ЦСАУ

91

:

k 1
x(k) A(n) x(k0 ), k k0.	(4.24)

n k0

(k,k0), ,

(4.23), , -

(4.24) :

k 1		k k0
	A(n),		;
(k, k0 ) n k0			(4.25)
	E,	k k0.

A(k) , A(k) = A0

),

:

(k k0 )	(4.26)
0 (k, k0 ) 0 (k k0 ) A0

:

x(k +1) = A0 x(k). (4.27)

(4.26), ,

-

.

,

A(k) , A(k)

A0 A1(k).

. :

x(k +1) = [A0 + A1(k)] x(k) = A0 x(k) + A1(k) x(k) = A0 x(k) + r(k). (4.28)

(4.28) -

A0 r(k) = A1(k) x(k).

(4.28)

, (4.23).

:

k 1

x(k) A(0k k0 ) x(k) A(0k n 1) r(n).

n k0

: r(n) = A1(n) x(n)

92

:

	k 1
x(k) A0(k k0 ) x(k0 ) A0(k n 1) A1 (n) x(n).		(4.29)
	n k0
(4.29) .
x(n) -
(4.29) :	k 1
x(k) A0(k k0 ) x(k0 ) A0(k n 1) A1 (n) A0(k k0 ) x(k0 )
	n k0
n 1
A0(n m 1)	A1 (m) x(m) .
m k0

:

x(k) = [E + S( 0 A1) + S( 0 A1 S( 0 A1)) + S( 0 A1 S( 0 A1

S ( 0 A1))) + …] 0 x(k0),	(4.30)
S , .
, (4.30),
:
(k, k0) = [E+ S( 0 A1) + S( 0 A1 S( 0 A1))+ …] 0,	(4.31)

0 (4.26).

(4.31) ,

. 1 -

, (k, k0)

.

(k,k0) -

.

.

1.-

:

	(k0, k0) = .	(4.32)
2.	(k2, k1) (k1, k0) = (k2, k0),	(4.33)

A(k) k, -

:

min(k0, k1, k2) k max(k0, k1, k2).

93

A(k) k N, -

(4.33) max(k0, k1, k2) N.

3.	(k1, k2) = –1(k2, k1),		(4.34)
A(k) :
k = k2 – 1, k2 – 2, … k1		(k2 k1),
k = k1 – 1, k1 – 2, … k2		(k1 k2).
4.	(4.33) :
	(k +n)= (k) (n).		(4.35)
5.	(4.34) :
	(k) = –1(–k).		(4.36)

– . (4.25)

n . (k,k0)

(4.31)

A1(k) 0 (

) . ,

-.

-

( ) .

-

. -

.

1. .

, (4.27) -

( ) , 0 -

.

		1		0	...	0
A			0	2	...	0
	0			...	...		.
		...				...
			0	0	...
						n

(k – k0) :

94

			(k k0 )		0	...	0
				1	(k k0 )
	(k k0 )			0		...	0
(k k0 ) A					2				.
			...		...	...	...
				0	0	...	(nk k0 )

[ i E – A] i, -

, , -

:

= –1 .

,

(k – k0) :

							(k k0 )			0	...	0
								1		(k k0 )
	(k k0 )		(k k0 )		1			0			...	0			1
(k k0 ) A									2							.
							...			...	...	...
								0		0	...	(nk k0 )
4.1.						A	0		1	. -
							2 3

, .

,

. -

:

	E A				1	( 3) 2 2 3 2 0 .
				2	3

( )

1=–1, 2=–2. -

Adj E A 1=–1, 2=–2:

Adj E A		3		1					2	1	,
	1		2
							1	2		1

					95
Adj E A		3		1					1	1
	2		2							.
							2	2		2

		,		:
1	1	1	2	1
M		. M		.
1	2		1	1

( )k		0	M 1	1	1	( 1)k		0		2	1
(k) M	1						0	( 2)k
	0	( 2 )k		1	2					1	1

	2( 1)k ( 2)k	( 1)k ( 2)k
		( 1)k 2( 2)k	.
2( 1)k 2( 2)k

2. . -

. ,

(n – 1)- , n – , -

. ,

(k – k0) :

(k k0 ) A(k k0 )				0E 1A 2 A2 ... n 1A(n 1) .
					i
:
(k k		)	2 ...
	0					(n 1) , i 1,2,...n,
i			0 1 i	2 i	n 1 i

i .

( , j

m) i

:

		s	n 1
s! kj	k0 s	d	i i	, s 0,1,2,...m -1.
		s
		d	i 0
				j

96

4.2. , 4.1.

								0	1
A
								2	3
. (2 2),
(k) 0E 1A , 0 , 1 -
	k		0	,
	1				1	1
	k		0				2	.
	2				1
, -
	( 1)k 0 1,
	( 2)k 0 2 1.
, 1,
							, 0:
0 2( 1)k ( 2)k , 1 ( 1)k ( 2)k .					-
:
1 0			0			1			1 0
(k) 0E 1A 0	1							(2( 1)k	( 2)k )
0 1			2			3			0 1

(( 1)	k	( 2)	k	0	1	2( 1)k ( 2)k		0
				)			0	2( 1)k ( 2)k
				2	3

	0	( 1)k ( 2)k			2( 1)k ( 2)k	( 1)k ( 2)k
		3( 1)k 3( 2)k				( 1)k 2( 2)k	.
2( 1)k 2( 2)k				2( 1)k 2( 2)k

, 4.1.

3. z- . z-

(4.27),

:

zX(z) – zx(0) = AX(z).

X(z), :

X(z) = (zE – A) –1 zx(0),

, :

97

x(k) = Z –1 {(zE – A) –1 z} x(0).

, :

		(k) = Z –1 {(zE – A) –1 z}.	(4.37)
4.3.
0	1	z- .
A
2	3

(4.37): (k) Z 1 (zE A) 1 z .

(zE–A):

z 0		0	1 z		1
zE A					.
0 z		2	3	2 z 3

:
1	z	1 1		1		z 3 1
zE A								z	.
				z(z 3)			2
	2	z 3			2

z

, :

z(z 3)

z

z 3

1

1

z

(z

1)(z 2)

(z 1)(z 2)

zE A

z

2

z

2z

z

2

.

(z 1)(z 2)

(z 1)(z 2)

(z 1)(z 2)

z -

,

. :

Z

1 z(z 3)

1

2

1

1

2z

1

1

Z

z (

)

Z

Z

z 1

(z 1)(z

2)

z 2

z 1

z 2

2( 1)k ( 1)k ,

Z

1

z

1

1

1

1

z

1

z

Z

z

(

) Z

Z

z 1

(z

1)(z 2)

z 2

z

1

z 2

( 1)k ( 2)k ,

Z

1

2z

k

k

( 2)( 1)

2( 2)

,

(z 1)(z 2)

										98
Z	1		z 2			1		2			1			1	2z				1		z
					Z		z (					)	Z					Z
			(z 1)(z 2)					z	2
											z 1				z 2					z 1

( 1)k 2( 2)k .

,

4.1 4.2.

(4.37) –

, . -

(zE – A) –1 -

.

(zE – A) –1 z = F. :

zF =zE + AF. (4.38)

(zE + A)

:

z2F= A2F+ zA +z2E.

(4.39)

(zE + A), :

z3F = A3F+ zA2 +z2A + z3E.	(4.40)
, :
z4F = A4F+ zA3+z2A2 + z3A + z4E,	(4.41)
………………..
…………………..
znF = AnF + zAn–1+… + zn–1A + znE.	(4.42)

(4.38)–(4.42) F = F

ai -

, :

a0F = a0F,

a1zF = a1AF + a1zE,

a2z2F = a2A2F + a2zA + a2z2E,

………..

an–1z(n–1)F = an–1A(n–1)F + an–1zA(n–2) + … + an–1z(n–1)E,

(4.43)

znF = AnF + zA(n–1) + … + z(n–1)A + znE,

ai –

anzn + an–1z(n–1) + … + a1z + a0 = 0 (an = 1).

99

(4.43), -

:

n	n	n	n	n
ai ziF ai AiF ai A(i 1) z ai A(i 2) z 2				... ai A(i n 1) z(n 1) z nE,
i 0	i 0	i 1	i 2	i n 1
				(4.44)

an = 1.

(4.44)

. -

(4.44) F, :

	n	n		n		n
	z j ai A(i j)			z j ai A(i j)
F	j 1	i j		j 1		i j			.	(4.45)
		n				zE A
		ai zi

i 0

(4.45) –

Adj(zE – A)z. F (4.45), -

:

(k) = Z –1(F).

4.2.2

, -

. -

, -

. L* -

L :

, Lx = L* , x ,

(4.46)

– , -

:

k 1
a, b aT (i 1) b(i).	(4.47)

i k0

, Lx (4.23), :

	100
k 1	k 2
, Lx T (i 1) x(i 1) A(i) x(i) T (k ) x(k) T (i 1) x(i 1)
i k0	i k0 1

k 1	k 1	T (i)
T (k0 ) x(k0 ) T (i 1) A(i) x(i)
i k0	i k0

T (k) x(k) T (k0 ) x(k0 ).

T (i 1) A(i) x(i)

(4.48)

-

:
T(k), x(k) = const,	(4.49)

k = 0, 1, 2, … .

(4.49):

T(k) x(k) = T(k0) x(k0),

(4.48) , :

L*( ) = T(i) – T(i+1) A(i),

:

T(i) = T(i +1) A(i).

, -

A(i), :
(i +1)= [A –1(i)] T (i).	(4.50)

(4.50) -

.

-

.

(k, k0) (4.23):

(k +1, k0) = A(k) (k, k0).

-

, :

[ –1(k +1, k0)] T = [A –1(k)] T [ –1(k, k0)] T. (4.51)

(4.51) (4.50) ,

[ –1(k, k0)] T (4.50) , , -

.

(4.49) -

, .