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6.5.3 Acoeca ycooc o ycooc oe aco oa coco ea opaac cocoe paoec, oopoo oo peapeo o eeo, apep, a oce ayxax oea epec ooe paoec (pc. 6.17). oooe opeeee oo ec eoyeoo e.

Pc. 6.17 opeee acoeco ycooc Ec p e pocpace o M eopaeo cac paoc x oopa (yi - y'i) 0, o oyeoe ee oceeo opaaec eoyeoy. Taoe ee aaec acoec yco.

ee aaec acoec yco, ec oo oopa aoe, o, ec, o oec ycoe p t.

oe acoeco ycooc oee yo, e oe ycooc o yoy. Ec ee acoec ycoo, o oo aepa ycoo o yoy. Ho opaoe yepee, ooe oop, ecpaeo. ee oe yco o yoy, o e c acoec yco.

6.6 HEOXOMOE COBE CTOBOCT B. 6.2 oyeo eoxooe ocaooe ycoe ycooc opaeoc ecex ace ope xapaepcecoo ypae , o eo, op o pacoaac cea o o oc.

Bx opypoax oe e oo pa ycooc, o a, cyoc, eo cceoa ycooc: eoxoo a op xapaepcecoo ypae poep, ea o eo oyococ e. Oao ao eo coepeo eaeae aae cceoa cy ceyx p.

1 aaa opeee ope xapaepcecoo ypae poco peaec oo ypae epoo opoo opa; cex pyx cyae pxoc ooac pa pe, cpaeo poo eoa.

2 opeee ycooc eoxoo a oo a ope, ooy opeeee ope pecae eyy pyoey paoy. Mey e e oya ox opy, o oop oo o cy o oeo ypae a ycooc cce, o eo o e, epy oepe, epecye poepoa cce aoaecoo peypoa.

aaa cceoa aco cac a opao, o eoxoo opee oe ypae, p oopx ccea a ycoa.

B pacope cceoae ec eo, ooe cy o ycooc cce o a aae yco ycooc, e pea xapaepcecoo ypae e axo eo ope. ep a ycoe, oopoe ceye paccope, ec eoxooe ycoe ycooc.

yc xapaepcecoe ypaee n- cee ee op s1, s2,..., sn. Toa o ypaee oo aca cey opao an (s s1 ) (s s2 ) (s sn) = 0. (6.26) Ec ccea ycoa, o op o o ece opae, o oeco-cope c opaeo eceo ac.

yc s1 = , > 0, oa s s1 = s + > 0.

yc s2,3 = i, > 0, oa (s s2) (s s3) = (s + i ) (s + + i ) = (s + )2 + 2 > 0.

Oca ceye, o oce pacp coo ce oe ypae yy ooe. x paccye ceye, o, oa xo o oeo xapaepcecoo ypae opaee, o ccea eycoa.

Ec ce oe xapaepcecoo ooa ai > 0, o oe eceoe ooeoe aee s, ocaeoe ypaee, e oe opa eo y , ceoaeo, e ec ope xapaepcecoo ypae. ooy p ai > eooo oee apacax coe, xapaepyx aepoecy eycooc, .e. aepoeca eycooc eooa. Oao oe oy oeaea eycooc, .e. oee pee cocax e oea c apacae ayo. o oae, oa cyecy oeco-copee op c ooeo eceo ac. ooy ycoe ooeoc oeo p ope cce oe yx ec eoxo ycoe, o e ocao, a ypae epoo opoo opa o ycoe ec ocao.

eceo:

a2 s2 + a1 s + a0 = 0; s1,2=.

Ec op oeco-copee, o a12 4 a0 a2 < 0, a1 > 0; a2 > 0. Ceoaeo, a0 > 0, a a a12 < 4 a0 a2.

6.7 AEPAECE PTEP CTOBOCT pep ycooc Payca ypa ooe o oea xapaepcecoo ypae e ce eo ope cea cyee o ycooc cce.

Coa ye A.Cooa, peoaa eap, oca epe eapc aeao ype aay axoe yco ycooc eo cce oo opa. Tay e aay oca Mace coe oae, a oopo pcycoa ac aea Payc. B peyae, eaco py o pya pax opax, Payc yp e epaeca, coee oopx ec eoxo ocao ycoe ycooc cce oo opa.

6.7.1 pep ycooc Payca pep, oop peo Payc, aoee poco ocec a. 6.1, e D(s) = a0 sn + a1 sn1 +... + an1 s + an (6.27) xapaepcec oo.

Taa 6.o Cpo Coe e ri a 1 2 3 1 a0 = c11 a2 = c21 a4 = c31 2 a1 = c12 a3 = c22 a5 = c32 r3 = a0/a1 3 c13 = a2 c23 = c31 c33 = c41 r3a3 r3c32 r3cr4 = a1/c13 4 c14 = c22 r4cr5 = c13/c14 5 ri = c1,i- ic1,i = c2,i-2 c2,i = c3,i-2 ric2,i-1 ric3,i-/c1,i-B epo cpoe acac ope opaca eco oe xapaepcecoo ypae, ee e ec, o opo - ee ec.

o pyo oe a opeeec a ck,i = ck+1,i2 ri ck+1, i1, (6.28) e ri = c1,i-2 /c1,i-1; k oep coa; i oep cpo.

co cpo a Payca pao cee xapaepcecoo ooa c ea (n + 1). oce aoe a oo cea ceyee cyee o ycooc cce coaco yco ycooc Payca.

oo, o ccea aoaecoo ypae a ycoa, eoxoo ocaoo, o oe epoo coa a Payca e o o e a, .e. p a0 > 0 ooe ca:

c11 = a0 > 0; c12 = a1 > 0; c13 > 0;...; c1,n + 1 > 0. (6.29) Ec e ce oe epoo coa ooe, o ccea eycoa, a co pax ope pao cy epee aa epo coe a Payca.

o pep oe yoe, oa aa cee ae oeo xapaepcecoo ypae, oe eo popapoa a BM ae pooe peee p cceoa a ycooc oeo ypae o oex apaepo cce.

6.7.2 pep ycooc ypa yp papaoa aepaec pep ycooc ope opeeee, cocae oeo xapaepcecoo ypae cce.

oeo xapaepcecoo ypae (6.27) cpo caaa a opeee ypa (6.30) o ceyey pay: o ao aoa opeee cea apao ca ce oe xapaepcecoo ypae o a1 o an ope opaca eco.

Co epx o ao aoa oo oea xapaepcecoo ypae c oceoaeo opaca eca, a co oea c oceoaeo ya eca. Ha eco oeo c eca oe n ee y poca y.

Oepa ao opeeee ypa aoae op, oy opeee ypa eo opa.

; ; ; (6.31) Hoep opeee opeeec oepo oea o aoa. Ca pep opypyec cey opao.

oo, o ccea aoaecoo ypae a ycoa, eoxoo ocaoo, o ce opeee ypa e a, oaoe co ao epoo oea xapaepcecoo ypae a0, .e. p a0 > 0:

1 > 0; 2 > 0; 3 > 0; ; n > 0. (6.32) Ec pacp opeee ypa ypae epoo, opoo peeo opa, o oyac ceye yco ycooc:

1) n = 1; a0 s + a1 = 0; yco ycooc: a0 > 0; a1 > 0.

2) n = 2; a0 s2 + a1 s + a0 = 0; yco ycooc: a0 > 0; a1 > 0; a2 > 0.

3) n = 3; a0 s3 + a1 s2 + a2 s + a3 = 0; yco ycooc: a0 > 0; a1 > 0; a2 > 0; a3 > 0; aa2 a0 a3 > 0.

pep ypa oo pe p n < 4.

Ta a n = an n-1, o p an > 0 poep ycooc eoxoo poep opeee o 1 o n-1.

Ec an = 0 n-1 = 0 p 1 > 0,..., o ccea axoc a pae ycooc, pe p an = 0 - paa aepoeco ycooc (o ope pae y); p an-1 = 0 - paa oeaeo ycooc (ec a oeco-copex op).

o oy pep oo opee pecoe aee apaepa, p oopo ccea axoc a pae ycooc.

6.7.3. pep ycooc eapa-apo p cceoa ycooc cce aoaecoo peypoa, ex opo xapaepcecoo ypae n 5, peoeyec cooa oy oa pep ypa, peoey 1914 .. eapo P. apo oey eop aoaecoo ypae a pep ycooc eapaapo, oop opypyec cey opao.

oo, o ccea aoaecoo ypae a ycoo, eoxoo ocaoo, o ooc eoxooe ycoe ycooc o opeee ypa c e eca ( c ee eca) ooe, .e.

a0 > 0, a1 > 0,..., an > 0; 2 > 0, 4 > 0, 6 > 0,... (6.33) a0 > 0, a1 > 0,..., an > 0; 1 > 0, 3 > 0, 5 > 0,... (6.33, a) B ao opypoe pep ycooc peyec pacpe eeo ca opeeee, e o pep ypa.

pep 6.1 cceoa a ycooc c oo pep Payca ccey, ec xapaepcecoe ypaee ee D(s) = 3s4 + 5s3 + 2s2 + 7s + 10 = 0.

oeo ypae cocaec aa Payca.

Taa Payca pepy 6.cpo coe k ri a k 1 2 1 a0 = 3 a2 = 2 a4 = 2 a1 = 5 a3 = 7 a5 = r3 = 0,6 3 a13 = 10 a23 = r4 = 2,27 4 a14 = a24 = 0 15,r5 = 0,14 5 a15 = 10 a25 = 0 Ccea e ycoa, a a a oeo epoo coa pa: a0 > 0, a1 > 0, c13 < 0, c14 > 0, c15 > 0.

pep 6.2 cceoa a ycooc c oo pep ypa, ec xapaepcecoe ypaee ee :

3s3 + 2s2 + 4s + 2 = 0;

a0 = 3; a1 = 2; a2 = 4; a3= 2;

1 = 2 > 0; ; >0.

Ccea ycoa, a a 1 > 0, 2 > 0, 3 > 0.

6.7.4 cooc ycaoac opeoc Ccea aoaecoo peypoa paccaec yco, o ycaoec pee oa oeceac aa opeoc, a epexo poecc poea o opao, .e. ccea oa ycoo (e "pacaac") epexo poecc oe ayxa c eee pee. B peax ayx Pc. 6.18 epexo poecc ycoo ccee ACP opaa c opaea, o cyae a xo cce ecye ca (t) = x(t) y(t). Paccapae aa ypae.

Ec a xo paccapaeo cce oaec cyeaa y x(t) = x0, o p ycoo ccee oce ooa epexooo poecca a ee xoe ycaaaec eoopoe ocooe aee yyc (pc. 6.18).

epexo poecc ocaec ypaee (3.8). B ycaoc pee ce pooe pa y ypaee pae :

a0 yyc = b0 x0, (6.34) oya (6.35) Paoc ys = x0 yyc = x0 (6.36) aaec ycaoc aee opeoc. Cce, ee ys 0, aac cmamuecuu, a ycaoac opeoc ys cao cce. oa paccapaec oocea opeoc oe caa S:

. (6.37) oce ao opeoc ycaoec pee eoxoo e ooe aee oea yce cce, o p ocaoo oo ae oceeo ccea caoc eycoo, .e. oae o ey peoae ycooc peoae ao opeoc. Peee o poe oo paccope a ceye pepe.

yc aaa ccea, cpyypa cxea oopo opaea a pc. 6.19.

Pc. 6.19 Cpyypa cxea cce aoaecoo peypoa Ha o cxee ; ;.

epeaoa y paoyo cce ye:

, e K oe yce cce K = k1 k2 k3.

ycaoeoc pea ypaee (6.34) pae (1 + K) yyc = K x0, oya yyc = K x0 /(1 + K), a ca cce oe caa, cooeceo:

ys = x0/(1 + K), S = 1/(1 + K).

Xapaepcecoe ypaee paccapaeo cce ee :

.

Ta a ce oe xapaepcecoo ypae peeo opa ooe, o coaco pep ycooc ypa ccea ye ycoa, ec oec epaeco:

, oopoo oo opee oe yce, .e.:

.

Bea aaec pee oeo yce. ycooc cce eoxoo ocaoo, o oe yce cce ee peeoo ae K < Kp. Ec T1= T2 = T3, o Kp = 8 , ceoaeo, K < 8. Ec e oye ao opeoc aa ca S < 0,01 (S < 1 %), o oyaec K > 100. Papeee oo oa ec oo ocox aa. y eo papee pa, a, apep, oo e ocoe pee T1, T2, T3 oc peyeoo ae oea yce.

Haoee o y papee aoo oa o eee cpyypo cxe, eee ooex ce.

B oe cyae ccea aaec acmamueco ooceo eoopoo oyaeo oec f, ec p f = const ycaoeec aee opeoc ys e ac o ae f. B ao ccee oo pcycoa eppyee eo.

caoac opeoc pee opao ocooo paccoacoa paa y.

6.7.5 Oac ycooc Ha ycooc cce aoaecoo peypoa oaa e apaep cce, o ao o o a pepe, paccopeo e.

eoepec opa acoc ycooc o apaepo cce aaec oac ycooc ee paccopee . A. Bepac. ocpoee oace ycooc ec o aoee ex pa peyao cceoa ycooc cce.

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