<<

. 14
( 15)



>>

.
A, B] = d2c(P, G, T) (C.4)
(C.3) -
T.
mag, phase] = dbode(a, b, c, d, iu, w)
DBODE { - -
( ) .

427
MAG, PHASE] = dbode(A, B, C, D, iu, W) -
z = e|! :
(C.1) i-
W ( ),
.
! : DBODE MAG PHASE ( -
y
),
length(W) .
MAG, PHASE] = dbode(NUM, DEN, W) -
, -

G(z) = NUM(z) (C.5)
DEN(z)
NUM DEN -
.
y, x] = dimpulse(a, b, c, d, iu, n)
DIMPULSE { ( ) -
.
Y = dimpulse(A, B, C, D, iu, n)
x n + 1] = Ax n] + Bu n] y n] = Cx n] + Du n] (C.6)
(- ),
i- . n ,
. DIMPULSE Y
yn
,
.
Y, X] = dimpulse(A, B, C, D, iu, n)
.
Y = dimpulse(NUM, DEN, n)
(C.5),
NUM , DEN -
.
y, x] = dlsim(a, b, c, d, u, x0)
DLSIM {
Y = dlsim(A, B, C, D, U) (C.6)
U. U -
,
u. U .
DLSIM Y,
y LENGTH(U) .
428
Y, X] = dlsim(A, B, C, D, U) -
.
dlsim(A, B, C, D, U, X0) ,
( ) .
Y = dlsim(NUM, DEN, U)
, -
(C.5), NUM DEN -
.
dlsim(NUM, DEN, U) lter(NUM, DEN, U).
y, x] = dstep(a, b, c, d, iu, n)
DSTEP { -
.
Y = dstep(A, B, C, D, iu, n)
(C.6) i- . n
( ). DSTEP Y,
y,
n.
Y, X] = dstep(A, B, C, D, iu, n)
.
Y = dstep(NUM, DEN, n)
(C.5), NUM DEN -
,
.
y, x] = impulse(a, b, c, d, iu, t)
IMPULSE { ( ) -
.
Y = impulse(A, B, C, D, iu, T) -
(C.1) i- . T -
,
. IMPULSE
Y ,
y LENGTH(T) .
Y, X] = impulse(A, B, C, D, iu, T)
.
Y = impulse(NUM, DEN, T) -
(C.2), NUM
DEN
.


429
L, P] = lqe(A, G, C, Q, R)
LQE { -
.
:
x(t) = Ax(t) + Bu(t) + Gw(t) -
_
z(t) = Cx(t) + Du(t) + v(t) -
-
:
Efw(t)g = Efv(t)g = 0 Efw(t)w T (t)g = QEfv(t)v T (t)g = R

lqe(A, G, C, Q, R) -
L ,
_
x(t) = A^(t) + Bu(t) + L(z(t) ; H^(t) ; Du(t))
^ x x
,
x(t).
L, P] = lqe(A, G, C, Q, R) { -
L
P,
.
K, S] = lqr(A, B, Q, R, N)
LQR {
.
K, S] = lqr(A, B, Q, R) -
K ,
u(t) = ;Kx(t)
Z1
(xT (t)Qx(t) + uT (t)Ru(t))dt
J= (C.7)
0

x(t) = Ax(t) + Bu(t):
_
, S{ -
T ;1 B T S + Q = 0
SA + A S ; SBR
K, S] = lqr(A, B, Q, R, N) -
T
2x (t)Nu(t), ux -
.

430
-
. K, S]
= lqr2(A, B, Q, R, N), -
, LQR.
K, S] = lqry(A, B, Q, R, N)
LQRY { -
.
K, S] = lqry(A, B, C, D, Q, R) -
K , u(t) =
;Ky(t)
Z1
(y T (t)Qy(t) + uT (t)Ru(t))dt
J= (C.8)
0




x(t) = Ax(t) + Bu(t)
_ y(t) = Cx(t) + Du(t):
, S{ -
SA + A S ; SBRT
;1 B T S + Q = 0
K, S] = lqry(A, B, Q, R, N) -
T
2y (t)Nu(t), uy
.
y, x] = lsim(a, b, c, d, u, t, x0)
LSIM {
.
lsim(A, B, C, D, U, T) -
(C.1) U. U
,
u. U ,
U length(T) .
Y=lsim(A, B, C, D, U, T) ( )
Y, y
length(T) .
Y, X] = lsim(A, B, C, D, U, T)
.
lsim(A, B, C, D, U, T, X0) ,
.

431
lsim(NUM, DEN, U, T) -
(C.2), NUM DEN
.
X = lyap(A, B, C)
LYAP { .
X = lyap(A, C) -
AX + XAT = ;C:
X = lyap(A, B, C) -
AX + XB = ;C:
. DLYAP.
Gm, Pm, Wcg, Wcp] = margin(mag, phase, w)
MARGIN {
.
Gm, Pm, Wcg, Wcp] = margin(MAG, PHASE, W)
Gm, Pm -
Wcg Wcp
( ), MAG, PHASE, W -
.
.
Am, Bm, Cm, Dm] = minreal(A, B, C, D, TOL)
MINREAL {
.
Am, Bm, Cm, Dm] = minreal(A, B, C, D) -
(A,
B, C, D).
.
Am, Bm, Cm, Dm] = minreal(A, B, C, D, TOL)
TOL , -
.
Zm, Pm] = minreal(Z, P), Z P{ - ,-
, , -
TOL = 10*SQRT(EPS)*ABS(Z(i)).
Zm, Pm] = minreal(Z, P, TOL) TOL.
NUMm, DENm] = minreal(NUM, DEN), NUM, DEN {
- -
, MINREAL .

432
NUMm, DENm] = minreal(NUM, DEN, TOL) -
TOL.
Ob = obsv(A, C)
OBSV { .
obsv(A, C)
Ob = C CA CA2 : : : CAn;1 ]T :
Abar, Bbar, Cbar, T, K] = obsvf(A, B, C, TOL)
OBSVF { .
Abar, Bbar, Cbar, T, K] = obsvf(A, B, C) -
.
Abar, Bbar, Cbar, T, K] = obsvf(A, B, C, TOL)
TOL.
(A C) r n,
T ,
0 Bbar = TB Cbar = CT 0 (T 0 = T ;1)
Abar = TAT -

Abar = Ano A12 Bbar = Bno Cbar = 0 Co ]
0 Ao Bo
Co(sI ; Ao);1 Bo
(Ao Co)
C(sI ; A);1 B:
K = place(A, B, P)
PLACE { -
( ).
K = place(A, B, P) -
K , A-BK
P. -
P -
. -
, -
.
"ndigits" (n )
, .
, -
A-BK P
.

433
,
10% P
.
num, den] = ss2tf(a, b, c, d, iu)
SS2TF { -
.
NUM, DEN] = ss2tf(A, B, C, D, iu)
(C.2) (C.5) (C.1), (C.6) i- .
DEN
s.
NUM, ,
y.
y, x] = step(a, b, c, d, iu, t)
STEP { -
.
Y = step(A, B, C, D, iu, T) -
(C.1) i-
. T
. STEP Y,
, y
length(T) .
Y, X] = step(A, B, C, D, iu, T)
.
Y = step(NUM, DEN, T)
(C.2), NUM
DEN
.
a, b, c, d] = tf2ss(num, den)
TF2SS { -
.
A, B, C, D] = tf2ss(NUM, DEN) -
(C.1) (C.6) (C.2), (C.5)
( ) . DEN
s.
NUM, ,
y.
. -
-
.

434
ss2df

function Ad,Bd,Cd,T]=ss2df(A,B,C)
n,m]=size(A)
v,p]=eig(A)
k=1
P= ]
while k<=n,
if all(imag(v(:,k))==0)
P= P v(:,k)]
k=k+1
else
P= P, 1/2*(v(:,k)+v(:,k+1)),...
1/2/j*(v(:,k)-v(:,k+1))]
k=k+2
end
end
T=inv(P)
Ad=T*A*P
Bd=T*B
Cd=C*P
tf2cf
SIMO-
function A,B,C,D]=tf2cf(num,den)
n=length(den)-1
l,r]=size(num)
if (r>n+1) | (den(1)==0)
error(' -
.')
end
dn=den/den(1)
nm= zeros(l,n-r+1),num/den(1)]
A= zeros(n-1,1) eye(n-1,n-1) -dn(n+1:-1:2)]
B= zeros(n-1,1) 1]
C=nm(:,r:-1:2)-nm(:,1)*dn(r:-1:2)
D(:,1)=nm(:,1)
435
tf2of
MISO-
function A,B,C,D]=tf2of(num,den)
n=length(den)-1
m,r]=size(num)
if (r>n+1) | (den(1)==0)
error(' -
.')
end
dn=den/den(1)
nm= zeros(m,n-r+1),num/den(1)]
A= zeros(n-1,1) eye(n-1,n-1) -dn(n+1:-1:2)]
C= 1, zeros(1,n-1)]
D(1,:)=nm(:,1)'
nm=nm-nm(:,1)*dn
B(1,:)=nm(:,2)'
for k=2:n
sm=0
for l=1:k-1
sm=sm+B(l,:)*dn(k-l+1)
end
B(k,:)=nm(:,k+1)'-sm
end




436
D. D.
SCILAB

1990- , -
-
Scilab,
MATLAB,
: Scilab -
, .. . Scilab -
- -
(INRIA)
( ) -
:
http://www-rocq.inria.fr/scilab/
-
-
Scilab. Scilab -
: .. ,
.. "
MATLAB Scilab" 10].
Scilab : -
, ( Scilab) -
.
, , Scilab { -
. ,-
, Netlib: http://www.netlib.org/
-
Scilab. -
Scilab, MATLAB, -
: -
, , , ,
, . Scilab
-
( , , -
)
. -
, .
-
437
MATLAB.
Scilab
: , -
, -
( ). -
Scilab ,
( -
, -
) ( , -
H1- , , -
.)
(LMI), ,
Metanet.
Scilab Scicos -
-
(
SIMULINK).
(
, ). -
Maple.
, Parallel
Scilab.
Scilab -
, :
{
( , " -
", -
Scilab
)
{ -
{ , -
{ ,
, .
Scilab.

.
a=1 {
438
1==1 {
'string' {
z=poly(0,'z') { z
p=1+3*z+4.5*zb2 { z
p = 1 + 3z + 4.5zb2
r=z/p {
r=
z
1 + 3z + 4:5zb2

A= a+1 2 3 0 0 atan(1) 5 9 -1] { 3 3-
b= %t,%f] { 1 2-
Mc= 'this','is' 'a' ,'matrix'] { 2 2-
Mp= p,1-z 1,z*p] { 2 2-
Mp =
1;z
! 1 + 3z + 4:5zb2 !
z + 3zb2 + 4:5zb3 !
! 1
F=Mp/poly( 1+%i 1-%i 1],'z') {

F=
;1
1 + 3z + 4:5zb2
! !
;2 + 4z ; 3z 2 ; 2z + zb2
b2 + zb3
z + 3zb2 + 4:5zb3 !
1
!
;2 + 4z ; 3zb2 + zb3 ;2 + 4z ; 3zb2 + zb3
Sp=sparse( 1,2 4,5 3,10], 1,2,3]) {
Sp =
( 4, 10) sparse matrix
( 1, 2) 1.
( 3, 10) 3.
( 4, 5) 2.

439
Sp(1,10)==Sp(1,1) {

L=list(a,-(1:5), Mp, 'this','is' 'a','list']) {
L=
L(1)
1.
L(2)
! - 1. - 2. - 3. - 4. - 5. !
L(3)
1;z
! 1 + 3z + 4:5zb2 !
b2 + 4:5zb3 !
! 1 z + 3z
L(4)
! this is !
! a list !
Lt=tlist( 'mylist','color','position','weight'],'blue', 0,1],10)
{
Lt('color')
{
A=diag( 2,3,4]) B= 1 0 0 1 0 0]
C= 1 -1 0] D=0*C*B x0= 0 0 0]
Sl=syslin('c',A,B,C,D,x0) {

Sl =
Sl(1) (state-space system:)
lss
! 2: 0: 0: !
Sl(2) = A matrix = ! 0: 3: 0: !
! 0: 0: 4: !




440
Sl(3) = B matrix =
! 1: 0: !
! 0: 1: !
! 0: 0: !
Sl(4) = C matrix =
! 1: ;1: 0: !
Sl(5) = D matrix =
! 0: 0: !
Sl(6) = X0 (initial state) =
! 0: !
! 0: !
! 0: !
Sl(7) = Time domain =
c
Sl("A"), Sl("C") { -
Slt=ss2tf(Sl) {
Slt =
;1
! ;21+ s ;3 + s !
Slt('num'), Slt('den')

v=1:5 W=v'*v { -
W(1,:) {
W(:,$) {
Mp'*Mp+eye {
ans =
column 1
! 3 + 6z + 18zb2 + 27zb3 + 20:25zb4 !
1 + 3z + 4:5zb2
! !
column 2
! 2 !
! 1 + 3z + 4:5z !
! 2 ; 2z + 2zb2 + 6zb3 + 18zb4 + 27zb5 + 20:25zb6 !
Mp1=Mp(1,1)+4.5*%i {
Fi=C*(z*eye-A)b(-1)*B {
F(:,1)*Fi {
441
ans =
1 + 3z + 4:5zb2 1 + 3z + 4:5zb2
4 ; 10z + 10zb2 ; 5zb3 + zb4 6 ; 14z + 13zb2 ; 6zb3 + zb4
;1
1
4 ; 10z + 10z b2 ; 5zb3 + zb4 6 ; 14z + 13zb2 ; 6zb3 + zb4
M= Mp -Mp Mp' Mp+eye] Fi, Fi(:,1)] {
F=syslin('c',F) Num=F('num') Den=F('den') {
.
inv(A) inv(Mp) {
inv(Sl*Sl') { -
w=ss2tf(ans) {
w = 18 ; 30bs + 18:5sb2 ; 5sb3 + 0:5sb4
6:5 ; 5s + sb2
w1=inv(ss2tf(Sl)*ss2tf(Sl')) { -
A=rand(3,3) B=rand(3,1) n=contr(A,B) {
K=ppol(A,B, -1-%i -1+%i -1])
poly(A-B*K,'z')-poly( -1-%i -1+%i -1],'z') { -

K = - 113.28616 111.62667 33.092441
s=sin(0:0.1:5*%pi)
ss= t(s(1:128),-1)
{
xbasc() plot2d3("enn",1,abs(ss)')
{

de (' x]=fact(n)','if n=0 then x=1,
else x=n*fact(n-1),end')
10+fact(5)

de (' f,g,ind]=rosenbro(x,ind)','a=x(2)-x(1)b2, b=1-x(2),
442
. D.1. Scilab.
{ ,{ .

f=100.*ab2 + bb2 , g(1)=-400.*x(1)*a , g(2)=200.*a -
2.*b ')
f,x,g]=optim(rosenbro, 2 2],'qn')

a=rand(3,3) e=expm(a)
de (' ydot]=f(t,y)','ydot=a*y')
e(:,1)-ode( 1 0 0],0,1,f)

s=poly(0,'s')
h= 1/s,1/(s+1) 1/s/(s+1),1/(s+2)/(s+2)]
w=tf2ss(h) ss2tf(w)
h1=clean(ans)
h1 = 1 1
! !
s 1+s
1 1
! s + s2 4 + 4s + s2 !
:
sl=syslin('c',1/(s*s+0.2*s+1))
instants=0:0.05:20
{ :
y=csim('step',instants,sl)
xbasc() plot2d(instants',y')
{ :
443
de (' in]=u(t)','if t<3 then
in=0 else in=1 end')
y1=csim(u,instants,sl)
plot2d(instants',y1')
{ :
yi=csim('imp',instants,sl)
xbasc() plot2d(instants',yi')
yi1=csim('step',instants,s*sl)
plot2d(instants',yi1')
{
dt=0.05 sld=dscr(tf2ss(sl),0.05)
{ :
u=ones(instants)
yyy= ts(u,sld)
xbasc() plot(instants,yyy)
{ :
u=0*ones(instants) u(1)=1/dt
yy= ts(u,sld)
xbasc() plot(instants,yy)
{
w1= w,w]
clean(ss2tf(w1))
w2= w w]
clean(ss2tf(w2))
{
z=poly(0,'z') horner(h,(1-z)/(1+z))
{ - (" ")
ans = 1 + z 1+z
! 2 2!
1;z 2
! 1 + 2z 2z z 1 + 2z + z 2 !
+
2; 9 + 6z + z




444
E.
1.!!!! -
/ .. , .. , ..
: . .. . .: ,
1988. 306 .
2.!!!! :
..
. .: . ., 1989. 263 .
3.!!!! ..
. .: , 1976. 424 .
4.!!!! -
..
. .: , 1997. 206 .
5.!!!! -
..
// .
. .: , 1977. C.
50{53.
6.!!!! -
. ., . ., ..
-
// . 1996. N 4.0
=

. 4{17.
7.!!!! . ., . ., . ., -
-
..
: . .: , 1981. 98 .
8.!!!! . ., ..
: . .: ,
1989. 88 .
9.!!!! -
. ., . ., ..
. // . 1988. N=
0

12. . 3{39.
10.!!!! -
. ., ..
MATLAB-5
Scilab. .
11.!!!! -
. ., . ., ..
. 2- . .: , 1959.
445
12.!!!! -
..
. .: , 1975. 240 .
13.!!!! . ., . ., ..
. .: , 1987, 599 .
14.!!!! -
. ., . ., .. .
( { ) -
// . 1996. N= 10. . 3{40.
0


15.!!!! -
. ., ..
. .: , 1975. 768 .
16.!!!! .
..
.: , 1971.
17.!!!! . .: ,
..
1994.
18.!!!! -
. ., ..
. .: -
, 1958.
19.!!!! -
..
. .: , 1973. 697 c.
20.!!!! : -
/.. ,
..
- , .. ,.. . .: -
, 1983. 200 .
21.!!!! -
. ., . ., ..
-
. .: , 1984. 215 .
22.!!!! . ., ..
. .: -
, 1992.
23.!!!! ..
. .: , 1987. 230 .
24.!!!! . ., . ., ..
. .: , 1988.

446
25.!!!! - -
..
. , 1997. 386 .
26.!!!! . ., ..
.. . , 1973.
151 .
27.!!!! . . 6-
/. : .. ( .). .: ,
1978{1980.
28.!!!! ..
. .: , 1991.
29.!!!! -
..
-
: , -
, // . 1993. N= 3,
0

.3{62.
30.!!!! . ., . ., ..
-
. .: , 1978. 400 .
31.!!!! ..
// -
/ ... -
, .. . .: , 1998. . 53{84.
32.!!!! . . MATLAB 5.2. -
Windows: . .:
, 1999. 288 .
33.!!!! . -
..
, 1995.
34.!!!! ..
. 2- . .: - , 1998.
35.!!!! .
/ . .. . .: , 1988.
36.!!!! . ., ..
. .: ,
1981. 216 .
447
37.!!!! -
. ., . ., ..
// -
. N 10. 1997. . 4{26.
0
=


38.!!!! -
..
. .: , 1974.
39.!!!! -
. ., . ., ..
// . 1996, N= 2 . 3{33.
0



40.!!!! -
..
. .: , 1967. 336 .
41.!!!! . ., ..
-
// .
. 1967. . 174, N= 1. . 47{49.
0


42.!!!! -
. ., ..
-
. // .
. 1978. . 241, N 2. . 301{304.
0
=


43.!!!! . ., ..
-
// -
. 1986. N 2, . 21{30.
0
=


44.!!!! .
., .
. .: , 1970. 704 .
45.!!!! . .: , 1979.
..
46.!!!! -
., ., .
. .: , 1971.
47.!!!! -
., .
. .: , 1986. 650 .
48.!!!! ..
. .: , 1990.

448
49.!!!! .
..
. , 1994. 344 .
50.!!!! ..
// .
. 1991. . 318. N 4. . 844{848.
0
=


51.!!!! -
. ., . ., ..
. .: -
, 1977. 272 .
52.!!!! . .: ,
..
, 1997. 496 .
53.!!!! . .: , 1978. 280 .
.
54.!!!! - ., .
. .: , 1964. 168 .
55.!!!! . ., . ., ..
. .: - , 1992.
56.!!!! . ., ..
. .: , 2000.
57.!!!! -
.
. .: , 1978.
58.!!!! -
..
. .: , 1984. 152 .
59.!!!! . -
.
. .: , 1991. 432 .
60.!!!! -
..
. .: , 1950.
61.!!!! { : .
., .
.: , 1990.
62.!!!! -
..
: . /. . .. , ..
, .. . .: , 1986.


449
63.!!!! -
..
. .: , 1990. 128 .
64.!!!! -
. ., . ., ..
-
. .: , 2000.
65.!!!! . .: , 1990.
.
66.!!!! -
: / ... .
.: , 1977. . 1. 366 . . 2. 455 .
67.!!!! -
..
.. . . 1. 1994 . 2. 1996.
68.!!!! -
. ., ..
. .: , 1987, 424 .
69.!!!! . ., ..
. // -
. 1994. N= 9. . 3{26.
0



70.!!!! -
., .
. .: , 1979.
71.!!!! . 1, 2/
. .. . .: , 1954.
72.!!!! : -
. 2- ./ . .. . . .:
1996. 192 .
73.!!!! ..
. .: , 1973. 321 .
74.!!!! . ., . ., . ., -
-
..
. { .: -
, 1972. 260 .
75.!!!! -
. ., . ., ..
- -
. .: , 1980.

450
76.!!!! ..
: . .: , 1986. 615 .
77.!!!! . .:
.
, 1988. 410 .
78.!!!! . .: , 1983.
..
79.!!!! ..
. .: , 1974, 332 .
80.!!!! . . ,
..
1981. 176 .
81.!!!! MATLAB. -
..
. .: - , 1997 { 350 .
82.!!!! ..
MATLAB 5 . .: - , 1998.
314 .
83.!!!! -
..
.: , 1970.
84.!!!! ..
. .: , 1979.
85.!!!! -
. ., ..
. .: , 1984.
86.!!!! -
. ., ..
. .:
, 1987.
87.!!!! -
. ., ..
: . . . { .: ,
1997. 320 .
88.!!!! -
.
. .: , 1983. 638 .
89.!!!! -
..
. .: , 1994. 464 .


451
90.!!!! ., ., .
. .: , 1980. 300 .
91.!!!! . .:
..
, 1996. 212 .
92.!!!! - . .: -
..
, 1973.
93.!!!!
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460
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109 - 376
365 186
330
393 242
309 309
182 382
303,
17
336
261
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168
309
311, 322
16
261
330
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309
418

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