<<

. 10
( 15)



>>

, , , -
( . 11.3)
A(p) (t) = ;B(p) (t) (11.45)
(t) = '( (t)) (11.46)
d{ A(p) B(p) {
p -
dt .
-
(; ) :
B(s)
W (s) = A(s) (11.47)

286
s 2 C, B(s) A(s) -
p A(p) B(p) s.
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30]. ( , ,
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, (11.48)
; ;
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-
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287
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288
11.5.2.
:
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1) k1 k2 -
k1 6= 1 k2 6= ;1 2)
(11.48)),
k0 , k1 k0 k2 -
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W (s) = A (s)
B ,
(s)
. . A (|!) 6= 0 ; 2 ;1 +1]
;! 4) !
Re 1+k1 W (|!) k2W (|!)+1 >0: 23
-
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c>0 ">0
jx(t)j
(11.2) t > t0
;"(t;t0 ) 30].
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289
11.5.3. ..
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: 1)
2) '( ) { -
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), 24 0 k )
! 2 0 1]
3) # ,
;
1 +Re 1+|!# W (|!) > 0
k
-
15, 30, 76, 83, 94].
-
.
W (|!) = U (!) + |V (!), U (!) = U (!) =
Re(W (|!)) V (!) = !V (!) = !Im(W (|!)): , -
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;k
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.
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V (x) = xT Hx + # '( )d : 30, 94].
0
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11.6. . -
11.6.1.
( )
(10.7):
x(t) = f (x(t) t)
_ (11.49)
, 30].
24




290
(10.8), (10.9):
x(t) = Ax(t) + B (t) (t) = Cx(t)
_ (11.50)
(t) = '( t):
,
102], -
u(t) : 25

x(t) = (x(t) u(t) t)
_ (11.51)
u(t) = U(x t):
, -
( (11.51)) .
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291
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30, 102].
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292
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102].
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11.6.2.
-
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(11.49) , -
, (x t) = 0 f( ) -
, . . 27
f +(x t) (x t) > 0
f(x t) = f ;(x t) (x t) < 0:
( x)
f (x t) x(t) = f0 (x t)
_ -
0

, ..
(x t) 0 .
, -
(x), v ;(x)
v +

, -
.
v(x)
: , -
27
,
(x t) = 0:

293
v0 (x t) = 0
.
30]
. (11.50) ,
= 0 (t) '( t) . -
'( t)
(t) (t):
0
30] -
(t) 2 '( t), '( t) -
(
) -
;
, ]:
+


(t) 2 (t) = 0 (t):
0
, -
, -
(11.49) -
x(t) 2 f (x(t) t) :
_
102] (11.51),
, -
(x t) = 0 . .
u+ (x t) (x t) > 0
u(t) = u; (x t) (x t) < 0
ueq (t) ( -
" "), -
(x t) = 0:
-
.
-
.. .
(11.49).
-
-
v ; (x)
, v (x)
+

x
v (x) -
0




294
. 28

.
30],
'( t) ,
, ,
. , , -
,
.
30],
(11.50), -
, ..
x(t) = Ax(t) + B (t)
_
(t) 2 '(Cx(t) t):
- (t) -
. x(t) T -
det B B 6= 0 (t)
. -
, -
,
;1
T
(t) = B B B (x(t) ; Ax(t)) :
_
, 30], -
'( ) (11.50) ( -
) -
.
,
dim (t) = dim (t) = m i- '
i- : 'i = 'i ( i ):
,
, ,
(t) = 0 (t): , -
- -

x(t) = Ax(t) + B (t) Cx(t) = 0 (t)
_
28
, -
f( )
. ,
30, 102].


295
(t) { .
0



;
D(s) = det sInC A B :
0
;
D(s)=det(sIn ;A)det ; C(sIn ;A);1 B : 29 -
-
-
, -
.
-
, (..
(t)) (11.50),
-
-
30].
-
( -
, -
,-
).
102] -
.
(11.51). -
(x)=0 ,
(x(t)) (11.51)
. , -
ueq (t)
_ (t) = 0:
(11.51), -
(x(t)) = 0: , -
- ,
:
29


AB
detA 6= 0 det C D = detA det(D ; CA;1 B)

det D 6= 0 det A D = detD det(A ; BD;1 C) 30]:
B
C


296
x(t) = (x(t) ueq (t) t)
_ (x(t)) = 0:
-
(11.50),
(t) 0: (11.50) (11.51),
0

x(t) = Ax(t) + Bu(t)
_ (t) = Cx(t):
;
_ (t) , _ (t) = C Ax(t)+Bu(t)
ueq (t) = ;(CB);1 CAx(t)
det CB 6= 0):
( -
;
x(t) = A ; B(CB);1 CA x(t)
_ Cx(t) = 0:
, -
;
D(s) = det(sIn ;A)det ;C(sIn ;A);1B
30] -
.
, , -
-
.
30, 102].

-
. ,
, -
{ ( ) -
8, 21, 101, 102, 191].
. 12.1.




297
12.

-
:-
.
.
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.
12.1. -
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( ) 8, 9, 40, 102, 191].
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101, 102],
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.


298
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8, 102, 191].
-
102]. -
2

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-
.
:
{
{ ,
.
-
, -
5, 21, 102].
-
.

x(t)2Rn u(t)2R:
x(t) = Ax(t) + Bu(t)
_ (12.1)
( ) -

n
X
(x(t)) = Cx(t) ci xi(t) (12.2)
i=1
, ,
2
, -
-
,
.

299
C = c1 c2 : : : cn ] { - ,
.
0( (x(t)) x(t)
t t
(12.2)).
-
-
8, 102]:
{ -
(12.2)
{ -
{ .
, -
_t ,
t
..
lim _ > 0 lim _ < 0: (12.3)
!;0 !+0
,

(x(t) t) _ (t) < 0: (12.4)
x 2 X t2R
( , 102])
.
-
(12.1), (12.2), -
n ; 1. , -
-
u :
B(s)
W(s) = (sIn ; A);1 B = A(s) (12.5)
, , ci -
(1 n- ) C:
-
,
.

300
, -
C -
( A, B). 3

, -
( ). -
{ -
( ) -
. -
.
-
101]
n
X
u(t) = ; ki (x(t))xi (t) (12.6)
i=1
-
(x) = 0
ki+ xi (x) > 0
ki(x) = k ; xi (x) < 0 i = 1 2 : : : n: (12.7)
i
(x) = x:
ki+ ki; { ,
. -
(12.4) t _t < 0: ,
101]
(sign ( B)) ki+ > j Bj;1 ai (12.8)
(sign ( B)) ki; < j Bj;1 ai i = 1 2 ::: n
ai { A = a1 a2 : : : an :
x=0 -
. ,
( )
, -
3
. 294. , -
( . 3, 4, 102],
B(s) = c1 +c2 s+ +cn sn;1
(x) = x
. 2.2.), (12.5)
(12.1).

301
, -
-
.
. , -
101]
n;1
X
u (x) = ; ki(x)xi ; 0 sign( (x)) (12.9)
i=1
= const > 0 { -
0
, sign( 0 ) = sign( B):
-
-

(sign ( B)) ki+ j Bj;1 ai ; ci ( an )
(sign ( B)) ki; j Bj;1 ai ; ci ( an ) i = 1 2 : : : n (12.10)
an < 0:
-

ul (t) = T x(t) (12.11)
{ (
n;1
). -
(12.1), (12.11) -
, (-
) .
-
u+ (x) (x) > 0
u(x) = (12.12)
u; (x) (x) < 0
u+ (x) u;(x) { .
102], ,
(12.1), (12.12) x=0 -
, (12.3)

Bu; (x) > Bul (x)
Bu+ (x) > Bul (x) (12.13)
302
ul -
, , (12.13)
, u - -
:
u(t) = ; (x(t))x(t) ; (x(t)) (12.14)
= 1 : : : k 0 : : : 0]
( B)xi (x) > 0
i
i(x) = ( B)xi (x) < 0 i = 1 2 : : : k
i
(x) = 0 sign( B (x))
; i:
0 > 0, i



u(t) = ; l (x(t))ul (x(t)) ; (x(t)) (12.15)

( B)ul (x) > 0
l
l (x) = ( B)ul (x) < 0
l
(x) = 0 sign( B (x))
;1 ;1:
l l
-
-
x(t) ( {
(x)).
, -
.
-
(. . 8. 3, 4, 8,
47, 102]). " "
.
-
, 5, 9, 21, 102] 12.6.3. -
,
.
(
) , -
,
303
2, 7, 106, 116] ( .
12.6.5. . 336).
-
.
-
122].
,
-
9, 119, 191].
9]
x(t) = Ax(t) + Bu(t) y(t) = Lx(t)
_ (12.16)
x(t)2Rn u(t)2R y(t)2Rl : -
( ) -
y=0 c{ l- .
-

u = ; sign (y) (y) = y (12.17)
> 0:
W(s) = B(s)
, A(s)
, B(s)
-
n;1
{ ( ) -
36, 106], n = degA(s). -
(MIMO- )-
64, 106].
(. , . 321) , -
Wu (s) u
Wu (s) = L (sIn ; A);1 B (12.18)
- ,
-
limt!1 x(t) = 0: 4



,
4
V (x) = j (y(x))j 106].

304
9] -
K 2Rl :
u(t) = ;K T (t)y(t) ; sign ( (y(t))) (y(t)) = y(t) (12.19)
_
K(t) = ; (y(t));y(t)
; = ;T > 0 >0{ .
12.2. -
-
-
. 8. . 181 .
-
. 8. ,
, {
5, 21, 22, 102]. -
, , , -
(
., , 9, 74] 12.5.),
. -
-
, , ,
.
.


x(t)=Ax(t)+Bu(t) y(t)=Cx(t) x(t)2Rn y(t)2Rl : (12.20)
_
(12.20) . -
, , rank C = l:
102], -
.
y(t) = C1 x1 (t)+ C2 x2 (t)
x(t) = colfx1 (t) x2 (t)g x2(t) 2 Rl det C2 6= 0: ,
,
, , rank C = l:
.
305
x(t) = colfx1 (t) y(t)g ( .
˜ . 8.3. . 187
). ,
x
˜
T = IC C gn ; l :
n;l 0
gl
1 2


˜x ˜ ˜ ˜
_ A = TAT ;1 B = TB:
x(t) = A˜(t) + Bu(t)
˜

x1 (t) = A11x1 (t) + A12 y(t) + B1 u(t)
_ (12.21)
y(t) = A21x1 (t) + A22y(t) + B2 u(t):
_
A = A11 A12 gn ; l B = B1 gn ; l :
˜ ˜
A21 A22 gl B2 gl
102], (A C)
(A11 A21 ) . .
x1 = A11x1
_ z = A21 x1:
. 102]
_
x1 (t) = A11 x1(t) + A12 y (t) + B1 u(t) ; Lv(t)
b b b (12.22)
_
y (t) = A21x1 (t) + A22 y (t) + B2 u(t) + v(t)
b b b
v(t) = Msign t = y(t); y (t) M>0{ -
b
t
" ", sign( )
.
(12.21) (12.22),
:
(
" = x1 (t) ; x1 (t): (12.23)
"(t) = A11"(t) + A12 t + Lv(t)
_ b
_ (t) = A21"(t) + A22 t ; v(t):
- v(t) -
, =0 -
. y(t) y (t):
^
102], -
( )M
.
L -

306
". -
( . . 297)
_t = 0
v(t) v = veq -
(12.23), 0: -
t
, veq = A21x1
b
"(t) = A11"(t) + LA21 :"(t)
_ (12.24)
(A11 A21 ) -
L , -
(12.24), , -
{
( . . 7.3. , . 174, . 8.5 . 185).
-
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(8.10). -
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5, 21, 22] , -
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.
12.3.
12.3.1.
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5
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307
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74, 75, 93, 103, 106]:
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308
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309
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311
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30, 66, 64, 76, 93, 103].
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B., . 413.

313
- .
MATLAB Simulink 32, 72, 81, 82], -
(. 10]).
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9, 103, 106] A. . 407). -
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2, 7, 8]. -
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( ), .
64].
, -
-
.
12.4.
12.4.1.
1.
( )
x(t) = (A + A)x(t) + (B + B)r(t)
_ (12.25)
" -
8
,"
.

314
x(t) 2Rn { r(t) 2
Rm { ( ) A, B { n n-
n m- A, B { n n-
n m- . -
{ x(t) -
xM (t) 2 Rn ( ) , -

xM (t) = AM xM (t) + BM r(t)
_ (12.26)
AM , BM { n n- n m- ,
( AM ).
-
-
( ) 41, 42, 74, 75]. , -
9, 103, 106].
Qt = 1 e(t)T Pe(t)
2
e(t) = x(t) ; xM (t) { T
P =P >0{
n n- , .
_ t = e(t)T P ;(A + A)x(t) + (B + B)r(t) ; AM xM (t)+
!(x t) Q
+BM r(t) : 9
r A !(x t) = Pe(t)x(t)TT (12.27)
r B !(x t) = Pe(t)r(t) :
-
(A.14)
d A(t) = ; Pe(t)x(t)T
dt (12.28)
d B(t) = ; Pe(t)r(t)T :
dt
-
( (x t) 0).
(
(12.25) ). ,
= colfAM ; A BM ; Bg:
,
9
A(t), B(t):

315
P = PT > 0 10

PAM + AT P = ;G G = GT > 0: ,
M
0>0

!(x t) = e(t)T PAM e(t) = ;0:5e(t)T Ge(t) ; 0Qt :
xM (t) . .
r(t):
-
. -
-
9, 103, 106].
( ) = ( ; ^)
()
d A(t) = ; Pe(t)x(t)T + ; A(t) ; ^
A
dt (12.29)
d B(t) = ; Pe(t)r(t)T + ; B(t) ; ^
B
dt
^^
A B{
( . 64]).
,
- (A.6),
(A.8), - -

d A(t) = ; Pe(t)x(t)T ; 1 d ;Pe(t)x(t)T
dt dt (12.30)
d B(t) = ; Pe(t)r(t)T ; 1 d ;Pe(t)r(t)T > 0:
dt dt 1


, (12.28), (12.30)
, . . A+ A(t)!
! AM B + B(t) ! BM t!1 -
colfxM (t) r(t)g , .. -
(12.26) -
( , r(t) n -
, ).
2.

x(t) = Ax(t) + Bu(t)
_ (12.31)
-
10
11.4.4. . 274.

316
x(t)2Rn { , u(t)2Rm { -
r(t)2Rm -
.
( ) .
Qt = 1 eT Pe e =
2
= e(t) = x(t) ; xM (t) P = P T > 0 xM (t) { -
(12.26).
,
;
_
Qt =!(x t)=e(t)T P Ax(t)+Bu(t);AM xM (t);BM r(t) :(12.32)
x xM 2Rn r2Rm
Ax(t) + Bu (t) ; AM xM (t) ; BM r(t) = AM e(t) (12.33)
u 2Rm : u (t)

u (t) = Kr r(t) + Kx x(t): (12.34)
Kr = B + BM Kx = B + (AM ; A) . . AM ; A L(B)
L(B) L(B) {
BM , -
B:

rankB = rankfB BM g = rankfB AM ; Ag: (12.35)
(12.35) , -
, -
".
P=
T
, !(x t) ;e(t)TGe(t)
=P >0 - u (t)
; 0Qt . . (A.10)
(Qt ) = 0 Qt : P

PAM + AT P = ;G G = GT > 0:
M


= colfKx Kr g.
rKx !(x t) = BTT Pe(t)x(t)TT (12.36)
rKr !(x t) = B Pe(t)r(t) :
317
41,
75]
u(t) = Kr (t)r(t) + Kx (t)x(t)
d Kx (t) = ; B T Pe(t)x(t)T (12.37)
dt
d Kr (t) = ; B T Pe(t)r(t)T :
dt
(12.37) 38, 41, 75].
, -
B(s)
W(s)= A(s)
,
n;k;1 k = degB(s): -
, .. B(s) ( -
. . 12.1. -
(12.18) . 304).
, 70- -
, -
39, 64, 69]. 12.6.5.
. , -
, -
. 12.7.
12.4.2.
74, 75, 170] -
.
.
-
(12.26). (12.31).
1 e(t)T Pe(t)
P = PT > 0
Qt = 2 T;
_
(12.32): Qt = !(x t) = e(t) P Ax(t) + Bu(t) ;
AM xM (t);BM r(t) : , P -
T T
PAM + AM P = ;G G = G > 0:
-
u(t) ( (t) u(t))
-
(A.15), (A.9).
ru !(x ) = BT Pe(t):
318
(A.15) ( = 0)
0
;
u(t) = ; sign B T Pe(t) : (12.38)
(12.38) -
, (
, ). -
-
- -
, .
12.4.3. -
- 43, 75]
-
.
-
, -
, .
, -
(12.31) (12.26).
: e(t) ! 0 t ! 1:
1
Qt = 2 e(t)T Pe(t): -
_
Q (12.32). ,
(12.35)
u 2 Rm x xM 2 Rn
(12.33)
m:

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