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224
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:
;
y(t) = F(u(t))
y(t) = F(u(t) t) ; :
,
(
. 72]).
( -
) -

x(t) = f(x u t) ;
_ (10.1)
y(t) = g(x u t): ; :
x(t) 2 Rn u(t) 2 Rm y(t) 2 Rl { ,
n g( )2Rl {
f( )2R -
.
f( ) g( ) .
MIMO- . SISO-
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n-
ny s
F y dy d y : : : d n u du : : : d u t = 0:
2

dt dt2 dt dt dts


dn y(t) =' y dy d2y : : : dn;1 y u du : : : ds u t : (10.2)
dtn dt dt2 dtn;1 dt dts
-
(10.1). ,-
n;1 y
x1(t) = y(t) x2 (t) = dy : : : xn (t) = d n;1 : -
dt dt
, , -
-


225
(10.2),
8
> x1(t)
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> x2(t)
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(10.3)
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dt dt
s
1 2 3

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1. . F(x)
x2R
F; F+ . . F; F(x) F+ : -
- ,
-
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8
< 1 x>1
sat(x) = : x ;1 x 1
;1 x < 1:
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x, ,
x 2 x; x+] x; < 0 < x+:
F(x) = 0 -
- ,
8
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;
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x<;
x+
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3. . -
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-
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< sat(x ; ) x >
;x
F(x) = : 0
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226
(" ") :
4. " .
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y(x) = c sign(x) ( ) sign(x)
-
8
< 1 x>0
sign(x) = : 0 x = 0
;1 x > 0:
c>0 { " ". 4

5. . -
. 2, 4:
8
<c x>
;
F(x) = : 0 x (10.4)
;c + x<; :
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:
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y(t) = F(u t0 t) 94].
t]
, sign(0) -
4
. ,
;1 1] y(x) 2 csign (x),
, sign(0) .
11.6.2. 30, 102].


227
2. -
y(t) = F(u(t) u(t) t).
_ -
-
(10.1), .
y(t) -
( ) 44].
, -
.
, -
, -

xp (t) = fp (xp(t) u(t) t) y(t) = gp (xp(t) u(t) t) (10.5)
_
xc (t) = fc (xc (t) y(t) t) u(t) = gc (xc(t) y(t) t) (10.6)
_
xp (t) 2 Rnp xc(t) 2 Rnc
y(t)2Rl {
,
, -
m{
, u(t)2R , -
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-
f( ), g( ) . u(t) y(t)
-
-
x(t) = colfxp (t) xc (t)g 2
R n n=n +n ,
p c
_
x(t) = f (x(t) t) y(t) = g (x(t) t) : (10.7)
1. -
, . -
, {
( , , ) .
,
-
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2.
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228
gp (xp (t) u(t) t) = 0
gc (xc(t) y(t) t) = 0:
-
- 72].
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,
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( -
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). -
.
-

x(t) = A(t)x(t) + B(t) (t) + r(t)
_
(t) = C(t)x(t) + D(t) (t) (10.8)
-

(t) = '( t): (10.9)
x(t) 2 Rn {
(10.8),
(t) 2 Rl {
(t) 2 Rm { (10.9).
r(t) 2 Rn
- '( ) t -
(10.8), (10.9)
( . 10.1).
-
.
(10.8), (10.9) A(t) 0n
, n
B(t) C(t) In D(t) 0n .. ,
n
{ ,
229
. 10.1. -
.
(t) -
(t)
x(t) = (t) (t) = x(t):
_ '(x t) f(x t) ,
x(t) = f(x t):
_
-
(t) 2 R ( -
), 5
( )
:
C(sI ; A);1 B
Wl (s) = + D:
, .
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.
,
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Wl (s) , -
(t) = ;'( t):
10.3.
, -
. -
, -
, -
5


( . 15, 76]).

230
. -
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.
10.3.1.
-
44].
. 1.1. y(t)
;
y(t) = S x(t0 ) u t0 t]
x(t0 ) { , u t0 t] { -
, t0 t] t > t0:
44]. ,
:
, ..
t0 t > t0 x(t0 ) = x0 u t0 t] v t0 t] k :
; ;
k S(x0 u t0 t] ) ; S(x0 v t0 t] ) = S 0 k(u t0 t] ; v t0 t] ) (10.10)
.. -
,
.
t t0 x0 (t0 )=x00
, ..
x00(t0 )= x00 k :
0
; ;
k S(x00 O) ; S(x00 O) = S k(x00 ; x00) O (10.11)
0 0

..
. 6

(10.10), k = 1 v = O,
S(x0 u t0 t]); S(x0 O) = S(0 u t0 t])
y(t) = S(x0 O) + S(0 u t0 t] ):
, { -

X
6 0
O{ , u(t) 0:

231
-
. -
S(x0 O) ,
x0 , -
;
S 0 ut t]
0
.
(10.11) , -
-
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(10.11) x00 = x0 x00 = 0
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k x0
S(kx0 O) = kS(x0 O):
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10.3.2.
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x(t) = x(t)2 ; x(t)
_ x(0) = x0 :
232
: x1 = 0 x2 = 1: -
, x0 < 1 x(t) x(t)
_ -
x1 x(t) ! 0:
, x(t) ! 1
x0 > 1 x(t) > 0
_
( x(t)
). , x1 { -
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X =R
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79]. , -
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10.3.3. .
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( -
5.1. 79, 93]).
.
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A(s) = 0 s=| . ( T = 2 = :)
, , -
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233
. 10.2. .

.
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x(t) = f x(t) :
_ (10.12)
79]. x(t) -
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> 0,
x0 2 X G -
G (x0 ), G (x0 ) <
, 0<
7

(10.12) .
2
G Rn
x
(x)
7
G
2G (jjx ; xG jj), jj jj {
G (x) = inf x ( ,
Rn :
)
G




234
, n=2
-
(10.12). ,
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T x + 2 T x + x = ku u = c signx
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(T = 0:1 c, = 0:25), -
79, 94].
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-
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8
, n = 2:

235
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55, 56, 76, 93])
. x(t) (t) (10.8), (10.9)
( -
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jjx(t)jj const 2) (t) -
t 2 0 1) 3) (t) -
; ] >0 > 0,
t 2 0 1): 2
10.3.4. .
-
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v .
x(t) = Ax(t)
_ v(x) = Ax -
fx g { det A 6= 0),
(
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( -
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fx g = N(A) N(A) X . 3 . 113.
(10.12),
x -
( {
xi x ):
f(x ) = 0: (10.13)
, -
fx g
(10.12),
.
(" "),
(" ..
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236
. 10.3 -
-
() ( ). 9




. 10.3. .

10.3.5. . -

12],
(10.12) , ,
, -
x00 2 X
x0 (
) L > 0 (L < 1) x0 x00

jjf(x0) ; f(x00)jj Ljjx0 ; x00jj: (10.14)

-
(10.12) 12, 79]. -
, f(x)
X -
, .
(10.14), , -
, . 5.1.
x + 0:5x + 5 sin x = 0
{ _ {
9
x + signx + x = 0:
_

237
.
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-
(10.12) .
, , -
f(x) ..10

f(x) -
, -
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,
76]. -
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(
10
L
(10.14)), -
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f(x) = x2
36]. , -
f(x) = sign(x)
, 0 .
(10.14) .

238
.
p
x(t) = sign(x(t)) jx(t)j
_ x(0) = x0 :
x0 = 0: , -
x1 (t) 0: ,
t2 x (t) = ; t2
, x2 (t) = 4 3 4 -
.
, -
.
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-
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10.3.6.
-
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-
.
,
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, (x) = 0
. 10.4, . , -
. ,
(x) = 0 -
.
.
-
, { -
239
. 10.4.

(10.12), f(x) -
( x) . -
( . 30, 102]).
. 11.6.
10.3.7.
, -
,
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, -
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,
(
), -
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15, 72, 76].
-
,
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. .
-

240
. -
, -
-
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( ) 113]. -
15, 76].
-
. -
,
-
. ,
-
,
-
113].
( ) -
-
.
, -
76, 94, 102, 106].




241
11. -

11.1.
.
{ . -

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242
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76, 94], -
103, 106] -
102, 191]. -
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243
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11.2.
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( , 1885)
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244
11.2.1.
-
-
.
-
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-
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( -
).
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11.2.2.
, ,
L -
L.
,
{ (-
) n = 2:
, 79].
G . -
L{ G.
,
x0 2 G x0 2 L { G L.
x0 2 L. x0 -
L
x00 2 L. x00 = (x0) -
,
1
-
, . 1.1.

245
, -
.
x0, ,
( ), . . -
x0 = (x0 ).
L
(x) { -
g = (x0) h = (x00):
O x:
, -
h = '(g): , g0 = (x0 ) -
, ,
g0 = '(g0 ): (11.1)

(g h) h = '(g): -
-
0 < d'(g)
. <1
dg g=g0
( )
d'(g) >1 ( -
dg g=g0
79]). -
-
.
. 11.3.4.
. 256.
, -
, . 72] . 13.3.
11.2.3. -

-
, -
94]. ,
, .
{


246
1( . ).
-
2

" ", " ", " "
N " "S 1, . . -
IP = N ; S = 1:
2( . ).

x1 (t) = f1(x1 x2 )
_
x2 (t) = f2(x1 x2 )
_
f1(x1 x2 ) f2 (x1 x2 )
P
x1 x2
@f1 (x1 x2 ) + @f2 (x1 x2 )
,
. @x1 @x2
3( ). -
P -
, -
, .
4( . ). G-
3
R @f1 (x1 x2 ) @f2 (x1 x2 )
, @x1 + @x2 dt < 0:
G

11.3. ( -
)
(
{ ) -
.
.
.. .. (1934), . .
(1957), . (1950), . . (1960). -
, -
.
( )
.
2
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3




247
, -
,
15, 76, 84, 94, 113].
11.3.1. ." "
-
, (. .
10.7)
;
x(t) = Ax(t) + B ; (t)
_ (t) = Cx(t)
(t) = '( (t)):
x(t) 2 Rn {
(t) 2 R { (t) 2 R { -
'( ). 4

(11.2) .

; B(s)
W (s) = C sIn ; A);1 B = A(s) (11.2)
(; ) :
5
d
p = dt (11.2) -

A(p) (t) = ;B(p) (t) (11.3)
(t) = '( (t))
A(p) B(p)
A(s) B(s) -
(11.2).
(11.3) -
( T = 2 = ).
" "
4
,
.
, , , -
5
n- ,
(11.2).

248
( ),
. , , (t) (t + T):
(t) (t + T):
,
, ( ) ,
, - -
H(!) = jW (|!)j -

H( ) H(k ) k = 2 3 4 ::: (11.4)
.. -
. 6

. -
, ,
-
2 3 ::: (
) ,
. -
-
(
). -
,
. ,
(11.2) -
, ,
,7
( ) .
, .
6
, ,
. a
H(!)
priori , -
, . 113].
, -
7
.
15, 76, 94, 113],
.


249
11.3.2.
, ,
-
: -
?
.
1. (t) = A sin t {
A 6= 0 : , -
, ,
A1 . . (t) = A1 sin t:
(t)
, sin t = A : -
(t)
(t) = A1 (t) = q (t) q = A1 { .
A A
, , -
(t) = q (t) q:
W (s) = q:
. -
, -
,
. , -
-
.
-
-
A1 A , , -
q
: q = q(A) q = q(A ):
W (s A) = q(A) W (s A ) = q(A ) {
A(
A ), . -
.
2.
(t) = A sin t , ..
(t) = A1 sin t+B1 cos t: (t)

250
, A 6= 0 6= 0
t
_ (t) : cos t = A1 _ (t):
cos t
(t) 1
(t) = A1 (t) + B1 1 _ (t): q = A1
sin t A A A
q 0 = B1 (t)
A
q 0 _ (t):
(t) = q (t) +
q, q 0 ,
, -
-
, -
q 0 = q 0(A) q 0 = q 0 (A
: q = q(A) q = q(A ) ):
-
q 0 (A)
: W (s A )=q(A)+ s: 8
" "
. -
(t) = A sin t:
(t) 15, 66, 76, 94]:
(t) '( (t)) '(A sin t) = A0 + A1 sin t + B1 cos t +
+ A2 sin(2 t) + B2 cos(2 t) + :
, -
, ..
(t) A0 + A1 sin t + B1 cos t
66]
1 Z 2 '(A sin )d
A0 = 2
Z2 0

A1 = 1 '(A sin ) sin d (11.5)
1 Z 2 '(A sin ) cos d
0

B=
1
0

q(A)
q0 (A) < 0
8
, , - .

251
q 0(A) ,
1 Z 2 '(A sin ) sin d
q(A ) = A (11.6)
Z0 2
q 0(A ) = 1
A 0 '(A sin ) cos d (11.7)
,
-
. ,
q(A) = 4c q 0 (A) = 0:
'( ) = c sign( ) A
,
(
), ( -
),
.
,
'( )
q0 = 0
(11.7) .
, -
(-
) -
: (t) = + A sin t:
0


( , ,
94, 113]).
-
(11.2).
11.3.3.
(11.2),
(11.3). -
( ) ,-
. 1.6.1. . 46, -
.
-
, -
( .(11.3)):
A(p) (t) = ;B(p) (t):
252
t
(t) = 0 e
2 C:
t:
(t) = e (t) (t)
0
, -
, . . A( ) 6= 0 A(s) {
s2C -
A(p) (t) , -
B( ) :
0=; ,
A( ) 0
(t) = ;W ( ) (t) W ( ) = W (s) s= W (s) { -
, (11.3) ( . (11.2)
(t) = ;W(| ) (t)
. 1.6.). =| :
W (| ) {
.
-
. , (t) =
; | t ;| t
cos t (t) = 20 e + e -
0
,
;
(t) = ; 20 W (| )e| t + W (;| )e;| t :
W (| )=H( )e| ( ) H( )=abs(W (| ))
{ , ( ) = arg(W (| )) { -
,
; |( t+ ( ) ;|( t+ ( )
) = ; 0 H( ) cos( t + ( ):
(t)=; 20 H( ) e +e
-
, (t) = q (t): - ,
(t) = 0 e| t 0 6= 0
(t) = ;W(| ) (t)
(t) = q (t):
(t)
,
t,
qW (| ) = ;1 (11.8)

. ,
253
W (s),
q , -
!= - "
" W (|!)
(;q 0): 9 , (11.8) -
(
0 0
(11.8) ). -
-
, { -
.
.
, ,
-
, -
. , ,

A(p) (t) = ;B(p) (t) (11.9)
0
(t) = q(A ) (t) + q (A ) _ (t)
q(A )
q 0(A ) (11.6), (11.7).
(t) = 0 e| t :
,
(t) = 0 W (| )e| t
_ (t) = ;| 0 W (| )e| t :

q 0 (A ) _ (t)= e| t W (| );q(A )+|q 0 (A ) :

<<

. 8
( 15)



>>