ЦСАУ

211

A(z) F(z) z –1, -

, E(z), (7.27), -

z. -

, -

.

, , F(z)

. F(z)

. , -

.

7.5.2

(7.26) ,

F(z). , (7.26)

M(z) (7.22)

:

G(z)	1		M (z)	.	(7.28)
W (z)			1 M (z)

W(z)

( ), G(z) ,

,

M(z), (7.22), -

. ,

. W(z) M(z)

z –1 ( -

):
W(z) = wn z – n + w n + 1 z – n – 1 + …;	(7.29)
M(z) = mk z – k + mk + 1 z –k – 1 + …,	(7.30)

n 0 k 0. (7.29) (7.30) -

(7.28), :

				m z k m			z (k 1)	...
	G(z)			k	k 1
		(w z n w		z (n 1)	...)(1	m z k m			z (k 1) ...)
										(7.31)
		n	n 1				k	k 1

dk n z (k n) dk n 1z (k n 1) ...

212

G(z) -

, (7.31)

z,

k n. , M(z)

z –1

W(z).

F(z).

F(z)=1. (7.28)

M(z) ,

7.1.

7.1

					N	M(z)
	1(t)				1	z –1
	t 1(t)				2	2z –1 – z –2
		t	2	1(t)	3	3z –1 – 3z –2 – 3z –3
		2

,

M(z) z –1, M(z)/(1 – M(z))

(7.28))

, .

W(z) , -

, (7.28) , -

G(z) ,

. -

F(z) = 1, F(z) -

z 1.

1 , N , -

,

, N+M–1, M

W(z).

W(z), -

, .

213

7.9. , -

. 7.22,

1

W (z) z 2 z 1.

,

-

.

, , M (z) z 1, .

. -

M (z) z 2 . -

(7.28) :

G(z) (z 2 z 1) 1 . z 2 1

-

. F(z) (7.25)

F (z) 1 M (z) 1 z 1 . 1 z 1

(7.28) , -

W(z). , , -

, -

,

, . F(z)

, z -

.

, -

z F(z).

, , -

W(z)

,

, F(z), ,

z –1.

214

7.5.3

-

-

. , ,

, ( )

. -

, -

, -

M(z).

W(z) -

( zipi ),

:

(1 zi z 1 )

W (z) i (1 pi z 1 ) Ag (z),

i

Ag(z) , -

.

(7.28) :

		(1 pi z 1 )		M (z)
	G(z)	i			.	(7.32)
		(1 zi z 1 )		Ag (z)(1 M (z))

i

G(z) -

zi pi, -

, M(z) 1 – M(z), -

M(z) 1 – M(z) (

(7.25)):

M (z) (1 zi z 1)(M m z m M m 1z (m 1) ...),	(7.33)
i
1 M (z) (1 pi z 1)(1 z 1)N (1 a1z 1 a2 z 2 ...),	(7.34)
i

215

m z –1

W(z) ( G(z)), N

. ,

W(z) (1 – z –1)

(7.34) ,

,

W(z) , .

(7.33) (7.34) -

-

(7.26) (7.28).

7.10. -

.7.22 :

W (z) 0,000392z 1 (1 2,78z 1 )(1 0,2z 1 ) . (1 z 1 )2 (1 0,286z 1 )

, -

.

z 2,78 z 1 ,

.

M (z) 1 M (z) -

(7.33) (7.34) . ,

M (z) (1 2,78z 1 )(M1z 1 ...) .

M(z) -

M1z 1 , -

G(z) 0,000392z 1. -

M1 .

– , .

2, 1 M (z) -

(1 z 1 )2 . – -

, W (z) , -

. W (z) -

216

z 1, (1 z 1 )2 1 M (z)

. ,

1 M (z) (1 z 1 )2 F (z) (1 z 1 )2 (1 a1z 1 ) ,

a1

z 2,78, , , F (z) .

, M(z) 3 ( .

), -

, M(z) M1z 1

M 2 z 2 .

,

M1 ,

M 2

a1.

M(z)

1 M (z) , :

M

1

z 1 (2,78M

1

M

2

)z 2

2,78M

2

z 3 (2

a )z

1 (2a 1)z 2

a z 3.

1

1

1

z, -

:

M1 2 a1,

2,78M1 M 2 2a1 1, 2,78M 2 a1.

, : M1 =0,723; M 2 = –0,46; a1=1,277.

,

M (z) 0,723z 1 1,554z 2 1,277z 3 0,723z 2 1,554z 1,277 . z3

, z-

	E(z) R(z) (1 M (z))	Tz 1	(1 0,723z 1 1,554z 2 1,277z 3 )
		(1 z 1)2

z 1 1,277z 2.

T=1. ,

.

(7.28) :

					217
G(z)	1			M (z)	1844	(1 0,286z 1 )(1 0,636z 1 )		.
	W (z)			M (z)		(1 0,2z 1 )(1 1,277z	1 )
		1

.

, -

,

. , , -

7.10

(127,7%).

(

, , z N). -

,

(

) -

.

, ,

, -

M(z) « ».

,

.

.

1 – M(z),

(1 – cz – 1):

1 M (z) 1 M (z) 1 cz 1 ,

– ,

– 1 < c < 1 M (z).

M (z)

, -

z N = 0 -

. , ,

-

218

.

: , -

, -

. , , .

7.6

, -

, -

).

-

( ) ( -

) . -

-

.

.

7.1.

:

x(k +1) = A x(k) + B u(k), (7.35)

:

	0	1		0	...	0
	0	0		1	...	...
A		...		...	...	...		,
...
a		a	2	a	...	a
	1			3			n

B :

B = [0 0 … 1] T.

:
u(t) = – G x(t),	(7.36)

G = [g1 g2 … g n].

7.23

.

			219
	r(k)	u(k)		x(k)
			x(k +1) = A x(k) + B u(k)

G

7.23

.

(7.36) :

			x(k +1) = (A – B G) x(k).											(7.37)
-
:				E – A + B G = 0
:
				1				0		...		0
	0							1		...		0
				...				...		...	...
...														0.
	0			0				0		...	1
a	g	a	2	g	2	a	3	g	3	...	a	n	g
1	1													n

, :

n + (a n + g n) n – 1 + (a n – 1 + g n – 1) n – 2 + … + (a1 + g1) = 0. (7.38)

-

(7.38) ( ,

). , -

, ai, -

gi, -

gi G -

(7.38). .

7.11. , -

.7.23, -

:

	220
	x(k +1) = A x(k) + B u(k),
0	1
A	,
1	2

0 B 1 .

1,2 1,1, . -

. -

1 0,4; 2 0,6 -

.	-
. 7.23			E A BG		0 .
G g1	g2 . ,
	2 (g2 2) (g1 1) 0 .

1 0,4; 2 0,6 , -

( 1 )( 2 ) 2 0,24 0 .

, -

G g1 g2 0,76 1 .

G

,

, -

.

.

7.2. -

:

x(k +1) = A x(k) + B u(k),

(7.39)

– (n n), u(k) – -

, – (n 1).

(7.39) ,

:

y(k) = P x(k),	(7.40)
x(k) = P –1 y(k),