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. 5
( 15)



>>

8
> dx1 = x (t)
< x1 (0) = x1 0
dt 2
(5.8)
> dx2 = ;a2x1 (t) ; a1x2 (t) x2 (0) = x2 0 :
:
dt
123
.
x2 (t)
x1(t)
" ", . . x1
( x2 > 0) x1
( x2 < 0):
, ( x1),
-
.
.
x1(t):
-
s1 6= s2 s1 s2 2 R:
. -
, -
x = colf1 s1 g
0
1
x = colf1 s2 g s1 6= s2: -
0
2


(si > 0 { )
(si < 0 { -
). , -
, . . 5.3
" ( ), s1 = 1 s2 = ;3
"
= ;0:2 = 1
" " ( ), s1 2 = |
(5.8). -
.
.)
:
al=-0.2 beta=1
{
A= 0 1 -(alb2+beta b2) 2*al]
{ A
x0= 0.5, 0.3]
{
t=0:0.05:15
{
u=zeros(size(t))
124
. 5.3. (5.8).

{
y=lsim(A,B,C,D,u,t,x0)
{
plot(y(:,1),y(:,2),'w'),grid
{
plot(t,y(:,1),'w'),grid
{ .
3 -
lsim.
-
. , -
-
( ) -
x2 : , -
,
, .
, . 5.3. -

125
-
, -
.
. -
T
, ( -
5.3.1.). -
T ;1 -
. , -
,
.
-
.
-
, xxx -
0 0 0
1 2 3
, h1 h2
( 4.3.). -
-
. -
12].
5.4.
1. ){ ) x = Ax
_
) A = 0:875 ;1:125 ) A = 1 ;4
1:125 ;1:675 2 ;5
;7:9 ) A = ;50:8 51:2
) A = ;22:9 2:6 26
;101
7:5
(.
121) , -
( . 123), , -
.
2. , -
. 5.4, -
(
).



126
. 5.4. 2.




127
6. -
.
6.1. .
, -

x(t) = A(t)x(t) + B(t)u(t)
_ x(t0 ) = x0 (6.1)
x(t)2Rn u(t)2Rm:
{ .. x(t)
x0 u(t): 1 -
.
6.1.1.

x(t) = A(t)x(t)
_ (6.2)
x(t) 2 Rn : n (6.2) -
t0 : xi (t0 ) = x0 i i = 1 2 : : : n : -
xi (t) n n- X(t) =
... x (t)... : : : ... x (t)]:
x1 (t) 2 -
n
12, 79, 97] (" -
"): W(t) =det X(t) 0
( t), W(t)6= 0 ( t). -
x0 i , X(t) -
t: X(t)
(6.2).
, - xi (t)
:-
x0 -
Pn
xi (t) : x(t) = i=1 ixi (t) -
i
x0 -
Pn
x0 i ( . . x0 = i=1 ix0 i ): -
X(t)
x(t)
, ,
1
y(t)
y(t) = C(t)x(t) + D(t)u(t):

128
T
: x(t) = X(t)C C= ::: n]
1
x0 = X0 C: , - X(t)

_
X(t) = A(t)X(t) X(t0 ) = X0 :
(t t0 ) = X(t)X0;1 -
, . , (t0 t0 ) =
,
In - . (t t0 ) -
, ,
x0 i
ei = 0 : : : |{z} : : : 0]T :
1
i
, -

_ (t t0 ) = A(t) (t t0 ) (t0 t0 ) = In : (6.3)
-
x0
x(i) x0
0
(6.2)
x(t) = (t t0 )x0 : (6.4)
,
.
,
(t t0 ):
(6.3) , (6.4)
.
.
25]. -
, ..
A(t) A: (t t0 )
= t ; t0 -
t0 ) = eA
= t ; t0
(t
1
X (A )k
(A )k +
A =I +A + (A ) +
2
en In +
+ k! : (6.5)
2 k!
k=1

129
, -

x(t) = eA x0 = t ; t0 : (6.6)
, -
t0 (
= t ; t0 ),
t0 = 0 (6.6)
x(t) = eAt x0 : (6.7)
-
,
. , A = diagfs1
eAt
s2 : : : sn g -
eAt esn t g:
diagfes1 t es2t : : :
: =
,
, A. -
-
( 6.5.), -
(6.1).
6.1.2.
, -
(6.1) x(t) = x (t) + x (t)
x (t) { { -
(6.2)
x (t) { - -
(6.1) .
3, 47, 94, 66] Z t
x (t) = (t )B( )u( )d :
t0
(6.4), -
: Z t
x(t) = (t t0 )x0 + (t )B( )u( )d : (6.8)
t0

Z t
x(t) = eA(t;t0 ) x eA(t; )Bu( )d
+ (6.9)
0
t0

130
Rt
t0 = 0 { x(t) = eAt x0 + eA(t; ) Bu( )d :
,
0
.
. -
,
W(s) = Ts1 1 : -
+
x(t) = Ax(t) + Bu(t) y(t) = Cx(t)
_
A = ;1=T B = 1=T C = 1: x0 = 0 u(t) 1
(6.9)
1 Z t e;(t; )=T d = e;t=T e =T t = 1 ; e;t=T
x(t) = T 0
0

-
.
6.1.3.
.
(t0 t0 ) = In:
1. t0
2. : t0 t1 t
(t t0 ) = (t t1 ) (t1 t0 ):

3. det (t t0 ) 6= 0 t0 t:
;1
4. (t t0 ) = X(t)X(t0 ) X(t) { -
.
5. (t t0 );1 = (t0 t) t0 t:
6.
_ (t t0 ) = A(t) (t t0 ) (t0 t0 ) = In:
(t0 t)T
7. -

d (t0 t)T = ;A(t)T (t t )T (t0 t0 ) = In :
dt 0




, " "
t0 :

131
(t t0 ) = T ;1˜ (t t0 )T ˜ (t t0 )
det T 6= 0
8.
(6.3),
˜
A(t) = TA(t)T ;1:
A(t)
,
eT ;1AT = T ;1eAT:
9.
(t t0 ) = eA(t;t0 ) = eAt e;At0 :
(t + t0 + ) = (t t0 )
-
.
6.2.
( ) w(t) ,
- -
15, 66, 76, 95].
-
( ), , -
{ , . -
, -
-
. -
- .
. -
diy
y(t), dti
u(t). w(t)
, ..
i
limt!0 d y
t < 0 y i (0; ) =
, u(t) 0 =0
t<0 dti
i = 0 : : : n ; 1: x(t) ,
(6.9), ,
u(t) . -
, (6.9)
-
. x(0) = x(0; ):
w(t) x0 = 0:
u(t) = (t) , -
x(0+ ) = limt!0 x(t) x0 : -
t>0
x(0+ ) - :
132
t=0 f(t) t 0 -
Rt
f( ) ( )d = f(0):
x(t) = eAtB:
0
(6.9) u(t) = (t) x0 = 0
y(t) = Cx(t) -
w(t) = CeAt B:
w(t) = CeAt B + D (t): 2
, x(t) -
-
x0 = B: ,
x(t) = Ax(t)
_ -
x0 = B w(t) = Cx(t)
, -
-
.
-
,
, -
.
6.3.
-
n- :
dn y(t) +a dn;1 y(t) + dm u(t) + + b u(t) (6.10)
+ an y(t)=b0 dtm m
dtn dtn;1
1


y(0; ) y(0; ) : : :
_
y n;1(0; ): x0 -

x(t) = Ax(t) + Bu(t)
_ y(t) = Cx(t) + Du(t) x(0) = x0 (6.11)

.
, u(t)2R
, -
2
,B { . -
i-
, -
B w(t)
wi(t) ui :

133
, u(t) 0
t<0 (t):
, t < 0 (6.11)
y(0; ) = Cx0 y(0; ) = x(0) = CAx0 : : : y n;1 (0; ) = CAn;1 x0 :
_ _
, n -
n x0
8
> Cx0 = y(0; )
>
< CAx = y(0; )
_ (6.12)
0
>
> n;1
: CA x = y n;1 (0 ):
;
0

(6.12) . -
2 3
C
6 CA 7
Q=6 7
4 5
CAn;1
z = y(0; ) y(0; ) : : : y n;1(0; )]T :
_
x0 = Q;1z:
(6.12) Qx0 = z
, , -
det Q 6= 0:
Q ,
7.3.,
(6.11). -
. , (6.11)
( . 2.3.), Q = In -
(6.10), , x0 = z:
, , -
y(0; ) = 0 y(0; ) = 0 : : : y n;1 (0; ) = 0
_ z=0 ,
, x0 = 0:
(6.10),
, x0 (
. 6.2. (t)).
6.4.
, -
, . -
-
. .
134
6.4.1.

x(t) = Ax(t) + Bu(t)
_ y(t) = Cx(t) + Du(t) t2R: (6.13)
-
: 3


x k + 1]=Px k]+Qu k] y k]=C 0 x k]+D0 u k] k=0 1 : : : :(6.14)
,
. ,
, u k] = u(tk ) tk = kT0 T0 = const {
, ,
y k] = y(tk ) { (6.13) (6.14)
tk = kT0:
,
.
1. .
-
,
, -
.
, -
,
" " .
2. .
-
-
.
( )
.
-
. ,
-
3
k = 0 1 ::: .
t2R -
, , , .


135
" " , -
- .
-
( ),
. ,
-
.
3. -
. -
. -

. -
-
. -
- -
, , -
-
. , ( -
t -
) T0 -
. -
4


- -
26].

" 36].
"
4. -
.
. -
-
,
.
, -
-
4
T0 < 0:05t -
" "
-
T0
. -
.

136
. ,
. -
,
, -
, , .
-
. , ,
, . -
, u(t) tk;1 t < tk tk = kT0 -
fu(ti)g k;1:
0
-
- -
, ,
. , -
u k] -
u(t) -
. , u k]
u(t): -
. ,
T0:
. -
-
u(t):
-
(" "),
u(t) = u(tk ) tk t < tk+1 tk = kT0 k = 0 1 2 : : : : (6.15)
6.4.2.
P Q C 0 D0 (6.14)
A B C D (6.13), -
. 6.4.1. -
u(t):
- -
(6.15). -
-
z- 15, 76, 95]. -
-
137
n W(s) o
(1 ; z ;1) Z Z
WD (z) = s
z- -
.
.
(6.9), -
(6.13) tk tk+1] u(t) u(tk )
x0 = x(tk ):
Z tk+1
) = eA(tk+1 ;tk ) x(t eA(tk+1; )Bu( )d =
x(tk+1 k) +
tk
Z tk+1
= eAT0 x(tk ) + eA(tk+1; )d Bu(tk ):
tk
=
Z Z
tk+1 T0
eA(tk+1; )d = eA d :
tk+1 ; : = tk+1 ; -
tk 0
(det A 6= 0),
A
Z T0
eA d = A;1(eAT0 ; In) ,
0

x(tk+1) = eAT0 x(tk ) + A;1(eAT0 ; In )Bu(tk ) det A 6= 0: (6.16)
(6.13), y(tk ) = Cx(tk ) + Du(tk ):
tk -
:
x k] = x(tk ) u k] = u(tk ) y k] = y(tk ):
(6.14) (6.16), , -
( det A 6= 0)
0 D0
PQC
P = eAT0 Q = A;1(P ; In ) B C 0 = C D0 = D: (6.17)
(6.14), -
1.5. :
WD (z) = C (zIn ; P);1Q + D: (6.18)

WD (z)
, -
. -

138
.
(6.17) Q -
A
. -
, -
.
6.5.
,
eAt -
,
.
, -
.
-
-
.
( -
).
6.5.1.
eAt -
, A -
, .. (6.13)
. ,
( ., -
, 3, 47]).
1. A -
.
A = diagfs1 s2 : : : sn g Imsi = 0 i = 1 : : : n: -
(6.5) ,
e At = diagfes1 t es2t : : : esn t g si t {
e .
2. A - -
.
0
A= ; 0 ,
, s1 2 = | |2 = ;1:

139
(6.5), ,
eAt = ;sin tt cos tt :
cos sin

A A= ;
( s1 2 = | ), A=
0
In + ; 0 : , -
, 5


0
0 t:
= e In t e ;
eAt
, . 1,2 , -

eAt = e t ;sin t t cos tt :
cos sin

3. A
. 2 3
010
A = 4 0 0 1 5 . . si = 0 i = 1 2 3:
000
,
2 3
001
A2 = 4 0 0 0 5 A3 = A4 = : : : = 0n :6
000
, (6.5)
2 3
1 t t2 =2
eAt = 4 0 1 t 5 :
00 1
eA+B = eA eB
,
5
, .. ,
AB = BA:
, ,
6
53, 115]. ,
.

140
-
2R
s1 = s2 = s3 =
2 3
10
A =40 15
.. .2
00
2 3
1 t t2 =2
eAt = e t 4 0 1 t 5 :
00 1
4. A -
.
I2
A
A 4 A= 0 A
22

2 2- A=; : A
s1 2 = s3 4 = |
. 2,
A = 0 I2 + A0 A0 : , -
00
-
.
2 3
cos t sin t t cos t t sin t
eAt = e t 6 ;sin t cos t ;tsin t t tsin tt 7 :
6 cos 7
40 5
0 cos
;sin t cos t
0 0
,
. ,
2J ::: 0 3 2 J1 t
03
e :. : :
At = 4 . .
...1 . . . ... 5 .. .. .. 5
e
A=4
0 : : : e Jl t
0 : : : Jl
J1 : : : Jl { .
A ,
-
˜ TAT ;1
T , A= { -
. , 8 ( . 6.1.3.),

141
eAt = T ;1eAtT: -
˜



( , A -
), -
.
-
.
-
3, 47, 94]. , R(s)
A ; ;1
: L(eAt ) = sIn ; A (.
10 . 35).
15, 66, 76, 94, 95].
6.5.2.
-
(6.5) , -
.
k, (6.5)

k
X (A )i
(A )k
(A )2 +
eA In + A + 2 In +
+ k! i! : (6.19)
i=1
, k=1
eA In + A (6.20)
.
7

(6.19) .
-
ex
. -
F (x)
ex G (x)
F G
.
7
6.10.2. .


142
(;
F (x) = 1 + ( + )1! x + ( + )( + 1); 1)2! x2 +
+ ( + )( ( +; 1) 1) 2 (1 + 1) ! x
; (6.21)
G (x) = 1 ; ( + )1! x + ( + )( ; 1); 1)2! x2 +
(
+
+ (;1) ( + )( ( +; 1) 1) 2 (1 + 1) ! x :
;
, x=A
eA F (A )G;1(A ) (6.22)
F (A ) G (A ) { (6.21).
(6.22)
( ).
(6.22).
, (6.19) -
(6.22) = 0: , -
(6.20) (1 0).
(0 1)
eA (In ; A );1 (6.23)
.
(1 1)
(. . 153)
eA (In + A =2) (In ; A =2);1 (6.24)
(2 2)
; ; ;1
eA 12In + 6A + (A )2 12In ; 6A + (A )2 : (6.25)
, (3 3)
(6.22),
F3 3 (A ) = 120In + 60A + 12(A )2 + (A )3
G3 3 (A ) = 120In ; 60A + 12(A )2 ; (A )3 : (6.26)

143
, ( k=
max( )) .
O( k )
(6.19) " -
" (6.22) ( ) ={ -
6= 0
O( 2 +1):
-
. 8

-
G (A ) .
, , , -
-
. -
G , ,
A , -
j
G ( ): (6.21) , = -
Re j > 0 j = 1 : : : : ,
det G (A ) 6= 0:
, G
-
:
! 0 G (A ) ! In ,
det G (A ) 6= 0:
-
(6.19) (6.22) ( -
)
(; ) 72]. -
k
eAT0 = eA T0 = k :
A { 53], -
.
;
+ an In
= ; a1
An An;1 + a An;2 +
2
ai {
det( In ; A) = n+ a1 n;1+ n;2 +
a2 +an :
, -
eA
. 6.10.
8



144
. . 6.7.
6.10. -
Q (6.14),
det A = 0:
6.6. Q
, (6.17) Q
det A 6= 0 :
, ,
Q A, -
, P=
eAT0 (6.19)
(6.22), (6.17) " " A:
;1
Q A . ,
(6.20) P = In + AT0 -
Q = BT0 (1 1) (6.24)
;1 BT :
Q = (In ; AT0=2)
(" ") { 0
-
(6.13). u(t)
tk t < tk+1 -
.
(6.17)
. PQ
" " .
-
u(t) = u(tk ) tk t < tk+1: -
(6.13)
x(t) = Ax(t) + Bu(t)
_ x(tk ) = x k] tk t < tk+1 (6.27)
u(t) = 0
_ u(tk ) = u k]:
(n+m)- x(t) =
colfx(t) u(t)g (n + m) (n + m)-
A= A B:
0m 0m
n m
(6.27)
_
x(t) = Ax(t) x(tk ) = colfx k] u k]g tk t < tk+1: (6.28)
( (6.14)) -

x k + 1] = P x k] (6.29)
145
P = eAT0 : A (6.5)
P , P
:
P 0 Q0 :
P= 0 I
m
(6.29) ,
x k + 1] = P 0 x k] + Q0u k]: (6.30)
(6.30) (6.14), , P Q (6.14)
P 0 Q0: ,9
P = eAT0 :
, .
6.5.1. , -
A ,
Q (6.17) . -
(6.9)
.
6.7.
-
-
. (6.15).
. 6.4.2. -
. -
k;1
fu(ti )g 0
u(t)
- .
z-
6.7.1.
-
, x k] y k] -
tk = kT0 ( 6.4.1.),
O ,
9 c2d
CONTROL SYSTEMS MATLAB 139].


146
tk " = (k + ")T0 0 " < 1: 10 , , , -
- (6.15),
,
x(tk ") x(tk+1 ") u(t) tk " t < tk+1 ":
, 6.4.2., (6.13)
tk " tk+1 "] (6.9).
Z tk+1 "
eA(tk+1 ";tk ")x(t eA(tk+1 "; ) Bu(
x(tk+1 ") = k ") + )d =
Z tk+1 tk "
eAT0 x(tk ") + eA(tk+1; ) d Bu(tk )+
=
tk "
Z tk+1 "
eA(tk+1 "; ) d Bu(tk+1):
+
tk+1
, (6.16) -
x(tk+1 ") = Px(tk ") + Q1u(tk ) + Q2u(tk+1)
y(tk ") = Cx(tk ") + Du(tk )
P = eAT0 det A 6= 0
- Q1 Q2 -

Q1 = A;1 (P ; P" ) Q2 = A;1 (In ; P" ) P" = eAT0 ":
x k] = x(tk ") y k] = y(tk ")

x k+1]=Px k]+Q1 u k]+Q2 u k+1]
y k]=Cx k]+Du k] k=1 2 : : : (6.31)
-
; P);1
WD (z ") = C (zIn (Q1 + Q2 z) + D:
Q1 Q2
, , (6.31) -
(6.14). -
(6.31) (6.14). -
x k] = x k] ; Q2u k]
˜ x k] = x k] + Q2 u k]
˜
6.4.1.
10
" = 0: "
z- ".


147
x k+1]=P x k]+(PQ2 +Q1 )u k] y k]=C˜ k]+(CQ2 +D)u k]:
˜ ˜ x
(6.14),
P =e AT0 Q = PQ2 + Q1 C 0 = C D0 = CQ2 + D:
Q1 Q2 -
, . 6.6. x(tk+1 "), -
(6.27) tk " tk+1]
x(tk ") = x k] u(tk ") = u k] -
tk+1 tk+1 "] x(tk+1) -
, u(tk+1) = u k + 1]:

x k + 1] = Px k] + P"Q1;"u k] + Q" u k + 1]
P" ˜ Q" Q1;" {
˜" = eAT0 " P" = eAT0 (1;") :
˜
P ,
˜

, P = P" P1;":
6.7.2.

u(tk ) tk t tk k = 0 1 2 :::
u(t) = (6.32)
0 tk < t kT0
0< 1{ , tk =
(k + )T0 : 11
(6.27). tk tk ]
x(tk ) = x k] u(tk ) = u k] -
tk tk+1] x(tk )
u(tk ) = 0: 6.6. 6.7.1.

x k+1]=Px k]+P1; Q u k] y k]=Cx k] k=0 1 : : : (6.33)
P1; Q {
˜ ˜
P1; = eAT0 (1; ) P = eAT0 :
˜ ˜


(6.15), 6.4.2. -
11
(6.32) = 1:




148
6.7.3.
-

u(t) = u(tk )e; tk t < tk+1 k = 0 1 2 ::: (6.34)
(t;tk )


{ . 6.6., -
, -

x(t) = Ax(t) + Bu(t)
_ x(tk ) = x k] tk t < tk+1 (6.35)
u(t) = ; u(t)
_ u(tk ) = u k]:
eAT
6.6., -
˜ 0
˜
, P
P0 Q0 :
P = 0 e; T0 I
m
P = P 0 Q = Q0
(6.14).
6.7.4.
, -
u(tk ) T0 :
8
> u(t ) 1; t;tk
>k tk t tk k=0 1 2 : : :
< T0
u(t)= > (6.36)
>
: 0 tk < t kT0
tk = (k + )T0 : x(tk ) -
, (6.13) (6.36)
A
6.6. .
tk t tk (6.36)
x(t) = Ax(t) + Bu(t)
_ x(tk ) = x k] tk t tk
u(t) = v(t)
_ u(tk ) = u k]: (6.37)
v(tk ) = ;u k]( T0 );1 :
v(t) = 0
_

149
. . 6.6. 6.7.2.
(n + 2m)- x(t) = col x(t) u(t) v(t)
(n + 2m) (n + 2m)-
2
0n m 3
A B
A = 4 0m n 0m Im 5 :
m
0m 0m 0m m
n m
(6.37) -
t 2 tk tk ],
P
2 3
P Q1 Q2
4 0m Im Im T0 5 :
P= n
0m n 0m Im
m

x(tk ) = P x(tk ) + Q1 u(tk ) +;Q2 v(tk ) = P x(tk ) + Q1 u(tk );
;Q2 u(tk)( T0 );1 = P x(tk)+ + Q1 ; Q2 ( T0);1 u(tk ):
;
Q = Q1 ; Q2 ( T0 );1
x(tk ): x(tk ) = P x(tk ) + Q u(tk ):
tk < t < tk+1 (6.36)
x(t) = Ax(t)
_ tk < t < tk+1
, x(tk ) = P1; x(tk ):
,
x k] (6.14),
P =e AT0 Q = P1; Q :
76] -
-
. -
eAT
R T0
0

eA(T ; ) ( )d () ,
0
0
.
6.8. -
, 6.4.2. (6.18), -
-
(6.14), -
(6.13).
W(s)
150
W(s) -
. 4.
.
, WD (z)
s W(s): -
" " ( ),
.
(6.20).
P = In + AT0 ,
(6.18) .
6.6. Q = BT0 :
WD (z) = C (zIn ; P);1 Q = C (zIn ; In ; AT0 );1BT0 =
z ; 1 In ; A ;1 B:
= C T0
W(s) = C (sIn ; A);1B , WD (z) -
W(s)
WD (z) = W(s) : (6.38)
;
s = zT 1
0



(6.23),
1
WD (z) = z W(s) z ; 1: (6.39)
s = zT
0

, (6.24) ( (1 1 )) -
:
2
WD (z) = z + 1 W(s) : (6.40)
s= T z ;1
2
0 z +1
-
T0
W(s). T0 -
. ,
. 6.10. (6.39) (6.40) -
( )
T0 > 0:


151
-
. , -
w- WD (z)
w = T0 | |2 = ;1 2 0 1) { -
2
15, 66, 76, 95]. , w- ,
z;1 = w a 2 = 1;w (6.40) -
z+1 z+1
WD (| ) = 1 ; T0 | W(| ):
2
, -
-
-
4'( ) = ;arctgT0 -
2
r
T02
1+ 4 . , -
2

,
-
" " -
.
! -
( )
T0: -
T0 0:3! ;1 :
. -
W(s) WD (z) -
zi = esi T0
, -
si zi : -
, -
-
(yi (t) = P(t)esi t yi k] = PD k]zik -
), , y(kT0) y k] ,
12

zi = esiT0 i = 1 : : : n: , -
W(s) WD (z)
P(t) PD k] { , -
12
si zi :

152
si zi : -
.
T0 -
.
WD (z) = W(s) 1: WD (z) -
s = T ln z
0
z, -
ln z: , -
z;1 z ; 1:
z ; 1 ln z
ln z ln z 2 z + 1
z
WD (z) = W T z ; 1
2
0 z+1
.
. 151 (6.40)
2 -
z+1
.
6.9. -
-
, -
, ..
.
-
. -
- -
,
. -
-
(1.5) WD (z): -
117, 118]. -
- -
. ,
Tj (j = 1 2 : : : N N{ -
)
T0
T0 = cj Tj cj
j = 1 2 : : : N: , Tj -
1
fj Tj = fj

153
1
f0 = T0 : -
-
,
T0
, -
fj = cj f0 :
6.9.1.
, -

x1 (t) = Ax1 (t) + Bu1 (t) y1 (t) = C1 x1(t)
_ (6.41)
x1(t) y1 (t) A B C { -
, u1 (t) -
u1 (t) = Ky1 (t) + g1 (k1 T1 ) k1 = E(t=T1) g(t) { -
, T1 = const { ,
E( ) .-
, (6.41),
,
-
T1 -
.
;
x1 (t) = A + BK x1 (t) + Bg1 (tk1 ) y1 (t) = Cx1 (t):
_ (6.42)
-
tk1 . 6.4.2. -
. :
x1 k1 + 1] = P1 x1 k1 ] + Q1g1 k1 ] y1 k1 ] = Cx1 k1 ] (6.43)
P1 = e(A+BK)T1 : g1 (tk1 ) -
-
T1 ( , tk1 g1 (tk1 ) g1 k1 ])
, -

x01 k1 + 1] = P10 x01 k1 ] + Q01g2 k1 ] g1 k1 ] = C1 x01 k1 ]
0 (6.44)

154
g2: -
x k1 ] = colfx1 k1 ] x01 k1 ]g: , -
(6.43), (6.44) :
x1 k1 + 1] = P1 x1 k1 ] + Q1g2 k1 ] y1 k1 ] = C1 x1 k1 ] (6.45)
P1 Q1 C1
P1 Q1C1 Q = 0
0
C1 = C ... 0]:
P1 = 0 P 0 Q01
1
1

, g2 k1 ]
tk2 tk2+T0 ] T0 =dT1
d: ,
d T1 (
tk2 = T0 k2 k2 = 0 1 2 : : :) g2 k1 ] = g2 k2 ]
k1 = E(t=T1 ) k2 = E(t=T0 ): x1 k2 + 1]
x1 k2 ] (6.45) ,
k1 k1 + 1 : : : k1 + d g2 k1 ] .-
(6.45),
x1 k1 +2]= P1 x1 k1 +1]+Q1 g2 k1 ]= P1 (P1 x1 k1 ]+Q1 g2 k1 ])+Q1 g2 k1 ]=
= P12 x1 k1 ] + (P1 + I)Q1g2 k1 ]
:::
x1 k2 + 1] = P1d x1 k2 ] + (P1d;1 + + P1 + I)Q1 g2 k2 ]:
, -
, -
k2 T 0 :

x1 k2 + 1] = P 0 1 x1 k2 ] + Q01 g2 k2 ] y1 k2 ] = C1 x1 k2 ] (6.46)
;
P 0 1 = P1d Q01 = P1d;1 + + P1 + I Q1 C1 = C ... 0]:
, g2 k2 ]
T0
x2 k2 + 1] = P2 x2 k2 ] + Q2g k2 ] g2 k2 ] = C2 x2 k2 ] (6.47)
, x k2 ] = colfx1 k2 ]
x2 k2 ]g (6.46), (6.47)

x k2 + 1] = P x k2 ] + Qg k2 ] y k2 ] = Cx k2 ] (6.48)
155
PQC :
P 0 1 Q01 C2 Q = 0 C = C1 ... 0]:
P= 0 P2 Q2
,
T0 :
,
,
T0:
(6.48), -
-
.
6.9.2.
-
(6.14) -
. -
, , , -
. -

- . -
tk = kT0 k = 0 1 : : : -

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