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, , .
a2∙y(2)(t) + a1∙y(1)(t) + a0∙y(t) = b2∙x(2)(t) + b1∙x(1)(t) + b0∙x(t) (1)
: y(t) ;
x(t) ;
y(j)(t) j- y(t);
x(j)(t) j- y(t);
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am, bm .
W(p) , :
W(p) = y/
(1) , d/dt , :
y(2)(t) = d 2y/ dt 2 = p 2y; y(1)(t) = d y/ dt = p y;
x(2)(t) = d 2x/ dt 2 = p 2x; x(1)(t) = d x/ dt = p x.
, :
a2∙p2y + a1∙py + a0∙y = b2∙p2x + b1∙px + b0∙x.(2)
(1) , (2) .
(2) :
∙(a2∙p2 + a1∙p + a0) = ∙(b2∙p2 + b1∙p + b0). (3)
(3) :
W(p) = y/ = (b2∙p2 + b1∙p + b0)/ (a2∙p2 + a1∙p + a0) (4)
(4) a0 b0, :
W(p) = (b0/a0)∙[(b2/b0)∙p2 + (b1/b0)∙p + 1]/[(a2/a0)∙p2 + (a1/ a0)∙p + 1],
W(p) = ∙(T2x∙p2 + T1x∙p + 1)/(T2y∙p2 + T1y∙p + 1) (5)
: T2x T1x ;
T2 T1 .
. , , , = 0. , (5), : W(p = 0) = K, : = ∙.
.
h(t) , x(t) = 1(t):
y(t) = h(t)∙1(t). (6)
g(t) ( ) , x(t) = δ(t) = 1′(t), , .. , 1:
y(t) = g(t)∙δ(t) = g(t)∙1′(t) (7)
[( = 0)] 1(t) δ(t).
. g(t) h(t): g(t) = dh(t)/dt.
.
, y(t) ( y′(t)) , ..:
y′(t) = K∙x(t). (8)
:
p∙y = K∙x. (9)
x(t) = 1(t) (8) : y′(t) = K, dy = K∙dt. , :
y(t) = h(t) = K∙t. (10)
(9) :
W(p) = y/x = K/p. (11)
10 , :
g(t) = h′(t) = K (12)
()
:
T∙ y′(t) + y(t) = K∙ x(t). (13)
, , y(t) 1(t).
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(13) :
T∙p∙y + y = y(T∙p + 1) = K∙ x. (14)
(14) :
W(p) = y/x = K/(T∙p + 1). (15)
, , L[g(t)] , , , .
L[g(t)] = W(p) = K/(T∙p + 1) = (K/T)∙1/(p + 1/T). (16)
f(t) L[f(t)] = 1/(p + 1/T), f(t) = ept, p , () , L[f(t)] : p + 1/T = 0, = - 1/T. , :
g(t) = (K/T)∙f(t) = (K/T)∙e-t/T (17)
h(t) (17), L[g(t)] (1/), , 1:
L[h(t)] = L[g(t)]∙1/ = (1/)∙ (K/T)∙1/(p + 1/T). (18)
h(t) (18) , .
(K/T)/[p∙(p + 1/T)] = A/p + B/(p + 1/T) =
= [A∙(p + 1/T) + B∙p]/[p∙(p + 1/T)],
K/T = A/T + A∙p + B∙p = A/T + p∙(A + B).
, :
K/T = A/T, = ;
+ = 0, = - = -;
:
(K/T)/[p∙(p + 1/T)] = K/p - K/(p + 1/T) =
= K∙[1/p 1/(p + 1/T)]. (19)
(19) , :
h(t) = K∙(1 e-t/T). (20)
(1/) f(t) ( = 0), :
f(t) = ept = e0t = e0 = 1.
, () ξ, , :
T2∙y′′(t) + 2ξ∙T∙y′(t) + y(t) = K∙ x(t). (21)
(21) :
T2∙p2∙y + 2ξ∙T∙p∙y + y = (T2∙p2 + 2ξ∙T∙p + 1)∙y = K∙ x;
W(p) = y/x = K/(T2∙p2 + 2ξ∙T∙p + 1). (22)
:
h(t) = K∙[1 (e-ξt/T/r)∙sin(rt/T + α)] (23)
: r = > 0 ;
α = arctg(r/ξ) ;
r/(2πT) = f .
g(t) , h(t):
g(t) = h′(t) = (K/T)∙e-ξt/T∙[(ξ/r)∙sin(rt/T + α) cos(rt/T + α)] (24)
, y(t) ′(t) , ..:
(t) = K∙ ′(t). (25)
:
y = K∙ p∙x. (26)
(26) :
W(p) = y/x = p∙K. (27)
, W(p) L[g(t)] , :
L[g(t)] = p∙K (28)
(28) , :
L[h(t)] = W(p)∙1/p = ∙1 (29)
, h(t) = ∙δ(t) x(t) = 1(t) :
(t) = h(t) ∙1(t) = K∙ δ(t) (30)
2.
, , tp. , h(t) 95% h(t→∞) = K.
, tp.
:
h(t) = K∙(1 - e-t/T).
, : tp ≈ 3,
h(t) = K∙(1 - e-t/T) = K∙(1 - e-tp/T) = K∙(1 - e-3T/T) = K∙(1 - e-3) ≈ 0,95K.
h(t), . , . : h′(t = 0) = (K/T)∙ e-t/T = (K/T)∙ e-0/T = K/T = tgψ, ψ h(t) t = 0. t = T h(t) = K∙(1 e-1) = 0, 632K.
, , ξ.
ξ< 1 ; ξ ≥ 1 .
tp , h(t) 95% h(t → ∞), .. h(t) 0,95 ≤ h(t) ≤ 1,05 .
tp , ξ. tp ξ = 0,707, h(t) , 1,05.
ξ .
[Δhmax = hmax h(t → ∞)] h(t → ∞), , δhmax.
δhmax = Δhmax∙100% / h(t → ∞) = [hmax h(t → ∞)]∙100% / h(t → ∞)
tp δhmax . , δhmax ≤ (30 40)%.
. , W(p).
, W(p), , 1.
W(p), , .
W(p) :
W(p) = W(p)/[1 + ∙ W(p)].
, :W(p) = W(p)/[1 + W(p)].
W(p) .
, (1/), , ..:
W(p) = Ko∙Wo(p). (1)
: Ko ;
Wo(p) , → 0 1.
, , , :
W(p) = K/(T∙p + 1) = K∙[1/(T∙p + 1)] = Ko∙Wo(p)
: = ; Wo(p) = 1/(T∙p + 1).
ε = ≠ 0 t → ∞.
:
ε = /(1 + ) (2)
(2) , ( = const) , .. . . .
, (1/ps), ..:
W(p) = Ko∙Wo(p)/ps, (3)
s .
, , , :
W(p) = K/p = K∙(1/p) = Ko∙Wo(p)(1/)
: = ; Wo(p) = 1; s = 1.
ε Ko. Ko .
(3), s . , , :
, , , , :
W(p) = (K1/p)∙[K2/(T∙p + 1)] = K1∙K2∙(1/p)∙[1/(T∙p + 1)].
: = K1∙K2; Wo(p) = 1/(T∙p + 1); s = 1.
:
S(p) = ε/x = (T∙p + 1)∙p/[(T∙p + 1)∙p + ] (4)
= = const ε = S(p = 0)∙ . (4) , S(p = 0) = 0, , , ε = 0 ≠ 0 ≠ 0.