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. 2
( 15)



>>

, , -
-
{ -
(1.25).
(1.24) -
z- 76, 66].
;
W( ) = C In ; A ;1B
.
W( ) -
.
, y(t) 2 R u(t) 2 R l = m = 1 W( ) -
: W( ) = A( ) :
B(
)
, -
Wi j ( ) = Ai j ( )
B i = 1 ::: l j = 1 ::: m
i j( )
ui yi :
2Y ( ) 3 2 W1 1( ) : : : W1 m( ) 3 2 U1 ( ) 3
1
4 ... ... 5 4 ... 5 :
5 = 4 ... ...
Yl ( ) Wl m ( ) Um ( )
Wl 1 ( ) : : :
X(s) x(t)
- -
10

L x(t) = R01 e;st x(t)dt:
;
s



35
1.5.2.
W( ).
-
; ;1
R( ) = In; A A:
;
In ; A
R( ) = adj In ; A)
det(
adj( ) -
; T
In ; A In ; A)
(
53]. -
n det( In ; A) = A( ) = n + a1 n;1 + a2 n;2 +
+an . -
A: , Wi j ( )
(1.25), ( -
) Ai j ( ) A( ):
A -
. -
, -
.
i i
A:
f ig 53].
(1.23) -
, , -
A -
C:
(1.24) A
C -
.
R( ) , -
In ; A , -
{ :
( , )
.
: { , -
47, 94],
,
. -
-
, -
36
. -
, -
-
-
1, 100]. 11
" " -
-
66]
. .
W( ) = CWx ( )+D Wx( ) = ; In ;A ;1B: ( ,
Wx ( ) n m-
Wx ( ) B Wx( ) =
).
w1( ) w2( ) : : : wm( )] B = b1 b2 : : : bm] wj ( ) bj
Wx( ) B:
j = 1 2 ::: m {
wj ( ), ,
( In ; A)wj ( ) = bj j = 1 2 ::: m (1.26)
n -
n -
wj ( ) = w1j ( ) w2j ( ) : : : wmj ( )] .
T
(1.26)
3, 53, 66],
wij ( ) = ij(( )) i = 1 2 : : : n j = 1 2 ::: m (1.27)
( ) = det( In ; A)
(1.27), -
A, ij ( ) , i-
In ; A bj :
,
Wx( ) .
C -
D .
,
A0 x(t) = A1x(t) + B1 u(t)
_ (1.28)
A n ,
11
i;j
aij aij = 0 2 (i j =
1 2 : : : n).

37
A0 ;n n- , detA0 6= 0:
A = A;1 A1 B = A;1B1 :
(1.23) 0 0
, -

( A0 ; A1 )wj ( ) = bj j = 1 2 ::: m: (1.29)
(1.29) ( ) = det( A0 ; A1 )
( )
A:
.
1.5.3. -

1. . -
. 1.4.1. . 25, RLC- . -
, (1.12)
y(t) = x(t) ( { )
;1
1 1= 1 :
W(s) = s + T -
T Ts + 1
15, 76] ( ). -
uR (t) , -
Ts
W(s) = Ts + 1 { -
.
(1.13), . 26, (1.26)
(s + RL;1 )w1 (s) + L;1w2 (s) = L;1
;C;1w1(s) + sw2(s) = 0
(Ls + R)w1 (s) + w2 (s) = 1 (1.30)
;w1(s) + sCw2(s) = 0
s 2 C w1 (s) w2 (s) {
x1 x2 : (1.30) (s) = LCs2 + RCs + 1 1 (s) = Cs
2 (s) = 1

w1 (s) = LCs2 +Cs + 1 w2 (s) = LCs2 + 1 + 1 :
RCs RCs
(1.13), -

W(s) = ;Rw1 (s);w2 (s)+1 = ;RCs ; RL ; 1 + LCs + RCs + 1 =
2

LCs + RCs + 1
2


38
q
p
Ks2 = R C:
= T 2 s2 + 2 Ts + 1 T = LC K = LC = T -
2
2L
-
( < 1)
15, 76] ( 1) -
. - -
( - ).
, -
-
. , -
D 6= 0
. .
.
(1.28), (1.29).
2. .
(1.14), .26.
(1.29)
8
< sw e(s) ; w! e(s) = 0
(Ls + R)wi (s) + C w! e (s) = 1
: ;C w (s)e+ Jw e(s) = 0
M ie !e
8
< sw M (s) ; w! M (s) = 0
: (Ls +w (s) + Jw Ce w! M (s) = 0
R)wi M (s) +
;C (s) = ;1:
M iM !M
wj k (s) j 2 f i !g k 2 fe Mg -
e(t) M(t)
(t) i(t) !(t): -
T
(t) i(t)] -
: w e (s) w M (s) wi e (s) wi M (s): -
(s) = s(JLs + JRs + Ce CM ) (s) e = CM
2

(s) M = ;(Ls + R) (s)i e = Js2 (s)i e = Ce s -
(1.14):
2 3
;(Ls+R)
CM
W(s)= 4 s(JLs +JRs+Ce CM ) s(JLs +JRs+Ce CM )5 : (1.31)
2 2
Ce
Js
JLs +JRs+Ce CM JLs +JRs+Ce CM
2 2

, -
( -
39
) , -
.
, ( , -
)
,
. 12

,
-
, -
.
-
-
{ .
JR 4LCe CM , -
2

{ , .
(
) ( ).
-
. .
1.8 -
(1.31).
3. .
.
, -
. 1.4.2. . 27. (1.15) (1.29),
sw (s) ; w! (s) = 0 (1.32)
Jx sw! (s) = 1:
(s) = Jx s2 (s) = 1 W(s) = K K = Jx
;1
s 2
..
.
. 28, -
(1.16) ( {
(1.17)).
:
12
l=m=1
{ (single input { single
SISO
output),
l>1 m>1{
{ (multi input { multi output).
MIMO
, , SIMO MISO.


40
. 1.8. .
-
.
,
, .
,
, ( ) 19,
23, 98]:
{ V (t)
{ x(t) H(t)
{
: (t) #(t) !z (t):
,
( )
. -
,
VxH (1.16) .
,
23]
8_
< (t) = (;ay + ay ) (t) ; ay #(t) + ay (t)
!z (t) = amz (t) ; a!zz !z (t) ; amz #(t) ; amz (t)
_ (1.33)
m
: #(t) = ! (t):
_ z
-
( ) ay -
41
, -
ay . ,
(t) = #(t) ; (t), -
19, 23, 98].
(1.33)
8
_ (t) = !z (t) + ay (t)
<
!z (t) = ;amz (t) ; a!zz !z (t) ; amz (t)
_ (1.34)
m
: _ = !z (t):
#(t)
, -
, , (1.29)

(s ; ay )w (s) ; w!z (s) = 0 (1.35)
;amz w (s) + (s + a!zz )w!z (s) = ;amz :
m


:

W!z (s) = s2 + (a!z ;amz (s ; ay ) + a a!z
mz + ay )s + amz y mz
W (s) = s2 + (a!z + a;am+ a + a a!z : (1.36)
z
)s mz y mz
mz y
(1.34) -
1 W! (s):
W# (s) = s z

( ) .
amz . amz > 0 -
,
{ . 13


(1.25)
MATLAB-5 82]. ,
W!z (s) W (s) (1.33)
.
13
-
(. 19, 98]).

42
syms a alpha y a delta y a alpha m a omega m
syms a delta m a thet y s al thet om del
{
I=eye(3,3) {
A= -a thet y+a alpha y, 0 ,-a alpha y ...
a alpha m, -a omega m, -a alpha m ...
0 1 0]
B= a delta y -a delta m 0]
C= 0, 1, 0 -1, 0, 1]
{ ABC (1.33)
W=C*inv(s*I-A)*B
{
num,den]=numden(W)
{ -
d1=collect(den)
lln1=collect(num)
{ .
-
W!z (s) W (s)

W!z (s) = ;s(amz s ; amz A(s) amz (ay ; ay ))
ay +

ay s2 + (ay a!zz + amz )s + amz ay
W (s) = ; m
A(s)
A(s) = s3 + (a!zz ; ay + ay )s2 + (amz + (ay ; ay )a!zz )s + ay amz :
m m
ay = 0 ay = 0
n2=subs(n1, a thet y,a delta y], 0 0])
d2=subs(d1, a thet y,a delta y], 0 0])
(1.36).
4. . -
-



43
-
(. 2 . 1.4.3. . 32).
(1.19) :
8
> sw1 (s) ; w2 (s) = 0
>
< k w (s) + m sw (s) ; k w (s) = 0
11 1 2 13
> sw3 (s) ; w4 (s) = 0
>
: ;k w (s) + (k + k )w (s) + m sw (s) = k :
11 1 2 3 2 4 2

w2 (s), w4 (s)
(m1 s2 + k1)w1 (s) ; k1 w3 (s) =0 (1.37)
;k1w1(s) + (m2s 1 2 3
2 + k + k )w (s) = k :
2

: (s) = (m1 s2 + k1 )(m2 s2 + k1 + k2 ) ; k1 =
(1.37) 2

= m1 m2 s4 + (k1 m1 + k1 m2 + k2 m1 )s2 + k1 k2 1 (s) = k1 k2
3 (s) = k2 (m1 s + k1 ): (-
2

W1 (s) = k1(s)k2
(1.19)),
m1 s2 (m2 s2 + k1) ( h (t) ; h (t)).
( h1 (t)) W2 (s) = ; -
(s) 2 3

, (s)
s1 2 = |!1 s3 4 = |!2 !1 6= !2 : -
, , -
.
, .
. 1.20 . 30 -
(1.20).
, -

s(JC s + C )WMC (s) ; H cos sWMC (s) = 1
0
M
H cos 0 sWMC (s) + s(JB s + B )W C (s) = 0
s(JC s + C )WMC (s) ; H cos MC (s) = 0
0 sW
M
H cos 0 sWMC (s) + s(JB s + B )W C (s) = 1
:
WMC (s) = JBsA(s) B
s+ WMB (s) = WMC (s) = ; HsA(s) 0
cos

WMB (s) = JCsA(s) C
s+ A(s)=JB JC s2 +(JB C + JC B )s+ B C +
+H 2 cos2 0 : , -

44
r
JB J2 2
T= -
C
B C + H cos 0
= p JB C + JC B 2 :
2 JB JC ( B C + H 2 cos 0)
91] !n T ;1:
, = =0 = 0: -
B C
-
. -
91].
5. . " -
"( ), -
x k + 1] = x k] + u k] y k] = x k]
1
W(z) = z ; 1 -
(1.25). y k] = x k] + u k]
z:
W(z) = z ; 1
T0.
, W(z) = z T0 1
, W(z) =
;
T0 z
z ; 1:
" "
{ (1.22), . 33, -
.
(1.22)
wi :
8
> zw1 (z) = 1
>
< ;w (z) + zw (z) = 0
1 2
> ;w2 (z) + zw3 (z) = 0
>
: ;w3(z) + zw4 (z) = 0
wi (z) = z ;i i = 1 : : : 4:
(1.22),
W(z) = 1 + z + z + z :
2 3
(1.38)
4z 4
-
.



45
1.6.
, -
, -
. -
.
1.6.1.
u(t) = ues0 t ,
(1.23).
u2Rm s0 2 C. (1.23)
x 2 Cn {
x(t) = xes0 t -
. u(t) x(t) (1.23)
s0 t = Axes0t + Bues0t (s0 I; A)xes0 t = Bues0t (s0 I; A)x = Bu:
s0xe
, : s0 -
A
det(s0 I ; A) 6= 0: x = (s0 I ; A);1 Bu ,
,
,
x(t) = (s0I ; A);1 Bues0 t : (1.39)
, -
.
x(t)
˜ (1.23). x(t) = x(t) +
˜
x(t) (1.23) , x(t) -
, (1.23)
u(t) 0: , (1.23)
(1.39)
.
x(t). -
x(t) ! 0
t!1
(1.39).
x(t) -
(1.23): y(t) = Cx(t)+Du(t) = C(s0 I;A) ;1 Bues0 t +
= (C(s0 I ; A);1 B + D)ues0 t = W(s0 )ues0t = W(s0)u(t):
Dues0 t
,
( { ), -
.
,
.. .

46
-
u(t) = u cos !t u 1 (e|!t + e;|!t ) !{ -
2
, |2 = ;1:
,
s0 = |! -
,
1;
y(t) = 2 W(|!)e|!t + W(|!)e;|!t u: (1.40)
W(|!) (! 2 R |2 = ;1)
. -
-
(1.23). 2
W(|!)
Wij (|!)
14


W(|!) = A(!)e|!+'(!) = U(!) + |V (!)
A(!) = jW(|!)j { -
( )
'(!) = argW(|!) { ( )
-
U(!) = ReW(|!) V (!) = ImW(|!) {
( ).
,
! 2 !0 !1 ]
W(|!)
!0 = 0 !1 = 1 )
( -
( ), .-
( -
, ),
L(!) = 20lgA(!) -
lg(!):
W( ) {
U(;!) = U(!) V (;!) = ;V (!)
,
. . W(;|!) = conj(W(|!)) A(;!) = A(!) '(;!) = ;'(!):
- -
tg'(!) = V (!) U(!) 6= 0:
, arctg( )
U(!) h i
;2 2 -
'(!) .
W(|!)
.
14




47
( )
Ql
r (|!)
QLi=1 i
W(|!)
i=l+1 ri (|!)
;
P
'(!) = L 'i(!) "+"
i=1
";"
i = 1 2 ::: l ( ),
{ i = l+1 i +2 ::: L ( ).
'i(!)
8 V
arctg Uii(!) Ui(!) > 0
>
>
>
> (!)
Ui(!) = 0 Vi(!) 6= 0
> 2 signVi(!)
<
'i(!) = > (1.41)
> arctg Vi (!) + signVi(!) Ui(!) < 0 Vi(!) 6= 0
>
>
> Ui (!)
: Ui(!) = Vi(!) = 0
Ui (!) = Re ri (|!) Vi(!) = Im ri (|!):
, -
( -
) i- - y(t) j-
uj (t): (1.40),
;
yi (t) = 1 W(|!)e|!t + W(|!)e;|!t uj =
2
1 A(!);e|('(!)+!t) + e;|('(!)+!t) = yi cos(!t + ')
=2
yi = A(!)uj { ( {
), ' = '(!) { " "
. ,
,
( -
).
.
1.6.2.
(1.24)
u 2 Rm
z0 2 C z0 6= 0.
uz k ,
u k] = (1.24)
0
x 2 C n:
xz k
x k] = 0 -
u k] x k] (1.24)
= Axz0 + Buz0 (z0 I ; A)x = Bu:
k+1 k k
xz0
48
det(z0 I ; A) 6= 0 x = (z0 I ; A);1 Bu
x k] = (z0 I ; A);1 Buz0 : k (1.42)
, (1.42) -
. (
c ) -
y k] = Cx k] + Du k] = C(z0 I ; A) ;1 Buz k + Duz k =
0 0
(C(z0 I ; A) ;1B + D)uz k = W(z0 )uz k = W(z0 )u k]:
0 0
" " u k] =
1 (e|!k + e;|!k )
u cos !k u 2 !{ -
. u k] -
15
1 (z+ + z; )
k k |! :
u k] = u 2 z =e ,-
|!
(1.42) z0 = e
;
y k] = 1 W(e|! )e|!k + W(e|! )e;|!k u:
2
W(e|! )
. -
,
!. 2
(1.24)
, . 1.6.1.
. 46, -
15, 47, 66, 76, 95]
A(!) = jW(e|! )j { -
( ), A(;!) = A(!)
'(!) = argW(e|! ) { ( ),
-
'(;!) = ;'(!)
U(!) = ReW(e|! ) V (!) = ImW(e|! ) { -
( ), U(;!) = U(!)
,
V (;!) = ;V (!):
-
2 : W(e |!+2 N ) =
W(e|! ) N = 1 2 3 : : : : ,
W(z)
z
z = e|! -
," " -
. " "
k
, -
15
2!
, ;1
ju k]j juj.
,

49
,
, 2N
,
(cos !k cos(! 2 N)k): -
!2 0 2 ) -
W(e|! )
,
! 2 0 ]: !N = -
.
! > !N ( )
.
1.6.3. -


, -
u k] , " "-
u(t) T0 : u k] =
u(t)jt=kT0 k = 1 2 3 ::: : -
! u(t): u(t) = u cos !t -
u k] = u cos(!kT0 ): ,-
, ! = !T0 : -
z = e|!T0 W(z) : A(!) = jW(e|!T0 )j
'(!) = argW(e|!T0 ) U(!) = ReW(e|!T0 ) V (!) = ImW(e|!T0 )
W(e|!T0 ) (
) !:
-
!N = =T0: !N
,
j!j !N :
" " , -
-
,
j j !N {
u(t):
-
T0



50
T0 . -
16

-
, -
. -
- -
-
.
1.6.4.
-
.
1. . .
1.5.3. . 38,
(1.13)
q
p
Ks T = LC K =LC=T 2 = R C :
2
W(s)= T 2 s2 +2 Ts+1 2L
s = |! ! 2 R -

K!
A(!) = jW(|!)j = p
2
=
(1 ; T 2! 2 )2 + 4 2 T 2! 2
LC! 2
=p :
(1 ; LC! 2 )2 + R2 C2 ! 2
(1.41), . 48, -
!0
8
> ; arctg RC! 2 LC! 2 < 1
> 1 ; LC!
<
LC! 2 = 1
'(!) = > 2
> RC!
: arctg ;1 + LC! 2 LC! 2 > 1:
,
( ). -
-
A(!) 0:5: , ,
2

], C= 10;5 ]. T = 6:32 10;3 ],
R= 800 ], L= 4
= 0:63 !c = 200
1/c]. . 1.9
,-
16
,
.

51
. 1.9. - -

MATLAB

L=4.0 R=800 C=10e-6
{
T=sqrt(L*C), xi=R/2*sqrt(C/L), K=L*C
{ TK
ommax=600 omega=0:ommax/100:ommax
{ !
s=j*omega % { s = |!
W=K*s.b2 ./(Tb2*s.b2+ 2*xi* T*s+ 1)
{ s W(s)
A=abs( W)
{
plot(omega, A, 'w'), grid
{ . 'w'
( . 72, 81, 139])
2. . -
( )
2 . 28 -
(1.16). -
(1.33), -
(1.36).
52
-
4]: ay = ;2:10 c;1 ] ay = 0:16 c;1 ]
amz = 29:4 c;2 ] a!zz = 2:18 c;1 ] amz = 60:7 c;2 ]:
m
,
:
W# (s) = s(s2 + 4:28s + 34:0) = s(T ;k#+ 2 + 1) 1)
;(60:7s + 127) (s
s Ts +
22


k# = 3:75 c;1 ]
= 0:48 c] T = 0:17 c] =
0:37 : ( )
. 1.10




. 1.10. .

MATLAB

a alpha y= -2.10 a delta y= 0.16
a alpha m= 29.4 a omega m = 2.18
a delta m= 60.7
ay ay amz a!zz amz
{ -
m
(1.33)
num=- a delta m* 1, - a alpha y]
den= 1, a omega m, -a alpha y, ...
53
a alpha m, -a alpha y*a omega m, 0]
{ -
W# (s) (1.34)
k=-n1(2)/d1(3), tau=n1(1)/n1(2)
T=sqrt(1/d1(3)), ksi=d1(2)/d1(3)/2/T
{ k# T
W# (s)
om=logspace(-1, 2)
{ !
mag, phase]=bode(num, den, om)
{ -
bode ( . )
lmag=20*log10(mag)
{
semilogx(om, lmag, 'w', 1/tau, 0, '+w', 1/T, 0, '+w'),
grid
{ ( ) . '+'
! 15, 76].

-
-
MATLAB- :
A= a alpha y, 0 , -a alpha y ...
a alpha m, -a omega m, -a alpha m ...
0 1 0]
B= a delta y -a delta m 0] C= 0 0 1] D=0
{ (1.33)
n, d]=ss2tf(A, B, C, D, 1)
{ -
mag, phase]=bode(A, B, C, D, 1, om)
{ .
, -
- ay
. -
(1.34) ,
B .

54
3. -
-
(1.19), . 32. W1 (s) W2 (s)
. 1.5.3. ( 4, . 38), -
:
A1 (!) = jm m ! 4 ; (k m + k1 k2 + k m )! 2 + k k j
k1m2 2 1
12 11 12

m1 ! 2 jk1 ; m2 ! 2j
A2 (!) = jm m ! 4 ; (k m + k m + k m )! 2 + k k j :
12 11 12 21 12
. 1.11 , -
126]: m1 = 500
], m2 = 400 ], k1 = 60 / ], k2 = 170 / ].




. 1.11. .

MATLAB-

k 1= 60e3 k 2= 170e3
m 1= 500 m 2= 400
{
ommax= 50 omega=0:ommax/500:ommax
55
{ !
A 1=k 1*k 2./abs(m 1*m 2*omega.b4-...
(k 1*m 1+ k 1*m 2+k 2*m 1).*omega.b2+ k 1*k 2 )
A 2=m 1*omega.b2 .*abs(k 1- m 2.*omega.b2)./...
abs(m 1*m 2*omega.b4-...
(k 1*m 1+ k 1*m 2+k 2*m 1).*omega.b2+ k 1*k 2)
{ A1 (!) A2 (!)
subplot(211), plot(omega, A 1, 'w'), grid
axis( 0 ommax 0 10])
subplot(212), plot(omega, A 2, 'w'), grid
axis( 0 ommax 0 10])
axis
{ .
.
-
.
2, ,
bode.
4. . . 1.5.3. . 45, -
-
(1.38) W(z) = 1 + z + z + z :
2 3
(1.22), -
4z 4
-
1 + e|! + e2|! + e3|!
z = e|! : W(e|! ) = :
4e4|!
W(z) -
;
W(e|! ) = 0:25 e;4|! +e;3|! +e;2|! +e;|!
=
; ;1:5|! ;0:5|! 0:5|! 1:5|! ;
0:25 e;2:5|! e = 0:5 e;2:5|! cos 1:5! +
+e +e +e
cos 0:5! = e;2:5|! cos ! cos 0:5!: A(!)
jW(e|! j = jcos(!)cos(0:5!)j: A(!) .
1.12.
MATLAB-

omega=0:0.005:2*pi z=exp(i*omega)
z = e|!
{
-
W= (1+ z+ z.b2+ z.b3)./z.b4/4
W(e|! )
{
56
A=abs(W)
plot(omega, A, 'w'), grid, axis( 0, 2*pi, 0, 1])
{ .




. 1.12.
. -
-
. -
-
= e|! { ;
= |! ( -
R( ) = In ;A ;1:
) ;
R( + | ) = ( + | )In ;
; ;
A ;1 = In ; A ; | In ( In ; A)2 + 2 In ;1 = U + |V
= Re = Im ( = |! =0
|!
=! =e = cos ! = sin !). -
U V:
CB
. -
, , -
.

57
1.7.
-
, -
. -
" ". , -
47].
1.7.1.
, -
(.
1.13, ). , -
, { -
.




. 1.13. .
Si i = 1 2 -

xi(t) = Ai(t)xi (t) + Bi (t)ui (t) yi (t) = Ci (t)xi (t)
_
Ai(t) Bi (t) Ci (t) , -
, ni ni ni mi li mi : ( )-
x(t) = col x1 (t) x2(t) 2Rn1 +n2
: u(t) =

58
col u1(t) u2 (t) 2 Rm1 +m2 y(t) = col y1 (t) y2 (t) 2
Rl1+l2 : , ,
-
(1.3)
x(t) = A(t)x(t) + B(t)u(t) y(t) = C(t)x(t)
_
A(t) B(t) C(t) -
:
A(t) = 0 1 (t) 0n1 (t)2 B(t) = 0B1 (t) 0B1 (t)2
A n nm
n2 n1 A2 n2 m1 2


C(t) = C1 (t) 0l1 (t)2 :
n
0l2 n1 C2
1.7.2.
S S1
u(t) u1 (t)
S2 y(t) y2 (t) S1
S2 , , l1 =
m2 u2 (t) = y1 (t):
( . 1.13, ).
x1(t) = A1(t)x1 (t) + B1 (t)u(t)
_ y1 (t) = C1 (t)x1 (t)
x2(t) = A2(t)x2 (t) + B2 (t)C1 (t)x1 (t) y(t) = C2 (t)x2 (t)
_
(1.3)
A(t) = B A1 (t)(t) 0n1 (t)2 B(t) = 0B1 (t)
n
2 (t)C1 A2 n1 m2
C(t) = 0l2 C2 (t) ] :
n1

1.7.3.
, ..
S2 ( ) -
S S1 :
S S2
( . 1.13, ). , , m1 = l2
m2 = l1 m = m1 l = l2 n = n1 + n2 u1(t) = u(t) y2 (t)
59
u2(t) = y1 (t): -

x1 (t)=A1 (t)x1 (t) B1 (t)C2 (t)x2 (t)+B1 (t)u(t) y(t)=C1 (t)x1 (t)
_
x2 (t)=A2 (t)x2 (t)+B2 (t)C1 (t)x1 (t)
_
(1.3)
A(t) = B A1(t)(t) B1 (t)C2 (t) B(t) = 0B1 (t)
2 (t)C1 A2(t) n1 m2
0l
C(t) = C1 (t) n2 ] :
1

-
, ,
. , -
,
(
) .
1.8.
. 1.1. . 15,
{ -
X -
,
.
. -
n n- T
det T 6= 0: ,
,
-
.
, - -
. , -
. -
.
n det T 6=
T{
0 x(t) 2Rn { .
x(t) = Tx(t):
˜ -
T x(t)
˜ x(t) -

60
x(t) = T ;1 x(t):
˜ -
17

(1.2) . x(t) =
_
_
T ;1x(t)
˜
_
x(t) = TA(t)T ;1x(t) + TB(t)u(t)
˜ ˜ x(t0 ) = x0 = Tx0
˜ ˜ (1.43)
y(t) = C(t)T ;1 x(t) + D(t)u(t)
˜
˜ ˜ ˜
A(t) = TA(t)T ;1 B(t) = TB(t) C(t) =
C(t)T ;1 : (1.43) (1.2):
˜x ˜
_
x(t) = A(t)˜ (t) + B(t)u(t)
˜ x(t0 ) = x0 = Tx0 (1.44)
˜ ˜
˜ x(t) + D(t)u(t):
y(t) = C(t)˜
(1.44)
(1.2) . ,
. 18

,

x(t) = Ax(t) + Bu(t) y(t) = Cx(t) + Du(t):
_ (1.45)
T
˜x ˜ ˜x
_
x(t) = A˜(t) + Bu(t) y(t) = C˜(t) + Du(t)
˜ (1.46)
˜˜˜
ABC . -
(1.46) (1.25)
: 19


W(s) = C ;sIn ; A ;1B + D = CT ;1;sIn ; TAT ;1 ;1TB + D =
˜ ˜ ˜˜
;
= C sIn ; A ;1T + D W(s):
, -
T . ,
x(t) = T ;1 x(t):
˜
17
T
T ;1 ( ).
-
18
˜
T = T(t). A(t)
˜ = ;T(t) + T(t)A(t) T ;1 (t):
_
A(t)
(AB);1 = B ;1 A;1 det(AB) = detA detB
19
.

61
-
. , -
-
,
.
˜
A A = TAT ;1 .
. ,
˜
: det(sIn ; A) det(sIn ; A) -
, . ,-
A1 = 0 1
, . , 00
A2 = 0 0 s1 2 = 0
00
.

˜
, A
˜
A = T ;1 AT:
,
, , ,
. , -
,
T: -
, 7. .166.
. . -
(1.15) ( . 1.4.2. . 28)
(1.3)
A= 0 1 B = J0 C = 1 0]:
;1
00 x

T= 1 1 (det T = 1):
01
.

;1
˜= 0 1 ˜ = Jx ˜
C = 1 ;1]:
A 00 B J ;1
x
" " (
)
62
_ y(t) = x1 (t) ; x2(t)
x1 (t) = x2 (t) + Jx ;1 u(t)
˜ ˜ ˜ ˜ (1.47)
_
x2 (t) = Jx ;1 u(t):
˜
, (1.47) -
(1.15), -
{ .
,
.
x(t)
{ ,
-
. -
, (1.15)
(1.47) , . 1.14




. 1.14. (1.15) ( ) (1.47) ( ).
, -
, -
.
, -
. - , -
( ) . -
,
, -
. (
), -
.
-
.
63
1.9.
1. , -
xi (i = 1 2 3 4),
174].
-
.
.
. , -
.
) , -
x2X =R
, 4


Ex k + 1] = Rx k] + r k = 0 1 2 3 : : :
ER r
.
)
x k + 1] = Ax k] + g:
. ,
(In ; B);1 = I + B + B 2 + + Bk +
B , -
.
) ( k = 0)
. x k]:
7.00 (.. k = 7)?
2.
.
.
-
, .
-
, .
, -
.

x1 (t) = x2 (t)
_
x2 (t) = 2x1 (t)3 ; u(t)x2 (t):
_
64
x1(1) = 1 x2 (1) = ;1
) u(t)
0: x(t): ( . t).
) ( . . 1.3.),
,
.
)
x1(1) = 1:5 x2 (1) = 0:5 u(t) 0:5.
4. -
. (1.5)
174]
Ex k + 1] = Ax k] + Bu k] (1.48)
n n- A E n m- B -
det E = 0
.
-
, " -
" . -
,
. -
.

T x k + 1] = C x k] + u k]
0 D v k]
x k] 2Rn T C { m n- (n ;
,D{
m v k]2Rn;m:
m) n u k]2R
T
n n- -
D
(1.5):
) x = Tx ,
˜
, x k]
x k] = H˜ k] ; Gv k]:
x
G H:
) ,

x k + 1] = R˜ k] + Bv k] + u k]:
˜ x
65
R B: ( , x k]
x k]
˜ )
5. 174, 188]
. Y k]
C k] I k] -
G k]: -
, -
.
Y k] = C k] + I k] + G k]
C k + 1] = mY k]
;
I k + 1] = C k + 1] ; C k]
(1.48)
2 3 2 3 23
1 1 ;1
000 0
E = 40 1 05 A = 40 0 m 5 B = 405
1; 0 0; 0 1
x = colfI C Y g:
4, -
(1.5) -
.
6. -
-
( 19, 23, 94], . . 28)
8
> !x (t) = ; Jz ; Jy !y !z + Mx (t)
>_
> Jx Jx
>
<
; My (t)
!y (t) = ; Jx Jy Jz !z !x + Jy
_
>
>

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