Колесников А.А. Синергетические методы управления сложными системами: механические и электромеханические системы
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W (s)=K
ψ
1+
1
T
ψ
s
K
ψ
=
T
r
4T L
m
T
ψ
=4T
r
W (s)=K
ω
1+
1
T
ω
s
• K
ω
=
J
2mpT
• T
ω
=8T
• r
s
=0, 03 r
r
=0, 0172 L
m
=0, 0154 L
s
= L
r
=0, 0158 J =
0, 968
·
2
m =3 p =2 P = 110 U = 220 I = 195, 5
• k =1 T =0, 002
• K
I
=1, 97 T
I
=0, 0682 K
ψ
=0, 0075 T
ψ
=3, 67 K
ω
=40, 33 T
ω
=0, 016
ψ
r0
=0, 987 ω
r0
= 150
M
c
= µω
r
+ M
c0
.
µ =0, 5 · ·
M
c0
=
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
400
· , t<0, 7;
1000
· , 0, 7 t<1, 5;
100
· , t 1, 5.
• M
c
= µω
r
+ M
c0
signω
r
µ =0, 5 · · M
c0
= 200 ·
• M
c
= µω
r
+ ξω
r
|ω
r
| + M
c0
µ =0, 5 · · ξ =0, 2 · ·
2
/
2
M
c0
=
200
·
• M
c
= µω
r
+ αω
3
r
+ M
c0
µ =0, 5 · · α =0, 0025 · ·
3
/
3
M
c0
=
200
·
•
•
•
•
•
r
s
=0, 067 r
r
=0, 032 L
m
=0, 0287 L
s
=0, 0294 L
r
=0, 0297
J =0, 621 ·
2
m =3 p =2 P =55 U = 220 I =99, 31
K =1 T =0, 002
•
•
• r
s
=0, 032 r
r
=0, 019 L
m
=0, 0215 L
s
= L
r
=0, 022 J =
1, 142
·
2
m =3 p =2 P =90 U = 220 I = 158, 5
• r
s
=0, 16 r
r
=0, 078 L
m
=0, 0489 L
s
=0, 05 L
r
=0, 051
J =0, 2245 ·
2
m =3 p =2 P =30 U = 220 I =54, 97
K =1 T =0, 002 K =1
T =0, 001
J
dω
r
dt
=
m
2
pk
r
ψ
r
i
sy
− M
c
;
dψ
r
dt
= r
r
k
r
i
sx
−
1
T
r
ψ
r
;
di
sy
dt
= −
1
T
∗
s
i
sy
− ω
ψ
i
sx
−
k
r
L
∗
s
ω
r
ψ
r
+
1
L
∗
s
u
sy
;
di
sx
dt
= −
1
T
∗
s
i
sx
+ ω
ψ
i
sy
+
k
r
T
r
L
∗
s
ψ
r
+
1
L
∗
s
u
sx
;
du
sx
dt
=
1
T
(k u
1
− u
sx
);
du
sy
dt
=
1
T
(k u
2
− u
sy
),
ω
r
= ω
r0
.
ψ
r
= ψ
r0
.
M
c
(t)
z
1
z
2
dz
1
dt
= η
1
Φ ;
dz
2
dt
= η
2
Φ ;
dω
r
dt
=
m
2J
pk
r
ψ
r
i
sy
− z
1
;
dψ
r
dt
= r
r
k
r
i
sx
−
1
T
r
ψ
r
− z
2
;
di
sy
dt
= −
1
T
∗
s
i
sy
− ω
ψ
i
sx
−
k
r
L
∗
s
ω
r
ψ
r
+
1
L
∗
s
u
sy
;
di
sx
dt
= −
1
T
∗
s
i
sx
+ ω
ψ
i
sy
+
k
r
T
r
L
∗
s
ψ
r
+
1
L
∗
s
u
sx
;
du
sx
dt
=
1
T
(k u
1
− u
sx
);
du
sy
dt
=
1
T
(k u
2
− u
sy
),
Φ = ω
r
−ω
r0
=0 Φ = ψ
r
−ψ
r0
=0
ψ
1
= β
11
(u
sx
− ϕ
1
)+β
12
(u
sy
− ϕ
2
);
ψ
2
= β
21
(u
sx
− ϕ
1
)+β
22
(u
sy
− ϕ
2
),
T
1
dψ
1
dt
+ ψ
1
=0; T
2
dψ
2
dt
+ ψ
2
=0.
B =
β
11
β
12
β
21
β
22
T
1
=1/λ
1
T
2
=1/λ
2
ψ
1
=0
ψ
2
=0
T
1
,T
2
> 0
u
1
=
T
k
K
1
(ϕ
1
− u
sx
)+
u
sx
T
+
dϕ
1
dt
+ K
3
(ϕ
2
− u
sy
)
;
u
2
=
T
k
K
2
(ϕ
2
− u
sy
)+
u
sy
T
+
dϕ
2
dt
+ K
4
(ϕ
1
− u
sx
)
,
K
1
=
β
11
β
22
λ
1
− β
12
β
21
λ
2
β
11
β
22
− β
12
β
21
; K
2
=
β
11
β
22
λ
2
− β
12
β
21
λ
1
β
11
β
22
− β
12
β
21
;
K
3
=
β
12
β
22
(λ
1
− λ
2
)
β
11
β
22
− β
12
β
21
; K
4
=
β
11
β
21
(λ
2
− λ
1
)
β
11
β
22
− β
12
β
21
.
ψ
1
=0 ψ
2
=0
dz
1
dt
= η
1
(ω
r
− ω
r0
);
dz
2
dt
= η
2
(ψ
r
− ψ
r0
);
dω
r
dt
=
m
2J
pk
r
ψ
r
i
sy
− z
1
;
dψ
r
dt
= r
r
k
r
i
sx
−
1
T
r
ψ
r
− z
2
;
di
sy
dt
= −
1
T
∗
s
i
sy
− ω
ψ
i
sx
−
k
r
L
∗
s
ω
r
ψ
r
+
1
L
∗
s
ϕ
2
;
di
sx
dt
= −
1
T
∗
s
i
sx
+ ω
ψ
i
sy
+
k
r
T
r
L
∗
s
ψ
r
+
1
L
∗
s
ϕ
1
;
ψ
3
= i
sx
− ϕ
3
; ψ
4
= i
sy
− ϕ
4
,
ψ
3
=0 ψ
4
=0
T
3
dψ
3
dt
+ ψ
3
=0; T
4
dψ
4
dt
+ ψ
4
=0,
T
3
=1/λ
3
,T
4
=1/λ
4
> 0
ϕ
1
= L
∗
s
λ
3
(ϕ
3
− i
sx
)+
i
sx
T
∗
s
+
dϕ
3
dt
− ω
ψ
i
sy
−
k
r
T
r
ψ
r
;
ϕ
2
= L
∗
s
λ
4
(ϕ
4
− i
sy
)+
i
sy
T
∗
s
+
dϕ
4
dt
+ ω
ψ
i
sx
+ k
r
ψ
r
ω
r
.
ψ
3
=0 ψ
4
=0
dz
1
dt
= η
1
(ω
r
− ω
r0
);
dz
2
dt
= η
2
(ψ
r
− ψ
r0
);
dω
r
dt
=
m
2J
pk
r
ψ
r
ϕ
4
− z
1
;
dψ
r
dt
= r
r
k
r
ϕ
3
−
1
T
r
ψ
r
− z
2
.
ψ
5
= ω
r
+ γ
1
z
1
; ψ
6
= ψ
r
+ γ
2
z
2
.
T
5
dψ
5
dt
+ ψ
5
=0; T
6
dψ
6
dt
+ ψ
6
=0
ϕ
3
=
1
r
r
k
r
γ
2
η
2
(ψ
r0
− ψ
r
) − η
2
(λ
6
γ
2
− 1)z
2
+
1 − λ
6
T
r
T
r
ψ
r
;
ϕ
4
=
2J
mpk
r
ψ
r
(γ
1
η
1
(ω
r0
− ω
r
) − η
1
(λ
5
γ
1
− 1)z
1
− λ
5
ω
r
) .
•
dz
1
dt
= η
1
(ω
r
− ω
r0
);
dω
r
dt
= γ
1
η
1
(ω
r0
− ω
r
) − λ
5
ω
r
− (1 + η
1
(λ
5
γ
1
− 1))z
1
;
•
dz
2
dt
= η
2
(ψ
r
− ψ
r0
);
dψ
r
dt
= γ
2
η
2
(ψ
r0
− ψ
r
) − λ
6
ψ
r
− (1 + η
2
(λ
6
γ
2
− 1))z
2
.
d
2
ω
r
dt
2
+(γ
1
η
1
+ λ
5
)
dω
r
dt
+ η
1
(1 + η
1
(λ
5
γ
1
− 1))(ω
r
− ω
r0
)=0;
d
2
ψ
r
dt
2
+(γ
2
η
2
+ λ
6
)
dψ
r
dt
+ η
2
(1 + η
2
(λ
6
γ
2
− 1))(ψ
r
− ψ
r0
)=0.
η
1
,η
2
> 0 λ
5
γ
1
> 1 −
1
η
1
λ
6
γ
2
> 1 −
1
η
2
ω
r
= ω
r0
ψ
r
= ψ
r0
dz
1
dt
= η
1
(ω
r
− ω
r0
);
dω
r
dt
= γ
1
η
1
(ω
r0
− ω
r
) − λ
5
ω
r
− η
1
(λ
5
γ
1
− 1)z
1
−
M
c
J
;
dz
2
dt
= η
2
(ψ
r
− ψ
r0
);
dψ
r
dt
= γ
2
η
2
(ψ
r0
− ψ
r
) − λ
6
ψ
r
− η
2
(λ
6
γ
2
− 1)z
2
.
ψ
r
= ψ
r0
d
2
ω
r
dt
2
+(γ
1
η
1
+ λ
5
)
dω
r
dt
+ η
2
1
(λ
5
γ
1
− 1)(ω
r
− ω
r0
)+
1
J
dM
c
dt
=0.
M
c
ω
r
= f(M
c
) M
c
=const
d
2
ω
r
dt
2
+(γ
1
η
1
+ λ
5
)
dω
r
dt
+ η
2
1
(λ
5
γ
1
− 1)(ω
r
− ω
r0
)=0.
ω = ω
r0
η
1
> 0,γ
1
>
1
λ
5
,
M
c
=const
M
c
= µω
r
,
µ
d
2
ω
r
dt
2
+(γ
1
η
1
+ λ
5
+
µ
J
)
dω
r
dt
+ η
2
1
(λ
5
γ
1
− 1)(ω
r
− ω
r0
)=0.
ω
r
= ω
r0
M
c
= M
c0
sign ω
r
M
c0
=const> 0
dM
c
dt
=
d
dt
(M
c0
sign ω
r
)=
d
dt
M
c0
|ω
r
|
ω
r
= M
c0
˙ω
r
ω
r
sign ω
r
−|ω
r
|˙ω
r
ω
2
r
=0
ω
r
=0
ω
r
= ω
r0
M
c
=const
M
c
= ξω
r
|ω
r
|.
d
2
ω
r
dt
2
+(γ
1
η
1
+ λ
5
+
2ξ
J
|ω
r
|)
dω
r
dt
+ η
2
1
(λ
5
γ
1
− 1)(ω
r
− ω
r0
)=0.
ω
r
= ω
r0
η
1
γ
1
M
c
= αω
3
r
.
ω
r
d
2
ω
r
dt
2
+(γ
1
η
1
+ λ
5
+
3α
J
ω
2
r
)
dω
r
dt
+ η
2
1
(λ
5
γ
1
− 1)(ω
r
− ω
r0
)=0.
ω
r
= ω
r0
•
•
•