, , . - , , . .
, , .
. , , . .
, , 1936 . 40- , , , .
, , ui, Xi, Yi , (. . 2.1).
.2.1
UT =[u1 u2....ur],
-
XT =[x1 x2....xm],
-
YT =[y1 y2....yn].
t Y (t0) U (t 0, t), ..
Y (t)= F [ Y (t0), U (t0,t)], (2.1)
F- .
t Y (t0) U (t 0, t)
X (t)= Y [ Y (t0), U (t0,t)]. (2.2)
2.1 2.2 . , , 2.1 2.2
d Y (t)/dt= F* [ Y (t0), U (t0,t), (2.3)
d X (t)/dt= Y * [ Y (t0), U (t0,t)]. (2.4)
, :
d Y/ dt =AY+BU, (2.5)
X=CY+DU, (2.6)
A - nxn, . ;
B - nxr, . ui yi.
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C - mxn. xi ( ) yi.
D - mxr, u i x i
.
, C C, J, , u. :
u =L di /dt + R i + C w (2.7)
( )
Jdw /dt= C i - (2.8)
,
y1=w y2 =i , x1 = C w. (2.9) (2.5) (2.6),
2.10-2.12 - .
- , m, k f u(t) - (. . 2.2).
,
m d2z/dt2 +f dz/dt + kz=u(t) (2.14)
- y1=z y2=dz/dt.
y2=dy1/dt d2z/dt2 = dy2/dt.
.2.2.
x1 z, .. x1= y1=z. , (2.14)
(2.15) (2.5)-(2.6), -
- , , , ..
(2.5).
p Y (p) = AY (p) + BU (p)
X (p)= CY (p)
Y (t), U (t) X (t), , p; A, B C - .
AY (p) p Y (p)= p 1Y (p) ( 1- nxn),
(p 1- A) Y (p)= BU (p)
Y (p)= (p 1- A)-1 BU (p)
X (p)= C (p 1- A)-1 BU (p)
G (p)=(p 1- A)-1 B, H (p)= C (p 1- A)-1 B,
Y (p)= G (p) U (p), X (p)= H (p) U (p) (*)
H (p) mxr (.. X (p) - m, U (p)- r) - . Hij j- i- . , i- (*), U (p)
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xi(p)=Hi1(p) u1(p) +....+Hij(p) uj(p) +....+Hir(p) ur(p)
(m=r=1), .
H(p)= X(p)/ U(p)
, .. .
G (p) nxr - .
H (p) . ,
T
j ij D ij .
- - .
( ) | |
, |
G (p)=(p 1- A)-1 B
H (p)= C (p 1- A)-1 B,
(p 1- A)-1, , .. , det(p 1-A), . A.
2 , (p 1-A), . .
. 3.1 , - (2.5) (2.6).
. 1/p .
.3.1.
.
. : , . , () - , -, , , . .
- . , A=a, B=b, C=c, D=d,
.3.2.
b=a=1/T, c=1, d=0
.3.3
, .
1. n 1/p, n - . , yi, - d yi /dt, ..
2. , (2.5).
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3. , (2.6).
(2.9)
. 3.4.
. 3.5.
, . , , , . 3.6.
. 3.6.
.
: , .
.
,
N(p) x= M(p) u,
Pi = di / dti
, (4.1)
(4.2)
: y1= x1 i=1, 2,.... n-1.
, , (4.1), ..
....................
....................
- ,
A = B= C =
, C . 4.1.:
. 4.1.
, (4.3)
(4.4)
x1=x2/ b0 (4.4) (4.2) ,
- , (4.1) (4.3) , , .
C =
- b0 , y1 x2. . 4.1.
(4.5)
(4.6)
Xi
,
, (4.5) bi, yi+1 xi. .
C =
M(p), ..
,
, ( m+1) c bi x. (. . 4.1.)
. .
|
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.
C =
1:
,
a0=1/T2, b0=1/T2, b1=T1/T2
:
. 4.2.
:
,
, : a2=7, a1=12, a0=0, b2=1, b1=3, b0=2.
. 4.3.
A = B = C T= D =0
:
l i - , a i -
p=l i
n , . 4.4.
. 4.4.
xi .
, . : l 1 =0, l 2=-3, l 3 =-4.
, a 1=1/6, a 2=-2/3, a 3=3/2. (.4.5.)
. 4.5.
A = B = C T= D =0
- , . , .. m=n,
bk lk - () (). n>m ( ), (n-m) . - , , .
. 4.6.
, n .
. 4.7.
,
A = B = C T= D =0
, , , . , .
.
, : , . ( ) . , , , , , . , . .
.
, . (, , , .., , . w 1 M, ().
, , .
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(. 5.1) J1, J2 J3 C12 C23.
. 5.1.
Cik = d Mi / d j k - . ¥.
. M Mc1. . Mc2 - Mc3.
, . , .
, .5.2, .
. 5.2.
,
5.1, ,
, , , , , , - . JS, M Mc. , , M.
(.5.2) , . w 1 = w 2 = w. M12=0. 5.1 5.1,
JS dw /dt= M- Mc, (5.3)
JS = J1+J2 Mc= Mc1 + Mc2
5.3 . . . JS Mc , , , .
, . , .
5.1, 5.2 5.3 .
T dM/dt = b (w 0-w 1) - M, (5.4)
b = M/(w 0-w 1) - , T- w 0 - . 6.4
, .
, 5.4 , . .
, (
s< s
. .
, . , .
T dw 0/dt= K u - w 0 (5.6)
u - ( ; w 0- ; T K - .
.
5.6, 5.4 5.2 , 5.7
dw 0 /dt= -(1/ T) w 0 +(K / T) u (5.7,)
dM/dt = (b / T) w 0- (b / T)w 1 -(1/ T)M, (5.7,)
dw 1/dt= - (1/ J1) M12 +(1/ J1) M -(1/ J1) Mc1 (5.7,)
dM12 /dt =C12 w 1 - C12 w 2 (5.7,)
dw 2/dt= (1/ J2) M12 - (1/ J2) Mc2 (5.7, )
Y =[ w 0 M w 1 M12 w 2 ], (5.8, )
-
U = [ u Mc1 Mc2 ] (5.8, )
,
,
. 5.3.
, 5- ( ). , , ,
. M . , , .
5.2 ,,.
dw 1/dt= - (1/ J1) M12 +(1/ J1) M -(1/ J1) Mc1
dM12 /dt =C12 w 1 - C12 w 2
dw 2/dt= (1/ J2) M12 - (1/ J2) Mc2
Y =[ w 1 M12 w 2 ], - U = [ M Mc1 Mc2 ] ,
.6.1 .
. 6.1.
A
det(p 1 - A)=0
det (p 1 - A) = det =0
,
, - (p1=0)
p23= j
W 0= , (6.1)
p23= jW 0 ,
det(p 1 - A)=
, :
- W 0;
- W 0, , ;
- .
.
C =[ 1 1 1],
H (p)= G (p) = (p 1 - A)-1 B =
M w 1 M w 2.
Ww 1M(p)= w 1(p)/M(p)= (6.2)
Ww 2M(p)= w 2(p)/M(p)= (6.3)
(6.1)
Ww 1M(p)= (6.4)
Ww 2M(p)= (6.5)
g = (J1+J2)/J1. Ww 1M(p)
Ww 1M(p)= (6.6)
, w 2=W 0. . 6.2
. 6.2.
- w 2=W 0.. 1/JS -20 /. (w ³ W 0 ) -20 /, g , .
90 . 180 , .. +90 w 2=W 0. -90 . .
, M JS, . , .
w 1 w =W 0 . Ww 1M(p) . , . :
1. J2< < J1. g 1, , ..
Ww 1M(p)=1/ JS p
2.. W 0 >> w, w - , . :
, J2< < J1 W 0> > w , , .
. 6.3 w 2 (6.5)
. 6.3.
, . W 0 -60 /. -270 .
Ww 2M(p) , . , . w > W 0 -60 / , g. :
, , , .
, (, ) . , .
, , , ,
, .
,
JS dw /dt= M- Mc,
JS = J1+J2 Mc= Mc1 + Mc2,
, - . . (. . 6.4,,).
.6.4
. , , , . 0 M Mc1 Mc2 .
w 2 , (6.5). M(p)= M/p
w 2(p)= ,
e - , B(p)=1
A(p)= p2[(1/W 02)p2+1] - p1,2 =0 p3,4= jW 0
w 2(t) = w (t) +w (t),
, - - .
n
x(t)=
w (t) =
w (t) =
p=0,
w (t)= e t.
, pi,i+1
x(t)= 2Re
A(p)=dA(p)/dp= -2jW 02
w (t)= e ´ 2Re = -(e /W 0) Sin W 0t
w 2(t) = w (t) +w (t) = e t -(e /W 0) Sin W 0t
w 2(t) .6.5.. , w 2 (t) e =M/JS e /W 0 W 0.
.6.5
, (6.4),
Ww 1M(p)= Ww 2M(p)+ ,
Ww 2M(p).
, w 1(t) , w (t) w"(t) ,
w (t) =L-1 { ´ M/p }
(P)=1 A(p)= p(2/W 012) p=j/W 0 A(jW 0)= 2j/W 0,
- . w (t) w 1(t) , w 2(t). w 1(t) , J2, W 0 , . . , .
, , ( 5):
T dM/dt = b (w 0-w 1) - M,
JS dw 1/dt= M- Mc.
T= JS /b,
dM/dt = -(1/ T)M - (b / T)w 1+ (b / T) w 0,
dw 1/dt= (1/ b T) M -(1/ b T) Mc.
(7.1) . 7.1 .
. 7.1.
. 7.2.
Y =[ M w 1 ],
-
U = [w 0 Mc]
,
A = ; B = ; C = .
det (l 1 - A)=0
TT l 2 + T l +1=0
, , A ( - )
.
d = 1/2 T
, - 3 :
- d > W ;
- d = W (l 1= l 2= - d);
- d < W - .
w 1 w 0 Mc, - M. , . , .
H (p)= C (p 1 - A)-1 B.
(p 1 - A)-1 = Adj(p 1 - A)/det(p 1 - A).
Adj(p 1 - A)= = .
det(p 1 - A)= (TT p2 + T p+1)/ TT
H (p)= [det(p 1 - A)]-1[1 1] ´ ´ .
H (p)= [TT/(TT p2+ Tp+1)] ´ .
,
Ww 1,w 0(p)=w 1(p)/w 0(p)= , (7.2)
Ww 1,Mc(p)=w 1(p)/ Mc(p)= , (7.3)
WM,w 0,=M(p)/ w 0(p)= , (7.4)
WM,Mc(p)=M(p)/ Mc(p)=