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Kursach po TAU Variant Kh 2011

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 ????????????
1.1. ????????? ??????????????? ?????? ??? ?????, ??????? ??????????? ?????????? ???????????????? ??????????
1.3 dy/dt+0.4y^2+0.7y=0.8x.(1)
? ??????????? ?????? ?????????????? ???????? y=y^0=0.5.
???????
??????? ????????? ??????????????? ????????? ??? ????? [1]:
+ (1.3 dy/dt+0.4y^2+0.7y)�|_(y=y^0 )=1.3�0+0.4(0.5)^2+0.7�0.5=0.8�x^0.(2)
???????????, ??? ? ??????????? ???????????? ???????? ?????????? y ?????????? ???, ??? ?? ?????????? ?? ??????????????? ???????? y0 ???????? ??? ????? ?????????? ?????.
????????? ????????? ?????????? ????? ?y. ????? ? ???????????? ????????
y(t)=y^0+?y.
???????? ???????, ??????? ? ????? ????? ????????? (1), ? ??? ?? ???????? ????????? ???? ????? ??????????. ????????? (1) ?????? ???
+ (1.3 dy/dt+0.4y^2+0.7y)�|_(y=y^0 )++ d(1.3 dy/dt+0.4y^2+0.7y)/(dy ? )�|_(y=y^0 )�?y ?++ d(1.3 dy/dt+0.4y^2+0.7y)/dy�|_(y=y^0 )�?y==0.8�x.(3) ????? ?? ????????? (3) ????????? ??????????????? ????????? (2), ??????? ??????? ??????????????? ????????? ????????:
+ d(1.3 dy/dt+0.4y^2+0.7y)/(dy ? )�|_(y=y^0 )�?y ?++ d(1.3 dy/dt+0.4y^2+0.7y)/dy�|_(y=y^0 )�?y=0.8�x-0.8�x^0,
1.3�?y ?++ (0.8y+0.7)�|_(y=y^0 )�?y=0.8�?x,
1.3�?y ?+1.1�?y=0.8�?x. (4) 1.2. ?????????? ?????????????? ???????? x=x^0.
???????
????????? x^0 ?? ????????? (2):
x^0=(1.3�0+0.4(0.5)^2+0.7�0.5)/0.8=0.5625.
1.3. ????????? ???????????? ??????? ???????????????? ?????. ??? ?????????? ????? ??????
???????
????????? ????????? (4), ????????? ???????? p:
1.3p?y+1.1?y=0.8?x.
???????????? ??????? ?????:
w(p)=?y/?x=0.8/(1.3p+1.1)=0.72/(1.2p+1).
?????? ????? ???????? ?????????????? ? k=0.72 ? T=1.2.
1.4. ??????? ?????????? ?????????????? (??????? ???????) ????? ?????.
???????
????????? w(t)?w(p), ??????? ??????? ????? ????? ? ??????? ??????? ??????????:
w(t)=+ 0.72/(1.2p+1)^' e^(-pt) �|_(p=-1/1.2)=0.72/1.2 e^(-t/1.2)=0.6e^(-0.83t).
?????????? ?????????????? ?????????? ?? ??????? 1.
1.5. ????? ?????????? ???????? ???????????????? ?????????, ??????? ?????????? ??????? ?? ?????? ???????????????? ????? ??? ??????????? ??????? ??????? x(t)=1(t).
???????
?????????? ??????? ????? ????????????? ?????????
h(t)?(w(p))/p??�?w(t)dt?=?�?0.6e^(-0.83t) dt?=0.6(e^(-0.83t)/(-0.83)+C).
????????? ????????? ? ?? ????????? ??????? (t=0): h(0)=0.6(e^(-0.83t)/(-0.83)+C)=0,
e^0/(-0.83)+C=0,
C=1.2.
????? h(t)=0.6(e^(-0.83t)/(-0.83)+1.2)=0.72(1-e^(-0.83t) ) (??????? 1).
??????? 1 - ??????? ? ?????????? ??????? ?????
??????????? ???????
2.1. ??????????, ????? ?????????? ?????? ????? ???????? ? ??????? ????? ? ???????????? ????????
w(p)=(-1.2p+1.1)/(1.3p^2+4p+0.9).
???????
??????????? ??????? ???????:
w(p)=(-1.2p+1.1)/(1.3p^2+4p+0.9)=(-1.2p+1.1)/(1.3�0.24�2.83(p/0.24+1)(p/2.83+1) )=(-1.36)/(4.2p+1)�p/(0.35p+1)+1.22/(4.2p+1)(0.35p+1) .
?????????? ??????? ???????????? ????? ???????????? ??????? ???????, ????????? ?? ??????????????? ??????????? ??????????????? ? ???????-????????????????? ??????? ? ??????????????? ????? ??????? ???????.
2.2. ???? ????? ??????????? ???????? ????? ????? ? ?????????????? ???????
???????
? ?????????????? ?????? ??? p=0 ???????
w(0)=k_(??.)=1.1/0.9=1.22.
2.3. ???????? ?? ????? ??????????? ???????
???????
??????? ????????? ?????????? ???????? ??? ??????????? ???????:
D(p)=1.3p^2+4p+0.9=0,
1.3(p+0.24)(p+2.83)=0,
[�(p_1=-0.24,@p_2=-2.83.)�
??? ????? ??????????????????? ????????? ?????? ????, ??????? ????? ?????????.
2.4. ???????? ?? ????? ??????????-????????
???????
????? ??????????? ???????????? ??????? ??????????? ????? ????????????? (p_1=-0.24, p_2=-2.83), ?????? ????? ?????????:
K(p)=-1.2p+1.1=0,
p=1.1/1.2=0.92>0.
?????? ????????? ???????????? ??????? ????? ? ?????? ?????????????, ??????? ????? ?? ???????? ??????????-???????.
2.5. ????????? ??????????????? ????? ????? ?????.
???????
????? ????????
w(p)=(-1.36)/(4.2p+1)�p/(0.35p+1)+1.22/(4.2p+1)(0.35p+1) .
?) ?????????? w_1 (p)=1.22/(4.2p+1)(0.35p+1) .
???? ???????????? ??????????
L_1 (?)=20lgA_1 (?),
??? A_1 (?)=1.22/(v((?1+(4.2?)?^2 ) ) v((1+?(0.35?)?^2 ) )).
?????????? L_1 (?) ????? ???????? ????????? ???????? ?20lgA?_1.1 (?) ? ?20lgA?_1.2 (?), ???
A_1.1 (?)=1.22/v((?1+(4.2?)?^2 ) ), ? A_1.2 (?)=1/v((1+?(0.35?)?^2 ) ).
??? ???? T_1=4.2, T_2=0.35, ????? ?_1=1/T_1 =0.24 ? ?_2=1/T_2 =2.86 - ?????, ? ??????? ??????? ?????? ??????.
??????????????? ???? ???????? ?? ??????? 2.
??????? 2 - ???? ??? w_1 (p)=(1.22)/(4.2p+1)(0.35p+1) ?) ?????????? w_2 (p)=(-1.36)/(4.2p+1)�p/(0.35p+1).
???? ???????????? ??????????
L_2 (?)=20lgA_2 (?),
??? A_2 (?)=1.36?/(v((?1+(4.2?)?^2 ) ) v((1+?(0.35?)?^2 ) )).
?????????? L_2 (?) ????? ???????? ????????? ???????? ?20lgA?_2.1 (?) ? ?20lgA?_2.2 (?), ???
A_2.1 (?)=1.36?/v((?1+(4.2?)?^2 ) ), ? A_2.2 (?)=1/v((1+?(0.35?)?^2 ) ).
??? ???? T_1=4.2, T_2=0.35, ????? ?_1=1/T_1 =0.24 ? ?_2=1/T_2 =2.86 - ?????, ? ??????? ??????? ?????? ??????.
??????????????? ???? ???????? ?? ??????? 3.
??????? 3 - ???? ??? w_2 (p)=(-1.36)/(4.2p+1)�p/(0.35p+1)
??????? ???? ????????? ????? ???????, ????????? ??? ???????????? ????? ???????????? ?????????? ??????? ? ????????????? ????????? w_1 (p)=1.22/(4.2p+1)(0.35p+1) ? w_2 (p)=(-1.36)/(4.2p+1)�p/(0.35p+1).
??? ?<1.22/1.36=0.9 w_2 (p)?0 ? w(p)�w_1 (p)=1.22/(4.2p+1)(0.35p+1) , ? ??? ?>0.9 w(p)�w_2 (p)=(-1.36)/(4.2p+1)�p/(0.35p+1), ?? ???? ?? ?????????? ??(0,0.9) ???? L(?) ????? ????, ??? L_1 (?), ? ?? ?????????? ??(0.9,+8), ??? L_2 (?). ??? ????????? ?????????? ????????? L_1 (?) ??? ?<0.9 ? L_2 (?) ??? ?>0.9 (??????? 4).
??????? 4 - ?????????????? ????
???????? ???? ????????? ????? ???????? ?? ??????? 5. ??????? 5 - ???????? ????
???????? ??? ?????, ?????????????? ???? ?? ??????????.
??? ??????? ? ??????? ????? ????? ??????????????
f_1 (?)=-arctg(4.2?)-arctg(0.35?),
f_2 (?)=p/2-arctg(4.2?)-arctg(0.35?).
??????? ?????????????? ??? ??????????? ???????? f_1 (?) ??? ?<0.9 ? f_2 (?) ??? ?>0.9 (??????? 6).
??????? 6 - ???
2.6. ????? ?????? ????? ???? ?? ??????? ???????? ?? ??????? ????????? ???????
? ????? ?????? ?????? ???? ???????????? ???????? ? ?????????????? ????? ????????????????, ????????????? ? ??????????? ???????.
????? ????? ??????? ?? r ???????????, m ???????????????? ? n ????????????? ???????, ? ??????????????? ??????????? ?????????????? A_1, A_2, A_3. ????? ???? ????????? ????? ????? ????? ???:
L=20lgA_1 A_2 A_3=20lgA_1 ?�k?_m ?^m�k_n/?^n =20lgA_1 ?�k?_m k_n ?^(m-n)=(m-n)20lgA_1 k_m k_n ?.
?? ??????? ??????? ??????? ? ???????? ???????? ???????????? (????????????) ????? ??????? ???????? ???????????? [1], ??????? ??? ??0 A_1=const. ?????
L=const+(m-n)20lg?,
?? ???? ?????? ?????????? (m-n)20 ??/??? ? ???????????? ?????? ???????????????? ? ????????????? ???????.
?? ??????? ???????? ??????????? ????? ???????????? ????? ????????? A_1=(a_0 ?^q+a_1 ?^(q-1)+...+a_q)/(b_0 ?^g+b_1 ?^(g-1)+...+b_g ), ??????? ??? ??8 ????????? ? a_0/b_0 lim?^(q-g).
????? L_1=(q-g)20lg a_0/b_0 ?, ??
L=(u-v)20lg? ??? ??8,
?????? ????? (u-v)20 ??/???, ??? u - ??????? ?????????, v - ??????? ??????????? ???????????? ??????? ?????.
??? ????????? ????? w(p)=(-1.2p+1.1)/(1.3p^2+4p+0.9) ?????? ????? ???? ??? ??0 ? -20 ??/??? ??? ??8.
2.7. ???????? ?????? ????? ????? ? ???? ????????????????? ?????????.
??????? w(p)=(-1.2p+1.1)/(1.3p^2+4p+0.9)=(Y(p))/(X(p)),
Y(p)(1.3p^2+4p+0.9)=X(p)(-1.2p+1.1),
1.3 (d^2 y)/(dt^2 )+4 dy/dt+0.9y=-1.2 dx/dt+1.1x,
1.3y �+4y ?+0.9y=-1.2x ?+1.1x.
2.8. ???????? ?????? ????? ????? ? ???????????? ?????????.
???????
????? w(p)=(-1.2p+1.1)/(1.3p^2+4p+0.9)=y/u=y/z�z/u, ??? y/z=w_1 (p)=(-1.2p+1.1)/1, z/u=w_2 (p)=1/(1.3p^2+4p+0.9).
w_1 (p) ????????????? ????????? y=-1.2 dz/dt+1.1z, ? w_2 (p) - u=1.3 (d^2 z)/(dt^2 )+4 dz/dt+0.9z.
?????? ?????????? ????????? x_1=z, x_2=dz/dt ? ??????? ??????? ?????????
{�((dx_1)/dt ?=x?_2,@(dx_2)/dt=-0.7x_1-3x_2+0.8u,@y=1.1x_1-1.2x_2.)�
?????? ??????? ???????????? ? ????? ?????? ? ???????????? ????????? ???????
{�(X ?=AX+Bu,@y=CX,)�
??? X=|�(x_1@x_2 )| - ?????? ?????????,
?=|�(0&1@-0.7&-3)|, B=|�(0@0.8)|, C=|�(1.1&-1.2)|, D=0 - ????????? ????????????.
2.9. ????????? ?????????? ?????????????? ????? ?????.
???????
????????? ??????? ??????????, ???????? ?????????? ??????? ?????
h(p)=w(p)/p=(-1.2p+1.1)/(1.3p^2+4p+0.9)p,
h(t)=+ ((-1.2p+1.1) e^pt)/(1.3p^3+4p^2+0.9p)^' �|_(p=0)++ ((-1.2p+1.1) e^pt)/(1.3p^3+4p^2+0.9p)^' �|_(p=-0.24)++ ((-1.2p+1.1) e^pt)/(1.3p^3+4p^2+0.9p)^' �|_(p=-2.83)=
=(1.1e^0)/0.9+(1.2�0.24+1.1)/(3.9�?0.24?^2-8�0.24+0.9) e^(-0.24t)+(1.2�2.83+1.1)/(3.9�?2.83?^2-8�2.83+0.9) e^(-2.83t)=
=1.22-1.75e^(-0.24t)+0.47e^(-2.83t).
??????? 7 - ?????????? ?????????????? ?????
3. ????????? ???????
3.1. ????? ?????? ?????????? ????? ???????????? ??????? W(s), ????????? - ???????????? ??????? K(s), ? ????????????? ??????? - ???????????? ??????? H(s). ????????? ??????? ????-????? ??????? ??????????????? ?????????????, ????????? ???????? ?????? tg(t), ?????? ?????????? u(t), ???????????? ?????? y(t), ??????? ?????????? w(t), ?????? ???????? ????? f(t), ?????? ?????? e(t).
??????? 8 - ????-????? ??????? ??????????????? ?????????????
3.2. ???????????, ??? K(p)=const ? H(p)=const, ????????? ???????????? ??????? (??): G(p) ?? ????? g(t) ? ?????? y(t); Gu(p) ?? ????? g(t) ? ?????? u(t); Ge(p) ?? ????? g(t) ? ?????? e(t); Gfe(p) ?? ????? w(t) ? ?????? e(t).
???????
????????? ???????????? ???????:
G(p) ?? ????? g(t) ? ?????? y(t): G(p)=(K(p)W(p))/(1+K(p)W(p)H(p));
Gu(p) ?? ????? g(t) ? ?????? u(t): G_u (p)=(K(p))/(1+K(p)W(p)H(p));
Ge(p) ?? ????? g(t) ? ?????? e(t): G_e (p)=1/(1+K(p)W(p)H(p));
Gfe(p) ?? ????? w(t) ? ?????? e(t): G_fe (p)=(-W(p)H(p))/(1+K(p)W(p)H(p)) (???????? ????? (??????? 9) ??????????? ? ????? ?? ??????? 10).
??????? 9 - ???????? ?????
??????? 10 - ????? ? ???????????? ???????? Gfe(p)
3.3. ????????? ???????? ???????, ??????????, ??? ????? ????????? k ? h ????????? ??????? ?????????.
???????
???????????? ??????? ????????? ???????: G(p)=(K(p)W(p))/(1+K(p)W(p)H(p)), ????? ?????????????????? ????????? ???????? ??????? ?????? ???:
D(p)=1+K(p)W(p)H(p)=1+kh (-1.2p+1.1)/(1.3p^2+4p+0.9)=0,
D(p) = 0 ??? 1.3p^2+4p+0.9+kh(-1.2p+1.1)=0,
1.3p^2+p(4-1.2kh)+(0.9+1.1kh)=0.
???????? ????????? ??????? ???????, ??? a_2=1.3, a_1 = 4-1.2kh, a_2=0.9+1.1kh, ????? ??????? ???????????? ??????? ?? ???????? ??????? ?????? ???:
{�(a_0>0,@a_1>0,@a_2>0;)� {�(1.3>0,@ 4-1.2kh,>0,@0.9+1.1kh>0;)�[�({�(h>0,@k<3.3/h,@k>-0.8/h,)�@{�(h<0,@k>3.3/h,@k<-0.8/h.)� )�
???????? ??????? ?????????? ????????????? k ? h, ??????????????? ??????? ???????????? ?? ???????? ???????. ????????? ???????? k ? h ????? ? ?????????????? ??????? ???????????? ????????? (??????? 11).
??????? 11 - ??????? ???????? k ? h
3.4. ?????? 1h, ???????? k ???, ????? ????? ???????????? ?? ????????? ??? ?? ????? 6 ??, ? ????? ?? ???? - ?? ????? 30o (??????????? ????? ??????????? ??????? ??? ??????????).
3.5. ????????? ?????????? ??????? ?? ?????? ??? ????????? ???????? k .
3.6. ??????? ????? ??????????? ???????? ? ?????????????????, ???????? ?? ?? ???????. 3.7. ???????? ?? ????????? ??????? ????????????? ???????
3.8. ??????????? ???????????????-???????????? ????????? (??-?????????) ? ???????????? ???????? K(s)=(k(p+?))/s ??? a=1. ? ??????? ???????? ??????? ??????????, ????? ??????????? ?????? ???? ???????? ?? k, ????? ??????? ???? ??????????. ???????? ??????????? k, ?????? ???????? ??????????????? ????? ??????????? ? ???????????? ??????????? ??????????.
3.9. ????????? ?????????? ??????? ?? ?????? ??? ????????? ??????????. ??????? ????? ??????????? ???????? ? ?????????????????, ???????? ?? ?? ???????.
3.10. ????????? ??????????? ????????? ?????????????? ?????????? ????????? ??????? ? ?????????? ?????????? ??????????????? M.
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