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Теории управления

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Oi?aaeaiea – ioiineony e iaoaiaoe/aneie oai?ee oi?aaeaiey
aeaeaeaieai oaoie/aneie nenoaiu.

Iaiaoiaeeii iaienaoue aeai?eoi, ii eioi?iio iaeioi?ay nenoaia
oi?aaeyaony n iiiiuueth yia?aaoe/aneiai aicaeaenoaey, iai?eia? :
eaoaoaeueiue aiia?ao oi?aaeyaony n iiiiuueth ?oeaaie iaoeiu. Ieacuaaaony
nicaeaoue oi?aaeaiea yoi ia i/aiue neiaeii e yoi iiaeii naeaeaoue
eiooeoeaii. Iaeiaei nicaeaoue iioeiaeueiia oi?aaeaiea /?acau/aeii
neiaeii.

Oai?ey iioeiecaoeee – yoi iaoea i iaeeo/oeo aeai?eoiao
(oi?aaeaiey) nicaeaiiuo ii iaeioi?iio e?eoa?eth ea/anoaa

E?eoa?ee ea/anoaa – nicaeaiea (aano?aeoiia) iaeioi?ie ooieoeee
?enea, eioi?ay aeieaeia auoue a i?ioeanna iioeiecaoeee ieieiece?iaaiia
(yeno?aiaeueiay caaea/a).

Oi?aaeaiea auaaao iioeiaeueiui e eaaceiioeiaeueiui.

Iioeiaeueiia – ia aoiaaa,

Eaaceiioeiaeueiia – ?aaeueiia, no?aieony e eaeaaeueiiio.

Oi?aaeaiea auaaao :

1) I?ia?aiiiia

N iiiiuueth io?eoeaoaeueiie ia?aoiie nayce

I?ia?aiiiia oi?aaeaiea –

o?aaoaony nicaeaoue i?ia?aiio, eioi?ay aeaao iioeiaeueioth o?aaeoi?eth
(caeiaeaia a YAI) aeaeaeaiey iaeioi?ie nenoaiu.

I?eia? 1 : Ia?aaiae eaoaoaeueiiai aiia?aoa ec oi/ee A a

oi/eo A.

E?eoa?ee – ieieiece?iaaoue ?anoiae ai?th/aai.

Aeey ?aaeecaoeee oaeie caaea/e nicaeaii aeaa nenoaiu – Novstar

(NOA) e Aeiiann (?inney), noieiinoue eo i/aiue auniea.

I?eia? 2 : Iaaei nicaeaoue oaeoth o?aaeoi?eth, /oiau oa?ee neaoeeny ec
oi/ee ‘A’ a oi/eo ‘A’ ca ieieiaeueiia a?aiy.

A

A – Iioeiaeueiay

A
A
o?aaeoi?ey

Oi?aaeaiea n iiiiuueth io?eoeaoaeueiie ia?aoiie nayce

Io?eoeaoaeueiie ia?aoiie naycueth – iacuaaaony ia?aaea/a yia?aee n
auoiaea ia aoiae iaeioi?ie oi?aaeyaiie nenoaiie

ao + Nenoaia auo

ia?aoiay naycue

Auaaao aeaa aeaea ia?aoiie nayce : Iieiaeeoaeueiay IN e io?eoeaoaeueiay
IN.

Io?eoeaoaeueiay IN oiaiueoaao aoiaeiia aicaeaenoaea ia nenoaio
i?iii?oeeiiaeueii auoiaeiiio ioeeeeo (aeaiioe?oao nenoaio a oeaeii).

Aaoiiaoeea – iaoea eco/athuay oai?eth aiaeeca e neioaca

nenoai oi?aaeaiey (ei??aeoe?iaea aeaeaeaiey, iioeiecaoeey
ia?aoiaeiuo i?ioeannia) e nicaeaiea iioeiaeueiiai oi?aaeaiey.

?aaeeiaaoiiaoeea – iaoea, eco/athuay aii?inu oi?aaeaiey

aeaeaeaieai ?aaeeioaoie/aneeo nenoai.

No?oeoo?iay noaia nenoaiu ?aaeeioi?aaeaiey :

?aaeei- ((( Ono?ienoai (-(( Iauaeo (( Aeao/ee

i?eaiiee Oi?aaeaiey Oi?aaeaiey

IIN

?aaeeii?eaiiia ono?ienoai – ono?ienoai auaeaeaiey neaiaea

ii iaeioi?iio ?aaeeieaiaeo.

Iniaaiiinoue auaeaeaiey neaiaea ninoieo a oii, /oi neaiae
auaeaeyaony ia oiia aioo?aiieo ooiia e iiiao.

Aioo?aiiea ooiu – oaieiaua ooiu, eioi?ua anaaaea eiatho

ianoi a ?aaeeii?eaiiii ono?ienoaa.

Oaeei ia?acii a ?aaeeiaaoiiaoeea neo/aeiua i?ioeannu eco/athony iniai
(ooi, iiiaoa, naia o?aaeoi?ey aeaeaeaiey)

Ono?ienoai oi?aaeaiey – eae i?aaeei – au/eneeoaeueiay nen-

oaia n i?eaiaeii e yia?aaoe/aneie

onoaiiaeie.

I?eaiae – i?aia?aciaaoaeue iaoaie/aneeo eieaaaiee a yeae-

o?e/aneea.

Iauaeo oi?aaeaiey – iaeioi?ay aeeiaie/aneay nenoaia.

Aeeiaie/aneay nenoaia – nenoaia, eioi?ay iienuaaaony ee-

iaeiuie e iaeeiaeiuie aeeooa?ai-

oeeaeueiuie o?aaiaieyie aunieiai

ii?yaeea.

Aeao/ee – ono?ienoai, eioi?ia ecia?yao iieiaeaiea eaoaoaeue-

iiai aiia?aoa a i?ino?ainoaa.

Aeaaa 1 Noioanoe/aneia oi?aaeaiea

A neo/aa noioanoe/aneiai oi?aaeaiey, oi?aaeyaiua i?ioeannu yaeythony
neo/aeiuie (noioanoe/aneeie). Ia/aeueiay oi/ea oi?aaeaiey A e eiia/iay A
ia ecaanoiu. A yoii neo/aa nai

oi?aaeyaiue i?ioeann iienuaaaony noioanoe/aneeie o?aaiaie-

yie, eioi?ua, eae i?aaeei, ai?ieneie?othony ia?eianeeie i?ioeannaie.

I?eia?u nenoai aaoiiaoe/aneiai oi?aaeaiey

Nenoaiu aaoiiaoe/aneiai oi?aaeaiey iiaeii iienaoue i?eaee-

aeaiii eniieuecoy eeiaeiua eee iaeeiaeiua aeeooa?aioeeaeueiua

o?aaiaiey (aeaoa?ieie?iaaiiue iiaeoiae aac o/aoa ooiia).Yoi

auei aei 60o aiaeia: ana iiaeoiaeu auee noioanoe/aneea eeiaeiua e
iaeeiaeiua aeeooa?aioeeaeueiua o?aaiaiey.

I?eia? 1 (aeaoa?ieie?iaaiiue)

Oi?aaeaiea aeaeaeaieai einie/aneiai aiia?aoa a a?aae-

oaoeeiiiii iiea caiee (caaea/a aeaoo oae).

A aaioeaio?e/aneie nenoaia eii?aeeiao

Z r – ?annoiyiea io oeaio?a caiee

C – oeaio? caiee (any aa ianna)

E.A.

r E.A. – einie/aneee aiia?ao

X Ia einie/aneee aiia?ao aeaenoaoao

C i?eoyaeaiea :

E.A. F2 – oi?aaeythuay neea

F3 – nii?ioeaeaiea n?aaeu

O?aoee caeii Iuethoiia :

Anee yoi o?aaiaiea ni?iaeoe?iaaoue ia ine ei-

i?aeeiao, oi iieo/ei neaaeothuea o?e o?aaiaiey :

(1)- nenoaia eeiaeiuo aeeooa?aioeeaeueiuo o?aaiaiee 2-ai ii-

?yaeea, eioi?ay iienuaaao aeaeaeaiea einie/aneiai aiia-

?aoa.

Neeu U1,U2,U3 – neeu oi?aaeaiey.

r(t) – o?aaeoi?ey

Ieacuaaaony, /oi a caaeneiinoe io ia/aeueiuo oneiaee e ia-

?aiao?ia K1,K2,K3 o?aaeoi?ey r(t) iiaeao auoue e?oaiaay,

yeeeinieaeiay, ia?aaiee/aneay.

I?eia? 2 : Iaeeiaeiay nenoaia. Iienuaaaony iaeeiaeiui
aeeooa?aioeeaeueiui o?aaiaieai.

Aaia?aoi? eieaaaiee :

Iiaeii iieacaoue, /oi i?ioeann

x(t) iienuaaaony aeeooa?ai-

x(t) oeeaeueiui o?aaiaieai 2-ai

M ii?yaeea n iaeeiaeiui

.

R

C Anee aieinoue aa?uee?iaaoue,

iiaeao noaoue io-

eai e oiaaea iu iieo/ei ne-

ionieaeaeueiia eieaaaiea:

x(t)=a sin((t+()

(aaoieieaaaiey)

– iieiaeeoaeueii, oi aiieeooaea eieaaaiee oaaee-

/eaaaony n oa/aieai a?aiaie.

– io?eoeaoaeueii – aiieeooaea eieaaaiee oiaiueoa-

aony n oa/aieai a?aiaie aei ioey.

Aeaaa 2

Iaoaiaoe/aneia iienaiea nenoai (aeaoa?ieie?iaaiiay oa?ey) (eaeaaeueiue
neo/ae)

Eeiaeiua nenoaiu, eioi?ua iienuaathony aeeooa?aioeeaeueiuie

o?aaiaieyie iacuaathony aeeiaie/aneeie nenoaiaie.

Anee nenoaia iienuaaaony aeaaa?ae/aneeie o?aaiaieyie –

– yoi iienaiea ninoiyiey ?aaiiaaney (noaoe/aneea nenoaiu)

(1)

(1)- eeiaeiia aeeooa?aioeeaeueiia o?aaiaiea n-ai ii?yaeea.

I?aaay /anoue – yoi aeeooa?aioeeaeueiia o?aaiaiea aic-

aeaenoaey. Anee Ly=0 (2) ,oi Ly=Px.

(2)- iaeii?iaeiia aeeooa?aioeeaeueiia o?aaiaiea – iienuaaao

eeiaeiua aeeiaie/aneea nenoaiu aac aicaeaenoaey ia

ieo. Iai?eia? eieaaaoaeueiue eiioo?.

I?aaay /anoue o?aaiaiey (1) iienuaaao aicaeaenoaea ia ee-

iaeioth nenoaio eee iacuaaaony oi?aaeaieai.

Ly=x – oi?aaeaiea.

Anee anoue /anoue Px – oi yoi neiaeiia oi?aaeaiea, o/eouaa-

thuaa nei?inoue, onei?aiea.

Ia?aaeaoi/iay ooieoeey eeiaeiie nenoaiu

Io aeeooa?aioeeaeueiiai o?aaiaiey (1) iiaeii ia?aeoe e eeiae-

iie nenoaia, o.a. e iaeioi?iio /aou?aoiiethnieeo.

Ao W(p) Auo

Yoio /aou?aoiiethniee iiaeii nicaeaoue ia yeaiaioiie aaca eee

niiaeaee?iaaoue ia YAI.

Io aeeooa?aioeeaeueiiai o?aaiaiey (1) e W(p) iiaeii ia?aeoe

aeaoiy iooyie – eniieuecoy neiaiee/aneee iaoiae e 2-a i?ai-

a?aciaaiea Eaieana.

Neaiiee/aneee iaoiae Oeae Naeaea.

I?eiaiea neiaiee/aneee iaoiae e (1) iieo/ei :

(3)

Oi?ioea (3) i?aaenoaaeyao niaie ioiioaiea aeaoo iieeiiiia –

iienaiea ia?aaeaoi/iie ooieoeee.

Eniieueciaaiea i?aia?aciaaiey Eaieana

– i?aia?aciaaiea Eaieana, p=j(

Anee iu i?eiaiei i?aia?aciaaiea Eaieana e eaaie /anoe (1)

, iieo/ei :

(4)

X(p) Y(p)

W(p)

Anee i?aaay /anoue ia?aaeaoi/iie ooieoeee i?inoaeoay –

, oi aicaeaenoaea iau/iia. Ia?aaeaoi/-

iay ooieoeey aoaeao eiaoue aeae :

, aaea ciaiaia-

oaeue ae?iae anoue oa?aeoa?enoe/aneia o?aaia-

iea.

I?eia? : Aeeooa?aioeeaeueiia o?aaiaiea 2-ai ii?yaeea iienu-

aaaony ia?aaeaoi/iie ooieoeeae :

(6)

Aeey iaoiaeaeaiey ?aoaiey aeeooa?aioeeaeueiiai o?aaiaiey nia/a-

ea iaiaoiaeeii ?aoeoue neaaeothuaa o?aaiaiea :

Ecaanoii, /oi aeeooa?aioeeaeueiia o?aaiaiea 2-ai ii?yaeea

eiaao ?aoaiea a aeaea eiiieaeniie yeniiiaiou eee aeaenoaee

iaae iae. (Yoi caaeneo io ei?iae oa?aeoa?enoe/aneiai o?aa-

iaiey). Anee ei?ie eiiieaeniua, oiaaea ?aoaiea aoaeao :

(t)

(7)(

(7) e (7)’ – ?aoaiea a aeaea ia?anoathuae eee caoooathuae neionieaeu,
eeai iau/iie neionieaeu, anee (=0.

Onoie/eainoue eeiaeiuo nenoai

Eeiaeiay nenoaia iieiinoueth iienuaaaony ia?aaeaoi/iie ooie-

oeeae, eioi?ay i?aaenoaaeyao niaie :

a eiiieaneiie ieineinoe

p=(+j( . Yoe iieeiiiu iieo/aiu ec aeeooa?aioeeaeueiuo o?aa-

iaiee iooai i?aia?aciaaiey Eaieana.

Noaaeony i?iaeaia: eae enneaaeiaaoue nenoaio n iiiiuueth W(p)

Ieacuaaaony, /oi yoi i?iua naeaeaoue /ai enneaaeiaaoue aeeooa-

?aioeeaeueiua o?aaiaiey. Enneaaeiaaiea ii W(p) i?iecaiaeeony n iiiiuueth
aiaeeca iiethnia e ioeae.

Iiethnii iacuaaaony oi cia/aiea ei?iy o?aaiaiey a ciaiaiaoaea, i?e
eioi?ii Q(p)=0.

Eiee/anoai ei?iae ii?aaeaeyaony noaiaiueth iieeiiia. Anee

)=0,

W(p)=( – iiethn.

Ioeyie W(p) iacuaathony oi/ee ia eiiieaeniie ieineinoe,

aaea iieeiii P(p)=0.

Eiee/anoai ioeae ii?aaeaeyaony ii?yaeeii iiee-

iiia.

j(

( > 0 iiethnu

( > 0

– iiethnu (ei?ie oa?aeoa?enoe/aneiai o?aa-

iaiey). Anee ei?ie eiiieaeniua, oi iie nii?yaeaiiua.

Auaiaeu :

1. Anee ei?ie oa?aeoa?enoe/aneiai o?aaiaiey Q(p)

iaoiaeyony a eaaie iieoieineinoe , oi nenoaia on-

((t(() – ?aoaiea aeey eiiieaeniuo

ei?iae.

((t(().

Nenoaia iaonoie/eaa.

?aniieiaeaiea ioeae ii?aaeaeyao ei??aeoe?othuea naienoaa nenoaiu,
o.a. ieacuaatho aicaeaenoaea ia ia?aoiaeiie i?ioeann

Anee ioee a eaaie iieoieineinoe, oi oaeay nenoaia iacuaaaony
ieieiaeueii oaciaie.

Anee ioee a i?aaie iieoieineinoe – iaeeiaeii oaciaay

nenoaia.

Anee iiethnu ia iieiie ine, o.a. (=0, oi nenoaia iaoi-

aeeony a eieaaaoaeueiii ?aaeeia (Nenoaia aac iioa?ue).

Ia?aaeaoi/iay ooieoeey eeiaeiie nenoaiu ia iieiie ine

A yoii neo/aa iinea i?aia?aciaaiee iieo/ei:

W(j()=A(()+jB(() –

Ia?aaeaoi/iay ooieoeey anoue eiiieaeniia /enei.

Caia/aiea: Ia iooaoue n ei?iyie ia iieiie ine.

Ieacuaaaony i/aiue oaeiaii enneaaeiaaoue W(j()ia iieiie ine ia n
iiiiuueth ioeae e iiethnia, a n eniieueciaaieai eiiieae-

niie ia?aaeaoi/iie ooieoeee.

Eiiieaeniay ooieoeey :

A*O

O*O

A*O iieacuaaao naeaeoeaiinoue nenoaiu ii

aiieeooaeiiio niaeo?o.

O*O iieacuaaao – eaeie naeaea oac iieo/aao ia

auoiaea oeeueo?a eaaeaeay aa?iiieea.

Caia/aiea: Ecaanoii, /oi niaeo? neaiaea (ii

Oo?uea) oaeiaii i?aaenoaaeyoue a eii-

ieaeniie aeaea, o.a. o niaeo?a anoue A*O (?an-

i?aaeaeaiea aa?iiiee ii aiieeooaea io /anoiou), e O*O (?an-

i?aaeaeaiea oac).

Auaiaeu: Eiiieaeniia i?aaenoaaeaiea niaeo?a eee ia?aaeaoi/-

iie ooieoeee W(p) i/aiue oaeiaii ?aaeeioaoieea. Yoi

iicaieyao eiiiaeoii caienaoue A*O e O*O.

Ia?aaeaoi/iay ooieoeey nenoai ?aaeeiaaoiiaoeee

1)

auo

Ia?aaeaoi/iay ooieoeey iineaaeiaaoaeueii niaaeeiaiiuo caaiue-

2)

Ia?aaeaoi/iay ooieoeey ia?ae-

eaeueii niaaeeiaiiuo caaiueaa:

ao auo

: :

: :

: :

3) y(t) Ia?aaeaoi/iay ooieoeey nenoaiu

(((( n ia?aoiie naycueth:

Oeiiaua caaiuey ?aaeeiaaoiiaoeee

1) Eia?oeeiiiia caaii

Ia?aaeaoi/iay ooieoeey :

C

W(() A*O

K

( (()= – arctgT( O*O

0
(

-45(

-90(

2) Eioaa?e?othuaa caaii

Ia?aaeaoi/iay ooieoeey :

3) Aeeooa?aioee?othuaa caaii

C

R

R L

W(() A*O Ia?aaeaoi/iay ooieoeey :

W(p)=Kp

A*O: W(()=K(

0 (

4) Oi?ne?othuaa caaii

W(() A*O

Ia?aaeaoi/iay ooieoeey:

0

( (()

0 (

5) Caiacaeuaathuaa caaii

=1 Ia?aaeaoi/iay ooieoeey :

((() O*O

A*O

1

Caiacaeuaathuaa caaii iacuaaaony eeieae caaea?aeee, aaea

5) Eieaaaoaeueiia caaii

Ia?aaeaoi/iay ooieoeey:

– ia?aiao? caoooaiey

1 – naiiaicaoaeaeathuayny

nenoaia

O*O

6) Iaieieiaeueii oaciaia caaii

Ia?aaeaoi/iay ooieoeey:

; W(()=1

A*O

O*O

Oeeo?iaua nenoaiu aaoiiaoe/aneiai oi?aaeaiey

Caaeai i?ioeann: Aoaeai ?anniao?eaaoue i?i-

y(t) oeann y(t) a aeene?aoiua ii-

iaiou a?aiaie.

Oaeie i?ioeann iacuaaaony n

aeene?aoiui a?aiaiai.

Cia/aiey yoiai i?ioeanna a

aeene?aoiua iiiaiou :

– cia/aiey

Nouanoaotho aeaa oeia i?ioeanna n aeene?aoiui a?aiaiai :

1)I?ioeann n aeene?aoiui a?aiaiai e iai?a?uaiui iiiaeanoaii

yaeyaony iai?a-

iai?a?uaia a

n?aaeiai eaaae?aoe/aneii).

ICN

– iai?a?uaiua ooieoeee

ICN – i?eai? n ca?yaeiaie naycueth

– eioa?aae aeene?aoecaoeee ai a?aiaie (eaaioiaaiea ii

a?aiaie)

Aeey oaeeo i?ioeannia ninoaaeythony ?aciinoiua o?aaiaiey :

– 1-a i?e?auaiea, eiia/iay ?aciinoue

– 2-y ?aciinoue

2) I?ioeann n aeene?aoiui a?aiaiai e aeene?aoiui iiiaeanoaii

ninoiyiee.

caienu-

aaaony a oeeo?iaii aeaea – aeene?aoiay ooieoeey, any aaca

enneaaeiaaiee ae?oaay. Eaaioiaaiea eaeao e ai a?aiaie e

ii o?iaith.

I/aiue /anoi aeaeaaony aeia?iia eaaioiaaiea 0;1. A yoii

neo/aa aiia?aoo?a neeueii oi?iuaaony.

Caia/aiea :

, oi auoiae-

, o.a. oaeie aea, ii aeene?ao-

iue.

– aeiiieaeueiia ?ani?aaeaeaiea.

Ieacuaaaony, anee /enei o?iaiae eaaioiaaiey ( 8,oi

eo iiaeii ioiaeaeanoaeoue n iai?a?uaiuie nenoaiaie.

I?aaenoaaeaiea aeeooa?aioeeaeueiuo o?aaiaiee, iienuaathueo

nenoaiu aaoiiaoe/aneiai oi?aaeaiey eiia/iuo ?aciinoae

– ia?aay ?aciinoue, aiaeia ia?-

aie i?iecaiaeiie

n – iai?a?uaiia a?aiy, iai?a?uaiia iiiaeanoai ninoi-

yiee.

– aiaeia 2e

i?iecaiaeiie

…………………………………

– aiaeia E-oie i?iecaiaeiie

Anee yoi iiaenoaaeoue a iai?a?uaiia aeeooa?aioeeaeueiia o?aa-

iaiea oi iieo/ei neaaeothuaa :

Anee iiaenoaaeoue a (2) ?aciinoe, oi iieo/ei :

– ?aciinoiia o?aaiaiea n aeene?aioiui a?aiaiai.

Z -i?aia?aciaaiey

Aiaeiae/iu i?aia?aciaaieth Eaieana. Yoi i/aiue oaeiaiue aiia?ao aeey
enneaaeiaaiey nenoai n aeene?aoiui a?aiaiai a

/anoioiie iaeanoe. Aeey yoiai aianoi yeniiiaiou (aeey oi?i-

– yoi anoue Z-i?aia?aciaaiea. Aeey

oiai, /oiau aaanoe Z-i?aia?aciaaiea eniieuecoaony nea-

aeothuee i?eai nayce iai?a?uaiiai i?ioeanna X(t)e aeene?aoii-

(1)

X(1),X(2) – auai?ea n aeene?ao-

?anniio?ei i?aia?aciaaiea Eaieana :

(2)

Oi?iaeueii aaaaeai iiaoth ia?aiaiioth :

(3)

Eniieuecoy (2) e (3) iieo/ei

(4)

(4) – iacuaaaony Z-i?aia?aciaaiey, iieacuaaao eae ia?aeoe

io ooieoeee n aeene?aoiui a?aiaiai (X(n)) e niaeo?o

ia Z-ieineinoe.(Iii i?iua i?aia?aciaaiey Eaieana,

ii eiaao oa aea naienoaa e aeey ?aciuo aeene?aoiuo

ooieoeee eiathony niaoeeaeueiua oaaeeoeu.

Onoie/eainoue nenoai n aeene?aoiui a?aiaiai

Nenoaiu n iai?a?uaiui a?aiaiai oa?aeoa?ecothony ia?aaeaoi/-

iie ooieoeeae (ioiioaiey 2o iieeiiiia), oiaea naiia a Z-i?a

,

(ia eeiaeiie ine)

P-ieineinoue
Z-ieineinoue

(Nenoaia

onoie/eaa)

– ie?oaeiinoue, neaaeiaaoaeueii eaaay eiiieaeniay iieo-

ieineinoue eaa/a i?aia?acoaony ai aioo?aiiinoue e?oaa

Anee iiethnu ia?aaeaoi/iie ooieoeee iaoiaeyony ai aioo?aiiin-

oe e?oaa, oi nenoaia onoie/eaa, anee iiethnu iaoiaeyony ia

naiii e?oaa, oi aoaeao eieaaaoaeueiue i?ioeann, anee aia

e?oaa – nenoaia iaonoie/eaay.

– onoie/eaay nenoaia – eieaaaoaeueiay

nenoaia

n

– iaonoie/eaay nenoaia

n

Aeaaa 3

Iaeeiaeiua aeeiaie/aneea nenoaiu

Iaeeiaeiua aeeiaie/aneea nenoaiu iienuaathony aeeooa?aioee-

aeueiuie o?aaiaieyie :

Anee eeiaeiua aeeooa?aioeeaeueiua o?aaiaiey eiatho ?aoaiey

(yeniiiaioeeaeueiua), oi aeey iaeeiaeiuo aeeooa?aioeeaeueiuo

o?aaiaiee iao iaueo ?aoaiee (ca ?aaeeei eneeth/aieai), ii

ana ?aaeueiua aeeiaie/aneea nenoaiu iaeeiaeiu, iaeioi?ua

ec ieo iaeuecy eeiaa?ece?iaaoue, eae auoue ?

Auoiae : 1) Oai,aaea aiciiaeii, aeaeaaony eeiaa?ecaoeey i?aaie

/anoe o?aaiaiey (1).

Eeiaa?ecaoeey – caiaia iaeeiaeiie ooieoeee ia eeiaeioth.

(2) f(x,t)=A(t)x + B(t) + S(x,t)

S(x,t) – iaei, ei iiaeii i?eiaa?a/ue.

Anee i?aaay /anoue (1) ia caaeneo io a?aiaie, oi nenoaia

Eeiaa?ecaoeey eniieuecoaony,eae i?aaeei, aeey i?iaa?ee

onoie/eainoe nenoaiu. Aeey enneaaeiaaiey naienoa iaeeiae-

iuo aeeiaie/aneeo nenoai, iau/ii eniieuecothony ea/anoaaiiua

e /eneaiiua iaoiaeu ?aoaiey iaeeiaeiuo aeeooa?aioeeaeueiuo o?aaiaiee.
Oai?ey iaeeiaeiuo o?aaiaiee /anoi iacuaaaony

oai?eae iaeeiaeiuo eieaaaiee.

I?eia? : Iaeeiaeiie aeeiaie/aneie nenoaiu o?aaiaiee Aaiaea?

Iiey.

– iaeeiaeiinoue.

= const

Aeeooa?aioeeaeueiia o?aaiaiea iacuaaaony iaeeiaeiui, anee

iii iaeeiaeii ioiineoaeueii ?acuneeaaaiie ia?aiaiiie (na-

iie ia?aiaiiie eee aa i?iecaiaeiie) (iaeeiaeiinoue ec-ca

eaaae?aoa)

O?aaoaony iaeoe ?aoaiea x(t) .

Nouanoaotho /eneaiiua iaoiaeu ?aoaiey oaeeo aeeooa?aioeeaeue-

iuo o?aaiaiee ( /eneaiiua iaoiaeu ?anniao?eaathony ia nao-

) . ?aoaiea iieo/aaony ia iai?a?uaiia , a

aeene?aoiia.

*eneaiiua iaoiaeu iienuaa-

t thony a eieaa: Yeuenaieueoe

‘Oai?ey aeeooa?aioeeaeueiuo

o?aaiaiee e aa?eaoeeiiiia

en/eneaiea’.

U

*eneaiiue iaoiae Yeea?a ( /eneaiiue iaoiae)

(5)

*eneaiiue iaoiae i?aaeiacia/ai aeey ?aoaiey ia-

eeiaeiuo aeeooa?aioeeaeueiuo o?aaiaiee.

,

iiaenoaaeyaony a i?aaoth /anoue o?aaiaiey (5) e

o.ae. Yoi iacuaaaony ?aeeo?aioiinoueth.

Ea/anoaaiiay oai?ey ?aoaiey iaeeiaeiuo aeeooa-

?aioeeaeueiuo o?aaiaiee (a i?eeiaeaiee e iaeeiaeiui nenoa-

iai)

A ioee/ea io /eneaiiiai iaoiaea (Iaoiae Yeea?a), eioi?ue

aeaao ?aoaiea a 1e oi/ea ( ia aeaao o?aaeoi?eth (ioaeii aea-

eaoue 1000 oi/ae, /oiau iieo/eoue o?aaeoi?eth)).

Ioai Ea?a a 19 aaea aeae ea/anoaaiioth oai?eth ?aoaiey aeeo-

oa?aioeeaeueiuo o?aaiaiee, iia eniieuecoaony aeey ?aoaiey ia-

eeiaeiuo aeeooa?aioeeaeueiuo o?aaiaiee a aeaea iaeioi?iai oa-

ciaiai ii?o?aoa (iaeioi?ue a?aoe/aneee iaoa?eae, ii ei-

oi?iio iiaeii aiaeece?iaaoue o?aaeoi?eth aeaeaeaiey aeeiaie/an-

eie nenoaiu, o.a. oaeoe/anee iieo/eoue ?aoaiea (1-ai ec

?aoaiee).

Ia i?eia?a X e Y :

f(x,y) – iaeioi?ay iaee-

( dy iaeiay ooieoeey

– iaeeiaeiay

ooieoeey

x

Iaeoe ?aoaiea icia/aao – iaeoe y=((x) (2),

eioi?ay oaeiaeaoai?yao (1).

Ioai Ea?a ?acaee iaoiae , eae iaeoe (2) i?yii ia

ieineinoe.

Iaoiae ecieeei

, ia e?eaie

f(x,y)=const ana i?iecaiaeiua eiatho iaeii e oiaea cia/aiea,

oaeay e?eaay iacuaaaony ecieeeiie. (tg(=const, (=const)

Iiaeii au/eneeoue iiiaeanoai ecieeei, yoi iiiaeanoai aeaao ii-

ea iai?aaeaiee. Eanaoaeueiay e yoiio iieth e anoue ?aoaiea,

o.i. yoi anoue o?aaeoi?ey, eioi?oth iu ?acuneeaaai.

y

– ?aoaiea aeeo. –
ecieeeia

o?aaiaiey

x

x

Aaee/eia ?aaeeona – cia/aiea i?iecaiaeiie, ethaay ie?oaeiinoue –
ecieeeia. ?aoaiea (eanaoaeueiay e iieth iai?aaeaiey) –

-anoue eanaoaeueiay e aaeoi?ai, ?aniieiaeaiiay ia ecieeeiao.

( – ecieeeia

( ?aoaiea

– O?aaiaiea Aaiaea? Iiey

x(t) – iai?yaeaiea ia eiioo?a aaoiaaia?aoi?a, oaciaay ia-

?aiaiiay

= const – ia?aiao?

– aoi?ay oaciaay ia?aiaiiay

O/eouaay yoi eiaai :

= 0


ecieeeia

– oaciaue ii?o?ao

– ?aoaiea aeeooa?aioeeaeue-

iiai o?aaiaiey Aaiaea?

Iiey – ie?oaeiinoue

= 0)

Anee ia aoiaeu X e Y inoeeeia?aoa iiaeaoue aeaa neionieaeu, oi

iieo/ei ie?oaeiinoue (oeao?a Eennaaeo), neaaeiaaoaeueii ie?oae-

iinoue aeaao ?aoaiey neionieaeaeueiiai eieaaaiey.

x Y

t t

( 0 (ni. o?-a (1)’) oaciaue ii?o?ao aoaeao 2o oe-

iia :

Y X(t)

X

t

Auaiaeu :

1) Aeeiaie/aneea nenoaiu ?aaeeiaaoiiaoeee iienuaa-

thony aeeooa?aioeeaeueiuie o?aaiaieyie 1, 2 e ai-

eaa aunieiai ii?yaeea ( iai?eia?: eieaaaoaeue-

iay nenoaia(nieia/iay nenoaia, aaoiaaia?aoi?,

iieao einie/aneiai aiia?aoa a iiea i?eoyaea-

iey caiee) iienuaaaony aeeo. o?aaiaieai 2-ai

ii?yaeea e auoa.

2) Eeiaeiua aeeiaie/aneea nenoaiu iienuaathony ee-

iaeiuie aeeo. o?aaiaieyie. Eeiaeiay aeeiaie/an-

eay nenoaia ninoaaeaiiay ec R,L,C – oeaii/ae e

aeoeaiuo yeaiaioia (o?aicenoi?ia e o.ae.).

Ethaay eeiaeiay nenoaia iooai i?aia?aciaaiey

Eaieana iiaeao auoue i?aaenoaaeaia a aeaea ia?a-

aeaoi/iie ooieoeee.(Aeeo. o?aaiaiea i?aia?acoao-

ny ii Eaieano). Ia?aaeaoi/iay ooieoeey caienu-

aaaony aeey oaeianoaa a eiiieaeniii aeaea, ia

iieiie ine p=j( iiaeii iaeoe A*O e O*O eeiae-

iie nenoaiu. Ia?aaeaoi/iay ooieoeey aeaao eioi?-

iaoeeth ia onoie/eainoe nenoaiu.

3) Iaeeiaeiua aeeiaie/aneea nenoaiu iienuaathony

iaeeiaeiuie aeeo. o?aaiaieyie, a yoeo nenoaiao

e ae?.), iaueo ?aoaiee e aiaeeca /a?ac ia?aaea-

oi/ioth ooieoeeth eae i?aaeei ia nouanoaoao, ii-

yoiio anoue aeaa iaoiaea :

a) /eneaiiue iaoiae (Yeea?a e ae?.) (ainnoaiiaea-

iea ii oi/eai)

a) ?aoaiea aeeo. o?aaiaiee iaoiaeii oaciaiai ii?o-

?aoa (ea/anoaaiiay oai?ey). (Yoi iaaeyaeiue

iooue auyniaiey iiaaaeaiey iaeeiaeiie nenoaiu)

Noioanoe/aneea nenoaiu

Noioanoeea – neo/aeiinoue.

Ii?aaeaeaiea: Aeeiaie/aneay nenoaia iacuaaaony noioanoe/an-

eie , anee iia iienuaaaony aeeooa?aioeeaeueiui

eee ?aciinoiui o?aaiaieai, a i?aaoth /anoue

eioi?iai aoiaeeo neo/aeiue i?ioeann.

Oaeoth nenoaio iiaeii i?aaenoaaeoue a aeaea eeiaeiiai eee ia-

eeiaeiiai /aou?aoiiethnieea, ia aoiae eioi?iai iiaeaaony ooi

Noioanoe/aneay

((t) nenoaia X(t)

((t)- ooi

X(t)- auoiaeiie i?ioeann

Ninoaaeaiea iiaeaee ethaie aeeiaie/aneie nenoaiu aeieaeii

a ?aaeueiuo oneiaeyo(iai?eia? aeaeaeaiea naiieaoa eee ?aea-

ou) ninoaaeyoueny n iiiiuueth i?aaeaa?eoaeueiuo yenia?eiaioia

iaae aeaeaeaieai ?aaeueiie nenoaiu. (Eae i?aaeei yoi aeeooa-

?aioeeaeueiua eee ?aciinoiua o?aaiaiey) e a yoe o?aaiaiey

anoaaeyaony iaeioi?ue ooi, eioi?ue yaeyaony neo/aeiui

i?ioeannii.

Aeey aeaeueiaeoaai ninoaaeaiey iiaeaee eniieuecoaony eaeai-

oeoeeaoeey iiaeaee ia iniiaaiee yenia?eiaioa eee yenia?e-

iaioaeueiuo aeaiiuo.

Eaeaioeoeeaoeeae iacuaaaony ioeaiea eiyooeoeeaioia ?aciino-

iiai o?aaiaiey e ioeaiea ia?aiao?ia ooia:

aeenia?nee, iao. iaeeaeaiey, eiaa?eaoeee e ae?.

Eaeaioeoeeaoeey neoaeeo aeey oiai, /oiau ?aaeueiue i?ioeann e

iiaeaeue auee aeecee.Iieo/ea iiaeaeue iu eiaai aiciiaeiinoue,

eniieuecoy yoo iiaeaeue, iieo/eoue aeeceoth e ?aaeueiie ea?oe-

ia neooaoeeth aeaeaeaiey nenoaiu e nicaeaoue oi?aaeaiea neooa-

oeeae ii iaoae iiaeaee.

Auaiae: Iiaeaeue ioaeia, /oiau ia YAI iao/eoueny i?iaeoe?iaaoue

oi?aaeyaiua aeeiaie/aneea nenoaiu aeey ethauo oaeoe-

/aneeo neooaoeee, ecaanoiuo ec i?aeoeee.

I?aaeeueii nicaeaiiay iiaeaeue – yoi iaeneioi oniaoa a i?iae-

oe?iaaiee yooaeoeaiie nenoa-

iu. Iinea nicaeaiey e io?aaioee iiaeaee noioanoe/aneie aee-

iaie/aneie nenoaiu nicaeaaony aiia?aoo?a ii yoie iiaeaee,

eioi?ay i?iaa?yaony ia aeeiaie/aneii noaiaea.

Aeeiaie/aneee noaiae – 2e yoai iiaeaee?iaaiey ?aaeueiie neoo-

aoeee oaea n aiia?aoo?ie.

3e yoai ninoieo a i?iaa?ea aiia?aoo?u ia iieeaiia.( Ia

ai?oo o?ainii?oiiai eee aiaiiiai n?aaenoaa).

Iiaeaee?iaaiea neo/aeiuo i?ioeannia n aeene?aoiui a?aiaiai

– auai?ea neo/aeiiai i?ioeanna n aeene?aoiui

a?aiaiai.

X(t) I?ioeann (1) a iauai aeaea i/aiue

o?oaeii aiaeece?iaaoue, yoio i?i-

oeann, eae i?aaeei, iieo/ai ec

yenia?eiaioa. Yoio ?aaeueiue

i?ioeann iau/ii aii?ieneie?oaony

ae?oaei i?ioeannii, eioi?ue iic-

aieeo iai iaoaiaoe/anee nicaea-

t aaoue iiaeaee, aeeceea e ?aaeueii-

io i?ioeanno.

Oaeia nicaeaiea iiaeaeae iacuaaaony – aii?ieneiaoeeae.

Nai aii?ieneie?othuee i?ioeann iacuaaaony aa?aaao.

Ia?eianeay aii?ieneiaoeey neo/aeiuo i?ioeannia

Ia?eianeei i?ioeannii iacuaaaony oaeie i?ioeann, o eioi?iai

iiiaiia?iay ieioiinoue aa?iyoiinoe

. Iaeioi?ua

cia/aiey oaciauo ia?aiaiiuo a n-ia?iii i?ino?ainoaa – yoi

iiiaiia?iay ieioiinoue aa?iyoiinoe

Aeaoia?iay ieioiinoue Iiiaiia?iay OIA ianao anth ei-

aa?iyoiinoe oi?iaoeeth i neo//aeiii i?ioean-

W(x,y) na. Aieueoa eioi?iaoeee ia no-

uanoaoao.

Iaeiaei eniieueciaaoue yoo iii-

aiia?ioth OIA /?acau/aeii nei-

aeii ia i?aeoeea, iiyoiio /an-

oi i?eaaaatho e iaeioi?ui ai-

i?ieneiaoeeyi i?ioeanna :

Y

X

Aii?ieneie?iaaoue – aua?aoue oaeea ion/aou

auee

iacaaeneiu, oiaaea iiiaiia?iay OIA oaeoi?ecoaony neaaeoth-

– oaeoi?ecaoeey.

Iaeiaei i?e oaeie oaeoi?ecaoeee iiaeao iioa?youeny eioi?ia-

oeey i neo/aeiii i?ioeanna. Anoue iioa?y eioi?iaoeee aeey

i?iecaieueiuo ion/aoia (ei?aeee?iaaiiinoue i?ioeanna).

Nouanoaoao 2e niinia aii?ieneiaoeee – ia?eianeee niinia

aii?ieneiaoeee. Aeey ia?eianeeo i?ioeannia iiiaiia?iay OIA

oaeoi?ecoaony oae :

– on-

eiaiay ieioiinoue aa?iyoiinoe.

Oaeoi?ecaoeey (2) iicaieyao neeueii oi?inoeoue iaoaiaoe/an-

eea aueeaaeee a caaea/ao oeeueo?aoeee e oi?aaeaiey.

Ii?aaeaeaiea : I?ioeann iacuaaaony ia?eianeei, anee auiie-

iyaony oneiaea (2)

Ieacuaaaony, nouanoaoao i/aiue iiiai aaia?aoi?ia ia?eian-

eeo i?ioeannia. Iu ia?aoiaeei e eo ?anniio?aieth.

I?ioeannu aaoi?aa?annee

I?ioeann aaoi?aa?annee – i?inoie aaia?aoi? ia?eianeiai

i?ioeanna.

1. Iaeiinayciay ?aa?anney

– io aaia?aoi?a aaeiai ooia

– ei??aeyoeey.

(0 eiaai

onoie/eaue i?ioeann.

a1 iui e aeinoaoi/iui oneiaeai

onoie/eainoe nenoaiu.

Onoie/eainoue iaeeiaeiuo nenoai

Iaeeiaeiay noioanoe/aneay nenoaia :

Onoie/eainoue iaeeiaeiuo aeeiaie/aneeo nenoai ii?a-

aeaeyaony ooieoeeae Eyioiiaa.

Ii?aaeaeaiea onoie/eainoe ii Eyioiiao aeey aeaoa?ieie?iaai-

iie nenoaiu.

Aaiaeeony niaoeeaeueiay ooieoeey, iacuaaaiay ooieoeeae Eyioii-

. Ooieoeey oaeiaeaoai?yao neaaeothuei

oneiaeyi :

=0

0,

Aeey noioanoe/aneie nenoaiu (3)

iau/ii ooieoeeth Eyioiiaa auae-

. A oneiaea

onoie/eainoe aeey nenoaiu (3)

aoaeao neaaeothuei:

,

i(( (anneiioioe/anee)

Aiaeec ea/anoaa ?aaiou noioanoe/aneeo nenoai ?aaeeiaaoiia-

oeee

Ea/anoai eeiaeiuo e iaeeiaeiuo noioanoe/aneeo nenoai ii-

?aaeaeyaony ?aaeueiui ea/anoaii oeeueo?a. (ni. auoa)

Neioac i?aaeiieaaaao, /oi iiaeaeue niioaaonoaoao ?aaeueiiio neo/aeiiio
i?ioeanno, eioi?ue iu oeeueo?oai. A yoii neo/aa

ea/anoai ii?aaeaeyaony neaaeothuei ia?acii :

I?eia?: Iaeiiia?iue oeeueo? Eaeiaia.

– ooi iaaethaeaiee

– aiinoa?ei?iay aeenia?ney

– eiyooeoeeaio oneeaiey

oeeueo?a Eaeiaia

i – aeene?aoiia a?aiy

Ea/anoai oeeueo?aoeee ii?aaeaeyaony aaeaeaaoiinoueth iiaeaee e
?aaeueiiai i?ioeanna. Eae i?iaa?eoue aaeaeaaoiinoue iiaeaee

?aaeueiiio i?ioeanno ? Naeaeaoue yoi

,

.

i

Oai?aia : I?ioeann oiaaea e oieueei oiaaea aaeaeaaoai iiaeaee,

eiaaea iaaycea yaeyaony aaeui ooiii.

I?iaeaia ea/anoaa ii?aaeaeyaony i?iaeaiie yeno?aiieyoeee.

PAGE

PAGE 54

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