ЦСАУ

221

(7.39) -

:

y(k +1) = A1 y(k) + B1 u(k),

1 – .

		0	1		0		...	0
		0	0		1		...	0
A1
	...		...		...		...	...
		0	0		0		...	1
	a		a	2	a	3	...	a
		1							n

:

A1 = P A P–1,

B1 = P B = [0 0 … 0 1] T.

:

	P1
	P A
P	1	,
...
P An 1
	1

P1 = [0 0 … 1][B AB … An –1B] – 1.

(7.41)

(7.42)

(7.43)

(7.44)

(7.45)

. , -

(7.40) (7.41) -(7.42) (7.43),

(7.44) (7.45). :

p11	p12 ...
p	p	...
P 21	22
...	... ...
p	p	...
n1	n2

p1n		P1
p		P		(7.46)
2n		2	.
...		...
p		P
	nn	n

y(k) (7.40) :

y1(k) = P1 x(k)

y1(k +1) = P1 x(k +1).

222

x(k+1)

(7.39). (7.42) (7.41) :

y1(k +1) = P1 A x(k) + P1 B u(k) = y2(k). (7.47)

(7.40), y(k) x(k), -

(7.47) P1 B = 0 :

y1(k +1) = y2(k) = P1 A x(k).

( )

, :

y1(k +2) = y2(k +1) = y3(k) = P1 A x(k +1) = P1 A2 x(k),

, P1 A B = 0.

n – 1 , : yn – 1(k +1) = yn(k) = P1 An – 1 x(k)

P1 An – 2 B = 0.

, -

:

	P1
	P
y(k)	2	x(k),
...
P An 1
	1

(7.40) (7.44).

(7.44) (7.43), :

		P1B					0
		P AB						0
		1
B		...						.
1							...
			n 1					1
	P1A			B

:

[P1 B P1 A B … P1 An – 1 B] = P1[B A B … An – 1 B] = [0 0 … 1].

P1 -

(7.45):

P1 = [0 0 … 1] [B A B … An – 1 B] –1 = [0 0 … 1] Q –1,

Q = [B A B A2 B … An – 1 B] – ,

(n n) (7.39) ( . 4.4.2).

(7.39)

, , , Q

223

n, .

(7.45) . .

,

, , (7.39)(7.41) .

(7.41) -

-

G1 ( 7.1). -

(7.39)

:

G = G1 P.

, ,

, u(k)) -.

7.3.

:

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

– (n n) , u(k) – -

(m 1), – (n m)

. -

, -

: u(k) = – G x(k),

G –

(n m).

7.12. -, . 7.23,

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

1	1
A	1	,
0

1 B 1 .

224

-

. Q:

1	0
Q B AB	.
1	1

, .

.

(7.45)

P 0 1 Q 1 1 1 .
1
(7.44)
P1		1	1
P P A		1	2 .
1

-

:

y(k +1) = A1 y(k) + B1 u(k),

A1 PAP 1		0	1
			,
	1		2

0 B1 PB .

1

, , -

7.11.

-

1 0,4; 2 0,6 ( . 7.11).

2

G1 g1 g2 0,76	1 .
	1 1	1 0,24 1,24 .
G G1P 0,76
	1	2

2 , 7.11 -

G , G1 .

225

8

( ).

,

, ) .

,

. -

, -

. -

, ,

-

.

, , –

.

8- ,

256 . ,

.

,

,

– ( , « -

»).

.

8.1 -

. -

( ).

.

226

u(t)

d

(t)

8.1

), ( -

).

) ( ) ( ) -

.

(t) d.

-

( ) . -

:

u(t) kPe(t) kI e( )d ,

(8.1)

e(t) = d – (t) – , kP kI – -

.

(8.1) :
x(t) t d ( ) d x(t0 ),	(8.2)
t0

t0 – .

(8.2) -

, , -

. , t = kT, t0 = (k – 1)T, :

kT	d ( ) d d T	T	( (kT ) (k 1)T )	(8.3)
(k 1)T	2

k = 1, 2, …

, (8.2)

t = kT d

227

(kT) [(k –1)T]. (8.3)

,

, , (kT) [(k –1)T],

t = kT, . , -

, -

. -

,

. ,

(8.3) -

t = (k + 1)T, (8.2)

:

x (k 1)T d T	T	(kT )	T	(k 1)T x(kT ).	(8.4)
2		2
(8.1) (8.4) :
u[(k +1)T] = kp[ d – (kT)] + kI x[(k +1)T].					(8.5)

(8.5) -

t=(k+1)T, k =0,1,2, … -

.

8.2

. -

,

.

. ,

,

u[(k+1)T] -

( -

u[(k+1)T]). u[(k+1)T] ,

-

. ,

,

.

(t), ( ),

« » . -

(kT) ( ) -

.

228

. -

, ( -

) . -

( -

), (kT) -

. -

u[(k+1)T] -

.

.

		(t)

d
u[(k+1)T]
		(kT)
	8.2

-

u[(k+1)T], -

,

, .

,

, -

229

,

. , ,

,

.

, ,

. -

,

,

-

. ,

, : -

, -

.

-

.

D(z).

-

,

7.4.

8.2

, -

,

,

.

8.3.

x(t)

( ),

y(t) – , x(t) -

q – . -

( ) -

:

q= 2 – N ,

N – ( ).

230

y

2q

q

q

2

q

x

2

–q

–2q

x(t)		y(t)

Q

8.3

, ( x(t) – y(t)) -

	q		q	.
2		2

, , -

8.3, -

, , ( . 8.4):

, .