Колесников А.А. Синергетические методы управления сложными системами: механические и электромеханические системы
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˙
θ(t)=ω;
˙ω(t)=(a
1
ϕ
1
ϕ
2
− M
c
(ω) −z
1
)a
2
;
˙z
1
(t)=f
1
(θ, ω, ϕ
1
,ϕ
2
,z
1
,...z
r
);
˙z
r
(t)=f
r
(θ, ω, ϕ
1
,ϕ
2
,z
1
,...z
r
).
ϕ
1
(θ, ω, z
1
) ϕ
2
(θ, ω, z
1
)
ϕ
2
(θ, ω, z
1
)
ψ
2
=0
ϕ
2
(θ, ω, z
1
)=Φ
∗
ψ
2
=Φ− Φ
∗
Φ
∗
ϕ
1
(θ, ω, z
1
)
ϕ
1
(θ, ω, z
1
) ϕ
2
(θ, ω, z
1
)
ω = ω
∗
=const
M
f
(t)=const
˙ω(t)=(a
1
i Φ − M
c
(ω) − z
1
)a
2
;
˙
i
(t)=(u − a
1
Φω −a
3
i )a
4
;
˙
Φ(t)=(u
− a
5
f
1
(Φ))a
6
;
˙z
1
(t)=β(ω −ω
∗
),
z
1
=
ˆ
M
f
(t)
u (ω, i , Φ,z
1
)
u (ω,i , Φ,z
1
) ω = ω
∗
Φ=Φ
∗
ψ
1
= i − ϕ
1
(ω, z
1
)=0;
ψ
2
=Φ− Φ
∗
=0.
˙ω(t)=(a
1
ϕ
1
(ω, z
1
)Φ
∗
− M
c
(ω) − z
1
)a
2
;
˙z
1
(t)=β(ω −ω
∗
).
ϕ
1
(ω, z
1
)
ϕ
1
(ω, z
1
)=
1
a
1
Φ
∗
M
c
(ω)+z
1
−
b
1
a
2
(ω −ω
∗
)
.
˙ω(t)=−b
1
(ω −ω
∗
),
b
1
> 0
ω = ω
∗
z
1
M
f
(t)
ψ
1
=0 ψ
2
=0
i =
1
a
1
Φ
∗
M
c
(ω)+z
1
−
b
1
a
2
(ω −ω
∗
)
Φ=Φ
∗
z
1
z
1
= β
(ω −ω
∗
)dt
˙ω(t)=−b
1
(ω −ω
∗
)+a
2
β
(ω −ω
∗
)dt − a
2
M
f
(t).
¨ω(t)=−b
1
˙ω(t)+a
2
βω(t) − a
2
βω
∗
− a
2
˙
M
f
(t).
˙
M
f
(t)=0 M
f
(t)=const b
1
> 0 β<0
w → ω
∗
M
f
(t)
ϕ
1
(ω, z
1
) ϕ
2
(ω, z
1
)=Φ
∗
h
1
(θ, ω, i , Φ,z
1
,...z
r
)=
β(ω − ω
∗
)
u = a
3
i +
1
T
1
a
4
1
a
1
Φ
∗
M
c
−
b
1
a
2
(ω −ω
∗
)
− i
−
1
a
4
a
1
Φ
∗
b
1
− a
2
∂M
c
∂ω
×
× (a
1
i Φ − M
c
) − β(ω −ω
∗
)
+
β
a
4
a
1
Φ
∗
1
T
1
+ b
1
− a
2
∂M
c
∂ω
(ω −ω
∗
)dt + a
1
ωΦ;
u
= a
5
f
1
(Φ) −
1
T
2
a
6
(Φ − Φ
∗
).
ψ
1
=0 ψ
2
=0
P
2
=46, 5
ω = 160 Φ =15 r =0, 035 r =59 L =0, 0017 c =88, 49
p =2 w = 1250 J =1, 2
2
r = var
(t =5, 10, 15, 20c)
r = r + δr
k
n
k =1/R T
n
T = L /R
T
M
= J/c
M
Σ
W
PT
(s)=k
PT
+
1
T
PT
s
,
k
PT
=
T
2T
n
k k
n
T
PT
=2T
n
k k
n
W
PC
(s)=k
PC
k
PC
=
T
M
4T
n
∆ω =
M
Σ
k
PC
c
.
W
PC
= k
PC
+
1
T
PC
s
k
PT
=
T
M
4T
n
,T
PC
=
32T
2
n
T
M
.
Φ=Φ
∗
Φ
∗
=Φ
ω
∗
ω
ω
∗
>ω
e = cωΦ
Φ=
⎧
⎪
⎨
⎪
⎩
Φ
, |ω| ω ;
u
− i r
c|ω|
,
|ω| >ω .
Φ=g
1
(ω)Φ + g
2
(ω)
u
− i r
c|ω|
.
g
1
(ω) g
2
(ω)
g
1
(ω)=
1+sign(ω
−|ω|)
2
;
g
2
(ω)=
1 − sign(ω
−|ω|)
2
.
sign g
1
(ω) g
2
(ω)
ω
∗
ψ
1
= i − ϕ
1
(ω, z
1
)=0;
ψ
2
=Φ− G(ω
∗
)=0.
G(ω
∗
)=
1+th(ω
− ω
∗
)
2
Φ
+
1 − th(ω
− ω
∗
)
2
u
− i a
3
a
1
|ω
∗
|
ϕ
1
(ω, z
1
)
Φ
∗
↔ G(ω
∗
)
ω
∗
= ω
ω
∗
= −ω
ω
∗
=1, 5ω
ω
∗
=0, 5ω
ω
∗
=2ω
|i | i
max
ψ
1
=0
ψ
1
= i − i
max
th(c
1
ω + c
2
z
1
)=0
ψ
2
=0
Φ=Φ ψ
2
=Φ− Φ
˙ω(t)=(a
1
i
max
th(c
1
ω + c
2
z
1
)Φ − M
c
(ω) − M
f
(t)) a
2
.
z
1
= β
(ω − ω
∗
)dt
¨ω(t)=a
2
∂M
c
(ω)
∂ω
−
a
1
i
max
Φ c
1
ch
2
(c
1
ω + c
2
z
1
)
˙ω(t)−
−
a
1
i
max
Φ c
2
ch
2
(c
1
ω + c
2
z
1
)
(ω − ω
∗
)+a
2
˙
M
f
(t)=0.
M
f
(t)=const
ω = ω
∗
a
i
Φ i
max
ch
2
(c
1
ω + c
2
z
1
)
∂M
c
(ω)
∂ω
0
c
1
< 0 βc
2
< 0
u =
i
max
a
4
ch
2
σ(ω)
a
1
i Φ−M
c
(ω)−β
(ω−ω
∗
)dt
a
2
c
1
+c
2
β(ω−ω
∗
)
+
+
i
max
T
1
a
4
th
σ(ω)+a
1
ωΦ+
a
3
−
1
T
1
a
4
i
;
u
= a
5
f
1
(Φ) −
1
T
2
a
6
(Φ − Φ
∗
),
σ(ω)=c
1
ω + c
2
β
(ω − ω
∗
)dt)
i
max
= i
∆M
f
= 140
ω
∗
=0, 5ω