ЦСАУ

101

(4.50). -

:

1 (k, k0 ) T k 1 A 1 (n) T , k k0 .	(4.52)
n k0	(k, k0)

k0 k « » (4.23),

:

x(n) = A –1(n) x(n +1)

n = k n = k0.

:

k 1
x(k0 ) A 1 (n) x(k),
n k0
:
k 1
(k0 , k) A 1 (n).		(4.53)
n k0
(4.53) (4.52)
(4.34):
–1(k, k0) = (k0, k),	k k0.

,

(4.25) -

(4.53). , -

A(k) ( .

4.1.3).

4.2.3

(4.5) (4.6).

-

, . -

-

.

A(k)

(4.51):

–1(k, k0) = –1(k +1, k0) A(k).

(4.54)

102

(4.5) –1(k +1, k0),

(4.54) – x(k), :

–1(k +1, k0) x(k +1) – –1(k, k0) x(k) = –1(k +1, k0) B(k) r(k).

k m

m k0 k –1 .

, , , -

:

k 1

1 (k, k0 ) x(k) 1 (k0 , k0 ) x(k0 ) 1 (m 1, k0 ) B(m) r(m).

m k0

(4.55)

(4.55) (k, k0) ,

1 (k0, k0) = – (

(4.33) (4.34)):

k 1
x(k) (k, k0 ) x(k0 ) (k, m 1) B(m) r(m).	(4.56)

m k0

(4.56) -

(4.5). -

,

, , -

.

A, B, C D -

(4.5) (4.6) x(k)

( ):

k 1
x(k) (k - k0 ) x(k0 ) (k m 1) B r(m).	(4.57)

m k0

y(k)

(4.56) (4.57) -

(4.6):

- :

k 1

y(k ) C(k) (k, k0 ) x(k0 ) C(k) (k, m 1) B(m) r(m) D(k) r(k),

m k0

(4.58)

- :

103

k 1

y(k) C (k k0 ) x(k0 ) C (k m 1) B r(m) D r(k).(4.59)

m k0

, -

(4.5) (4.6) -

z- . z- -

(4.5) (4.6)

X(z) Y(z):

X(z)= (zE – A) –1 z x(0) + (zE – A) –1 BR(z),			(4.60)
Y(z)= C(zE – A) –1 z x(0) + [C (zE – A) –1 B + D] R(z).			(4.61)
z- (4.60)
(4.61), x(k) y(k).
4.4. :
0	1	0
x(k 1)		x(k) r(k),
2	3	1

y(k) 1 1 x(k).

–

1 k 0, r(k) 1(k)

0 k 0.

: y(0)=0, y (1)=0.

k=2,3.

(4.59).

4.1.

(4.59) :

	0			1		0		1 0 , D			0 .
	A			, B		, C
	2			3		1
, :
	2( 1)	k	( 2)		k	( 1)	k	( 2)	k		x	(0)
y(k) 1 0											1
2( 1)k				2( 2)k		( 1)k 2( 2)k					x2	(0)

k 1		2( 1)k m 1 ( 2)k m 1	( 1)k m 1 ( 2)k m 1 0
1 0			( 1)k m 1	2( 2)k m 1		.
m 0	2( 1)k m 1 2( 2)k m 1					1

k=0, :

						104
y(0) 1	0	1	0	x1 (0)			1 0	x1		(0)			x	(0).
													1
		0	1	x2	(0)			x2		(0)
k=1, :
0	1	x	(0)			1			x			(0)				x2 (0)
y(1) 1 0		1		1			0				2				1		1.
2	3	x2 (0)					2x1		(0) 3x2 (0)

y(k), -

x(k): x1 (0) 0, x2 (0) 1.

-

, k

x(0) . k=2, 3 -

y(2) 1	2 3 0				1	0			1	0	3		1 4,
	0	6				0
			7 1			2			3	1
y(3) 1	6		7 0		1	0	2 3 0
	0							6
	14		15 1						7 1
1 0	0	1	0	3	1 9.
			7
	2	3	1

4.3

4.3.1

z-

.

-

:
	Y1 (z)		Y(z) = W(z) R(z),	(4.62)
	Y (z)
Y(z)	2		z- m- -
	...
	Ym (z)

y(k),

		105
R1 (z)
R (z)		z- p- -
R(z) 2
...
Rp (z)

r(k), :

W11 (z)			W12	(z)	...	W1 p
W (z)			W	(z)	...	W
	21		22			2 p
W(z) ...
...
W	m1	(z)	W	(z)	...	W
			m2			mp

(z)

(z)

(4.63)

(z)

, Wij(z) –

j i .

(4.63) .

z-

(4.60) (4.61):

X(z) = (zE – A) –1 z x(0) + (zE – A) –1 B R(z),

Y(z) = C (zE – A) –1 z x(0) + [C (zE – A) –1 B + D] R(z).

(4.62) , -

x(0) = 0,

W(z) =C (zE – A) –1 B + D.	(4.64)
(4.64) (4.37) :
w(k) = C (k –1) B + D (0),	(4.65)

w(k) – ( ) ,

1, t 0,(t)

0, t 0.

-

(4.65) :

C (k 1) B,	k 1,
	k 0,	(4.66)
w(k) D,
0 ,	k 0.

106

, -

, k0, (4.66) :

C (k k0 1) B,	k k0 1,
	k k0 ,			(4.67)
w(k k0 ) D,
0 ,	k k	0	.

, , -

(4.5) (4.6), -

:

C (k,k0 1) B,	k k0		1,
	k k0 ,
w(k, k0 ) D,
0 ,	k k	0	.

4.3.2

, -3.4 ,

, -

. -

.

( -

– ) -

-

. -

,

. :

, ,

.

,

. , -

, 4.3. 1. ( . 4.3, ):

x2(k) = ax1(k),

X2(z) = aX1(z).

107

2. ( . 4.3, ):

x3(k) = x1(k) + x2(k), X3(z) = X1(z) +X2(z).

3. ( . 4.3, ):

x2(k) = x1(k +1), X2(z) = zX1(z) – zx1(0)

, X1(z):

X1(z) = z –1X2(z) + x1(0).

	x1(k)			x2(k)	x1(k)
			a
					)
					x1(k)

x1(k)

x3(k)

x2(k)

x2(k)

)

	x2(k)			x1(k)	x2(k)
			1

)

4.3

a x2(k)

1

x3(k)

1

x1(0)

1

z –1

x1(k)

( -

) :) ;

) ;) ;

108

) .

4.5.

:

y(k +2)+ 5y(k +1) + 6y(k) = r(k).	(4.68)
(4.68) y(k + 2):
y(k + 2) = r(k) – 5y(k +1) – 6y(k).	(4.69)

-

, y(k). , -

4.3, , -

(4.68). 4.4.

y(1) x2(0) y(0)

				z –1							y(k)
r(k)	1 y(k +2)						y(k +1) z –1
											Y(z)
R(z)	x2(k +1)			–5			x2(k)			x1(k)
							X2(z)			X1(z)
							–6

4.4

.

-

, z –1 :

x	(k 1)	0	1			x		(k) 0		r(k),	(4.70)
1						1
x2 (k 1)		6	5			x2 (k)			1
	y(k) 1 0				x1		(k) .				(4.71)
					x2		(k)

(4.70)

, -

X1(z) X2(z), – R(z), x1(0), x2(0):

								109
X	1	(z)					1	z	1		x	(0)	1		z	2
				1 1 5z							1						R(z), (4.72)
						6z 1
X	2 (z)							1			x2	(0)	z 1

= 1 + 6z –2 + 5z –1.

(4.72) ,

,

(4.37).

z- (4.72)

.

-

:

	W (z)	Y (z)		z 2		z 2			1	.
		R(z)				1 6z 2 5z 1		z 2	5z 6

, -

(4.68).

, -

, ,

.

: , -

.

z –1:

		b	b z 1	... b	z m		Y (z)
	W (z)	0	1	m				,	(4.73)
		a0 a1z 1 ... an z n					R(z)

110

n m – , -

– a0.

(4.73)

X(z):

Y (z)

(b

b z 1

... b

z m ) X (z)

0

1

m

.

R(z)

(a0 a1z 1 ... an z n ) X (z)

, :

Y(z) = b0X(z) + b1z –1X(z) + … + bmz –mX(z),

(4.74)

R(z) = a0X(z) + a1z –1X(z) + … + anz –nX(z).

(4.75)

(4.75) ( -

R(z) X(z)) X(z), :

X (z)

1

R(z)

a1

z 1 X (z) ...

an

z n X (z).

(4.76)

a0

a0

a0

(4.74) (4.76) ,

.

m = n = 3

4.5.

b0

b1

b2

R( z )

1

X ( z )

z –1

z –1

z –1

b3

a0

Y(z)

x3(k)

x2(k)

x1(k)

a1 a0

a2 a0

a3 a0

4.5