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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 Ch(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 tT t1 tt 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) TCyece 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 Ck h(s) = = + +.

(T1s +1)(T2s +1) s s s +1/ T1 s +1/ Ta) ) ) 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 kT22Te C0 = k; C1 = ; C1 =.

T1 - T2 T2 -Tepexoa 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 Tpa 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-TW (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- TAa 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 2T4T2 + Tw(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 xPc. 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.

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