeTCH

TaHlKe

M

neoco6ennou.

II03TOMy

pemeHHe

U

O

(or) ef}uH-

cmeennoe

II: orrpe.n;eJIJIeTCH

paBeHcTBoM

(21.31):

, 1 [

l~

(1

- T)

+-

19]

UO

(t) =

pO

II

h

O

(T)

II

' (21.35)

l~

(1

- T) +-

l~

rp;e ·MunUMa.n.bnllJl.

WYHH~JI

kO

('r)

=

{k~

(T),

kg

(Tn

=

{l~

(1

-

or)

+

zg,

zg

(1

-

T)

+

z~}

HaXO,D;I!TCJI

Ita

yCJIOBMJI (21.27). <I>YHHqMll h

(T)

=

B'

(T)

s

(r),

cpep;H

KOTOpbIX

MbI

~eM

kO

(T),

YlKe

BbIqMCJIeIlbI

BbIlIIe,

B § 18 (CM. CTp. 127).

TaKMM

o6paaoM,

aap;aqa

(21.27)

CBOP;MTCJI

K

CJIe.o;yIO~eMY:

HaiiTH

1

pO

=

~in

}

([lI(1-

T) +-12

]21'-

Us

(1

~

T1

+l4P}'f' dT (21.36)

rrpH

ll1'-

'2

g

ls

-+

gl4

= 1. (21.37)

Herrocpe.n;cTBeHHoe

pemeHHe

aa.n;aqH (21.36), (21.37) aaTpy.n;HMTeJIbHo.

IIo3TO-

My

1JHcJIa'

z~

(i

= 1'00',

4)

6y.n;eM

HCK8Tb,

HCrrOJIb3yH

.n;orrOJIHHTeJlbHO

MexaHll-

qeCHMe

coo6palKeHHH.

3aMeTllM,

rrpem.n;e

Bcero,

qTO

BBll.n;y

cTaqHoHapHocTH

CH-

CTeMbI

(3.10) H

cHMMeTpllM

CllJIOBOrO rrOJIJI

rrpH

aaMeHe

BpeMeHH

t = 1 -

t'

MC-

xo.n;HaJI aa.n;aqa

TpaHclf/opMHpyeTcH

B

TaKyIO

lKe

aa.n;aqy, HO

rrpH

yCJIOBllM,

qTO

TO'IKa

rrepe6paCbIBaeTCH

11:8

COCTOHHHH X (0) =

{O,

0,

O}

B COCTOHHHe x (1) =

=

{-1,

0, 0,

OJ,

8

He

Hao6opOT,

KaK

3TO 6bIJIO

paHbme.

OTCIOp;a B

CHJIy

ep;HH-

CTBeHHOCTll

yrrpaBJIeHHH

UO

(t) (21.35)

BbITeKaeT,

qTO

OllTHMaJIbHaJl

TpaeHTopHH

XO

(T)

=

{x~

(T),

x~

(T)}

cllMMeTplI:1JHa OTHOCHTeJIbHO

rrpJIMoii

Xl

= - 1/2 B

IIJIOCKOCTll

{Xl'

x

a

}·

Ho

Tor.n;a

rpalf/HH

COCTaBJIJIIO~eii

ug

(T)

Ha

rrJIOCHOCTll

{T, u2}

TaHlKe

6YP;eT CMMMeTpHqeH OTHOCHTeJlbHO

rrpJIMoii

T = 1/2, a

rpalf/llI{

COC-

TaBJIHIO~eii

u~

(T)

Ha

rrJIOCKOCTH {T,U

l

} 6y.n;eT

aHTHCII:MMeTpllqeH

OTHOCHTeJIbHO

3Toii

me

npHMon.

fl3

KOJIJIHHeapHOCTH

BeKTopOB

u

O

(T)

H k

O

(T)

CJIe.n;yeT Tor.n;a, qTO

H

rpa!).HKH

/IJYHH~H

k~

(T)

==

l~

(1

-T)

+

zg

H

kg

(T)=

z~

(1

-

T)

+

l~

JIBJIJI-

IOTCH COOTBeTCTBeHHO

aHTHCHMMeTpHlJIlbIM

H CHMMeTpll1JHbIM OTHOCHTeJIbHO

rrpJI-

Mon

l'

= 1/2

(Ha

nJIOCKOCTJIX {T,

k~}).

YKaaaHHoe

cBoiicTBO

WYHHqHii

k~

(t') M

kg

(T)

HMeeT

MecTo,

OqeBHp;HO,

JIIDIIb

Torp;a,

Korp;a

z~

= -

2zg,

z~

=

O.

YqllTbI,

BaJI

3TH

paBeHCTBa

H yCJIOBHe CBHaH (21.37),

MbI

MomeM

(BbIpalKan

[2'

l4

qepea

l1)

clf/OPMYJIllpOBaTb

8ap;aqy

(21.36), (21.37)

TaK:

HaiiTlI

r (1 '

)2

(1

-

l1

)2]'/2

pO

= min

J[l~

2'

- T

f-

-g-

dt'. (21.38)

I,

q . ,

6'"

__

,

__

'-.C"--';,.c.:..

'--'-''-'--'-~_

.",;,-

.........

,.,;;

••.

:

',',','

...

';'.'.,

''y!'

'J')'

'doc

_~~

~~.

·

..

'

p

....

~

.

~

.'

,--"'

C

CB09:CTBA

OllTHMAJIbHOrO

YllPABJIEHHP.:

[rJI.6

164

Pernall

3TY

sa,n;a'IY, nOJIOlRIlB g = 10, aaxo,n;HM:

l~

= 0,1116;

l~

=~0,0558;

l~

= 0;

l~

= 0,0888;

pO

= 0,0944.

Tor,n;a

HS

(21.35) nOJIY'IaeM, 'ITO

° 0,0558

(1

- 2t)

u (t) -

::--:-:-:---=r='==========

I . - 0,0944 l'O,0124t

2

-

0,0124t

-+

0,0120

(21.39}

"j

o 0,0888

u

2

tt) = 0,0944 YO,0124t2 _ 0,0124t +0,0120

Ha

pHC. 21.4

Hso6pameaa

OllTHMam,aall TpaeKTopHll

xO

(t) =

{x~

(t),

x~

(t)}

TO'lKH;

;t:J;JIll

aarJIll,n;aOCTH Ha

lITOM

pHcyHKe

npe,n;CTaBJIeHO

TaKme ynpaBJIeHHe

u

O

=

{u~,

u~},

COOTBeTCTBYIOm;ee

pas-

JIH'IHLIM TO'lKaM nyTlil. RpOMe

Toro,

.x

J

MacrnTa6bI

no

OCHM

Xl

H X

2

BbI6pa-

HbI paSJIH'IHbIMH.

§ 22. YnpaBJIeUHe B cJIy'lae

_.

RBa3HJIHUeHUOro

oi'h.eKTa

,

.

-/

.x,

Co,n;eplliaHHeM

HaCToam;eH

MOHorpa<p1u1

aBJmeTCa

HCCJle-

,n;OBaHHeaa,n;a~

06

ynpaBJleHHH

JlHHeHHhIMH

06'beKTaMH.

06m;ee

HCCJle,n;OBaHHe

np06JleMbI

yn-

PRC.

21.4.

paBJleHHa

B

HeJlHHeHHbIX

CH-

CTeMax

BbIXO,n;HT ,n;aJleKO

aa

paMKII

,n;aHHoH

KHlIrH.

O,n;HaKo,

~T06bI

OTMeTIITb

CBaab

paccMaTpllBae-

MblX

HaMil

JlIlHeHHbIX

np06JleM

H

HCnOJlbayeMbIx

MeTo,n;OB C

aa,n;a~aMH

60Jlee

06m;ero

po,n;a,

nOJleaHO

06Cy,n;HTb

BoaMOlliHbIH

nepexo,n;

01'

Ha-

mHX

CJly~aeB

K

HeJlHHeHHbIM

cHTya~HaM.

IIepBhIM

marOM

TaKoro

rre-

pexo,n;a

aBJlaeTCa

HCCJle,n;OBaHHe

KBaaHJlHHeHHOH

CHCTeMbI,

ypaB-

HeHHa

,n;BII}KeHHa

KOTOPOH

OTJlH~aIOTCH

01'

ypaBHeHHH

mmeHHbIX

JlHmb

MaJlhIMH

HeJlHHeHHbIMH

,n;06aBKaMII.

YKaaaHHbIH

KBaaHJlIlHeHHbIH

CJly'laH

II

COCTaBJlaeT npe,n;MeT

,n;aHHoro

naparpa<pa.

IIpll

3TOM ,n;JIa

onpe,n;eJleHHOCTII

MbI

OrpaHH'lHMCa

Jlllmb

aa,n;a'leH,

B

KOTOpOH

pecyp-

CbI

ynpaBJleHHa

o~eHHBaIOTca

BeJlH'lIlHOH

x

[u]

BH,n;a (21.2).

IIMeHHo

no

3TOH

npll'lllHe

npe,n;JlaraeMbIH

a,n;eCb

MaTepHaJl

CJle,n;yeT

cpaay

aa

§ 21, r,n;e

6bIJla

Hay'leHa

COOTBeTCTBYIOm;aa

JlIlHeHHaa

np06Jlei1a.

B

,n;pyrHx

TlInH'lHblX

cJlyqaax

x

[u]

(HJlH

x

[dU])

nepexo,n;

01'

JlHUeHHOH

clITya~IIH

K

clITya~HH

KBaaHJlIlHeHHOH

ocym;ecTBJlaeTCa

no

aHaJJ;O-

rll'luoMy

IIJlaHY

H

n03ToMY

Ha

UIIX

Mbl

AaJlee

OCTaHaBJlIlBaTbCa

He

6y-

p;eM.

OTMeTIIM

Jlllmb,

~ITO

II

B

Ima3l1JlIIHeHHOM

CJIy'lae

Hall60Jlee

yrw6-

HOH ,n;Jla

KOHl\peTHblx

Bbl'lIlC.1IeUHH

Ol\aabIBaeTca

BeJIlI'IIIHa

x

[uJ

BH-

Aa

(18.1).

§

22]

ynp

ABJIEHHE

RBAaHJIHHE9:HbIM

OB'I>ERTOM

165

IITaK,

nycTb

nOBeAeHHe

ynpaBJlaeMOH

CHCTeMbI

onllCbIBaeTca

ypaBHeHlleM

1; = f (t,

x)

+g (t,

x)u.

(22.1)

3,n;eCb x =

{XI'oo"X

n

} -

n-MepHbIH

BeKTOp

<paaOBbIX

Koopp;HHaT

06'I>eKTa;

f (t, x) H g (t, x)

--

n-MepHbIe

BeKTOp-<pYHK~IIH;

U -

CKa-

JlapHOe

yrrpaBJlHIOm;ee

Boap;eiicTBlIe.

By,n;eM

npep;nOJlaraTb,

'ITO

f (t, x)

H g (t,

x)

HenpepbIBHbI

no

t

Ha

paCCMaTpllBaeMOM

OTpeaKe

BpeMeHll

Ita,

tfJl

II

aBJlaIOTCa

aHaJlHTH'IeCKIIMH

<pYHK~lIaMII

nepeMeHHbIX

Xi

B

HeKOTOpOH

OKpeCTHOCTII

TO'lKH

x = 0,

T.

e.

npllMeM,

'ITO

3TH

<PYHK-

~IIH

paaJlararoTca

B CXO,n;am;HeCa

CTeneHHbIe

paAbI

00

f (t, x) =

~

t<m)

(t, x), g (t, x) =

~

g(m)

(t,

x),

(22.2)

m=l

m=O

r~e

j<m)

H

g(m)

- <pOPMbI

m-ro

nopaAl\a

no

Xi'

,I.J;onycTHM,

'ITO

BeJlll-

'1HHbI

Xi

(t)

MOlliHO

C'IHTaTb

p;OCTaTO'lHO

MaJlhIMlI

(HaCTOJlbKO

BO

Bca-

I\OM

CJly~ae,

~T06bI

CXOP;IIJlIlCb

paCCMaTpllBaeMble

HaMil

PH,n;bI).

IIHa-

'Ie

rOBopa,

B

AaHHOM

CJly'lae

pe'lb

lI,n;eT

06

ynpaBJleHHII

06'beKTOM

(22.1)

Jlllmb

B OKpeCTHOCTH

HeKOToporo

ero

p;BlIllieHlIa,

npll'leM

BeK-

TOp

x

(t)

IIMeeT

KaK

paa

CMbICJl

OTKJlOHeHlm

<paaOBoro

Bel\TOpa

01'

31'0-

ro

,n;BHllieHHa,

KOTopoe

B

KOOpp;IIHaTaX

Xi

aanllCblBaeTca

TaKHM

o6paaoM

B BHp;e

Xi

(t)

= °(i =

1,oo.,n)

(CM.

no

nOBOp;y

COCTaBJleHIIH

TaKIIX

ypaBHeHHH

B § 3).

Hac

6yp;eT

HHTepeCOBaTb

BbI60p

ynpaBJleHHa

U (t),

KOTopoe

npllBOAHT

06'beKT

K

yKaaaHHoMy

p;BHllieHHIO

(IIJlIl

K

COCToa-

HHIO

paBHoBeclIa)

x = 0.

MbI

OrpaHH'IIIBaeMCa

npH

3TOM

CKaJlapUbIM

ynpaBJle.HHeM

Jlllmb

C

~eJlbIO

ynpom;eHHa

BbIKJlap;OK.

IITaK,

CeH'IaC

,n;Jla CHCTeMbI (22.1), (22.2)

MbI

YKallieM

npocToii:

cnoco6

nOCJlep;OBa-

TeJlbHorO

nocTpoeHHa

ynpaBJleHlIa

u (t),

KOTopoe

OKaabIBaeTca

6Jllla-

KHM

K

OnTHMaJlbHOMY

(B CMbICJle

aap;a'lH

13.1)

rrpll

KpaeBbIX

YCJlOBHaX

x

(ta)

= x

ilt

H X

(tfJ)

=

x(3

= 0,

Korp;a

HHTeHCIIBHOCTb

x

[a]

onpeAe-

JleHa

paBeHCTBOM

(21.2)

II

Ha~aJlbHbIe

BoaMym;eHHJl

,n;OCTaTO'lHO

Ma-

JlbI.

IIpH

3TOM

6yp;eM

onllpaTbca

Ha

lIaBeCTHoe

HaM

lIa

§

21

pemeHlIe

aHaJlOrH'lHOii

aaAa'l1l

p;Jla

JlHHeHH@

CIICTeMbI

nepBoro

np1l6JlH}HeHHa

1; = A (t) x +b (t) u, (22.3)

r,n;e

A (t) x =

{fi

l

)

(t,

x),

,

f~)

(t, x)},

b (t) =

{gin)

(t), ,

g~)

(t)}.

,ll;JlH

,n;aJlbHeiimero

y,n;06HO

npep;'baBIITb

O,n;HO

Tpe60BaHue

K

Munu-

Ma/wnoa

<PYHI\~IIH

h

O

('t),

KOTopaa

nOJly'laeTca

npll

pelTIeHIIJI

Yl\aaaH-

HOH

JlHHeHHOH

aap;a'lll

no

npaBIIJlY

JlwnU.Mw,ca 17.1.

QT06hI

C<POPMY-

JlHpOBaTb

31'0

Tpe60BaHlIe,

npe,n;nOJlOmHM

,n;Jla onpe,n;eJleHHOCTJI,

'ITO

B:eKTOp c (15.2) yP;OBJlCTBOpHeT YCJlOj:JHIQ

c~.=1=

0,

II,

CJle,n;OBaTeJlbHO,

166

P

'-.:c>'

[rJI.

6

CBOnCTBA

onTHMAnhHoro

ynPABnEHHfl

no

(21.4)

HMeeM:

n-l

In

=

+-

(1-

~

lici)

.

n

i=1

B

aTOM

CJlY'Iae

sa)J;a'la

(21.4)

Ha

YCJlOBHbIH

aKCTpeMYM

CBO)J;HTCfl K

np06J1eMe

6esycnoBHoro

MHHHMyMa

~YHK~HH

t/'l

n-1

'iJ

(l1,

...

, In-I) =

~

I

~

lih(i)

('t')

+

+-

(1-

.2j

lici)

h(n)

('t')

I

d't'.

tIL

i=l

n

i=l

lloaToMy

'IHCJla

IOl,

...

,I~_ll

onpe)J;eJlHIO~He

h

O

('t),

6Y)J;yT

y;UOBJleTBO-

pHTh

CHCTeMe

ypaBHeHHH

t/'l

n-1

~t

=

~

[h(i)

('t')

-

;i

h(n)

('t')]

sgn

[2}

lih(i) ('t) +

t

t"

n

i=l

n-1

+

+-

(1-

.2}

lici)

h(n)

('t')]

d't'

= 0

- n

~=1

(i=1,

...

,n-1).

YnOMflHYToe

BhIille

Tpe60BaHlle

COCTOHT B

CJle)J;YIO~eM:

6YAeM

- "

)J;aJlee

Bcer)J;a

npe)J;nOJlaraTb

Ha'laJlhHDle

)J;aHHDle

Xi

TaKHMH,

'ITO

rpa-

n

<p

HK

MHHHMaJlbHOH

~YHK~HH

hO

('t')

=

~

l~h(i)

('t')

nepeCeKaeT

OCb 't nO)J;

i=l

a'll

) I

HeHYJleBhIMH

yrJIaMH

H

flK06HaH

0

7fT""'~

0

(Ill'

..

, In-i)

(

1

n-1

OTJIH'IeH

01'

Hynfl

npH

Ii

=

It

't'

= 't'i,

T.

e.

a~

\\n-

1

0

d~t

alial 1

=1=

0

n'pH

Ii

=

Ii,

0

't'

=

't'i

(I )

II

le

(i,k=1,

...

,n-1;

j=1,

...

,s).

3)J;eCh

s

~

=

2}

[h(i) ('t'i)

_2h(n)

('t'1)]

a"j

al.ar..

. c

al,.

t"

3=1

"

H

CHMBOJlbI

'tj

=

'tj

(I) (j = 1,

...

,s)

OSHa'laIOT

MOMeHThI

BpeMeHH,

B

n

lWTOphIe

<PYHK~HH

h

('t')

=

~

lih(i)

('t')

06pam;aeTCH

B

HyJIb.

AJIfl

HaC

i=l

lJ8,JRHO,

'ITO

npH

yKaS8,HHOM

YCJlOBlln

B

OKpeCTHOCl'll

'l'01JKH

I =

to

/~.

(L·

§ 22j

YTIPABnEHHE

HBA3HrtHHEnH~M

OB~EHTOM

161

cy~eCTBYIOT

rrpOHSBO)J;HhIe

.

a"j 0 0

7il

npH

/1

=

II,

...

,

In

=

In

k

(j=1,

...

,s;

k=1,

...

,n),

KaK

aTO CJle)J;yeT

HS

HSBeCTHhIX

TeOpeM

0

HeHBHhIX

<PYHK~HHX

(CM.,

HarrpHMep,

[23*],

T.

I,

CTp.

447).

BOJlee

Toro,

npH

)J;aHHOM

YCJlOBHH

H3

TOH

me

TeopeMhI

0

HeHBHDIX

~YHK~HHX

BhITeKaeT,

'ITO

BeJlHtIHHhI

l~

6Y)J;yT

HerrpepbIBHO-)J;H<p<pepeH~HpyeMhIMH

cPYHK~HHMH

BeJlHtIHH

C

1

,

...

,C

n

,

rro

KpaHHeH

Mape

npH

)J;OCTaTO'lHO

MaJlhIX

H3MeHeHHHX

Ci'

OTCIO)J;a

CJle)J;yeT,

'ITO

npH

MaJlbIX

HilMeHeHHHX

I1xf

H I1xr,

T.

e.

rrpH

MaJlbIX

H3MeHeHHflX

I1Cb

6Y)J;yT

rrpOHCXO)J;HTh

MaJlble

HSMeHeHHH

BeJlHtIHH

l~,

rrpHtIeM

6Y)J;YT

crrpaBe)J;JlHBhI

o~eHKH

I

L\l~

I<

u>d

I1c

II,

jl1v

O

I=

\11

;0

1<

U>2

~

I1c

II,

(22.4)

r)J;e

U>1

H w

2

-

rrocToflHHhIe,

3aBHcfl~He

JlHillb

01'

UCXOOUMX

3Ha'leHHH

X"

H

t",

tfJ.

OrrHpaHcb

Ha

3TH

o~eHKH

H

HCrrOJlb3YH

Te

CBOHCTBa

orr-

THMaJlbHOrO

yrrpaBJleHHH

UO

(t),

KOTophIe

6hIJlH

HaH)J;eHDI B § 21,

He-

TPY)J;HO

rrpHHTH

K

CJle)J;YIO~eMY

BhIBO)J;Y.

llYCTb

MhI

H3MeHHeM

HCXO)J;-

Hble

3Ha'leHHH

XIX

H

xfl

HaCTOJlbKO

MaJlO,

'ITO

BeKTOp

C (15.2)

TaKme

rrpeTeprreBaeT

)J;OCTaTO'lHO

MaJlOe

H3MeHeHHe

I1c

..

Tor)J;a

rrpH

Bcex

't,

sa

HCKJlIO'IeHHeM,

MomeT

6bITb,

JlHillb

3Ha'leHHH,

JIema~HX

B

HeKOTO-

pbIX

8rOKpecTHocTflX

TO'leK

'til

6YileT

BhIrrOJlHHTbCfl

HepaBeHcTBo

Il1u

o

(t) I<W

a

~l1c

II,

(22.5)

rrpHQeM

cyMMapHafl

rrpOTflmeHHOCTh

yrroMHHYTbIX

8rOKpecTHocTeH

(o603Ha'lHM

ee

(;[138j]

= a (@» 6Y)J;eT

y)J;OBJleTBopflTb

yCJIOBHIO

a

(@)

< w

4

11l1c

II.

(22.6)

3)J;eCh

l1u

o

(t) -

HSMeHeHHe

OrrTHMaJlbH.oro

yrrpaBJleHHH,

BbI3-

BaHHoe

H3MeHeHHflMH

11

XIX

H

I1

x

/'l,

a

BeJlHtIHHhI

U>a,

W

4

-

CYTb

rrOCTOHH-

Hble,

3aBHCH~He

JlHillb

01' HCXO)J;HhIX

3Ha'leHHH

x", t

lX

H

t-fJ.

-

llepeH)J;eM

Tenepb

K

rrocTpoeHHIO

yrrpaBJleHHH

u

(t)

)J;Jlfl

CHCTeMhI

(22.1); (22.2).

BY)J;eM

rrpe)J;rrOJlaraTb,

'ITO

CHCTeMa

rrepBoro

rrpH6J1H-

meHHH

(22.3)

HBJlHeTCH

eno/me

ynpae.MI-eMoU

(CM.

§20).

llycTb

U(l)

(t)

-

OrrTHMaJlhHOe

yrrpaBneHHe,

pa3pemaIO~ee

sa)J;a'lY

13.1

rrpH

x

[u]

(21.2) )J;Jlfl

JlHHeHHOH

CHCTeMbI

rrepBoro

rrpH6J1HmeHHfl

(22.3), H

X(l)

(XIX,

t)(22.3)

-

)J;BHmOHHe

CHCTeMhI

(22.3)

rrpH

U=U(l)

(t).

llpH

aTOM,

KaK

MbI

3HaeM,

yrrpaBJleHHe

U(l)

(t)

orrpe)J;eJlHeTCfl

paBeHCTBOM

(21.9).

lis

MaTepHaJla

§

21

HeTpy)J;HO

BhIBeCTH

TaKme

o~eHKY

I

U(l)

(t) I<W

5

II

x"'ll

(22.7)

168

______

~~~~~~~~_'_

••

~.'_'.'_._'_.~'__'_

__

'.,::,.

····::_···j·;,.;·Vi{·

._'_.'_.'_.~_'__'.'__.__';_.:....l..:....'

•.

_'_'.'~'~,.__

··:'·

";;'>'

C

c'

,

~

"

~

I.

CBOfl:CTBA

OIITHMAJIbHOrO

YIIPABJIEHlUf

[rJI.

6

II

y6eAIIThCH

IIO::lTOMY

B

TOM,

qTO

ABIDReHHe

X(l)

(XIX,

t)(22.8)

YAOBJIeT-

BOpHeT

HepaBeHcTBy

II

X(l)

(X",

t)(22.3)

II

<

06/1

XIX

II,

(22.8)

rue

0

5

,

06

-

nOCTOHHHhIe,

OnpeAeJIHeMlile

JIHIIIb

CBOHCTBaMH

CHCTeMIil

nepBOrO

npII6JIHmeHlIH

(22.3).

IIpeAnOJIOmHM

Tenepb,

qTO

ynpaBJIe-

HlIe

U(l) (t)

npHJIOmeHO

K

HeJIIIHeHHOH

HCXOAHOH

CHCTeMe

(22.1).

8TO

ynpaBJIeHlIe

u =

U(l)

(t)

npHBeAeT

ABIImeHlIe

X(l)

(x",

t)(22.l)

AaHHOH

CHCTeMIil

(22.1) K

HeKoTopoMy

COCTOHHHIO

X(l)

(X

cx

,

t(3)(22.l)

=

=y(2)(tf).

lIaBeCTHO,

qTO

peIIIeHlIH

HeJIHHeHHIilX

CHCTeM

AIItPtPepeH-

~HaJIbHhIX

ypaBHeHlIH

(22.1) C

aHaJIHTHqeCKOH

no

x

npaBoH

qaCTbIO

MomHO

paaJIaraTb

B

PHAbI

no

CTeneHHM

HaqaJIbHhIX

AaHHbIX

XIX

(npII

tPHKcHpoBaHHoM

u (t)).

OnIIpaHcb

Ha

8TO

CBOHCTBO

paccMaTpHBaeMoH

HeJIHHeHHOH

CHCTeMIil,

MomHO

npOBepHTb

Tenepb,

qTO

C

TOqHOCTbIO

AO

qJIeHOB

BToporo

nOpH):t;Ka

MaJIOCTH

no

II

X

CX

II

HMeeM

([12*1,

CTp.

56,'

174):

t

y(2)

(t)

=~

X

[t,

-rl

Z(2)

("1')

d-r,

(22.9)

tel

rAe

z<2)

("') -

/(2)

('"

x(l)

(x""1')

) +

gel)

("1'

x(l)

(x""1')

)

U(l)

("1')

(22 10)

• -

.,

,(22.3)

,

,(22.3)

.•

lIa

(22.7) - (22.10)

CJIeAyeT

Tenepb,

qTO

~

Z(2)

("1')

11<071Ix"IF

(0

7

-

const).

(22.11)

PaccMoTpHM

CHOBa

aaAaqy

13.1 B

nepBOM

npH6JIHmeHHII

0

npHBeAe-

HlIH

CHCTeMIil

(22.3),

HO

Tenepb

yme

He

B

TOqKY

x (tf) = 0, a B

TOq-

Ky

X (tf) = -

y(2)

(tf).

IIYCTb

U(2)

(t) -

peIIIeHHe

::lTOH

HOBOH

aa-

Jl;aqH.

B

TaKOM

CJIyqae

BeKTOp

(15.2)

onpeAeJIHeTCH

yme

paBeHcTBoM

c = - X

[tf),

tell X

CX

-

y(2)

(tf)

II

OTJIHqaeTCH

OTj

aHaJIOrHqHOrOBeK-

Topa

c

AJIH

npeAIilAy~eH

aaAaqH

Ha

BeJIHwHy

~C(2)

= -

y(2)

(til).

ITO::lTOMY

B

CHJIy

o~eHOK

(22.4) - (22.6), (22.11)

6YAeT

enpaBeAJIII-

BO

HepaBeHCTBO

I

~u21

= I

U(2)

(t)

-.U(l)

(t)

1<

03]\

y(2)

(t{3)

II

<

1.

2

II

X

CX

11

2

(22.12)

npH

Bcex

t,

aa

IICKJIIOqeHIIeM,

MomeT

61ilTb,

JIIIIIIb~

MHomeCTBa

@2

aHaqeHlIH

t,

JIema~IIX

Ha

IIHTepBaJIaX,

cyMMapHaH

npOTHmeHHOCTb

(IIJIII

Mepa)

KOTOplilX

a

(@2)

YAOBJIeTBOpHeT

HepaBeHcTBy

a

(@2)

=

(6411

y(2)

(t{3)

II

<

~

1.

2

11

x"1I

2

.

(22.13)

Wa

PaCCMOTpIIM

ABIImeHHe

X(2)

(X

CX

, t)(22.l)

CIICTeMIil

(22.1),

COOTBeT-

CTBYIO~ee

ynpaBJIeHIIIO

u =

U(2)

(t).

MomHo

npOBepIITb,

qTO

C

TOq-

HOCTbIO

no

TfJIeHOB

BToporo

nopHAKa

no

II

XIX

II

::lTO

ABIImeHlIe

YAOBJIeT-

§ 22]

ynPABJIEIUlEHBA3IIJIIIHEi'lHbIM

OB'bEHTOM

169

BopHeT

HpaeBoMy

yCJIOBHIO

X(2)

(XIX,

tf)(22.l) =

O.

ECJIII

me

Tenepb

yqeCTb

qJIeHbI

TpeTberO

nopHAKa

no

II

x"

II,

TO

X(2)

(XIX,

tf)(22.l) =

=

y(3)

(til) H

t

y(3)

(t)

=

~

X

[t,

-r]

Z(3)

(-r)

d"1',

(22.14)

t

IX

rAe

Z(3)

("1')

=

jC3)

(-r,

x(l)

(XIX,

-r)(22.3» +

n

/(2)

(1)

cx

+

2j

8 ('t', x

8;~

.'t'\22.3»

LlX~2)

(XIX,

-r)

+

i=l

'

+gel)

('t,

Llx(2) (x

cx

,

-r))

U(l)

(-r)

+g(l}

(-r,

X(l)

(XIX,

-r)(22.8)

LlU2

(t) +

+

g<2)

(-r,

X(l)

(x«, 't)(22.8»)

U(l)

(-r).

(22.15)

X

CX

3Aecb

~X(2)

(x

iZ

,

t)

~

qJIeHhI

BToporo

nopHAHa

no

1\

~

B

peIIIe-

HHH

CIICTeMbI

(22.1).

lIMeeM:

t

LlX(2)

(x",

t)

=

~

X

[t,

"1']

(Z(2)

('t)

+b

(-r)

LlU2

(-r))

d-r. (22.16)

t

IX

lIa

(22.14) - (22.16)

BIilTeKaeT

HepaBeHcTBo

CX

3

II

y(3)

(t)

II

<

08

1\

X

11

•

(22.17)

CHOBa

peIIIIIM

aaAaqy

13.1 B

nepBOM

npII6JIIImeHHII,

HO

yme

HanpaB-

JIHH

Tenepb

ABIImeHlIe

(22.3) B

TOqHY

x (tf) = -

y(2)

(tf) -

y(3)

(tf).

OnTlIMaJIbHOe

ynpaBJIeHIIe,

paapeIIIaIO~ee

'j

::lTy

TpeTbIO

aaAaqy,

060aHaqHM

qepea

U(3)

(t).

Tenepb

~C(3)

= -

y(3)

(tf)

II

B

CHJIy

(22.4) -

(22.6), (22.17)

6YAyT

cnpaBeAJIHBbI

o~eHHII:

3

I~U3

(t)1

= I

U(3)

(t) -

U(2)

(t)

I <0

3

~

yeS) (tf)

II

<

1.

3

11

x<X

11

,

(22.18)

8

a

(@3)

<

(()411

y(8)

(til)

II

<

~

1.

3

11

X"

11

.

(22.19)

(03

IIpOAOJIIKaH

paccymAaTb

aHaJIOrMqHbIM

o6paaoM,

nOJIyqMM

Ha

k-M

~are:

k

C = - X [ttl, t

iZ

] x

IX

_

~

y(j) (t{3),

LlC(k)

= -

y(k)

(t{3),

j=l

npnqeM

6YAeT

BbIIIOJIHHThCH

HepaBeHcIBo

ILlu

k

(t)

I= I

U(k)

(t)

- U(k-l)

(t)

1<

0311

y(k)

(t{3)

II

< I.k

\1

x"l\k (22.20)

IIpH

Bcex

t,

aa

MCHJIIOqeHMeM,

MomeT

OhITb,

MHomeCTBa

@k'

aHaqeHHR

Красовский Н.Н. Теория управления движением

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