Колесников А.А. Синергетические методы управления сложными системами: механические и электромеханические системы
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δ
5
=
k
δ
5
1
u
2
− k
δ
5
2
u
6
− k
δ
5
3
u
4
+
k
δ
5
4
u
2
+ k
δ
5
5
u
6
+ k
δ
5
6
u
3
−
− k
δ
5
7
u
6
− k
δ
5
8
u
5
− k
δ
5
9
u
2
− k
δ
5
10
1
−k
δ
5
11
+ k
δ
5
12
u
3
− k
δ
5
13
u
4
;
δ
6
= −k
δ
6
1
u
3
,
δ
1
= δ δ
2
= δ δ
3
= δ δ
4
= δ δ
5
= δ δ
6
= δ
u
i
i =1, 2,...,6 k
δ
i
j
i =1, 2,...,6
α =0
◦
∆
i
i = 1, 6
•
•
P
δ M
z
δ
u =
&
P (δ
) M
z
(δ )
'
V
∗
H
∗
Ψ
1
Ψ
2
Ψ
1
= V − V
∗
, Ψ
2
= ω
z
− ϕ
1
,
V
∗
ϕ
1
T
i
˙
Ψ
i
(t)+Ψ
i
=0,i=1, 2.
T
i
T
1
> 0 T
2
> 0
Ψ
1
=0 Ψ
2
=0
˙
Θ(t)=
P
mV
∗
sin(ϑ − Θ) +
L
mV
∗
−
g
V
∗
cos Θ;
˙
ϑ(t)=ϕ
1
;
˙
H(t)=V
∗
sin Θ.
ϕ
1
Ψ
3
ϕ
2
Ψ
3
=sin(ϑ − Θ) − ϕ
2
.
Ψ
3
T
3
˙
Ψ
3
(t)+Ψ
3
=0.
T
3
> 0 Ψ
3
=0
˙
Θ(t)=
P
mV
∗
ϕ
2
+
L
mV
∗
−
g
V
∗
cos Θ;
˙
H(t)=V
∗
sin Θ.
ϕ
2
Ψ
4
Ψ
4
= V
∗
sin Θ + A(H − H
∗
),
H
∗
A
Ψ
4
T
4
˙
Ψ
4
(t)+Ψ
4
=0.
T
4
> 0
Ψ
4
=0
˙
H(t)=−A(H − H
∗
).
ϕ
2
ϕ
1
M
z
P
ϕ
2
ϕ
2
ϕ
1
ϕ
1
P M
z
P =
D + mg sin Θ
cos(ϑ −Θ)
− (V − V
∗
)
5m
T
1
cos(ϑ −Θ)
M
z
m =6, 5
g =9, 8 /
2
I
z
=0, 074 ·
2
T
1
=4 T
2
=0, 5 T
3
=1, 3 T
4
=0, 1
A =1 V
∗
=14 H
∗
=5
L D V (t)
u
r
(x
1
,...,x
5
) r =1, 4, 5
n =5
m =3 u
k
k =1, 4, 5
dim z = n − λm,
z n m
λ
n =5
(λ =1,m=3)
˙x
2ψ
(t)=−x
1ψ
x
3ψ
+ x
5ψ
1 − x
2
2ψ
− x
2
3ψ
,
˙x
3ψ
(t)=x
1ψ
x
2ψ
− x
4ψ
1 − x
2
2ψ
− x
2
3ψ
.
ψ
1,4,5
=0
ψ
1
=0 ψ
4
=0 ψ
5
=0
u
1
u
4
u
5
ν
k
ν
1
= x
1ψ
ν
4
= x
4ψ
ν
5
= x
5ψ
˙x
1
(t) ˙x
4
(t) ˙x
5
(t)
u
1
u
4
u
5
u
1
u
4
u
5
˙x
1
(t) ˙x
4
(t) ˙x
5
(t)
x
1
x
4
x
5
ψ
1,4,5
=0
u
1
u
4
u
5
ψ
1,4,5
=0 x
1ψ
x
4ψ
x
5ψ
ψ
1,4,5
=0
n =5
n − m =2
ν
1
ν
4
ν
5
u
1
u
4
u
5
ν
1
ν
4
ν
5
˙x
2ψ
(t)=−ν
1
x
3ψ
+ ν
5
1 − x
2
2ψ
− x
2
3ψ
,
˙x
3ψ
(t)=ν
1
x
2ψ
− ν
4
1 − x
2
2ψ
− x
2
3ψ
.
ν
1
ν
4
ν
5
ψ
1,4,5
=0
x
2ψ
= x
3ψ
=0 ν
1
ν
4
ν
5
ψ
1,4,5
=0
u
1
u
4
u
5
ψ
1,4,5
=0
ψ
1,4,5
=0
ν
1
ν
4
ν
5
ν
1
x
1ψ
= ν
1
x
1s
ψ
1
= x
1
T
1
˙
ψ
1
(t)+ψ
1
=0
u
1
u
1
=
A − C
B
x
4
x
5
−
1
T
1
(x
1
− x
1s
).
T
1
˙x
1
(t)+x
1
= x
1s
,
x
1
(t)=ν
1
=(x
10
− x
1s
)e
−t/T
1
+ x
1s
.
T
1
˙ν
1
(t)+ν
1
=0,
˙x
2ψ
(t)=−ν
1
x
3ψ
+ ν
5
1 − x
2
2ψ
− x
2
3ψ
,
˙x
3ψ
(t)=ν
1
x
2ψ
− ν
4
1 − x
2
2ψ
− x
2
3ψ
.
ν
1
x
1s
x
1s
=0
ν
4
= ρx
3ψ
ν
5
= −γx
2ψ
,
T
1
˙ν
1
(t)+ν
1
=0,
˙x
2ψ
(t)=−ν
1
x
3ψ
− γx
2ψ
1 − x
2
2ψ
− x
2
3ψ
,
˙x
3ψ
(t)=ν
1
x
2ψ
− ρx
3ψ
1 − x
2
2ψ
− x
2
3ψ
.
V =0,5ν
2
1
+0,5x
2
2ψ
+0,5x
2
3ψ
,
˙
V (t)=−
1
T
1
ν
2
1
− (γx
2
2ψ
+ ρx
2
3ψ
)
1 − x
2
2ψ
− x
2
3ψ
.
T
1
> 0,γ > 0,ρ> 0 x
2
2ψ
+ x
2
3ψ
< 1
x
2ψ
x
3ψ
γ = β = α
˙x
2ψ
(t)x
2ψ
+˙x
3ψ
(t)x
3ψ
= −α(x
2
2ψ
+ x
2
3ψ
)
1 − x
2
2ψ
− x
2
3ψ
.
x
2
(t)=r(t)sinϕ(t) x
3
(t)=r(t)cosϕ(t) r
2
=
x
2
2ψ
+ x
2
3ψ
˙r(t)=−αr
√
1 − r
2
;
˙ϕ(t)=1.
r(t)=
1
√
1+ce
αt
; ϕ(t)=ϕ
0
+ t.
r(t) α>0 t →∞ α<0
r
s
=1
ψ
4
= β
2
x
2
+ β
3
x
3
+ β
4
x
4
+ x
5
;
ψ
5
= α
2
x
2
+ α
3
x
3
+ α
4
x
4
+ x
5
.
ψ
4
=0 ψ
5
=0
ψ
45
=(β
2
− α
2
)x
2ψ
+(β
3
− α
3
)x
3ψ
+(β
4
− α
4
)x
4ψ
=0,
ψ
45
=(β
2
α
4
− α
2
β
4
)x
2ψ
+(β
3
α
4
− α
3
β
4
)x
3ψ
− (β
4
− α
4
)x
5ψ
=0.
x
4ψ
= ν
4
=
β
3
− α
3
α
4
− β
4
x
3ψ
= ρx
3ψ
α
2
= β
2
> 0,
x
5ψ
= ν
5
=
β
2
α
4
− α
2
β
4
β
4
− α
4
x
2ψ
α
3
β
4
= β
3
α
4
α
2
= β
2
x
5ψ
= ν
5
= −α
2
x
2ψ
= −γx
2ψ
.
γ = α
2
> 0; ρ =
β
3
− α
3
α
4
− β
4
> 0,
α
2
= β
2
> 0 α
3
β
4
= β
3
α
4
α
3
< 0 β
4
< 0 T
1
> 0
γ =0 ρ =0
x
2ψ
= x
2s
x
3ψ
= x
3s
u
4
u
5
T
4
˙
ψ
4
(t)+ψ
4
=0,T
4
> 0,
T
5
˙
ψ
5
(t)+ψ
5
=0,T
5
> 0.
(β
4
− α
4
)u
4
= − (β
3
− α
3
)x
1
x
2
+(β
3
− α
3
)x
4
1 − x
2
2
− x
2
3
+
+(β
4
− α
4
)
C − B
A
x
1
x
5
−
1
T
4
ψ
4
+
1
T
5
ψ
5
;
(β
4
− α
4
)u
5
=α
2
(β
4
− α
4
)x
1
x
3
− α
2
(β
4
− α
4
)x
5
1 − x
2
2
− x
2
3
+
+(β
4
− α
4
)
B − A
C
x
1
x
4
+
α
4
T
4
ψ
4
−
β
4
T
5
ψ
5
.
u
4
u
5
ψ
4
=0 ψ
5
=0
ψ
4
= ψ
5
=0 x
4ψ
= ρx
3ψ
x
5ψ
= −γx
2ψ
u
1
u
4
u
5
u
k
sup
= u
k
max
µ
k
(x
1
,...,x
5
),k=1, 4, 5.
|µ
k
| <u
k
max
|µ
k
| u
k
max
·
2
A =7 B =8 C =9
T
1
= T
4
= T
5
=1 α
2
= α
4
= β
2
= β
3
=1 α
3
= β
4
= −1
ν
1
ν
4
ν
5