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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 CACA 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) yA x xx 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 x3.4 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, acy(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 ayoaoa 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 bxapaepcy y = x eycaoec (epexoo) pee p o ope axooo 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.

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