q (s) = s n + q n-1 s n-1 ++ q 0, (1.3.23)
p (s) = p n-1 s n-1 + p n-2 s n-2 ++ p 0.
-
X (s) = ,
(1.3.22)
Y (s) = = (1.3.24)
= Y 1(s) + Y 2(s) + Y 3(s).
y (t) = y 1(t) + y 2(t) + y 3(t), (1.3.25)
y 1(t) y 2(t) , y 3(t) .
1.3.5. , . 1.3.3, :
T 1 ;
T 2 ;
D T (s) = T 2 T 1 ;
0 ;
1 ;
Q ;
P (s) ;
R ;
G (s) ,
G (s) = .
t , .
. 1.3.3
. 1.3.2
G (s) = , (1.3.26)
t = RC 1/(1+ RC 0 Q). , D T (t) @ exp(- t / t).
.
. , . 1.3.4 .
|
|
. 1.3.4
Y 1(s) = G11 X 1(s) + G12 X 2(s), (1.3.27)
Y 2(s) = G21 X 1(s) + G22 X 2(s),
G km m - k - . , (1.3.27)
. 1.3.5.
. 1.3.5
. , , . 1.3.6.
. 1.3.6.
,
E (s) = X (s) Z (s) = X (s) R (s) Y (s),
Y (s) = G (s) E (s).
. (1.3.28)
(1.3.28) .
, .
1.3.6. G (s) R (s) > 1. R (s) G (s)
. (1.3.29)
, R (s) , , .. Y (s) = = X (s), , (1.3.28),
|
|
R (s) = 11/ G (s). (1.3.29)
1.4. .
.
, , . , , (1.3.27) . 1.3.5 . 1.4.1.
. 1.4.1.
. , (), () . , , . , . , , , . , . .
, . ,
a 11 x 1 + a 12 x 2 + u 1 = x 1, (1.4.1)
a 21 x 1 + a 22 x 2 + u 2 = x 2.
u 1 u 2 , x 1 x 2 . , (1.4.1), . 1.4.2.
. 1.4.2.
(1.4.1)
(1- a 11) x 1 - a 12 x 2 = u 1, (1.4.2)
- a 21 x 1 + (1- a 22) x 2 = u 2
,
x 1 = , (1.4.3)
x 2 = ,
D = 1 a 11 a 22 a 12 a 21 + a 11 a 22. (1.4.4)
a 11, a 22 a 12 a 21 a 11 a 22. , a 11 a 21 a 12 , , a 22 a 12 a 21.
x 1 u 1 (1.4.3) (1.4.4) u 1 x 1. x 1 u 2 (1.4.3) a 12, .. .
X (s) Y (s)
, (1.4.4)
k , , , ; D ; Dk k - . ,
D = 1 ( ) + ( 2 ) ( 3 ) + .
1.4.1. . 1.4.3 . , , . X (s) Y (s) : 1 1 = G 1 G 2 G 3 G 4 2 2 = G 5 G 6 G 7 G 8.
|
|
. 1.4.3
:
L 1 = G 2 R 2, L 2 = G 3 R 3, L 3 = G 6 R 6, L 4 = G 7 R 7.
L 1 L 2 L 3 L 4,
D = 1 (L 1 + L 2 + L 3 + L 4) + (L 1 L 3 + L 1 L 4 + L 2 L 3 + L 2 L 4).
1(2) D , 1(2)
D1 = 1 (L 3 + L 4), D2 = 1 (L 1 + L 2).
1.4.2. . 1.4.4.
. 1.4.4
:
1 = G 1 G 2 G 3 G 4 G 5 G 6, 2 = G 1 G 2 G 7 G 6, 3 = G 1 G 2 G 3 G 4 G 8.
:
L 1 = - G 2 G 3 G 4 G 5 G 6 R 2, L 2 = - G 5 G 6 R 1, L 3 = - G 8 R 1, L 4 = - G 7 R 2 G 2,
L 5 = - G 4 R 4, L 6 = - G 1 G 2 G 3 G 4 G 5 G 6 R 3, = - G 1 G 2 G 7 G 6 R 3,
L 8 = - G 1 G 2 G 3 G 4 G 8 R 3.
L 5 L 4 L 7; L 3 L 4; . :
D = 1 (L 1+ L 2+ L 3+ L 4+ L 5+ L 6+ L 7+ L 8) + (L 5 L 7+ L 5 L 4+ L 3 L 4).
:
D1 = D3 = 1 D2= 1 L 5 = 1 + G 4 R 4.
,
1.5. .
, . 1.5.1.
. 1.5.1
u (t) (.. ) e (t) (.. ) () . ( , ).
( ). , . .
1. ( , , -)
u (t) = k e (t) = k [ y (t) y (t)], (1.5.1)
,
.. ( k ).
, () (e = e 0 = const) , .. .
. , . () , .. u ¹ 0 ( ), (1.5.1) , m ¹ 0. , ( ). , () , , ( ) .
2. , (, ) ( , -, -)
|
|
, (1.5.2)
.
, ( ). , (.. ), , . , (t = 0) F m a = F / m, v = (at) t =0 s = (at 2/2) t =0 . , . - ( ). . , , . e = e 0 = const - d e /d t = 0, .. ( ).
-, k () . T , (1.5.2), , .
3. ( , , -)
(1.5.3)
.
. , ( ), . e = 0 ( d u /d t = 0 u = const).
, ( t ∞) ( ).
T , (I.5.3), .
4. , ( , - , -)
(1.5.4)
.
: , , , . (.. d u /d t = 0) ( e = 0 d e /d t = 0).
, , ( ).
(1.5.4) k , T ( ).
5. , , ( , - , - , -)
|
|
(1.5.5)
.
e = 0, . . .
, (.. ), . (I.5.5) : k , .
u (e), .
(-), u 0,
(1.5.6)
; sign .
. , . , .. , e ( ): e = +D u 0 + u 0, e ( ) e = D.
(-)
(1.5.7)
.. , 1/ T , , . . , .
, , . , (, , .. ; , ; ..). , , , ( ), . , ( ) (. 1.5.4).
. I.5.1 e u t = 0 - .
. 1.5.1
- ( ).
- u : ( ), e, .. ( e) t = 0 u , . , ( ) .
- e = const, , ( ). , e = const t = 0 t = var, , , u (t) e (t) u (t). , .. , . - , (, , ). .
|
|
- u , e = const. u (t) - t = 0 . e (t) u (t) , -; ..
- - t = 0, -.
- - u (e).
- e (-) e.
.
, , , .
, , ( ) .
()
= . (1.5.8)
T (1.5.8) T y (t) ( ) x (t)= q (t), . 1.5.2.
e ¥ .
( ) , G R,
. (1.5.9)
, X (s) = 1/ s
. (1.5.10)
, - e ¥ 0 (. . 42).
. 1.5.2
1.6. .
, , . (, ) () :
- ;
- ;
- ;
- , ;
- ;
- .
.
.
,
G (s) + D G (s). (1.3.28) (G >>D G 0)
. (1.6.1)
. (1.6.1) ,
. (1.6.2)
, , ( = 1) (R).
.
, . . . 1.6.1 , v (t), , n (t) ().
. 1.6.1
G (s), G (s), G (s), G (s), y(t) ,
, (1.6.3)
,
N (s) V (s) . G (s) G (s) G (s) G (s) >> 1. ,
, (1.6.4)
, G @ 1.
(1.5.9)
.
(1.6.5)
(1.6.4) , G (s), / . , .. G (s) G (s) >> 1, .
! , , .
:
1. (s) D s x (t).
2. , (s).
3. G (s) G (s) .. .
, , . .
1.5 - , .
(1.5.5) -
G (s) = K 1 + K 2 / s + K 3 s, (1.6.6)
K 1 = k, K 2 = k / T , K 3 = k T .
- K 1, K 2 K 3.
- , . , , X (s) = 1/ s N (s) = N 0/ s 2 s 0 (-) (1.6.5) (1.6.6)
, .. e ¥ 0 N 0 0. (1.6.7)
.
. 1.6.8.
. 1.6.8
:
- Tr , 10% 90% , . 1.6.8;
- M ;
- Tp ;
- ;
-
(M C)/ C;
- Ts , , , , . 2d 10% 4 % C.
d = 2% t, .. Ts = = 4 t.
1.6.1.
, (1.6.8)
w , z .
( )
. (1.6.9)
(. 1.3.13)
y (t) = 1 (1/ b) exp( t / t) sin(bw t + q), (1.6.10)
b = , q = arctg(b / z), t = 1/ zw .
1.6.1, ,
, . (1.6.11)
.
, G (s)
x (t) = A cos(w t). (1.6.12)
. (1.6.13)
, (1.6.14)
G* (s) G (s) . (1.6.14). , , .. s = jw
Y (jw) = k 1 = , (1.6.15)
Y ( jw) = k 2 = .
G (jw)
G (jw) = ê G (jw)½exp(jj). (1.6.16)
(1.6.14)
Y (t) = k 1 exp(jw t) + k 2 exp( jw t) =
= (A /2) ê G (jw)½ exp(jj) exp(jw t) +
+ (A /2) ê G ( jw)½exp( jj) exp( jw t) =
= (A /2) ê G (jw)½ {exp[ j (w t + j)] + exp[ j (w t + j)]} =
= A ê G (jw)½cos(w t + j),
ê G ( jw)½= ê G (jw)½.
1.
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