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Ts +B W (s) = K +.

Ts K C W (s) =.

Ts +4 pa paoa aoo ea ee h t A ceoo.

B Aepoecoo epoo opa.

C Aepoecoo opoo opa.

5 aoe eo ocaec ypaee T y'(t) + y(t) = k x'(t) A Aepoecoe epoo opa.

B eaoe epepyee.

C Peaoe epepyee.

6 a ypaee ocaec oeaeoe eo A T y'(t) + y(t) = k x(t).

B T1T2 y (t) + (T1 + T2 ) y (t) + y(t) = k x(t).

C Tk2 y (t) + T y (t) + y(t) = k x(t).

7 ay py paoa ee eo coo aaa h A h h B C k 0 0 t t t 8 aoe eo ee ecoy y w t K T A Aepoecoe epoo opa.

B Peaoe epepyee.

C epaoe.

9 ay ecoy y ee aepoecoe eo epoo opa w A B C w w K t T T K t - t T 10 aoe eo ee opaey e AX i Im() k Re() A ceoe.

B epaoe.

C oeaeoe.

11 aa AX cooecye ey coo aaa i Im() A i Im() B i Im() C Re() Re() Re() 12 aoe eo c cooecye epeaoo ye oocc pye ocox ee k A W (s) =.

Ts +k B W (s) =.

Ts -ks C W (s) =.

Ts +13 aoe coeee aaec apae B A WW WWC WWW14 Bao apae pao ocyece epeoc ya epe eo y x W(s) A B C x x x W(s) W(s) W(s) y y y 1/W(s) W(s) 15 ao ao peypoa ee poopoa peyop A xp = -S1 y(t).

B xp = -S2 y (t).

C xp = -S1 y(t) - S1 y (t).

16 ay AX ee -peyop A B C Im Im Im = = Re = 0 0 Re Re 17 ay epeaoy y ee -peyop A W (s) = -S1 - S2 s.

SB W (s) = - - S1 - S2 s.

s SC W (s) = - - S1.

s 18 ao epexo poecc ye ACP c -peyopo t 19 ao aoo peypoa aoee pacpocpae a pae A -ao.

B -ao.

C -ao.

20 ao aoo peypoa ee p acpoex apaepa A -ao.

B -ao.

C -ao.

6 CTOBOCT HEHX CCTEM Bca ccea aoaecoo ypae oa opao yopoa p ec a ee cyax oex, yo , ecop a ece pax ocopox oye, oa oa paoa ycoo. B c c peao a ec oe o ycooc aaoo pea pao cce. ex cce aoaecoo ypae aa peo po cocoe paoec.

6.1 OHTE CTOBOCT EE OPEEEHE B pocee cyae oe ycooc cce cao co cocooc cce opaac cocoe paoec oce ceoe ex c, oope e ee oo coco. Ec ccea eycoa, o oa e opaaec cxooe cocoe.

Ta opao, paa p a cce:

1) ycmoue - cce, oope oce c oye opaac cxooe cocoe paoec;

2) empae - cce, oope oce c oye opaac cocoe paoec, ooe o cxooo;

3) eycmoue - cce, oopx e ycaaaec paoece oce c oye.

Hao ycooc paoec pecaec cey pcya (pc. 6.1).

A0 A1 ) ) A1 a) AA0 APc. 6.1 cpa o ycooc:

a ycoa ccea; eycoa ccea; epaa ccea ooee paoec apa xapaepyec oo A0. p ooe ooee A epo cyae ap cpec ooe A0, o opo e cpec oy ooe, pee -- cocoe apa epao.

pepo ycox cce oy cy ce oe e, poe eppyeo, oopoe ec epa oeo. epexoe poecc, cooecye yc xo caa, aepoecoo ea epoo opa eppyeo cey opao (pc. 6.2):

x a) x ) t t y y t t Pc. 6.2 epexoe poecc p yco oye:

a - aepoecoe eo epoo opa; - epaoe pepo eycoo cce oe cy oe, oxae ooeo opao c. Ta, eoope xece peaop, oopx pocxo oepece pea, c eyco oea, a a p oe eepayp copoc xeco pea yeaec, o co oepe po yee ee ea pea oe eepayp. B eex cceax oo pye coco.

Paccop cey pep (pc. 6.3):

a) ) B AAPc. 6.3 oyycoe coco paoec Cocoe paoec (pc. 6.3, a) ycoo o ex op, oa ooee e o a eoopy pay, opeeey, apep, oo B. B a ee, ap ye e epec oy A. Bopo cya (pc. 6.3, ) xapaepye pao oooe cocoe paoec eex cce, oopoe aaec oyyco.

Paccampua euee cucme, om nomu ycmouocmu " ao", " oo" u " eo":

- ccea ycoa " ao", ec ocapyec a a oac ycooc, o pa ee e opeee;

- ccea ycoa " oo", oa opeee pa oac ycooc, .e.

opeee pa oac aax ooe, p oopx ccea opaaec cxooe cocoe;

- ccea, oopa opaaec cxooe cocoe; p x aax ooex, aaec ycoo " eo". eoopoo acca cce ycooc " eo" aaec acoo ycooc.

Cya, opae a pc. 6.1, a, cooecye ycooc " eo", a a pc.

6.3, a o " oo", o " ao". B paccopeo pepe c apo opoc o ycooc peaec poco, o oe cyae e cea co, p ax ycox paoecoe cocoe cce ye ycoo.

a ye eoopao oeaoc, ea ccea aoaecoo peypoa oe cyae ocaec e epea ypaee c oco oea (3.8) aa yco (3.9).

Peypyea ea y(t) pecae coo peee ypae (3.8):

y(t) = yc(t) + y(t). (6.1) Ooceo cocax yc(t) y(t) pee (6.1) opoo oopoc . 3.4.

p paccope opoco ycooc epec ae oo cooa cocaa, opeeea o peee oopooo epeaoo ypae (3.8) e pao ac. ec cc o cocae aaec o, o o a pa o peee, oopoe oo o y oo eee epexooo poecca ceae p ycaoec pee. Byea cocaa xoo e, aca o a eeo oec pao ac epeaoo ypae (3.8), a ycooc cce e e.

Cocoe paoec cce opeeec peee ypae (3.8). Ta a epeaoe ypaee ee eceoe peee, o cocoe paoec eceo.

Maeaecoe opeeee o "ycooc" opypyec cey opao. Ccea ec ycoo, ec cooa cocaa epexooo poecca c eee pee cpec y, .e.

yc(t) 0 p t (6.2) p o xoa oopaa cce ye cpec yeo cocae, opeeeo e oece pao ac ypae (3.8).

Ec cooa cocaa eopaeo opacae, .e.

yc(t) p t, (6.3) o ccea eycoa.

oe ycooc pacpocpaec aoee o cya - ee cce.

6.2 CTOBOCT HEHOO EPEHAHOO PABHEH COCTOHHM OEHTAM a eco, oeee cce oce c oye, .e. coooe ee, ocaec peee oopooo epeaoo ypae c oco oea:

(n) an y (t) + an-1 y(n-1)(t) +... + a1 y'(t) + a0 y(t) = 0 (6.4) aa aa yco.

C ypaee ca xapaepcec oo:

n D(s) = an s + an-1 sn-1 +... + a1 s + a0. (6.5) e opae ooc peoo, o op oo ooa pa, oa peee ypae acaec e n s t j y(t) = e. (6.6) C j j cceye xapaep pee. yc, apep, ope s1 - ece, oa oo a cya:

a) s1 < 0. B o cyae cocaa C1es1t ee po, acoec paec oc accc t (pc. 6.4, a).

eceo, p s1 < 0 ee eco ycoe y1 = C1es1t 0, t.

Ta opao, ec ce op - ecee opaee, o ce caaee yy cpec y, a, ceoaeo, x cya.

y1 a) y1 ) Im s1 Re t t y1 ) Im y1 ) s) yc o ope ecee ooee, s1 > 0, oa acoa s2 Re ea caaeoo C1es1t ye epao opaca p t (pc. 6.4, ), .e. C1es1t t t y1 ) Im y p t. B e) o cyae y ae o cyae, oa ce ocae caaee pee cpec y p t.

Re t t Im y1 = c1 es1tt s1 Re Im sRe sIm ss2 Re Pc. 6.4 opaee cocax pee epeaoo ypae:

a - op ecee opaee; - op ecee ooee; op oeco-copee c opaeo eceo ac; - op oeco-copee c ooeo eceo ac; - op e; e yeo ope ) yc ypaee (6.5) ee oeco-copee op. ec ae oo a cya. ep cya, ec s1,2 = i, pe < 0, oa peee pecae coo ayxae oea c acoo (pc.

y1 = C1eS1t + C2eS2t = Cet sin(t + ) 6.5, ), a a p, , ceoaeo, ce paee ae cpec y e 0 t p opaca t.

Ec oeco-copee op e opaey ecey ac, o cooecye e pee cpec y p.

t ) yc > 0. Bo cyae peee c oea c apacae ayo (pc. 6.4, ), a a p, ceoaeo,.

et t y1 = C1eS1t + C2eS2t = Cet sin(t + ) ) oyc eep, o ypaee (6.5) ee e op, .e. s1,2 = i, oa peee ye e : = Csin(t + ), .e. eayxae y1 = C1ei + C2e-i = oea (pc. 6.4, ).

e) yc ypaee ee yeo ope s1 = 0, o cyae, .e. peee y1 = C pecae coo ocay.

Cocay pee yc(t) ae oee peee ypae e pao ac, oopy aco aa epexoo cocae pee. coa ccea xapaepyec e, o yc(t) 0 p t. Ec e o ycoe e coaec, o ccea eycoa, ec yc(t) = const, o ccea epaa, a ec yc(t) pecae coo eayxae oea, o ccea axoc a pae ycooc. Ta opao, ccea ycoa oa oo oa, oa ce op xapaepcecoo ypae e opaey ecey ac. o pao oyo aae - pa ycooc.

ycooc cce eoxoo ocaoo, o ce op xapaepcecoo ypae e opaee ec-ee ac.

eoepeca eppea oo paa oaaa a pc. 6.5.

Oca eae ceya opypoa paa ycooc: ycooc cce eoxoo ocaoo, o ce op xapaepcecoo ypae axoc eo oyococ oeco epeeo s. Ec xo o ope e cpaa o o oc, o ccea eycoa. Ec e xo o ope e a o oc, ccea axoc a pae ycooc. Ma oc i ec pae ycooc. Ec xapaepcecoe sPc. 6.5 eoepeca eppea paa ycooc:

a - ce op c opaeo eceo ac;

- ac ope ee ooey ecey ac ypaee ee oy apy x ope, a ce ocae op axoc eo oyococ, o ccea axoc a oeaeo pae ycooc. Ec e ypaee ee yeo ope, o ccea axoc a aepoeco pae ycooc.

6.3 OPAEHE BEH B AOBOM POCTPAHCTBE 6.3.1 oe aooo pocpaca p paccope ycooc e peao oe oaaoc eee eoopx ax o pecae eoepecoo xapaepa. Oco x ec oe aooo pocpaca, eeoe aaeo Apoo.

ao pocpaco aaec aoe pocpaco, oopo poyo oopaa o c e, opeee oeoe cocoe cce, aaee ao oopaa.

Meo aooo pocpaca pe a ex, a eex cce.

oe epeaoe ypaee n-o opa oo aca e cce n ex epeax ypae epoo opa:

dy1(t) / dt = a11y1(t) + a12 y2(t) +... + a1n yn (t) + x1(t);

dy2(t) / dt = a21y1(t) + a22 y2(t) +... + a2n yn (t) + x2(t);

...

dyn(t) / dt = an1yn(t) + an2 yn(t) +... + ann yn(t) + xn(t), oca epexo poecc p a oye.

x(t) = x1, x2,..., xn B aece aox oopa pa xoy oopay cce ee pooe.

Toa aooo pocpaca (pc. 6.6), cooecya coco cce a oe pee t, aaec uopaae moo (M).

eee coco cce o pee ye cooecoa e opaae o aoo pocpace o opeeeo paeop, oopa aaec aoo mpaemopue.

aoy epexooy poeccy ccee cooecye co opeeea aoa paeop aoo pocpace aoopo.

Meo aooo pocpaca oy aoee pacpocpaee p cceoa cce opoo opa. Bo cyae ao pocpaco ec ococ. Ccea epeax ypaPc. 6.6 aooe pocpaco e (6.7) cce opoo opa oe cyae acaec e:

dy1(t) = f1(y1, y2 );

(6.8) dydt(t) = f2 (y1, y2 ).

dt aoe mpaemopuu cucme mopoo nopa oaam ceyuu cocmau.

1 B ao oe aoo ococ oo poec ecey acaey aoo paeop, .e. epe ay oy aoo ococ poxo oo oa paeop. cee cocae aao oopa: y1 = 0, y2 = 0, oopoe cooecye coco paoec. paee coco paoec:

dy1(t) = 0;

dydt(t) = 0.

dt Hapaee acaeo aae oopa eopeeeo, ooy aao oopa, cooecyee coco paoec cce, aaec ocoo moo.

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