Колесников А.А. Синергетические методы управления сложными системами: энергетические системы
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i
d
i
q
u
d
u
q
i
fd
i
fq
U
fd
,U
fq
i
1d
i
1q
ψ
d
ψ
q
ψ
fd
ψ
fq
ψ
1d
ψ
1q
x
d
x
q
x
fd
x
fq
x
1d
x
1q
ψ
d
=(x
d
i
d
+ x
ad
(i
fd
+ i
1d
))/ω
0
;
ψ
q
=(x
q
i
q
+ x
ad
(i
fq
+ i
1q
))/ω
0
;
ψ
fd
=(x
fd
i
fd
+ x
ad
(i
d
+ i
1d
))/ω
0
;
ψ
fq
=(x
fq
i
fq
+ x
ad
(i
q
+ i
1q
))/ω
0
;
ψ
1d
=(x
1d
i
1d
+ x
ad
(i
d
+ i
fd
))/ω
0
;
ψ
1q
=(x
1q
i
1q
+ x
ad
(i
q
+ i
fq
))/ω
0
.
dδ
dt
= ω − ω
0
;
dω
dt
= c
3
(M
T
− x
ad
/ω
0
(i
q
(i
fq
+ i
1d
) − i
d
(i
fq
+ i
1q
)));
di
d
dt
= c
1
(a
1
a
2
i
d
+ a
2
n
2
i
q
ω − a
4
i
fd
+ a
5
ω(i
fq
+ i
1q
) − a
7
i
1d
+ a
3
ω
0
U
fd
);
di
q
dt
= c
2
(−a
8
n
1
ωi
d
+ a
9
i
q
− a
10
ω(i
fd
+ i
1d
) − a
11
i
fq
− a
13
i
1q
+ b
1
ω
0
U
fq
);
di
fd
dt
= c
1
(−a
14
i
d
− a
3
n
2
i
q
ω + a
16
i
fd
− a
17
ω(i
fq
+ i
1q
) − a
19
i
1d
− a
15
U
fd
);
di
fq
dt
= c
2
(b
1
n
1
ωi
d
− b
2
i
q
+ b
3
ω(i
fq
+ i
1d
)+a
21
i
fq
− a
23
i
1q
− a
20
U
fq
);
di
1d
dt
= c
1
(−a
25
i
d
− a
6
n
2
i
q
ω − a
26
i
fd
− a
24
ω(i
fq
+ i
1q
)+a
27
i
1d
− a
18
ω
0
U
fd
);
di
1q
dt
= c
2
(a
12
n
1
ωi
d
− a
28
i
q
+ a
29
ω(i
fd
+ i
1d
) − a
30
i
fq
+ a
31
i
1q
+ b
4
U
fq
);
dM
T
dt
= c
4
−M
T
−
ω − ω
0
σ
+ U
T
,
n
1
= x
d
+ x
n
n
2
= x
q
+ x
n
c
1
=(−x
2
ad
(x
d
+ x
1d
+ x
fd
+ x
n
− 2x
ad
)+x
1d
x
fd
n
1
)
−1
c
2
=
(−x
2
ad
(x
q
+ x
1q
+ x
fq
+ x
n
− 2x
ad
)+x
1q
x
fq
n
2
)
−1
c
3
= Tj
−1
c
4
= T
−1
c
a
1
=(r
a
+ R
n
)ω
0
a
2
=
x
2
ad
−x
1d
x
fd
a
3
=(x
ad
−x
1d
)x
ad
a
4
= a
3
r
fd
ω
0
a
5
= x
ad
a
2
a
6
=(x
ad
−x
fd
)x + ad a
7
= a
6
r
1d
ω
0
a
8
= x
2
ad
−x
1q
x
fd
a
9
= a
1
a
8
a
10
= x
ad
a
8
b
1
=(x
ad
−x
1q
)x
ad
a
11
= b
1
r
fd
ω
0
a
12
=(x
ad
−x
fd
)x
ad
a
13
= a
12
r
1q
ω
0
a
14
= a
1
a
3
a
15
=(x
2
ad
− x
1d
n
1
)ω
0
a
16
= a
15
r
fd
a
17
= a
3
x
ad
a
18
= x
ad
(x
ad
− n
1
)
a
19
= a
18
r
1d
ω
0
b
2
= a
1
b
1
b
3
= x
ad
b
1
a
20
=(x
2
ad
− x
1q
n
2
) a
21
= a
20
r
fq
a
22
=(x
2
ad
− x
ad
n
2
)ω
0
a
23
= a
22
r
1q
a
24
= a
6
x
ad
a
25
= a
6
a
1
a
26
= a
18
r
fd
ω
0
a
27
=(x
2
ad
− x
fd
n
1
)ω
0
r
1d
a
28
= a
12
a
1
a
29
= a
12
x
ad
b
4
=(x
ad
− n
2
)x
ad
ω
0
a
30
= b
4
r
fd
a
31
=(x
2
ad
− x
fq
n
2
)r
1q
ω
0
U
fd
U
fq
ω − ω
0
=0
i
fd
− i
fd0
=0,
i
fq
− i
fq0
=0.
δ = δ
0
U = U
0
m =3
ψ
1
= i
fd
− i
fd0
=0;
ψ
2
= U
2
− U
2
0
=(−r
a
i
d
− x
q
i
q
− x
ad
(i
fq
i
1q
))
2
+(−r
q
i
q
+ x
d
i
d
+ x
ad
(i
fd
+ i
1d
))
2
− U
2
0
=0;
ψ
3
= M
T
+ ϕ
5
=0.
T
1
˙
ψ
1
(t)+ψ
1
=0;
T
2
˙
ψ
2
(t)+ψ
2
=0;
T
3
˙
ψ
3
(t)+ψ
3
=0.
i
fd
= i
fd0
M
T
ψ
s
=0 s =1, 2, 3
dδ
dt
= ω − ω
0
;
dω
dt
= c
3
(−ϕ
5
− x
ad
/ω
0
(i
q
(i
fq
+ i
1d
) − i
d
(ξ + i
1q
)));
di
d
dt
= c
1
(a
1
a
2
i
d
+ a
2
n
2
i
q
ω − a
4
i
fd0
+ a
5
ω(ξ + i
1q
) − a
7
i
1d
+ a
3
ω
0
U
fd
);
di
q
dt
= c
2
(−a
8
n
1
ωi
d
+ a
9
i
q
− a
10
ω(i
fd0
+ i
1d
) − a
11
ξ − a
13
i
1q
+ b
1
ω
0
U
fq
);
di
1d
dt
= c
1
(−a
25
i
d
− a
6
n
2
i
q
ω − a
26
i
fd0
− a
24
ω(ξ + i
1q
)+a
27
i
1d
− a
18
ω
0
U
fd
);
di
1q
dt
= c
2
(a
12
n
1
ωi
d
− a
28
i
q
+ a
29
ω(i
fd0
+ i
1d
) − a
30
i
fq
+ a
31
i
1q
+ b
4
U
fq
),
ξ =
(−r
a
i
q
− ω
0
(x
d
i
d
+ x
ad
(i
fd0
+ i
1d
)))
2
− 2ω
2
0
x
ad
(x
d
i
d
(i
fd0
+ i
1d
)+x
ad
i
fd0
i
1d
)+U
2
0
U
fd
=
U
fd
(i
d
,i
q
,i
1d
,i
1q
,δ,ω) U
fq
= U
fq
(i
d
,i
q
,i
1d
,i
1q
,δ,ω)
ψ
s
=0 s =1, 2, 3
ϕ
5
δ = δ
0
ψ
4
= ω − ω
0
+ γ
5
(δ − δ
0
),
T
4
˙
ψ
4
(t)+ψ
4
=0.
ϕ
5
=
i
d
ω
0
ξ − r
a
i
2
d
/ω
0
+ i
d
i
q
x
q
+ i
q
x
ad
(i
fd0
+ i
1d
)+
1
c
3
T
4
(1 + T
4
γ
5
)(ω − ω
0
)+
γ
5
c
3
T
4
(δ − δ
0
).
dδ
dt
= −γ
5
(δ − δ
0
);
di
d
dt
= c
1
'
a
1
a
2
i
d
+(a
2
n
2
i
q
+ a
5
(ξ + i
1q
))(ω
0
+ γ
5
(δ − δ
0
)) − a
4
i
fd0
− a
7
i +1d + a
3
ω
0
U
fd
(
;
di
q
dt
= c
2
'
−(a
8
n
1
i
d
+ a
10
(i
fd0
+ i
1d
))(ω
0
+ γ
5
(δ − δ
0
)) + A
9
i
q
− a
11
ξ − a
13
i
1q
+ b
1
ω
0
U
fq
(
;
di
1d
dt
= c
1
'
−a
25
i
d
− (a
6
n
2
i
q
+ a
24
(ξ + i
1q
))(ω
0
+ γ
5
(δ − δ
0
)) − a
26
i
fd0
+ a
27
i
1d
+ a
18
ω
0
U
fd
(
;
di
1q
dt
= c
2
'
(a
12
n
1
i
d
+ a
29
(i
fd0
+ i
1d
))(ω
0
+ γ
5
(δ − δ
0
)) − a
28
i
q
− a
30
ξ + a
31
i
1q
+ b
4
U
fq
(
,
U
fd
= U
fd
(i
d
,i
q
,i
1d
,i
1q
,δ) U
fq
= U
fq
(i
d
,i
q
,i
1d
,i
1q
,δ)
ψ
4
=0
U
fd
U
fq
U
T
T
i
> 0,i=1, 2, 3, 4,γ
5
> 0.
P Q Tj =7
T
c
=4 x
d
= x
q
=0,537 x
ad
=0,443 x
fd
=0,561 x
fq
=0,35 x
1d
=0,275 x
1q
=0,475
x
n
=0,6 R
n
=0,8 r
a
=0,007 r
fd
=0,0004 r
fq
=0,0006 r
1d
=0,006 r
1q
=0,02 σ =0,05
ω
0
=1 T
1
= T
2
=4 T
3
= T
4
=2 γ
5
=1 i
fd0
=1 U
0
=1
δ
0
= π/3
•
•
•
•
T
s
uT
s
U
out
L r R
C (1 − u)T
s
u 0 ...1
K(u) u
di
L
dt
=
E
L
u −
U
out
− ri
L
L
;
dU
out
dt
=
1
C
i
L
−
U
out
R
− M(t),
M(t)
T
s
RC.
0 ...1
E
L r C R
L
C uT
s
L
E
K(u)=
1
1 −u
.
di
L
dt
=
E − i
L
r
L
− (1 − u)
U
out
L
;
dU
out
dt
=(1− u)
i
L
C
−
U
out
RC
− M(t).
uT
s
E L
C (1 − u)T
s
L
C C
di
L
dt
=
uE − i
L
r +(1− u)U
out
L
;
dU
out
dt
= −
U
out
RC
− (1 − u)
i
L
C
− M(t).
K(u)=−
u
1 − u
.
uT
s
L
1
C
1
L
2
C
2
R
(1 −u)T
s
L
2
C
1
di
1
dt
=
E − i
1
r
1
− (1 −u)U
1
L
1
;
dU
1
dt
=
i
1
− u(i
1
+ i
2
)
C
1
;
di
2
dt
=
uU
1
− U
out
− i
2
r
2
L
2
;
dU
out
dt
=
i
2
C
2
−
U
out
RC
2
.
U
out
= U
c
M(t)
z
di
L
dt
=
E
L
u −
U
out
− ri
L
L
;
dU
out
dt
=
1
C
i
L
−
U
out
R
− z;
dz
dt
= η(U
c
− U
out
),
U
c
η