517.958:532:519.711.3 22.181 96
.. , .

()

. , . .
: , , , , , .

, . (., , [1, 2]). , , - . . , , , - .
, , . , ( , ), - . . , , .
(, ) . .
, . . . . ,
, , [1]. .
. . (1 , . ) [3, 4]. .. [5]. [6, 7] , .. [., , 5]. [8] .
. . , [6-8].
- , , , . , .

(, , , , , .), , . . [1, 5].
:
- ,
- ,
- ,
- ,
- , , .
- . . , , .
. :
- ,
- ,
- .
, , [5].
, [9-12 .] . . -
1(), . , .
, , ,
- .
. . - .
, : , , . , .
, , . : 1) ; 2) , ; 3) .
(-)
(, -) [3, 4] . . .
- , ї . , ї ( ) ї , . , .
, , , . - , . , , ї , , ї .
- .
1. .
= /() (= 1 dx), (1)
_) - (, ) .
(1) ( ) (, ) (). , [0, 1] (, ):
-
| () = |/()() . (2)
0 0
(2) , :
1 1
( )()| -1( ( 11()( . (3)
0 0
(() - , ,
( 0) = ( 1) = 0. (3) :
-
|y(x)p(x)dx = |f (x)p(x)dx .
(4)
(4) (1). , (4) ї () ї () .
2.
/ = f (x) (y" = d2 y / dx2),
(5)
fx) - (a, b) .
1, (5) (x) [x0, x1] (a, b) :
xi xi
Iy"(x)p(x)dx = If (x)p(x)dx . (6)
x0 x0
(6) :
(7)
'(x)P(x)| , - jy'(x)p'(x)dx =y'(x)p(x)| *
y(x)p(x)| x0 - jy(x)p(x)dx
( x) , ( x0) = ( x0) = 0 ' (x0) = ( x0) = 0, (6), (7) :
11 |y(x)p(x)dx = |f (x)p(x)dx .
(8)
(8) (5). 3.
(n) = f (x) (y (n) = dny / dxn).
(n) _ An,
(9)
x
x
0
0
x
x
0
x
x
0
0
1 2, (9) () ( ) [0, 1] (, ) . ї (),
* = 0, = 0, (*=0,1,...,-1)
( (
, (9):
(-1) | ( ) ( = 1 /()( )(.
(
0 0

1-3 .
4.
= ( = ( / (X). (10)
(10) - , >0, - , <0.
(10) ( ) (, ) () [0, 1] (, ) . :
1 1
| ()( = |()( . (11)
0 0
(11) , :
1 1
()()| - |()'()( = |()()( . (12)
|0
0
() - , ,
( 0) = ( 1) = 0. (12) :
1 1
- |'()( = |()(. (13)
0 0
(13) :
1
|'( )(
= -------------.
1
|( )(
X
X
5.
= 2 (/ = / ). (14)
4, (14) (
) (, ) ()
[0, 1] (, ). :
1 1
|() = |2() . (15)
0 0
(15) , ї ( ) , ( 0) = ( 1) = 0,
1 1
|() = |2(). (16)
(16) :
1
|'( )
=
| 2( )
0
6. 2
" + 2 = 0 (" = 2 / 2). (17)
, 2,
1 1
|() = -2 |() . (18)
0 0
(18) 2:
1
| ( )
2 0
=0---------------.
1
|( )
0
- .
7. 2
2 2 2
-- = ( = (, )). (19)

(19) 1 () 2 () 0 < < 1, 0 < < 1:


0

0

1 1 2 1 2
, ()2 () = 2 [ [ 1 ()2 ().
2 1 2 2 1 2
1 1 2 2
00

00
(20) :
1 2
|1 () |2 () = 2 2 () | ^ 1 ().
0 0
00
(21)
1 () 2 () , 1 (0 ) = 1 (1 ) = 0 , 2 (0 ) = 2 (1 ) = 0 , 1 (0 ) = 1 (1 ) = 0 , 2 (0 ) = 2 (1 ) = 0 . (22)
2 2
I 2() I 1() -
* 2
2
. :
1_ 2 1 ^2

1 2 1~ 2( ) = I
22 ()
1 2
2
.
}1( ) = |
1() 2
(23)
(21)-(23)
|1 () | 22() = 2 |2 () | ∞ ) ,
1 2
1( )
0 0 1 1
2 24 2
0 0
1 1
| |1 ()2( ) = 21 |!( )2 ( ).
0 0 0 0
(24) 2 :
1 1
| |1 ()2( )
(24)

2 0 0
11
| |!( )2 ( )
0 0
8.

2 2 +
2 2


(25)
(25) 1 () 2 () 3 (). 0 < < 1, 0 < < 1, 0 < < 1:

1 1 2 1 2
() |2() I 1() + () |() I - 2 ( )
0 0
1 1
0 0 0
2
= |2 () |1 () 1 ^ ().
(26)
0
00

0
0


0
( (), 2 (), 3 () ,
1() = 1(1) = 0, 2 ( 0 ) = 2(1) = 0, (^0) = 3() = 0, !(0) = 1(1) = 0, 2(0) = 2(1) = 0 ,
(26) [ ^ 1()

2
0
\ 2 . .. \ ...
I 2 () , I-3 () - , :
2 ;
0 ^ 0
1 1 1

1_1_1_ *1.11.
0 0 0 0 0 0
= - jjj123 . (27)
0 0 0
(27) :
1 1 1
11 \123
0 0 0
=
0 0 0 0 0 0
-^-1 -^-1
11 |23 +11 \\

. , . , (, ..).
- .
9. ()
= () (, ) . (28)
() , : () = 0+1 + 2 2 +--------------+ + ( ), (29)
() - .
+1 (29) (28), :
' =0 + ! + 22 + + . (30)
(30) i () (I = 0,1,..., ) [0, 1] (, ) :
()& = ( (() + ( 0 () + 2 2 0 () + + 0 () );
0
1
' ()< = ( < () + ^1 () + 2 2 () + + ());
0 0

' () = () + () + 2 2 () + + ())
0 0
(3) :

' () = ^ (^ ( = 0,,..., ).
(3)
(32)
() ,
(0) = (1) = 0 0' = n),
(32) ,
- 0()^ = 0 0 () + |0(^ + 2 |20()$ +------ |0()<;
0 0 0 0 0

|() = 0 | () + | () + 2 |2 () +-+ | ();
(33)
^ ^ -1
I' () = 0 | () + 1 | () + 2 |2 () + + | ().
0 0 0 0 0
(33) -
0, 1, , .
(, ), - (' = 0,1,...,) (29).
, (33) , , , () .
10. ()
' + () ' + () = / ( ).
(34)
, () g(y) , () . (34) - () g(y) _).
9, () (34) () (' = 0,1,..., ), , :
=0


0





0
0
) I()^ + |'() +-----------+ |'()() =
0 0 0

= I/()0() - I()0() -10(); 0 0 0
0 |() + |() +-+ |() =
0 0 0
x
= I/()() - I()() - I'();
0 0 0
xi
0 I' () + I' () + + I () =
0 0 0

= I/()() - I() () - I'()
0 0 0
() ,
(0) = () = ^ (0) = () = 0 ( = )

|0 () ,
0

|' ( = 0,,...,) - ,
0

0 I0() + I0() +-----+ I0() =
0 0 0

I0() - If ()0() + I(0(); 0 0 0

0 I() + I() +-----------+ I[() =
0 0 0
x
= I()- I/()() + I()(); (35)
0 0 0
xi
0 I () + I () + + I () =
0 0 0

= I"()- I/()() + I()()



0
0
, 9, (/ = 0,1,...,). , ().
.
11. ()
_

(, )
2
+ -
_

(, )
2


(36)
- (, ) , () - , /I - .
, (,)
(, ) = 0+1 + 2 + 3 + 4 2 + 5 2 + (37) (36)
(37)
[( !)' + [ ( !)' + [ ( !)' + , ( !)' + 4 2( !)' + 5 2( 2 )' + ...] +
_3_

+ - [( 2)' + 1( 2). + [ ( 2). + 3 ( 2) + 4 ( + 5 2) + .] =
= 21.
^

{)' + ,((% )', + 2 ( 2)1 +...}+
+ {( + (% + ,((2)) + ...}= 2}2) '
(38)
(39)
(39) (),(),^)
2 1 (/ = 0,1,...,) ( () () - , (^) - )
0 < < 1, 0 < < 1, 10 < t < ^ . :


+
+
||0 )() 0 |( 2) ( ^ + 1 |(( 2)' )( + 2 |( 2) 1 ()^
|_ 0 0 0
1
I \ ( ()^^ I( 2)" ( + 1 I( 2) ( + 2 I (( 2 )' )' (
_
1 1 1
= 2 11()() |() ( = ,1,...,).
+

+

(), (), (),




() = (1) = ^ () = = ^ () = (1 ) = ^
() = (1) = 0, () = (1) = ^ ( = )
, (2)X., (2), , (2)X,
(2) , (2)" - ,
'
'o 0 h_
+
,
I p (t)p ()dydt k0 IP2p"(x)dx + kl IP2 (xp'(x)) dx + k2 |yP2p"(x)dx + ...
o L x0 x0 x0
h ,
I p(t)P(x)dxdt k0 IP2p( + kl IxP2p( )dy + k2 IP2 (yp( )) dy
'o Xo L 0 0 0
' xl
-2p 11 IPp (x)p (y)p(t)dxdydt, (i = 0,1,..., n).
'o 0 Xo
(40) :
' xl
-0 ff [P2[(x)p(y) + p"(y)p(x)"|p(t)dxdydt +
+
(40)
ko\\]P 2 \( x)P (y) + p( yp (x)] (t )dxdydt
to 0 Xo h xl
+kl UfP 2 [(xp( x))p (y)+xp( y )p(x )](t )dxdydt+
to 0 Xo h
+ k2 ]]\P 2 [(y p (y ))p ( x) + yp( x)P (y)]p (t )dxdydt +
to 0 Xo
h xl
+ = -2^ ff [Pp(x)p(y)p(t)dxdydt, (i = 0,1,..., n).
(41)
k(xy) 0, ,..., ,.... (41). k(xy) (37).
. , , <( x) = (x - x0)(x1 - x) , (( x) = (x - x0)2( x1 - x)2 - .

, , :
((x) = (x) exp ( - " - ) , (42)
(x xo )(xi x)
a(x), ^(x) - , xo xj.
(42) (x0, x1) ( ). a(x), ^(x) ї . (42) . 1.
. 1. .
, .
(.. , , ) [0, 1] , :

( ) = ( )( - 0 )2 (1 - )2, ( ) = ( ) sin2------( - 0),
(1 - 0)
() - 0, 1 .

( - 0)2( 1 - )2 sin2-( - 0) (22). -
(1 - 0)
() [0, 1]. , ()=1.
, , :
( ) = ( )( - 0)(1 - ), ( ) = ( ) sin ∞).
(1 - 0)

, .
1. - .
2. - - .
3. , -, , , , - . , . , .
4. .
, - ( ) .
1. .., ., -
.. . , -, . - -
: -
, 2004. - 368 .
2. ., . . // . - 2008. - 4(32). - .18-24.
3. Loeb J., Cahen G. Extraction, a partik des enregistrements de mesures, des parametres dynamiques d um system. // Automatisme. -1963. - 12. - PP. 17-28.
4. Loeb J., Cahen G. More about process identification. // Trans. on Automatic Control. - 1965. - PP. 359-361.
5. .. . . - : , 1971. - 328 .
6. ., . . . - .: ,1976. - 407 .
7. . ., . .
. . . - . . . -. ., 1976, . 9. - . 111-114.
8. ., . - // . , 1985. - . 80-85.
9. .., ..,
..
-
- , . - -
. . 1 2. - .,
1975.
10. -
. / .., ., . . -.: , 1984. - 295 .
11. . .
. .: , 1989. - 334 . 12. . ., . . - . - . . . . . - ., 1990.