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A coy eopey o oeo ae y lim y(t) = lim sy(s), t sW (s)A e y(s) = W (s)X (s) =, oo aca, o s sW (s)A lim y(t) = lim = AlimW (s).

t s0 ss p eo cyeao oec A = 1 oa blim y(t) = limW (s) =.

t sa4.9 TPEHPOBOHE AAH 1 Ocoo acoo xapaepco ec ayo-aoa xapaepca (AX), oopo aaec oopoe oopaee o oc ococ ope xapaepcecoo ypae a ococ AX. Ayo-aoa xapaepca ec oeco ye oe acaa oaaeo ope W (i) = M () ei () aepaeco ope W (i ) = Re() + i Im(), e M() aaec ayo-acoo xapaepco (AX); () ao-acoo xapaepco (X); Re() eeceo-acoo xapaepco (BX); Im() o acoo xapaepco (MX). Mey xapaepca cyecye c.

A Copypye ocoe coca oopoo oopae.

B Ec ec AX X, o a opao opeeec BX MX C a epe o BX MX AX X 2 acoe xapaepc oy oye cepeao peyae oa a xo oea apoecoo caa, a ae eopeec epeaoo y oecoo apa-epa s a i.

A ae acoe xapaepc oya cepeao B aaa epeaoa y W (s) =, ae ayo-aoy s + xapaepcy oaaeo aepaeco ope.

C aao epeaoe ypaee oea ypae y (t) + 4y (t) + 4y(t) = 3x(t), ae ayo-aoy xapaepcy.

3 Ayo-aoa xapaepca caa c py aec xapaepca.

A a opee ecoy y o ayo-aoo xapaepce B a opee AX o epexoo y C aaa ecoa y w(t) = e-t, ae AX.

4.10 TECT 1 B cooec co coca oopoo oopae epexo A B .

B Boy.

C B peyo.

2 Ayo-aoa xapaepca ec A Cyao ye.

B oeco ye.

C eeppoao ye.

3 a cepeao oya acoe xapaepc oae a xo oea A apoecoo caa x(t) = Asin t.

B -y x(t) = (t).

C Eoo cyeaoo caa x(t) = 1(t).

4 a epe o epeaoo y aco xapaepca oo A s = i.

B s =.

C s = eit.

k 5 Ec epeaoa y oea ypae W (s) = + s, o AX oaaeo s ope aec k A W (i) = e-i arctg.

k - 2 -i B W (i) = e.

k k - 2 -i 2 +arctg C W (i) = e.

6 Ec epeaoa y oea ypae W (s) = 3e-4s, o ayo-acoa xapaepca aec a A M () = 3e-4.

B M () = 3 sin 4 + cos4.

C M () = 3.

7 Ec epeaoa y oea ypae W (s) =, o ao4s (s + 3)(5s + 2) acoa xapaepca aec a A (x) = - - arctg - arctg.

2 3 B (x) = + arctg - arctg - arctg.

2 3 C (x) = -arctg - arctg - arctg.

3 8 Ec epeaoa y oea ypae W (s) = 4 + s, o eeceoacoa xapaepca aec A Re() = 16 + 2.

B Re() = 4.

C Re() =.

9 Ec epexoa y h(t) = t, o AX acaec -i A W (i) = e.

B W (i) =.

-i C W (i) = e.

10 Ayo-acoa xapaepca pecae coo A Ooee xooo caa xooy cay.

B Ooee a xooo xooo cao.

C Ooee ay xooo caa aye xooo.

5 CTPTPH AHA HEHX CCTEM 5.1 BEHO HAPABEHHOO ECTB p cceoa cce ypae epoceeoe aee popeae xapaep peopaoa cao oex eeax, ex. aece cce, epeaoe y oopx e pocx poe, aac o eeap e. o poe oe pecaec e cax ey coo ox ee. x ocoy cocae eo apaeoo ec, ocooe coco oopoo aaec o, o xoa ea y(t) ac o xoo e x(t), o opaoe oece xoa a xo ocycye. pcoeee xoy aoo ea pyoo ea e ee epeaoo y epoo ea.

eca ppoa ea apaeoo ec oe o. Xapaepyec oo cooecy ypaee e, oopoe opeee ope eeapoo ea.

Paa ceye e: yceoe, eppyee, eaoe peaoe epepye, opcpyee, coo aaa, epoo-opcpyee, aepoece epoo opoo opa, oeaeoe, oope o py ox aooepoce oo pae a ceye py:

1 Caece e, y oopx caeca xapaepca oa o y, e ooay c ey xoo xoo epee caeco pee. ooc yceoe, aepoecoe, oeaeoe e, y oopx epeao oe ca c epeaoo ye coooee k = W (s). poe oo, s=caece e c pa o aco, cee cocae yceoe eo.

2 epepye e, y oopx caeca xapaepca paa y, o eaoe peaoe epepye e; x epeaoy y cea xo cooe s, ooy W (s) = 0. epepye e c pa s=coo aco, o oc ooee aoe c.

3 Acaece e e, e ee caeco xapaepc, oocc eppyee eo, epeaoy y oopoo oaeo xo cooe, ooy W(0) =. eppye e c pa o s aco.

5.2 TOBE HAMECE BEH 5.2.1 ceoe eo ceoe eo aa ae caec (eepo). pepo eo oe cy aa c eapoao xapaepco cceax peypoa, pae yce, pae epea, peyop .. o eo oeo e cae ocpoo xoy ey a xoe.

paee e yceoo ea ee y(t) = kx(t), (5.1) e k - oe yce.

epeaoa y yceoo ea oyaec peyae peopaoa o aacy eo ypae y(s) = kx(s), oya y(s) W (s) = = k. (5.2) x(s) ocaoa s = (i) ae paee AX W(i) = k, (5.3) oca AX:

M() = k; (5.4) X:

() = 0. (5.5) pa acox xapaepc (AX, AX) pecae a pc. 5.1.

acoe xapaepc yceoo ea e ac o aco, pe X oeceo paa y, .e. apoecx oeax, oax a xo, eec oo aya k pa. Ayo-aoa xapaepca ec ooe ece co, ee pa pecae coo oy a ooeo e eceo oc.

i Im() M a) ) k k 0 Re() Pc. 5.1 acoe xapaepc yceoo ea:

a AX; AX x x a) ) (t) 0 t t w h k k (t) 0 t t Pc. 5.2 pa peex xapaepc yceoo ea:

a epexoa y; ecoa y Bpeee xapaepc oo oy eocpeceo ypae (5.1). Ec xoo ca x(t) = 1(t), o oya ypaee epexoo y h(t) = k1(t), (5.6) oa paa ocoo ee oey yce ea. Ec e x(t) = (t), o oya ypaee ecoo y w(t) = k(t). (5.7) pa peex xapaepc opae a pc. 5.2.

5.2.2 eppyee eo paee e eppyeo ea ee t y(t) = x()d, T y(t) = x(t) ; y(0) = 0, (5.8) T T ocoa pee ea.

Bxoo ca eppyeo ea pae epay o pee o xooo caa, yoeoy a oe.

T pepo eppyeo ea c ce, cypye pacxo eeca ep a opeee poeyo pee, ypoe eoc ..

epeaoa y eppyeo ea oyaec peyae peopaoa o aacy (5.8):

Tsy(s) = x(s) W (s) =. (5.9) Ts i Im() M a) ) ) 0 -/2 Re() W(i ) Pc. 5.3 acoe xapaepc eppyeo ea:

a AX; X; AX acoe xapaepc opayc peyae ocao s = i; x pa opae a pc. 5.3:

- AX -i 1 W (i) = = e ; (5.10) Ti T - AX M () = ; (5.11) T - X () = - / 2. (5.12) Ayo-acoa xapaepca eppyeo ea ec epoeco ye aco, a ao-acoa e ac o aco paa -. B o cyae AX ec o ye aco, ee oopa ooex aco coaae c opaeo e o oc.

epexoe xapaepc, pa oopx opae a pc. 5.4, opee ypae e (5.8) ocaoo xooo caa x(t) = 1(t) x(t) = (t) cooeceo oye pae:

- epexoo y t 1 h(t) = = t; (5.13) dt T 0 T - ecoo y t 1 w(t) = (5.14) (t)dt =.

T 0 T w h a) ) T t t Pc. 5.4 epexoe xapaepc eppyeo ea:

a epexoa y; ecoa y Ta opao, p oae a xo eppyeo ea ocooo eceaeo oye xoa oopaa yeaec o ecoeoc c ocoo copoc, .e. oeo ocoeoc ec o a, o epexoa y e ee ycaoeoc (p t ) oeoo ae. o coco ec po paoo o acaecx cce aoaecoo peypoa, coepax eppyee eo, o caecx cce, oope e coepa oo ea.

Pea a -y ec cyeao ye c ayo.

T 5.2.3 eaoe epepyee eo paee eaoo epepyeo ea y(t) = kx(t), (5.15) .e. eee xoo oopa poopoao copoc ee xoo oopa. B oepaopo ope ypaee ee y(s) = ksx(s), oya epeaoa y Y (s) W (s) = = ks. (5.16) X (s) acoe xapaepc, pa oopx pecae a pc. 5.5:

AX W(i) = k i = kei/2; (5.17) AX M() = k; (5.18) X () =. (5.19) a) ) Im ) M = 0 0 Re Pc. 5.5 acoe xapaepc eaoo epepyeo ea:

a AX; X; AX Ta opao, AX po poopoaa acoe, a X e ac o aco paa. Ceoaeo, oopa AX p > 0 coaae c ooeo e o oc.

epexoa y eaoo epepyeo ea ee :

h(t) = k1(t) = k(t), (5.20) .e. pecae coo -y c oa, pao k.

Becoa y pecae coo pooy o -y:

w(t) = k(t). (5.21) B ppoe eao epepyx ee e cyecye, a a p M(), a o pea oe paec pye apoece ca c acoo, oe aco cpea aoo oea. Heocyecoc eaoo ea a ae epexoo y, oopa paa -y ecoo y, pao pooo -y.

5.2.4 Peaoe epepyee eo Bcpeac e, oope peapy oo a copoc ee xooo caa. O ocac ypae ceyeo a aac pea epepy:

Ty(t) + y(t) = Tx(t). (5.22) pepo aoo ea ec RC-eoa (pc. 5.6).

R Ux eo J Ux J C Pc. 5.6 RC-eoa M a) i Im() ) ) T T Re() = 0 T 0 T Pc. 5.7 acoe xapaepc peaoo epepyeo ea:

a AX; X; AX epeaoa y ee :

Ts y(s) W (s) = =. (5.22) x(s) 1+ Ts acoe xapaepc, pa oopx pecae a pc. 5.7:

AX Ti T W (i) = = ei( / 2-arctgi) ; (5.23) 1+ Ti T 2 + AX T M () = ; (5.24) T 2 + X () = - arctgT. (5.25) peaoo epepyeo ea p yee aco ayo-acoa T xapaepca opacae, o ee epx pee opae eo.

T ao-acoa xapaepca p yee aco yeaec o o y.

T ooex aco W(i) pecae coo oyopyoc aepo c T T epo oe. oaaeca ae W(i) poyox oopaax 2T Ti(1- Ti) TT2 T W (i) = Re() + i Im() = = + i.

2 (1+ Ti)(1- Ti) 1+ T 2 1+ T T oyee ae Re() Im() oca ypaee opyoc payca c 2T T epo oe :

2T 2 T T + [i Im()]2 = Re() - 2T Re() 2T 2 TT2 T T T - + =.

2 2T 1+ T 2 2T 1+ T 2 Pacpa co, oyae oeco, oopoe oaae, o AX eceo pecae coo oyopyoc.

coy aoc aecx xapaepc, oyae ypaee epexoo y oepaopo ope o (3.39):

Ts T 1 h(s) = =.

1 + Ts s T 1 + s T pe opaoe peopaoae aaca oceey pae, oyae ypaee epexoo y o peeo oac:

T h(t) = e-t / T. (5.26) T Becoa y axoc a pooa o epexoo y T w(t) = - e-t / T. (5.27) T pa epexox xapaepc opae a pc. 5.8.

x x a) ) (t) 0 t t w h TA T t t TA T Pc. 5.8 epexoe xapaepc peaoo epepyeo ea:

a epexoa y; ecoa y Ha pc. 5.8, a cpae oaa epexoe y eaoo 1 peaoo epepyx ee. B cy ep peax ee eee xoo oopa epexoo y pocxo oceeo, a e cao, a cyae eaoo ea. oo, o p coca peaoo ea coca eaoo, eoxoo oopeeo yea oe epea T yea ocoy pee T a, o x poeee TT ocaaoc oco.

5.2.5 opcpyee eo opcpy eo aaec eo, ocaeoe ypaee dx(t) y(t) = k x(t) + T. (5.28) dt Taoe eo oe oyeo peyae apaeoo coee yceoo eaoo epepyeo ee. Oo xapaepyec y apaepa: oeo epea k ocoo pee T.

epeaoa y W(s) = k(1 + Ts). (5.29) aea (5.28) s = i ooe oy acoe xapaepc opcpyeo ea, pa oopx oaa a pc. 5.9:

AX W (i) = k(1+ iT ) = k 1+ (T )2 ei arctgT ; (5.30) AX M() = k 1+ (T )2 ; (5.31) X () = arctgT. (5.32) a) ) ) M i Im() W(i ) k / = 0 k Re() Pc. 5.9 acoe xapaepc opcpyeo ea:

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