ЦСАУ

81

(3.43) -

:

E*(s) = R*(s) – WH*(s)E*(s),

Y*(s) = W*(s) E*(s), (3.44)

WH*(s), Y*(s), E*(s) W*(s) , , -

, (2.25) (3.9).

(3.44) ,

, ,

». 3.18.

– WH*(s)

R*(s)		W*(s) Y*(s)

E*(s)

3.18

3.-

. , , -

3.18, Y*(s)

E*(s) :

Y * (s)			W * (s)		R * (s),
			WH * (s)
1
E * (s)			1		R * (s).
	1 WH * (s)

4.

, , -

-

.

-

.

3.19.

82

– WH*(s)

R*(s)	1			W*(s)		Y*(s)
				1
R(s)		1 E(s)				W(s)		1	Y(s)

E*(s)

– H(s)

3.19

, -

,

. ,

3.19 :

Y (s)	W (s)
			R * (s),
	1 WH * (s)
E(s) R(s)		W (s) H (s)		R * (s).	(3.45)
	1 WH * (s)

.

z-

(3.45).

-

, , , -

(Sedlar M. and Bekey G.).

z-

.

z- z ,

83

. .

, –

. -

, -

– . -

-

.

, , -

, , -

. ,

, z- .

( . . 3.20).

				y
			.
		3.20

W(s), , , – -

H(z). , -

-

. ,

,

– , -

,

H(z), – -

, -

. ( ),

, 3.21.

84

e*(t)

u(t)

y(t)

r(t)

e(t)

u*(t)

1

e Ts

H(z)

W(s)

s

3.21

:

F (s) 1 e Ts W (s). s

y(t) :

Y(s) = F(s) U*(s). (3.46)

, -(3.46) :

Y*(s) = F*(s) U*(s),

z- z = e Ts:

Y(z) = F(z) U(z).

,

3.22.

R(z)

E(z)

U(z)

Y(z)

H(z)

F(z)

r(kT)

e(kT)

u(kT)

y(kT)

3.22

,

-

.

85

4

, -

,

. -

-

( ), -

–

. -

, ,

.

-

:

-;

--

;

-;

.

-

,

) ,

.

, -

g, :

x(kT + T) = (x(kT), r(kT)),	(4.1)
y(kT) = g (x(kT), r(kT)),	(4.2)

y(kT) – kT, r(kT) –

kT.

86

(4.1) , -

(4.2) – .

4.1.1,

,

, (4.1), (4.2) -

:
x[(k + 1)T] = A(kT) x(kT) + B(kT) r(kT),	(4.3)
y(kT) = C(kT) x(kT) + D(kT) r(kT),	(4.4)

x(kT) – n, r(kT) – p, y(kT)

– m; A(kT), B(kT), C(kT), D(kT) – -

.

kT (k = 0, 1, 2, …). (4.3), (4.4) -

, A, B, C D .

(4.3), (4.4) ,

, T = 1:
x (k + 1) = A(k) x(k) + B(k) r(k),	(4.5)
y(k) = C(k) x(k) + D(k) r(k).	(4.6)

(4.5) (4.6)

, k « » «

»,

. (4.5)

(4.6) :

x(kj+1) = A(kj) x(kj) + B(kj) r(kj),	(4.7)
y(kj) = C(kj) x(kj)+ D(kj) r(kj).	(4.8)
, ,	,

(4.5), (4.6), 4.1.

87

D(k)

x(k)

y(k)

r(k)

B(k)

C(k)

T

A(k)

4.1

, -

–

). 4.2.

																					y1(t)
	r1(t)			r1*(t)									u1(t)
													.							.
													.							.
													.							.
													.							.
	rp(t)			rp*(t)									up(t)								ym(t)
								4.2
	ui(t), i						= 1, 2, …, p

, -

:

ui(t) = ui(kT) = ri(kT), kT t (k + 1)T.

(4.9)

88

-

:

	dx(t)	A(t) x(t) B(t) u(t),	(4.10)
	dt
	y(t) = C(t) x(t)+ D(t) u(t).		(4.11)
[1], (4.10)
:
		t
x(t) (t, t0 ) x(t0 ) (t, )B( )u( )d ,			(4.12)
		t0

(t, t0) – ( ) , -

A(t).

(4.12) t t0,

(4.12) t0=kT t=(k+1)T.

(4.9) u( ) ,

(4.12) :

x[(k +1)T] = [(k + 1)T, kT] x(kT) + [(k +1)T, kT] r(kT), (4.13)

(k 1)T

	[(k 1)T , kT] [(k 1)T , ] B( )dt.	(4.14)
	kT

(4.13) , ,

, 4.2.

-

. -

(4.11), t = kT

:

	y(kT) = C(kT) x(kT) + D(kT) r(kT).	(4.15)
	(4.13) (4.15) -
.
	(4.13) (4.15)
, = 1:
	x(k +1) = (k +1, k) x(k) + (k +1, k) r(k),	(4.16)
	y(k) = C(k) x(k) + D(k) r(k),	(4.17)
	k 1
	(k 1, k) (k 1, ) B( )d .	(4.18)
	k

89

, ,

(4.11) (4.12) t0 = tk, t = tk+1, tk tk+1 –

(

-

):

	x(tk+1) = (tk+1, tk) x(tk)+ (tk+1, tk) r(tk),	(4.19)
	y(tk) = C(tk) x(tk)+D(tk) r(tk),	(4.20)
	tk 1
	(tk 1, tk ) (tk 1, ) B( )d .	(4.21)

tk

, -

(4.5) -

(4.13). ,

, (4.12)

t t0,

(t, t0)

A(t). , (4.12)

t t0 ( « » ), t t0 ( « -» ).

(4.13), (4.16) (4.19) (4.12),

.

, , (4.16)

. (4.16)

x(k):

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

, -

:

x(k) = (k,k +1) x(k +1) + ( k,k +1) r(k),

	k
	(k, k 1) (k, ) B( )d .	(4.22)
	k 1

(4.22)

» k + 1 « » k. -

90

, (4.12) t0 = k + 1

t = k.

(4.13), (4.5) -

A(k) B(k) -. , A(k) k

. A(k)

k N, (4.5) ,

k, N.

:

x(k) = –1(k) x(k +1) – –1(k) (k) r(k), k N.

4.2

(4.3)–(4.8) (4.13)–(4.20) -

, ,

, , (4.5), (4.6) A(k)

(k+1,k), B(k) (k+1,k),

(4.16) (4.17) . -

, -

(4.3), (4.5), (4.7), (4.13), (4.16) (4.19).

(4.5)(4.6) . ,

-

.

, -

(4.5):

x(k +1) = A(k) x(k). (4.23)

x(k0), (4.23)

:

x(k0 +1) = A(k0) x(k0).

:

x(k0 + 2) = A(k0 +1) A(k0) x(k0).