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. 7
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(. . 11 . 180), (8.8)
,
(A11 A21 ). , -
-
E:
.
. -
( 174]). 6

l
(.. -
). -
'(t), -
,
u(t) cos ' + l'(t) = g sin '(t) (8.11)
g 9:81 { u(t) { -
{
( . 8.2). ,

x(t) = u(t) + l sin '(t): (8.12)
-
, (8.11), (8.12) -
x(t)
;
x(t) = g x(t) ; u(t) :
l
v(t) = x(t):
_

v(t) = 0 gl;1 v(t) ; gl;1 u(t):
_
x(t)
_ 10 x(t) 0
, -
6
( . . 30)
,
- 19], -
, 23, 19, 98],
.

189
. 8.2. .


;gl;1 0 :
Q = 0 ;gl;1
det Q 6= 0
-
.
.
x(t) (u(t) -
). y(t) = 0 1]x(t)
y(t) { . -
Q= 0 1 det Q = ;1:
10
, -
( { ). -
(8.3)

_
v(t) = (1 ; l1)^ (t) + l1 x(t) ; gl;1u(t)
^ x (8.13)
_
x(t) = gl;1v (t) + l2(^ (t) ; x(t)):
^ ^ x
l1 l2 { , -
-
det(sI ; A ) = s + l2 s + g(l1 ; 1)l;1 :
2



190
(8.13) u(t) x(t) -
v (t) x (t):
^^ -
,
,
(8.10).
, -
(8.8) -
, -
(8.10). -
( ) A11 = A22 = 0 A21 = 1 A12 =
B1 = ;gl;1
gl;1 (8.10)
_
v (t) = ;e^(t) + (gl;1 ; e2 )x(t) ; gl;1u(t):
^ v
e{ . -
det(sI ; A ) = s + el;1 : e = ;s1 l s1 {
-
.
3] -
. ( -
3] )
(n ; 1)- -
( , rank C = 1).
.
1. ( A B C) -
(.
. 77, "" 2 . 97).
2. -
i

(det In;1 ; A ) = sn;1 + 1 sn;2 + + n;1:
3. P
2 3 2 3
; n;1 n;1
In;1 In;1
;
6 n;2 7 6 n;2 7
6 7 6 7
P =6 ::: 7 P ;1 = 6 ::: 7:
6 7 6 7
;15
4 4 5
1
0 ::: 0 1 0 ::: 0 1
, A


191
,
2 3
0 0 : : : 0 ; n;1 ( 1 ; 1) n;1 ; n
6 1 0 : : : 0 ; n;2 ( 1 ; 1) n;2 ; n;1 + n;1 7
6 7
˜
A = PAP ;1 = 6: : : : : : : : : : : : : : : 7
:::
6 7
; ( 1 ; 1) 1 ; 2 +
4 5
0 0 ::: 1 1 2
; 1+ 1
0 0 ::: 0 1
2 3
b1 ; n;1bn
6 b2 ; n;2bn 7
6 7
˜
B = PB = 6 : : : 7 :
6 7
bn;1 ; 1bn
4 5
bn
˜ n;1
4. A A
˜
A,
˜
n-1 B
n;1
( - ) B: -
˜
A an :
:
_
x(t) = Ax(t) + an y(t) + Bu(t): (8.14)
-
( , ),
x(t) = colfx(t) y(t)g:
˜
x(t) = P ;1 x(t):
^ ˜
8.4.
(8.4), ( -
)
. -
, , -
,
.
-
.
, ,
(" -
", "internal model of disturbances").

192
, -
( ) -
-
.
. 7

, ,
-
.

. -
,
.
8

-

" " ("internal model
-
principle").
, ,
-
, ,
(" "), -
.. 9

, -
{-
N
X
2C {
e i t Pi (t) ,
i
i=1
Pi (t) - . -
, ,
,
-
. -
-
.
7
(. 12 93, 76, 103, 106]).
,
8
.
-
9
, 6.6.

193
.
f(t) v(t)
, -

x_s(t) = As(t)xs (t) ys (t) = Cs xs (t) xs(t0 ) = xs0 t t0 : (8.15)
xs(t) 2 Rns { ", ys(t) 2 Rn+l
"
- {
ys (t) = colff(t) v(t)g
, Cs = Cf Cf Cv {
As Cs { -
Cv
n ns l ns -
xs(t)
f(t) v(t) (8.1). xs0 -
(8.15), (8.1), . -
(" ")
x(t) = (x(t) xs(t)) 2 Rn+ns : (8.1), (8.15),

_
x(t) = Ax(t) + Bu(t) y(t) = Cx(t) x(t0 ) = x0 t t0 (8.16)
ABC
:
A = 0 A Cf B = 0 B C = C Cv ]:
ns n As ns m
(8.16)
n = n + ns -
(8.3).
.
-
()
d2 = u(t) + M(t)
Jx dt2 (8.17)
Jx { ,
(t) { , u(t) { , M(t) { -
. -
!x (t) = _ (t): u(t) -
. -
M(t): -
M(t) = M0 + V t M0 V { .
194
-

_
M(t) = V (t) (8.18)
_
V (t) = 0
M(0) V (0):
" { " x(t) = !x (t) M(t)
T
V (t)] : y(t) = !x (t): ,
(8.16),
213 213
0 Jx 0
4 0 0 1 5 B = 4 J0x 5 C = 1 0 0]:
A= (8.19)
000 0
.
(8.3) (8.17), (8.18)
n = 3: x1 (t) x1 (t) x1(t)
!x (t) M(t) V (t) -
. L-
det(sI3 ;
l l
A + LC) = s3 + l1 s2 + J2 s + J3 :
x x
A (s) = s3 + 2 0 s2 + 2 2 s + 3 li :
0 0
l1 = 2 0 l2 = 2Jx 0 l3 = Jx 3 : -
2
0
0
. -
5= 0 : -
(8.3)
8_ ^
> ! x (t) = ;l1 !x (t) + M(t)=Jx + l1 !x (t) + u(t)=Jx
<^ ^
_
^ ^
M(t) = ;l2 !x (t) + V (t) + l2 !x (t) (8.20)
^
>_
: V (t) = ;l ! (t) + l ! (t):
^ 3 ^x 3x

. 8.3 ( )
(8.17), (8.20) -
:
10;3
Jx=54.3 , M(0)=0.25 ,V =5 /.
2

0 =0.5 1/c.




195
. 8.3. .

MATLAB-


.
x0 = 0, 0, 0]' x0= 0 0.25 0.005]'
x = colf x M V g
{
A= 0 1/J 0 0 0 1 0 0 0]
B= 1/J 0 0] C= 1 0 0]
{ ABC
Om=0.5 l1=2*Om l2=2*J*Omb2 l3=J*Omb3
{ 0
l1 l2 l3
L= l1 l2 l3] A =A-L*C B =B
{ (8.20)
Ae= A, zeros(3,3) L*C, A ] Be= B B] Ce= C, -C]
{ (8.17), (8.20)
xe0= x0 x0 ]
{ (8.17), (8.20)
t=0:0.01:30 u=zeros(size(t))
{ -
y,xe] =lsim(Ae,Be,Ce,0,u,t,xe0)
{
plot(t,xe(:,2),'w',t,xe(:,5),'.w'),grid
plot(t,xe(:,3),'w',t,xe(:,6),'.w'),grid
196
{ .
-
( . 8.3. . 187).
(8.17), (8.18) (8.19). -
10

. 191 , -
. ,
(3.13) ( . . 88, 3.2.3.) -
˜˜
A, C .
2 3
000
˜6 7 ˜
A=6 1 0 0 7 C = 0 0 1]:
4 5
010
˜
Q Q{
, -
T:
2 3 2 3
10 0 001
6 7 ˜6 7
Q = 6 0 Jx ;1 07 Q=6 0 1 0 7
4 5 4 5
Jx ;1
0 0 100
2 3
0 0 Jx ;1
6 7
˜
T = Q;1 Q = 6 0 Jx ;1 0 7 :
4 5
10 0

-
p
A (s) = s2 + 2 0 s + -
p
2
0
0
. =2 =: 2
1 0 2 0
P .
8.3.
2 3 2 3
1 0 ; 02 1 0 02
p p
6 7 6 7
P =6 0 1 ; 2 0 7 P ;1 = 6 0 1 2 0 7:
4 5 4 5
00 1 00 1
10
(8.19) .

197
p
2 3 2 3
0 ; 02 ; 2 03 ; 0 2Jx ;1
p 7 B = 6 ;p2 0 Jx ;1
6 7 7
˜
A = P AP ;1 = 6 1 ; 2 0 ; 0 2 6 7
4 5 4 5
p Jx ;1
0 1 2 0


A A , -
B
- B:
"p
" # # " #
0 ; 02 ; 0 2Jx ;1
; 2 03
p B= p
A= a3 = :
1;20 ; 02 ; 2 0 Jx ;1

_ 1 (t) = ; 0 2 x1p ; p2 0 3 !x (t) ; 0 2 Jp u(t)
;1
x (t)
x2 (t) = x1 (t) ; 2 0 x2 (t) ; 0 2 !x (t) ; 2 0 Jx ;1 u(t): (8.21)
x
_
x(t) = colfx(t) !x (t)g: -
(8.17), (8.18)
Tb { xx ˜ {
x: 2 3
00 1
p
6 7
Tb = T ;1 P ;1 = 6 0 Jx Jx 2 0 7:
4 5
Jx 0 Jx 0 2
^ ^
x(t) = Tbx(t)
^ !x (t) M(t)
^ V (t).
-
p
^
M(t) = Jx x1 (t) + Jx 2 0 x2 (t) (8.22)
^
V (t) = Jx !x (t) + Jx 0 2x2 (t):
(8.21), (8.22) -
.
( ) -
MATLAB
82].


198
syms Jx Om
{ Jx 0
A= 0 1/Jx 0 0 0 1 0 0 0]
B= 1/Jx 0 0 ]'
C= 1 0 0 ]
{ (8.19)
A = 0 0 0 1 0 0 0 1 0]
C = 0 0 1]
˜˜
{ AC
Q= C C*A C*A^2]
Q = C C *A C *A ^2]
{
T=inv(Q )*Q
B =T*B
˜
{ T B
bet1=sqrt(2)*Om
bet2=Om^2
{ 12
P= 1 0 -bet2 0 1 -bet1 0 0 1]
{ P
Atil=P*A *inv(P)
Btil=P*B
Abar=Atil(1:2,1:2)
{ AA
abar=Atil(1:2,3) bbar=Btil(1:2,1)
{ ab
Tbk=inv(T)*inv(P)
{ Tb
jx=54.3 om=0.5
{ Jx 0
Tb=double(subs(Tbk, Jx Om], jx om]))
Ao=double(subs(A,Jx,jx))
Bo=double(subs(B,Jx,jx))
Co=C
{
Af=sym( A zeros(3,2) abar*C Abar])
Bf=sym( B bbar])
199
Cf=Tbk* 0 0 0 1 0 0 0 0 0 1 C 0 0 ]
Df=zeros(3,1)
{ -
- - -
af=double(subs(Af, Jx Om], jx om]))
bf=double(subs(Bf, Jx Om], jx om]))
cf=double(subs(Cf, Jx Om], jx om]))
df=Df
{
stl=ss(af,bf,cf,df)
{ , -
( . 82])
t=0:0.1:30
u=zeros(size(t))'
x0= 0 0.25 5e-3 0 0]'
{ , -
y,ti,x]=lsim(stl,u,t,x0)
plot(ti,x(:,2),ti,y(:,2)),grid,title('M(t)')
gure
plot(ti,x(:,3),ti,y(:,3)),grid,title('V(t)')
{ .
-
. 8.3
, -
-
.
. , -
( -
{ ), , , 8, 47, 88].
8.5.
1. ,
, -
Mf : M(t) = M0 + V t + Mf sin f t:
f
,
v(t), v

200
v0 : , ..
y(t) = !x (t) + v0 sin f t: -
^
"M (t) = M(t) ; M(t) ( -
)
) n- (8.20)
) (8.21), (8.22).
-
:
0
2. . 42
. , -
#(t),
-
(!z (t) (t)).
3. , 3]
2 3 23
111 0
C= 0 0 1
A = 40 1 05 B = 415 100
011 1
.
4. , 3]
2 3 23
0111 0
60 0 1 07 607
A = 60 0 0 17 B = 607 C = 2 0 1 0]
4 5 45
1000 1

s1 = s2 = s3 = ;1.
S1 {
5. -
S2
, u(t) y(t) 174]).
S1 , S1
.
S = fS1 , S2g
, -
u(t):




201
9. -

9.1.
-
si . 1

, y(t) -
Pn
n- y(t) = i=1 Ci yi (t)
Ci , -
yi (t) = esi t -
yi (t) (" ")
si t {
si yi (t) = Pi(t)e ( Pi (t) { -
, ).
" " -
-
. -
-
. ,
,
.
9.2.
-
. -
, ,
u(t)2R:
-

x(t) = Ax(t) + Bu(t):
_ (9.1)
x(t) (9.1) -
.
u(t) = ;Kx(t) (9.2)
K{ n l- -
( m = 1):
-
x(t) = (A ; BK)x(t):
_ (9.3)
.
1



202
( -
K)
, -
det(sIn ;A+BK) = D(s) = sn +d1sn;1 +: : :+dn;1 +dn
di: -
7.2. 4. 2

.
, (9.1)
, ..
AB
2 3 23
0 1 0 ::: 0 0
6 07 607
0 0 1 :::
6 7
7 B = 6 ... 7
...
A=6 67 (9.4)
6 7 67
4 15 405
0 0 0
:::
;an ;an;1 ;an;2 : : : ;a1 1
det(sIn ;A) = sn +a1 sn;1 + +an : -
(9.2) K = k1 k2 : : : kn ]
A ; BK
,
(9.3) -
det(sIn ; A + BK) =
n +(a1 +kn)sn;1 +: : : +(an;1 +k2)s+an +k1:
=s -
di
:
8
= dn ; an
> k1
>
>k = dn;1 ; an;1
>2
<
(9.5)
>k
> n;1 = d2 ; a2
>
>
:k = d1 ; a1 :
n

, . -
(9.1).
, 8 (. .
7.2.), T , -
.3
, -
2
D(s)
,
.
T
3
.

203
˜ ˜
, , A = TAT 1 B = TB
˜
det(sIn ; A) det(sIn ; A):
(9.4),
˜˜ ˜
(A B) K
(9.5). -
. , x(t) = Tx(t)
˜
˜ x(t) = ;KTx(t) = ;Kx(t)
˜
u(t) = ;K˜
˜
K = KT: (9.6)
, -
-
. :
{ -
{
( )
{
(9.5), (9.6).
,
( -
) 4 11. ,
-
. 8.2.
.
3].
-
,
.
-
-
.
-
. -
-
, -
, n .
9.3. . -
,
x(t)
204
y(t): ( -
). -
x(t), x(t)
^
( . 9.1).

x(t) = Ax(t) + Bu(t) y(t) = Cx(t) x(t0 ) = x0 (9.7)
_
u(t) = ;K^(t)
x (9.8)
_
x(t) = (A ; LC)^ (t) + Bu(t) + Ly(t) x(t0 ) = x0 : (9.9)
^ x ^ ^
(9.8), (9.9) ,
y(t), {
u(t): (9.2) -




. 9.1. -
.
,
(9.7).
-
76]. 4

:
(9.7){(9.9), -
? -
.
-
4
p = rankC:


205
-
. -
"(t) = x(t) ; x(t):
^ -
x(t) = x(t) ;"(t)
^ (9.7) { (9.9)

x(t) = Ax(t) + Bu(t) x(t0 ) = x0
_ (9.10)
u(t) = ;Kx(t) + K"(t) (9.11)
"(t) = (A ; LC)"(t) "(t0) = x0 ; x0 :
_ ^ (9.12)
(9.7){(9.9) (9.10){(9.12)
(9.7){(9.9) x(t) =
˜
col x(t) x(t)
^ x(t) = col x(t) x(t);^ (t) = col x(t) "(t) ,
x
, , .
_
x(t) x(t) = Ax(t)
A :
A = A ;0BK A BK : ; LC
A ,
-

det(sIn ; A) = det(sIn ; A + BK) det(sIn ; A + LC):
(9.10){(9.12) -
(9.7){(9.9), -
(9.7){(9.9)
. , -
.
3, 47]. -
, -
,
" " -
(9.2) -
(8.5) (9.9).
(9.7){(9.9)
-
.-
, (-
K) ( L)
2
.
206
, -
,
. 8.3. 3].
(9.10){(9.12) ,
-
(9.7) { (9.9) -
(9.2), (9.3)
. -
K"(t) (9.11).
"(t) . -
t
.
, SISO- (l = m = 1)
(9.8), (9.9)
. -
5


,
-
.
, -
, 76]. , ,
-
. -
,
-
D(s). -
, D(s) = A(s) + B(s) A(s) B(s) { -
.
A(s) = A0(s)R(s) B(s) = B0(s)R(s) . . -
;
D(s) = R(s) A0(s) + B0(s)
. -
A0(s) B0 (s)
D(s)
R(s): -
,
,
.
9.5.1. .
5




207
9.4.
(.
167, . 7.1.),
( ) -
. -
, ..{
t ! 1: -
-
. -
, , 3, 20].
, , - , -
19, 20],
23], -
. -
6


-
. 7.2. ( . 10, . 172). ,
, -
x(t) = Ax(t) + Bu(t)
_ x0
x1
= t1 ; t0 > 0
3, 30, 83].
;
T
u(t) = B T eA (t1 ;t) W( );1 x1 ; eA x0 (9.13)
Z T
eA BBT eA d :
W( ) = (9.14)
0
,
( -
), , -
,
x0 x1 .
, " " -
6
, ,
. " " -
2, 3, 23, 93].

208
(9.13) -
3]. -
r(t) -
R1
t
eA(t ; )Br( )d = 0: ,
1
t0
( (6.9), . 130)
r(t): -
u(t) u(t) + r(t) . 3], -
u(t) (9.13) , x0 x1,
(..
R1
t
u(t)T u(t)dt).
t0
W(t0 t1) 3, 47]:
Zt1
(t0 t)B(t)B(t)T (t0 t)T dt:
W(t0 t1) =
t0
nn ,
{ W(t0 t1 ) = W(t0 t1 )T
1)
2) t0 t1 t0
3) -
( ): 7


_
W(t t1 ) = A(t)W(t t1 ) + W(t t1)A(t)T ; B(t)B(t)T
W(t1 t1) = 0: (9.15)
!1
,
W( ) W
AW + WAT ; BB T = 0)
4)
W(t0 t1 ) = W(t0 t) + (t0 t)W(t t1 ) (t0 t)T :
(9.14)
.
-
7
. 11.4.4. . 274
.

209
w(t) = eAtB: 6.2. ( . 132), -
(m = 1) w(t),
, -
x(t) = Ax(t)
_ x(0) = B: m > 1,
bi
B = b1 ...b2 ... : : : ...bm ] w(t) m .
-
.
-
.
, -
. , -
,{ -
, x1 = 0: ;
C = W( );1 x1;
(9.13).
;eA x0 C 2R
T (t ;t) T (t;t T
u(t) = B T eA = B T e;A eA = t1 ; t0 :
C 0)
1

8

_ (t) = ;AT (t) (t0) = eA : (9.16)
T
(t) = e;A (t;t0 ) (t0 ) =
,
eAT (t ;t) : (t) -
1

u(t), ,T
u(t) = B (t) (t) -
T
(9.16), (t0) = eA C: ,-
,
8
<_
> x(t) = Ax(t) + Bu(t) x(t0 ) = x0
u(t) = B T (t) (9.17)
>_ AT
: (t0 ) = e C:
T
(t) = ;A (t)
,
(
x(t) = Ax(t) + BB T (t) x(t0 ) = x0
_
(9.18)
T
_ (t) = ;AT (t) (t0) = eA C:
(t) -
8
x(t) -
T
x(t) (t) = const: -
_ = ;A T
x = Ax
, _ 3].

210
-
3, 47, 88]. (t) (t) =
S(t)x(t) S(t) { - .
(t) -
_
Sx + S x = ;AT Sx:
_
_
, Sx+ SAx+
T T
SBB Sx = ;A Sx: -
x, S(t)
:
_
S(t) + S(t)A + AT S(t) + S(t)BB T S(t) = 0: (9.19)
S(t0 ), , -
x1 = 0:
T
C = ;W( );1 eA x0 (t0) = ;eA W( );1 eA x0 : (9.20)
(t0) = S(t0 )x(t0 ) -
T
W( );1 eA .
S(t0) = ;eA , ,
x(t0 ) = x0 -
> 0,
u(t) = B T S(t)x(t) (9.21)
S(t) (9.19).
-
, -
2, 3, 23, 93, 47]. C. . 423
MATLAB-
.
9.5. -
9.5.1.

(8.17), . 194 ( . -
1.4.2.). -
.

u(t) = ;k! !x (t): (9.22)
211
,
( { -
)
. -
^
, M(t) -
(8.20).
9


^
u(t) = ;k! !x (t) ; km M(t): (9.23)
k! -
. -
D(s) = s + k! k! = ;Jx s1 s1 {
Jx
D(s): , -
km = 1: -
. 195
;1
s1 = 0:2 . 9.2.




. 9.2. .
, -
. -

. -
u(t) " " -
,
^
M(t)
, -
9
, ( ) -
(8.20), (9.23).

212
;M(t): . 9.3 ( )
(8.17), (8.20), (9.23).




. 9.3.
(8.20), (9.23).
-
( La = 12
o ).
, ' = 33 -
(8.20), (9.23) !x
u -
, -
W(s) = k ss2+ 2 +s1) 1
+
2
k = 1:13 ;2 , T = 0:83
(Ts
, = 5:3 . -
(
. 196)
Km=1 C = Kw Km 0]
Ac=A -B* Bc=L
{
num,den]=ss2tf(Ac,Bc, Kw Km 0],0,1)
{
nf=num/J df=conv( 1,0],den)
{ -
om=logspace(-1,1)
mag,ph]=bode(nf,df,om)
{ .




213
9.5.2.
, -
( , ),
. -
52, 162]. -
-
( ) N
8 ;
'1(t) + ! 2 sin '1 (t) = k '2 (t) ; '1 (t) + u(t)
>
>
>
> ::: ;
<
i+1(t) ; 2'i (t) + 'i+1(t)
' (t) + ! 2 sin ' (t) = k ' (9.24)
>i i
: ' (t) + ! 2 sin ' (t): = k ;' (t) ; '2 (t): : : N ; 1)
> :: (i = 3
>
>
N N N;1 N
'i(t) (i = 1 2 : : : N) { u(t)
{ , -
, !k{
(! { -
,k{ ).
, -
, .
8 ;
> '1(t) + ! 2'1 (t) = k '2 (t) ; '1 (t) + u(t)
>
> :::
>
< ;
'i(t) + ! 2'i (t) = k 'i+1(t) ; 2'i (t) + 'i+1(t) (9.25)
> :::
>
: ' (t) + ! 2 ' (t) = k ;' (i =; ' :(t) N ; 1)
2 3 ::
>
>
(t) :
N N N;1 N

x(t)2R2N x(t) = colf'1 '1 '2 '_2 : : :
_
'N '_N g:
x(t) = Ax(t) + Bu(t)
_ (9.25)
2 3 23
A1 A12 0 ::: 0 0 B1
::: 0 0 7
6 A12 A2 A12 607
6 ... 0 0 7 67
60 A A 7
B=6 0 7
67
A = 6 .. 7
6 ... 7
6 ... 7
...12 . . 2 . . . ...
6. 7
. 67
6 7 405
0 0 0
4 : : : A2 A12 5
0
0 0 0 : : : A12 A1

214
A1 = ;! 20; k 1 A2 = ;! 20; 2k 1 A12 = k 0
0
0 0 0
T
B1 = 0 1 ] :
" " -
, -
. -
-
. -
. 9.4. .
(9.13).




. 9.4. .
N = 10 k = 5 ;2, ! = 0:4
;1 'i (0) = 0 'i(0) = 0 (i = 1 2 : : : N), = 50 c, 'i( ) =
_
(;1)i+1 30 ., 'i ( ) = 0
_ . 9.4, 9.5.
-
( ).
9.5 '9 (t) '10(t)
t 2 30 50] .
u(t)
, ( )-
,
, -
.
x( ) ,
t = + ( > 0), -
,
+ ]:
215
. 9.5. .

.
MATLAB-

k=5 om2=(0.2*2*pi)^2 N=10
{ k! N
x0=zeros(2*N,1) x1=zeros(2*N,1)
{
t0=0 t1=50
{ t0 t1
xmax=30/57.3
{
A=zeros(2*N)
{ A
B=zeros(2*N,1) B(2)=1
{ B
A1= 0 1 -om2 0]
216
{ A1
sig=1
for in=1:N
{ A x1
x1(2*(in-1)+1)=xmax*sig
sig=-sig
(;1)i
{
for in1=1:N
l=2*(in1-1)+1
A(l:l+1,l:l+1)=A1
A(l+1,l)=A(l+1,l)-k
if ((in1>1)&(in1<N))
A(l+1,l)=A(l+1,l)-k
end
if in1<N
A(l+1,l+2)=k
A(l+3,l)=k
end
end
end
{ A x1
C=eye(2*N,2*N) D=zeros(2*N,1)
C = I2 D = 0
{
lsim ( . C. . 431)
Th=t1-t0
= t1 ; t0
{
t=0:Th/1000:Th
Nt=length(t)
w0=B u=zeros(size(t))
{ w(0) = B
u(t) 0 w(t)
w=lsim(A,B,C,D,u,t,w0)
{ w(t)
W=zeros(size(A))
W( )
{
for k=1:Nt
W=W+w(k,:)'*w(k,:)
end
W=W*Th/Nt
W( )
{

217
c=inv(W)*(x1-expm(A*Th)*x0)
{ C (9.20)
for k=1:Nt
u(k)=B'*expm(A'*(t1-t(k)))*c
end
{ -
(9.13)
y,ti,x]=lsim(A,B,C,D,u,t,x0)
{ (
).
9.6.
1. -
, -
.
2. . 194 -
, , -
, -
T0 , -
-
.
, -
.
3. 3]
,
2 3 23
111 0
C= 0 0 1 :
A = 40 1 05 B = 415 100
011 1
4.
9.25, . 214, , -
i- (1 i N):
5. 3]
x1 (t) = x1 (t) + x2(t) + u(t)
_
x2 (t) = x1 (t) y(t) = x2 (t)
_

, -
s1 = ;3 s2 3 = ;2 |:
,
218
6. x(t) + x(t) = u(t)
y(t) = x(t) + { -
3], ,
x(t) ! 0, y(t) ! 0 t ! 1:
_
.
7. 6 3] -
(t) = 0 sin !t !.




219
10.

10.1.
. -
.
, { ,
{ -
( )
-
. -
1

, , -
( )
u(t) 2 Rm y(t) 2 Rl
y = Ku K{
l m- , t .
-
, -

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