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

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ecoa y Ta opao, p oae a xo eppyeo ea ocooo eceaeo oye xoa oopaa yeaec o ecoeoc c ocoo copoc, .e. oeo ocoeoc ec o a, o epexoa y e ee ycaoeoc (p t ) oeoo ae. o coco ec po paoo o acaecx cce aoaecoo peypoa, coepax eppyee eo, o caecx cce, oope e coepa oo ea.

Pea a -y ec cyeao ye c ayo.

T 5.2.3 eaoe epepyee eo paee eaoo epepyeo ea y(t) = kx(t), (5.15) .e. eee xoo oopa poopoao copoc ee xoo oopa. B oepaopo ope ypaee ee y(s) = ksx(s), oya epeaoa y Y (s) W (s) = = ks. (5.16) X (s) acoe xapaepc, pa oopx pecae a pc. 5.5:

AX W(i) = k i = kei/2;

(5.17) AX M() = k;

(5.18) X () =. (5.19) a) ) Im ) M = 0 0 Re Pc. 5.5 acoe xapaepc eaoo epepyeo ea:

a AX;

X;

AX Ta opao, AX po poopoaa acoe, a X e ac o aco paa. Ceoaeo, oopa AX p > 0 coaae c ooeo e o oc.

epexoa y eaoo epepyeo ea ee :

h(t) = k1(t) = k(t), (5.20) .e. pecae coo -y c oa, pao k.

Becoa y pecae coo pooy o -y:

w(t) = k(t). (5.21) B ppoe eao epepyx ee e cyecye, a a p M(), a o pea oe paec pye apoece ca c acoo, oe aco cpea aoo oea. Heocyecoc eaoo ea a ae epexoo y, oopa paa -y ecoo y, pao pooo -y.

5.2.4 Peaoe epepyee eo Bcpeac e, oope peapy oo a copoc ee xooo caa. O ocac ypae ceyeo a aac pea epepy:

Ty(t) + y(t) = Tx(t). (5.22) pepo aoo ea ec RC-eoa (pc. 5.6).

R Ux eo J Ux J C Pc. 5.6 RC-eoa M a) i Im() ) ) T T Re() = 0 T 0 T Pc. 5.7 acoe xapaepc peaoo epepyeo ea:

a AX;

X;

AX epeaoa y ee :

Ts y(s) W (s) = =. (5.22) x(s) 1+ Ts acoe xapaepc, pa oopx pecae a pc. 5.7:

AX Ti T W (i) = = ei( / 2-arctgi) ;

(5.23) 1+ Ti T 2 + AX T M () = ;

(5.24) T 2 + X () = - arctgT. (5.25) peaoo epepyeo ea p yee aco ayo-acoa T xapaepca opacae, o ee epx pee opae eo.

T ao-acoa xapaepca p yee aco yeaec o o y.

T ooex aco W(i) pecae coo oyopyoc aepo c T T epo oe. oaaeca ae W(i) poyox oopaax 2T Ti(1- Ti) TT2 T W (i) = Re() + i Im() = = + i.

2 (1+ Ti)(1- Ti) 1+ T 2 1+ T T oyee ae Re() Im() oca ypaee opyoc payca c 2T T epo oe :

2T 2 T T + [i Im()]2 = Re() - 2T Re() 2T 2 TT2 T T T - + =.

2 2T 1+ T 2 2T 1+ T 2 Pacpa co, oyae oeco, oopoe oaae, o AX eceo pecae coo oyopyoc.

coy aoc aecx xapaepc, oyae ypaee epexoo y oepaopo ope o (3.39):

Ts T 1 h(s) = =.

1 + Ts s T 1 + s T pe opaoe peopaoae aaca oceey pae, oyae ypaee epexoo y o peeo oac:

T h(t) = e-t / T. (5.26) T Becoa y axoc a pooa o epexoo y T w(t) = - e-t / T. (5.27) T pa epexox xapaepc opae a pc. 5.8.

x x a) ) (t) 0 t t w h TA T t t TA T Pc. 5.8 epexoe xapaepc peaoo epepyeo ea:

a epexoa y;

ecoa y Ha pc. 5.8, a cpae oaa epexoe y eaoo 1 peaoo epepyx ee. B cy ep peax ee eee xoo oopa epexoo y pocxo oceeo, a e cao, a cyae eaoo ea. oo, o p coca peaoo ea coca eaoo, eoxoo oopeeo yea oe epea T yea ocoy pee T a, o x poeee TT ocaaoc oco.

5.2.5 opcpyee eo opcpy eo aaec eo, ocaeoe ypaee dx(t) y(t) = k x(t) + T. (5.28) dt Taoe eo oe oyeo peyae apaeoo coee yceoo eaoo epepyeo ee. Oo xapaepyec y apaepa: oeo epea k ocoo pee T.

epeaoa y W(s) = k(1 + Ts). (5.29) aea (5.28) s = i ooe oy acoe xapaepc opcpyeo ea, pa oopx oaa a pc. 5.9:

AX W (i) = k(1+ iT ) = k 1+ (T )2 ei arctgT ;

(5.30) AX M() = k 1+ (T )2 ;

(5.31) X () = arctgT. (5.32) a) ) ) M i Im() W(i ) k / = 0 k Re() Pc. 5.9 acoe xapaepc opcpyeo ea:

a AX;

X;

AX a o pao, ayo-aoa xapaepca pecae coo py, apaey o oc epeceay ecey oc oe Re = k.

epexoe xapaepc oya eocpeceo ypae (5.28):

- epexoa y xoo ca x(t) = 1(t), a xoo ca h(t) = k(1(t) + T(t));

(5.33) - ecoa y xoo ca x(t) = (t), a xoo ca w(t) = k((t) + T(t)). (5.34) paec opa ooo oo epexoy y, oopa pecaea a pc. 5.10.

h k t Pc. 5.10 epexoa y opcpyeo ea 5.2.6 eo coo aaa pepo ea coo aaa ec pacopep (pc. 5.11) v L Pc. 5.11 Cxea pacopepa Ec a xoy oopay p pacxo aepaa aae pacopepa, a a xo pacxo aepaa oe pacopepa, o xoo ca ye oop xoo ca x(t) c aaae, pa pee e aepaa o eca L opy o eca py, pe =. paee ea coo aaa v y(t) = x(t ). (5.35) epeaoa y oyaec peyae peopaoa aaca (5.35):

W(s) = es. (5.36) acoe xapaepc:

AX W (i) = e-i ;

(5.37) AX M () = 1;

(5.38) X () = -. (5.39) pa acox xapaepc opae a pc. 5.12.

M ) a) i Im() ) W(i ) arctg() = Re() Pc. 5.12 acoe xapaepc ea coo aaa:

a AX;

X;

AX Ta a M() = 1, a ocaae o ae xox oea po poopoao acoe c oeo poopoaoc pa pee coo aaa, o oopa AX pecae coo opyoc eoo payca c epo aae oopa.

epexoe xapaepc oyac ocaoo cooecyx xox cao ypaee ea (5.35):

- epexoa y h(t) = 1(t ), (5.40) - ecoa y w(t) = (t ). (5.41) pa epexox xapaepc opae a pc. 5.13.

x x a) ) 0 t t w h 0 t t Pc. 5.13 epexoe xapaepc ea coo aaa:

a epexoa y;

ecoa y 5.2.7 Aepoecoe eo epoo opa Aepoecoe eo epoo opa aaec ae epo. Oo ocaec epea ypaee epoo opa ee e oeae xapaep epexooo poecca. pepo ax ee oe cy a epeca e, aa copoee eoc, eoe oe.

eoe epeaoe ypaee ee Ty(t) + y(t) = kx(t), (5.42) e T - ocoa pee ea;

k - oe yce, k > 0, T > 0.

ocoa pee xapaepye epooc ea ac o e acc copoe eoc - e oe acca, copoee eoc, e oe epooc ea oe T.

epeaoy y oya ypae (5.42) y(s) k W (s) = =. (5.43) x(s) Ts + acoe xapaepc, pa oopx pecae a pc. 5.14:

- AX k k W (i) = = e-iarctgT ;

(5.44) Ti + T 2 + - AX k M () = ;

(5.45) T 2 + - X () = -arctgT. (5.46) ) ) a) M Im k k Re Pc. 5.14 acoe xapaepc aepoecoo ea epoo opa:

a AX;

X;

AX Ayo-acoa xapaepca aepoecoo ea epoo opa a yeo acoe paa oey yce k, c yeee aco oa oooo yeaec, acoec cpec y.

ao-acoa xapaepca p yee aco o 0 o eec o 0 o -.

Ceoaeo, oopa AX > 0 eo e eepo apae pecae k coo oyopyoc aepo k c epo oe, oopa ocaec ypaee 2 k k +[Im()]2 =. (5.47) Re() - 2 oaaeco oceeo oeca aaoo oaaecy oooo pae peaoo epepyeo ea. ae eceo o ace AX aec x ope pae k kT Re() = ;

Im() = 2 1+ T 2 1+ T ocac (5.47).

paee epexoo y oya a peee ypae p x(t) = k(t) oepaopo ope k 1 C0 C h(s) = y(s) = = +.

Ts +1 s s s + T T epexo opay, oya paee epexoo y o peeo oac h(t) = k[1- e-t / T ]. (5.48) Becoy y oo oy a pooy o epexoo y h w k k T T = t2 t T t1 t t t Pc. 5.15 epexoe xapaepc aepoecoo ea epoo opa:

a epexoa y;

ecoa y k w(t) = e-t / T. (5.49) T pa epexox xapaepc opae a pc. 5.15.

a o pao, epexoe xapaepc peca coo oooe y pee, o oo opee ae apaep, a oe yce, pa ycaoeyc ae h();

ocoy pee, pay epay pee T o o aca epexoo y o o epecee acaeo c ee acoo (pc. 5.15, a).

5.2.8 epoo-opcpyee eo epoo-opcpyee eo aa ae epo--epepy ypy eo, ocaec oo epea ypaee epoo opa Ty (t) + y(t) = k[T0 x (t) + x(t)]. (5.50) T Cyece apaepo ea ec oe =. Ec < 1, o eo o T co coca paec eppyey epooy e, ec e > 1, o eo e epepy e.

epeaoa y ea:

T0s + W (s) = k. (5.51) Ts + acoe xapaepc oya peyae ae s = i:

AX T0i +1 T022 + W (i) = k = k ei(arctgT0-arctgT) ;

(5.52) Ti + T 2 + a) ) ) M Im k k = = k k Re Pc. 5.16 acoe xapaepc epoo-opcpyeo ea > 1:

> a AX;

X;

AX a) ) ) M Im k k k = = Re k Pc. 5.17 acoe xapaepc epoo-opcpyeo ea < 1:

a AX;

X;

AX AX T022 + M () = k ;

(5.53) T 2 + X () = arctgT0 - arctgT. (5.54) pa acox xapaepc > 1 < 1 opae cooeceo a pc.

5.16 5.17.

coy aoc aecx xapaepc, acac ypae epexoo ecoo y, cooeceo T h(t) = k1+ -1 e-t / T ;

(5.55) T k T w(t) = - -1 e-t / T, (5.56) T T x pa > 1 < 1 opae a pc. 5.18. 5.19.

a) ) h w k > t k 1- k T t Pc. 5.18 epexoe xapaepc epoo-opcpyeo ea > 1:

a epexoa y ;

ecoa y a) ) h w k k 1- T < k t t Pc. 5.19 epexoe xapaepc epoo-opcpyeo ea < 1:

a epexoa y;

ecoa y 5.2.9 Aepoecoe eo opoo opa paee aepoecoo ea opoo opa yoo aca e T1T2 y (t) + (T1 + T2 ) y (t) + y(t) = kx(t), (5.57) e T1, T2 ocoe pee;

k oe yce;

T1, T2, k > 0.

oce peopaoa (5.57) o aacy [T1T2s2 + (T1 + T2)s +1]y(s) = kx(s), oya epeaoa y ea paa:

k k W (s) = =. (5.58) (T1s +1)(T2s +1) T1T2s2 + (T1 + T2)s + Aepoecoe eo opoo opa oo cpyypo peca e oceoaeoo coee yx ee epoo opa c oco pee T1 T (pc. 5.20), ooy oo e oocc cy eeapx. op xapaepcecoo ypae ecee.

k 1 y x T1s +1 T2s + Pc. 5.20 Cpyypa cxea aepoecoo ea opoo opa acoe xapaepc, pa oopx opae a pc. 5.21:

AX k k W(i) = = e-i(arctgT1+arctgT2) ;

(5.59) (T1i+1)(T2i+1) (T122 +1)(T222 +1) AX k M () = ;

(5.60) (T122 +1)(T222 +1) X () = -(arctgT1 + arctgT2). (5.61) cpae ypo oaa xapaepc ea epoo opa.

Ayo-acoa xapaepca p ee aco o 0 o eec o k o 0. ao-acoa xapaepca eec o 0 o . oopa ayo-aoo xapaepc e 4- 3- apaax. Cpaa acoe xapaepc ea epoo opa, o, o oaee opoo ea epoo opa yeae epooc oea, yeae oy yeae ocaae o ae.

paee epexoo y oepaopo ope ee C0 C1 C k h(s) = = + +.

(T1s +1)(T2s +1) s s s +1/ T1 s +1/ T a) ) ) Mk Im k k = Re -/ Pc. 5.21 acoe xapaepc aepoecoo ea opoo opa:

a AX;

X;

AX a) ) w h k t t Pc. 5.22 epexoe xapaepc aepoecoo ea opoo opa:

a epexoa y;

ecoa y epexo opay, oya h(t) = C0 + C1e-t / T1 + C2e-t / T2, (5.62) kT12T2 kT22T e C0 = k;

C1 = ;

C1 =.

T1 - T2 T2 -T epexoa y pecae coo eoeaey py, ey oy oy epea acoec cpeyc y() = k.

paee ecoo y:

C1 C w(t) = h (t) = - e-t / T1 - e-t / T2. (5.63) T1 T pa epexox xapaepc opae a pc. 5.22.

5.2.10 oeaeoe eo oeaeoe eo, a aepoecoe, ec eo opoo opa ocaec epea ypaee opoo opa, oopoe yoo aca e T2 y (t) + T y (t) + y(t) = kx(t). (5.64) Xapaepcecoe ypaee oeaeoo ea T2s2 + Ts +1 = oo e apy oeco copex ope, a o ye oo o cyae, ec T T < 2. Ec e 2, o op ypae ecee eo ye aepoec T T opoo opa. Xapaepc oeaeoo ea e :

- epeaoa y k W (s) = ;

(5.65) T2s2 + Ts + acoe xapaepc, pa oopx opae a pc. 5.23:

AX T -iarctg k k 1-T W (i) = = e ;

(5.66) (-T22 +1) + iT (1 - T22)2 + T AX k M () = ;

(5.67) (1 - T22 )2 + T X T () = -arctg. (5.68) 1- T Aa ayo-acoo xapaepc oaae, o p ax aex aco, oa 4 << 2, aaec eoopoe yeee AX o cpae c T aepoec eo, pe p ox aex a pae AX oec T acy. Bpeee p T = 0 AX ep pap opoo poa p ae p =.

T epexoa y oepaopo ope:

k h(s) =.

s T2s2 + Ts + B opaoe peopaoae aaca, oya a) ) ) M Im = k k Re p = p Pc. 5.23 acoe xapaepc oeaeoo ea:

a AX;

X;

AX a) ) w h k t t Pc. 5.24 epexoe xapaepc oeaeoo ea:

a epexoa y;

ecoa y h(t) = k[1+ Ae-t sin(t - )], (5.69) T T 2T e A = ;

= ;

= ;

= arctg.

A A 2T 4T2 + T w(t) = -Ae-t sin(t - ) + Ae-t cos(t - ) = (5.70) = Ae-t (cos(t - ) - sin(t - )).

pa epexox y opae a pc. 5.24.

pepo oeaeoo ea oy cy ypya exaeca ccea c cyece e acc, epoe a peyopa aco pae aa a e eepa pye.

ac cyae oeaeoo ea ec ocepaoe eo, oa xapaepcecoe ypaee ee co e op. B o cyae epeaoa y ea peopayec y k W (s) =. (5.71) T s2 + Ayo-aoa xapaepca k W (i) = (5.72) 1- T ec eceo ye c oye k M () = (5.73) 1- T ao 0, < 1/ T;

() = (5.74) -, > 1/ T, oopa oopo pacooe a eceo oyoc (pc. 5.25).

a) ) ) M Im k p = p= k T T = 0 1 Re p = T Pc. 5.25 acoe xapaepc ocepaoo ea:

a AX;

X;

AX Bpeee xapaepc:

- epexoa y h(t) = k 1- cos t ;

(5.75) T - ecoa y k w(t) = sin t (5.76) T T peca coo apoece oea (pc. 5.26). acoa p = aaec T peoaco acoo.

a) ) w h k t t Pc. 5.26 y ocepaoo ea:

a epexoa;

ecoa 5.2.11 Ocoe e Opeeee ao-aox cce (ee) o ao paee. Bce paccopee e oocc ao-ao e. Oao a pae cpeac eao-aoe e, y oopx xo o y oc epeaoo y ee ooey eecey ac. pepa ax ee c eo coo aaa, a ae e c epeao y k k(1- T0s) k W (s) = ;

W (s) = ;

W (s) =. (5.77) Ts -1 (Ts -1) T1T2s2 - (T1 + T2)s + Ocoeoc eao-aox ee o cpae c ao-ao ec o, o ee, ex oaoe AX, y x ocaae o ae oe.

Hapep, cpaa a ea aepoecoe epoo opa eo c epeaoo k ye W (s) =, ex AX oox cyax Ts - k M () =, T 2 + o X epo cyae () = -arctgT eec o y o -, a o opo () = - + arctgT eec o o -.

Heao-aoe e cpeac epecx cxeax p epeax ocox coeex.

ac cyae eao-aox ee c eycoe e, oa oo oc e ooey eecey ac. Paccopeoe e eo ec eao-ao eyco eo, aoee pacpocpae cpe eycox ee, aaec aepo eo. eycox ee e cyecye ycaoeoc pea, c eee pee p o xoo cae xoa ea cpec ecoeoc.

5.3 OCHOBHE COCO COEHEH BEHEB 5.3.1 Cpyype cxe p aae cee cce aoaecoo ypae poo coyec cpyyp aa, oco o oopoo cya ceye.

Cpyypa cxea ec paec opaee epeaoo ypae oea oaae a ococo oo paecoo pecae aoc.

a) eo ) x c x x ) x ye x1 x1 + x2 x1 x1 x2 cyaop ) x2 x Pc. 5.27 coe ooae eeo cpyypo cxe ee cpyypo cxe aac e, a ye eco, opaac e poyoo, yp oopx acaec epeaoa y ea.

Baoc ey e opaaec uuu cu co cpea, yaa apaee epea caa. Ha e cac ycooe ooaee caa.

Toa a c, oopo pocxo paeee , aaec yo.

Aepaecoe coee ecox cao opaaec e pya a c aaec cyamopo.

opae ocox eeo cpyypx cxe coyc ceye ycoe ooae (pc. 5.27):

Cocaee cpyypo cxe ec o epx ao cceoa cox oeo ypae, oa oe cocaea a ocoa aeaecoo oca, a ae cxo eco oe oea.

ao coo a cpyypa cxea, e cea pcycy oo p a coee: apaeoe, oceoaeoe c opao c. aae paccope o coee ec oyee coooe ey epeaoo ye coee epeao y oex ee.

5.3.2 apaeoe coeee ee p apaeo coee (pc. 5.28) ca xoa cex ee oao pa cay xoa cce x(t), a xo y(t) pae cye cao xoo ee.

y x W y2 y x x W y x W Pc. 5.28 apaeoe coeee ee aoo ea oepaopo ope oo aca:

y1(s) = x(s)W1(s);

y2 (s) = x(s)W2(s);

...;

yn (s) = x(s)Wn (s), oa xo ce cce n y(s) = y1(s) +... + yn(s) = x(s)[W1(s) +... +Wn(s)] = x(s) (s), (5.78) W i i= oya epeaoa y apaeoo coee n y(s) Wc (s) = = (s). (5.79) Wi x(s) i = Ta opao, nepeamoa yu cucme napaeo coeuex ee paa cye nepeamox yu omex ee.

Bpeee xapaepc, acoc, epexoy y oo oy (5.79):

n n hc (t) = L-1[h(s)] = L-1 (s) = (t), (5.80) hi hi i = i= .e. epexoa y apaeoo coee paa cye epexox y oex ee.

acoe xapaepc apaeoo coee oya cey opao:

n n Wc (i) = (i) = Re () + i Im () ;

W j j j j =1 j = n n Rec () = ();

Imc () = (). (5.81) Rei Imi i=1 i = a o (5.81), ayo-aoa xapaepca apaeoo coee oe oyea peyae coe ecex x ace AX oex ee o pay coe eopo. Ha pc. 5.29 peea cpa oye AX yx apaeo coeex ee, aax co AX.

a) ) ) Im Im Im k1 k2 k1 + k W1(i 1) 0 0 Re W2(i 1) Re Re W2(i 1) W1(i 1) W(i 1) Pc. 5.29 ocpoee AX apaeoo coee:

a AX epoo ea;

AX opoo ea;

AX apaeoo coee epoo opoo ee pepo exooecoo oea, eeo ooy cpyypy cxey, oe cy eoa apaeo paoax oox peaopo (pc. 5.30).

Cpe poy Cop Pc. 5.30 pep exooecoo oea apaeoo coee 5.2.3 oceoaeoe coeee ee p oceoaeo coee xo peyeo ea oaec a xo oceyeo (pc. 5.31).

pae xox cao oce aoo ea oepaopo ope e :

y1(s) = x(s)W1(s);

y2 (s) = y1(s)W2 (s);

...;

yn (s) = yn-1(s)Wn (s).

y1 y2 yn x W1 W2 Wn Pc. 5.31 oceoaeoe coeee ee Bxoo ca oceeo ea ec xoo ce cce: y(s) = yn(s), a epeaoa y cce coaco opeee ee y(s) yn (s) Wc (s) = =.

x(s) x(s) poo oceoaey ocaoy, oya epeaoy y oceoaeoo coee n Wc (s) = W1(s)W2(s)... Wn (s) = (s). (5.82) Wi i= Ta opao, nepeamoa yu cucme noceoameo coeuex ee paa npoueeu nepeamox yu omex ee.

acoe xapaepc eo oya (5.82), a a n i () j n n j Wc (i) = (i) = ()e, W j M j j =1 j = oa n n Mc () = ();

c () = (), (5.83) Mi i j =1 j = .e. ayo-acoa xapaepca oceoaeoo coee paa poee AX oex ee, a ao-acoa cye X oex ee.

cpa ocpoe AX yx oceoaeo coeex ee, aax co AX, peea a pc. 5.32.

epexoy y oya cey opao. Ec xoo ca x(t) = 1(t), o a xoe epoo ea ee eo epexoy y h1(t), oopa oaec a xo opoo ea. Ha xoe opoo ea oya epexoy y yx oceoaeo co a) ) ) Im Im Im k1 k2 k1 k 1+2 Re Re Re M M M1 M Pc. 5.32 ocpoee AX oceoaeoo coee:

a AX epoo ea;

AX opoo ea;

AX oceoaeoo coee epoo opoo ee eex ee. Ec cocea epexoa y opoo ea h2(t), o epexoa y coee opeec epe epa ae:

t dh1(t - ) hc (t) = () (5.84) h d d + h1(0)h2(t).

pooa paccye ae, oo oy paee epexoo y oo ca oceoaeo coeex ee.

Ceye oe, o ce oyee yepe cpae oo ee apaeoo ec.

pepo exooecoo oea, eeo cpyypy cxey oceoaeoo coee, ec o exooec poecc, oopo oee ca yac pecac e cooecyeo ea.

5.3.4 Coeee c opao c Opao c aa epeay caa c xoa ea a eo xo, e ca opao c aepaec cypyec c e cao. Cpyypa cxea coee c opao c opaea a pc. 5.33.

Ec x1 = x + xoc, o c aaec ooeo, ec e x1 = x - xoc, o opaeo.

oa epeaoo y coee c ooeo opao c xoe ca aoo ea oepaopo ope acac a:

y(s) = x1(s)W p (s);

x1(s) = x(s) + xoc(s);

xoc(s) = y(s)Woc(s).

ca oyeo cce x1(s) xoc(s), oya y(s) = x(s)W p (s) + y(s)Woc (s)W p (s) ;

y(s)(1-Woc (s)W p(s)) = x(s)W p (s), oya nepeamoa yu coeueu c nooumeo opamo c ee y x1 Wp(s) x xoc Woc(s) Pc. 5.33 Coeee c opao c W p(s) y(s) Wc (s) = =. (5.85) x(s) 1-W p (s)Woc (s) coeueu c ompuameo opamo c epeaoa y oc aao opao opeeec ooaeo e paee W p (s) Wc (s) =. (5.85) 1+ W p (s)Woc(s) Ha pae aoee pacpocpae c cce c opaeo opao c, oocc, apep, ce oooype cce aoaecoo peypoa, pe po e pacooe oe, a opao peyop.

5.3.5 epeaoe y ayo cce Cpyypa cxea oooypo cce aoaecoo peypoa peea a pc. 5.34.

f0(t) ya y(t) Wp(s) Wo(s) y Pc. 5.34 Cpyypa cxea oooypo cce B paceax cce aoaecoo peypoa coy p ocox a epeaox y. y opeec cey opao.

ao epeaoo ye ec nepeamoa yu no aay peyupoau ya - y(t), f0(0) = 0 :

Wo (s)Wp (s) W (s) =. (5.86) 1+Wo (s)Wp (s) epeamoa yu aymo cucme ouu, .e. o aay ya - (t), e (t) = ya(t) - y(t) oa peypoa f0(t) = 0:

W(s) =. (5.87) 1+ Wo (s)Wp (s) epeamoa yu aymo cucme no oyaey oecmu, .e. o aay f0(t) - y(t)= 0 :

Wo (s) W (s) =. (5.88) f 1+Wo (s)Wp (s) Aa epeaox y ayo cce oaae, o aeae y x o o e, a ce pa. ayo cce oo aca e p pyx epeaox y, apep, o o oyaey oec.

Xapaepcecoe ypaee ayo cce axoc aeaee epeaoo y acaec e 1+Wo (s)Wp (s) = 0. (5.89) op oo ypae pa oca si epeaoo y ayo cce.

aece coca poecco, poeax ayo ccee, cyeceo oac o aox paoyo e, cocoe ex e cax ee. Ta a epeaoa y paoyo e acaec e Wp.c(s) = Wo(s)Wp(s), o aa epeaoa y oe acaa a Wp.c (s) W (s) =.

1+Wp.c (s) 5.3.6 paa peopaoa cpyypx cxe Peae oe oaa coo cpyypo. poee oa epeaox y cox oeo cxeax ocaec a ce peopaoa x cpyypx cxe pe oco a coee.

pep paoc peopaoa cpyypo cxe aaec o, o xoe xoe ca peopayeoo yaca o oce peopaoa oao.

Ha pae peo cpeac cxe, oopx oo cpay e e o o coee, a pao, ec, a aaee, epepece c. B o cyae oae eoxooc epecao epeoca cyaopo yo.

Hapep, peyec ocyec epeoc ya epe eo o apae pacpocpae caa (pc. 5.35, a).

peopaoa oe yaco, ee ypo, oop ee o xoo ca x(t) a xox x(t) y (t). Tpeyec epeec ye "1" epe eo "2" c epeaoo ye W(s).

poco epeoc po cxee, opaeo a (pc. 5.35, ). a cxea e cooecye cxoo, a a ocycye xoo ca x(t), o eec a caa y(t), pe y(s) = x(s) W(s), ceoaeo, pee cxe cxoo eoxoo ooy e a xoe y(t) eo c epeaoo ye. Toa W (s) oya cxey (pc. 5.35, ), cooecyy cxoo.

a) ) (2) (2) y (1) y (1) x x W(s) W(s) y x ) (2) y (1) x W(s) 1/W(s) y Pc. 5.35 pep epeoca ya epe eo:

a o peopaoa;

epaoe peopaoae;

oce peopaoa Ta opao, epeoc ya epe eo c epeaoo ye W(s) o apae pacpocpae caa copooaec oee ooo e ea, eeo epeaoy y.

W (s) Paccope pep ec oaaeco paa epeoca ya epe eo.

Ocae paa epeoca poc e oaaeca cey opao.

1 epeoc ya epe ye ocyecec e ooex peopaoa (pc.

5.36).

a) ) x (1) (2) x x x (2) (1) x x x x Pc. 5.36 epeoc ya epe ye:

a o epeoca;

oce epeoca 2 epeoc cyaopa epe cyaop pooc e ooex peopaoa (o epee ec caaex cya e eec) (pc. 5.37).

x1 + x2 x1 + x2 + x3 x1 + x3 x1 + x2 + x x1 x x2 x3 x x Pc. 5.37 epeoc cyaopa epe cyaop:

a o peopaoa;

oce peopaoa 3 p epeoce ya epe cyaop o apae caa ooo e peopaoaoo yaca oec ooeoe eo c epeaoo ye (-1) (pc. 5.38).

) a) x1 x1 + x2 x1 x1 + x x x x x Pc. 5.38 epeoc ya epe cyaop:

a o peopaoa;

oce peopaoa 4 p epeoce cyaopa epe ye o apae caa ooo e oec eo c epeaoo ye +1 (pc. 5.39).

) x1 + x x1 + x2 a) x x x1 + x x x x1 + x Pc. 5.39 epeoc cyaopa epe eo:

a o peopaoa;

oce peopaoa 5 epeoc ya epe eo o apae caa po oe ooeoo ea c epeaoo ye (pc. 5.40).

W (s) a) ) x x W(s) x W(s) x W(s) W(s) x 1/W(s) x Pc. 5.40 epeoc ya epe eo:

a o peopaoa;

oce peopaoa 6 p epeoce ya epe eo po apae caa oec ooeoe eo c epeaoo ye W(s) (pc. 5.41).

a) ) x W(s) x W(s) x x W(s) W(s) x W(s) W(s) x W(s) Pc. 5.41 epeoc ea epe ye:

a o peopaoa;

oce peopaoa 7 epeoc cyaopa epe eo o apae caa copooaec oee ooeoo ea c epeaoo ye W(s) (pc. 5.42).

a) ) x1 + x2 W(s) W(x1 + x2) x1 x1 W(s) W(x1 + x2) x W(s) x Pc. 5.42 epeoc cyaopa epe eo:

a o peopaoa;

oce peopaoa 8 epeoc cyaopa epe eo po apae caa po oe ooeoo ea c epeaoo ye (pc. 5.43).

W (s) ) a) x1 W(s) x1W + x x1 W(s) x1 W + x 1/W(s) x x Pc. 5.43 epeoc ea epe cyaop:

a o peopaoa;

oce peopaoa 9 Becee eea po c po oe ooex ee, po e ooeo W2(s) (pc. 5.44).

W (s) a) ) W1(s) W1(s) 1/W2(s) y y x x W2(s) W2(s) Pc. 5.44 Becee eea po c:

a o peopaoa;

oce peopaoa 10 Becee eea py c copooaec oee oo opo px ex ee c epeaoo ye W2(s) ooeo e ea c epeaoo ye (pc. 5.45).

W (s) a) ) W1(s) W1(s) W2(s) y y x x 1/W2(s) W2(s) Pc. 5.45 Becee eea py c:

a o peopaoa;

oce peopaoa 11 Becee eea opao c copooaec oee po e eea c epeaoo ye W2(s), a ooeo e ea c epeaoo ye (pc. 5.46).

W (s) a) ) x y W1(s) x y 1/W2(s) W2(s) W1(s) W2(s) Pc. 5.46 Becee eea opao c:

a o peopaoa;

oce peopaoa 12 Becee eea opay c copooaec oee opao c ea c epeaoo ye W2(s), po e ea c epeaoo ye, ooeo ea c epeaoo ye W2(s) (pc. 5.47).

W2 (s) a) ) x y x y W1(s) W2(s) 1/W2(s) W1(s) W2(s) Pc. 5.47 Becee eea opay c:

a o peopaoa;

oce peopaoa puep 5.1 aca epeaoy y coee, opaeoo a pc. 5.48.

W2(s) y x W1(s) W3(s) W5(s) W4(s) W6(s) Pc. 5.48 Cpyypa cxea eoopoo oea W5(s) W (s) = W1(s)[W2(s) + W3(s) + W4 (s)].

1-W5(s)W6(s) puep 5.2 peopaoa cpyypy cxey (pc. 5.49) aca epeaoy y a) 1 1 2 W1(s) W3(s) W2(s) ) 2 1 W1(s) W3(s) W2(s) 1/W3(s) Pc. 5.49 Cpyypa cxea eoopoo oea c epepec c:

a o peopaoa;

oce peopaoa W1 s W2 s W3 s W s =.

1+W1 s W2 s + W2 s W3 s + W1 s W2 s W3 s 5.3.7 opya Mecoa p oe epeaox y cox cpyypx cxe e cea ae yoo ooac paa peopaoa. B 1953 . Mcoo o peoeo pao ce epeaoo y ey y aa ya. o pao paaec ceye opyo b r np (s) pi (s)) W (1+W j j =1 i= Wmn (s) =, (5.90) b pi (1+W (s)) i= r e Wmn(s) epeaoa y ey ya m n;

(s) cya r epeaox Wnp j j = y pax px ye ya m ye n;

Wpi (s) - epeaoa y paoyoo oypa, a co ao, cooecy opaeo opao c;

- poeee, aee ce aye oyp cce;

* - a ooaae cee co cex eo, coepax poee epeaox y ox ex e ee, a e c W(s) = 1.

puep 5.3 aca epeaoy y cce (pc. 5.50) o aay (x y).

y x W1(s) W2(s) W3(s) W4(s) Pc. 5.50 Cpyypa cxea exooecoo oea B cpyypo cxee oea o aay (x y) eec o po y (r = 1) c epeaoo ye Wp1(s) = W1(s) W2(s) a ayx oypa (b = 2) c epeao y paoyx ee c opae opa c:

Wp 1 (s) = W1(s)W2(s)W3(s)W4 (s) ;

Wp 2 (s) = W1(s)W4(s).

oca oyeoe paee (5.90), oya:

W1(s)W2(s)(1+W1(s)W4(s)) (1+W1(s)W2(s)W3(s)W4(s)) Wx-y (s)=.

(1+W1(s)W4(s))(1+W1(s)W2(s)W3(s)W4(s)) Pacpa co ca e, coepae epeaoe y ox ee, ooaeo oya:

W1(s)W2(s) Wx-y (s) =.

1+W1(s)W2(s) +W1(s)W2 (s)W3(s)W4(s) 5.4 TOBE AOH PEPOBAH aoo peypoa aaec ypaee, ocaee acoc ey xoo peyopa y(t) = y(t) - ya eo xoo xp(t).

Bce ao peypoa opaec a pocee: poopoa (), epa (), epea () poee: poopoao epa (), poopoao-epea (), poopoao epao-epea ().

He poc xapaepca cex aoo peypoa c o pe x aecx coc.

5.4.1 poopoa ao peypoa poopoa ao peypoa ocaec ypaee xp (t) = -S1y(t), (5.91) e S1 apaep acpo.

a () opaae o a, o peyop aec ccey o py opaeo opao c.

poopoa peyopo oe cy ooe yceoe eo c ee oeo yce, eoe opaey opay c o ooe oey. B c c aece xapaepc -peyopa ocoo coaa c xapaepca yceoo ea e :

- epeaoa y W (s) = -S1;

(5.92) - acoe xapaepc, pa oopx opae a pc. 5.51:

AX W (i) = -S1 = S1ei ;

(5.93) AX M () = S1;

(5.94) X () =. (5.95) a) ) ) M Im s -s 0 0 Re Pc. 5.51 acoe xapaepc -peyopa:

a AX;

X;

AX epexoa y (pc. 5.52, a):

h(t) = xp(t) = -S11(t) (5.96) Becoa y (pc. 5.52, ):

w(t) = -S1(t) (5.97) y y a) ) (t) 0 t t w -h s s1 (t) 0 t t Pc. 5.52 epexoe xapaepc -peyopa:

a epexoa y;

ecoa y oo, o c eoca ococa oo oo aoa peypoa, eoxoo paccope epexo poecc ayo cce.

epexo poecc ACP c -peyopo, opae a pc. 5.53, xapaepyec e, o eec caeca oa pey poa, paa yyc ya. eceo, o eopee o oeo ae y oo aca:

y yyc ya t Pc. 5.53 epexo poecc ACP c -peyopo lim y(t) = lim sy(s) = lim sx(s)Wc (s), t s0 s 1 Wo (s) Wo (s) a a x(s) = ;

Wc(s) = =, s 1+Wo (s)Wp(s) 1+Wo (s)S o Wo (s) ko s lim y(t) = = =, (5.98) t s 1+Wo (s)S1 1+ ko S ec limWo (s) = ko.

s Ta opao, caeca oa peypoa ac o oea yce oea apaepa acpo peyopa. pe caeca oa e ee, e oe aee apaepa acpo S1. oo, o a oa ocycoaa, .e. yyc = 0 p ko 0, eoxoo, o S1. Ceoaeo, auue cmamueco ouu peyupoau emc opauecu eocmamo ACP c nponopuoa peymopo.

5.4.2 epa ao peypoa epa ao peypoa ocaec ypaee t xp (t) = -S0 y()d, (5.99) xp (t) = -S0y(t) (5.99) e S0 - apaep acpo peyopa.

epa peyopo oe cy eppyee eo c epee epeao oeo, eoe opaey opay c oey.

aece xapaepc -peyopa e :

- epeaoa y S W(s) = ;

(5.100) s - acoe xapaepc, opaee a pc. 5.54:

-S AX W (i) = = (S0 / )ei(- / 2) = (S0 / )ei / 2 ;

(5.101) i S AX M () = ;

(5.102) X () =. (5.103) a) ) Im M ) / 0 Re Pc. 5.54 acoe xapaepc -aoa peypoa:

a AX;

X;

AX y y a) ) 0 t t -h -w s t t Pc. 5.55 epexoe xapaepc -aoa peypoa:

a epexoa y;

ecoa y epexoe xapaepc, pa oopx pecae a pc. 5.55:

epexoa y h(t) = S0t;

(5.104) ecoa y w(t) = S0 (5.105) y ya t Pc. 5.56 epexo poecc APC c -peyopo epexoo poecc ACP c -peyopo, opae a pc. 5.56, xapaepyec ocyce caeco o peypoa, ao aee ooe peypyeo e o ycaoeoc ae o cpae c py aoa peypoa, ao peee peypoa.

a ococo epaoo peyopa ec ocyce caeco o peypoa. eceo:

1 Wo (s) lim y(t) = lim sy(s) = lim s = 0.

t 0 s0 s0 S s 1+ Wo (s) s 5.4.3 epea ao peypoa epea ao peypoa ocaec ypaee xp (t) = -S2y (t), (5.106) e S2 apaep acpo, oopoe ec ypaee eaoo epepyeo ea. Ha pae epea ao oe peaoa peo opeeeo epae aco. epeaa cocaa oc ao peypoa oo, o ye cpoece peyopa, a a o cyae peyop peapye e a acooe aee peypyeo e, a a copoc ee ee. epea peyop e peec peypoa, a a p o ocoo ae peypyeo e xoo ca aoo peyopa pae y.

aece xapaepc -aoa peypoa:

epeaoa y W (s) = -S2s ;

(5.107) acoe xapaepc, opaee a pc. 5.57:

i AX W (i) = -S2i = S2e ;

(5.108) AX M () = S2 ;

(5.109) X () =. (5.110) Im a) ) ) M = Re Pc. 5.57 acoe xapaepc -aoa peypoa:

a AX;

X;

AX y h a) ) s2 (t) 0 t t Pc. 5.58 epexoa y -aoa peypoa:

a eoe oece, epexoa y epexoe xapaepc:

epexoa y h(t) = S2 (t);

(5.111) ecoa y w(t) = S2(t), (5.112) pa oopx opae a pc. 5.58.

epeaa cocaa yacye oo cox aoax peypoa yye aeca epexooo poecca.

5.4.4 poopoao-epea ao peypoa poopoao-epea ao peypoa o-caec ypaee. (5.113) xp (t) = -[S1y(t) + S2y (t)] o peyop o cyecy coco yx apaeo ex cocax:

poopoao epepye.

aece xapaepc -peyopa:

epeaoa y ;

(5.114) W (s) = -(S1 + S2 s) acoe xapaepc, pa oopx opae a pc. 5.59:

2 AX (5.115) W(i) = -(S1 + S2 i) = S1 + S2 2 ei(+arctg(S2/ S1));

2 AX ;

(5.116) M () = S1 + S2 X. (5.117) () = arctg(S2/ S1) + a) ) ) M Im W(i ) s s = Re Pc. 5.59 acoe xapaepc -peyopa:

a AX;

-- X;

- AX epexoa y, pa oopo opae a pc. 5.60:

h(t) = -S11(t) - S2(t). (5.117') Becoa y:

. (5.118) w(t) = -S1(t) - S2 (t) h(t) s t Pc. 5.60 epexoa y -peyopa C o pe aeca poecca peypoa ayo ACP poopoao epea peyop oaae ocoeoc oox aoo peypoa (pc.

5.61). Hae oec o pooo o y(t) yeae cpoece peyopa, aoap ey yeaec aeca oa o cpae c poopoa peyopo.

B ycaoxc peax, oa y' = 0, peyop ee ce a o peyop. Bea caeco o ocaec ao e, a cyae pee peyopa, eceo:

1 Wo(s) Ko (5.119) lim y(t) = limsy(s) = lims = t s0 s s 1+Wo(s)(S1 + S2s) KoS1 + y yyc ya t Pc. 5.61 epexo poecc ACP c -peyopo 5.4.5 poopoao-epa ao peypoa poopoao-epa ao peypoa ocaec ypaee t (5.120) xp (t) = -(S1y (t ) + S y ()d) pecae coo apaeoe coeee poopoao epao cocax. aece xapaepc -pey-opa:

epeaoa y S W(s) = - S1 + ;

(5.121) s acoe xapaepc (pc. 5.62):

S AX W(i) = - S1 + ;

(5.122) i 2 S1 2 + S AX M()= ;

(5.123) S X. (5.124) () = + arctg 2 S epexoa y (pc. 5.63, a):

h(t) = -(S1 1(t) + S0 t). (5.125) Becoa y (pc. 5.63, ):

w(t) = -(S1(t) + S0). (5.126) M a) ) ) Im W(i ) s s1 Re Pc. 5.62 acoe xapaepc -peyopa:

a - AX;

- X;

AX h a) w ) s s 0 t t Pc. 5.63 epexoe xapaepc:

a epexoa y;

ecoa y poopoao-epa peyop coeae cee ococa - -aoo peypoa, a eo: poopoaa cocaa oeceae ocaooe cpoece peyopa, a epaa cocaa pye caecy oy peypoa. epexo poecc ACP c peyopo opae a pc. 5.64.

B aae poecca peypoa ocoy po pae poopoaa cocaa, a a epaa cocaa ac e oo o acooo ae, o o pee. C yeee pee opacae po epao cocae, oeceae ycpaee caeco o, .e.

1 Wo (s) y(t) = sy(s) = s = lim lim lim t -> s->0 s->0 s 1+ Wo (s)(S1 + S0 / s) (5.127) sWo (s) = = 0.

lim s-> s + Wo (s)S1s + S oopo apaepo acpo S0 S1 oo e ye ec ao cocae. B acoc, p S0 = 0 oyaec -peyop, a p S1 = 0 - peyop.

y t Pc. 5.64 epexo poecc ACP c -, - -peyopa 5.4.6 poopoao-epao-epea ao peypoa poopoao-epao-epea ao peypoa ocaec ypaee t xp (t) = -(S1y(t) + S y()d + S2y(t)). (5.128) aece xapaepc -peyopa:

epeaoa y S W(s) = (S1 + + S2s). (5.129) s acoe xapaepc (pc. 5.65):

S AX W(i) = (S1+ + S2s);

(5.130) i S1 2 + (S0 - S22) AX ;

(5.131) M ( ) = S X. (5.132) ( ) = + arctg S0 - S epexoe xapaepc:

epexoa y, p t > h(t) = (S1 + S0t + S2 (t));

(5.133) ecoa y w(t) = (S1 (t) + S0 + S2 '(t)). (5.134) a) ) ) M Im 3 / s s1 0 Re / s2 s Pc. 5.65 acoe xapaepc -peyopa:

a AX;

X;

AX pa epexoo y -peyopa pecae a pc. 5.66.

h s 0 t Pc. 5.66 epexoa y -peyopa -peyop coeae cee ococa cex pex pocex aoo peypoa: cooe cpoece aoap a yca o pooo o y(t) ocyce caeco o, oopoe oeceae epaa cocaa (pc. 5.67).

y t Pc. 5.67 epexoe poecc ACP c pa aoa peypoa Heoxoo oe, o peee peyopo c epea coca, ecop a x ococa, e cea eecoopao, a oa eoyco. Ta, oeo c o aaae o aay peypoa ecoeo o oece o pooo o peypyeo e, a a o yc ye ocya peyop o cee pee coo aaa oce pxoa oye, a oopoe oee oy aoc oe ooe. oee oo, ax cyax -peyop oe "pacaa" oe ccea oepe ycooc.

5.5 TPEHPOBOHE AAH 1 e aac oee ee cce, oopx pocxo peopaoae xox cao xoe. Ec epeaoa y ea ee poco po, o aoe eo oocc pye ox eeapx ee, ypae oopx oo oy epeaoo ypae a2 y (t) + a1 y (t) + a0 y(t) = b1 x (t) + b0 x(t), ppaa e e oe y.

Paa ceye e: yceoe, epaoe, eaoe peaoe epepye, coo aaa, aepoecoe epoo opa, aepoecoe opoo opa, oeaeoe. aoe epecex ee paccapaec c o aaa x aecx xapaepc.

A ae e ocac ooe epea ypae?

B oey eaoe epepyee eo ec e peayeo?

C Ha ae py ec oe e?

2 p aae cee cce aoaecoo ypae poo coyec cpyyp aa. B o cpyypo cxee oy pcycoa oo p a coee: oceoaeoe, apaeoe, coeee c opao c. aee epeaox y oex ee ooe aca epeaoe y coee ocpo x acoe xapaepc.

Peae oe oaa coo cpyypo, x ec, a aaee, epepece c, oope eoxoo paa, coy paa peopaoa cpyypx cxe.

A ae epeaoe y oo aca oooypo cce aoaecoo peypoa?

B aa epeaoe y ee W1(s) = k;

W2 (s) = 4. aca acoe Ts xapaepc oceoaeoo apaeoo coee.

C epeoc ax eeo p peopaoa cxe pooc e ooex peopaoa?

3 eea oooypo cce aoaecoo peypoa c oe peyop. Bce ao peypoa opaec a pocee:

poopoa, epea, epa poee: poopoao epa, poopoao-epea, poopoao-epao- epea. Bce ao peypoa paccapac c o pe x aecx coc.

A ao aoo peypoa ec e peayec?

B o ae eee ao peypoa epeao cocae?

C epeaoe y peyopo acac co ao "". ay opa ae o a?

5.6 TECT 1 ae e oocc pye caecx ee?

A Caeca xapaepca oa o y.

B Caeca xapaepca e cyecye.

C Caeca xapaepca paa y.

2 epeaoa y aoo ea ee W (s) = ?

T s A ceoo.

B Peaoo epepyeo.

C epaoo.

3 epeaoa y aepoecoo ea epoo o-pa Ks A W (s) =.

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 W W W W C W W W 14 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.

S B W (s) = - - S1 - S2 s.

s S C 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) A A0 A Pc. 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 A A Pc. 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 s Re s Im s s2 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 s Pc. 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 ypa Pc. 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.

2 Hapaee e a paeop oea cpea. ee opaae o o aoo paeop pocxo o acoo cpee opy aaa oopa.

3 Boax y1 = 0, y2 = 0, .e. ocox oax, pocxo ocaoa e.

4 B cceax opoo opa aoe paeop epecea oc accc o p yo, a a p y2(t) = 0, =, a y1(t) = y(t) ocae coeo acya.

5 B epxx apaax oopao ococ opaaa oa ec dy ( t ) cea cea apao, a x - cpaa aeo, a a p y ( t ) = > 0 epeea dt dy ( t ) y1(t) = y(t) opacae, a p y ( t ) = < 0 epeea y1 (t) = y(t) yae.

dt 6 B o oe aoo ococ, e epeea y2(t) y f2(y1, y2) e pa y, aoa paeop ee oo oo opeeeoe apaee, cooecyee pooo ao oe, oya ceye, o aoe paeop e epeceac.

Haae yco epexooo poecca opee oopa aao o M a aoo paeop.

Cooyoc aox paeop, cooecyx ce oo ao ccee aa yco, aaec ao nopmpemo cucme.

6.3.2 aoe oppe ex cce opoo opa oye ypae, ocax ao oppe cce opoo opa, eoxoo ccee epeax ypae (6.8) opoe ypaee oe a epoe c paccope pe t, peyae eo oya:

.

Peee oo ypae ae ceeco epax px a aoo ococ, o oop cpoc aoe paeop cce.

aoe oppe ex cce opoo opa accpyc o a ocox oe.

ea ccea opoo opa ocaec epea ypaee a (6.9) e y(t) - xoa oopaa cce;

a0, a1, a2 - ocoe oe.

dy1(t) Ooa y(t) = y1(t), a = y2(t), oa dt, ypaee (6.9) oo aca e cce epeax ypae:

(6.10) Pae opoe ypaee a epoe, oya (6.11) peee oopoo ye ypaee aox paeop y2 = f(y1, c1, c2), (6.12) e ci - ocoe eppoa.

Boo ec pax cyae aox paeop acoc o ope xapaepcecoo ypae a2 s2 + a1 s + a0 = 0.

Cya op - e p a1 = 0, a0 > 0, a2 > 0: s1,2 = +i;

=. Ccea axoc a pae ycooc.

paeue cucme: a2y1"(t) + a0y1(t) = 0, eo peeue ueem u y1(t) = Asin(t + ), (6.13) oya y2(t) = y1'(t) = A cos(t + ). (6.14) pa y1(t) oaa a pc. 6.7.

Pc. 6.7 ao oppe a ep:

a - ococ ope xapaepcecoo ypae;

- epexo poecc;

- ao oppe oye ypae aoo paeop pae (6.13) (6.14) oo apa caa, peyae oya ypaee:

. (6.15) Bpaee (6.15) pecae coo ypaee ca c oyoc A A. aaa pae A, oya ceeco aox paeop, oope e e epeceac e o ep aae oopa (pc. 6.7, ).

Hapaee e opaae o M ao ooe aoo ococ opeeec o ay y2. p ooeo ee y1 oe oo yeac, a p opaeo y2 - yeac, ceoaeo, ee opaae o a aoo ococ pocxo o acoo cpee, ooy eayxa epoec oea ccee cooecye a aoo ococ ayma aoa mpaemopu.

Ocoa oa cce ec eoepec epo aox paeop oc aae ep, a caa ccea aaec ocepamuo (.e. ccea e paccea ep, e pe).

Cya 2 op - oece e opaee eecee ac p a12 < 4a0a2;

a1 > 0, a2 > 0, a0 > 0:

S1,2 = - i (pc. 6.8, a), = -a1/2a2, = (1/2a2) - ccea ycoa.

Peee ypae (6.9) ee :

y1(t) = Ae-t sin(t + ). (6.16) Pc. 6.8 ao oppe a yco oyc:

a - pacooee ope xapaepcecoo ypae;

- epexo poecc;

- ao oppe Oya y2 (t) = y'(t) = Ae-t cos(t + + ), (6.17) e ;

.

pae (6.16) (6.17) a aoo ococ apaepecoe ypaee cpae (c apaepo t). C a oopoo, cooecy ooy epoy oea, opaaa oa paec aay oopa, a a ae y1 y a epo oea caoc ee, .e. epexo poecc ee xapaep ayxax oea.

Ocoa oa aaec ycmou oyco.

Cya 3 op - oece e ooee eecee ac p a21 < 4a0a1;

a0 > 0, a1 < 0, a2 > 0: s1,2 = + i.

o cya cooecye pacxoc oea ccee, .e. ccea ec eycoo. Peee ypae (6.9):

y1(t) = Aet sin(t + ). (6.18) Oya y2(t) = y'(t) = Aet cos(t + + ). (6.17) aoa oa, ac o aoo paeop, eopaeo yaec o aaa oopa.

Coco eycooo paoec cce cooecye ocoa oa, oopa aaec eycmou oyc (pc.6.9).

Ec peyae co yoo aoo oye ccea e coco paoec, o oa ye eopaeo yac o Pc. 6.9 ao oppe a eyco oyc:

a - pacooee ope xapaepcecoo ypae;

- epexo poecc;

- ao oppe - eo o cpa aoo paeop, .e. ccee oae oeae poecc c opacae ayo.

Cya 4 op eecee opaee p a21 > 4a0a2, a1 > 0, a2 > 0, a0 > 0:

s1,2 = ;

.

o cya cooecye aepoecoy poeccy ccee, caa ccea ycoa.

Peee ypae (6.9) y1(t) =. (6.20) Oya y2(t) =. (6.21) pae oac c epexo poecca a 1 2 cya pe c ypae y2 = s2 y1 y2 = s1 y1, oope oyac (6.20), (6.21) p s1 = 0 s2 = 0 (opaee ooo ope y).

Bce aoe paeop ac aao oopa - ocoy oy, aaey ycmou yo (pc. 6.10). Bpe e coco paoec eopeec pao ecoeoc.

Pc. 6.10 ao oppe a yco ye:

a - pacooee ope xapaepcecoo ypae;

- epexo poecc;

- ao oppe Cya 5 op - eecee ooee p a12 > 4a0a2, a1 < 0, a2 > 0, a0 > 0:

s1,2 = .

B ccee ye aepoec poecc, oa eycoa. Peee ypae (6.9):

y1(t) =. (6.22) Pc. 6.11 ao oppe a eyco ye:

a pacooee ope xapaepcecoo ypae;

- epexo poecc;

- ao oppe Oya:

y2(t) = y'(t) =. (6.23) aoe paeop apae o aaa oopa ecoeoc, .e. ec ccee eec ooee o coco paoec (aao oopa), o c eee pee oo ye eopaeo opaca.

Ocoa oa oc aae eycmou ye (pc. 6.11). o aao co cyae p epexooo poecca a 1 cooecy aoe paeop a 1, e pae paeop opeec ypae y2 = s1y1 y2 = s2y1. p epexooo poecca cooecy aoe paeop a 2.

Cya 6 op -- eecee e pae a p a1 > 0, a2 > 0, a0 < 0:

s1 = -1, s2 =. B o cyae ye eycoa ccea (p a0 = 0 - paa ycooc).

epexo poecc ccee ee aepoec xapaep, o ao oppe ee coepeo pyo .

ac ec cya, oa a1 = 0, , ya, o a0 < 0, ypaee (6.9) aec e ;

(6.24) eppoae oo ypae ae:

2 y y 1 - = 1. (6.25) c ( c w ) Pc. 6.12 ao oppe a ceo:

a pacooee ope xapaepcecoo ypae;

epexo poecc;

ao oppe Bpaee (6.25) pecae coo ypaee ceeca paocopox epo, oeceoe a oc. Acoa epo: y2 = y1.

aa aco coco pex aox paeop, .e. ocoa oa paccapaec a oa aox paeop.

Ocoa oa oc aae ceo, a aco a aoo ococ aac ceapapca cea (pc. 6.12).

o y ceapapca opaaa oa paec coco paoec, a o y py yaec o eo.

ac o o aoo paeop, opaaa oa o cee ocaoo ooo pee yaec o coco paoec a co yoo ooe paccoe.

Ceo ec eyco cocoe paoec, ae oa aae yco oo cooecy oe a ceapapce, aeee oyee po oy, o opaaa oa, oa a coce paeop, ye eopaeo yac o e o coco paoec.

6.4 OHTE CTOBOCT BEH Teop ycooc e a coaa aae aeo ea e pycc aeao Aecapo Mxaoe yo (1857 1918) c c aaa eeco exa.

a ccea, y oa eao (ec a ee e ecy ae oye) peao, ocaec epea ypae, peee oopx opeee paeop ee e.

ee aaec eoye, ec oo oyeo peyae paccope eapoao cce.

ee c yeo oye, oax peao ccee, aaec oye.

Heoyeoe ee aaec ycmou, ec ocaoo ae oye co yoo ao oo oyeoe ee o eoyeoo. Ec e oyeoe ee aeo ooec o eoyeoo p co yoo cax oyex, o oo aaec eycmou.

B eop ycooc cyecy pae o (ep), a o: opaa ycooc (ycooc o paeop), ycooc o yoy, acoeca ycooc ..

pee e epe opeee x o, eoxoo yo, o oaec o a oye. e oye oo pae a a a.

1 yce oye.

Pc. 6.13 ece ycoo oye:

a ycoe oyee;

ee aoo pocpace Boyee aaec yc, ec oo ecye eee opooo poeya pee ( t) (pc. 6.13, a). yc ca oe, ec a pe t oopaa e yceae aeo ec. Bo cyae eo e aaec oeo ce opaae o M0 cce aaoo ooe M0 eoopoe pyoe ooee M0'. Tpaeop eoyeoo e cxo o M, a oyeoo M0' oaec o epo (pc. 6.13, ). Be yca caaec a ce e cce, xo o ecoa oo p pee t.

Ooa epe yi0 oopa o M0, i = 1, , n;

epe y'i0 M'0,. p ao ce paoc oopa aa o acoo ee, .e. yoeope yco, e eoopoe ocaoo aoe ooeoe co.

Pc. 6.14 Hepepo ecye oye Ma oyeue aaec aoe ycoe oyee, oopoe ae a c aaoo ooe opaae o cce.

Ma oye cooecy ae, e ee, e ee oye.

2 Hepepo ecye oye.

Tae oye ecy a ccey e oo aa oe pee, o oceye (pc. 6.14). Ha ep aec, o ye ax oye ceae oee o o, a a o e oee oy opy, e yce. Ho a pae oaaec e a. Cce, ycoe p ycx oyex, yco p epepx;

eycoe p epo e - eyco p opo.

po oo ec o a, o epepoe oyee oo peca e oceoaeoc yco, .e. papea ec pa x(t) a yc eoc dt, ooy aee paccapac yce oye.

6.5 OCHOBHE B CTOBOCT 6.5.1 Opaa ycooc Boc oe -opecoc eoyeoo e. C o e paccapaec paeop eoyeoo e M0M cpoc poe p payco, oc oopoo ec a paeop.

Caec, o paeop oyeoo e ao ooec o paeop eoyeoo e, ec oa eo e -opecoc eoyeoo e ( - ao). Boyeoe ee cxo o M '0 (pc. 6.15).

cooc - o coco e, eee aece, a e oece xapaep. ooy p opypoe o ycooc aa paa oooc oopa co aoe, o pa oyeoo e e a o-pecoc eoyeoo e p o ae. Ec aa oooc cyecye, o ee ycoo, ec oa ocycye, o eycoo.

Pc. 6.15 o "Opaa ycooc" oop, ccea oaae opao ycooc, ec p o oo oopa aoe ooe o y aee pae, o paeop oyeoo e e a -opecoc eoyeoo e, o oceee aaec yco. Ec e oopa aoe e, o eoyeoe ee eycoo.

oe opao ycooc ee cyece, pa eocao, opaa pee eo peoc. p opao ycooc oyeoe ee oe aeo oac o eoyeoo.

Ec ae paeop , o o M M ' yc c pa copoc, o c eee pee paccoe ey oe oaac o (pc. 6.16), .e.c yi oopa o M, a y'i - M ', o p a opao ycooc oe oaac, o e (yi - y'i) cay o. B c c oc oe ycooc o yoy.

Pc. 6.16 o ycooc o yoy 6.5.2 cooc o yoy ee aaec yco o yoy, ec oo > 0 oo yaa co = () > 0 aoe, o epaeca p t = t0 ceye epaeco cex t > t0.

Cc o ycooc o yoy coco o, o ee ycoo, ec p ocaoo ao aao ce M'0 o M0 oa M' oceye e ocaoo a M (pc. 6.16). Ec e oopa aoe () e, o ee eycoo.

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 r3c r4 = a1/c13 4 c14 = c22 r4c r5 = 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;

a a2 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 eapa apo, 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.

Oac ycooc cpoc pocpace apaepo, o oop oaec pocpaco, oopaa oopoo c apaep cce. oeco apaepo oe , o paecoo opae aoee pacpocpae c a.

yc xapaepcecoe ypaee cce ee s3 + A s2 + B s + 1 = 0, (6.38) e A B apaep cce.

ycooc cce, cxo pep ypa, eoxoo ocaoo, o AB> 1, oya paa oac ycooc ye AB= 1.

Pc. 6.20 epoa Bepacoo B ococ apaepo A B paa oac ycooc pecae coo epoy, aaey epoo Bepacoo (pc. 6.20). Oac ycoo pao oeea pxoo.

pa oac ycooc oy ae, ec ppa y oe a0, an xapaepcecoo ypae peoce opeee ypa:

a0 = 0;

an = 0;

n-1 = 0. (6.39) Bopa x pa cooecye a yeoo op xapaepcecoo ypae, a pe - a co x ope. pae (6.39) paa pocpaco apaepo a p oace, oopx ycoo ye a oac, e opeee ypa 1,..., n-2 ooe.

6.8 ACTOTHE PTEP CTOBOCT acoe pep ycooc oo cy o ycooc cce aoaecoo ypae o y x acox xapaepc. pep oo cceoa ycooc cce cooo opa e pocy eoepecy eppea.

6.8.1 p apyea B ocoe acox pepe ycooc e cece ecoo eop y oecoo epeeoo pa apyea. yc a oo n- cee (6.27):

D(s) = a0 sn + an-1 sn-1 +... +an.

o oo cooec c eopeo ey oo peca e D(s) = a0 (s s1) (s s2)... (s sn), (6.40) e sj = j + ij op ypae D(s) = 0;

j = 1, 2, , n.

a ope eoepec oe opae eopo, poee aaa oopa oe sj (pc. 6.21, a). a eo paa oy oecoo ca, a yo, opaoa eopo c ooe apaee eceo oc, apyey ae oecoo ca.

Be (s sj) eoepec opaac eopo, poee o sj pooo oe s (pc. 6.21, ).

p s = i, apep, oya:

D(i) = a0 (i s1) (i s2)... (i sn), (6.41) o cex eopo yy axoc a o oc (pc. 6.21, ).

Paccapa eop D(i), oya, o oy eo pae D(i ) = a0i s1i s2...i sn, (6.42) a apye. (6.43) Ec p a ooeoe apaee ocea yo pae po acoo cpe, o p ee aco o o + a eeap eop oopaaec a yo, ec ope pacooe cea o o oc, a - ec cpaa (pc. 6.21, ).

Pc. 6.21 p apyea Ec oo ee m pax ope (n m) ex, o p ee o - o + eee apyea eopa D(i ) pao cye yo oopoa eopa (i sj), .e.

(6.44) Oya eae ceyee pao: eee apyea D(i ) p ee aco o o + pao paoc ey co ex pax ope ypae D(s) = 0, yoeo a.

p ee aco o 0 o eee apyea eopa D(i ) ye oe ee. (6.45) o pao ooeo ocoy cex acox pepe.

6.8.2 pep Mxaoa o pep o cyecy ec eoepeco eppeae pa apyea copypoa 1938 . coec ye Mxao.

Paccapaec xapaepcec oo (6.27):

D(s) = a0 sn + a1 sn-1 +... + an.

aea s = i, po oecoy ooy, aaeoy ye Mxaoa.

D(i ) = a0 (i )n + a1 (i )n-1 +... + an = U() + i V() = D() ei(), (6.46) e ;

aa cooeceo eeceo o y Mxaoa;

D() oy D(i);

() aa D(i ).

p ee aco oe eopa D(i ) ye oca eoopy py oeco ococ, oopa aaec oopao Mxaoa.

p ee aco o 0 o yo oopoa eopa D opy aaa oopa pae (6.45):

oca co pax ope ooa (6.47) .e. m = 0, ec. (6.48) oceee ec eoxo ycoe ycooc, o eocao. oo, o oy eoxooe ocaooe ycoe ycooc, eoxoo c op, eae a o oc, .e. oo oc ycoe:

D(i ) 0. (6.49) opy (6.48 6.49) peca coo aeaecoe paee pep ycooc Mxaoa. oo, o ccea aoaecoo ypae a ycoa, eoxoo ocaoo, o oopa Mxaoa D(i ) p ee o o oepyc, e poxo epe y, opy aaa oopa po acoo cpe a yo, e n opo xapaepcecoo ypae.

ycox cce oopa Mxaoa aaec p = 0, a eeceo oyoc, .e. D(0) = an;

poe oo c poco aco aa oa oooo opaca, .e.

eop oe oopaac oo po acoo cpe, a a opaca a eeapx eopo (i sj), ec caae a eopa D(i ).

Bc c pep ycooc oo copypoa cey opao:

nouoa aeamee nepeamoo yuu ACP (xapamepucmuecoo nouoa) opayemc yu Muxaoa. moo, mo cucmea amoamuecoo ynpaeu a ycmoua, eoxouo u ocmamoo, mo oopa Muxaoa npu ueeuu acmom om 0 o, auac npu = 0 a eecmeo nooumeo noyocu, oxou moo npomu acoo cmpeu noceoameo n apamo oopuamo nococmu, e n nopo xapamepucmuecoo ypaeu.

oopa Mxaoa ycox cce ee ay cpaey opy yxo ecoeoc o apae, oep oopoo pae cee xapaepcecoo ypae (pc. 6.22).

Pc. 6.22 oopa Mxaoa pao eycooc cce ec apyee ca oceoaeoc poxoe apao.

pep oopaa Mxaoa eycox cce pecae a pc. 6.23:

Pc. 6.23 oopa Mxaoa eycox cce:

a aaec a opaeo eceo oyoc;

e oxo n-apao oopao ococ;

e oxaae aao oopa Pc. 6.24 oopa Mxaoa epax cce:

a, ccea oe ycoa;

ccea eycoa epax cce oopa Mxaoa opae a pc. 6.24. B epx yx cyax eoe eopa o ccey a ycooc, ocee e ccea eycoa.

ocpoee oopaa Mxaoa paec pooc o eoo opox oe, o eoo cooaex oopao. ep eo coc opeee pa oe oopaa Mxaoa, cooecyx cpoa ae aco. p opo eoe opeec oopa oex ee, pe paa coe yoe eopo, cpo co oopa.

Aapy oopa Mxaoa, oo ycao ceyee: oa oopa Mxaoa oceoaeo poxo apa, o eecea a oc epeceac ooepeo. B oax epecee c eeceo oc opaaec y a y V(), a oax epecee po c o oc ecea y U().

aco, p oopx pocxo epeceee oce, opeec op ypae (6.50) To epecee px U() V() c oc accc a aee ope ypae (pc. 6.25) U() = 0: 1, 3, 5, ;

Pc. 6.25 ecea a cocae y Mxaoa:

a ycoa ccea;

eycoa ccea V() = 0: 0, 2, 4, B o cyae ycoo cce oaeo coee epaeca 0 < 1 < 2 < 3 < 4 <...

Bc c oo pec ceyy opypoy pep ycooc:

Cucmea amoamuecoo ynpaeu yem ycmoua moa u moo moa, oa eecmea U() u ua V() yuu Muxaoa, npupaee y, uem ce ecmumee u nepeeauec opu, npue oee uco mux ope pao nopy xapamepucmuecoo ypaeu n, u npu = 0 yoemopemc ycoue U(0) > 0;

V'(0) > 0.

6.8.3 pep Haca o aco pep papaoa 1932 . aepac ye Haco, o ooe cy o ycooc ayo cce o y AX paoyo cce.

yc epeaoa y paoyo cce ee ,.

epeaoa y ayo ACP o aay ypae:

.

Xapaepcecoe ypaee paoyo cce (n-o opa) opeeeo, a.

Xapaepcecoe ypaee ayo cce (n-o opa) paaec, a Paccop, o pecae ce paee 1 + W(s):

, (6.51) e xapaepcece oo, cooeceo, ayo paoyo ACP. oca s = i, oy AX paoyo cce (pc.6.26).

Beop, ceoaeo, ae ce coca ayo paoyo cce, o oy, a ee ce W(i ) ooc Pc. 6.26 AX paoyo cce eo (1, i0) oo cea o o ycooc ayo cce. B aee paccapaec AX, cooecya ooe acoa.

Be p cya coco paoec paoyo cce: ycoa, epaa eycoa.

1 c ya ccea paoyo coco ycoa. Toa eee apyea xapaepcecoo ooa paoyo cce coaco pep ycooc Mxaoa ye pao (6.48):

.

oo, o aya ccea a ycoa, oo oc paeco (6.48):

.

Oca ceye, o ppaee apyea eopa pao y:

(6.52) Coooee (6.52) oaae, o ycooc ayo cce eoxoo, o eop, aao oopoo axoc oe (1, i0), a oe, co o AX paoyo cce, e oxaa oy (1, i0) p ee o 0 o (pc. 6.27).

Pc. 6.27 AX:

a paoyo cce;

y H(i ) Ta opao, pep Haca ac:

Ecu paoyma cucmea amoamuecoo ynpaeu ycmoua, mo ayma cucmea amoamuecoo ynpaeu yem ycmoua, ecu anumyo-aoa xapamepucmua paoymo cucme e oxamaem moy (-1, i0) npu ueeuu om o.

2 c ya ccea paoyo coco eycoa.

p paccope oooypx oooypx cce peypoa, coepax eycoe e, paoya ccea oe oaac eycoo.

yc paoyo coco ccea eycoa, p o xapaepcecoe ypaee paoyo cce ee m ope pao oyococ. Toa coaco py apyea (6.25):

Ec opeoa, o ccea ayo coco a ycoa, o oo oc paeco (6.48):

Bo cyae yo oopoa eopa H(i ) = ye pae (6.53) oceee oop o o, o AX y H(i ) p ee aco o 0 o oxaae aao oopa ooeo apae pa.

co oopoo eopa H(i ) opy aaa oopa pao cy oopoo eopa AX paoyo cce W(i ) opy o (1, i0). Ha ocoa oo eae ceya opypoa pep Haca.

Ecu paoyma cucmea amoamuecoo ynpaeu eycmoua, mo moo, mo ayma cucmea amoamuecoo ynpaeu a ycmoua, eoxouo u ocmamoo, mo AX paoymo cucme W(i) npu ueeuu acmom om 0 o oxamaa moy (-1, i0) nooumeo anpaeuu pa, e m - uco npax ope xapamepucmuecoo ypaeu paoymo cucme.

Pc. 6.28 AX:

a H(i );

W(i ) p m = Ha pc. 6.28 opae aece pepa AX H(i ) AX paoyo cce, cooecye ycoo ayo ccee, oopa paoyo coco eycoa m = 2.

p coo ope W(i ) oy oy apye p opeee ca ee oopoo opy o (1, i0). B o cyae yoo pe "pao epexoo", peoeoe . . Haoe epexo W(i ) epe eecey oc p opaca ooe, ec o pocxo cepxy , opae, ec o pocxo cy epx. Ec W(i ) aaec aaaec a oc, o oa coepae oepexoa. Toa pep Haca oo copypoa cey opao.

Ec paoya ccea aoaecoo ypae eycoa, o oo o aya ccea aoaecoo ypae a ycoa, eoxoo ocaoo, o paoc ey co ooex opaex epexoo AX paoyo cce W(i ) epe opeo eeceo oc (, 1) p ee aco o 0 o a paa, e m co pax ope xapaepcecoo ypae.

B aece pepa a pc. 6.29 opaea AX paoyo cce: co pax ope m = 2;

co epexoo a oo Pc. 6.29 AX paoyo cce p m = ex, o opae, x paoc paa 1 =, ceoaeo, aya ccea ycoa.

3 c ya ccea paoyo coco epaa.

B o cyae ccea oa coepa eppye e, oa xapaepcecoe ypaee paoyo cce ee op, pae y, acaec e (6.54) e opo acaa;

A1(s) oo, e e ope, pax y.

Ayo-aoa xapaepca paoyo cce acaec e. (6.55) p = 0, W(i ) = AX peepeae pap, ooy pea opoc o ycooc ayo cce pyo, a a eco, oxaae AX oy (1, i0) e.

o coxpa opypoy pep ycox paoyo coco cce, p ocpoe oopaa Mxaoa p ee aco o o + oxo aao oopa o oyopyoc ecoeo aoo payca r. Toa yee op ay ao e yo oopoa, a ee op, .e. a eopo oepec a yo (pc. 6.30).

Pc. 6.30 Oxo aaa oopa o ye ecoeo aoo payca r Oxoy aaa oopa o ao ye rei cooecye epeaoa y paoyo cce (6.56) p r 0, payc R, a apye eec o o p ee o o.

Pc. 6.31 AX epao paoyo cce:

a c acao epoo opa, = 1;

c acao opoo opa, = Ta opao, p e o oyopyoc ecoeo aoo payca ococ ope AX paoyo cce caa W(i ) oe pecaea e eopa ecoeo oo , oopaaeoc a oeco ococ o acoo cpee a yo, pa .

p ee o 0 o, .e. r 0, 0, W(i ) eec o ye ecoeo ooo payca, oca yo o 0 o (pc. 6.31). pep Haca opypyec cey opao.

Ccea aoaecoo peypoa, epaa paoyo coco, ycoa ayo coco, ec AX paoyo cce c eo ooee ecoeoc e oxaae oy (-1, i0) p ee o 0 o.

a o pc. 6.31, ec paoya ccea ee aca epoo opa, o aya ccea ycoa, a a oa (1, i0) e oxaaec, ec e aca ye opoo opa, o aya ccea eycoa oa (1, i0) oxaaec AX paoyo cce.

ococa pep Haca c:

1) peoc p eecx ypaex eoopx ee paoyo cce;

2) oooc cceoa ycooc cce c aaae.

pep 6.3 cceoa ycooc cce pepe Mxaoa, ec xapaepcecoe ypaee cce ee D(s) = 2s4 + 4s3 + 2s2 + 5s + 1 = 0.

ae s = i, axoc ecea a y Mxaoa:

D(i ) = 2(i )4 + 4(i )3 + 2(i )2 + 5(i ) + 1, oya U() = 24 22 + 1;

V() = (42 + 5).

oopa Mxaoa opae a pc. 6.32. Eo aa oaae, o ccea eycoa. Ec cooa cece, o U() = 0;

V() = 0.

Peee x ypae ae: 21,3 = 1 i;

0 = 0;

2,4 = .

Pc. 6.32 oopa Mxaoa pepy 6. Ta a ec oeco-copee op, o ccea eycoa.

pep 6.4 cceoa ycooc cce aoaecoo peypoa (pc.

6.33), ec W1(s) = ;

W2(s) = e-2s.

Pc. 6.33 Cpyypa cxea ACP B paoyo coco ccea aoaecoo peypoa ycoa.

Ayo-aoa xapaepca paoyo cce acaec:

opaea a pc. 6.34.

Pc. 6.34 AX paoyo cce pepy 6. Ta a ayo-aoa xapaepca paoyo cce e oxaae oy c oopaa (1, i0), o aya ccea ycoa.

6.8.4 peee pepe cceoa ycooc cce Cpaee paccopex pepe ycooc ooe cea cey o ooceo x peoc.

pep ycooc ypa eecoopao pe, oa xapaepcecoe ypaee ee cee e e epex (n < 4).

pep ycooc Payca ae cp oe p ceo aax oeax, eecoopao ooac, oa n > 4.

pep ycooc Mxaoa eecoopao pe p cceoa cox oooypx cce, oa eoxoo c e ee cpyyp cce cpec caa a ee ycooc.

pep ycooc Haca eecoopao pe p cceoa cox cce. o pep oaaec eceo pe, oa ac ce xapaepc oex eeo cce aa cepeao, pe p aae cce, ocaex aaec y.

oo coeo poo aae acoe pep ycooc oy cooa oe apaepo cce a ee ycooc.

Ha pc. 6.35 opae oopa Mxaoa ycoo cce. Opeo OM pae ae eopa D(i ) (6.35) p = 0 pae ae oea an xapaepcecoo ypae.

Moo oaa, o oe yce cce e oo a coo e an xapaepcecoo ypae. ooy p eo yee ye yeac oo oe an, o cyae Pc. 6.35 oopa Mxaoa ycox cce 3-o opa Pc. 6.36 Cpyypa cxea cce c pe e ce eop D(i ) oya oaooe ooeoe eceoe ppaee, c pa Mxaoa e eopa epeaec apao, apep, ooe 1 ooee 2 (pc. 6.35). Ec yea oe yce ae, o p eoopo eo peeo ae oopa Mxaoa poe epe aao oopa, ccea e a pay ycooc. aeee yeee oea yce ceae ccey eycoo.

ec ooo opaoe peee aa, a eo, axoee peeoo oea yce. Opeo OM''0 (pc. 6.35) cooecye peeoy ae oea (an)p, aee oopoo oo oca o epoaaoy ooe po Mxaoa opeo M2M0.

Oe e apaepo cce a ee ycooc, oo oyc pepe Haca. B aece pepa e paccopea ccea peeo opa c pe epo e (pc. 6.36), oopo Ayo-aoe xapaepc paoyo cce pax ae oea yce k = K1 K2 K3 opaea a pc. 6.37, a.

Pc. 6.37 AX caeco cce peeo opa:

a pax oeo yce;

epae opax ee e acaa Bce xapaepc oy oye "epoaao" ye ee acaa, pe yoee e epa xapaepcy c o acao, a e aca opa eee e acaa. B o cyae ocaoo epa oy AX pa acea yea paep opea OA, paoo ee, o coo e pa, o coo yeaec oe yce. p o oa A ye epeeac pao (pc. 6.37, ). p ao ae oea yce k cce aca e OA e, oa A axoc ooe A1. B o cyae AX paoyo cce e oxaae oy A1, , ceoaeo, aya ccea ycoa. p yee oea yce k aca e yeaec, peca oa ec apao p k = kp aae ooee A2, ccea axoc a pae ycooc.

p k > kp peca oa pooae epeeac apao, aae ooee A3, ccea caoc eycoo.

Be oea yce a ycooc, coy pep Haca, oo poce cce cooo opa, acoc, c "oopa" xapaepca (pc. 6.38, a). B o cyae p ao ae oea yce peca oa axoc ooe A1, aya ccea ycoa. eee oea yce epeae oy ooee A2, k = kp1, ccea xo a pay ycooc. aeee yeee oea yce po ccey eycooc, a a peca oa aae ooee A3 oxaaec AX.

ooee A4, oopo k = kp2, ec pae ycooc, a ooee A peco o ycoo, a a e oxaaec AX. Ta opao, oo cea cey o.

Ccea ycoa p ax aex oea yce k < kp1 p ocaoo ox k > kp2, ee e pa ycooc p k = kp1 k = kp2, eycoa p kp1 < k < kp2.

Pc. 6.38 AX cce cooo opa:

a "oopaa" AX epoo opa;

"oopaa" AX opoo opa Pc. 6.39 AX pocx cce:

a AX cce epoo opa;

AX cce opoo opa Aa ayo-aoo xapaepc paoyo cce, opaeo a pc.

6.38, , oaae, o ccea ee p peex ae oea yce k1p, k2p, k3p, cooecye oa A2, A4, A6 pae ycooc. p aex oea yce k < kp1, kp2 < k < kp3 ccea ycoa (o A1, A5), a p aex kp1 < k < kp2, k > kp3 ccea eycoa (oa A3, A7).

peee pep Haca cceoa oee pocx cce - cce epoo opoo opa oaae, o ec paoya ccea ec cceo epoo opa e aaa, o a ec apaep cce, AX paoyo cce cea ye pacoaac eepo apae (pc. 6.39, a) , ceoaeo, aya ccea cea ye ycoo.

paoyx cce opoo opa AX pacoaaec e oyococ , ceoaeo, a ec ee apaep, AX oa e oxaae oy ( 1, i0), cceyea aya ccea cea ye ycoo.

Tae c oo pepe ycooc Mxaoa Haca oy pee opoc caa cce. B acoc, o cocoo caa ec eee o opaeo c.

6.8.5 Aa ycooc o oapec aco xapaepca B eepo pae oa aa ycooc poo o oapec aco xapaepca, ocpoee oopx poe, e ayo-aoo xapaepc. Ec poce acoc ey oeee AX paoyo cce oapeco ayo-acoo oapeco aoacoo xapaepca, o oo copypoa pep Haca peeo oapec aco xapaepca.

oo, o ccea aoaecoo ypae a ycoo, eoxoo ocaoo, o paoc ey co ooex opaex epexoo oapeco aoaco Pc. 6.40 acoe xapaepc:

a AX;

oapece acoe xapaepc o xapaepco px (2j + 1), e j = 0, 1, 2,... o cex oacx, e oapeca ayo-acoa xapaepca ooea, a paa, e m co pax ope xapaepccoo ypae paoyo cce.

Ha pc. 6.40 pee AX paoyo cce cooecye e AX X.

Aa acox xapaepc oaae, o paoc ey co ooex opaex epexoo paa y, o ec aya ccea ye ycoa oo o cyae, ec pae op yy ocycoa, .e. paoya ccea oa ycoo.

6.9 -PAEHE B . 6.7 o paccopeo ocpoee oace ycooc c cooae pep ypa aece pepa ocpoea epoa Bepacoo. Ha pae coyc pye oee oe eo cceoa pax apaepo cce a ee ycooc, .e. papaoa ceye ceae eo ocpoe oace ycooc:

1) ye aaa epeee ope xapaepcecoo ypae ococ ope eo opeoo oopaa;

2) ye aaa ca ope xapaepcecoo ypae, eax pao oyococ, pocpace apaepo cce eo -paa pocpaca apaepo, oop peoe papaoa 1948 . Heapo.

6.9.1 oe -pae Paccop xapaepcecoe ypaee ayo cce n-o opa, oopoe cea oe peeo y:

D(s) = sn + a1 sn-1 +... + an = 0 (a0 = 1). (6.57) peca cee oopaoe pocpaco, oc oopoo c oe ypae, oo oyo aae pocpaco oeo. ao oe oo pocpaca cooecy opee cee ae oeo ypae cooecy oo n- cee, oop ee n ope, acx o cex ae oeo ai. Ec e oe, o op yy epeeac oeco ococ ope oo ypae Paccop ypaee peeo opa D(s) = s3 + a1 s2 + a2 s + a3 = 0 (6.58) cooecyee ey pocpaco oeo a1, a2, a3 (pc. 6.41).

ao oe pocpaca cooecye oe opeee oo oe opeeee p op.

Hapep, oa M ee oopa {a1M, a2M, a3M}, ceoaeo, xapaepcec oo acaec e D(s) = s3 + a1M s2 + a2M s2 + a3M ee op S1M, S2M, S3M.

oa o ope pae 0 +i, oa oa pocpaca ye yoeop ypae D(i ) = (i )3 + a1(i )2 + a2(i ) + a3 = 0.

p < < oy ypae cooecye eoopa oepxoc Q.

Pc. 6.41 C ope xapaepcecoo ypae pocpaca oeo:

a ococ ope xapaepcecoo ypae;

pocpaco apaepo Ec op e, o oa pocpace oeo oaae a y oepxoc Q. p epecee ee op epexo oo oyococ pyy.

Ta opao, oepxoc Q paee ce pocpaco a oac c pa oeco pax ex ope, x ooaa D(m), e m co pax ope xapaepcecoo ypae.

Paee pocpaca apaepo a oac c oao co pax ope yp ao oac eee cpe oyex oace oac ycooc aaec eoo -pae.

ypae peeo opa oo e 4 oac D(3), D(2), D(1), D(0), oce ye oac ycooc.

Ec ec e ce oe, a ac x, apep, a1 a2, p a3 = const, o eco oepxoc oy , oopa ec ceee oepxoc Q paee ococ oeo a1, a2 a oac c oao co pax ope (pc. 6.42).

Pc. 6.42 paa -pae ococ oeo paee pa -pae oya xapaepcecoo ypae cce aeo s = i.

D(i ) = (i )n + a1 (i )n-1 + + an = 0. (6.59) pay -pae oo cpo e oo pocpace oeo epeaoo ypae, o pocpace apaepo cce.

6.9.2 -paee o ooy apaepy yc peyec c e a ycooc aoo-o apaepa v, eo xoeo xapaepcecoe ypaee. o ypaee oo pec y D(s) = M(s) + v N(s) = 0. (6.60) paa -pae opeec a D(i ) = M(i ) + v N(i ) = 0, (6.61) oya v = = X() + i Y(). (6.62) aa ae o o, oo c X() Y() ocpo pay pae, pay cpo oo > 0, a < 0 oya epa oopaee (pc. 6.43).

Ec ococ oecx ope ac o o oc p ee o o pxoa ee cea, o ococ apaepa v oy e ye cooecoa ee o pae -pae, oopy ae pxy cea. Ec e ococ v epecea pay -pae o apae pxo (1) (pc 6.43), o oy cooecye epexo op pao oyococ ey, ec e po pxo o ope epexo eo oyococ pay. Ec pxoa oa, o y oc epecea a op.

opeee oac ycooc ocaoo a pacpeeee ope p ao-o oo ae apaepa v. epexo ococ v o ooo apaepa pyoy, o cy epecee pa -pae, apae cy pxoo oo opee aee D(m).

peeeo a oac ycooc ec oac, yp oopo apaea pxoa oopa cooecye oac c ao co ex ope. B pao oac epec aee apaepa v o oy pepe ccea poepec a ycooc.

Pc. 6.43 -paee o ooy apaepy Ta a v eeceoe co, o oyeo oac e oo opeo eeceo oc, eae oac ycooc, apep, opeo AB.

6.9.3 -paee o y apaepa Ha pae aco peyec c e a ycooc yx, a e ooo apaepa.

Xapaepcecoe ypaee o cyae poc y:

D(s) = N(s) + M(s) + L(s) = 0, (6.63) oca s = i, oya ypaee pa -pae D(i) = N(i) + M(i) + L(i) = 0. (6.64) Ec ooa (6.65) o ypaee pa oo pa a a:

N1 () + M1 () + L1 () = 0;

(6.66) N2 () + M2 () + L2 () = 0.

oce ccea peaec ooceo apaepo :

= ;

=, (6.67) e aaa pae ae aco o - o, aoo ee ae o apaepec ypae opeec e cpoc paa -pae.

p o oo ceye p cya.

1 p aao acoe opeee 0;

1 0;

2 0 o o y. Bo cyae ccea coeca, ypae (6.66) peca coo pe ococ - (pc. 6.44, a).

Pc. 6.44 cpa cyecoa pee cce ypae (6.66):

a - peee cyecye;

oex pee e;

- peee eopeeeo 2 p eoopo ae = 0, a 1 0;

2 0. Toa ccea (6.66) ecoeca, oex pee e. pe 1 2 apae (pc. 6.44, ).

3 p eoopo ae ce opeee pa y, oa caoc eopeee. pe 1 2 cac py c pyo, o cyae oya e oy, a, a aaey, ocoy py (pc. 6.44, ), ypaee oopo:

N1() + M1() + L1() = 0. (6.68) Ocoa pa e oocc po -pae, a a ce ee oa cooecye oo o e aee aco, apaee e o e ycao eooo.

B ocoo ocoe pe oa p = 0 =, o o cyae, oa an = 0 o a0 = 0, cooeceo. Ec a0 an e ac o , o ocoe pe ocycy.

oce ocpoe pa -pae ocox px eoxoo x apxoa, oyc cey pao: p opaca o - o paa pae pxyec cea, ec > 0, cpaa, ec < 0.

Ta a c e y, o pa -pae ooex opaex aco coaa, ooy py -pae oxo a, oa cea pxyec oo pxoo.

pxoa ocox , a pao, oapa pxyec a, o ecax cope c -pae apxoae eapxoae copo po po apae py pyy (pc. 6.45 a, ).

B ex cyax, oa ocoa pa ee eco p eoopo oeo ae aco = 0 p o poxo epe y ee a, ocoa pa pxyec coaco pay, o oo pxoo (pc. 6.45, ). Ec e e ee a, o ocoa pa e pxyec paccope pacaec (pc. 6.45, ).

oce aece pxo opee oac, peeyy a oac ycooc, .e. oac, yp oopo apaea pxoa.

epeceee pa -pae apxoao o eapxoay cooecye epexoy yx oeco-cope-x ope eo oyococ ope pay, aoopo. epeceee ocoo po c oo pxoo cooecye epexoy ooo op.

Pc. 6.45 pao pxo ocoo po p -pae o y apaepa:

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