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????? ?? ???????? ??????
?? ??????????:
"?????? ?????? ??????????"
(??????? 2???)
????????: ??????? ??. 5-78-1,
??????: ????????? ?.?.
??????, 2012
???????? ??????
??????? 2
?????? ????? ?
????? ????????? ?
p1 = -10
p2 = -5+4j
p3 = -5-4j
?????? ????? ?
k1 = 10
k2 = 100
T1 = 0.1c
1. ??? ????? ? ???????? ???????????? ??????? ?? ??????? ? ???????. ?? ????????? ???????? ???????????? ???????? ????????? ??? ??????? ??????? ?? ??????? ? ??????. ????????????? ??????? ?? ?????? ????? ??????? z,x,x1,y1 ????? W2'(p) = W2(p) - 1. ?? ????????? ???????????? ??????? "W(p)=" "Y(p)" /"X(p)" ????????:
Y(p)=Y1(p)+F2(p)
Y1(p) = X1(p)*W1(p)
X1(p) = X(p)+F1(p)
X(p) = G(p) +Z(p)
Z(p) = Y1(p)*(-W2'(p))
?????? ???????? Y1(p):
Y1(p) = W1(p)*( Y1(p)*(-W2'(p))+ G(p)+ F1(p))
Y_1 ("p" )"= " (W1(p)*(G(p)+ Y_1 (p)))/(1+W_1 (p)*W_2'(p)) " " "Y(p)=" (W1(p)*(G(p)+ F1(p)))/(1+W_1 (p)*W_2'(p)) "+ F2(p)" ???
"Y(p) = G(p)" "W" _"g" "(p)+" "F" _"1" ("p" ) "W" _"f1" "(p)+" "F" _"2" "(p)" ?.?.
"W" _"g" ("p" )"= " ("W" _"1" "(p)" )/("1+" "W" _"1" ?"(p)W'" ?_"2" "(p)" ) - ???????????? ??????? ?? ??????? g(p).
"W" _"f1" ("p" )"= " ("W" _"1" "(p)" )/("1+" "W" _"1" ?"(p)W'" ?_"2" "(p)" ) - ???????????? ??????? ?? ?????? f1(p).
2. ?? ???????? ?????? ??????????????????? ????????? ???????? ????? ??????? ??????????? ????????????????? ?????????, ?????????? , ??????? ???????????? ?????? ?? ????????? ? ?? ?? ???????????? ???????? ???????? ??????????? ???????? (??????????, ????????????). ?????????? ???????????? ???????.
????? ??????? ???????????????? ????????? ???????????? ?? ???????:
"y" _"???" "(t) = " ?_"i=1" ^"n" �?"C" _"i" "e" ^("p" _"i" "t" ) ?
?.?. ??????? ???? ????? ?????? ??????, ?? ????????? ????? ????? ???:
y???(t) = ?1e-10t + C2e(-5+4j)t + C3e(-5-4j)t = ?1e-10t + Ae-5tsin(4t+?) ???????
"y" ("?")"=" ?"lim" ?-"t??" ??"y" _"???" ? "= " ?"lim" ?-"t??" ?"(?1e-10t + Ae-5tsin(4t+?) ) = 0" ???????? ??????????? ???????? ????????????. ????? ??????????????????? ????????? ????????? ? ????? ?????????????. ??????? ?????????.
3. ??? ????? ?? ???????? ??????? ????????? ???????????? ??????? ???????, ??????????? ?? ???? ????????? ????????????? ???????? ?????, ???????????? ??????? ????????? ???????, ?????????????????? ????????? ??????????? ? ????????? ??????. ?). ???????????? ??????? ??? ??????????? ???????:
"W" _"???" "(p)= " "W" _"1" "W" _"2" "= " ("k" _"1" "k" _"2" ("Tp+1" ))/"p" ^"2" ?????????????????? ????????? ??????????? ???????:
A(p) = p2=0.
?). ???????????? ??????? ????????? ???????:
"W" _"???" "(p)=" ("W" _"1" "(p)" )/("1+" "W" _"???" "(p)" ) "=" (k_1/p)/(1+("k" _"1" "k" _"2" ("Tp+1" ))/"p" ^"2" ) "=" (k_1 "p" ^"2" )/(p("p" ^"2" +"k" _"1" "k" _"2" ("Tp+1" ))) "=" (k_1 p)/("p" ^"2" +"k" _"1" "k" _"2" ("Tp+1" ) ) " " ?????????????????? ????????? ????????? ???????:
"D" ("p" )"=" "p" ^"2" +"k" _"1" "k" _"2" ("Tp+1" )"=0 " .
4. ?????????? ???????? ?????? ??????????????? ?????? ? ??????? ????????????? ??????
???????????? ??????? ?????? ?? ??????? g(t):
"W" _"??" ("p" )"=" "1" /("1+" "W" _"???" ("p" ) ) "=" p^2/("p" ^"2" +"k" _"1" "k" _"2" ("Tp+1" ) )
???????? ????????? ?? ???????????:
p^2 "k" _"1" "k" _"2" +"k" _"1" "k" _"2" "Tp+" "p" ^"2" "p" ^"2" "+" p^3 "T+" "p" ^"4" /("k" _"1" "k" _"2" ) " " "p" ^"2" /("k" _"1" "k" _"2" ) "-" (p^3 "T" )/("k" _"1" "k" _"2" )
-p^3 "T"-"p" ^"4" /("k" _"1" "k" _"2" )
???????? ????????????:
?0 = 0
?1 = 0
?2 = "1" /("k" _"1" "k" _"2" )
?3 = "T" /("k" _"1" "k" _"2" ) ? ?.?.
"x" _"g" ("?")"=" "c" _"0" "g" ("t" )"+" "c" _"1" "dg(t)" /"dt" "+" "c" _"2" ("d" ^"2" "g(t)" )/("d" "t" ^"2" ) "+...=" "1" /("k" _"1" "k" _"2" ) ("d" ^"2" "g(t)" )/("d" "t" ^"2" ) "+" "T" /("k" _"1" "k" _"2" ) ("d" ^"3" "g(t)" )/("d" "t" ^"3" ) "+..." ????? g(t) = g0 = const, ????? "x" _"g" ("?")"=" 0
????? g(t) = at, ??? a = const, ????? "x" _"g" ("?")"=" 0
????? g(t) = a2t, ????? "x" _"g" ("?")"=" "1" /("k" _"1" "k" _"2" ) "a" ^"2" ????? g(t) = a3t, ????? "x" _"g" ("?")"=" "1" /("k" _"1" "k" _"2" ) "a" ^"3" "t+" T/("k" _"1" "k" _"2" ) "a" ^"4" 5. ??????????? ???????????? ??????????? ? ????????? ??????? ?? ???????.
?). ??? ??????????? ???????:
A(p) = p2 = 0
?.?. ??????? p, ?? ??????? ?????????? ?????????.
?). ??? ????????? ???????:
D(p) = "p" ^"2" +"k" _"1" "k" _"2" ("Tp+1" ) = 0
?.?. ??? ???????????? ?????? 0, ?? ??????? ?????????.
6. ??????????? ???????????? ??????????? ? ????????? ??????? ?? ???????. ?). ??? ??????????? ???????:
A(p) = p2=0 ??????? ? ???????????? ???????? A(p) ?????????? ?????????.
?). ??? ????????? ???????:
D(p) = "p" ^"2" +"k" _"1" "k" _"2" "Tp+" "k" _"1" "k" _"2" a0 = k1k2=1000>0
?1 = a1 = 100>0
"?"_"2" "=" |�("a" _"1" &"0" @"a" _"0" &"a" _"2" )|"=" "a" _"1" "a" _"2" "-" "a" _"0" "0=100*1-1000*0=100>0" ??????? ?????????, ??? ??? ??? ???????????? ?????? 0.
7. ??????????? ???????????? ??????????? ? ????????? ??????? ?? ?????????
?). ??? ??????????? ???????:
A(p) = p2 ?(jw) = -w2
?????? ??? ????? ???:
??????? ?????????:
?"arg" ?-"0<w<?" ?"A(jw)" "=n*" "?"/"2" ? ?????? ?????? n=2 ? ?????? ??? ???????? ????? ???? ????????, ??????? ??????? ? ?????????????????? ?????????? A(p) ???????????. ?). ??? ????????? ???????:
D(p) = "p" ^"2" +"k" _"1" "k" _"2" "Tp+" "k" _"1" "k" _"2" D(jw) = -w2+k1k2+jw"k" _"1" "k" _"2" "T" =X(w)+jY(w)
?????? ??? ????? ???:
??? ????? ??? ?????????? ??????? w ?????? ??? ???????? ?????? ??????? ??????? ? ???????? ??????????????? ????? ??? ????????. ?????????????, ??????? ?????????.
8. ??????????? ???????????? ????????? ??????? ?? ????????? (?? ??????????????? ??????????????? ??????????? ???????).
"W" _"???" "(jw)= -w2 " "L" ("w" )"= 40lgw " ??????? ?????????, ?.?. ?????? ??? ????? ???? ? ?? ?????????? -?.
9. ??????????? ???????????? ????????? ??????? ?? ????????? (?? ??? ??????????? ???????). ??? ????? ???:
??? ?????, ??????? ?????????, ??? ??? ?????? ??? ?? ?????????? ??????????? ?????(-1;0).
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