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.., .. () - . NUMERICAL METHOD OF SOLUTION OF PROBLEM STOCHASTIC LINEAR DIFFERENCE EQUATIONS PARAMETERS IDENTIFICATION IN CASE HANDICAPES ARE AVAILABLE IN INPUT PREDICTORS Pechekhonov A.N., Kacuba O.A. (Samara) The article is touching upon the problem of numerical algorithm designing for solution of the problem of stochastic linear difference equation parameters identification on the basis of non-linear method of least squares. () , , . : , , , ,
399


3. (II)

.. , , , . . . , , (). - [1], . « » - , . , , . , : i ( ) = C n ( i ) Gn (i ) + ei , ­ , ­ , ­ , C n ( ) ­ ,
400

{ei }

n


.. . -- -10, 2002, .399-401

G ­ . ( , ). , , . : Gn (i ) r: zi -
m =1



r

( b0m ) z

i-m

=

m =1



r1

( a0m ) xi

-m

+ 1 (i) , yi = zi + 2 (i )
^ ^

[2] b( N ) a( N ) b0 a0 (b) , :


b ~ B CR a

min

-1 (b)U 1 (b, a), N
r + r +1 1



U 1 (b, a) = N


i =1

N

T T ( yi - yr (i )b - xr1 (i )a) 2 ,

(1)

(b) = 1 + + bT b ,
yr (i ) = ( yi -1 , K, yi - r )T Rr , xr1 (i ) = ( xi ,K, xi
- r1

)T Rr1 +1 ,

2; 2 , . . (1) (, ..) -

=

2 1

401


3. (II)

, . (1), :
VN (b, a, ) = U 1 (b, a ) - (b), R1 , N VN ( ) =
^ b ~ B CR a ^ ^ ^

min

VN (b, a, ).
r + r1+1

U 1 (b, a) = (Y - AY b - xa, Y - AY b - xa) N
VN (b, a, ) = Y Y - (1+ ) +
^ T ^ ^ ^ T T T b AY AY - I r AY T T a x AY x ^

x x



T b AY Y b - 2 T = a x Y a

T

T T = Y T Y - (1+ ) + bT ( AY AY - I r )b + 2bT ( AY x)a + aT (xT x)a - 2(Y T AY )b - 2(Y T x)a .

y

0

Y = ( y1 K y N ) , AY =
y
^

T

L

y1- M

r

M
N -1

,

Ly

N -r

x1 x= M xN

x1-r1 M. L x N -r1 L

VN (b, a, ) b, a , :
^ T AY AY - I x T AY r T AY x x T x T b AY = T a x

Y . Y
-1

(2)

(2)
^ ^ b ( N , ^ ^ a ( N , ^ T ) AY AY - I = x T AY )

r

T T AY x AY Y x T x xT Y



, ,

402


.. . -- -10, 2002, .399-403
^ AT Y V N ( ) = Y T Y - (1 + ) - Y xT Y ^ T T AY AY - I x T AY T T AY x AY Y T T x x x Y

r

.

. VN ( ) , (1), : 1).
V N ( ) = 0
^

^

(3)
min

( ) ; 2). (3) [0, 1 ( N ) ,
^

( N ))

det( xT x) 0

,

min

(N ) ­

3). xT x [0, min ( N )) (1).
b ~ B CR a

[(


T T AY AY ) - ( AY x)( xT x) -1 ( xT AY )

]

min

(b)U (b, a ) =
r + r +1 1

-1

1 N

U 1 (b( N ), a( N )) N

^

^

(b( N ))
T T AY x AY Y TT x x x Y
-1

^

,


^ T ^ b( N ) AY AY - 1 ( N ) I = ^ xT AY a( N )
r

.

(4)

. , , ^ T T A A - I r AY x det( xT x ) 0 , det Y Y = 0 xT AY xT x
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3. (II)
^ T T det ( AY AY ) - ( AY x)( xT x) -1 ( xT AY ) - I r = 0 , VN ( ) [0, min ( N )) , min ( N ) 0 T T ( AY AY ) - ( AY x)(xT x)-1(xT AY )) . , ^^ ^ & VN ( ) = -(1 + + bT ( )b( )) -1, (-, min ( N )) .

[

]

(- ,
T

min

( N )) VN ( )
^

^

, ; , VN ( ) 0 ( AT AT A AT x I N - Y Y Y Y xT xT A xT x Y
^ ^
-1 T AY xT - ) ,

, VN ( ) > 0 , (-, 0) . 1), 2) 3). 3) . . : ^ ^ ^ ~ ~ V (b, ) = U 1 (b) - (b), R
N N 1 ^ ~^ VN ( ) = min VN (b, ) . ~ bCBR
r

~ U 1 (b) = (Y - AY b, Y - AY b) ; N
^ ^ ^ ~ T VN (b, ) = Y T Y - (1 + ) + bT ( AY AY - I r )b - 2(Y T AY )b .

VN (b, ) b ,
T T ( AY AY - I r )b = AY Y ^

^

(5)
404


.. . -- -10, 2002, .399-405
T T (5) b( N , ) = ( AY AY - I r ) -1 ( AY Y ) , ^ ^ ~^ T T VN ( ) = Y T Y - (1 + ) - Y T AY ( AY AY - I r ) -1 AY Y , ^ ^ ^

~^ . VN ( ) , (1), : 1).
~^ VN ( ) = 0

(6)
min

( ) ; 2). (6) [0; 1 ( N ) , T AY AY
^

( N ))

^

min

( N ) -

3). 1 ( N ) [0, min ( N )) (1). ~1 ^ ~1 (b) = U N (b( N )) , -1 min (b)U N ~ ^ bCBRr (b( N ))
T T b( N ) = ( AY AY - 1 ( N ) I r ) -1 AY Y . . . ^ ^

. 1. .., . . ­ ., 1999. ­ 147 . 2. Peshekhonov A.N. Piecewise linear approximation of nonlinear dynamic systems at presence of hindrances in output signals
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3. (II)

//The International Conference of Intelligent systems and information technologies in control: Theses of reports June 19-23, 2000. ­ St. Petersburg / Pskov, Russian Federation. ­ P. 24-27. ­ P. 206-208.

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