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

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a, - oapa pxoa;

- oa pxoa;

- e pxyec 6.10 CTOBOCT CCTEM C AABAHEM CCTEM C PPAOHAHM BEHM Bce peae cce aoaecoo peypoa c ccea c aaae. Heoxo ocao ycoe ycooc ex cce c oco aaae ec pacooee cex ope xapaepcecoo ypae eo oyococ.

Heocpeceoe axoee ope xapaepcecoo ypae apyeo, c c eo paeeoc, ooy pe pep ycooc. Oao oo ope pe oo pep ycooc Haca.

Ec Wp.c(i) - ayo-aoa xapaepca paoyo cce e aaa, a Wp.c. (i) - ayo-aoa xapaepca paoyo cce c aaae, o oo aca:

Wp.c.(i) = Wp.c(i)e-i;

M() = M();

() = () -.

pa AX paoyx cce e aaa c aaae pecae a pc. 6.46. a o paa, AX paoyo cce c aaae apyaec, a a aa p ee aco o 0 o + eec o 0 o -.

Ec e pe aaa, o oo a, a aaeoe, pecoe aee, p oopo ccea ye axoc a pae ycooc.

Pc. 6.46 AX paoyo cce c aaae oo pecoo cya cpaea ac Wp.c. (ip) = = 1. (6.68) coooe (6.68) oo aca ae aoacoo xapaepc, p oopx epeceaec opaea ecea oc, .e.

(ip) = (p) pp = (2j + 1), (6.69) e j = 0, 1, 2,..., oya (6.70) Maoe pecoe pe aaa ec pa opeeec p j = 0:

(6.71) Eo oo opee paec cocoo, oo pooc opyoc eoo payca a ococ AX, ee epeceee c AX paoyo cce e aaa opeee (p), a c aaae ooe opee p cooeceo p.

6.11 TPEHPOBOHE AAH 1 Bca ccea aoaecoo ypae oa paoa ycoo. o ycooc oaec cocooc cce opaac epoaaoe cocoe oce c oye, .e. y(t) 0 p t. Heoxo ocao ycoe ycooc ec opaeoc eceo ac cex ope xapaepcecoo ypae.

A aa ccea aaec epao?

B ye ccea aoaecoo ypae ycoo, ec op xapaepcecoo ypae:

S1 = -2;

S2,3 = -3 + 4i;

S4 = 5?

C ye ccea aoaecoo ypae ycoo, ec op xapaepcecoo ypae pacooe cea o o oc?

2 oea a opoc o ycooc cce aoaecoo ypae coyc pep ycooc, ooe cy o ycooc, e axo eo ope. ep ec eoxooe ycoe, coaco oopoy ce oe xapaepcecoo ypae o ooe. Ceye pyo pepe c aepaece pep ycooc, pee ceo, o pep Payca pep ypa.

A ax cce aoaecoo ypae eoxooe ycoe ycooc ec ocao?

B Ec xapaepcecoe ypaee cce 3S3 + 4S2 + 2S + 1 = 0, o cooec c pepe ypa a ccea a) ycoa;

) eycoa;

) axoc a pae ycooc.

C a cxo a eoxoo pacoaa, o cceoa ycooc oo o pe pep Payca?

3 cceoa ycooc poo pec acoe pep ycooc. B cooec c pepe Mxaoa cpoc oopa Mxaoa, oop ycox cce oe aac a eceo ooeo oyoc, oxo oceoaeo, yxo ecoeoc, e e opaac y, n apao oopao ococ, e n - opo xapaepcecoo ypae.

Bop aco pepe ec pep Haca, oo cy o ycooc ayo cce o AX paoyo cce, pe paoya ccea oe ycoo, eycoo epao, o aya ccea p oe opeeex yco oe o cex cyax ycoo A Copypye pep Haca cya, oa paoya ccea e ycoa.

B ye ycoa ccea aoaecoo ypae cooec c pepe Mxaoa, ec ecea y Mxaoa U() = 2 - 32;

a y Mxaoa V() = + 33?

C yc paoya ccea ycoa ee AX:

ye aya ccea ycoo?

6.12 TECT 1 aa ecx cce ye ycoo?

2 aa ccea aaec ycoo, ec oce c oye A Ccea e opaaec cxooe cocoe.

B pae ooe ycaoeec cocoe, ooe o epoaaoo.

C Ccea opaaec cxooe cocoe.

3 aa cce, ocaex ypaee, ye eycoo?

A y''(t) + 2 y'(t) +3 y(t) = 0.

B y'''(t) + y''(t) +4 y'(t) + 3 y(t) = 0.

C y''(t) - y'(t) + y(t) = 0.

4 Oe ee xapaepcecoe ypaee a3s3 + a2s2 + a1s + a0 = 0. ao opeeee ec opeeee ypa:

A ;

B ;

C.

5 Coaco aepaecoy pep ypa ccea ycoa, ec A Bce aoae op aoo opeee ypa ooe.

B a opeee ypa ooee, a aoae op opae.

C aoae op aoo opeee ypa eoo opa ooe, eeoo opae.

6 aa cce coaco pep Mxaoa ye ycoo, ec oopa Mxaoa ee 7 aa cce coaco pep Mxaoa ye axoc a pae ycooc:

8 a o op xapaepcecoo ypae ycoo cce?

A C opaeo eceo ac.

B C ooeo eceo ac.

C oeco-copee c opae ooe ece ac.

9 aa cce ye ycoo, ec ecea a y Mxaoa e 10 yc paoya ccea ycoa, o aa ayx cce ye ycoa, ec AX paoyo cce ee :

11 yc paoya ccea epaa, o aa aya ccea ye ycoa, ec AX paoyo cce ee :

12 yc paoya ccea e ycoa, o aa aya ccea ye ycoa, ec AX paoyo cce ee :

7 OECEEHE CTOBOCT 7.1 CTOBE HECTOBE BEH COEHEH Bce e cce aoaecoo peypoa opaec a ycoe eycoe. Ta, eeape e, a ye oeaoc, c yco, cee cocae eppyee eo, ooceec pye epax ee.

Heycoe e e oc pao oyococ aoee pacpocpae pepo ax ee ec aepooe eo.

Ha ycooc cce oaa e apaep peypyeoo oea. oo, o ccea a cao, eoxoo oece peye aac ycooc, pe, ec apaep opeee peo oy ec poecce cyaa cce, o aac ycooc ceye aa o, e p oo ycaoex eex apaepax. ocee ycooc ooo ocyec ae opo cooecyx eeo cce peypoa. B acoc, ceye pa ae acpo peyopo, o ccea a ycoo.

ae ceo opee acpo peyopo, p oopx op xapaepcecoo ypae ayo cce axoc a o oc (ACP axoc a pae ycooc) oo, o ae o ec eoa coa ycoy ACP c aa coca.

7.2 CHTE CTOBX CCTEM Ce ycox cce aoaecoo peypoa coc, a yoyo e, opy acpoe peyopo a opao, o aya ccea aoaecoo peypoa a ycoo.

Coaco pep Haca paa ycooc opeeec ypaee Wo(i)Wpe(S0, S1, S2, i) = -1, (7.1) eoepec opaa a poxoe AX paoyo cce epe oy (-1, i0). ec Wpe(S0, S1, S2, i) - AX -peyopa;

S0, S1, S2 - acpo peyopa. a eco -aoa peypoa oo oy pae ao peypoa. Paccop ce ycoo oooypo cce peypoa c pa a peyopo.

7.2.1 ocpoee pa ycooc cce c -peyopo paa ycooc, opeeea o ypae (7.1), cce c -peyopo aec a Wo(i)Wpe(S0, S1, i) = -1. (7.2) oceee ypaee oo aca e cce ypae, coy ayo-acoe aoacoe xapaepc:

(7.2, a) eecee e acoe xapaepc:

(7.2, ) B ococ apaepo acpoe S0, S1 -peyopa cpoc paa ycooc (pc. 7.1) o ypae (7.2), oopx o aao acoe opeec acpo S0 S1. oyea pa ec pae ycooc, e o po pacoaaec oac ycoo pao, a e - oac eycoo pao cce peypoa.

To 1 2 a po cooecy pae ycooc - -peyopo.

Pc. 7.1 paa ycooc cce c -peyopo 7.2.2 pa ycooc cce c -peyopo Ec ccee aoaecoo peypoa coyec -peyop c epeaoo ye Wpe (s) = -S1, o ccea ypae (7.2) pae :

(7.3) opoo ypae cce (7.3) opeeec paoa acoa p (pc. 7.2), cooecya pae ycooc, o oopo epoo ypae opeeec peeoe aee acpo S1:

(7.4) Pc. 7.2 Opeeee aco pa ycooc cce c -peyopo peeoe aee acpo -peyopa S1 oo opee paec eoo, coy coooee Wo(i) S1 = 1. Ec p, o S1 = 1, o opeo d a opaeo eeceo oyoc ooc opeeec AX oea cooecye ee eceo ac p paece o y. B o cyae AX paoyo cce coaae c AX oea.

Pc. 7.3 paecoe opeeee peeoo ae acpo -peyopa eee acpo S1 po oy, o AX paoyo cce aae yeac oceae a eeceo opaeo oyoc opeo r = dS1.

aeee yeee S1 po oy, o p ao-o ae S1 AX paoyo cce poe epe oy (-1, i0), .e. ccea e a pay ycooc r = 1. o aee S1 ye c pee opeec coooe dS1pe = 1, ceoaeo, S1pe=, .e. opeee acpo ocaoo ocpo AX oea ep opeo d.

7.2.3 pa ycooc cce c -peyopo cooa ccee aoaecoo peypoa - peyopa ccea ypae (7.2) opeee pa ycooc acaec e:

. (7.5) a cyae cooa -peyopa, opoo ypae cce (7.5) opeeec paoa acoa (pc. 7.4), o oopo epoo ypae opeeec peeoe aee acpo S0:

S0pe =. (7.6) p paeco opeee peeoo ae apaepa acpo S0 ccea ypae (7.5) acaec e Wo(i) e-i/2 = -1.

Cpoc AX oea, a ae AX paoyo cce p S0 = 1 (pc. 7.5). ocpoe ocee eop AX oea eoxoo paepy a yo, a eo oy pae a. Bpeyae ocpoe opeeec opeo d, oceae AX paoyo cce a opaeo eeceo oyoc. eee ae acpo S0 po oy, o AX paoyo cce "pacyxae" oceae ye a opaeo eeceo oyoc opeo r, opeee a r = S0d.

aeee yeee S0 po oy, o AX paoyo cce poe epe oy (-1, i0), ceoaeo r = 1, a oca peeoe aee acpo peyopa opeec a S0pe =.

Ta opao, oo, o cepoa ycoy ccey, eoxoo pa acpo - -peyopo ee peex ae, a -peyopa oac, pacooeo e pa ycooc.

7.3 OEHA AACA CTOBOCT Ce ycox cce, axoxc o pa ycooc e oaax eoxo aaco ycooc, e yoeope oy peay ccey, a a oe eee epeex, ae eaeoe, oe ec ccey ycooo pea. B c c eoxoo oeceo oe aac ycooc. Haoee pacpocpae oea oceeo c ceye oe.

7.3.1 opee eo oe aaca ycooc a eco, pae ycooc ococ ope xapaepcecoo ypae ec a oc, ooy, e e op xapaepcecoo ypae pacoaac o oc, e e ccea axoc pae ycooc. Ceoaeo, oe aac ycooc oo o pacooe ope xapaepcecoo ypae. Tao oeo ec cee ycooc, oopa opeeec paccoe o o oc aeo op (pc. 7.5, a).

Pc. 7.5 opee oaae oe aaca ycooc:

a - cee ycooc;

- cee oeaeoc;

- oopeeoe cooae cee ycooc cee oeaeoc Ec aac ycooc ye aa epe oaae a, o ccea oa e cee ycooc oe pay aao a, oac pacooe ope ye axoc cea o po = a (pc. 7.5, a).

py oaaee o py ec cee oeaeoc m - oy aoo ooe eceo o ace op sj xapaepcecoo ypae o cpae c py op (pc. 7.5, ):

. (7.8) C eoepeco o pe cee oeaeoc ec aeco ya, aeoo ey ya OA OB, poee epe aao oopa aoee yaee op o oc, .e. tg = m = arctg m. op, axoec a x yax, pacooe a opao, o ce ocae op, ea cea o x ( ceope AOB).

oecee aaca ycooc eoxoo, o cee oeaeoc ccee a oe paa aao m > ma, a oac aaoo aaca ycooc o cyae opeec ceopo AOB (pc.7.5, ).

B pe cyae oe aaca ycooc oo cooa oopeeo oa paccopex oaae - cee ycooc cee oeaeoc. Bo cyae oac oecee aaoo aaca ycooc opeeec oac ABCD (pc. 7.5, ).

7.3.2 acoe eo Cpe acox eoo oe aaca ycooc pee ceo ec eo, cae c ayo-aoo xapaepco paoyo cce, o aac ycooc o oy aac ycooc o ae.

anac ycmouocmu no oy opeeec a a opea d, paoo pacco o o epecee AX paoyo cce c opaeo eeceo oyoc o o (-1, i0) (pc. 7.6, a).

Pc. 7.6 acoe eo:

a - aac ycooc o oy;

- aac ycooc o ae ceo aac ycooc o oy oaae, a coo oe ec oy AX paoyo cce, o ccea a a pay ycooc.

anac ycmouocmu no ae - o yo, ea ey eeceo opaeo oyoc yo, poee aaa oopa oy epecee AX c eo opyoc c epo aae oopa (pc.7.6, ).

ceo aac ycooc o ae oaae, a coo oo yec ocaae o ae paoyo ccee p eeo oye AX, o ccea a a pay ycooc. a pao, oaae coy ece.

paoococooc cce peyec, o aac ycooc o oy ae e ee eoopx aax e: d > da;

> a.

O ocox acox eoo oe aaca ycooc ec noaame oeameocmu, oop a oee aac ycooc o oy aac ycooc o ae. Oaaec, o cee oc ayo cce pae ycooc oo opee o ee acya ayo-acoo xapaepc paoyo cce. o acy aaec noaamee oeameocmu M, ec M(0) = 1 (pc. 7.7).

Pc. 7.7 AX ayo cce:

1 - eoeaeo;

2 - oeaeo;

3 - a pae ycooc e oe acy ee AX ayo cce, e e AX paoyo cce oe (-1, i0) , ceoaeo, e ee aac ycooc ee ccea a o oy, a o ae. a eco, AX ayo cce opeeec epe AX paoyo cce cey opao, oya AX ayo cce, cooeceo, paa.

Aa AX paoyo cce oaae, o, a o pc. 7.8, ee oy pae e opea OB, .e. Wp.c(i) = OB.

Beop 1 + Wp.c(i) opeeec a paoc eopo OA OB, .e.

.

Pc. 7.8 Opeeee oaae oeaeoc Ceoaeo, M.c () =.

Ec e acoy o 0 o, o ooee aae opacae, a ae aae yeac, ceoaeo, AX ayo cce aae ye opaca, a ae yeac, .e. ye e acy. oo, o o acy e aay ey, a, ceoaeo, aa oaae oeaeoc, eoxoo, o eoepec a ococ AX paoyo cce ooee opeo OB AB eo ocoy ey (pc. 7.8):

= M = const. (7.9) Ec aa oaae oeaeoc, o aa aac ycooc (acy AX ayo cce e oe pea eoopo apaee aao e), paac eoepec aa a ococ AX paoyo cce opyoc payco, c epo a opaeo eeceo oyoc a pacco, oopy e oa epecea ayo-aoa xapaepca paoyo cce (pc. 7.9).

Pc. 7.9 pyoa apaa 7.4 AHA CCTEM HA AAC CTOBOCT 7.4.1 Pacpee acoe xapaepc a eco, ayo-aoa xapaepca ec oop oopaee o oc ococ ope xapaepcecoo ypae a ococ AX, exao ee oye Pc. 7.10 Pacpea acoa xapaepca o cee ycooc:

a - ococ ope xapaepcecoo ypae;

- acoe xapaepc ec aea epeaoo y oecoo apaepa s a i. Beee paccopee aaca ycooc paoco epeocy pa ycooc.

Ec aac ycooc xapaepyec cmene ycmouocmu, o o cyae paa ycooc a caec eo a ey a (pc. 7.10, a).

Oopaee oo pa ycooc, xapaepyec aao cee ycooc, a ococ AX ac eoop oopa, oop oy aae pacpeo ayo-aoo xapaepc. B ococ ope xapaepcecoo ypae a oa a po aao cee ycooc opeeec a s = -a + i. Ceoaeo, oye pacpeo acoo xapaepc eoxoo epeaoo y oec apaep s ae a (-a + i).

oopa pacpeo ayo-aoo xapaepc (PAX) W( a + i) o cpae c oo AX ca a pe ("pacyx") (pc. 7.10, ), c c e a xapaepca oya aae pacpea. Coaco coca oopoo oopae p = 0 a PAX xo o yo 90 eceo oyoc.

Ceya pacpea acoa xapaepca xapaepyec aao cee oeaeoc. Bo cyae paa ycooc opeeec ya AOB (pc. 7.11, a).

Oopaee o pa a ococ AX ae oopa pacpeo ayo-aoo xapaepc o cee oeaeoc m.

Ha yax AOB apaep s ee oopa (-, i), oope ca coooee = m, oa s = - + i = -m+i, ceoaeo, o Pc. 7.11 Pacpea acoa xapaepca o cee oeaeoc:

a - ococ ope xapaepcecoo ypae;

- acoe xapaepc ye PAX ocaoo epeaoo y oec apaep s ae a (-m + i). oopa paccapaeo PAX - W(-m + i) a ococ AX pe, e oopa oo AX, p = 0 xo o yo (pc. 7.11, ).

Pacpee ayo-aoe xapaepc oy aca epe pacpee ayo- aoacoe xapaepc:

W(- + i) = M(, )e-i(,);

W(-m + i) = M(m, )e-i(m,). (7.10) pep 7.1 ocpo pacpee acoe xapaepc, ec :

a) poo aey s = - m + i, ee AX: ;

AX: ;

X: ;

pa acox xapaepc opae a pc. 7.12.

Pc. 7.12 Pacpee acoe xapaepc:

a - AX;

- X;

- AX Cpaee AX PAX oaae, o o aco ae M(m, ), (m, ) oe o acoo ee, e M(), (), ooy oopa W(-m + i) pe, e W(i).

) poo aey s = - + i, ee AX: ;

AX: ;

X: (, ) =,.

pa acox xapaepc opae a pc. 7.13.

Pc. 7.13 Pacpee acoe xapaepc:

a - AX;

- X;

- AX 7.4.2 Aau cucme a anac ycmouocmu aaa cce a aac ycooc coyec aao pep Haca.

Coaco pep Haca aya ccea axoc a pae ycooc, ec AX paoyo cce poxo epe oy (-1, i0). pe o pep cceoa cce a aac ycooc, ceye, o ec paoya ccea oaae aaco ycooc pacpea ayo-aoa xapaepca e oxaae oy (-1, i0), o aya ccea ee aac ycooc e ee, e aa.

Ec aac ycooc oeaec cee ycooc, o aaa cce aao pep Haca oe cooa ceye opypoe. Ec paoya ccea ee cee ycooc a, o aya ccea ye oaa aao cee ycooc, ecPAX paoyo cce W(-, i) poxo epe oy (-1, i0). Ec PAX paoyo cce W (-, i) e oxaae oy (-1, i0), o cee ycooc ayo cce ye e aao a.

coe, coaco oopoy aya ccea ye oaa aao cee oeaeoc m, opypyec cey opao. Ec paoya ccea ee cee oeaeoc ma, o aya ccea ye oaa aao cee oeaeoc, ec PAX paoyo cce W(-m + i) poxo epe oy (-1, i0).

Ec PAX paoyo cce W(-m + i) e oxaae oy (-1, i0), o cee oeaeoc ayo cce ye e ma.

p aae cce a aac ycooc o oy o ae eoxoo ocpo AX ayo cce opee cceyee aac ycooc paec, coaco x opeee.

p oee aaca ycooc o oaae oeaeoc M cpoc AX paoyo cce opyoc (pc. 7.9) payca c epo oe. aya ccea oaae aaco ycooc e aaoo, ec AX paoyo cce e axo yp o opyoc. Ec AX acaec o opyoc, o aya ccea oaae aa aaco ycooc.

7.5 CHTE CCTEM, OAAX AAHHM AACOM CTOBOCT B. 7.2 paccope ce ycox cce. Teep eoxoo poec ce cce, oaax aa aaco ycooc, apep, aao cee oeaeoc ma. o ceo ao cyae ye oa pace acpoe peyopo ayo oooypo ccee peypoa.

a eco, oo, o aya ccea oaaa aa aaco ycooc - aao cee oeaeoc, eoxoo ocaoo, o PAX paoyo cce W(-m + i) poxoa epe oy (-1, i0). Ha ocoa oo oo aca Wo(-mp + ip)Wp(-mp + ip ) = -1. (7.11) paee (7.11) oo cec ccee yx ypae, opaax c ey aco xapaepca oea peyopa:

M (m, P )Hp (m, p, S0, S1, S2 ) =1;

o (7.12) (m, P ) + p (m, p, S0, S1, S2 ) = -, o e S0, S1, S2 - apaep acpoe peyopo. Ccea ypae (7.12) ooe opee paoy acoy apaep acpoe peyopo, a ccea oe acaa ae e:

7.5.1 Cucmea c -peymopo Pacpea ayo-aoa xapaepca -peyopa acaec e:

Wp(-m + i) = -S1 = S1e-i, oa ccea ypae (7.12) cce aoaecoo peypoa c -peyopo peopayec y:

(7.13) Pc. 7.14 Opeeee acpo -peyopa, oeceae aay cee oeaeoc:

a - opeeee paoe aco p;

- opeeee ae PAX oea p paoe acoe p opoo ypae cce opeeec paoa acoa p. oce oo opee paec, eo ceye ocpo pacpey aoacoy xapaepcy oea py, pay - (pc. 7.14, a), epeceee oopx ae p.

Hacpoa -peyopa opeec o coooe, (7.14) e aee pacpeo AX oea oo opee a aaec, a paec (pc. 7.14, ).

7.5.2 Ccea c -peyopo Pacpea ayo-aoa xapaepca -peyopa ee .

Cyeo o xapaepc ccea ypae (7.12) opeee acpo S0 paoe aco acaec e:

(7.15) Pc. 7.15 Opeeee acpo -peyopa, oeceae aay cee oeaeoc:

a - opeeee paoe aco p;

opeeee ae PAX oea p paoe acoe p Peee cce ypae (7.15) oe poeeo a aaec, a paec. paecoe peee opoo ypae c e opeee paoe aco pecaeo a pc. 7.15, a.

Ha pc. 7.15, pecaeo opeeee ae PAX oea p paoe acoe.

Hacpoa S0 -peyopa, oeceaa aay cee oeaeoc, opeeec coooee. (7.16) 7.5.3 Ccea c -peyopo Pacpea ayo-aoa xapaepca -peyopa:

, oya peyopa PAX ;

PX .

-peyop ee a apaepa acpoe S0 S1, oope ece c p oea pacey. Ccea ypae (7.12) acaec e:

(7.17) oyea ccea ooe opee oo a eecx, a ao p, ooy oa ee ecoeoe oeco pee.

oye x pee ccea papeaec ooceo ae acpoe:

(7.19) M* (m, p) = ;

* (m, ) = - (m, ).

S1, S m = m m < m m > m.

.

, :

- yx ae ocoo pee T11 (coa), T12 (y-pa), T12 > T11. a o pao, yeee ocoo pee ee yee aaca ycooc o ae o o 1, ec copaa acoa 1 = pacoaaec eee aco cpea c. Ec e copaa acoa pacooc paee aco cpea, o yeee ocoo pee aepoecoo ea ye aac ycooc.

o pao cpaeo aepoecx, oeaex ee. opcpyeo ea e ocoo pee pooooo, a pa cpyyp, ex epeaoy y paoyo cce, apep,, oo e oec, .e. c yeee T1 aac o ae yeaec.

py aoee pacpocpae ye oecee ycooc aaca ycooc ACP ec eee ee py e ooex ee, pe eee ooo oo e ea acoc o cpyyp apaepo cce ae pae peya. ooy pa op ooeoo ea oo cea, ec eca cpyypa apaep cce.

7.7 CTPTPHA HECTOBOCT Cpyypo-yco aac cce, oope p ax-o aex x apaepo oy ca yco. Cpyypo-eyco aac cce, oope e oy ca yco p ax oax ae x apaepo.

Bopoc cpyypo ycooc oa p ee ooex ee, .e. oyaea ccea oa , epy oepe, cpyypo-ycoo. B pe cyae o y cpyypo cxe oo opee, ec ccea cpyypo ycoo cpyypo-eycoo.

Ccea ec cpyypo-ycoo, ec ee coca xo oo ycoe epoe oeaee e. Xopoe eoepeco eppeae ec paccopee oopaa Mxaoa.

yc ccea coco ooo eppyeo ycox epox oeaex ee. Bo cyae oopa Mxaoa ee , opae a pc. 7.19, a. Aa oo oopaa oaae, o p ocaoo ax oyex ec oopa caec eoo pao ccea caoc ycoo, ceoaeo, ccea c o eppy eo cpyypo-ycoa.

Ccea, cocoa yx eppyx ee oo ca ycox epox oeaex ee, cpyypo-eycoa. oopa Mxaoa o cce opae a pc. 7.20, , oopoo o, o a oye e yacc cy oopa pao a opao, o ccea caa ycoo.

Pc. 7.19 oopa Mxaoa c e opeee cpyypo ycooc cce, cocoe ycox epox, oeaex ee:

a - ooo eppyeo ea;

- yx eppyx ee 7.8 BHE MAX APAMETPOB HA CTOBOCT p papaoe aeaecoo oca cce epeo occ e e oye, aaec peepee a apaepa cce. oceee ee oe opa epeax ypae o ycooc cy o pe "poe" ypae co y ye. Oao opex cyae oo oe e ax apaepo a ycooc.

yc a apaep xo eo xapaepcecoe ypaee cce, .e.

o ypaee acaec cey opao D(s) = D1 (s) + D0(s) = 0, (7.20) e - a apaep;

D0(s) - oo opa n;

D1(s) - oo opa N = m + n.

ec oo p xapaepx cya:

1 opo ce y a ey e opa aeae, m = 1. B o cyae o ope xapaepcecoo ypae s = p > 0, yxo ecoeoc o opaeo eeceo oc. p ocaoo ax aex ccea ye ycoo, ec op poeoo ypae D0(s) = 0 ee.

2 opo ce y a a opa e opa aeae, m = 2.

B o cyae ycoe ycooc cce ec ycooc pee poeoo ypae D0(s) = 0 oee epaeca.

3 Paoc opo ce aeae m > 2. Bo cyae opaca ae apaep p cceoa ycooc eoyco.

Bcpeac cya, oa a apaep xo ypaee cce e ooa. cooc ao cce opeeec e, a pacoaac yxoe ecoeoc op: cpaa cea o o oc. Pacooee x ope opeeec, a aae, cooae ypaee. oo, o cxoa ccea p ocaoo ax a ycoo, eoxoo ocaoo, o poeoe cooaeoe ypae, aoe opo, yoeop yco ycooc.

7.9 OPPETPE CTPOCTBA a ye eoopao oopoc, o peo oecee ycooc aaca ycooc cce ec eee ee ooeoo eea, oop cpae, oppepye coca cxoo cce, aaec oppepy eeo.

Ec o ee ocaoo coe, o o aaec oppepy ycpoco.

Ta opao, oppepyee ycpoco - o yoa ee cce aoaecoo peypoa o ooe, oecea eoxoe aece coca o cce. Bac ee ccey pa opao.

7.9.1 oceoamea oppeu oppepyee ycpoco aec py e cce oo oce aa e peapeoo yce. Ha pc. 7.20 opaea cpyypa cxea cce aoaecoo peypoa c oceoae oppepy ycpoco W(s).

peee oceoaex oppepyx ycpoc aoee yoo cceax, y oopx ca ypae pecae coo apee ocooo oa.

B aece oppepyx ycpoc oy pa ceye:

ueaoe uepeupyee eo W(s) = T s;

(7.21) ueaoe uepeupyee eo c coecm eeue npouoo u omoeu W(s) = k(T s + 1);

(7.22) uepuoe uepeupyue e W(s) = (7.23) ueaoe umepupyee eo W(s) = ;

(7.24) uepuooe umepupyee eo W(s) = (7.25) Pc. 7.20 Cpyypa cxea cce c oceoaeo oppee cooae oppepyeo eea c epeaoo ye (7.21) ee oepe opa o ee ooe peypyeo e. Bo cyae eoxoo ya a cao ooee, a eo pooy, .e. oppepyee ycpoco oo pac e (7.22). Oao epeaoa y oppepyeo ycpoca oa pac e (7.23).

cooae eppyx ee (7.24), (7.25) oae opo acaa cce, o ee yxye ycooc, ooy oopeeo eoxoo oaoc o ooex cpecax oppe c e oe ycooc.

Beee poox ec o cocoo ao oppe.

7.9.2 apaea oppeu p apaeo oppe oppepyee ycpoco oaec apaeo ooy eco oco e (pc. 7.21), p o ooa oppe yx o: ypeaa pa c (pc. 7.21, a) opaa c (pc. 7.21, ). B ayo ccee paa ey a apaeo oppe caoc ycoo coc oy, ae e cac "oxae" ao c.

Oao a pae ae ceo coy opaey opay c.

B acoc o a oppepyeo ycpoca paa ceye opax ce:

ecma opama c W(s) = k = const, (7.26) e k - oe eco opao c;

uepuoa ecma opama c ;

(7.27) Pc. 7.22 Cpyypa cxea apaeo oppe:

a - pa c;

- opaa c ueaa ua opama c (uepeupya) W(s) = Ts;

(7.28) uepuoa ua opama c (uopoa) ;

(7.29) uepuoa oppemupya opama c (acmamueca oppeu) W(s) = ;

(7.30) - ouupoaa opama c (uopoa c ocmamoo epaoepocm) (7.31) Aa pee pax oppepyx ycpoc ooe cea eoope o peoea ooceo x cooa. ooea eca opaa c (7.26) cy yee oea yce, o p o eoxoo ce a ocoo pee, oopa ae yeaec, ccea oe ca eycoo. Opaea eca opaa c (7.26) coyec yee epooc cce. Ta a ooee opae c ey a coo oep ycooc, o aee e ceax ooopo ye cac, o opaa c opaea. ece opae c aypy eppye coca, a e c coxpa aca. Oxa eco opao c pepaae acaece c caece. B paeco pee aoee pacpocpaee oya epoa a opaa c.

7.10 TPEHPOBOHE AAH 1 Ha ycooc cce aoaecoo ypae oaa e apaep peypyeoo oea. ocee ycooc oo ocyec opo cooecyx eeo cce peypoa, acoc, opo acpoe peyopo. Ta opao, ce ycox cce coc opeee acpoe peyopo, p oopx ccea ye axoc a pae ycooc.

A Ha ao pep apyec ce ycox cce?

B a opao cpoc paa ycooc cce peypoa c peyopo, e a acpoex apaepa?

C a cepoa ycoy ccey c- -peyopo?

2 Ccea aoaecoo ypae e oo oa ycoo, o oaa eoop aaco ycooc. oce oo oe c oo opex acox eoo.

A ae opee eo oe aaca ycooc a ec?

B ao ec cc ee oaae oeaeoc?

C C oo ax acox xapaepc oc paccopee oe aaca ycooc?

3 aaa cce a aac ycooc coyec aao pep Haca, a oopo ocoa ce cce, oaax aaco ycooc. p o, a p cee ycox cce, eoxoo opee acpo peyopo, p oopx ccea oaae aa aaco ycooc.

A Ec aac ycooc oeaec oaaee oeaeoc, o a oe aac ycooc ayo cce?

B Ooao e peaec aaa opeee acpoe - -peyopo a aa aac ycooc?

C o oaae ep "cpyypa eycooc"?

7.11 TECT 1 peeoe aee acpoe -peyopa, p oopo ccea axoc a pae ycooc opeeec o coooe:

A ;

B ;

C.

2 a opee cee ycooc?

A ;

B ;

C.

3 a oy pacpey ayo-aoy xapaepcy?

A aeo epeaoo y S = i.

B aeo epeaoo y S = i + M.

C aeo epeaoo y S = -m + i.

4 ao oaae opeee aac ycooc?

A Cee ayxa.

B oaae oeaeoc.

C Bpe peypoa.

5 a oaaee opeeec aac ycooc p pacee acpoex apaepo eoo PAX?

A aac ycooc a ae.

B oaae oeaeoc.

C Cee oeaeoc.

6 p pacee peyopo a aa aac ycooc x acpo pac A Be po aaoo aaca ycooc.

B Ha po aaoo aaca ycooc.

C Byp oac, opaeo po aaoo aaca ycooc.

7 cce peypoa c -peyopo paa aaoo aaca ycooc cpoc oopaax A S1 - S0.

B Re(m, ) - Im(m, ).

C Re() - Im().

8 ae acoe xapaepc coyc p aae cce peypoa a aac ycooc o oy ae?

A AX oea AX peyopa.

B AX paoyo cce.

C AX ayo cce.

9 paa aaoo aaca ycooc opeeec o ypae A Wo(-mp, ip) Wo(-mp, ip, S0, S1) = 1.

B Wo(ip) Wo(ip, S0, S1) = 1.

C Wo(-mp, ip) = Wo(-mp, ip, S0, S1).

10 p cee cce a aa aac ycooc paoa acoa - o A acoa, p oopo ccea axoc a pae ycooc.

B acoa, p oopo ccea axoc a pae aaoo aaca ycooc.

C acoa, p oopo ccea axoc oac eycoo pao.

8 CCEOBAHE AECTBA POECCOB PEPOBAH Oo poe, oax p ocpoe cce aoaecoo peypoa, apy c poeo ycooc, ec aeco peypoa, xapaepyee ooc aoc poea epexooo poecca.

Ccea aoaecoo peypoa aaec aeceo, ec oa yoeope opeee exooec peoa: apep, a ye ec pea cce, ec a ee xo ecy paoo poa oye a o aay ypae, a o aay oye, .e. oeceaec paa oooc pxoa cce eoopoe ycaoeec cocoe. Taoe oe aeca aoaeco cce oxaae ee caece aece coca, paee oeceo ope oye aae oaaee aeca ypae.

8.1 OAATE AECTBA 8.1.1 pe oaae Haoee pacpocpae p oaae pep aeca, pee cceax ypae, c:

1 Caeca oa peypoa yc, opeeea a paoc ey ycaoc aee peypyeo epeeo ee aa aee (pc. 8.1), .e. yc = yyc - ya.

2 aeca oa peypoa y, opeeea a aoee ooee epexoo poecce peypyeo epe-eo o ee ycaoeoc ae (pc.

8.2).

3 Bpe peypoa Tp - pe, a oopoe paoc ey ey aee peypyeo epeeo ee aa aee ( ycaoc) caoc ee (pc. 8.1, 8.2), |ya(t) - y(t)| <.

4 epepeypoae, epeoe % paoe ooe epoo acaoo ooe peypyeo epeeo o ee ycaoeoc ae oy ycaoeyc ae (pc. 8.3):

. (8.1) aeco peypoa caec yoeope, ec epepeypoae e peae 30 - 40 %.

5 Cee ayxa, epea %, cy oeceo oeo ecoc ayxa oeaex poecco opeeec a ooee paoc epo pee ay epo aye (pc. 8.4):

. (8.2) ecoc ayxa oea ccee caec yoeopeo, ec cee ayxa cocae 75 % e, eoopx cyax oycaec opa 60 %.

oo, o ccea aoaecoo peypoa yoeopa peyeoy aecy eoxoo, o pe oaae aeca peypoa o cce ee pa aa, .e.

.

oa peoa o aecy peypoa oy oee ece, apep, epexo poecc oe ooo ooo e epeo.

pe oaae aeca yoo cooa ex cyax, oa eec pa epexooo poecca y(t), oop oe oye cepeao peao ccee peypoa ye oepoa a BM. Ec e ao oooc e, .e. e yaec a opao oy py epexooo poecca, o oyc oce oaae aeca, oope cc e ocpoe paa epexooo poecca o oea ypa-e o aco xapaepca.

8.1.2 ocee oaae aeca Ocoy pyy cpe ocex oaaee aeca coca opee oaae aeca peypoa, oop oocc cee ycooc cee oeaeoc. oaae ye cooa opeee oe aaca ycooc (. 7.3, e o ao x opeeee). Co pe aeca peypoa oo cea ceye o.

1 Cee ycooc, opeeea o opye (7.7), xapaepye ecoc ayxa aoee eeo ayxae eoeaeo cocae epexooo poecca, oopa opeeec a y(t) = Ce-t. yc paccapaea ccea ocaec epea ypaee opoo opa, xapaepcecoe ypaee oopoo ee a ecex pax op s1 = 1, s2 = 2 < 2 (pc. 8.5, a). oce cooecy e eeape cocae coooo e cce (pc. 8.5, ):

a o pao epexox poecco, e ee acooe aee op xapaepcecoo ypae, e eeee ayxae cooecya ey cocaa. Peypy epexo poecc y(t) = yi(t). Eo ayxae opeeec aoee eeo ayxae cocae, .e. ae o acooy ae ope xapaepcecoo ypae.

Pc. 8.5 Opeeee aeca ooox epexox poecco o cee ycooc:

a - pacooee ope xapaepcecoo ypae;

- cocae epexooo poecca Pc. 8.6 Opeeee aeca oeaex epexox poecco o cee ycooc:

a - pacooee ope xapaepcecoo ypae;

- epexoe poecc Ec e xapaepcecoe ypaee cce ee oece copee op, o cocaa epexooo poecca yi(t) ye e oeae xapaep yi(t) = Cie-tcost, ecea ac op, a aec cee ycooc, a a =, xapaepye oay (pc. 8.6).

a o pc. 8.6, a oeaex epexox poecca pao aco e oaoe oae, .e. yo = e-t. Ho p oaoo cee ycooc aeco x epexox poecco cyeceo oaec py o pya. Ceoaeo, a cee ycooc oe aeca oeaex epexox poecco eocaoo.

Cee ycooc oe cooaa oe pee peypoa ooox epexox poecco. acaea = e-t oe t = 0 oceae a oc accc opeo (pc. 8.5, ). Bpe peypoa o cyae opeeec a Tp <. (8.3) Ec peyec ye pe peypoa, o, a ceye (8.3), cee ycooc ao yea. p oee pee peypoa acoa e yaec.

2 Cee oeaeoc a e, a cee ycooc, coyec oe aaca ycooc oe aeca peypoa. Cee oeaeoc, opeeea cooec c (7.8), xapaepye ayxae aoee eeo ayxae cocae, oopa opeeec a y(t) = Ae-mtsint, oya ceye, o eee aco ee eee ay oea.

Cee oeaeoc ooao caa co cee ayxa. eceo, oe pee t0 aya cooo cocae opeeec a y1=, a oe pee t0 + T, .e. epe epo, y3 =. B o cyae cee ayxa, coaco (8.2), aec:

, a a, o = 1 - e-2m. (8.4) Cee ayxa eec o 0 o 1, a cee oeaeoc - o 0 o.

Haoee aco coyc ceye x ae: m = 0,141 ( = 0,61);

m = 0,221 ( = 0,75);

m = 0,366 ( = 0,9);

m = 0,478 ( = 0,95).

3 Oea caeco o oe oyea o peeo eopee:

yc =, (8.5) e W.c(s) - epeaoa y ayo cce o aay o;

X(s) opaee aaeo oec, oce cyae x(t) = C = const oa X(s) =. Cyeo ecaaoo yc =.

Hapep, cce c epa peyopo caeca oa ocycye, a cce c poopoa peyopo paa.

Ec Wo(s) oe epea pae k, o.

oceeo coooe o, o cceax c -peyopo caeca oa yeaec c yeee ae apaepa acpo peyopa. B peax cceax epec acao oooe aee S1, cxo oecee aaca ycooc.

B aee ceye ae, o aeca oa ope eoa e oeaec.

8.1.3 epae pep aeca epae pep aeca peca coo opeeee epa o pee peeax o 0 o o eoopo y epexooo poecca y(t) o (t) cc eocpeceo, o o epexo y cce, o oea epeaoo y cce. e cooa x pepe ec oyee oe oe cpoec ooe peypyeo e o ycaoeoc ae. epa pep aeca pec a peoa: a) pocoa ce epaa;

) ecooc pae epe oe epeaoo ypae.

8.1.3.1 ue umepa pumepu (8.6) cy oe aeca eoeaex poecco. eoepec o pep xapaepye oa, aey ey po epexooo poecca oc accc (pc. 8.7, a), o yae a pe peypoa, a ey aecx ooe. Ec eeca pa epexooo poecca, o eca epeaoa y ayo cce W.c(s) xoa epeea x(t) = 1(t), o aee eoo epaoo pep opeeec c cooae eope o oeo ae y. eceo, opyy (8.6) oo aca ae:

oa e epa pep aeca oo c py eoa.

Hapep, ec a epeaoe ypaee aae yco:

(n-1) an y(n)(t) + an-1y (t) + +a0y(t) = 0, y(n - 1)(0), , y(0), y(0), o, poeppoa eo, oy ycox cce y(i)() = 0 i = 1, 2, , n.

(n - 2) Toa any(n - 1)(0) - an - 1 y (0) - a0J = 0, oya J =, a p yex aax ycox J =.

Cyecy oa eoo epaoo pep, oope pec ex cyax, oa aa yaco epexooo poecca ec eee oece, apep,..

Bee opyy, ooy c ao pep. oo poepepye o s y, ocyecy peopaoae o aacy y y(t):

.

Ec epe peey p s 0, o oy.

Ceye oe, o ce ax pepe e peyec ae epexooo poecca. e ee aee eoo epaoo pep, e ye aeco poecca peypoa. Oao cooae aoo a pepe aoepeex epexox poecco e ae oeo ap, a, apep, eayxae cyco J = 0. ooy oe aeca peypoa ax poecco coy -epae oe, aoepeeoc oepao y oopx ycpaea a-o cocoo.

pep 8. Tpeyec c J* cce c Peee.

Hae y(s):.

.

Pc. 8.7 epae oe aeca peypoa:

a - ea;

- oya;

- apaa 8.1.3.2 Moy umepa pumepu (8.7) peec oe aeca oeaex poecco, a eoeaex poecco o coaae c e epa pepe. eo ce peyec ae epexooo poecca. Ha pae o pep coyec p ceo cceoa cce a oex c peee ceo ex, .e. a, e oepa oy e pecae pyoc. eoepec pep pae oa, aeo ey po y(t) oc accc (pc. 8.7, ). B eoopx cyax coy oa oyoo epaoo pep:

(8.8) oopa pae o ec ae epexooo poecca eo oe.

8.1.3.3 mepa apamu pumepu, (8.9) ec aoee pacpocpae pepe aeca pecae coo oa o po y2(t) (pc. 8.7, ). a o (8.9), pae o ee opa epexooo poecca xo pep c pa eco, o po oy, o aa yaco epexooo poecca popeae aoee aee, e eo "xoc", oop paec e e a apa pep. Cpec poa (8.9), aec py aoe ooe peypyeo e, ooy ae ae pep cea cooecy oeae poecca c a ayxae. Ce ycpae oo eocaa pe yyey apay oey:

, (8.10) oopa, poe cax ooe, yae c eco oeo x pooy.

Becoo oe paec pa eaeoy pee apaca peec peeax T, (8.11) e Tp - eaea eoc epexooo poecca.

apa pep, a e, oo c e ocpoe epexooo poecca o acoo xapaepce ayo cce peopaoa o ype o xooo caa.

coy opyy Pee, oya:

B aee ceye oe, o acoe ae o epao oe ca o cee e peca epeca. O cya coocae pax apao acpo oo o e cce, aae opeee apaepo acpo cce.

8.2 ACTOTHE METO AHAA AECTBA PEPOBAH B eepo pae poo coyc acoe eo cceoa cce ypae. B acoc, pya eoo, papaoaa B. B. Coooo, ooe oe aeco peypoa o eece aco xapaepca, ocpo epexoe poecc, aae cepoa oppepye ycpoca.

8.2.1 acoc ey epexoo aco xapaepca oe aeca peypoa eoxoo ycao c ey epexo aco xapaepca. B acoo oac epexoa xapaepca acaec epe peopaoae ype:

(8.12, a) epe AX cce opaee xoo epeeo o ype c pyo copo y(i) = W(i) X(i). (8.12, ) coy opaoe peopaoae ype ocee coooe, epexoo poecc (epexoa xapaepca) opeeec cey opao:

(8.13) p oec a xo eo cyeao ye x(t) = 1(t), opaee oopo x(i) = 1/(i), coooee (8.13) epexoo y aec a.

peca AX epe ecey y ac W(i) = Re() + iIm() paaa eit o opye epa, paee epexoo y peopayec oee yooy y c cooae BX - Re():

. (8.14) MX - Im():

. (8.15) Ha pae coyec opya (8.14), oopo BX pecae coo coy y eppoae ooo oo peo: ce eoa c peee BM o ye peapeo apoca coo xapaepc Re() ycoo-e y - cyo pae cyo peyoo, o ooe oy ocaoo yoe pae.

Ec a ccey ecye poooe oyee, o epexo poecc opeeec o ooe eeceo o xapaepca:

Reo() = Re[W(i)X(i)], Imo() = Im[W(i)X(i)], (8.16) p o eoxoo, o oc y W(s)X(s) pacoaac cea o o oc.

8.2.2 Coca eeceo-acox xapaepc cooecyx epexox poecco Ocoe coca BX epexox poecco cey (8.14).

1 Cocmo ueocmu: ec BX oo peca cyo (8.17, a) ao cocae cooecye epexo poecc, (8.17, ) o epexo poecc y(t) oe pecae cyo cocax. (8.17, ) Pc. 8.8 Cooece acao o oc opa:

a BX;

nepexoe npoecc 2 Cooece acao o oc opa Re( ) u y(t).

Ec yo Re() a oco oe, o cooecyee aee y(t) oe yoaec a o oe (pc. 8.8).

3 Cooece acao o oc accc Re( ) u y(t).

Ec apye cooecye pae acoo xapaepc yo a ocooe co, o apye cooecye pae epexooo poecca ye ec a o co (pc. 8.9), .e.

(8.18) Pc. 8.9 Cooece acao o oc accc:

a - BX;

- epexoe poecc 4 Haaoe aee BX pao oeoy aeu nepexoo xapamepucmuu (puc. 8.9). (8.19) Haaoe aee MX Im(0) = 0.

5 oeoe aee BX pao aaoy aeu nepexoo xapamepucmuu. (8.20) epec peca pap epepoc e-eceo-acoo xapaepce.

yc p = 1 BX ee pap epepoc (pc. 8.10, a) Re(1) =, p o xapaepcecoe ypaee cce ye e ope s1 = i1, .e. ccee ycaaac eayxae apoece oea, ec ocae op ee.

Pc. 8.10 Pae BX:

a - c papa;

- c co ocp yo Bco ocp BX, a oop Re() epexo epe y p acoe o 1, cooecye eeo ayxa oea (pc. 8.10, ).

6 mo epexoa xapaepca ea nepepeyupoaue 18 %, BX oa m nooumeo eopacmae yue acmom, m.e. Re( ) > 0,.

Pc. 8.11 BX, cooecya epexoo xapaepce c 18 % 7 co ooooo poea epexooo poecca.

mo nepexoo npoecc ue oomo xapamep, ocmamoo, mo coomemcmya ey BX Re( ) ac nooume Pc. 8.12 co ooooo poea epexooo poecca:

a - BX;

nepexoo npoecc o, enpepo yue acmom c ompuameo, yae, no acomo euue npouoo (puc. 8.12) Re( ) > 0, < 0.

8 Opeeee aoeo ae nepepeyupoau nepexooo npoecca max no acuyy BX (puc. 8.13), (8.21) e Remax - acaoe aee;

Re(0) - aaoe aee.

Pc. 8.13 opeee aoeo ae epepeypoa 9 Ec BX a paeeao, .e. oe apocpoaa paee c aaoo aco 0 - 2 oeo aoa =, o pe peypoa epexooo poecca cce aeo peeax.

Pc. 8.14 Apoca BX paee 8.3 BCTBTEHOCT ABTOMATECX CCTEM p aae ycooc aeca aoaecx cce peoaaoc, o ae apaepo oea ypaeo ycpoca ocac poecce cyaa cce oco. B eceoc e apaep cce ocoo ec o pa pa, o a aaeoe, cyaaooe eee. poe oo, ae apaepo oy e papoc cece oyco a ooee eye ae epeex oac o pacex. B c c oae aaa opeee papoca ee apaepo cce a caece aece coca poecca ypae.

B apa apaepo cce a ee caece aece coca aac apaepec oye, a oae p o ooe xapaepc cce o pacex ae - apaepec opeoc (oa).

oe cee papoca ee apaepo cce coy oe - yceoc cce. yceoc - o coco cce e co xoe epeee oaae aeca p ooe oo oo ee apaepa o cxooo paceoo ae. ooae poooooo coca coyec oe "pyoc" cce, coxpae co coca p x apaepecx oyex, aac py poac.

oece oea yceoc c:

-- y yceoc;

-- oe yceoc.

ye yceoc aaec aca pooa ao-o aeco xapaepc aoo-o oaae o eeyc (appyeoy) apaepy ki. Hapep, epeaoo y W(s, ki), ace o apaepa ki, y yceoc opeeec a (8.26) epexoo y h(t, ki) o ooe apaepy ki:

(8.27) e ki0 - paceoe aee apaepa ki.

Ha pae aco coy oocey y yceoc, oopa cooeceo (8.26), (8.27) aec:

Ta, oooypo cce aoaecoo peypoa, cocoe oea Wo(s) = k0 (s) peyopa Wp(s), oocea y yceoc o ooe apaepy k0 opeeec coooee, c yeo (5.86) oo peopayec y, (8.28) oopoe oaae, o yceoc oo cce pey-poa ee coc oea ooc opeeec oo epeaoo ye paoyo cce. e ee aee y yceoc, .e. e pyee ccea, e ee oo-eoe ooee xoo epeeo , ceoaeo, ye aeco cce.

Ec y yceoc paaec co, o oa aaec oeo yceoc. C oo oea yceoc oeaec yceoc cox oaaee aeca, apep, oaae oeaeoc, epepeypoae. Oea ee xoa poecca o ooe oye pooc o opye y(t) = Vkg(t)k.

o ooe eco apaepec oye pe p cyepo, oop oo pocppoa cey pepo.

yc ccea ypae ocaec epea ypaee epoo opa Ty'(t) + y(t) = k x(t) , oopo oc e y yceoc Vky(t) = ;

.

Ec poepepoa cxooe ypaee o apaepa k T poec oyex paex aey epe y yceoc, o oya ypae yceoc paccapaeo cce:

Opee , oo a eee xoa poecca ypae a ce ee apaepo k T:

y(t) = k + T.

y yceoc pe poepoa cce c ae eee aecex oaaee p ooe ae apaepo cce o paceo.

8.4 OHTE O PABEMOCT HAAEMOCT OETA p poepoa cce ypae eoxoo peapeo oea ae cpyype coca oeo a ypaeoc aaeoc.

Oe aaec ooc ypae, ec eo c oo eoopoo opaeoo ypaeo oec oo epeec eee oeoo epaa pee oo aaoo coco aaoe oeoe cocoe. ocyece aoo epeoa oea eoxoo, o e ocaoo, o aa oopa coco acea xo o oo cocax ypaeo oec.

e caoap oe aaec ooc aae, ec o peyaa ae (epe epe ce) xox oopa oo opee (occao) peye ae oopa coco. oo aaeoc occaaaeoc oea eoxoo (o e ocaoo), o aa oopaa coco a caa o ee epe c o aaex cao.

8.5 TPEHPOBOHE AAH 1 Hapy c poeo ycooc p cee cce aoaecoo peypoa ec poea aeca peypoa, xapaepya ooc aoc poea epexooo poecca. oe aeca peypoa oeceo ope coyc oaae aeca, oope opaec a pe, ocee, acoe, epae.

A ae oaae aeca aac p oey?

B ao ocex oaaee aeca peypoa coy oe aeca oeaex epexox poecco?

C o ec ooe ao cooa epax pepe aeca peypoa?

2 B eepo pae poo coyc acoe eo cceoa cce ypae, oope oo oe aeco peypoa o eece aco xapaepca.

A Ec BX pecaea cyo, o o pecae coo epexo poecc?

B Ec BX o oc oopa ye pa, o a oee ce epexo poecc?

C a opee aaoe oeoe ae epexooo poecca?

8.6 TECT 1 ao oaae oocc pye px oaaee aeca peypoa?

A Cee ycooc.

B Bpe peypoa.

C Haaoe ooee.

2 ao oaae aeca aaec caeco oo?

A Macaoe ooee o aaoo ae.

B Ooee o aaoo ae ycaoec coco.

C Paoc ey aca a ae epexooo poecca.

3 Cee ayxa opeeec a A %;

B %;

C %, e y1, y2, y3 - ay xox oea.

4 Ec - cee ycooc, o pe peypoa aepoecx epexox poecco opeeec a A ;

B.

C.

5 Oea caeco o oe oyea a A ;

B ;

C.

6 epa apa pep aeca peypoa - o A.

B J = y2(t)dt;

.

C.

7 p aae epexooo poecca c oo BX eoxoo pec cooece aca o oc oopa. Ec BX yeac pa, .e. caa Re(), o epexo poecc A ec pa - y(t);

B ec pa - ;

C Coec pa -.

8 coe ooooc poea epexooo poecca ec A Re() > 0;

;

B Re() > 0;

;

C Re() < 0;

< 0.

9 CHTE CCTEM ABTOMATECOO PEPOBAH 9.1 AA CHTEA Paccopee e aa oocc aaa aaa aoaecx cce.

aa cea oo paccapa a opae aaa aaa. O opaec a a a: o-epx, peyec opee cpyypy, o-opx, apaep cce o aa oaae aeca.

Ce ec ae ao poepoa ocpypoa cce, oco aoee a poee peyao, oyex eope aoaecoo ypae. p pee aa ooo cea eoxoo opee aopecy yoay cpyyp cce.

Aopecy cpyypy cce axo p oo aeaecx eoo a ocoa peoa, acax aeaeco ope. Bc c poeypy oca aopeco cpyyp aa eopeec ceo aaec ocpypoae cce ypae.

Ce yoao cpyyp aaec ope opex eeo cce coacoa x xapaepc. o a poepoa e ee oa cpoo aeaeco oco oocc oac eepoo cycca. oceoaeoc pee aa ooo cea oe pao.

B pocx cyax aay oa yaec pe c eoooeco o pe eao oceoaeoc. p poepoa cox poex cce ypae pe ay oceoaeoc, a pao, oaaec eooo, ooy oce cyae aay cea pea cey opao.

Baae, cxo peoa aae cce ya yco ee pao, o aaoa cepoo oopyoa pa yoao eoxoe ee:

peypy opa, coeoe ycpoco, a, oope ece c oeo ypae opay eey ac cce. ae a ocoa peoa caec aec coca cce opee ee eey ac, aopeca cpyypa oopo axoc c yeo coc pax yoao eoxox eeo. Texeca e peaa ocyecec c cooae caapx ypoax peyopo pax oppepyx oecpyx ycpoc. poecc opeee aopeco yoao cpyyp cce ypae eco epeeac ey coo, x pxoc o o ecoo pa. Ooaeoe peee o cpyype cce paec a ocoe opocca ey ooc aeco pao cce, c oo copo, pocoo aeoc - c pyo.

ae ao poepoa cce ypae ec pace acpoex apaepo paoo peyopa. B paee 7 oeaoc, o o ceo ycox cce oaoc opeeee apaepo acpoe peyopo p eco cpyype. He poc eo pacea acpoex apaepo oooypo cce aoaecoo ypae.

B acoee pe papaoao oo eoo pacea acpoe peyopa, o x c oee o, o pyoe, pye - poc, o pe. Bo cex eoax eoxoo oece poecc peypoa, a pao, yoeop y pa pep, o oopx ooe oece aa aac ycooc, a opo - oece aeco peypoa.

9.2 BOP OTMAHX HACTPOE PETOPOB METOOM HEATXAX OEAH Meo eayxax oea, peoe ye epo Hoco, ec pe eoo opeee oax acpoe peyopo, oecea eoxo aac ycooc, eoopy cee ayxa eoy aecy oy.

Pacem peymopo c ou napaempo acmpou pooc o a ocoaec a pacee pecoo ae acpo poopoao cocae, p oopo ACP ye axoc a pae ycooc. paee pacea o acpo oc pep ycooc Haca, o oece aac ycooc. eoopoo ae aco p oo oc coooee Wp.c.(ip) = -1.

Ta opao, -peyop paccaec o o aco xapaepca oea. pae pacea pecx ae acpo S1p aco p e :

o(p) = -;

(9.1) S1p=. (9.2) Oaa acpoa -peyopa:

S1o= 0,55 S1p. (9.3) Pace peyopo c y oee apaepa acpo pooc a aa: a epo - opeeec pecoe aee poopoao cocae;

a opo oeceaec cee ayxa = 0,8 0,9.

Oae acpo peyopo axo o cey opya:

-- -peyop S1o = 0,45 S1p ;

(9.4) S0o = 0,086S1pp;

-- -peyop S1o = 0,6 S1p;

S0o = 0,192S1pp;

(9.5) S2o = 0,471.

9.3 AOPTM PACETA OACT HACTPOE TOBX PETOPOB METOOM PAX Meo pacpex acox xapaepc oca paee 7 cooa p cee cce c aa aaco ycooc.

Meoa pacea oax acpoe peyopo eoo PAX aaoa. o oa acpoa ao eoe oa acpo peyopa, oeceae aay cee oeaeoc ma poecca peypoa p ye epaoo apaoo pep J. B c c pace acpoex apaepo peyopa pacaaec a a aa: opeeee acpoe, oeceax aa aac ycooc aay cee oeaeoc, opeeee acpoe, oeceax aeco peypoa, oeaeoe o epaoy apaoy pep.

ep a opoo oca paee 7. Pace peyopo c o acpoe apaepo (- -peyop) oec o (ep) a. peyopo c y acpoe apaepa a epo ae paccaec pao cee oeaeoc ococ apaepo acpoe S0, S1. Ha opo ae eoxoo pa oo oy apy acpoe S0o, S1o, cooecyy aoy ae epaoo apaoo pep aeca. Pace oo pep pax poecco peypoa oaae, o eo yy -peyopa cooecye oa a po pao cee oeaeoc, pacooea ecoo paee ep (pc. 9.1, a). Tao oo ec oa 3. Pa oa a po pao cee oeaeoc cooecy pae poecc peypoa (pc. 9.1, ).

Pc. 9.1 Bop oax acpoe -peyopa:

a - pa pao cee oeaeoc;

- pa epexox poecco peypoa pax acpoe peyopa B oe 1 ocycye poopoaa cocaa, peyop paoae a epa, ocoeoc oopoo ec aoa aeca oa. Boax 2 3 peyop paoae a -peyop, pe cpae x yx poecco o, o c o pe aaoo aeca peypoa epexo poecc oe ye, e oe 2. Ta a p e o po pao cee oeaeoc poopoaa cocaa opacae, opacae paoa acoa, ceoaeo, yeaec aeca oa peypoa, o c eoopoo oea (oa 2) aae yeac ea acpo epao cocae S0, oopa opeee copoc ycpae caeco o. e ee ea S0, e eeee paec caeca oa, .e. aaec aae "xoca" epexooo poecca (oa 4). B oe 5 ocycye epaa cocaa, peyop paoae a poopoa, eo ocoeoc ec ae caeco o peypoa.

Oae acpo peyopa paccac o yy J. x opa eoxoo pacca pep J cex ap acpoe peyopa o po pao cee oeaeoc. a poeypa pyoea a pae pea eepo eoe opeee ecoaxoe o 3. Paoa acoa opeeec, cxo coooe p = 1,2 0 p 0,8 , e 0 - acoa, cooecya epe po m = ma;

- acoa, cooecya poopoaoy aoy pey-poa. oce oo o opya (7.18) paccac,.

poeypa pacea oax apaepo acpoe -pey-opa aaoa pacey -peyopa. B ococ apaepo S1 S2 cpoc pa aao cee oeaeoc (pc. 9.2, a). p e o po pao yeaec epeo-aa cocaa S2 acoa. Ceoaeo, e oe S2, e ee aeca oa peypoa. Bea acpo, poopoaa cocae S1, caaa yeaec, a ae yeaec, pe, e oe S2, e ee caeca oa. Becaaoe xopoo cppyec paa poecco pey-poa pax acpoe peyopo, opaex a pc. 9.2, .

Pc. 9.2 Bop oax acpoe -peyopa:

a - pao cee oeaeoc;

- pa poecco peypoa pax acpoe -peyopa Oae acpo S1*, S2* opeec yco ya J, oopoy a po pao cee oeaeoc cooecye oa, pacooea a ee epe.

9.4 PAOAHATEC METO CHTEA CCTEM Paccapae eo oocc pye paoaaecx eoo, papaoax B. . Poae, ocoy oopoo aoe ceye ooe.

Bo-epx, caec, o ccea peypoa oaae eoxo aaco ycooc, ec ee oaae oeaeoc e peae e M = 1,1 1,6, .e. o pepe oaoc ec oeceee aaoo oaae oeaeoc Ma.

Bo-opx, ey ccey peypoa oo paccapa a coeopa aco p, epe oop poxo cocae apo xox oec.

B acoc o aecx coc ACP apo c pa acoa peepea pae ee, .e. aya aa xooo caa yy pye, e a xoe.

eao cceo peypoa caec ccea, oaaa aco py coca. Ayo-acoa xapaepca ao cce ooceo oyax oec paa y o ce aaoe aco o 0, a ooceo ypaeo oec oa paa 1, .e. M() = 0;

My() = 1.

aaa opa oax apaepo acpoe cce aaec o, o aoe cee p AX peao cce AX eao cce. Ta a peax cceax paec eooo oc, o ooc ycoe M() = 0, o apaep acpo o pac a opao, o ccea aoee eco poaa "oace" apo. Ta a poocee oe c oaco po, o eecoopao pa ao eo, oop apapoa ayee pee acox xapaepc cce opecoc o c yeo acoo. pee peao cce eao ocyecec ye paoe p Teopa. coe oaoc oo aca e:

- ooceo oyaeo oec M(0) = 0;

;

(9.6) - ooceo ypaeo oec My(0) = 1;

. (9.7) pae (9.6), (9.7) cya opeee oax apaepo acpoe cce. Pace pooc ceye ope.

1 B npocmpacme napaempo acmpoe peymopa onpeeemc paua oacmu, omopo cucmea oaaem ocmamo anaco ycmouocmu.

2 B mo oacmu onpeeemc moa, yoemopa uuyy omoeu acmomx xapamepucmu peao cucme om xapamepucmu ueao.

cxou au mc acmome xapamepucmuu oema, acmocmu, anumyo-aoa.

nocmpoeu pau aaoo anaca ycmouocmu ucnoyemc ceyu noxo. a uecmo, anac ycmouocmu oem onpeemc y ucou euuau: anaco ycmouocmu no oy u anaco ycmouocmu no ae, xapamepuyuu cmene yaeu AX paoymo cucme om "onaco" mou (-1, i0). Ho oaaemc, mo cmene yaeu AX paoymo cucme om mou (-1, i0) oem m onpeeea no euue acuya anumyo-acmomo xapamepucmuu aymo cucme (c. 7.3.2).

Ta opao, peoae, o acy AX ayo cce e pea eoopo apaee aao e, coc peoa, o AX paoyo cce e axoa yp oac, opaeo payco r c epo a pacco R o aaa oopa, pacooeo a opaeo eeceo oyoc.

oce opeee oac aaoo aaca ycooc pooc opeeee o o oac, cooecye oa acpoa peyopa.

9.4.1 -peymop epeaoa y -peyopa acaec e W(s) = kp.

Ayo-aoa xapaepca paoyo cce c -peyopo:

Wp.c.(i) = kp Wo (i).

Opeeee oao acpo kp* pooc ceye ope.

Pc. 9.3 opeee ae oea epea -peyopa Cpoc AX paoyo cce p kp = 1, o cooecye W(i) = Wo(i), .e.ocpoe AX peypyeoo oea (pc. 9.3). aee, aaa oopa pooc y o yo (9.8) opaeo eeceo oyoc.

Bepaec opyoc c epo a eeceo opaeo oyoc, acaac oopeeo AX oea oo ya:

(9.9) B oce cyae pace cce aoaecoo peypoa pooc a oeceee oaae oeaeoc M = 1,62, o apapye aac ycooc o oy d = 0,38 o ae = 36o, a cee ayxa epexooo poecca oe aeo ee M.c(0) = 1: = 0,9. B cooec c opy (9.8) (9.9) pa (9.10) Haeoe aee oea epea ec oa aee.

9.4.2 Ȗpeymop epeaoa y -peyopa:

.

Ayo-aoa xapaepca paoyo cce:

Pc. 9.4 Opeeee peeoo oea epea -peyopa Pace -peyopa pooc a aa:

1 o AX peypyeoo oea cpoc AX paoyo cce kp = 1 eoopoo ae ocoo pee Tp, ea oopo paec o, yoo ocpoe xapaepc:

oce yoo cpo, oopaa a eop AX oea a yo 90 o acoo cpee yea eo y Tp pa (pc. 9.4).

2 pooc o yo opaeo eeceo oyoc epaec opyoc c epo, pacooe a o oc, acaac oopeeo ocpoeo AX Wp.c1(i). Bea oea epea kp, oeceaa aay ey acya AX ayo cce (aa oaae oeaeoc Ma opeeec o opye (9.9)), , ceoaeo, ea peeoo oea epea -peyopa, oopa ec eo oa aee, opeeec a. (9.11) Ec, o ;

(9.12) 9.4.3 Ȗpeymop epeaoa y -peyopa:

Ayo-aoa xapaepca paoyo cce:

.

Pace -peyopa pooc ceye ope:

1 Cpoc ceeco AX paoyo cce p kp = 1 eoopx pax aex pee opoa Tl (l = 1, 2, 3,...), paex pooo, o c o pe yoca ocpoe:

Pc. 9.5 opeee oeo epea -peyopa pax Tl opeee pa oac ycooc -peyopa epoaao epaec AX oea W(i), oopy ocaoo e peeax III apaa oeco ococ W (pc. 9.5).

Ha o xapaepce pac o A1, A2, A3,... c acoa 1, 2, 3,..., oope coec c aao oopa opea OA1, OA2, OA3,.... opea oax A1, A2, A3,... occaaac epeyp. aee opeec ooe oe Bj AX paoyo cce. C o e a occaoex epeypax oaac OAj ope AjBj, opeeee, a Aj B =. Coe o Bj c peee opoa Tl j Tl j ao po, oya AX paoyo cce. Aao opao cpoc AX paoyo cce pyx ae Tl.

2 pooumc uu no yo eecmeo ompuameo noyocu u cmpomc opyocmu c empo a mo ocu, acauec AX paoymo cucme paux Tul u mo npo. aoo aeu Tul onpeeemc npeeoe aeue ouuema nepeau ecu M = 1,62, o = 38, kpl =.

a r l 3 B nococmu napaempo acmpoe kp Tu cmpoumc paua oacmu, omopo acuy AX aymo cucme omocumeo ynpaeo oecmu e npeaem aao euu. C mo e ucnoymc noyee ae kpl, Tul (puc.

9.6).

Pc. 9.6 Opeeee oao acpo -peyopa Oa acpoa peyopa cooecye oa, oopo ooee ye acao, a a eo e oec ycoe (9.7). Tao oo ec oa aca acaeo pae oac oycoo aaca ycooc, poeeo epe aao oopa. eceo, a pya pa, xoa aaa oopa c o ooee, oopoe opeee yoo oe, e ye poxo epe oac oycoo aaca ycooc, ooy oy oy ey ooe ao ccee eooo e yee ee ycooc e eoxoo e.

9.5 TPEHPOBOHE AAH 1 Bae ao poepoa ocpypoa cce ec ce, oa eoxoo opee aopecy yoay cpyypy. Ec cpyypa eca, o ce coc opeee apaepo acpoe peyopo. Bce eo pacea ocex opaec a oe, o pyoee poce, o pee.

Haoee pacpocpae c eo eayxax oea, eo PAX paoaaec eo.

A Be aaec ce yoao cpyyp?

B ae eo pacea apaepo acpoe peyopo oocc o eoa?

C a aaec ce, aac pacee apaepo acpoe peyopo?

2 O ox eoo pacea apaepo acpoe peyopo ec eo PAX, ocoa a aaoe pep Haca. Pace pacaaec a a aa:

opeeee acpoe, oeceax aa aac ycooc, opeeee acpoe, oeceax aeco peypoa.

A ae apaep acpoe peyopo aac oa coaco eoy PAX?

B a oaaee oeaec aeco peypoa eoe PAX?

C a pac oae acpo eoe PAX peyopo c y acpoe apaepa?

3 Bop o eoo pacea oax acpoe peyopa ec paoaaec eo, ocoa a cooa AX peypyeoo oea.

A a oaaee oeaec aac ycooc paoaaeco eoe?

B a paoaaeco eoe oeaec aeco peypoa?

C a opee oae acpo -peyopa?

9.6 TECT 1 Bop aopeco cpyyp cce aoaecoo peypoa aaec ope A yoax eeo x xapaepc.

B Cpyyp cce aoaecoo peypoa.

C apaepo acpoe ox peyopo.

2 Bop oax acpoe peyopo eoo eayxax oea ooc A To eoa.

B Cya eoa.

C pe eoa.

3 p ope oax acpoe -peyopa paoa acoa opeeec a A p = 0,8 .

B p = .

C p = 1,2 , e acoa, cooecya poopoaoy aoy peypoa.

4 Toa, cooecya oa acpoa -peyopa, pacooea a po aao cee oeaeoc ococ apaepo acpo peyopa S2 S1:

A Cea o acya.

B Bepe.

C Cpaa o acya.

5 a epa pepe oeaec aeco peypoa eoe PAX pacea oax apaepo acpoe peyopo?

A e epa pepe.

B Moy epa pepe.

C apa epa pepe.

6 cox ao caaec pace oax apaepo acpoe peyopo - -peyopo?

A ooo.

B yx.

C pex.

7 B paoaaeco eoe pacea oax apaepo acpoe peyopo caec, o oecee aa aac ycooc, ec AX paoyo cce opyoc payca c epo a opaeo eeceo oyoc A epeceac.

B acac.

C He axo py a pya.

8 p ao ae oea epea K, oop e ece, cpoc AX paoyo cce?

A K = 0.

B K = 1.

C K = Ko.

9 Ec oaae oeaeoc M = 1,62, o oe epea pae A K = 1/r.

B K = r.

C K = 1/r + 1.

10 p pacee oax acpoe apaepo -peyopa ocoa pee Tp paec A Tp = 1.

B pooo.

C Tp = To.

10 PEEHE TPEHPOBOHX AAH Pae 1 A Cooyoc execx cpec, ox eoop poecc, aaec oeo ypae. pepo oea ypae, apep, ec poecc pea, epya p.

B Bxoe epeee c ypa, ec o cya oepa ypaeo epeeo cooec c eoop aoo ypae.

C epeea, oopy eoxoo oepa cooec c eoop aoo ypae, aaec ypaeo.

2 A B ACP, opaeo a pc. 1.2, peaoa p peypoa o ooe o oye.

B Ec peyop ee peypyee oece p ooe peypyeo epeeo o aaoo ae (y(t) = y(t) ya), o aoe peee aaec peypye o ooe, y(t) aaec ooee oo ypae.

C Haoee eo ec opoaa ccea peypoa.

3 A ea ccea oocc accy cce o oco a ypae a poecco ypae.

B acc "xapaep yopoa" ec a:

a) cce caa;

) cce popaoo peypoa;

) cee cce;

) cce oaoo ypae;

) aae cce.

C acc "xapaep oa cao" opaeec a:

a) epepe cce;

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

Pae 1 A Ca aaec peyp, ec eo aeaec pecaee ec apaee aaa y pee.

B Cyecy peeoe acooe pecae cao.

C oco a peypx cao oocc: epoec, o epoec eepoec.

2 A peopaoae ype aaec oepaop.

B Xapaep coca cepa epoecoo caa c:

a) cep cea cpe, aco ocox apo pa ocoo acoe;

) e oe epo caa T, e "ye" cep;

p T oya eepoecy y;

) c yeee eoc yco p ocoo epoe ay apo yeac, a cep caoc "ye";

) ec c yeee eoc poyox yco yea ay o aoy A0 = 1/, o x oceoaeoc cpec oceoaeoc ea-y, a ay cep ocooy cex aco ae An = 1/T.

C Cepao xapaepco eepoeco yaaec ea, e A ecoeo ae ay epoeco y.

3 A ea-ye aaec y, yoeopa yco:

.

B Ca e eoo caa a cceyeo oee oa ye peoo op e, o pacxo oaaeoo eeca ec cao a ey.

C apoec ca xapaepyec ayo, epoo ao.

Pae 1 A pae ca aac ypae, ocae oeee cce peypoa ycaoec pee p ocox oecx.

Caeco xapaepco oea (cce) aaec acoc xoo e o xoo caeco pee.

B pae a aac ypae, ocae oeee cce peypoa p eycaoec pee poox xox oecx.

C paec peepyap, epeca eoc, epep xec peaop ooo epeea ocac ooe epea ypaee c oco oea epoo opa.

2 A oaaeca eoc cce poo cepe, coco pex oo:

1 onm: a xo cce oaec xoo ca x1(t) opeeec xoa oopaa y1(t) ycaoec pee;

2 onm: a xo cce oaec pyo ca x2(t) opeeec oopaa y2(t);

3 onm: a xo cce oaec ca, pa cye xox cao x3(t) = x1(t) + x2(t), opeeec xoa oopaa y3(t). aee poepec oee cooec y3(t) = y1(t) + y2(t) oo oea pee. Ec oo oec, o oec p cyepo, ccea, ceoaeo, ec eo.

B Oco aec xapaepca, coye eop aoaecoo ypae, c: epeaoa y, epeaoe ypaee, epexoa y, ecoa y, aco xapaepc:

ayo-aoa, ayo-acoa, ao-acoa, eeceo-acoa.

C Cxea pacea a c oo peex xapaepc coco ceyx ao:

1) paec caap ca a xoe (t) = ((t), , n(t));

2) xoo ca pooo op pecaec a cyepo caapx cao x(t) = 1(t) + 22(t) + + nn(t);

3) opeeec pea cce a caape ca ;

4) xoo ca y(t) opeeec a cyepo xox cao yi(t):

.

3 A peopaoae aaca aaec peopaoae y x(t) epeeo t y x(s) pyo epeeo p oo oepaopa.

Oco coca peopaoa aaca c ceye:

a) eopea eoc ;

) eopea oo ;

) eopea ayxa ;

) eopea aaa p.

B.

C epeaoo ye oea aaec ooee peopaoaoo o aacy xooo caa y(s) peopaoaoy o aacy xooy cay x(s) p yex aax ycox.

Pae 1 A Oco coca oopoo oopae c:

a) oo oeco ococ oopaaec pyo oeco ococ;

) ecoeo a yo oopaaec ao e ecoeo a yo, y coxpac;

) peyo oo oeco ococ oopaaec ao e oo peyo pyo oeco ococ, apaee oxoa coxpaec;

) ype oac ooo peyoa peopayec o ype oac pyoo peyoa.

B Re() = M() cos ();

Im() = M() sin ().

C ;

.

2 A cepeao oya AX X. AX pecae coo ooee ay xooo caa aye xooo caa. X paoc a xooo xooo caa.

B ;

.

C.

3 A Becoa y pecae coo opaoe peopaoae ype o AX.

B Ec h(t) epexoa y, o W(i) = (i) h(i).

C.

Pae 1 A Ooe epea ypae ocac aepoecoe eo epoo opa, aepoecoe eo opoo opa, oeaeoe eo.

B peax ee AX M() 0 p. eao-epepyeo ea M() p, eo eocyecoc ae a peex xapaepc, a a h(t) = (t), a w(t) = '(t).

C Toe e opaec a:

a) caece, y oopx caeca xapaepca oa o y;

) epepye, y oopx caeca xapaepca paa y;

) acaece, y oopx caeca xapaepca e cyecye.

2 A oooypo cce aoaecoo peypoa oo aca epeaoe y o aay peypoa, o aay oye, o aay o.

B p oceoaeo coee:

;

;

.

p apaeo coee:

;

;

.

C e ooex peopaoa pooc epeoc ya epe ye epeoc cyaopa epe cyaop.

3 A ec e peaye -ao peypoa.

B Beee ao peypoa epeao cocae yeae cpoece peyopa.

C a "" y epeaox y peyopa yae o a, o peyop aec ccey o py opaeo opao c.

Pae 1 A Ccea, oopa oce c oye pae ooe cocoe paoec, ooe o epoaaoo, aaec epao.

B Ccea aoaecoo ypae e ycoa, a a o ope S ooe.

C Ccea aoaecoo peypoa, y oopo op xapaepcecoo ypae pacooe cea o o oc, ycoa.

2 A Heoxooe ycoe ycooc ec ocao cce, ocaxc ooe epea ypae epoo opoo opa.

B B cooec c pepe ypa ccea ycoa: 1 = 4 > 0;

2 = 5 > 0;

3 = 5 > 0.

C cceoa ycooc c oo pep Payca eoxoo pacoaa ypaee, oopoe ocae ccey aoaecoo ypae.

3 A Ec paoya ccea e ycoa, o oo, o aya ccea a ycoo, eoxoo ocaoo, o AX paoyo cce oxaao oy (-1, i0) m/2 pa, e m co pax ope xapaepcecoo ypae paoyo cce.

B B cooec c pepe Mxaoa ccea aoaecoo ypae e ycoa, a a op e c ece epeyc ey coo.

C B cooec c pepe Haca ccea ycoa, a a AX paoyo cce e oxaae oy (1, i0).

Pae 1 A Ce ycox cce apyec a pep ycooc Haca.

B paa ycooc cce aoaecoo peypoa, c peyopa, e a acpoex apaepa, cpoc ococ apaepo acpo s s0 (-pe-yop) s2 s1 (-peyop) o ypae cooeceo.

C aaa cea cce peypoa c - -peyopo peaec ooao, a a ec a ypae a eecx p s1 ( s0).

2 A ope eoa oe aaca ycooc oocc cee ycooc cee oeaeoc.

B oaae oeaeoc o acy AX ayo cce.

C opee oe aaca ycooc oc paccopee epe pacpee ayo-aoe xapaepc.

3 A Ec aac ycooc oeaec oaaee oeaeoc M, o aya ccea oaae aa aaco ycooc, ec AX paoyo cce acaec opyoc payca r = M/(M2 1) c epo oe l = M2/(M2 1).

B aaa opeee acpoe peyopo a aa aac ycooc peaec eooao. aaa ee ecoeoe oeco pee.

C Cpyypo-eyco aac cce, oope e oy ca yco p ax oax ae x apaepo.

Pae 1 A p oaae aeca c oaae, oope oo eocpeceo o po epexooo poecca oea aeco peypoa.

oocc caeca oa peypoa, aeca oa peypoa, pe peypoa, epepeypoae, cee ayxa.

B oe aeca peypoa oeaex epexox poecco coyec cee oeaeoc.

C ooe aopo cooa epax pepe aeca ec oyee oe oe cpoec ooe peypyeo e o ycaoeoc ae.

2 A Ec BX pecaa cyo ao cocae cooecye epexo poecc, o epexo poecc pecaec cyo cocax.

B Ec BX a oc opa yeaec pa, o epexo poecc yeaec pa.

C oeoe aee epexooo poecca pao aaoy ae BX;

aaoe aee epexooo poecca pao oeoy ae BX.

Pae 1 A Ce yoao cpyyp aaec ope opex eeo coacoa x xapaepc.

B o eoa pacea apaepo acpoe peyopo oocc eo PAX paoaaec eo.

C Pace apaepo acpoe peyopo aaec apaepec ceo.

2 A Oa apaepa acpoe peyopo coaco eoy PAX c acpo, oeceae aay cee oeaeoc y apaoo epaoo pep.

B aeco peypoa eoe PAX oeaec apa epa pepe.

C peyopo c y acpoe apaepa oa acpoa cooecye oa, eaa a po aao cee oeaeoc ococ acpoex apaepo, oopo apa epa pep ae.

3 A B paoaaeco eoe aac ycooc oeaec oaaee oeaeoc.

B aeco peypoa paoaaeco eoe oeaec c oo pep oao pa, aaeoc aye pe AX peao cce AX eao cce a x acoax , acoc, p = 0. co oaoc acac e:

ooceo oyaeo oec ;

;

ooceo ypaeo oec ;

.

C Toa, cooecya oa acpoa -pey-opa, axoc oe aca acaeo, poeeo aaa oopa po aaoo aaca ycooc ococ apaepo acpoe T K (pe opoa oe epea).

CCO TEPATP 1 Aecee A. A., ae . X., y H. H., oe B. . Teop ypae:

e. C.: T, 1999. 435 c.

2 Coea . H., Coe A. . Teop aoaecoo ypae. M.: MXM, 1975. 165 c.

3 Cop aa o eop aoaecoo peypoa ypae / o pe. B. A. eceepcoo. M.: Haya, 1978. 512 c.

4 Teop aoaecoo ypae. . 1 / o pe. A. A. Bo-pooa. M.: Bca oa, 1986. 367 c.

5 Teop aoaecoo ypae. . 2 / o pe. A. A. Bo-pooa. M.: Bca oa, 1986. 504 c.

6 epoac A. A. ypc eop aoaecoo ypae: eoe ocoe yo. M.: Haya, 1986. 616 c.

7 yac B. A. Teop aoaecoo ypae. M.: Hepa, 1990. 416 c.

8 oo B. . Teop ex cce peypoa ypae. M.: Haya, 1989. 304 c.

9 Poa B. . Pace a poex aoaecx cce peypoa. M.: ep, 1973. 440 c.

10 Teop aoaecoo ypae. Cop aa opox opoco / Coc. . H. Coea. M.: 1974. 92 c.

11 eay A. A., yoc A.. Meo eop aoaecoo ypae. M.: Haya, 1971. 744 c.

12 Poa B. . Teop aoaecoo ypae eoepeec poecca. M.: epoaoa, 1985. 296 c.

13 yo E.. Oco aoaecoo peypoa eox poecco.

M.: ocepoa, 1956 264 c.

14 Cea E.. Oco pacea acpoe peyopo eoepeecx poecco. M.: epoa, 1982. 352 c.

15 oo E.. Teop ex cce aoaecoo peypoa ypae: eoe ocoe yo. M.: Haya, 1989. 389 c.

16 Teop aoaecoo ypae. . 1 / o pe. A. B. He-ya. M.:

Bca oa, 1978. 424 c.

17 Teop aoaecoo ypae. . 2 / o pe. A. B. Heya. M.:

Bca oa, 1972. 432c.

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