<<

. 13
( 15)



>>

-
-
142, 143, 176]. -
-
128].
-
, ( )

x = f(x)
_ (13.58)
393
( ){
z = f(z) + u(t)
_ (13.59)
x z u { n- .
u(t)
u(t) = ;Ke(t) (13.60)
e = x;z { ,K>0{ ,

e = f(x(t)) ; f(x(t) ; e) ; Ke
_ (13.61)
x(t) { , -
A(x) = @f (x)
(13.58). -
@x
,
(13.58){(13.60), K > 0, -
T (x);2KI ,
A(x)+A n
In { n n- ,
x2 . , 34], (13.58){(13.60)
t!1
: , ,
. e(t) 0
(13.61), -
. , (13.58) (13.59){(13.60)
,
.
.
-
: 1) -
( -
y = h(x)) 2) ( u(t)
m- ,m<n u(t) (13.59)
Bu(t), B{ m m- ).
(
-
) 143, 153, 176].
.
, (13.58)
{(13.60) -
394
. -
, -

x = f1(x u t) y1 = h1 (x)
_ (13.62)
z = f2(z u t) y2 = h2 (z)
_
( ):
w = W(w y1 y2 t) u = U(w y1 y2 t):
_ (13.63)
-
-
16, 17]. -
, ..
u(t) .
, -
, . (1673 .),
, -
16, 17]. -
(13.63)
(13.62) { (13.63) 16]
,
e(t) = x(t) ;z(t), . . -
56]. ,
(13.63), -
. -
( ).
, , -
,
, -
112, 135, 153].
13.5.2.
( ),
:
( P
m
xd = Axd + '0(yd ) + B i 'i (yd )
_ (13.64)
i=1
yd = Cxd
395
xd 2 Rn { yd 2 Rl {
( ) = col ( 1 : : : m ) { -
. , -
'i ( ), i = 0 1 : : : m, AC B
::: ,
m
1
, .
-
^i , i = 1 : : : m -
,
-
yd (t).
z = F(z yd )
_ (13.65)
^ = h(z yd ) (13.66)

lim ^(t) ; =0 (13.67)
t!1
^(t) = col ^1 (t) : : : ^m (t) .
-
( A B C)
:
!
m
X
^i 'i(yd ) + ^0 G(yd ; y)
x = Ax + '0 (yd ) + B
_
i=1
y = Cx (13.68)
_
^i = i(yd y) i = 0 1 ::: m (13.69)
x 2 Rn yd 2 Rl 2 R G 2 Rl -
0
.
(13.69) ; ^^ -
. z = x 0 1 :::
: : : ^m , (13.65) (13.68), (13.69).
(13.68) (13.64) , -

lim e(t) = 0 (13.70)
t!1
e(t) = x(t) ; xd (t) .

396
(13.70)
(13.67),
(13.69).
:
8

m
<
i 'i (yd ) + ^0 G˜
e = Ae + B
_ y (13.71)
i=1
: y = Ce
˜
˜i = ^i ; i = 1 ::: m { . -
i
-

_
^i = ; i (y ; yd )'i (yd ) i = 1 : : : m (13.72)
_
^0 = ; 0 (y ; yd )2 (13.73)
13.5.3.
-
.
1 106]. x = Ax + Bu y = Cx
_
W( ) = C( I ; A);1 B, u y2
Rl 2C - -
(.. '( ) = det( I;A) det W( )
-
), C B = lim !1 W( ) -
.2
, l=1 n- - -
, {
n;1 -
, .. -
- , 12.
2. f : 0 1) ! Rm
- -
0 1) ,
()
0 1) >0T >0
, Z t+T
I
f(s)f(s)T ds (13.74)
t
t 0 2. 3

. A.
3



397
1 64, 103]. ABCG -
n n, n m, l n, m l. rank(B) = m: -
n n-
P = P T > 0 l m- ,
PA + AT P < 0 P B = C T GT A = A + B GC
, x = Ax + Bu y = GCx
_
- - .
1
u = y+v -
v Gy . -
{ ,
" { -
", . 14, 64, 119].
f˜:
2 64, 103]. -
˜(t)
0 1) ! Rm . , - -
_
, ˜(t) ! 0 ˜(t) ! 0
t!1 f { . ,
t ! 1, ˜(t) f(t) ! 0
T
t ! 1.
.
1. , -
(13.64) -
W( ) = GC( I ; A);1 B - --
. (13.68), (13.72),
(13.73) (13.70). , ,
-
('1 (yd ) : : : 'm (yd )) , -
(13.67).
. -
m
1 eT Pe + 1 X jj ^ ; jj2 = +
V (x ^0 ^ t) = 2 2 i=0 i i i
+jj ^0 ; 0 jj2 = 0 (13.75)
P = PT > 0 .
0
_ _ e 6= 0
V , V <0
, :
(
_
^0 = ; 0 eT PBGCe
(13.76)
_
^i = ; i eT PB'i (yd )
398
_ e 6= 0
1, , V <0 -
, (13.72), (13.73)
x = Ax + Bu y = Cx
_ - -
V (t) = V (x(t) ^0 (t) ^(t) t)
. -
. ( 'i (yd (t)) i = 1 : : : m -
^i (t).
), e(t) , -
_ ; ke(t)k
(13.76) , V =e T (PA + AT P)e 2

> 0. R
V (t);V (0) ; 0t ke(s)k2 ds.
0 t] Rt
ke(s)k2 ds.
, V 0, : V (0) 0

Z1
ke(t)k2 dt < 1: (13.77)
0

'i (yd ) i = 1 : : : m , (13.71), e(t)
_
. (13.77) ( . 64],
2.1) , (13.70).
(13.67) ,
(13.70)
_ (t) ! 0
˜ t ! 1.
(13.72) , -
e ˜ 'd y ^0
(13.71), ˜ -
, e(t) . -
eT(t) ! 0 t ! 1.
, _
˜(t) 'd (t) ! 0 t ! 1.
(13.72) , ,
(13.67) 2.
. 1
(13.75)
V (x ^0 ^ t) > 0 e 6= 0 (13.78)
V (x ^0 ^ t) < 0
_ e 6= 0:
, , -
(13.75) (13.78).
-
-
.
13.5.4.

( . 13.3).
399
xd1 = p(xd2 ; xd1 + f(xd1 ) + sf1(xd1 ))
_
xd2 = xd1 ; xd2 + xd3
_ (13.79)
xd3 = ;qxd2
_
f(z) = M0 z + 0:5(M1 ; M0 )f1(z) f1(z) = jz + 1j ; jz ; 1j
M0 M 1 p q { . s = s(t) { -
, . -
, yd (t) = xd1 (t)
p q.
M0 , M1 a priori ,
-
. , -

x1 = p(x2 ; x1 + f(yd ) + c1 f1(yd ) + c0(x1 ; yd ))
_
x2 = x1 ; x2 + x3
_ (13.80)
x3 = ;qx2
_
c0 c1 { .
(13.72), (13.73)
c0 = ; 0(yd ; x1 )2
_ (13.81)
c1 = ; 1(x1 ; yd )f1 (yd )
_
0 1{ .
(13.80), (13.81)
. -
1, , s(t) = const. 4


(
e1 = p(e2 ; e1 + (c1 ; s)f1 (yd ) + c0 e1 )
_
e2 = e1 ; e2 + e3 (13.82)
_
e3 = ;qe2
_
ei = xi ; xdi , i = 1 2 3. (13.82), ,
(13.71),
h1i
;p p 0
A = 1 ;1 1 B= 0 C = 1 0 0]
0 ;q 0 0
s(t) { , , ,
4
1, , . ,
, , , -
,
.

400
^1 = c1, = s, = c0.
1 0




. 13.14. (13.79).

+ +q
2
W( ) = : (13.83)
3 + (p + 1) 2 + q + pq

, n = 3, , {
2 q>0 -
p. , - --
q>0 p M0 M1 .
, 1 -
x(t) :
e(t) ! 0. , yd (t) ; x1 (t) ! 0. ,
s(t) -
c1 (t) ; s ! 0
s. 1, -
,( . 2),
tZ+T
0

f12(yd (t)) dt (13.84)
t0
T > 0, >0 t0 0.
(13.84) , (13.84) ,
401
xd (t)
t ! 1.
xd1 = 0 , (13.79) -
. ,
xd1 (t) (;1 1) ( f1(z) ),
, , tk , k = 1 2 : : :.
tk = tk+1 ;tk tk -
,
xd1 = 0.
(13.84):
1 Z f 2(x (t)) dt:
T
0 = lim T!1 (13.85)
T 1 d1
0

-
0
. -
( ., , 36]) > 0,
0
c1(t) ; s ! 0 -
> 0. -
10 1
,
x21
d (13.86)
0


x21 { x21 (t)
d d
= sup jxd1 (t)j.
x2
-
. p=9 q=
14:286 M0 = 5=7 M1 = ; 6 : -
7
(13.79)
( . 13.14). -
xd (0) = 0:3 0:3 0:3]. -
x0,
c0(0) c1 (0) . -
-
20 (" ", " ", ),
.. s(t) 1 0 t 20 .
( . 13.15)
( . 13.16) , -
c1 (t) ; s
,
. c0(t)
402
.
, , -
" ":
s(t) = s0 + s1 sign sin 2T t (13.87)
0

s0 = 1:005, s2 = 0:005.
T0 = 5:0 , 1 = 1:0 . 13.16, 13.17.
, y(t) -
yd (t) (
yd (t) ; y(t) . 13.16 ). -
, -
s(t):
, s(t) -
.
-
1,
1 = 5:0 ( . 13.18, 13.19). , -
, .. ,
-
.
, -
-
.
-
. -
-
.




403
. 13.15. " -
"( 0=1 = 1).
1




. 13.16. .


404
. 13.17. -
1 = 1.




. 13.18. s .


405
. 13.19. -
1 = 5.




. 13.20. s = 5.
1




406
A. A.

.

x(t) = f(x t)
_ (A.1)
x(t) 2 Rn { (t) 2 Rm {
( ) , f( ) { xt
1

- , :
( ) -
;
x(s)ts=0 (s)ts=0
(t) = (A.2)
, (A.1),
(A.2) t0 -
x(0) (0):
, -
, , , -

Qt ! 0 t ! 1: (A.3)

Qt tt (A.4)
;
Qt = Q x(s)ts=0 (s)ts=0 { -
), t 2R < 1.
(
106]
1.
;
Qt = Q x(t) t Q( )2R
" " .
1
-
, .
, ( -
), , , , -
. (A.1)
.

407
2.
Zt ;
Qt = q x(s) (s) s ds q( )2R:
0


. , -
-
;
lim q x(s) (s) s = 0: (A.5)
t!1
-
9, 103, 106].

d ( + (x t)) = ;;r !(x t) (A.6)
dt
; = ;T > 0 { m m- !(x t) { -
(A.1) 2 , (x t) {
- , -
78]:
(x t)T = r !(x t) 0: (A.7)
, (x t)
(x t) = ;1r !(x t) (A.8)
;
(x t)T = ;1 sign r !(x t) (A.9)
;i = ;T > 0 { m m- (i = 1 2) ;2 { .
i
(A.6) -
() .
-
-
-
106]. :
!(x t) = @Q +
2
@t
T
;
rx Q f(x t) { !(x t) =
q(x t):

408
v 2Rm
{
= (x v t) + (x t) = v
f(x t), rx Q(x t), (x t), r !(x t)
{
t 03
kxk ! 1
{ inf t 0 Q(x t)
{ !(x t)
2Rm
{ (Q) ( (Q) > 0
Q > 0) , xt
t) ; (Q):
!(x (A.10)
,
0 = (x ):(Im ;; ;)( 0 ; )=0
+
;
Q x(t) ! 0 t!0 ..
> 0:
-
9]
V (x t) = Q(x t) + 1 k ; + (x t)k2 + (A.11)
2 ;

(A.1),
(A.6),
_
Vt = !(x(t) (t) t) + vtT ;r !(x(t) (t) t) (A.12)
;
!(x t) (A.6), vt = t (t)
, v0 2 L(;) L(;)
+ (x(t) (t) t):
{ ;+:
(A.6), dvt 2 L(;): , v0 2 L(;) t
dt +
0 ; ;vt = vt (;+ ; -
_t =
L(;)). , (A.12) V
T
!(x(t) (t) t) + vt r !(x(t) (t) t): ;
_
Vt ; Q(x(t) t)
, 0:
, V (x(t) (t) t) V (x(0) (0) 0) -
(A.1), (A.6). ,
R
1;
Q(x(t) t) dt < 1 -
0
(. ., 103, 64]) ,
limt!1 Q(x(t) t) = 0 .
kxk+ k k t 0:
3




409
(A.6) c -
( ).
:
v 2Rm
{
= (x v t) + (x t) = v
f(x t) r !(x t) (x t)
{ -
{ q(x t) xt
{ !(x t)
2Rm
{ ,
!(x t) 0 (A.13)
{ .
x(0) (0) (A.1), (A.6) -
(A.4), (A.5)
= Q0 + 0:5 0 ; ; (x0 0 0 :
2
;+
+ (x t) = v
(x t)
L < 1:
, ,
Qt (A.1), (A.6) -
x(t):
!(x t) Qt -
, . 4


(A.10), (A.13), -
.
(A.6)

d = ;;r !(x t): (A.14)
dt
(A.6) {
,
= 0; (x t) (A.15)
a 2 Rn f(x) (x 2 Rn )
,
4

y 2Rn f(x + y) f(x) + aT y -
f(x) x @f(x) 78].
rf(x) @f(x):
x

410
>0{ ( ).
(A.15)
(x t) -
: >0 1 ,
r !(x t)
(x t)r !(x t) (A.16)
9, 64, 106]
,
{ (A.15)
{ !(x t)
{ = (x t)
(A.10)
(x t) r !(x t) ;1 ; (x t) (A.17)
0

{ (A.16).
(A.6), (A.15)
(A.4).
-
.
,
{ (A.15)
{ !(x t)
{ (A.16)
(A.6), (A.15)
(A.5).


2 Rm (A.10) " -
" , =
.
..
(A.1), (A.2)
lim (t) = : (A.18)
t!1

, " " -
. , (A.2) -
, (A.1),
(A.2) (A.18) 103].
411
, 36, 59, 74, 103], -
. -
" " (A.18)
.
9, 36, 103].
. (t) mN
t>0 - -
, t0 > 0 >0 L>0 ,
0
t>t0
Z
t+L
(s) (s)T ds Im: 2 (A.19)
0
t
, (t) -
t!1
RN :
(A.14) -
:
{ (x t) 0 :
v 2Rm = (x v t) -
f(x t), rx Q(x t), (x t),
+ (x t) = v
r !(x t)
2Rm
!(x t)
(Q) ( (Q) > 0 Q > 0) , xt
!(x t) (Q) , ,
{ inf x Q(x t) x (t)
x (t) (A.1) x(t) = f(x t)
_
@f(x t) @ f(x t) @ f(x t) r Q(x t)
2 2
{ -
@ @x@ x
@2
;
(t) = @f x t {
{ - .
@
(A.14)
x(t0 ), (t0 ) -
colfx (t) g (A.1), (A.14) -
-
x(t0 ) (t0 ) t0 103].
,
(" ")
n . -
" " "-
. 64, 106].
",

412
B. B. - -

1966 . . . -
-
.
103], .
fg{ ,
2fg
{ -
2R
k;1 )
. k( 0
2 R,
k;1
k( 0) k = 0 1 2 ::: { ,
k;1 k;1 =
0 0
0] 1] : : : k ;1] : ( -
k;1 () () ).
0



k( )0 (B.1)


k( )0 (B.2)
(B.1)
( ), (B.2)
{ .
f g0
,
0]
; k k k
k + 1] = Tk k = 0 1 2 ::: (B.3)
0 0 0

k
0


= f 0( ) : : : k=f
k
k( )g ( ) ::: k( )g (B.4)
0
0 0

k + 1]:
k] k ( ) k( ) k = 0 1 2 ::: :
, (B.1), (B.2) (
k),
k + 1] (B.3).
.

413
1. 103] (B.3) -
( )
- -
(B.1) { (B.2) { ,
) k (B.2) = k]
) k kk
(B.1) k] -
: k ] = k + 1] = k + 2] = : : : :
, k] -
, , (B.3) -
.
r = k] -
(B.1), ( ) -
2
.
- -
.
1. " -1"

f g:
(' k] ) + k] " k] k = 0 1 2 : : : : (B.5)
k] 2 R " k] 2 R (' )
'( -
(' k] ) ' k]T ).
( )k (B.5) -
f g.
, :
1. ">0 k0 -

" k] " ' k] : (B.6)
2 0 1)
2. ,
k0
(' k] ) + k] " k]: (B.7)
, 2"
(B.5)
:
414
103].
1, 2
> min(1 2 ) 0]
8
j k]j " k]
> k]
>
>
> k] ; k] ' k]
> j k]j " k]
>
< k' k]k2
k + 1] = > (B.8)
k]' k] ; " k]sign k]
> k] ;
>
> k' k]k2
>
>
: " k] < j k]j < " k]
;
k] = ' k] k] + k] (B.9)
- -
(B.5).
=1 -
(B.8)
8
j k]j " k]
> k]
>
<
k + 1] = > k] ; k]' k] ; " k]sign k] (B.10)
k' k]k2
>
:
" k] < j k]j < " k]:
< 0:5 <1 (B.8)
8
j k]j " k]
> k]
>
<
k + 1] = > k] ; k] 2 ' k] (B.11)
k' k]k
>
:
j k]j " k]:
, -
0
( ) k] 0 < k]
00 < 2:
8
j k]j " k]
> k]
>
<
k + 1] = > k] ; k] k]' k] ; " k]sign k] (B.12)
k' k]k2
>
:
" k] < j k]j < " k]:
, -
, (B.5)
.
415
2. " -2"

k] (' k] ) + k] " k] k = 0 1 2 ::: (B.13)
k] 2 R (
{ -
).
, .
1. C >0 ">0
0< k] C " k] " ' k] (B.14)
2fg 2 0 1)
2.
= (B.13)
" ", . .
;
k] ' k] + k] " k]: (B.15)
- -
(B.13) 103]
( k] k] = 1
k]sign( k])' k]
k + 1] = k] = ;1 (B.16)
k] ; k] k' k]k 2

:
;
k] = sign " k] ; k]
; (B.17)
k] = k] ' k] k] + k]
k] 00 < 2(1 ; )C ;1 :
0 00 k] 0< 0
(B.6) (B.14) -
. , (B.5) (B.13)
. -
.
- , { -
103].
1. (B.7) k] -
(B.9). 0]
(B.8) ( > min(1 2 )), (B.10), (B.11) (
0 k] 00 < 2)
< 0:5), (B.12) ( 0 < -
;
"0 k]:
' k] + k] (B.18)
416
2. (B.15) k] k]
(B.17). (B.16)
-
;
k] ' k] + k] "0 k] (B.19)
"0 k] "0 k' k]k "0 > 0 0 < k] < C :
3.

(' k] 0 (B.20)
(B.20)
" ":
(' k] " > 0: (B.21)
0]
;
k] ;' k] k] 0
k + 1] = k] + k]' k] ' k] k] < 0 (B.22)
k] = k] ; k] (' k]k]kk]) 0 < 0 k] 00 k] 0
k' 2
- (B.20) 103].

(' k] + k] 0: (B.23)
(B.23) " ":

(' k] + k] " > 0:
j k]j C k] > 0 k] ! 0 k ! 1 P1 k] =
1
1 0 k] 2 0] r 0]
n k + 1] = k] r k + 1] = r k] k] 0
k + 1] = k] + k]' k] r k + 1] = r k] + 1 k] < 0
k] = (' k] k]) + k]
k]
k] = r k] ; k] k' k]k2
- (B.23) 103].
417
4.

,
( ) -
, { :
1. ( ) (B.1)
r k( ) 2 C
C k, .
2f g 8k
2. " >0 ,
=
k( ) ":
3. 8k k( ) :1
4. k] :
1
X
i] = 1:
k] > 0 k!1 k] = 0
lim
i=1
0] r 0] = 0
(
k] r k + 1] = r k] k ( k]) 0
k + 1] = k] + k]r k ( k]) r k + 1] = r k] + 1
k ( k]) < 0 (B.24)
k] = r k] ; k] kr (k] k])k2
k
- -
(B.1) 103].
k] , ,
C
k] = C +1r k] C1 > 0 C2 > 0:
2

(B.1)
(B.2),
D k] f g 2 D k] k) 2
f(x) x ,
1

8x x (x ; x ) rx f(x 0 ) ; f(x00 ):
00 T
f(x
00 )
0 00 0
DR n ,
2
, . . 8x0 x00 2 D,
,
1 ) x0 + (1 ; )x00 2 D: f(x)
0
D = fx : f(x) "g " 78].

418
PD k] .3 -
PD k] (B.24) -
D k] D
(B.1), (B.2),
- .
-
( . . . 12.5, 13.4,
36, 103, 106]).




D fg -
3
kPD ; k = inf k ; k.
PD , 0
2D
0




419
C. C.
MATLAB

MATLABR ( 72, 59, 81, 139]),
-
.


c = conv(a, b)
CONV { .
C = conv(A, B) A B. -
length(A)+length(B)-1.
AB ,
.
. XCORR, DECONV, CONV2.
x, cnt] = fmins(funfcn, x, tol, prnt)
FMINS { -
, .
X = fmins('f', x0) x0
x, f(x). 'f'
{ , ,
m- 72, 139].
X = fmins(F, X, tol) tol -
;3 :
. 10
X = fmins(F, X, tol, 1) -
. X, cnt] = fmins(F, X, : : : )
.
xf, termcode, path] =fsolve(fvec, x0, details, fparam,
jac, scale)
FSOLVE { .
X = fsolve('f', X0) X0 -
X, f(x) = 0. 'f' { , -
, ,
- m- .
y = logspace(d1, d2, n)
LOGSPACE {
.

420
logspace(d1, d2) -
10d1
50
10d2 : 10d1
d2 =
. logspace(d1, d2, N) N .
. LINSPACE ":".
LTIFR {
.
G = ltifr(A, b, s) -
G(s) = (sI ; A);1 b
s: - b
, A: G
size(A) length(s) .
x, y] = meshdom(x, y)
MESHDOM { XY -
.
XX, YY] = meshdom(X, Y) ,
XY XX YY,
3- -
.
tout, yout] = ode45(FunFcn, t0, t nal, y0, tol, trace)
ODE45 { -
- 4 5- -
.. ODE23.
T, Y] = ode45('yprime', T0, T nal, Y0) -
,
m- YPRIME.M T0 T nal -
Y0.
T, Y] = ode45(F, T0, T nal, Y0, TOL, 1) -
TOL
.
:
F{ ,
. :
yprime = fun(t, y)
F = 'fun'. t { ( ),
y{ ( - ), yprime {
: yprime(i) yi (t):
_
t0 { t.
421
t nal { t.
y0 { - .
tol { . : tol =
10;6 .
trace {
. trace = 0.
:
T- - ( -
).
Y- , -
.
c = poly(x)
POLY { .
A { n n- , poly(A) -
n+1 , -
det( In ; A): V{ ,
poly(V) , -
V .
ROOTS POLY { .
y = polyval(c, x)
POLYVAL { .
V{ ,
, polyval(V, s) ,
s. S{ ,
S.
. POLYVALM .
coe s, poles, k] = residue(u, v, k)
RESIDUE { -
.
R, P, K] = residue(B, A) , -
(.. )
B A:
B(s) = r1 + r2 + r
+ s ;np + k(s):
A(s) s ; p1 s ; p2 n
BA -
s. - R,
- P, {
- K.
422
B, A] = residue(R, P, K)
, B/A- .
r = roots(c)
ROOTS { .
roots(C) , -
C. C N+1 -
XN
, C1 + : : : + CN X + CN+1 :
. ROOTS1 POLY.
y = table1(tab, x0)
TABLE1 { .
Y = table1(TAB, X0) TAB -
- ,
X0 TAB.
. X0
TAB. X0 .
. TABLE2.


X = are(F, G, H)
ARE {
X = are(F, G, H) ( )
:
F T X + XF ; XGX + H = 0
G=GT 0, H=HT :
Ab, Bb, Cb]=balreal(A, B, C)
BALREAL { -
. 1

Ab, Bb, Cb] = balreal(A, B, C) -
(A B C):
Ab, Bb, Cb, G, T] = balreal(A, B, C) -
G, -
T -
, (A B C)
(Ab Bb Cb): ,
, , , -
1
, 53].

423
G ,
.
mag, phase] = bode(a, b, c, d, iu, w)
BODE { - ( -
) .
MAG, PHASE] = bode(A, B, C, D, iu, W) -

x(t) = Ax(t) + Bu(t)
_ y(t) = Cx(t) + Du(t) (C.1)
i- s = |!. W -
( ),
. BODE MAG PHASE
( ), -
y, length(W) .
MAG, PHASE] = bode(NUM, DEN, W) -
, -

G(s) = NUM(s) (C.2)
DEN(s)
NUM DEN -
.
. LOGSPACE
.
Co=ctrb(A, B)
CTRB { .
ctrb(A, B)
Co = B, AB, A2 B, : : : , An;1 B ].
Abar, Bbar, Cbar, T, K] = ctrbf(A, B, C)
CTRBF { .
Abar, Bbar, Cbar, T, K] = ctrbf(A, B, C) -
.
Abar, Bbar, Cbar, T, K] = ctrbf(A, B, C, TOL)
TOL.
Co(A B) rank r n,
T ,


424
Abar = TAT 0 Bbar = TB Cbar = CT 0 (T 0 = T ;1) -

Abar = Anc Ac
0 0
Bbar = Bc Cbar = Cnc Cc ]
A21
Cc(sI ; Ac);1Bc
(Ac Bc)
C(sI ; A);1 B:
P, G] = c2d(a, b, t)
C2D {
.
P, G] = c2d(A, B, T)

x(t) = Ax(t) + Bu(t)
_ (C.3)


x n + 1] = Px n] + G u n] (C.4)
-
( , sample time) T.
Wn, Z] = damp(A)
DAMP { -
.
Wn, Z] = damp(A) Wn Z, -
A. A
:
1) A , ,
"A"
2) A{ - , , -
-
3) A{ - , , -
, DAMP
.
G = dgram(A, B)
DGRAM { -
.
425
dgram(A, B) -
.
dgram(A', C') -
. GRAM.
L, M, P] = dlqe(A, G, C, Q, R)
DLQE { -
.

x n + 1] = Ax n] + Bu n] + Gw n] ;
z n] = Cx n] + Du n] + v n] ;
:
Efwg = Efvg = 0 EfwwT g = Q Efvv T g = R
dlqe(A, G, C, Q, R) -
L :
{
x n + 1] = Ax n] + Bu n]
{
x n] = x n] + L(z n] ; Hx n] ; Du n]):

( ) x x:
L, M, P] = dlqe(A, G, C, Q, R) -
L,
M
P = Ef(x ; x)(x ; x)T g:
K, S] = dlqr(A, B, Q, R)
DLQR { -
.
K, S] = dlqr(A, B, Q, R)
K , -T
P T
u = ;Kx J= x Qx + u Ru

x n + 1] = Ax n] + Bu n]:
426
S -

S ; AT SA + AT SB ;1 (R + B T SB)BS T A ; Q = 0:
X = dlyap(A, C)
DLYAP { .
X = dlyap(A, C)
AXAT + C = X:
. LYAP.
Ab, Bb, Cb, Db] = dmodred(A, B, C, D, ELIM)
DMODRED { .
Ab, Bb, Cb, Db] = dmodred(A, B, C, D, ELIM) -
, ,
ELIM.
X1, , X2, ,
A = A11 A12 B = B1 C = C1 C2 ]
A21 A22 B2
x n + 1] = Ax n] + Bu n] y n] = Cx n] + Du n]:
X2 n+1] X2 n],
X1.
LENGTH(ELIM)
,
ELIM .
. DBALREAL, BALREAL MODRED
a, b] = d2c(phi, gamma, t)
D2C { -

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