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. 11
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>>

r2R u
u = KxM xM + Kr r + us (12.39)
KxM = B + (AM ; A) Kr = B + BM us = B + (AM ; A):
, -
P = PT > 0 - u (t)
(12.39) ,
(A.10). P -
T
G = GT > 0:
PAM + AM P = ;G
(12.39) -

u(t) = KxM (t)xM (t) + Kr (t)r(t) + us (t) (12.40)

319
KxM (t) Kr (t) us (t) { , -
(t) = colfKxM (t) Kr (t) us (t)g:
; -
2
0 03
1 Imn
0 2 Imm 0
;=4 5
0 00
;
u(t) = KxM (t)xM (t) + Kr (t)r(t) ; sign B T Pe(t)
d KxM (t) = ; 1 B T Pe(t)xM (t)T (12.41)
dt
d Kr (t) = ; 2 B T Pe(t)r(t)T
dt
> 0:
1 2



, -
(12.35).
(MIMO) -
74, 93]. -
,
(. 12.1.)
. -
,
.
12.5. -
-
, " "{
( -
), " " " -
-
".
.
, -
- -
. ,
, , -
, . 12.1.
320
12.5.1.
-

x(t) = Ax(t) + Bu(t) y(t) = Cx(t)
_ (12.42)
u(t) = K(t)T y(t)
x 2 Rn u 2 R y 2 Rl K(t) 2 Rl {
.
Qt = 1 x(t)T Px(t) P=
2
T
= P > 0: . ,
;
_
Qt =!(x t)=x(t)T P Ax(t)+BK T y(t) r !(x t)=x(t)T PBy(t):
x(t)T PBy(t)
(
PB = C T g
),
l- g: -
, 36]
;
_
K(t) = ; y(t) ;y(t) (y) = g T y (12.43)
; = ;T > 0:
-
(A.10).
, K ,
T T
x PA x < 0 A = A + T C:
BK ,
P =P >0 K ,
PA + AT P < 0 PB = C T g A = A + BK T C: (12.44)
(
) 64,
104, 106].
. P = PT > 0 -
K (12.44), ,
T
g W(s) - - . 11
B(s)
, W(s) = A(s)
11

, B(s) { ( )
- -
n ; 1, n = degA(s)
36, 106].

321
;
W(s) = C sIn ;A ;1B {
.
,
T
x(t) !
g W(s)
0 K(t) ! const t ! 1:
(12.43) -
, (t) -
( ,
). (t) 0
T
g y(t) = 0 " ",
.
12


r(t) 7, 103]. 120] -
()
- -
-
.
12.5.2. -
,
" "( -
) (t)
(t) = g T x(t)
x(t) = Ax(t) + Bu(t)
_ (12.45)
x(t)2Rn u(t) (t)2R:
limt!1 x(t) = 0:
Qt = 1 (t)2
2
Qt = 0 t t:
12.1. -
.
,
;
_
Qt = !(x t) = g T x g T Ax + g T Bu(t) : (12.46)


u(t) = K(t)x(t) + us (t) (12.47)
12.1. n ; 1 -
12
(t) 0
;1 30, 106].
T
g W(s) n-

322
(t) = colfK(t) us (t)g:
;
rK !(x t) = ;gTT B gTT xxT (12.48)
rus !(x t) = g B g x:
-
(A.15)
u(t) = K(t)x(t) ; sign(g T B) (t) (12.49)
K(t) = ; 1(g T B) (t)x(t)T :
, (12.45), (12.48) T
( . . 12.1.) g x=T 0 -
, limt!1 g x(t) = 0 -
-
T; ;1
W(s) = g sI ; A B.
.-
(A.6) -
In 0
;= > 0:
1
00 1



;
u(t) = K(t)x(t) + Kr (t)r(t) ; sign g T B (t)
d K(t) = ; 1;g T B (t)x(t)T (12.50)
dt
> 0:
1 2


K(t) - -
9]
d K(t) = ; 1;g T B (t)x(t)T ; 2 d (g T B) (t)x(t)T : (12.51)
dt dt
- , -
y(t) 119], 12.1. ( -
(12.19), . 305).
12.5.3. . -


( ). -
323
. 12.1. -
.

1.5.3. . 42 -
,
13

W# (s) = s(s2 + (a!z +ma (s ; aa ) + a a!z ) :
y
z
y )s + mz
mz y mz
-
#(t) ( ) #(t) -
!z ,
( amz , ay amz
amz , ay :) -
(12.43) -
( ) -
.
# !z :
. -
, -
- .
, -
,
W# (s) { . , -
ay < 0: -
-
13
, W# (s) " ".

324
19, 23]. , g
(12.43) ( amz > 0).
( . 103, 106],
" = # ;#
64])
(12.43) -

_
k# (t) = ; (t)"(t) ; (k# (t) ; k# ) k# (0) = k#
o o
_
k! (t) = (t)!z (t) ; (k! (t) ; k! ) k! (0) = k!
o o
(12.52)
(t) = !z (t) ; "(t) "(t) = #(t) ; #(t)
(t) = k# (t)"(t) ; k! (t)!z (t):
>0{ , >0{ -
,( ),
>0{ .
oo
k# k! -
" " . ,

.
= !z ; " g (12.43)
g = 1 ]T : -
. (t) 0
_
#(t)# (t) = #(t) :
SIMULINK-
. 12.1 { 12.3,
{ . 12.4, 12.5.
-
-
: ay = ;1:3c ;1 , a = ;12:5c;2 , a!z = 0:5c;1
m mz
;2 = 150 = 0:02 z= 0:5c. 14
amz = 15:2c , -
(t)
. -
, -
.
.
, -
14
12.1 . 350.


325
-
(12.9), -
. 12.1.
= !z ; " (" = # ; #),
(12.52).
- -
(t) = !z (t) ; "(t) "(t) = #(t) ; #(t)
(t) = k# "(t) ; k! !z (t)
k# = k# "0
+
;
k#
k! = k! "0
+
;
k! :
SIMULINK -
. 12.6.
-
-
. 12.7.
k# = 2 k# = ;1 k! = 0:5c k! = ;0:25c
; ;
: +
15 +

= 0:5c.
-
- , ,
,
-
" ". -
-
(. ).
,
, -
.
-
12.7. -
.
-
, -
-
. -
.
, - -
15
. 302 (12.10).

326
. 12.2. - -
.




. 12.3. - -
.


327
. 12.4. # -
.




. 12.5. -
.

328
. 12.6. -
.




. 12.7. # -
.

329
12.6.
12.6.1.
-
,
. -
, -
( -
) -
. , -
, -
-
, .
-
103].
2, 7, 9, 23, 93, 106, 116].
-
-
2, 59, 72, 93, 87], -
2, 9, 93, 106, 103].
,
-
-
( -
).
. -
,
.
, , -
-
, -
, ( -
), { -
2, 8, 59, 93].
, -
, -
" -
", " ". -

330
" "
, -
-
59].
8, 59, 72, 87]: 1) , 2)
- , 3)
.
-
, -
,
. ( -
) -
.
-
-
,
. -
, . -
-
,
,
. -
, -
, -
.
( ) -
, ,
-
. , -
" "
, " ", -
,
, " ".
-
.

331
-
, -
. -
( ) {
, -
.
-
, -
-
. -
:{
{
{ " " ,
{ -
, -
.
, -
. -
72, 81, 82, 87, 139].
,
, -
.
,
-
.
-
.
12.6.2.
-

x(t) = A x(t) + B u(t)
_ (12.53)
x(t) 2 Rn u(t) 2 Rm :
,
;
xM (t) = GxM (t) + A(t) ; G x(t) + B(t)u(t)
_ (12.54)

332
xM (t)2Rn { ,G{ n n-
.
limt!1 Qt = 0
1
Qt = 2 e(t)Pe(t)T e(t) = xM (t) ; x(t) P = P T > 0: -
,
; ; ;
_
Qt =e(t)T P Ge(t)+ A(t);A x(t)+ B(t);B u(t) : (12.55)
, -

d A(t) = ; Pe(t)x(t)T d B(t) = ; Pe(t)u(t)T : (12.56)
dt dt
(A.10) P
T
PG + G P < 0:
x(t) ( , ,
) -
Qt ! 0:
-
:
A(t) ! A B(t) ! B (12.57)

. 13.5.3. ( A.),
( -
) -
colfx(t) u(t)g: , (12.57) ,
16

(12.53) ,
u(t) n .
12.6.3.
(12.53) -

˜ ˜
_
x(t) = Ax(t) + Bu(t) + v(t)
˜ (12.58)
˜˜
AB
x(t) u(t) v(t)
,
16 -
, (.
. 412).

333
,
= x(t) ; x(t):
=0 ˜
Qt = 1 T
2
;
_ ˜ ˜
Qt = T (A ; A(t))x(t) + (B ; B(t))u(t) ; v(t) (12.59)

˜˜
= colfA B vg: -
-
d A(t) = ; (t)x(t)T
˜
dt
d B(t) = ; (t)u(t)T
˜ (12.60)
dt
v(t) = ; 1 sign (t) (t) = x(t) ; x(t)
˜ 1 > 0:

,-
, 102].
-
.
12.6.4.
-
, ,
, ,
(,
, " -
").
-
,
,
.
-
.
SISO- ( -
).
SISO ,
- -
B(s)
, .. W(s) = A(s)
A(s) B(s):
334
, -

A(p)y(t) = B(p)u(t) (12.61)
d
p = dt { , -
P P
n;1 m
A(p) = pn + ai pi B(p) = bipi (m n)
i=0 i=0
ai bj i = 0 : : : n ; 1
n+m+1
j = 0 : : : m:
,-
u(t) y(t):
" "{
1
Wf (s) = G(s) G(s) { , -
degG(s) n ; 1:
Wf (s) u(t) y(t)
, " " uf (t) yf (t)

G(p)yf (t) = y(t) G(p)uf (t) = u(t): (12.62)
"( )
"
^ ^
( t) = A(s )yf (t) ; B(s )uf (t): (12.63)
-
ai bj Ti = 0 : : : n ; 1 j = 0 : : : m . .
: : : a0 ^m : : : ^0 : ^ ^
= ^n;1
a b b A(s ) B(s ) -
. -
,
.( ,
).
n;1 m
X X^
^ ^
= pn + ^ i pi bi pi
A(p ) a B(p ) = (12.64)
i=0 i=0
,
171]:
d ^i(t) = ; (t)yf (t)T i = 0 : : : n ; 1
dt a
(i)
(12.65)
d ^j (t) = ; (t)u(j) (t)T j = 0 : : : m:
dt b f

335
>0{ , (t) -
(12.63), uf (t), yf (t) { -
(12.62). , -
uf (t), yf (t), -
yf (t), uf (t) "
(i) (j)

" .
12.6.5.

-
( -
) ( -
).
8.2. ( . 183) , -
(-
)
.
2, 7, 116] , -
. 17

.
SISO- .
( . 1.8.), -
ABC
2 ;a ::: 1 3 2b 3
1 1
1 1
6 ;a2 ; 2 0 ::: 0 7 6 b2 7
6
: : : 0 7 B = 6 b3 7
7
A = 6 ;a3 0 ;. 3 67 (12.66)
6 .. 7 6 .. 7
... . . . ... 5
..
4. 4.5
;an 0 0 ::: ; n bn
C = 1 0 0 ::: 0 :
i (i = 2 3 : : : n) { aj bj (j =
1 2 : : : n) { -
.
17
134, 165, 166, 167, 173, 179].

336
-
;^
_ ^
x(t) = A(t) ; LC x(t) + B(t)u(t) + Ly(t)
^ ^ (12.67)
^ ^
A B(t) 2 R { T
A(t) { B
L = a1 (t) ; 1 a2 (t) : : : an (t)
- ^ ^ ^ 1{
n{
(t)2R
, .
^^
A(t) B(t)
-
2].
,
12.6.4.
171] 7].
, ,
:
(p+ 1 )(p+ 2 ) (p+ n )y(t)+ n (p+ 1 ) (p+ n;1 )y(t)+
(12.68)
+ 1 y(t)= n (p+ 1 ) (p+ n;1 )u(t) + + 1u(t)
d
i > 0(i = 1 2 : : : n) { p = dt .
(12.68), -
( m < n)
j, j -
(12.68)
B(s)
W(s) = A(s)
P P
n;1 m
A(s) = sn + ai si B(s) = bisi ( . (12.61)). -
i=0 i=0
-
sn
3] 1 s s ::: -
2

1 s+ (s + 1 )(s + 2) : : : (s + 1 )(s + 2) (s + n )
1
, , -
.
, -
(12.68), -
j j
^ j (t) ^ j (t): (12.63) -
,
G(s) = (s + 1)(s + 2) (s + n):
337
(t) = y(t) + s^+(t)n y(t) + (s + ^n;1 (t) ) y(t) + : : :
n
n )(s + n;1
+ (s + )^ 1 (t)(s + ) y(t) ; s^+(t)n u(t);
n (12.69)
n 1
; (s + ^n;1(t) ) u(t) ; : : : ; (s + )^1(t)(s + ) u(t):
)(s +
n n;1 n 1

(12.69) -
" "
u(t) y(t) (" ")
1:
s+ i
-
Qt = 1 ( )
2
2
d
dt i = ; (t)˜i (t)
y
(12.70)
d
dt i = ; (t)˜i (t)
u i = 1 2 ::: n
yi (t) ui (t) {
˜˜ -
.
(12.69), (12.70) (12.62){(12.65) -
-
( yi (t) ui (t)).
˜˜
(12.62){(12.65) yi (t)
˜ i-
y1 (t)
˜ (12.69) yi (t)
˜ y(t) -
i .
, (12.69) -
^i, ^i
. , ab -
-
W(s) (12.70)
^ j (t) ^ j (t) , (12.65)
^i , ^i
ab .
-
,
, .
- , ,
, -
. -
, " -
", , -
338
. -
. -
, -
. ,
, -
-
, ( ) -
, -
, -
.
12.7.
.
-
(" ") -
/ - -
.
-
, -
, -
( ), . 39, 69]. -
(
) ,
. -
69]
64]. -
107, 120, 122] , -
( -
, " "). -
- -
( ) (
64, 107, 164]). 124, 125]
,
-
.
122] -
339
,
.
12.7.1.

, -

xp (t) = Ap xp (t) + Bp u(t) yp (t) = Cp xp (t)
_ (12.71)
xp (t) 2 Rn , u(t) 2 R , yp (t) 2 R.
(12.71)
B(s)
Wp (s) = Cp (sIn ; Ap );1 Bp = A(s) (12.72)
s2C{ , deg A(s) = n deg B(s) = m k = n ; m
{ . , Wp (0) >
0 k > 1.
.
, ,
y(t) ( ). , -
(. 74, 124])
Am (p)yp (t) = KB(p)r(t) (12.73)
r(t) { ( ) ,p{
d
(p = dt ) Am(s) {
n K = Am (0) . (12.73)
B(0)
-
104, 120]
, .
K .
(12.73)
yf (t), -
- , -
. -
102]. -
, - (.
340
103, 104] 12.1. . 304) -
,
. -
. -
-
(" ", . 123, 164, 177] ), -
-
, -
.

B0(s) deg A0(s) = n0 :
Wc (s) = A0(s)
y(t) = yp (t)+ yc (t): -
uy
F(s)
W(s) = Wp (s) + Wc (s) = A(s)A0(s) (12.74)
F(s) = A(s)B0 (s)+A0(s)B(s) .
r(t) , -
y(t)
yp (t) of y(t) yf (t)
yp (t). -
- . Wr (s)
r(t) yp (t) , y(t) yf (t). (12.74)
,
B(s)A0 (s)
Wr (s) = Wf (s) F(s) (12.75)
Wf (s) { - . (12.73),
(12.75) ,
y(t) yf (t) Wf (s)
Wf (s) = A KF(s)(s) (12.76)
m (s)A0

K = Am (0) :
B(0)
, (12.76)
.
341
(12.76) -
- ,
T
xf (t) = Af xf (t) + Bf r(t) yf (t) =
_ (t)xf (t) (12.77)
xf (t) 2 RN (t) 2 RN { -
T
N = n + n0 .
: (t) = !1 (t) !2 (t) : : : !N (t)]
Af Bf
( , . 74). (t)
(12.76) -
T
; Af );1Bf :
Wf (s) = (sI P
F(s) = N !i sN;i . -
i=1

: !i i = 1 : : : N
N
X
!i sN;i = K(A(s)B0 (s) + A0 (s)B(s)): (12.78)
i=1
,
, 12.7.2.
-
107]:
" ("s + 1)k;2
Wc (s) = > 0: (12.79)
(s + )k;1

(12.74) (12.79) 107].
1. Wp (s) (12.72) { - (B(s) {
), k>1
Wp (0) > 0: 0>0
"0 ( ) > 0 , W(s) = Wp (s) +
Wc (s) { - ( ) >0
0 < " < "0( 0 ).
2. Wp (s) { (A(s) { -
), k > 1 Wp (0) > 0:
">0
, W(s) = Wp (s) + Wc (s) { 0.
0
, (12.79)

342
deg(As (s)) = k ; 1 = n ; m ; 1, -
" -
(12.74) -
.
2, -
(12.79),
(, , - ) .
(12.79) Wc (s) = s + :
, (12.79) -
(12.74)
. -
102, 189]:
x1 (t) = A11 x1 (t) + A12 x2(t)
_
x2 (t) = A21 x1 (t) + A22 x2(t) + bu(t)
_ (12.80)
y(t) = Cx(t)
x1(t) 2 RN;1 , x2 (t) 2 R y(t) = c1x1 (t) + c2 x2 (t) { -
, c2 b > 0 A11 xA12 A21 A22 b {
, C = c1 c2 ] :
, -
u(t) (t)
(12.77) , -
k , -
(12.73).
. -
.
u(t), -
(t) = y(t) ; yf (t)
.
12.7.2.
,
-
. , -
.
343
(12.72)
y (n) (t) + a1 y (n;1) (t) + + an y(t) = (12.81)
= b0 u(m) (t) + b1u(m;1) (t) + + bm u(t)
a1 : : : an b0 : : : bm { ( -
n n- -
).
y (n) (t) = 'T (t) (12.82)
'(t) = y n;1 (t) : : : y(t)T y(t) um (t) : : : u(t)]T = ;a1
_
;a2 : : : ;an b0 b1 : : : bm] '(t) 2 Rn+m+1. -
y (t) '(t)
˜˜
D(p)˜(n) (t) = y (n) (t) D(p)'(t) = '(t)
y ˜
D(p) = pn + d1pn;1 + + dn { -
d
, p dt . (12.82) ,
y (n) = 'T (t)
˜ ˜ (12.83)
y (t) '(t)
˜˜

_(t) = Ad (t) + bd y(t) _ (t) = Ad (t) + bdu(t)
(t) (t)2R n Ad bd
, det(sI ; Ad ) = D(s). , ,
. ,
(t)]T
'(t) =
˜ n (t) ::: (t) 1 (t) m+1 (t) :::
2 1

n
X
y (t) = y(t) ;
˜ dn;i+1 i (t) :
(n)

i=1
122]
_(t) = ;;(t)˜(t)'T (t)( T (t) ; (t)) =
'T ˜ T (12.84)
= ;;(t)˜(t)' (t) (t) + ;(t)'(t) ˜(t)
'˜ ˜
1
_
;(t) = ;;(t)˜(t)'T (t);(t) + (;(t) ; k0 ;2 (t))
'˜ (12.85)

344
˜(t)
k0 I > ;(0) = ;(0)T > 0 y (n) (t).
˜
'(t)
˜
, -
u(t) .
-
, . -
, 23, 106],
_ (t) = ;;(t)˜(t)'T (t) T (t) + ;(t)'(t) ˜(t)
' ˜T ˜ (12.86)
_
;(t) = ;;(t)˜(t)' (t);(t) + ;(t)

>0{ .
.
12.7.3.
, -
= y ; yf = 0. ,
(12.80),
_ (t) = c1x1 (t) + c2 x2 (t) ; yf (t) =
_ _ _
= c1A11 x1(t)+
+c1A12 x2(t) + c2A21 x1 (t) + c2 A22x2 (t) + c2bu(t) ; yf (t):
_
,
x2 (t) = c1 ( (t) + yf (t) ; c1 x1 (t)) (12.87)
2

(12.87) (12.87)
(c2 b);1 _ (t)=Lx1 (t)+a1 (t)+a1 yf (t);(c2 b);1y_f (t)+u(t) (12.88)
L { 1 (N ; 1)-
L = (c2 b);1 c1 A11 + c2 A21 ; c1A12 c c2A22 c1
+
2

1
a1 = c (c b) (c1 A12 + c2 A22):
22
x1 (t).
(12.87) (12.80),
x_1 (t) = A x1 (t) ; A12 (t) + A12 yf (t) (12.89)
c2 c2
345
A = A11 ; A12 c1 : (12.77), (12.88), (12.89) -
c2
. , -
. 191], A{ -
c2b > 0: , yf (t)
(jyf (t)j y f ) Af
r(t), f (t).

u(t) = ;ks (t) ; sign( (t)) (12.90)
ks { . (12.90)
-
= 0: , -
-
191]:
V1 = 1 (cb);1 2 + 1 xT Px1 V2 = 1 (cb);1 2:
2 21 2




. 12.8. .
-
. 12.8.
12.7.4. . -

-
-
( ) . , -
- ,
346
. , -
,
, -
.
-
23] ( . . . 1.4.2. . 29
1.5.3. . 41)
8
_ (t) = !z (t) + ay (t) + ay (t)
<
!_z (t) = ;amz (t) ; a!zz !z (t) ; amz (t) (12.91)
m
: _ = !z (t)
#(t)
#(t) !(t) { , (t) {
ay amz a! a
, (t) { !
amz { . -
-
. -
. ,
-
, (t): -
, -
#(t): (12.91)
(12.72), deg A(s) = 3 deg B(s) = 1 k = 2 :

Am(s) = p3 + am p2 + am p + am .
(12.73), 1 2 3
(12.81) -
(12.91)
a1 = a!zz ;ay a2 =amz ;a!zz ay a3 =0 b0 =;amz b1 = amz ay + ay amz :
m m
, a3
, -
.
-
k=2 (12.79)
Wc (s) = s + >0 > 0:
-
(12.76)
x# (t) = Ad x#(t) + bd #(t)
_ x (t) = Ad x (t) + bd (t) (12.92)
_

347
x#(t) x (t)2R3 Ad bd -
, det(sI;Ad ) D(s) = s3 +d1s2 +d2 s+d3 : -
'(t)
˜ '(t) = x# 3 (t) x# 2(t)
˜
˜(t) (12.84)
T
x 2(t) x 1 (t)] :
x# 3(t)
_ (12.92) x# (t) #(t) -
(t) 2 R
. 4


;a1 ;a2 b0 b1: ;(t) { 4 4,
;(0) = k0I:
(12.84), (12.85):
_(t) = ;;(t)˜(t)'T (t) T (t) + ;(t)'(t) ˜(t)
'˜ ˜
1
_
;(t) = ;;(t)˜(t)'T (t);(t) + ;(t) ; k0 ;2(t) : (12.93)

.
ya (t) = #(t)+yc (t)
yc (t) { -
(t) = ;ks (t) ; sign (t) (12.94)
(t) = y(t) ; yf (t) yf (t) { - (12.77).
(12.73) -
- , -
-
-
K = Am(0)
(12.77) (12.76), B(0)
F(s) = Ap (s)B0 (s) + A0(s)Bp (s) :
F(s) = s3 + ( a1 + b0)s2 + ( a2 + b0 + b1)s + b1 (12.95)
Wf (s) (12.76) { -
:
Am(s)A0 (s) = s4 + ( + d1)s3 + ( d1 + d2 )s2 + ( d2 + d3)s + d3 :
-




348
- (12.77):
xf 1(t) = xf 2(t) xf 2(t) = xf 3 (t)
_ _
xf 3(t) = xf 4(t)
_
xf 4(t) = ; d3xf 1(t) ; d3xf 2 (t) ; ( d1 + d2)xf 3 (t);
_
;( + d1)xf 4P4 + r(t)
(t)
(12.96)
yf (t) = K(t) i=1 !i (t)xf i (t)
!1 (t) = 4 (t) !2 (t) = 4 (t) + 3 (t) ; 2 (t)
!3 (t) = 3 (t) ; 1 (t) !4 =
K(t) = d(t) :3
4

-
. -
(12.74) F(s)
(12.95) -
. , -
19, 23]. -
W , b0 < 0 b1 < 0:
,
:
< 0 b0 + a1 < 0 b0 + a2 + b1 < 0 (12.97)
b2 + a1b0 + 2 a1a2 + b0b1 ; b1 > 0:
0

-
(12.94), (12.92),
(12.96), (12.79) (12.86) -
, -
. 12.1.
, -
3 -
.
-
:
{
Am (s)=s3 +am s2 +am s+am am =14:2 c;1 am =51 c;2 am =90 c;3:
1 2 3 1 2 3

: = 10 c;1 = ;2
{
(12.76): d1 = 20 c;1 d2 = 200 c;2
{
d3 = 103 c;3
349
12.1.

amz a!zz ay amz a1 a2
ay b0 b1
m
c;1c;2 c;1 c;1 c;1 c;2 c;2 c;2 c;3
N0
1 ;1:10 15:5 1:20 0:09 33:0 2:3 16:8 ;33 ;35
2 ;0:86 5:81 0:18 0:06 9:15 1:0 6:0 ;9:2 ;7:5
3 ;1:34 ;12:5 0:45 0:07 15:2 1:8 ;12 ;15:2 ;21:2


{ :
ks = 10 = 3
{ :
(0) = 0 0 0 ;10:]
;(0) = k0I k0 = 103 =5




. 12.9. = 0:5.
(12.97) -
, .. ,
(12.94). . 12.9
" " , -
(12.74)
F(s) = 0:5: 18
n
, -
18
= ;maxi (Resi) i = 1 2 : : : n 15, 95, 76].

350
. 12.10. .




. 12.11. .

ay amz amz : -
;1 ;2 :
a!zz = 0:18
ay = 0:07 . 12.9
m
, . 12.1.
. 12.10, 12.11.
-
r(t) # (t)
# (t) = #0 sign(sin(0:2 t))
#0 = 5 . -
12.1 . 12.10. . 12.11
1( -

351
). -
. -
, -
" " ,
( . (12.73), . 340) , -
, ay .
12.8.
1. ) , (12.16), (12.17) ( . 304), -
>0 -
, limt!1 = 0
Wu (s) -
.
.T
T
V (x) = x Px P = P > 0 , -
0 = fx : V (x) V0g ( -
, . 321).
) (12.19).
.
V (x) = xT Px + (K ; K )T ;;1 (K ; K )
K 2Rl P = P T > 0:
2. n- P" ( ) = "An ( )+
degAn ( ) = n degBn;1 ( ) = n ; 1 Bn;1 (0) > 0:
+Bn;1 ( )
"!0 n;1
, P" ( )
Bn;1 ( )
;1:
3.
( . 12.1) -
( . 12.5) -
103]. -
,
, -
.
)
.
) . 12.5.1, -
.
352
4. ,
. . 12.1 12.5 -
. , -
( ), (
), ( ).
MATLAB-
rand,
.
5.
12.5.1, (12.43)
;
_ (y) = g T y
K(t) = ; y(t) ;y(t) ; (K ; K) (12.98)
; = ;T > 0 - ,K- -
" " -
, >0- ( ).
;?




353
13. -

13.1.
(.. -
) . -
, ,
, .
, { -
.
, -
, -
, -
27]. : -
-
27] , -
- -
58],
.
,
-
, .. . -
(. ., 45, 11, 24, 55, 84].
, -
,
.. .
, .
XX . -
,
, ,
, . -
-
, XXI .
. -
-

354
, "
"
1994{1999 . 72, 128, 145, 150, 153].
-
.
1. ( ). -
, -
, ,
.. -
,
27, 76, 155]. -

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