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Mcepco opaoa Poccco eepa Taoc ocyapce exec yepce T. . aapea, . . Mapeo HEHE CCTEM ABTOMATECOO PEPOBAH onyeo

eo-emouecu oeueue yo no opaoau oacmu amoamuupoaoo auocmpoeu (MO AM) aecme yeoo nocou cmyemo cux yex aeeu, oyauxc no anpaeu "Texoou, oopyoaue u amoamuau auocmpoumex npouocm" u cneuaocm: "Texoou auocmpoeu";

"Memaopeyue cmau u ucmpyem";

"cmpyemae cucme umepupoax auocmpoumex npouocm" (anpaeue noomou unoupoax cneuaucmo "ocmpymopco-mexoouecoe oecneeue auocmpoumex npouocm");

"Amoamuau mexoouecux npoecco u npouocm ( auocmpoeuu)" (anpaeue noomou unoupoax cneuaucmo "Amoamuupoae mexoouu u npouocma") Tao aeco TT 681. 965.73- Pe e e :

aepa "Bcee ycpoca aoaa" Taocoo ceo aaooo eepoo cya;

oop execx ay, poeccop Mococoo ocyapceoo yepcea eepo oo B. C. aaupe aapea T. ., Mapeo . .

ee cce aoaecoo peypoa: eoe ocoe. Tao: o Ta. oc. ex. y-a, 2001. 264 c.

ISBN 5-8265-0149- B yeo oco oe ocoe p eo eop aoaecoo ypae: ocpoee cce ypae, eo x aeaecoo oca, pep oe ycooc aeca peypoa ex epepx eeppoax cce.

peaaeo cyeo cx yex aee, oya-xc o apae 651900 657900, o ce cce caooo opaoa.

681. 965.73- aapea T. ., Mapeo . ., Taoc ISBN 5-8265-0149-9 ocyapce exec yepce (TT), BBEEHE Teop aoaecoo ypae ec ocoo oepoeccoao co apae ooo opoaoo ceaca "Aoapoae exoo pooca".

Ocoo e aoaa ec cee eocpeceoo yac eoea ypae pooce poecca py exec oea.

B acoee pe aoaa exooecx poecco pecae coo oo aex cpec poca eoc pooca, eca pa apooo xoca. Ta opao, aaa ye c "Teop aoaecoo ypae" coco ye ocox po ocpoe yopoa aoaecx cce ypae a ae copeex aeaecx eoo execx cpec.

ye eop aoaecoo ypae oe pec cce oxo, pey paccope cce ee eococ, a e poco yea aopo, x a cocoe oex eeo.

eoe ocoe acao cooec c peoa ocyapceoo opaoaeoo caapa ypca "Teop aoaecoo ypae". Ocooe x coepae coca aeaecoe ocae aoaecx cce, oco acooo cpyypoo eoo cceoa cce, ycooc, oeceee ycooc, aeco peypoa, apaepec ce ex cce aoaecoo peypoa.

1 OA XAPATEPCTA OETOB CCTEM ABTOMATECOO PABEH 1.1 PATE CTOPECE CBEEH Bepe cee o aoaax oc aae ae p paoax epoa Aecapcoo "eaa" "Mexaa", e oca aoa, coae ca epoo eo yee ece: eoaoa op epe xpaa, oo opa, aoa poa co o p. e epoa aeo oepe co e e a pee eo oxy.

B cpee ea aeoe pae oya a aaea "apoa" aoaa, oa exa coa p aoao, opaax oe ec eoea, , o yc eaee, opeae paa aoaa eee cxoco c eoeo aa x "apoa", .e. eoeooo.

B XIII . ee oco-cxoac ax Aep o oa ocpo poo opa apa epe.

Beca epece apo coa XVII XVIII . B XVIII . eapce aco ep po eo c Ap coa exaecoo ca, exaecoo xyoa p. pepac eap aoao coa XVIII . pycc exao caoyo y. Eo eap, xpac pae, oee "acax o yp".

Ha pyee XVIII XIX ., oxy poeoo epeopoa, aaec o a pa aoa, ca c ee epee poeoc. oc epe aoaece ycpoca, oop oocc peyop ypo oyoa (1765 .), peyop copoc apoo a aa (1784 .), ccea popaoo ypae a cao aapa (1804 1808 .) .. o ooeo aao peyopocpoe.

B 1854 . ac pycc exa epoex . ocao peo cooa apox aax "epoa peyop copoc pae", a A.

aoc 1866 . papaoa peyop, e oay oa oy cooeceo ee ae apa oe. B 1879 . . Booc .

Bopo epe ocyece p pepcoo peypoa p ypae ae oa oo.

Ec epe peyop ca c apoo ao, o co opo oo XIX .

cyecey po peyopocpoe aa pa opeoc epeco ocee. Ta, 60-e o paoax B. oaea epe pee epec ae. A 1874 . o peo ocyec eo peypoa, coca ocoy copeeo epoao aoa.

o o epo pa aoa epo peyopocpoe, c ce oyopa coe, cpa opoy po exe. B o pe ee eeo cyo aa oppoac aee p aoa: p peypoa o ooe oyoa-aa, pac oe opax ce;

p peypoa o apye, ocy ocoo eop apaoc p. Haa c ypca poeccopa eepypcoo yepcea . oa 1823 ., eop peyopo xo coca eeo ypc oopa o exae apo aa.

Oa eop peyopo a papaoaa, ocoo, 1868 1876 . paoax .

Macea . Bepacoo. Ocoooaa pya Bepacoo c:

"O oe eop peyopo", "O peyopax epoo ec". Bx paoax oo a co copeex eepx eoo cceoa ycooc aeca peypoa.

oco pooaee ea . Bepacoo coa eep A.

Cooa, pao oopoo oce cceoa ycooc pa cxe peypoa, acoc, epoo peypoa c eco opao c. B o e epo copypoa aepaece pep ycooc Payca ypa.

yp poc poeoc opaaec a pa pao oac eop peypoa. B oe XIX . aae XX coe coac oe epoexaecx peypyx popo ae, a popae peyop, cee cce cxe oaypoa. Ta, 1877 . A. ao papaoa poe epo cee cce, coepae epece ee, peaaeo aoaecoo pa opy aeaeo ya oe cooec c eee pacco o e, oopa a poeocppoaa 1881 .

B 1882 . a poeo-xyoeceo cae Moce oaa poo copeeoo popaoo peyopa, papaoaoo H. axapo. o acoeo pee coyec p "ycaoe oycx peex ae peypyeoo apaepa", peoe 1884 . . Ceypo. B o e epo paaec apaepecoe peypoae: papaoa epea peyop B.

oae cxea oaypoa eepaopo M. oo-opooc.

ooe aee pa eop peypoa e cceoa A. yoa.

Eo py, oyoa 1892 ., "Oa aaa ycooc e" c ao exo pa eop ycooc. B o paoe A. yo a epoe cop ay aeaec cpooe opeeee ycooc e, a ae papaoa a eoa pee aa o ycooc. ep aaec oocoa ycaoe ox pa peoc aaa ycooc, ocoaoo a ex epeax ypaex, a opo ooe cceoa ycooc e oo p ecoeo ax ooex "ycooc ao", o p oex ooex "ycooc oo".

py a eop ec H. yoc, oop coa eop opao ycooc a ocoe apaox po a, a ae a aeaecoe ocae poecco x pyopooax, paccope e cyxoo pe peyopax, cceoa eoope poecc ycoo peypoa. aca ep pycc ye "Teop peypoa xoa a" (1909 .).

aay XX . epo eo ece eop aoaecoo peypoa oppyec a oa ca c po pax paeo. Ocoeo eo c o eop peypoa a ce oeexecoo xapaepa pooc paoax .

Boececoo (1922 1949 .) pyooe oo pyx coecx o o oac, oop 1934 . epe y p aoooo peypoa.

oo eo acyo ec papaoa oeo eoa pae poecca peypoa c eco peypye ea a p aoox poecco.

Ceye oe p epecx opee oo epoa: "cpoco oye ocooo oa c oco apee p epeeo ce oopoo eepaopa" .

eepa, "Coco oe yceoc peypoa ca oopoo ae" B.

Booa M. capeo p. a epo ae xapaepyec pae opoco aoaecoo peypoa pooca pacpeee epeco ep.

ooe aee e pao C. eeea . aoa oac ycooc epocce.

B pae o XX . coac oee ee eo cceoa, acoc, acoe. oc pao X. Haca (1932 .), coepae pep ycooc paoexecx ycee c opao c, A. Mxaoa (1938 .) "apoec eo eop peypoa", oope o pay oceoee o. B 1946 ..

oe . Mao e oapece acoe xapaepc.. pay, A. Xo, .

ee,. eca, B. Coooo aep papaoy acox eoo cea pacea cce, pa opy, yoy eepx paceo.

B 40 50 o papaaac oco eop eex cce, cooc oopx coco ocyc eoo oeo aeaecoo aapaa. ec ceye oe pao o ycooc A. ype (1944 1951 .), A. eoa (1955 .). aepa ao oo apae caec papaoa eop acoo ycooc, yo A. ype B. oco (1944 .), oee eao copypoao M.

Aepao (1949, 1963 .) oeeo o oo pee pyc ye B.

oo (1959 .).

ooe aee aeceoo cceoa eex cce e eo aoo ococ aooo pocpaca, oco oopx aoe A. Apoo eo oo 1930 1940 .

. papaoa oco eop peex (1955 .) ycx (60-e o) cce c pa a oy. H. po H. ooo (1934 .) papaoa eo apoecoo aaca opeee apaepo aooea yco x ooe.

B oceoee o eop aoaecoo ypae paaac ooopo, yoy oo cex apaex aopax poco eooo. Bo eoope x: eop aoaecoo peypoa o oye, eop oeca oye apaoc papaoa pyax. aoa, B. yeaa, . epoa p.;

p cpeaoo ypae eop oca cpeya papaoa B. aaee. A.

eayo, A. pacoc. B e o coac oco eop oaoo ypae . op. A. eo, H. pacoc p.

B acoee pe aee eop aoaecoo ypae epepoco pa oo execx cce. aece ypaee poecc e eco x opaax, ooecx opaaox eoeo-ax cceax, x e cyeceo oa o x po py oep.

aeee pae ycoee cce aoaec peo coa aoapoax cce ypae (AC) exooec poecca (ACT), pooco (AC) opac (ACO). o eoo ocpoe cce ocaoo ey coo, xo y exece cpeca, a oopx peayc AC, xapaep peaex aa cyeceo oac.

1.2 OCHOBHE OHT OPEEEH aaa aoaa coco ocyece aoaecoo ypae pa exec poecca.

o exooec poecc oo pace a p oee pocx epaoax cocax, o cax ey coo poecco. Bc c oop, o exooeco poecce e paoue onepauu, .e. ec, eocpece peyao oopx ec peyea opaoa aepaa, ep, opa, onepauu ynpaeu, oeceae pae ye oe yx peo, apae ..

Paoe oepa cope c apaa ep, , ec o oc eoeo, o a x oee apaaec eo eca ca. Ha oepa ypae apaaec eeya py eoea, oepa pey opeeeo aa coe.

aea pya eoea paox oepax paoo a exao aaec exauaue.

Cooyoc oepa ypae opaye poecc ypae. Ta opao, o ynpaeue oa ay opaa oo oo poecca, oopa oeceae ocee opeeeo e.

aea pya eoea oepax ypae ec execx ypax ycpoc aaec amoamuaue. Texecoe ycpoco, oee oepa ypae e eocpeceoo yac eoea, aaec amoamuecu ycmpocmo.

Cooyoc execx cpec, ox a poecc, ec oemo ynpaeu. Cooyoc cpec ypae oea opaye cucmey ynpaeu.

Ccea, oopo ce paoe oepa oepa ypae o aoaece ycpoca, aaec amoamueco. Ccea, oopo aoapoaa oo ac oepa, pya e x ac coxpaec a , aaec amoamuupoao (acmuo amoamueco).

ac cyae ypae ec peypoae. p peypoa oopa poecca (aee, eepaypa, pacxo, ooee p.) oepac a aao ae c oo ceax ycpoc aoaecx peyopo. Cooyoc peypyeoo oea aoaecoo peyopa opaye cucmey amoamuecoo peyupoau. Oe peypoa ypae o coe eco ppoe eca paoopa, o p ocpoe cce ypae eo x cceoa o e e.

aoo cxeaecoo opae cce aoaecoo ypae (peypoa) coy cpyype cxe, oopx oee ee cce opaac e poyoo, a c ey eea co cpea, oaa apaee epea caa (pc. 1.1).

a) ) Oe 1 2 Pc. 1.1 pep cpyypx cxe:

a o ee cce;

ecoo eeo cce Oco eea cce aoaecoo peypoa c oe peypyee ycpoco (peyop).

) a) u(t) u1(t) un(t)...

x1(t) y1(t) y(t) x(t) Oe Oe xl(t) ym(t) Pc. 1.2 pep opae oeo c xo xo caa:

a ooc xapaepyec ae eopo, ex o oo oopae;

ooc xapaepyec eco aoca oopaa o ee cce xapaepyec xoo oopao (cao) x(t) xoo oopao y(t), oopa ac o xooo caa. B co oepe xoa oopaa oe oc oya ypa (peypy) xapaep.

Boyaee oece (oyee) x(t) ae ooee ypaeo (peypyeo) oopa o aaoo ae. paee u(t) (peypyee xp(t)) oece cy oepa ypaeo (peypyeo) oopa y(t) cooec c eoop aoo ypae (oepa peypyeo oopa a aao ypoe) (pc. 1.2).

Oea ypae c poeccax xeco exoo exa, a aapa, oopx poea exooece poecc (eee, epeeae, pcaa, cya p.);

pooca cepo co, aoox ..;

pep ao, ap ee opac xeca, eeepepaaaa ..

1.3 PH PEPOBAH ep poe peyop, a ye oopoc paee, opee 1765 . .

oyo coao apoo a. paa cxea peyopa peea a pc. 1.3.

aae peypoa ec oepae apoo oe ocooo ypo.

Peyop pecae coo oao 1, ca cceo pao c peypye acoo 2. p yee ypo oao oaec epx, peyae eo acoa oycaec, epepa pyopoo yea oay o oe. p yee ypo oao oycaec, o po yee oa o , ceoaeo, oe ypo.

...

...

G G H Pc. 1.3 Peyop oyoa paec oopeeo c . oyo 1784 . ec a cocpypoa epoe peyop ca oopoo aa apoo a (pc. 1.4.) n G Pc. 1.4 Peyop aa p ee ca oopoo aa py 1 o ece epoeo c e coe ooee, o po epeee peypyeo opaa 2 ee oa apa. o co oepe ae eee ca oopoo aa, o apae, pooooo cxooy.

Cpae aa paccopex peyopo oaae, o oa o ocpoe o eoy py, oop ao poec a cpyypo cxee, pecaeo a pc. 1. x a) x ) G H G n Oe Oe (xp) (y) (xp) (y) Peyop Peyop Pc. 1.5 Cpyype cxe cce peypoa:

a oyoa;

aa B paccapaex pepax oco eea cce aoaecoo peypoa c: oe apoo oe apoa aa;

peypyee ycpoco oao epoea ya c peypy acoa, cooeceo, peyopax oyoa aa.

Bxoe oopa, o e peypyee epeee ypoe H co oopoo n;

peypye epeee oaa o apoo oe G pacxo apa apoy ay G, oyae oec aee apa oe, pacxo oa, eo eoopa cocooc epo cyae o opo apya a ay apoo a, aee apa pyopooe.

p, o oopoy ocpoe peyop oyoa aa, coco o, o peyop ee peypyee oece p ooe peypyeo epeeo o aaoo ae eaco o p, ax o ooee. Ta opao, acoc o ae xooo caa oea peyop ee eo xoo ca.

peaa aopa peypoa ocpy cce oc c, oya aae opao c, ooy o o e pocxo epeaa caa c xoa oea a eo xo o apae, opaoy apae epea ocooo oec a oe. Oe peyop opay ayy ccey, aaey aoaeco cceo peypoa (ACP). Ec ca opao c caaec c oco cao, o c aaec nooumeo, ec aec ompuameo. B aoaecx cceax ypae c cea opaea.

Cxe c opao c ocyec ynpaeue no omoeu (pc. 1.6) oaae poecca xoo oopa y(t) o aaoo ae ya;

y = y(t) ya aaec omoeue ouo ypae.

y x Oe y ya Peyop Pc. 1.6 Cpyypa cxea peypoa o ooe Paccopea ccea ypae c opao c oocc accy cce amoamuecoo peyupoau no omoeu.

Ta opao, aoaeco cceo peypoa o ooe aa ccey, oopo epec ooee peypyeo e o aaoo ae acoc o epeoo ooe oaec aoe oece a peypy opa, oopoe yeae ey ooe a, o y 0 p t.

poe peypoa o ooe ooe pyo coco peypoa o peypoae o oye oeca oye. Bo cyae peypyee oece paaaec peyopo acoc o e oye. Cce peypoa o oye c paoy ccea, a a x ocycye opaa c (pc. 1.7). e oo cocoa aaec o, o, ec coe oecpoa ce oye ccee, o peypyea ea e ye ooc o aaoo ae. Ceye ae, o oeca ocaec oo o epe oye.

Paccapae p peypoa epe peoe 1830 .

payc eepo . ocee p papaoe eop epoex peyopo xoa a o apye a ay a, ec o ocox oye oee, o peaoa coe peoee a pae ey e yaoc, a a aece coca a e oyca eocpeceoo cooa pa oeca.

x Peyop xp y x Oe Pc. 1.7 Cpyypa cxea peypoa o oye B 1940 . peoe p apaoc ocee eacoc ypaeo oopa o oye, paeca peaa oopoo a oyea oo 50-e o.

Heocao cce, ocpoex o py oeca oye oee.

oecpoa ce ooe oye oee yaec pae peo, a ae ax oye a oeae coco aocep, capee aaaopa, ooee coe aapae, .e. poooe eee coc oea, ooe e oe oeca.

Hapep, oacoc cooa pa ocee p peypoa ypo oc eoc, oa po oc cooocc c ee pacxoo, aaec o, o cece ee pacxox xapaepc ee a poe pacxoe, cape oc, ee peaa .., eoc oe epeoc, o oyce.

Peypoae o ooe eo oo eocaa, ec oeca ooe peypyeo oopa o aao pocxo eaco o oo, a pa ao o ooee, o o oopeeo yco ooc cpoec pyo. aco oee ooc cpoec cce po ee epaoococooc.

Haoee e ccea peypoa c opoae ACP, coeae oa paccapaex pa (pc. 1.8).

B x cceax aoee ce oye oecpyc cea peyopo, a oyp peypoa o opao c ycpae ooe peypyeo oopa, ae py oye.

Ta opao, ocoe ocpoe cce aoaecoo peypoa ea oe yaeae p peypoa, opeee, a opao ocyecec oepae peypyeo e a aao ypoe cooec c pa, a ee ooee o oo ypo. B acoee pe eco x Peyop xp y x Oe ya Peyop Pc. 1.8 Cpyypa cxea opoao ACP coy a yaeax pa peypoa: npuun peyupoau no omoeu npuun peyupoau no oyeu.

1.4 PMEP CCTEM ABTOMATECOO PEPOBAH BXMECO TEXHOO pep 1 Peypoae eepayp poya oyxopyao eooee.

oaaee eoc peypoa ec oepae eepayp poya a xoe eooea a aao ypoe.

B paccapaeo pepe eepaypa poya ec xoo peypyeo oopao. Caa eepayp eo ocyec, coy aece xooo peypyeo oec pacxo opeo eooce G. (pc. 1.9). Aa oea oaae, o ycpa oy ac oyax oec eooo. B c c peaaec ccea peypoa o ooe eepayp poya ye ee pacxoa opeo eooce.

poy a) Teooce coe exa Tepoapa Peyop Teooce poy ) T G. Oe Ta Peyop Pc. 1.9 Ccea peypoa eepayp poya eooee:

a exooeca cxea;

cpyypa cxea pep 2 Peypoae ae epxe ac peaoo oo.

ap a) Boyx Peyop ae c G P ) Oe Pa Peyop Pc. 1.10 Ccea peypoa ae epxe ac oo:

a exooeca cxea;

cpyypa cxea B ayyx peaox ooax aee (papee) oo peypyec eee oa oyxa epoo aa ey eeaopo apo (o) eopo (pc. 1.10). ec peypyeo eo ec papee, a peypye pacxo oyxa.

B paccopex cyax cpyype cxe cce aoaecoo peypoa oc ypoe xapaep. B o peao ACP oo e ceye cocae ee: oe peypoa, yce ee (apep, epoapa), yceo-peopaoaeoe ycpoco, peyop, coe exa (apep, epa coe exa), peypy opa (apep, acoa). oa cpyypa cxea opaea a pc. 1.11.

x y x Oe Peypy- a yce ee opa co peopao e ae exa Peyop aa Pc. 1.11 Cpyypa cxea ACP B aee coyc oo ypoee cxe, ycoo ooc a (yceo ee), peopaoae, coe exa, peypy opa oey. oooe ypoee ocec e, o xapaepc aa peypyeo opaa c coe exao, ycaaaex eocpeceo a oee, e ec poecce cyaa cce yac p poepoa ACP ece c xapaepca oea.

1.5 ACCA CCTEM ABTOMATECOO PABEH Bce cce aoaecoo ypae peypoa ec o pa paa a ceye ocoe acc.

1 o oco a ypae a poecco ypae:

a) ee cce;

) eee cce.

2 B acoc o oeo ypae a ypae a ee, a eee cce opaec a:

a) cce, ocaee ooe epea ypae c oco oea;

) cce, ocaee ooe epea ypae c epee oea;

) cce, ocaee ypae acx poox;

) cce c aaae, ocaee ypae c aaa apyeo.

3 o xapaepy pecae cao paa:

a) epepe cce;

) cpee cce, cpe oopx e yce, peee, poe.

4 o xapaepy poecco ypae:

a) eeppoae cce cce c opeee epee poecca;

) coxacece cce cce co cya epee poecca.

5 o xapaepy yopoa.

B acoc o oo, o aoy aoy eec aaoe aee peypyeo e, cce aoaecoo ypae opaec a:

a) cce caa, oepae ococo peypyeo e, .e.

ya(t) = const;

) cce popaoo peypoa, oopx aaoe aee peypyeo e eec o opeeeo apaee peeo popae;

) cee cce, oopx aaoe aee peypyeo e eec cooec c cocoe eoopoo aaoo eopa epeex o pee;

) cce oaoo ypae, oopx oaae eoc ac e oo o eyx ae oopa, a cpeao peypoa, o ae o xapaepa x ee poo, acoe yye, paaec eoop yoao. Haxoee oaoo ypae peoaae peee ocaoo coo aeaeco aa cooecy eoa, poe oo opaeco cocao ac cce ec oep;

) aae cce, oopx aoaec ec ae ya, cocee apaep cpyypa p epeex eex ex yco a ocoa aaa coco oee cce a, o coxpaoc aaoe aeco ee pao. Cce c eee aaoo ae peypyeo e aa cpea, c eee apaepo caoacpaac, c eee cpyyp caoopayc.

1.6 TPEHPOBOHE AAH 1 Ha pc. 1.12 opae oe c xo xo caa.

A o aoe oe ypae? pee ope pep.

B ae ee epeee c ypa?

C aa epeea ec ypaeo epeeo?

u1(t) un(t)...

x1(t) y1(t) Oe xl(t) ym(t) Pc. 1.12 Oe 2 Ha pc. 1.13 opaea cpyypa cxea cce aoaecoo peypoa.

x Peyop xp y x Oe ya Peyop Pc. 1.13 Cpyypa cxea ACP A ae p peypoa peaoa CAP, opaeo a pc. 1.13?

B o a peypoae o ooe?

C aa ccea peypoa ec aoee eo?

3 Ha ae ocoe acc ec cce aoaecoo peypoa?

A aoy accy oocc ea ccea?

B Ha ae oacc ec acc "xapaep yopoa"?

C o pecae coo acc "xapaep oa cao"?

1.7 TECT 1 ao poecc aaec exaae?

A Cooyoc oepa ypae.

B aea pya eoea paox oepax paoo a exao.

C aea pya eoea oepax ypae.

2 Ccey ypae opay:

A Cooyoc cpec ypae oea.

B Cooyoc cpec ypae.

C Oe ypae.

3 e xapaepyec o ee cce?

A Bxoo oopao.

...

...

B Bxoo oopao.

C Bxoo xoo oopaa.

4 ao p peypoa peaoa epo poeo peyope ypo oe apoo a, opeeo . oyo.

A Peypoae "o ooe".

B Peypoae "o oye".

C opoaoe peypoae.

5 aa ccea peypoa aaec aoaeco?

A Bce paoe oepa oepa ypae o aoaece ycpoca.

B ac oepa ypae o aoaece ycpoca, pyy ac oe eoe.

C Paoe oepa o a exa, a oepa ypae eoe.

6 eeppoae cce ypae opaa:

A Xapaep oa cao.

B Xapaep poecca ypae.

C Xapaep yopoa.

7 p acca cce ypae o xapaepy yopoa ccea aoaecoo peypoa oe :

A Cceo popaoo peypoa.

B Cceo c pacpeee apaepa.

C Coxaceco cceo.

8 Ccea aoaeco caa o ccea, oopo oepaec:

A ya(t) = const.

B ya(t) = f(t).

C ya = f(x).

9 o oco a ypae a poecco ypae cce opaec a:

A Hepepe cpee.

B eeppoae coxacece.

C ee eee.

10 B oax cceax ypae oaae eoc ac o:

A Teyx ae oopa.

B Teyx ae oopa, a ae xapaepa x ee poo, acoe yye.

C Cocex apaepo cce cpyyp.

2 PEPHE CHA X XAPATEPCT B eop aoaecoo ypae p paccope ex x cce e eco pae oec ca. Aa ce opex aoaecx cce cyeceo ypoaec, ec ooac papaoao ae x oec cao. Maeaec pecaee cao ec eoopa y pee, opeea ao eo ee, aoe e eaco o eco ppo. B acoc o xapaepa ee caa o pee, op aeaecoo pecae paa peype eeppoae epeype cyae ca.

2.1 OPEEEHE PEPHOO CHAA Ca aaec peyp, ec eo aeaec pecaee ec apaee aaa y pee, .e. o ocaec opeo ye pee.

Pea e ca paccapaec a cya poecc, opeee epooc xapaepca, a a e apaee pee eo eee o pee.

Bpaee peypoo caa, opeeeoo ye pee, aa pee npecmaeue cuaa. opa ac x y paa. Oo op ac ec pecaee e pooepecoo pa, a e oopoo ec pocee apoeco ye pee ocyc cyc. y oy aae apou, aa oopx xapaepyec ayo, acoo ao.

Moeco ay, aco a aa cnempo paccapaeo y pee.

oooe pecaee caa aaec aco. Bpeeoe acooe pecae caa coepeo aea. Bop oo oo pecae ac o ocoeoce ocao paccapaeo aa.

2.2 OCHOBHE T PEPHX CHAOB.

EPOECE HEPEPBHE CHA oco a peypx cao oocc epoec, o epoec eepoec ca.

epoec ca pecae coo y pee, yoeopy yco f (t) = f (t + T ), (2.1) f(t) T T t Pc. 2.1 pep epoeco y e t o oe pee a epae - < t < ;

T eoopa ocoa ae oe poeyo pee, yoeop yco (2.1), aaec epoo y f(t).

epoeca y f(t) oa eca oo peeax poeya pee, paoo epoy T, aee oa ooc oopec a poe aoo epoa.

epoec ca ec eocyec, a a pea ca e oe pooac ecoeo, o ee aao oe. Oao eopeecx cceoax oe epoecoo caa coyec poo ae peya, cooecye aae eceoc.

epoeca y poooo a, yoeopa yco pxe:

opaea ycoo-epepa, ee oeoe co cpeyo a epoe, oe pecaea po A f (t) = + An cos(nt - n ), (2.2) n= e A0 ocoa cocaa, An aya;

n = n acoa;

n aaa aa n- apo.

Ta opao, epoec ca oo paccapa a peya aoe py a pya ecoeoo oeca apo ocoo cocae.

o epoec ca pecae coo y, cocoy cy apoecx cocax c poo acoa. p ypae e poecco cpeac ca, aco oopx e axoc pocx pax coooex, o peopeee cooae o epoecx cao.

Oco coco ocex ec o a, o x oe opeee pe epo (o epo).

Heepoec cao aaec peyp ca, opeee eepoeco ye, aao peeax oeoo (t1 t t2 ) oyecoeoo (t1 t < ) poeya pee, e f(t) t1 t t Pc. 2.2 pep eepoecoo caa oopoo oa oeceo paa y. opa caa oe paec o.

Heepoec ca oo peca epoeco y-e pee c ecoeo o epoo (pc. 2.2).

Maeaec eo pecae cox cao a epoecx, a eepoecx e cooyoc eeapx apoecx cocax aaec apoec aao.

2.3 PEOPAOBAHE PE, EO OCHOBHE CBOCTBA xapaepc cepo cao coyec peopaoae ype. p peopaoae ype aaec oepaop F(i) = f (t)e-itdt, (2.3) opa peopaoae ype it F(t) = (2.4) F(i)e d.

peopaoae ype ca o aoe cooece a oeca y ( f (t) F(i)) : epoe oeco f(t) y eceoo apyea t;

opoe oeco F(i) y oo apyea i. poe peopaoae ype (2.3) ooe o aaoy opay f(t) a eo opaee F(i), opaoe peopaoae (2.4) ooe, aoopo, o aaoy opae F(i) a opa f(t).

Oco coca peopaoa ype c:

1 Coco eoc.

n Ec f (t) = fi (t), o i= n F(i) = (i), (2.5) Fi i= e f(t), f1(t),..., fn(t) eoope y;

F(i), F1(i),..., Fn(i) opae cooecyx y.

2 Teopea aaa.

Ec f(t) F(i), o f (t - ) e-i F(i). (2.6) 3 Teopea cee cepa.

Ec f(t) F(i), o ei0 f(t) F(i ( 0)). (2.7) 4 Pa xapaep y f(t).

Ec y f(t) ea, o ee opaee ec eeceo ye, eo ooceo opeeec a F(i) = F() = 2 f (t)costdt. (2.8) Ec y f(t) eea, o ee opaee ec co o ye, eeo ooceo :

F(i) = -i f (t)sin tdt (2.9) Oee oeco coc peopaoa ype opao oe, o eo peee e (2.5) (2.9) coyc p cceoa peypx cao.

2.4 CETP CHAOB a ye o caao, epoec ca pecaec po ype (2.2), cpyypa eo cepa ooc opeeec aya aa apo, .e. oye An apyeo n, n = 1, 2, Cep ay epoecoo caa, coco paoocox , a oopx poopoaa aya An cooecyx apo, pee a pc. 2.3.

An A1 A A A 1 2 1 3 Pc. 2.3 Cep epoecoo caa Hepepa pa, coea o cepa, aaec oae cepa ay. Ha pae aco yoa pee oeca opa pa ype:

f (t) = Aneint, (2.10) n= e A oeca aya, n t An = f (t)e-intdt. (2.11) T t cepa x epoecx cao oo ycao xapaepe coca:

1 Cep cea cpe, o coepa oo apo, aco oopx pa ocoo acoe. Heoope apo oy ocycoa.

2 e oe epo caa T, e ee epa = ey coce T acoa , ceoaeo, "ye" cep. p T oya eepoecy y, cep oopo caoc co, o p o ay yeac.

3 C yeee eoc yco p ocoo epoe ay apo yeac, a cep caoc "ye".

4 Ec c yeee eoc poyox yco yea ayy o aoy A0 =, o x oceoaeoc ye cpec oceoaeoc ea T y, a ay cep ocooy cex aco ae A =.

T eepoecx cao oc oe cnempao nomocmu, oopa pecae coo dA F(i) =, (2.12) d e A ecoeo ae ay eepoeco y, T / A = lim f (t)e-intdt. (2.13) T T -T / Bey F(i) aa ae cepao xapaepco eepoeco y, a oy F(i) = F() cepo.

ocoy cepaa xapaepca oeca ea, o ee oo peca e F(i) = a() + ib() = F()e-i(), b() 2 e a() = f (t)costd ;

b() = f (t)sin td ;

F() = a() + b() ;

() = arctg.

a() - Cpyypa cepa epoecoo caa ooc opeeec oye ao cepao xapaepc.

acoc oy a cepao xapaepc eepoecoo caa aa cooeceo cepo ay cepo a eepoecoo caa.

Ocoeoc cepax coc eepoecoo caa coco ceye:

1 Cep cea epepe xapaepyec ooc ay apo, pxoxc a epa [0;

].

2 p yee eoc yca eo cep pacpec o oc, a ae ooc ay yeac.

3 Ec oopeeo c yeee eoc poyooo yca yea eo ayy o aoy An =, o yc cpec ea-y, a T cepaa ooc ocoo ee, pao ee o ce aaoe aco (-;

).

2.5 PACPEEEHE HEP BCETPAX CHAOB B cyae epoecoo caa pe ey o pacpeee ooc eo cepe, oopa opeeec a R R 2 Pcp = A0 + An, (2.14) 4 n= e A0, An oe pa ype cooecyeo epoecoo caa;

R copoee eea yaca, epe oop poxo ca.

Pacpeeee ep cepe epoecoo caa pecaec e cy ecoeo ax caaex, cooecyx ecoeo a yaca acooo cepa:

W = (2.15) F() d.

Bpaee F() d pecae coo ep, eey cepa coca caa, pacooe ooce aco d opecoc aco, aaec epeeco cepao ooc eepoecoo caa. opya (2.15) aaec opyo Pe paeco apcea coyec opa acao aco poyca p yco, o ocoe cocae cepa poycac e ee.

2.6 PATECA PHA CETPA CAEH CHAOB p epeae epoecx cao epe peae cce ypae oe epeao opeeeoe oeco apo x ecoeoo ca. p o ao epea apoece cocae c ooceo o aya. B c c oc oe paeco p cepa caa, o oopo oaec oac aco, peeax oopo ea apoece cocae caa c aya, pea aepe aay ey. p ope paeco p cepa caa eoxoo ya peoa cay c epeeco o pe c o pe coxpae eo op.

B cyae eepoecoo caa a e, a cyae epoecoo caa, eaeo epeaa cocae caa co ae aya. C epeeco o pe paeca pa cepa oeaec o oac aco, peeax oopo cocpeooea oaa ac ce ep caa, c o e pe oycx cae op caa opee paecy py cepa e pecaec oo. pecaee o xapaepe cae caa acoc o p cepa oe oyeo p cceoa poxoe cao epe cce c aa xapaepca.

2.7 PECTABEHE CHAOB Ca oy pecae pa opao, p o xoo ca cea ec epep, a pecae oe ca a xoe.

O o e ca oe e pay ecy ppoy epecy, yoy, ceoy ..

B eop ypae aoee pacpocpaee oyo aeaecoe pecaee cao. Bce aeaecx pecae cao ec a p ocoe py:

1) epepoe pecaee xoo ca opeee o oe pee (pc. 2.4, );

a) x(t) y(t) Oe ) ) ) y y y 0 0 t t t Pc. 2.4 B aeaecx pecae cao:

a o-cxea cce;

epepoe;

cpeo-epepoe;

cpeoe 2) cpeo-epepoe pecaee xoo ca ec aoa o pee epepo eec oo o ypo (pc. 2.4, );

3) cpeoe pecaee xoo ca aoa a o pee, a o ypo (pc. 2.4, ).

B peyae aoa caa o pee p cpeo-epepo cpeo pecaex oe poo oep opa, a a ocac ae caa oo cpee oe pee. Oao aoap ooy coc peax cce x p opeeex ycox coxpaec oa opa o cae, ec oce ece cpee oe pee. o coco eco a meopea omeuoa: ca, ocae ye c opae cepo, ooc opeeec co ae, oca epe epa pee t = Fc, e FC pa cepa caa.

Cc eope oeoa coco o, o, ec peyec epeaa ca, ocae ye f(t) c opae cepo, o ocaoo epeaa oee oee ae, ocae epe oe poeyo pee t = Fc. o ae epep ca oe ooc occaoe a xoe cce.

Maeaece pecae cao a pae ae ceo peayc e oy. o oyue oa eee ooo apaepo aoo-o ecoo poecca o aoy pecaeoo cooe. Ta, cceax c epec caa o oye oa eee ooo apaepo cooacooo epecoo caa o aoy epeaaeoo oacooo cooe. B cyae oy apoecoo caa paa a ocox a oy: aya oy yoa oy, oopa opaeec a acoy aoy. Ha pae ae ceo cpeac ceae oy ayo-aoa ayo-acoa, p o o o oy ec oe, pyo apa.

2.8 CHA. X B Haoee aco eop aoaecoo ypae coyc ceye ca.

1 E cao (pc. 2.5):

0 p t < 0;

x(t) = 1(t) = (2.16) 1 p t 0.

1(t) aaec ae ye Xecaa. Cpoo oop, y Xecaa ec epeayea, oao, ec, pepy, a cceyeo oee peo op e, peyae eo pacxo oaaeoo eeca ec cao c F1 o F2, o oop, o a xoe oea peaoa caoopa ca eo F2 F1, ec oce paoc paa ee, o a xoe peayec e cao.

Cepaa xapaepca eoo caa:

-i F(i) = e.

2 Ea yca y ea-y (pc. 2.6) o y, yoeopa cey yco:

0 p t 0;

1) (t) = pt = 0;

(2.17) 2) (t)dt = 1.

x t Pc. 2.5 E cao x t Pc. 2.6 E yc ea-y aa ae ye paa, oa oocc accy cypx y. y ec ae epeayey y oo peca a yc ecoeo ao eoc ecoeo oo ay, .e. a pee, oopoy cpec poyo yc c ocoae t oa, pao ee (pc. 2.7, a), ec t 0 a, o oa yca coxpaac pao ee.

Tae -y oo peca a pee eoopo y (pc. 2.7, ):

(t) = lim (t,) = lim. (2.18) (2t +1) x a) x ) = = = 0 t t Pc. 2.7 pecaee ea-y:

a poyo yc;

(, t)-y oco coca ea-y oo oec ceye paeca:

0+ (2.19) (t)dt =1;

0 -y ec eo ye:

(t) = (t);

(2.20) x(t)(t)dt = x(0), (2.21) .e. epepo y oo pea oy opay.

oceee coooee, coy paccopee ye coca -y, oaaec cey opao:

0- 0+ 0+ x(t)(t)dt = x(t)(t)dt + x(t)(t)dt + x(t)(t)dt = x(0) = x(0).

(t)dt - - 0- 0+ 0 Cepaa xapaepca ea-y: F(i) = 1.

Mey ye Xecaa ye paa cyecye c, paaea coooee:

(2.22) (t)dt = 1(), (t) = 1[t].

Ha pae caec, o a xo oea oaa -y, ec pe ec poyooo yca aoo ee pee epexooo poecca.

3 apoec ca (pc. 2.8, a) x(t) = A sint (2.23) coyec p cceoa cce aoaecoo peypoa aco eoa.

x a) x ) T b b c c a a t t d d x ) t t Pc. 2.8 apoec ca:

a o ca;

pecaee apoecoo caa paee eopa;

apoec ca co co a Cycoa apoec ca oo peca a paee eopa o A opy aaa oopa (pc. 2.8, ) c eoopo yoo copoc, pa/c.

a) ) x x 0 0 t t Pc. 2.9 Cye eeape y apoec ca xapaepyec a apaepa, a aya A;

epo T;

aa .

Mey epoo yoo copoc cpae coooe 2 = T =. (2.24) T Ec oea aac e y, o o xapaepyc ao oea (pc.

2.8, ), oopa o peeo oac xapaepyec opeo t, o oo ay paa paaax (pc. 2.8, ). epeo ocyecec o opye 2t =. (2.25) T Ha pae oye apoecoo caa coyec eepaop cycoax oea.

4 Cye eeape y.

y oocc y Xecaa paa c aaae, .e. 1(t ) (t ) (pc. 2.9), 0, t ;

(t - ) = pe, t =.

Bce coca -y coxpac, o acac e:

A + (t - )dt = 1;

(t - ) = ( - t) = (-(t - ));

x(t) (t - ) dt = x().

5 Ca pooo op x(t) (pc. 2.10, a).

a) ) ~ ) x x x ti ti 0 0 t t t Pc. 2.10 Ca pooo op:

a xoo epep ca;

yc x(i);

cyepo yco, opeex ca x(t) o ca pooo op oo peca c oo -y. C o e eec poo oe pee t, cpoc co coo x(t) (pc.

2.10, ), cooecy ae caa oe pee t = ti ocoae ti.

~ o yc oo pa epe pey ea-y (t ti):

oa paa 1;

~, (t - ti ) = pa paa ti ;

coa paa ti, ~ .e..

xi (t) = x(ti )ti(t - ti ) ae y x(t) aopo yco (pc. 2.10, ), oo aca:

n ~(t) = )ti~ - ti ) x (t.

x(ti i= ~ Ec eep n, ti d, (t - ti ) (t - ), o t+ x(t) = x()(t - )d. (2.26) Ca pooo op oo peca epe ee y, eo paee (2.26) ceye poeppoa o ac, coy coooee (t - ) = 1 (t - ), peyae eo o-ya ceyee coooee t + x(t) = x(0) 1[t] + x () 1(t - )d. (2.27) 2.9 TPEHPOBOHE AAH 1 B cceax aoaecoo ypae aac pae oec ca. ypoe aaa cea opex cce oyc papaoao ae x oec cao.

A ao ca aaec peyp?

i x ( t ) B ae cyecy pecae cao?

C ae ca oocc oco a peypx cao?

2 xapaepc cepo cao coyec peopaoae ype. Cep epoecx cao xapaepyec opeee coca. eepoecoo caa oc oe cepao ooc.

A aoe peopaoae aaec peopaoae ype?

B a xapaep coca oaae cep epoecoo caa?

C o aoe cepaa xapaepca eepoeco y?

3 B eop aoaecoo ypae coyc a aaee caape ca, oop oocc e cao, ea yca y ea y, apoec ca.

A aa y aaec ea-ye?

B a a cceyeo oee oa ca e eoo caa?

C a apaepa xapaepyec apoec ca?

2.10 TECT 1 Ca aaec peyp, ec eo aeaec pecaee ec:

A apaee aaa y pee.

B apaee aaa y aco.

C apaee aaa y pee aco.

2 Ca aaec epoec, ec o pecae coo:

A y pee yoeope yco f(t) = f(t + T), - t.

B y pee yoeope yco f(t) = f(t + T), t1 t t2.

C y aco yoeope yco f() = f( + W), -.

3 aoe peopaoa aaec peopaoae ype?

A F i = f (t)eitdt.

B F = f (t)e-itdt.

C F i = f (t)e-itdt.

4 Cepao ooc eepoecoo caa aaec ea 1 dA A F i =.

d 1 d B F i =.

dA dA C F i =, d e A ecoeo ae ay eepoeco y.

5 ye Xecaa aaec y:

0 p t < 0;

A x(t) = 1 p t 0.

B x(t) = 1 p t.

0 p 0 > t > t2;

C x(t) = 1 p 0 t t1.

6 ea-ye aaec y, yoeopa yco:

p t = 0;

A (t) = 0 p t 0.

0 p t 0;

B (t) = (t)dt = 1.

p t = 0;

0 p t 0;

C (t) = (t)dt = 0.

p t = 0;

7 aa y oocc cy eeap y?

A x(t).

B x(t ).

C x(t) + x().

8 Ca pooo op oo peca a:

t A x(t) = x() h(t - ) d.

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

t C x(t) = x() h(t - ) () d.

9 Ca aaec apoec, ec A x(t) = Ah(t)sin t.

B x(t) = A(t)sin t.

C x(t) = Asin t.

10 Mey ye Xecaa ye paa cyecye c, paaea coooee A 1[t] = '(t).

B (t) = 1'[t].

C 1[t]dt = (t).

3 MATEMATECOE OCAHE ABTOMATECX CCTEM 3.1 OCHOBHE COCO MATEMATECOO OCAH. PABHEH BEH Maeaecoe ocae aoaeco cce ypae o ocae poecco, poeax ccee a e aea.

ocpoee oe cce ypae aaec c ye oea ypae cocae eo aeaecoo oca. B aece oea oe cya aapa, exooec poecc, pooco, pepe opac. Pae aeaecx oee oeo oycaaec x aaee. oe oca pae pe pao oea cce ypae oy oye o cocoo:

cepea, aaec, opoa cepeao aaec.

p cepeao cocoe ypae oee oya ye ocao ceax cepeo (eo aoo cepea) ye caceco opao peyao eo pecpa epeex oea ycox eo opao cyaa (eo accoo cepea).

p aaeco oca ypae oee oya a ocoa o xecx aooepoce poeax poecco.

p cepeao-aaeco oxoe ypae oee oya aaec ye c ocey yoee apaepo x ypae cepea eoa.

p papaoe aeaecoo oca aoaecx cce ceye ya ocoe eoooece ooe eop aoaecoo ypae. o pee ceo cce oxo pee aa ypae, paccapa oeee oea peyopa poecce peypoa epapo aoc;

oooc pee eoo eop aoaecoo ypae ccea cao paoopao eco ppo cece acpapoa aeaecx oee o opex ecx cce. poe oo, ccea paccapaec a e aoecyx ec opaoo eeo oaae cocooc epeaa ece oec opaoe ca oo, cpoo opeeeo apae;

a e ee cce paccapaec a peopaoae xooo oec xoy pea. Maeaecoe ocae a oex eeo, a cce eo cocaec, a pao, c po oye ypoe, yaoc oopx ac o y a cceoae cce ao oac, eo y oaeo oe cepeao poepe.

B oe cyae ypaeu ameamueco oeu oema uu cucme ynpaeu, ycmaauaue auoc ey xou u xou nepeeu, aamc ypaeuu ueu.

pae, ocae oeee cce peypoa ycaoec pee p ocox oecx, aac ypaeuu cmamuu.

pae, ocae oeee cce peypoa p eycaoec pee p poox xox oecx, aac ypaeuu uauu.

Bce oe peypoa oo pae a a acca: oe c cocpeooe oopaa, aa oopx ocaec ooe epea ypae, oe c pacpeee oopaa, aa oopx ocaec epea ypae acx poox. B aee paccapac oo oe c cocpeooe oopaa.

B aece pepa oo paccope oe c cocpeooe oopaa, ocae epea ypaee opoo opa (pc. 1.2) F(y, y', y", x, x') + f = 0, (3.1) e y xoa epeea;

x, f xoe epeee;

y', x' - epe pooe o pee;

y" - opa pooa o pee.

p ocox xox oecx x = x0;

f = f0 c eee pee xoa ea pae ocooe aee y = y0 ypaee (3.1) peopayec y:

F( y0, 0, 0, x0, 0) + f0 = 0. (3.2) oeoe ypaee (3.2) ec ypaee ca.

Caec pe oo xapaepoa c oo caecx xapaepc.

Cmamueco xapamepucmuo oea (cce) aaec acoc xoo e o xoo caeco pee.

Caecy xapaepcy oo ocpo cepeao, ec oaa a xo oea ocoe oec aep xoy epeey oce ooa epexooo poecca. Ec oe ee ecoo xoo, o o xapaepyec ceeco caecx xapaepc. B co oepe, caa caeca xapaepca dy xapaepyec oeo k, oop opeeec a k =. oeo c dx eeo caeco xapaepco oe yce ec epeeo eo, oeo e c e caec xapaepca oe yce ea ocoa (pc. 3.1).

y = arctg k y k = y x y x 0 x x a) ) Pc. 3.1 Caeca xapaepca oeo:

a eeoo;

eoo 3.2 PMEP PABHEH OETOB PABEH B eop aoaecoo ypae poo coyec eo aeaecx aao, coaco oopoy pae o eco ppoe oe ocac oo aeaec acoc.

Paccop eoope pep cocae ypae ca a pax o eco ppoe oeo.

3.2.1 paec peepyap pepo poceeo oea aoaecoo ypae ec paec peepyap, oopo eec po co oc. paa cpyypa cxe pecae a pc. 3.2.

Ocoo oopao, xapaepye cocoe paccapaeoo oea, ec ypoe oc H, oop paec aece xoo peypyeo e. Bxo cooeceo peypy oece ec copoc poa o peepyap Q, e oyee pacxo o peepyapa G. p ocoo cee op pocce a poe oc, ypoe a) ) Q (x) G Q H Oe H (x) (y) G Pc. 3.2 paeca eoc:

a paa cxea;

cpyypa cxea opeeec paoc (Q G). o yco pao oea ea poa Q eec pooo o pee.

paee a, ocaee acoc ypo H epexoo pee o Q, cooec c aoo pa acaec e dH S = Q - G, (3.3) dt e S oa oepeoo cee peepyapa.

paee (3.3) pecae coo aeaecoe ocae oea peypoa paeco eoc ec ooe epea ypaee 1-o opa.

3.2.2 epeca eoc epeco eoc aaec e, cocoa copoe R eoc C (pc. 3.3).

a) ) R q Ux Ux C Oe (x) (y) Pc. 3.3 epeca eoc:

a paa cxea;

cpyypa cxea Bxoo oopao aoo oea oe pa ap q a oaax oecaopa, a xoo apee a xoe e Ux.

epeaoe ypaee oe oyeo a ocoe aoa pxoa:

dq q R + = Ux. (3.4) dt C Ta opao, aeaec ocae epeco eoc ec ooeoe epeaoe ypaee 1-o opa.

3.2.3 Xec peaop ooo epeea yc peaope poeae xeca pea a A B (pc. 3.4). p oe ypae p ceye oye:

1) peaope ocyecec eaoe epeeae peaoo cec, .e.

oepa o cex oax peaopa oaoa;

2) eoeoc peaoo cec ocoa paa eoeoc cxooo peaea;

3) pea poeae oepecx ycox, .e. eepaypa peaope ocoa.

a) A ) CA0 CA CA B Oe (x) (y) CA Pc. 3.4 Xec peaop:

a paa cxea;

cpyypa cxea p x oyex peaop oe paccapac a oe c cocpeooe apaepa, aepa aac oopoo ee cey :

ee o-o o-o o-o e o-a peaea A, eeo eeca e-eca = ocy eeca A A, A eo peaopa cy peaope peaop o eo xoo pea ooe dCA V = q(CA0 - CA) -VKCA, (3.5) dt e V oe peaopa;

CA oepa eeca A;

t pe;

q oe pacxo peaea A;

CA0 xoa oepa eeca;

A, K ocaa copoc pea.

Ta opao, ocae xecoo peaopa eaoo epeea, oopo poeae pea a A B, ec ooe epea ypaee epoo opa.

a o x pex pepo, aece coca pax o eco ppoe oeo oaa eoop o epa, aoap ey ce paccopee oe ocac oo ypae ooe epea ypae epoo opa.

3.3 OPEEEHE HEHO CTAOHAPHO CCTEM. PH CEPO B eop ypae e ccea oo ooc e cce, oopx poeae poecc c caoap ocac e epea ypae c oco yoao ac o pee oea. Ba coco ax cce ec x cooece py cyepo. Bc c opeeee eo cce, a pao, aec ceye apae: e aac cce, oec py cyepo, oop aaec o, o pea oea a cyy xox cao (t) paa cye pea a a ca oeoc x xi(t).

xi Maeaeca ac pa cyepo coco yx coooe:

y (t) = yi (t) ;

(3.6) xi i i y(cx(t)) = cy(x(t)). (3.7) Bao oe, o eoc caecx xapaepc ec eoxo, o e ocao ycoe eoc, a a oee pa cyepo eoxoo e oo cae, o ae. B o e pe caeca xapaepca, ocaea ypaee po y = a x + b, e oeae py cyepo. oae o a pepe y y = 2 x + 3. oo poee cepe, oop oo pocppoa ocaoo e eee pex oo.

1 onm: a xo oea oa ca x1 = 2 opee xoy oopay o ece oo caa y1 = 7 (pc. 3.5, a).

2 onm: a xo oea oa pyo ca x2 = 3, opee cooecyee ey eee xoo oopa y2 = 9 (pc. 3.5, ).

3 onm: a xo oea oaec ca, pa cye epx yx oax, x3 = 5 opeeec xoo ca y3 = 13 (pc. 3.5, ).

Bcece oo, o y3 y1 + y2 (13 16 ), oo yepa, o ao y p cyepo e oec. ycpae aoo a eeoc ceye epeec aao oopa a opao, o yeoy xoy cooecoa yeo xo.

Ta a oco oeo ypae c ee, o p opeeex ycox eee xapaepc oy peo aee e xapaepca, .e. pooc ueapuau eex acoce.

a) ) 1 o 2 o y1(t) y2(t) x1(t) x2(t) Oe Oe ) 3 o y3(t) x3(t) = x1(t) + x2(t) Oe Pc. 3.5 cpa cepea o poepe oea y y = k x y y = f(x) y A x x x Pc. 3.6 eapa eeo caeco xapaepc O aoee pacpocpaex cocoo eapa ec paoee eeo y p Teopa opecoc aao o cee eex eo paoe.

yc caeca xapaepca ocaec eeo n pa epepyeo, e n oe aypaoe co, ye y = f(x), oopy eoxoo eapoa opecoc o (x0, y0) (pc. 3.6).

Ec peeax acao oox ooe y x o x0 y0 f(x) ao oaec o eo y, o oo f(x) ae ee pee y = f (x).

y f(x) axoc pa Teopa:

f (x0) f (x) = f (x0) + (x - x0) +... ;

1!

y - y0 = f (x) - f (x0) f (x0)(x - x0).

epexo oo ccee oopa, x = x - x0;

y = y - y0, oy eapoaoe ypaee oea dy y = kx, e k =.

dx x 3.4 HAMECOE OBEEHE HEHX CCTEM o cceo aee ye oac oe oeco eeo (oe oe ee), opayee eoopoe eocoe eco eooceo y, oope o o, .e. o oe oe, peyop, ccea peypoa ..

Ccea aaec aeco, ec oa ocaec epea, epa o oe ypae, ac y(t) x(t) Pc. 3.7 Cpyypa cxea cce o pee, aaec caeco, ec ee oca ocycye apaep pee.

Hao epec pecae yee aecoo oee eo cce, oopa oe cyae pecaea a pc. 3.7.

Ocoo aae ye aecoo oee eo cce ec oyee oooc pacca xoo ca y(t) oo ecoo xooo caa x(t).B c c eoxoo pacoaa aeaec aapao cceoa eo cce (pc. 3.8).

aece xapaepc Bpeee epeaoe epeaoa xapaepc ypaee y epexoa Becoa acoe y y xapaepc AX PAX AX AX AX BX PAX X X MX PX Pc. 3.8 aece xapaepc Oco aec xapaepca, coye eop aoaecoo ypae, c epeaoa y, epeaoe ypaee, peee xapaepc: epexoa y, ecoa y;

acoe xapaepc: ayo-aoa xapaepca (AX), pacpea ayo aoa xapaepca (PAX), oapece acoe xapaepc (AX).

Coca ocox acox xapaepc c epeaoe Bpeee ypaee xapaepc epeaoa y acoe xapaepc Pc. 3.9 Baoc aecx xapaepc ayo-acoa xapaepca (AX), ao-acoa xapaepca (X), eeceo-acoa xapaepca (BX), a acoa xapaepca (MX) cooeceo pacpee PAX, PX oapece AX, BX.

Mey xapaepca cyecye c, oopy cppye cxea, opaea a pc. 3.9.

P aecx xapaepc oo oy cepea ye, a eoope c eopeec. Ha pae cepeao oya peee xapaepc acoe, oee, AX X, ye a ocoe x acac epeaoe ypaee, epeaoa y, a ae pacpee oapece acoe xapaepc. Ta opao, o oe aecoe oeee eo cce eoxoo oaoc co ce aec xapaepca.

3.5 HAMECE POECC B CCTEMAX Oco aeaec aapao p ye cceoa cce ypae ec aapa epeax ypae. py paccapaex oeo ye opeee o ee oe c cocpeooe oopaa. p o paa caoape oe, oe epeax ypae oopx e ec o pee, ecaoape oe, yoopx oe ec c eee pee, apep, eee eopoooc, capee aaaopa p.

oco oeo peypoa c ecaoap oea, oao, copoc ee x coc aoo ee copoc peypoa, ooy ae oe p pacee cce peypoa oo peo paccapa a caoape eee opeeeoo poeya pee, a oop coca oea e ycea cyeceo ec.

aee yy paccapac ee caoape oe (cce) c cocpeooe oopaa, oope ocac ooe epea ypae c oco oea:

an y(n)(t) + an-1y(n-1)(t) +...+ a1y (t) + a0 y(t) = bmx(m)(t) + bm-1x(m-1)(t) +...

...+ b1x (t) + b0x(t). (3.8) paee (3.8) ocae oeee oea, oop ee caecy b xapaepcy y = x eycaoec (epexoo) pee p o ope a xooo caa x(t).

ac cya ypae (3.8) c ypae an y(n)(t) + an-1y(n-1)(t) +...+ a1y (t) + a0 y(t) = bmx(m)(t) + bm-1x(m-1)(t) +...

...+ b1x (t), (3.8, a) an y(n)(t) + an-1y(n-1)(t) +...+ a1y (t) = bmx(m)(t) + bm-1x(m-1)(t) +...

...+ b1x (t) + b0x(t). (3.8, ) oeo, ocaex ypaee (3.8, a), caeca xapaepca cyecye, o ec poeo, a a b0 = 0. oeo e, ocaex ypaee (3.8, ), caeca xapaepca e cyecye.

Oe, ee caecy xapaepcy, aac cmamuecuu, a e ee caeco xapaepc, aac acmamuecuu.

B oce cyae, a ye oeaoc e, ypae cce aoaecoo peypoa oaac ee, ooy, ec o ooo, poo eapa x ypae p oo pa Teopa ye paoe eex y eoopx epeex o cee ax ppae x epeex, x opecoc x ae, cooecyx ycaoeyc pey. B peyae oya eapoae ypae ooex. Ta opao, oce cyae epeaoe ypaee (3.8) ec ypaee ooex, oopoe ocae oe ccey peypoa oo opecoc ycaoeoc pea. ex cce ypae ooex cxoe ypae coaa.

oye pee ypae (3.8) eoxoo aa aae yco, o oop oaec cocoe poecca oe pee, po a eo aao t = 0:

( y(0) = y0;

y (0) = y0,..., y(n-1) (0) = y0n-1). (3.9) Oee peee ypae (3.8) pecaec e:

y(t) = yc(t) + y(t). (3.10) B pae (3.10) yc(t) ec o peee cooecyeo oopooo ypae y(t) acoe peee eoopooo ypae (3.8). Ceoaeo, yc(t) cooecye e cce ocyc xooo caa x(t) 0, .e. coceoy coooy e cce, opeeec coca cao cce, oope poc cocax ope xapaepcecoo ypae. Ec op pa, o n yc(t) = eit, (3.11) ci i= e i op xapaepcecoo ypae;

ci pooe ocoe, opeeee aax yco.

acoe peee y(t) ac o a y x(t), opeee xooe oece a ccey, cooecye yeoy e (coco) cce.

Peee (3.10) ypae (3.8) opeee aec poecc ccee, pocxo c oea oa xooo oec, oop p a aao ocea pee, ooy ee cce (epexoo poecc) paccapaec oo p t 0, t < 0 o p oeceo pa y.

Bxoo ca y(t), oyac eee aoo poecca, ec aoee oo xapaepco aecx coc cce, ooy opeeee oo caa, a ye oeaoc, ec ocoo aae eop peypoa. ec caoc ayao e ye aecx coc cce c oo peex xapaepc.

3.6 EPEXOHA BECOBA H 3.6.1 epexoa y oye epexoo y aece caapoo caa coyec ea y pee (2.16). Taoo poa oec cooecye, apep, cpoc ee apy cceax peypoa (oa oopa ccee peypoa).

h ) x(t) a) h() qx S t qx t Pc. 3.10 epexoa xapaepca xecoo peaopa:

a cyeaoe oece;

pa paoa epexoo yue aaec aaecoe paee pee eoo epeaoo ypae (3.8) p xoo cae x(t) = 1(t) yex aax ycox, .e.

an y(n)(t) + an-1y(n-1)(t) +...+ a1y (t) + a0 y(t) = b01(t), (n-1) y(0) = 0;

y (0),..., y (0) = 0. (3.12) puo paoa aaec pea oea (cce) a eoe cyeaoe oece p yex aax ycox.

Ha pae pa paoa opeeec cepea ye coyec aece cxox ax aaa cea cce aoaecoo ypae cceyeoo oea.

ec ceye ec o po opao aa. pa aaa (aaa o) aaec opeee pee epeaoo ypae c aa aa yco. B opao aae peyec occao oe epeaoo ypae o eco epao po, apep, epexoo y. Peee opao aa pecae aey cooc cece ee eoppeoc ec cyecye cea aeaec aapa. Ta, apep, ec peoo, o epexoa y ocaec peee ypae epoo opa a1y (t) + a0 y(t) = b0x(t), x(t) = 1(t), y(0) = 0, Ty (t) + y(t) = kx(t), b0 a e k = ;

T =, o opeee oea k oe yce T ocoa a0 a pee.

y() Bcae y'(t) = 0 , ceoaeo, y() = k x(), oya oe yce k =, x() a a x() = 1;

y() = h(), o k = h().

opeee ocoo pee T cxooe ypaee eppyec peeax o 0 o :

T y (t)dt = - y(t)]dt = - h(t)]dt.

[kx(t) [h() 0 0 paa ac oceeo pae ec e o oe, a oa S o cepeao co po paoa (pc. 3.10, ), oa oo aca: T h() = S, S oya T =.

h() 3.6.2 Becoa y oye ecoo y, ee ae aa unyco nepexoo yue, aece caapoo caa coyec -y (2.17):

0 p t ;

(t - ) = (t)dt = 1.

p t = ;

Ta opao, ecoo ye w(t) aaec pea cce a -y p yex aax ycox.

Ha pae ecoy y oex cyax oo oy cepea ye eca peo. Ca, o a xo oea oaa y, ec pe ec yca aoo ee pee epexooo poecca.

pepo oe cy cepe o c ecoo y xecoo peaopa (pc. 3.4), eoc oeo cceoa. B aece xooo caa peaop ao aec op paceo eeca (apep, ep). epe eoopoe pe o eeco oc a xoe, pe eo oepa epoaao opacae, a ae yae pacee eeco aec (pc. 3.11).

oaae a xo yc pecae coo pey ea-y, a a eo oa oa o e paa S. ooy oye ecoo y cepeao c epexo poecc oppy ye ee eo opa a ey oa xooo oec S.

x a) ) w S S t t t t Pc. 3.11 epexoa xapaepca xecoo peaopa:

a -y;

ecoa y Mey pee xapaepca: epexoo ecoo y cyecye aoe ooaoe cooece, oopoe opeeec cey opao:

t w(t) = h (t);

h(t) = w()d.

Becoy y oo oy a peee epeaoo ypae an y(n)(t) + an-1y(n-1) (t) +...+ a1y (t) + a0 y(t) = b(t);

y(t) = y (0) =... = y(n-1) (0) = 0.

p pee oox ypae ea-y epeo aae yco, b ec n = 2, o a2 y (t) + a1y (t) + a0 y(t) = 0;

y(0) = 0;

y (0) =.

a 3.7 HTEPA AME epa ae coyec opeee xoa oea y(t) p pooo xoo cae x(t) ecx h(t) o w(t).

peoaaec, o a xo oea, ocaeoo ecoo ye w(t), oaec ca x(t) (pc. 3.12, a), opooe ocae oopoo ao . 2.8.

Ec pea oea a (t ti) ooa epe w(t ti) (ecoa y), a pea ~ ~ a (t - ti ) epe w(t - ti ) (pea ecoa y), o a ocoa pa ~(t) cyepo oo aca xoo ca a yc x :

~ ~ yi (t) = w(t - ti )ti x(ti ).

~ x a) y ) ti yi ti 0 t t Pc. 3.12 pecaee xooo (a) xooo cao () aea xooo caa x(t) aopo yco, coa oopx coaae c cooecy oopaa (pc. 3.12), ooe aca pea a cyeay ~(t) y x a ocoa pa cyepo n n ~(t) = y (t) = ~ y ~i w(t - ti )ti x(ti ).

i=0 i= ~ ~ Ec eep ycpe ti 0, p o ti ;

n ;

(t - ti ) (t - );

w(t - ti ) w(t - ), a ti d, e epep apaep, oaa c aoo yca, o ooaeo oyae:

y(t) = - )x()d. (3.13) w(t oceee ypaee aaec epao ae (ypaee cep), opaa c ey xoo, xoo oea eo ecoo ye.

o cy ea ecoa y ec a oea, oopa oaae, a oo a co e a oe ycoe oyee, oaoe a eo xo oe pee = 0.

ecoo cca ecoo y epx pee eppoa oe aee a t, a a eooo peca peay ccey, oopo a xoy oopay aco oe pee oaa e oye, oope oc oceye oe pee.

Ec poec aey opye (3.13) t = =, d = d, o oo aca cepy opyy y(t) = - )w()d. (3.14) x(t Ec pecae xooo caa cooa e opyy (2.26), a (2.27), o epa ae acaec epe epexoy y:

t dx() y(t) = x(0)h(t) + - ) d, (3.15) h(t d t dx(t - ) y(t) = x(0)h(t) + h()d.

d 3.8 PEOPAOBAHE AACA Oco aeaec aapao, oop coyec eop aoaecoo ypae, ec cea eo paoo aaa, a aae oepao eo, ocoe oopoo e yoaoe peopaoae aaca.

3.8.1 Opeeee peopaoa aaca peopaoae aaca aaec peopaoae y x(t) epeeo t y x(s) pyo epeeo s p oo oepaopa, opeeeoo coooee L{x(t)} = x(s) = x(t)e- stdt, (3.16) e x(t) opa y;

x(s) opaee o aacy y x(t);

s oeca epeea s = + i.

opya (3.16) opeee poe peopaoae aaca. Booo a aaeoe opaoe peopaoae aaca, ooee o opae a opa. Oo opeeec coooee c+i L-1{x(s)} = x(t) = x(s)estds, (3.17) 2i c-i e c accca cxooc y x(s).

oca y, cpeaxc a pae, cocae a cooec ey opaa opae. opae eoopx aoee aco cpeaxc y eop ypae pee a. 3.1. Ec e y ocycye ae, o ee opaee oo oy eocpeceo, oyc coooee (3.16).

pep 3.1 Tpeyec a peopaoae aaca o y x(t) = eat.

Coaco opeee peopaoa aaca (3.16) ee 1 -at -(s+a)t x(s) = e-stdt = e e dt = - s + a e-(s+a) = s + a.

0 Ta opao, e-at.

s + a Taa 3. Taa peopaoa aaca Opa opae Opa opae e e 1 (t) 1 8 sint s2 + s 2 1 cost s s2 + 3 t e-t sint (s + )2 + s2 tn s + n!

4 (n = 1, 2, e-t cost (s + )2 + sn+1 ) (1- e-t ) 1 e-t s(s + ) s + 1 1(t - a) e-as t et (s + ) 3 s tn e-t (s + )n+ pooe peee peopaoa aaca oycoeo e, o opaee eoopx y oaaec poe x opao p oepa, ax a eppoae, epepoae a opae poe, e cooecye oepa a opaa.

3.8.2 Coca peopaoa aaca p cooa peopaoa aaca eoxoo a pe eo coca, eoope x opypyc cey opao.

1 Teopea eoc: x ecex oe-cx ocox A B eo oa opao cooecye aa e oa opae (3.18) Ax1(t) + Bx2 (t) Ax1(s) + Bx2(s), e x1(t) x1(s);

x2(t) x2(s).

2 Teopea oo: yoee apyea opaa a oe ocooe ooeoe co po ee apyea opae x(s) a o e co :

1 s x ( t ) x. (3.19) 3 Teopea ayxa: yoee opaa a y eat, e a oe eceoe oecoe co, ee a coo ''ceee" eaco epeeo s:

eat x(t) x(s - a). (3.20) 4 Teopea aaa: oo ocooo > x(t - ) e-sx(s). (3.21) 5 Teopea epepoa o apaepy: ec p o ae r opay x(t, r) cooecye opaee x(s, r), o f (t, r) f (s, r). (3.22) r r 6 Teopea epepoa opaa: ec x(t) x(s), o x (t) sx(s) - x(0), (3.23) .e. epepoae opaa coc yoe a s eo opae a x(0).

B acoc, ec x(0) = 0, o x'(t) s x(s). pe eopey eoxooe oeco pa, oya x(n) (t) snx(s) - sn-1x(0) - sn-2x (0) -...- x(n-1) (0). (3.24) Ec x(0) = x (0) =... = x(n-1) (0) = 0, o x(n) (t) sn x(s), (3.25) .e. p yex aax aex n-paoe epepoae opaa coc yoe a sn eo opae.

7 Teopea eppoa opaa: eppoae opaa peeax o 0 o t po ee opae a s:

t x(s) x(t)dt. (3.26) s 8 Teopea epepoa opae: epepoae opae coc yoe opaa a (-t) :

-tx(t) x (s). (3.27) 9 Teopea eppoa opae: eppoa opae peeax o s o cooecye eee opaa a t, .e. ec epa x(z)dz cxoc, o s x(t) x(s)ds. (3.28) t s 10 Teopea yoe opae: ec x(t) x(s), y(t) y(s), o cepe y t x y = y(t - ) d (3.29) x() cooecye poeee opae xy x(s) y(s). (3.30) 11 Teopea yoe opao: poee opao cooecye cepa opae +i y(t) x(t) = y(s)x(s) = x(z) y(s - z)dz, (3.31) 2i -i e = Re z.

12 Teopea o oeo aao aex y:

lim x(t) = lim sx(s) ;

(3.32) t s lim x(t) = lim sx(s). (3.33) t0 s 3.8.3 Peee epeax ypae O aex pee oepaooo cce peopaoa aaca ec peee ex epeax ypae c oco oea, oop a pa ocac paccapaee cce aoaecoo ypae.

Peee epeaoo ypae o cyae caaec ceyx ao:

1) peopaoae ypae o aacy;

2) ocae pee oac oecoo epeeoo s;

3) epexo oac eceoo epeeoo ye opaoo peopaoa aaca.

pep 3. a2 y (t) + a1y (t) + a0 y(t) = b01(t) ;

y(0) = y'(0) = 0.

peopaye aoe ypaee o aacy:

a2s2 y(s) + a1sy(s) + a0 y(s) = b0 1/ s, oya b y(s) =.

s(a2s2 + a1s + a0 ) yc oo a2s2 + a1s + a0 = 0 ee op s1 s2, oa, a ye oaao e, oo aca C0 C1 C y(s) = + +, s s - s1 s - s e C0, C1, C2 eoope oe, opeeee eoo eopeeex oeo:

b0 b0 b C0 = ;

C1 = ;

C2 =.

s1s2 s1(s1 - s2 ) s2 (s2 - s1) oyc aa opaoo peopaoa aaca, axo y(t) = C0 + C1es1t + C2es2t.

oyeoe paee y(t) ec peee eoo ooeoo epeaoo ypae opoo opa p xoo cae x(t) = 1(t), .e. e , a epexoo ye eoo oea opoo opa.

3.8.4 Paee a pocee po a o pepa 3.2, peee epeaoo ypae, oyeoe c cooae peopaoa aaca, pecae coo paoay po. oee opaoo peopaoa oyey po eoxoo pao a pocee po, oyc cey pao.

po n-1(s) M (s) = (3.34) n(s) aaec pao paoao po, ec opo ce ee, e opo aeae. paoe po (3.34) eoxoo a op ypae n (s) = 0.

Ec ope ece, o ey cooecy po a A.

s - s Ec op ecee paoc k, o cooecye cya poe A1 A2 s Ak sk - + +... +.

s - s1 - s1)2 (s - s1)k (s Ec op oeco copee, o A1s + B.

(s2 + as + b) Ec op oeco copee paoc k, o A1s + B1 A2s + B2 Ak s + Bk + +... +.

(s2 + as + b) (s2 + as + b)2 (s2 + as + b)k Ta opao, po (3.34) oo peca e n-1(s) A1 A2 Ak = + +... + + n(s) (s - s1) - s1)2 (s - s1)k (s B1 B2 Bm + + +... + +... + (s - s2) - s2)2 (s - s2 )m (s (3.35) C s + Dp C1s + D1 C2s + D p + + +... + + p (s2 + a1s + b1) (s2 + a1s + b1)2 (s2 + a1s + b1) Fqs + Eq F1s + E1 F2s + E + + +... + +...

(s2 + a1s + b1) (s2 + a1s + b1)2 (s2 + a1s + b1)q oe A1,..., Ak;

B1,..., Bm;

C1,..., Cp;

D1,..., Dp;

F1,..., Fq;

E1,..., Eq axoc eoo eopeeex oee. B o cyae paa ac (3.35) poc oey aeae oyaec paeco yx poe, y oopx aeae pa, ceoaeo, o pa ce. paeca ocex cocaec ccea aepaecx ypae opeee eecx oeo, oopa peaec ec eoa pee ex aepaecx cce.

p opeee opaa o oyeoy opae oyc cey opya cooec:

A Aes1t ;

s - s A k - A t es1t ;

(k -1)!

(s - s1)k a - t As + B B - Aa / e sin t b - a2 / 4.

Acost b - a2 / 4 + s2 + as + b b - a2 / s2 + pep 3.3 Ha opa, ec opaee.

(s +1)3(s - 2) aoe opaee pacaaec a pocee po:

s2 + 2 A1 A2 A3 B = + + +.

s +1 s (s +1)3(s - 2) (s +1)2 (s +1)3 - paa ac oceeo pae poc oey aeae, yco paeca cee oya:

s2 + 2 = A1(s +1)2(s - 2) + A2(s +1)(s - 2) + A3(s - 2) + B(s +1)3.

paeca oeo p cooecyx ceex s eo pao acx acaec ccea aepaecx ypae:

A1 + B = 0;

A + 3B = 1;

A3 - A2 - 3A1 + 3B = 0;

- 2A3 - 2A2 - 2A1 + B = 2, peee oopo ae A1 = 2/9;

A2 = 1/3;

A3 = 1;

B = 2/9. Ta opao, s2 + 2 2 1 1 = - + - +.

9(s +1) 9(s (s +1)3(s - 2) 3(s +1)2 (s +1)3 - 2) pe opaoe peopaoae, acaec paee opaa:

s2 + 2 2 1 1 L-1 = - e-t + te-t - t e-t + e2t.

(s +1)3(s - 2) 9 3 2 3.9 EPEATOHA H Oo ocox xapaepc oea ypae, coyeo eop aoaecoo ypae, ec epeaoa y, acaea epax peopaoa aaca.

epeamoo yue oea aaec ooee peopaoaoo o aacy xoa oea y(s) peopaoaoy o aacy xoy x(s) p yex aax ycox.

epeaoa y opeeec oo ype coca cce, ec ye oecoo epeeoo ooaaec:

y(s) W (s) =. (3.36) x(s) epeaoa y xapaepye ay oea oo o opeeeoy aay, caey ope xo oea ope xo (pc. 3.13).

Ec oe ee ecoo xoo xoo, o o xapaepyec eco epeao y, opee oope oo eocpeceo, oyc opeeee (3.36).

) a) x1 W1(s) y y x W(s) x W2(s) ) W11(s) y x x2 y W12(s) W22(s) Pc. 3.13 pep pax oeo:

a c o xoo o xoo;

y xoa o xoo;

y xoa y xoa pep 3.4 yc a xo oea oaec ca x(t) = 1(t), a a xoe caec ca, ocae ye y(t) = 2 e2t.

1 opeee epeaoo y eoxoo opee x(s) = ;

y(s) = s s + 2s oa epeaoa y W (s) =.

s + a epeaoe ypaee, epeaoa y ooc xapaepye ay eoo oea. Ec aao epeaoe ypaee oea, o oye epeaoo y eoxoo peopaoa epeaoe ypaee o y(s) aacy oyeoo aepaecoo ypae a ooee.

x(s) B oe cyae epeaoe ypaee oea pecaec e an y(n)(t) + an-1y(n-1)(t) +...+ a1y (t) + a0 y(t) = = bmx(m)(t) + bm-1x(m-1)(t) +...+ b1x (t) + b0x(t), (3.36, a) e an, , a0;

bm, , b0 ocoe oe.

oce peopaoa o aacy p yex aax ycox oya:

ansn y(s) + an-1sn-1y(s) +... + a1sy(s) + a0 y(s) = = bmsmx(s) + bm-1sm-1x(s) +... + b1sx(s) + b0x(s), (ansn + an-1sn-1 +... + a1s + a0)y(s) = (bmsm + bm-1sm-1 +... + b1s + b0)x(s), oa y(s) bmsm + bm-1sm-1 +... + b1s + b W (s) = =. (3.37) x(s) ansn + an-1sn-1 +... + a1s + a Ec eca epeaoa y oea, o opaee xoa oea y(s) pao poee epeaoo y a opaee xoa x(s):

y(s) = W(s) x(s). (3.38) oce ac ec e o oe, a oa opa ac pee epeaoo ypae oepaopo ope.

Ta opao, epeaoa y paa ooe yx ooo:

B s W s =, A s e B(s) = bmsm + bm-1sm-1 +... + b1s + b0 ;

A(s) = ansn + an-1sn-1 +......+ a1s + a0 y.

peax ecx oeo oo oe a xapaepy ocoeoc o a, o cee ooa B(s) cea ee paa cee ooa A(s), .e. m n, a o lim W (s) = 0.

s epeaoa y ae ao ooao caa c pee xapaepca.

Ec eec paee epexoo y, ceoaeo, xoo ca x(t) = 1(t) x(s) =, xoo ca y(t) = h(t) y(s) = h(s), oa epeaoa y s paa h(s) W (s) = = sh(s). (3.39) x(s) (3.39) oe oyeo paee epexoo y epe peopaoae aaca:

W (s) h(s) =. (3.40) s Ec eco paee ecoo y, o xoo ca x(t) = (t) x(s) = 1, xoo ca w(t) , ceoaeo, w(s) W (s) = = w(s), (3.41) x(s) .e. paee epeaoo y ec e o oe, a peopaoae aaca o ecoo y.

pep 3.5 yc oe ocaec epea ypaee y (t) + 3y (t) + 4y(t) = 2x(t);

y(0) = y (0) = 0. Ha h(s) w(s).

pe peopaoae aaca: s2 y(s) + 3sy(s) + 4y(s) = 2x(s), opeee epeaoy 2 2 y W (s) =. epexoa y h(s) = ;

h(t) = L-1 2.

s2 + 3s + 4 s(s2 + 3s + 4) s(s + 3s + 4) 2 Becoa y w(s) = ;

w(t) = L-1.

s2 + 3s + 4 s2 + 3s + 3.10 TPEHPOBOHE AAH 1 Maeaeca oe oea ypae cce ypae ycaaae aoc ey xo xo epee. Paa ypae ca ypae a. caoeo, o pae o eco ppoe oe ypae oaa eoop o epa ocac oo ypae c o pe aea.

A ae ypae aac ypae ca?

o pecae coo caeca xapaepca?

B ae ypae aac ypae a?

C a ypae ocac oe ypae: paec peepyap, epeca eoc, epep oepec xec peaop ooo epeea?

2 O acco cce, oope paccapae eop aoaecoo ypae o ee caoape cce, oec py cyepo. Ocoo aae ye aecoo oee x cce ec yee pacca xoo ca oo ecoo xooo caa, .e. pacca ay cce. C o e coyc aece xapaepc. Oco pee xapaepca, oope, a pao, oya cepeao, c epexoa y ecoa y.

A a oaa, o ccea ec eo cceo?

B ae xapaepc oocc aec xapaepca?

C o pecae coo cxea pacea a c oo peex xapaepc?

3 Oco aeaec aapao, coye eop aoaecoo ypae, ec peopaoae aaca, c oo oopoo acaec ocoa aeca xapaepca oea ypae epeaoa y.

A ae opeeee peopaoa aaca. Copypye ocoe coca.

B ae epax peopaoa aaca epeaoe ypaee 4y (t) + 2y (t) + y (t) + 2y(t) = sin t, y(0) = 0, y (0) = 1, y (0) = 0.

C aa xapaepca aaec epeaoo ye?

3.11 TECT 1 aoe ypae ec ypaee a?

A F y, y, 0, x, x + f = 0.

B F y, y, y, x, x + f = 0.

C F y0, 0, 0, x0, 0 + f = 0.

2 a epea ypaee ocaec aa ax oeo ypae, a epeca eoc, xec peaop ooo epeea?

dy(t) A T + y(t) = k x(t).

dt d y(t) dy(t) B T12 + T2 + y(t) = k x(t).

dt dt dy(t) dx(t) C T + y(t) = k.

dt dt 3 Maeaeca ac pa cyepo coco ceyx coooe y (t) yi xi (t) ;

xi A i i y x(t) y x(t).

y (t) = yi xi (t) ;

xi B i i y x(t) y x(t).

y (t) = yi xi (t) ;

xi C i i y x(t) = y x(t).

4 aa aeca xapaepca aaec epexoo ye?

A Pea cce a e cyea ca.

B Pea cce a -y.

C Pea cce a apoec ca.

5 a coooee ycaaaec c ey epexoo ye ecoo ye?

A h(t) = w (t).

t B h(t) = w(t)dt.

C h(t) = w(t) + w (t).

6 ay c ycaaae epa ae?

A Mey xo xo cao pooo op.

B Mey epexoo ye ecoo ye.

C Mey xo cao pooo op xo cao.

7 aoe peopaoae aaec peopaoae aaca?

A x(s) = (t)e-stdt.

x -st B x(s) = x(t)e dt.

-it C x(s) = x(t)e dt.

8 aa xapaepca aaec epeaoo ye?

A Ooee peopaoaoo o aacy xooo caa peopaoaoy o aacy xooy cay.

B Ooee xooo caa xooy p yex aax ycox.

C Ooee peopaoaoo o aacy xooo caa peopaoaoy o aacy xooy cay p yex aax ycox.

9 Ec eca epeaoa y, o epexoa y opeeec a W (s) A h(l) = L-1.

s B h(l) = L-1 sW (s).

C h(l) = L-1 W '(s).

10 ao epa aaec epao ae?

A y(t) = - ) x(t)d.

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

w(t C y(t) = x()d.

w(t) 4 ACTOTH METO CCEOBAH HEHX CCTEM 4.1 EMEHT TEOP H OMECHOO EPEMEHHOO oec co aaec co, opeeeoe coooee z = a + i b, e a b cooeceo ecea a ac ca. Taa opa ac oecoo ca aaec aepaeco. Ha oeco ococ, oopaax Re (ecea ac) Im (a ac), oecoe co eoepec pecaec eopo (pc. 4.1);

oo oe opaeo ae opx oopaax M (oy) (aa) acao oaaeo ope: z = Mei, e M a eopa, coeeo aao oopa c oo z;

- yo ey ooeo e eceo oc eopo z, pe ooe apaee caec apaee ocea po acoo cpe.

Im z b M a Re Pc. 4.1 opaee oecoo ca Tpe opa ac oecoo ca pooepeca, a a ei = cos isin, z = M cos iM sin.

Bce cocae oecoo ca ca ey coo cey coooe (pc. 4.15):

b M = a2 + b2 ;

= arctg ;

a = M cos;

b = M sin.

a p ce a (apyea) ca eoxoo ya, ao apae axoc oa z. He poc opy, o oop cee a coc Im z opeee ocpoo ya, paoo arctg (pc. 4.2).

Re z b I apa: z1 = a + ib, 1 = arctg ;

a b b a II apa: z2 = -a + ib, 2 = arctg = - arctg = + arctg ;

- a a 2 b -b b 3 a III apa: = -a - ib, arctg = + = - arctg ;

z3 3 = arctg - a a 2 b -b b 3 a IV apa: z4 = a - ib, 4 = arctg = -arctg = + arctg.

a a 2 b Im z2 z b 1 a a Re b z z Pc. 4.2 Opeeee a acoc o pacooe eopa z ypoe oepa a oec ca oeo a, o 1 = ei0;

-1 = ei;

i = ei / 2;

- i = e-i / 2.

Ha oec ca poo e e apeece oepa (coee, ae, yoee, eee), o a ece. Coee ae oee yoo poo a oec ca, aca aepaeco ope:

z3 = z1 z2 = (a1 ib1) (a2 ib2) = (a1 a2 ) i(b2 b1), a yoee eee a ca, aca oaaeo ope:

z3 = z1z2 = M1ei1 M ei2 = M1M ei(1 +2 ) ;

2 z3 = z1 / z2 = M1ei1 / M ei2 = M1 / M ei(1 -2 ).

2 Ec apye y oecoe co, o y ec ye oecoo epeeoo. Hapep, y W(s), s = + i.

Ta opao, oo caa, o ye oecoo epeeoo aaec eoop oepaop (pao), coaco oopoy a) i Im ) W(s) s W(1) W(0) Re Pc. 4.3 opeee y oeco epeeo oe oo ococ oecoo epeeoo cac cooece oa pyo ococ oecoo epeeoo (pc. 4.3).

Ec y oocc accy aaecx y (epepa, aa, o cy epepyea), o aa y oec pa oopoo oopae, oco coca oopoo c ceye:

1 oo oeco ococ s oopaaec pyo oeco ococ W(s) (pc. 4.4).

2 ecoeo a yo oopaaec ao e ecoeo a yo, y p o coxpac (pc. 4.4).

3 ecoeo a peyo oopaaec ao e pa ey ecoeo a peyo. Hapaee oxoa yo coxpaec. Bype oac ooo peyoa peopayec o ype oac pyoo peyoa (pc. 4.4).

i Im ) b a) c a s = + i W(s) A C B Re Pc. 4.4 oopoe oopaee 4.2 ACTOTHE XAPATEPCT Bay po p oca ex cce pa acoe xapaepc, xapaepye pea oea (cce) a apoec ca.

Ocoo acoo xapaepco ec ayo-aoa xapaepca (AX), oopa oe opeeea epe oopoe oopaee.

i Im W(s) S 1 = 0 1 = Re W(i ) 2 > Pc. 4.5 opeee AX Ayo-aoo xapaepco aaec oopoe oopaee o oc ococ ope xapaepcecoo ypae a oecy ococ ayo aoo xapaepc (pc. 4.5), pe caa a oc oopaaec oopa AX, paa e oyococ ope xapaepcecoo ypae oopaaec o ype oac AX.

Ayo-aoa xapaepca ec oeco ye, ooy oa oe , a a oeca y, pecaea oaaeo ope W (i) = M ()ei() (4.1) aepaeco ope W (i) = Re() + i Im(). (4.2) Moy M() oaaeo ope ac AX aaec anumyo-acmomo xapamepucmuo (AX), a aa apye () aaec ao-acmomo xapamepucmuo (X).

ecea ac ayo-aoo xapaepc Re() aaec eecmeo acmomo xapamepucmuo (BX).

Ma ac ayo-aoo xapaepc Im() aaec uo acmomo xapamepucmuo (MX).

Mey ce aco xapaepca cyecye c (pc. 4.1). a o x, oo opee pye, .e.

M() = Re2() + Im2(), (4.3) Im() () = arctg, (4.4) Re() Re() = M () cos (), (4.5) Im() = M ()sin (). (4.6) 4.3 CB PEOPAOBAH AACA PE a eco, a ea caoapa ccea aoaecoo ypae ocaec ooe epea ypaee, oopoe oepaopo ope ee (ansn + an-1sn-1 +... + a1s + a0)y(s) = (bmsm +bm-1sm-1 +... +b1s +b0)x(s), (4.7) e y(s) = y(t)e-stdt - peopaoae aaca y y(t).

peopaoae ype y y(t) opeeec paee y(i) = y(t)e-itdt, pe o oc yco, o y(t) = 0 p t < 0 y(t)dt cyecye.

Cpaa peopaoa aaca ype, o, o opao oo oe oyeo peopaoa aaca poco aeo s a i, o -a opoo yco peopaoae ype oec oee opaeoo acca y. ae ypae (4.9) s a i, oyae:

(an(i)n + an-1(i)n-1 +... + a1(i) + a0 )y(i) = = (bm (i)m + bm-1(i)m-1 +... + b1(i) + b0 )x(i), oya x(i) bm (i)m + bm-1(i)m-1 +... + b1(i) + b W (i) = =. (4.8) y(i) an (i)n + an-1(i)n-1 +... + a1(i) + a poo aa pae (4.8), oo aca, o B() + i B1() M ()ei () W (i) = = A() + i A1() M ()eiH () M () cea o: ayo-acoa xapaepca M () = ec eo M () ye;

ao-acoa xapaepca () = () () eeo ye;

eecea acoa xapaepca Re() eo ye;

a acoa xapaepca Im() eeo ye (pc. 4.6 4.7).

a) M Re ) Pc. 4.6 Coco eoc acox xapaepc:

a AX;

BX ) Im a) Pc. 4.7 Coco eeoc acox xapaepc:

a X;

MX Ayo-aoa xapaepca ae oe paccapac a opaee ype o ecoo y:

-it W (i) = (4.9) w(t)e dt.

Ta a e-it = cost - isin t, o (4.9) oy oye opy opeee eeceo o xapaepc:

W (i) = w(t){cost - i sin t}dt, , ceoaeo, Re() = cos tdt, (4.10) w(t) Im() = - (4.11) w(t)sin tdt.

ocex opy ceye, o Re() = Re(-), Im() = - Im(-), (4.12) i Im() < = Re() > Pc. 4.8 oopa AX a o ceecye o o, o AX p opaex acoax ec epa oopaee AX ooex aco ooceo eeceo oc (pc. 4.8).

p paecx paceax oo opaac ocpoee AX oo ooex aco. coy opyy opaoo peopaoa ype, oo o AX oy ecoy xapaepcy:

w(t) = (i)eitd. (4.13) W e-s pep 4.1 yc aaa epeaoa y oea W (s) =, peyec s2 + 2s + opee acoe xapaepc.

ae s a i, acae paee AX:

e-i e-i W (i) = =.

(i)2 + 2(i) + 3 (3 - 2) + 2i Ta a paccapae oe ee caoape, o, pe p cyepo, ee:

AX (pc. 4.9, a) M () = ;

(3 - 2)2 + X (pc. 4.9, ) () = - - arctg.

3 - oopa ayo-aoo xapaepc opae a pc. 4.9, .

a) ) M 1/ 1/ 0 1 2 i Im() ) 1/ = Re() Pc. 4.9 pa acox xapaepc:

a AX;

X;

AX Beecey y acoe xapaepc oo oya yoee ce aeae a paee, copeoe aeae:

e-i cos - i sin (3 - 2 ) - 2i W (i) = = = (3 - 2 ) + 2i (3 - 2 ) + 2i (3 - 2 ) - 2i (3 - 2 ) cos - 2sin - i (3 - 2 ) sin + 2 cos, = (3 - 2 )2 + oya eeceo-acoa xapaepca:

(3 - 2)cos - 2sin Re() = ;

(3 - 2)2 + a acoa xapaepca:

(3 - 2)sin + 2cos Im() =.

(3 - 2 )2 + 4.4 CB EPEHAHOO PABHEH C ACTOTHM XAPATEPCTAM Peee epeaoo ypae (3.36, a) ee y(t) = yc(t) + y (t), (4.14) e y(t) yeoe ee, ocaeoe ac peee;

yc(t) cooe e, ocaee o peee oopooo ypae.

ycaoe c ey AX epea ypaee paccapac yee e p xoo apoeco oec a: x(t) = 2A cost, oopoe oo peca o opye epa x(t) = Aeit + Ae-it paccapa a cyy xox cao, .e..

x(t) = x1(t) + x2(t) B o cyae acoe peee epeaoo ypae cy pa cyepo ae pecaec e cy y(t) = y1 (t) + y2 (t), e y1 (t) y2 (t) opeec cooeceo o x1(t) x2(t). B c c pee yy cac e y1 (t) = AW (i)eit ;

y2 (t) = AW (-i)e-it, e W(i), W(-i) eoope eece y, e ace o t, oeae opeee.

axoe W(i) y1 (t) epepyec n pa, a x1(t) - m pa ocac cxooe epeaoe ypaee, peyae oya AW(i)eit[an(i)n + an-1(i)n-1 +... + a1(i) + a0] = (4.15) = Aeit[bm(i)m + bm-1(i)m-1 +... + b1(i) + b0].

oyeoe paee (4.15) ooc coaae c oye paee paee (4.8) AX ee pa oepae o a, o ayo-aoa xapaepca oe oyea poco aeo epeeo s a i.

y W(i) oyaec aao opao o opye (4.15) aeo i a ( i).

aca oyee pae oecx y W(i) W(i) oaaeo ope W (i) = M ()ei();

W (-i) = M ()e-i(), acoe peee ypae (4.7) peopayec y y (t) = AM ()[ei()eit + e-i()e-it ] = 2AM () cos[t + ()].

Cpaee y(t), ocaeo ycaoec oea a xoe oea, c xo cao x(t) oaae, o ooee ay xox xox oea 2AM () pao = M (), a o a pa ec ayo-acoa xapaepca;

paoc 2A a [t + ()]- t = () - ao-acoa xapaepca.

C eee aco oea ayo- ao-acoe xapaepc ec o opeeeoy aoy acoc o ecx coc oea.

Oao ce peae ece cce oaa o o coco, oopoe aaec o, o p yee aco xox oea e eoopoo peea (aco cpea) cp oe paec e peapye a oea, .e.

aya xox oea paa y. Ta opao, oo peaoo oea lim M () = 0.

4.5 EC CMC ACTOTHX XAPATEPCT ec cc acox xapaepc ycaaaec p x cepeao opeee.

yc a xo eoo oea oaec apoec ca a x(t) = Asint. Ha xoe oea ycaoec pee (coceoe ee pepaoc) cy pa cyepo ye aac ae apoec ca c acoo, pao acoe xox oea, cy ooceo x o ae pyo ay (pc.

4.10), .e. y(t) = Bsin(t + ).

Cee pa ey apaepa xox xox apoecx cao e ac o ay a xooo caa, a opeeec oo aec coca caoo oea acoo oea, ooy aece aecx xapaepc oea ec coyc paccopee e acoe xapaepc. oye ocex cepea ye pooc p oo, oopx coyec aapaypa cocae eepaopa apoecx oea c peypyeo acoo ycpoca epe ay a oea.

B peyae poeex cepeo acoe xapaepc opeec cey opao.

Ayo-acoa xapaepca (AX) - ooee ay xox oea aye xooo caa:

B M () =. (4.16) A ao-acoa xapaepca (X) - paoc a xox xox oea:

() = x x (4.17) t 2, () = T e t - pe ca.

Ta opao, ayo-aoa xapaepca (AX) oe opeeea a oeca y, oopo AX ec oye, a X apyeo. ocee coooe a pa opee ec cc acox xapaepc.

e coe pacope ayo-aoy xapaepcy, cy cepeao, xoo ca, oo aca xoo ca. Hapep, AX aaa oopao (pc. 4.11), a xo oaec ca x(t) = 2 sin0,5t + 3 cos0,1t 0,8 sin10t.

a) y(t) x(t) Oe ) ) x x 0 t t T1 = 2/ T2 = 2/ x(t) = A2 sin(2t) x(t) = A1 sin(1t) ) yx yx ) t t t T t2 T y(t) = B1 sin(1t + 1) y(t) = B2 sin(2t + 2) Pc. 4.10 cepeaoe opeeee acox xapaepc:

a oe;

xoo ca aco 1;

xoo ca aco 2;

xoo ca aco 1;

xoo ca aco i Im() / 4 / Re() M = 1, M = = - = 0, = 0, Pc. 4.11 oopa AX Bxoo ca y(t) paccapaeo cyae oo aca, coy p cyepo, a cyy pex cao y1(t) = 22sin(0,5t /2);

y2(t) = 33sin(0,1t + /2 /4);

y3(t)= 1,50,8sin(10t 3/2);

y(t) = 4sin(0,5t /2) + 9 sin(0,1t /4) 1,2 sin(10t (3/2)).

4.6 MHMAHO-AOBE CCTEM Ayo-aoy xapaepcy cce oo aca e e (4.8), a, ocooac eopeo ey, a A A B B m ) (i - q j j= W (i) = k, (4.18) n ) (i - s j j= e qj y, a sj - oc epeaoo y.

ce y (4.18) pecae coo poeee cooee (i qj ).

eoepec a paoc ec eopo, aao oopoo e oe qj, a oe a o oc oe i (pc. 4.12). Cpaee yx eopo(i qj) (i qj), o oopx qj e eo oyococ xapaepyec ao, a pyo qj pao oyococ xapaepyec ao, oaae, o p oo o e oye cea <, .e. eopa, eaeo eo oyococ, aa ee.

i Im() i Im() ) a) i i q i qj q qj Re() Re() Pc. 4.12 opeee ao-aox cce Cce (e), ce y oca epeaox y oopx ea eo oyococ (ecea ac ye oco ec opaeo eo Re qj < 0;

Re sj < 0), aac uuao-aou.

Cce (e), y oopx xo o y oc epeaoo y e pao oyococ (ecea ac ye, oco ec ooeo eo Re qj > 0;

Re sj > 0), aac euuao-aou.

Moo oaa, o ao-aox ee cyecy acoc:

1 Im() Re() = - du;

u - 1 Re() Im() = - du;

(4.19) u - 1 dL () = - cth d, d - u e L(u) = ln M(u);

= ln ;

u - epeea eppoa.

acoc oaa, o ayo-aoa xapaepca ao aoo cce (ea) ooc opeeec ee BX, MX AX. o ooe aeo ypoc aa aaa cea paccapaex cce, opaac yee x BX AX.

Heao-aoy ccey pocee cyae oo peca e oceoaeoo coee ao-aoo cce ea, eeo o y pao oyococ , cooeceo, xapaepyeoc AX:

i - q q - i j W (i) = = e. (4.20) i + q q + i Ayo-acoa xapaepca oo ea M() = 1, a ao-acoa () = - arctg. Ta opao, paccapaeoe eo coxpae ayy xooo q apoecoo caa pao aye xooo caa p o acoe, aa e p ee aco o 0 o eec epae o o 0, .e. ee ea c AX W(i) po oae ooeoo ca a (), oop p i pae yeaec p opaca aco.

ooe e a pae coyc oppepoa aox xapaepc ee, oe ycooc ..

4.7 OHTE O OAPMECX ACTOTHX XAPATEPCTAX poe paccapaex e acox xapaepc, oa coy, a aaee, oapece acoe xapaepc (X). x oye paee AX (4.15) acaec e bm (i)m +... + b0 b W (i) = = k0M0()ei() a0 an (i)n +... + a oappyec lgW (i) = lg k0 + lg M0() + i()lg e.

oe ooe yx e coyec oapeca ea ee.

C ey co ee S eoop co N aec opyo S = 20lg N = LmN.

Xapaepca L() = Lm[k0M0()] = Lmk0 + LmM0() = 20lg M () (4.21) aaec oapeco ayo acoo xapaepco (AX).

p ocpoe oapecx acox xapaepc o oc accc oaaec acoa oapeco acae lg, ooy oapeca aya acoa xapaepca cpoc oopaax L();

lg, oapeca aoa acoa xapaepca (X) - ();

lg (pc. 4.13). oapece acoe xapaepc aa ae apaa oe.

L a) ) 20 lg k 0 lg lg / Pc. 4.13 oapece acoe xapaepc:

a AX;

X 4.8 BAMOCB HAMECX XAPATEPCT Ocoo aeco xapaepco oea cce ec epeaoe ypaee. poe eo oy pec:

1) epeaoa y;

2) acoe xapaepc: ayo-acoa, ao-acoa, ayo aoa;

3) epexoe xapaepc: epexoa y, ecoa y.

a x xapaepc oe opeeea, ec eco epeaoe ypaee oea. Ho, ecop a o, ceye ee pa ocaoc a x aoc.

B aece pepa paccop aoc ey epexoo ye py xapaepca.

Ec eca epexoa y h(t), o o opye (3.39) opeeec epeaoa y oea W(s) = s h(s), aeo s = i oopo, co oepe, oy oye acoe xapaepc: W(i) = (i) h(i).

Ta a (t) ec pooo o eo cyeao y, o ex cce ecoa y ec pooo o epexoo y, .e. w(t) = h(t).

epeaoe ypaee o cepeao co po paoa oya c oo pax eo, oox opee eo oe.

C ey oco xapaepca peea a. 4.1.

Taa 4. Bae cooec aecx xapaepc epeaoe an y n (t) + an-1y n-1 (t) +... + a1y (t) + a0 y(t) = ypaee = bmx m (t) + bm-1x m-1 (t) +... + b1x (t) + b0x(t) p yex A(s) Y(s) = B(s) X(s) aax ycox A(s) = ansn + an-1sn-1 +...+ a1s + a0;

B(s) = bmsm + bm-1sm-1 +...+ b1s + b cxoe epe ae aoe W(s) h(t) w(t) Xapa ypaee epca B(s) epeaoa W (s) = W(s) = s h(s) W(s) = w(s) y W(s) A(s) W (i) = B(i) ocaa W (i) = AX W(i) W(i) = i h(i) -it A(i) s = i = dt w(t)e t W (s) Peee .

epexoa h(t) = L- h(t) = ypae p s w()d y h(t) x(t) = 1(t) Becoa y Peee .

w(t) = L-1{W (s)} w(t) = h(t) w(t) ypae p p aae aecx xapaepc o oax opoco ec opeeee oea yce oea, o oop oa ooee xoo epeeo xoo ycaoec pee:

y() K =, (4.22) A o, a a y() = lim y(t), o t lim y(t) t K =.

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

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

t s a 4.9 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 ao 4s (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 eeceo acoa 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;

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