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. 4
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0 ;1
40 1 5 41 05
1 ;1 ;12:6 ;12:6 ;1
0 1 0
˜
C = CT ;1 = 0 0 ;1:18 0]: -
, -

;1:18s
W(s) = s3 + s2 ; 11:6s ; 11:6 : (3.15)
, -
, -
.
93
7. ,
, -
SISO- . -
, -
. , Q (3.12),
2
;1:188 0 1:188 0 3
Q = 6 ;13:8 ;1:188 13:8 1:188 7
60 7
0 rankQ = 3 detQ = 0:
4 05
1:188
;15:0
0 0 13:8
-
-
.
3. .
. 32 (1.19) -
2 3 2 3
0 1 0 0 0
A = 6 ;k10 1 0
6m 07 6 7
;1 0 k2m;1 B = 6 0 7:
7
1
4 15 405
0
0 ;(k1 + k2 )m;1
k1 m;1 k2 m;1
0
2 2 2

, : m1 =
500 , m2 = 400 , k1 = 60 / , k2 = 170 /. -
. .
12

-
A , : s1 2 = 25:8|
s3 4 = 5:19|
2
;0:024 3 2
;0:024 3 2 3 2 3
0:182 0:182
6;0:527| 7 0 6 0:527| 7 60:947| 7 0 6;0:947| 7
x1 = 4 0:033 7 x2 = 6 0:033 7
06
x3 = 40:050 7 x4 = 6 0:050 7 :
06
5 4 5 5 4 5
;0:849| ;0:259|
0:849| 0:259|
, . 3.2.3. . 87,
12
, { .


94
2
;0:0243 2 3 2 3 2 3
0 0:182 0
h1 = 6 0:033 7 h2 = 6;0:527 7 h3 = 60:050 7 h4 = 6 0:947
607 6 7 607 6 7
7:
4 5 405 4 5 40 5
;0:259
0 0:849 0
2
;7:12 0 26:0 0 3
T = 6 4:68 ;0:276 2:91 1:01 7 :
60 7
0
4 05
0
0 0:903 0 0:560
2 3 2 3
0 25:8 0
0 0
6 ;25:8 0 7 6 428 7
0
07
˜4 ˜4 5
A=6 0 B=6 0 7
5:20 5
0 0
0 ;5:20 0
0 238
C = ;0:020 0 0:182 0 :
˜
0:033 0 0:050 0
, ( -
D) -
"
"( ). , -
-

W(s) = ; s2 +1668 s2 225 + s2 +126:9 ;s222534:7 :
+ 305 +
. -
-
˜
A, A
2 3
0 100
˜6 0 1 07:
0
A=6 7
4 0 0 15
0
;1:8 104 0 ;695 0
( . . 7.)
2 3 2 3
0 0 0 14 104 00 0 1
60 0 7 Q= 60 0 1 0 7:
0 14 104 7˜6
Q= 6 0 40 1 0 ;6957
;24 10
4 45 5
425 0
;24 104 1 0 ;695 0
425 0 0
95
2 3
7 10;6 0 0 0
6 7
7 10;6
T = 6 ;8:30 104 0 0 7
4 5
2:35 10;3
0 0
;8:3 10;4 2:35 10;3
0 0

1:44 105 0 0 0 :
C = 5:1 104 0 425 0
. 435 MATLAB-
ss2df, . 3.1.2. -
-
.
.
. 1.5.
3.3.
1.
8
< x1 (t) = ;2x1 (t) + 2u(t)
_
: x2 (t) = 4x1 (t) ;;2x (t) + u(t)
_ x (t) + 4x3 (t)
x3 (t) = ;4x2 (t) 3
_
y(t) = x2(t) + 3x3 (t):
-
, , . -
.
.
.
MATLAB- : 13
A= -2 0 0 4 -1 4 0 -4 -1]
B= 2 0 1] C= 0 1 3]
Ad,Bd,Cd,T]=ss2df(A,B,C)
num,den]=ss2tf(A,B,C,0,1)
Ac,Bc,Cc,Dc]=tf2cf(num,den)
C.
13 tf2cf, tf2cf.


96
Ao,Bo,Co,Do]=tf2of(num,den)
Ac,Bc,Cc,Dc]=tf2cf(num,den)
Ao,Bo,Co,Do]=tf2of(num,den)
2. -
1, 174]:
) .
T
A = AF ATF { (2.10) . 74,
B = 1 0 : : : 0] :
) {
(2.12) ( . . 2.2. .
75).
) {
(2.15) ( . . 2.3.
. 76).
) .
T
A = AF C = 0 : : : 0 1]:
, T -
. ) T = Q;1
:
Q{ (3.8), . 86 . ) T=
2 qn 3
6 qn A 7
6.7 qn { -
4 .. 5
qn An;1
Q (3.8) . ) T=Q Q{ -
(3.12) ( . . ,
. ) T = sn sn A : : : sn An;1 ]
. 89)
S = Q;1 Q {
sn {
(3.12).
3. -
( . ., , -
) - ,
. (2.16), . 78, -
.
4.
P . 82
(3.3) (1.18), . 31.
5. x(t) = Ax(t)+
_
B1 u(t) + B2 u(t) y(t) = Cx(t)
_ (1.45).


97
4.

, , -
-
n- 12, 66, 79].
,
, -
, .
SISO- .
,
r + 1 sr;1
W(s) = sn +ba ssn;1b+ a s + + + br;1a + br+ a = A(s)
s B(s) (4.1)
0
+ n;1s n
2 n;2
1

, . . r < n. 1
1.8., -
. -
-
-
. ,
. -
.
, -
-
( . 1.4. ).
A B C D (1.45) -
-
.
.
, -
.
r=n (1.45),
1
D 6= 0: r>n ,
(1.2).
.


98
, ,
, .. -
, -
-
X( ,{ -
(1.45)). , -
(
) . 2

SISO- , -
, -
degA(s)
. ,
-
( ) -
. , , ,
-
.
, , -
n: -
A
, -
3

X,
dimX
: x 2 X = Rn :
n-
-
. -
.
4.1.
(2.12)
AB (2.10), (2.11).
2
7.
, (4.1)
3
.




99
(4.1).
8
>_
> x1(t) = x2(t)
> x (t)
> _2 = x3(t)
>
<
...
>
>
> xn;1(t) = xn (t)
_
>
>
: xn (t) = ;an x1(t) ; an;1 x2(t) ; ; a1 xn (t) + u(t)
_
y(t) = br x1 (t) + br;1x2 (t) + + b0 xr+1(t)
x(t) = Ax(t) + Bu(t)
_
2 3 23
0 1 0 ::: 0 0 0
60 1 ::: 0 0 7
0
6 7 67
0 : : : 0 0 7 B = 6 0 7 (4.2)
60 0 6 ... 7
A = 6 ... ... 7
6 7 67
:::
6 7 405
40 0 ::: 0 1 5
0 1
;an ;an;1 ;an;2 : : : ;a2 ;a1
C = br br;1 : : : b0 0 :{z: 0 :
|:}
n;r;1
SIMO- ,
W(s) l1
2 B (s) 3
1

W(s) = A(s) 6 B2...(s) 7
16 7
4 5
Bl (s)
A(s) (4.1), Bj (s) -
rj < n j = 1 : : : l:
(4.2), 1 n- C l n-
2 3
b1 r1 b1 r1;1 : : : b1 0 : : : 0
C=4 5:
:::
bl rl bl rl ;1 : : : bl 0 : : : 0
4.2.
(2.15), , l = m = 1: -
A (2.10), -
, ,
100
. B -
A(s) B(s):

T
B= n;1 n ] : (4.3)
1 2

i = 1 ::: n -
i
.

j;1
X
= bj;1 ;
= b0 iaj;i j = 2 3 : : : n: (4.4)
j
1
i=1
bi (4.4) -
bi B(s) i = 0 1 ::: r
.
" -
" .
8
> x1 (t)
>_ = x2 (t) + 1 u(t)
> x (t)
> _2 = x3 (t) + 2 u(t)
>
<
... (4.5)
>
> xn;1 (t) =
>_ xn (t) n;1 u(t)
>
>
: xn (t) = ;anx1(t) ; ; a2xn;1(t) ; a1xn(t) + nu(t)
_
y(t) = x1(t):
, r = 0 B(s) = b0
( ,
b0
C = b0T 0 : : : 0] {
B = 0 : : : 0 b0 ] ).
MISO- ,
W(s) 1m

1
W(s) = A(s) B1(s) B2(s) : : : Bm(s) ]
A(s) (4.1), Bj (s) -
rj < n j = 1 : : : m:

101
(2.15), n 1- B n l-
2 ::: 1m 3
11 12
...
B=4 5
n2 :::
n1 nm
ij i = 1 ::: n j = 1 ::: m
(4.4) Bj (s):
, ,-
" " ,
, -
,{ -
. MIMO- -
. ,
,
( ) -
A:
4.3. -
SISO- ,
(4.1). -
, .
W(s)
, ..
P
q+r
(2.5), W(s) = Wi (s)
i=1
8
Ki
>
> i = 1 ::: q
< s ; si
Wi (s) = > d0j s + dj
j =i;q
> i=q+1 : : : q+r:
: s2 ; 2 j s + 2 + 2
j j
( -
) A,




102
(2.4)
2 3
s1 00 ::: ::: 0
s2 0
60 07
::: :::
6. ... 7
...
6 .. 7
:::
6 7
60 07
sq 0
::: 0 :::
6 7
A=6 0 0 7:
::: ::: 0 :::
6 7
1 1
60 07
0 ;1
::: ::: :::
6 7
6 .. ... 7
1
... ...
6. 7
6 7
4 r5
0 0 ::: 0 0 r
0 ;r
0 0 ::: 0 r
n 1- B 1 n- C
, -
(2.5) -
Wi (s) = s Kis
. 4
;i
bici = Ki : -

d0j s + dj
Wj (s) = s2 ; 2 s + 2 + 2
j j j
d0j dj
BC
dj = c1(b2 j ; b1 j ) ; c2 (b1 j + b2 j ):
d0j = c1 b1 + c2 b2
,
. ,-
, :
0
j dj + dj
c2 = d0:
b1 = 0 b2 = 1 c1 =
j

4.4.
-
, , -
. -
k k-
A
, -
4
.

103
, { 2k: W(s)
(2.5), (2.9). -
W(s) Wi (s)
A (2.6), -
-
(2.7), { (2.8).
BC
.
, s1j = s2j = : : : = skj
( kj ) Wj (s)
kj
X Kji :
Wj (s) =
i=1 (s ; skj )
i

B
b1 = b2 = : : : = bkj ;1 = 0 bkj = 1
C -
ci = Kji : { c1 = 1
{ . -
Kji , -
bi :
A(s) 94].
1.
- -
,
, -
, ,
.
2. -
, degB(s) = degA(s):
-
,
˜
B(s) B(s)
W(s) A(s) = d + A(s) :
d 1 1- D (1.45), -
˜
B(s)
˜
W(s) = A(s)
. -
A B C:
104
3. -
( -
), ( ). -
, ,
. , -
-
.
, . 1.7.
-
. , -
-
" ",
-
. -
-
1.7. .
4. -
, -
-
-
.
C. . 435,
SIMO-
tf2cf) MISO-
( (
tf2of).
4.5.

MIMO- , -
(m > 1 l > 1): -
, -
.5
1. 22

s1 s0:
1 1
W1(s) = 1 1 W2 (s) = 0 1
s s
5
1, 88, 174].

105
, -
W1 (s) :
x(t) = u1(t)
_ y1 (t) = x(t) + u2(t) y2 (t) = x(t) + u2(t):


A1 = 0 B1 = 1 0] C1 = 1 D1 = 0 1 :
1 01
W2(s) -

x1 (t) = u1 (t)
_ y1 (t) = x1 (t)
x2 (t) = u2 (t)
_ y2 (t) = x2 (t)


A2 = 0 0 B2 = 1 0 C2 = 1 0 D2 = 0 0 :
00 01 01 00
, :
-
. ,
MIMO- -
" " -
,
A(s) -
Wi j (s):
2. 88].
2 0:7 3
0
W(s) = 6 9s + 1 7:
4 5
2:0 0:4
8s + 1 9s + 1
s1 = ; 1 s 2 = ; 1
9 8

W(s) = 9s 1 1 0:7 0:4 + 8s 1 1 0 0 = 9s 1 1 M1+ 8s 1 1 M2:
0
+0 + 20 + +

106
. 4.1. 2.
-
M1 M2:
, -

M1 = 0:7 0:4
0 rankM1 = 2 M2 = 0 0 rankM2 = 1
0 20
n = dimX = 3: A -
:
21
;9 0 0 3
A = 4 0 ;1 0 5 :
9
0 0 ;1 8

3 2- B 2 3-
W(s):
C ,
,
2 0:7 3
0
C= 1 0 0 :
9
B=4 0 0:4 5
011
9
0
1
4

2 -
, -
88].
( ., -
, 1], 2 106]) -
.
107
4.6.
1. ,
W(s) = k( 1ss+ + T 2ss+ 1)
1)( + 1)
s(T 22
2
1

k = 10c;1 = 5c = 0:2c T1 = 1c T2 = 5:2c -
1 2
(2.12), (2.15)
(2.1). -
?
2. -
. 4.1 94].
3. , r=n (4.1)
D 6= 0 ( .
(1.45), 1
98).




108
5. -

-
.
.
, .. ,
. ,-
, -
( ),

x(t) = Ax(t)
_ x(0) = x0: (5.1)
5.1. -
-
( ).
, { -
12, 15, 79], .
x0
x(t) (5.1). t -
x
X: ,
(5.1),
. -
, -
( )
t:
-
,
.
-
.
-
12, 79].
(5.1) -
. -
109
, " "
= ;t: dx=d = ;Ax( ): -
, (5.1) -
t 2 (;1 1):
,
{ , -
x0 t1 t 2 -
.
:
-
. ,
-
.
-
, .. .
. -
.
, .
,
. ,
-
" " , -
. , -
1

, (
), , ,
.
( ) ,
, .
t 2 R : x(t) = x
,
x 2X t: -
T >0 ,
t x(t) = x(t + T)
( ) .
1


.
R.

110
jt1 ; t2j < T xi (t)
6
xi(t1 ) = xi (t2 ):
-
, -
. x(t)
,
T:
5.2. .
5.2.1.
, x(t)
X: t0
x0: -
, ,
x0 : v = x(t) t=t0 :
_ v -
-
, t0 ,
. -
(5.1), ,
x
v(x) = Ax:

x(t) ,
,
,
.
X -
, -
. , -
(5.1), x v(x) = Ax: -
, -
-



111
. . 5.1 -
2

.
( 94]
x + 2x ; 3x + 4 sat(x) = 0 sat( ) {
_ , ..
226).




. 5.1. .

5.2.2.
-
, ,
v(x) = 0: ,
( ) -
12, 79]. , x0 -
v(x0 ) = 0 x(t) x0 :
{ x(t) x0 -
x0 : -
,
, "
"( " ").
(5.1). -
X 0 = fx0g
,
, -
2


x = f(x)
_ . -
.

112
Ax0 = 0 (5.2)
A { n n- , x0 { n- .
53, 66, 115], (5.2) -
x0 = 0 -
: detA 6= 0:
, A ,
-
. A(s), . . -
-
A(s) = det(sIn ;A) n detA:
, A(0) an = (;1)
,
A:
, (5.1) -
. an = 0 -
.
.
.
x0 2 X 0 Ax0 = 0
X A , X = N(A):
-
0 3 0

N(A)
, 53, 115], -
X: -
N(A) -
X A : dimN (A) = n ; rankA: -
, A( , -
)
f0g, , , -
, ,
.
5.2.3.
, . 3.1.2. -
. .
53, 115].
X
.
X1 X2 : : : Xm (
X = X1 X2 Xm), :
) N (A)
( -
3 -
fxg x 2 N(A)
A ,
f0g 2 N(A).
Ax = 0 53, 115]. ,

113
x2X x = x1 + x2 +
x1 2 X1 : : : xm 2 Xm :
+ xm
.(
, :
x1 2 X1 : : : x1 2 Xm
x = x1 + x2 + + xm = 0
x1 = x2 = = xm = 0): 2
, -
X1 : : : Xm
f0g:
X
A
X1 : : : Xm . . x 2 Xi
Ax 2 Xi i = 1 2 : : : m, X -
,
-
- . -
:
X
( - ) , -
(
) .
f(s) ( . 3.2.) A
- -
X
: f(s) = f1(s)f2 (s)
X = X1 X2 -
A:
f(s) A
Q
m
f(s) = (s ; si)ri si { ( )
i=1
X
, ri { ,
X1 : : : Xm
m
A -
(si I;A) ri :
-
, f(s) A

q
m
Y Y
(s ; si) (s2 ; 2 j +
ri + j2 )pj
f(s) = 2
j
i=1 j=1
si { ,
sj j+1 = | { , -
j j
X

114
q
m
X X
X= Xkr Xjc:
k=1 k=1
-
A -
(2.1.3.).
X
, -
L
XiA, x2X
.. P
xi 2 XiA i = 1 2 : : : L
x = L i xi
i=1
( 3, 53]). -
, -
.
A -
, -
-
( )
( - -
). -
-
Gi ( )
2R{
si . . x0 = i xi , x0 {
00 0
i i
, si :
v(x0 ) = Ax0 = 0 six0 :
ii
. -
4

. -
Gi
,
5

x(t): , -
(5.1), x(t) = six(t) x(0) = 0 x0:
_ -
ii
si t x(0):
x(t) = e -
i(t) 2 R
.
, , 12].
4
, ,
5
Gi
, . -
Gi
,
:
f0g: si = 0
, ,
Gi .

115
_ i(t) = si i (t) i(0) = i: x(t) = i (t)x0 : ,
0
i
i (t) = esi t 0 : ,
i
Gi i (t): -
si : si < 0
f0g, si > 0 {
f0g, si = 0 { x(t) x0 ,
.6
, ,
k .
, k -
, ,k 3, 53, 115].
, -
, -
. -
: x(0) -
, -
x x : : : xk -
0 0 0
Pk
1 2
, Pk : x(0) 0 i=1 ixi :
= 0

x(t) : x(t) = i=1 i (t)xi i(t) =
esit 0 :
i
-
, -
-
. . 3.1.2.
, -
A:
, ,
.
,
-
.
, -
,
-
.
-
, -
6
.

116
.
5.3. -
, X = R2 :
.
7

.
s1 s2 A -
, s1 6= s2 :
f0g -
G1 G2 : si < 0 -
Gi ,
si > 0 { . si = 0
Gi . , ,
-
, ,
, -
.
-
s1,s2 :
1. . s1 < 0 s2 < 0
f0g
{
{ (. . 5.2, ).
.
,
K
W(s) = (T s + 1)(T s + 1) (T1 > 0 T2 > 0):
1 2

2. . s1 > 0 s2 > 0 -
, -
.
, -
7
, . -
A
, " -
" . ,
.
.


117
. 5.2. .

. {

K
W(s) = (T s ; 1)(T s ; 1) (T1 > 0 T2 > 0):
1 2

3. . -
, , s1 > 0 s2 < 0
G1 , -
G2 { (. . 5.2, ).
, ,
, -
. {

K
W(s) = (T s ; 1)(T s + 1) (T1 > 0 T2 > 0):
1 2

4. , -
.
, s1 = 0 s2 6= 0: G1
-
G1:
. ,
s2 < 0 -
G1 { .

118
-

W(s) = s(T K+ 1) W(s) = s(T K; 1)
s s
2 2

(T1 > 0 T2 > 0).
5. -
.
, -
.
A
(.. ), -
. -
-
. -
,
, -
. (
),
-
.
W(s) = K : ,
s2
- ,
{ .
6. -
.
-
s1 = s2, -
. -
, -
x(t) = es1 t
(5.1) -
, .
s1 = s2 < 0
, s1 = s2 > 0 { -
.
, -
.
, -
, -
, -
119
, ( . 5.2,
).
{
s 1 = s2 < 0 {
s1 = s2 > 0: -
-
, ..

W(s) = (TsK 1)2 W(s) = (TsK 1)2 (T > 0):
;
+
- -
s1 2 = | > 0:
f0g: :
6= 0
7. . , -
- .
<0
( ), >0{ (-
)( . . 5.2, ).


W(s) = T 2s2 + K Ts + 1 (0 < < 1 T > 0 )
2
{
W(s) = T 2 s2 ; K Ts + 1
2
( ).
8. . =0 -
.
-
2= { . -

W(s) = T 2 sK+ 1 :
2



. ( -
) ( . 2.1.) -
( . 2.2.), .
120
5.3.1. ( ) -
A
, A
, -
. , -
. -
.
1. ,
. .
. 5.3. ,
s1 s2 2 R s1 6= 0 s2 6= 0 s1 6= s2:
{
A = diagfs1 s2g . (.
3.1.1.), -
-
. ,
x = e1 = 1 0] , x = e2 = 0 1] : T T
(5.1)
1 2
0 0

8
> dx1 = s x (t) x (0) = x
<
dt 11 1 10
(5.3)
> dx2 = s2x2 (t) x2 (0) = x2 0:
:
dt
(5.3) ( , "-
s1 6= 0)
"
-
. x1
x2 :
dx2 = s2x2 dx2 = s2 dx1
dx1 s1x1 x2 s1 x1
s
lnjx2 j = s2 lnjx1 j + C1 -
1

s2
jx2j = Cjx1 j s1 (s1 6= 0 x1 6= 0): (5.4)
(5.4) ,
. , -
" ", {" ".
( -
), { .
121
C (5.4)
s2
; s1
C = jx2 0 j jx1 0 j : -
,
C:
(5.4) ,
( { s2) . (5.4)
.
(s1 < 0) . (5.4)
A = diagfs1 s1g:
, -
" " , .
.
A= s1 s
1 -
01
79] (5.4) .
2. -
.
5 . 5.3. -
, -
A= 0 1 :
00
8
> dx1 = x2 (t) x1 (0) = x1 0
<
dt (5.5)
> dx2 = 0
: x2 (0) = x2 0:
dt
x1(t) = x1 0 + x2 0 t x2 (t) = x2 0: -
{ , ,
" " x2 0 > 0 " -
" x2 0 > 0:
.
3. -
. .
s1 2 = |
> 0: -
A= ; {
8
> dx1 = x (t) + x (t) x (0) = x
<
dt 1 2 1 10
(5.6)
> dx2 = ; x1 (t) + x2 (t) x2 (0) = x2 0 :
:
dt
122
=0 t -
x2 + x2 = C C 0( ).
1 2
( > 0) -
.
, , , -
x1 > 0 x2 = 0:
6= 0 .
,
. -
0{
, = jxj ' { .
79]
(0) = jx(0)j
(t) = (0)e t (5.7)
' = '(0) ; t
-
. t( ,
), -
. <0
( ),
>0 {" " ( ).
{
.
5.3.2. -

-
( -
). A
(2.10).
0 1
, A = ;a ;a a1 a2 { -
2 1
A(s)=s2 +a1 s+a2:

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