.


:




:

































 

 

 

 


-




, , . - , , . .

, , .

. , , . .

, , 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.

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)

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).

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.)

. .

.

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, ().

, , .

(. 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)=

, :

  1. W 0;
  2. W 0, , ;
  3. .

 

 

 

.

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)=





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