Колесников А.А. Синергетические методы управления сложными системами: механические и электромеханические системы
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γ =1 m =3
u
1
˙x
1
(t)=−x
2
− x
3
;
˙x
2
(t)=x
1
+ ax
2
+ u
2
;
˙x
3
(t)=bx
1
+ x
1
x
3
− cx
3
.
ψ
2
= x
2
+ λ
1
x
1
− λ
2
x
m
1
+ ηx
3
,
T
2
˙
ψ
2
(t)+ψ
2
=0,
u
2
= −(1 + ηb)x
1
+
λ
1
+ mλ
2
x
m−1
1
(x
2
+ x
3
) −
1
T
2
ψ
2
.
ψ
2
=0
˙x
1ψ
(t)=λ
1
x
1ψ
− λ
2
x
m
1ψ
+(η − 1)x
3ψ
;
˙x
3ψ
(t)=bx
1ψ
+ x
1ψ
x
3ψ
− cx
3ψ
.
λ
1
< 0 λ
2
> 0 m =2, 3 η =1 x
1ψ
→ 0 x
3ψ
→ 0
x
1s
= x
2s
= x
3s
=0
λ
1
= −1 λ
2
=1 m =2 η =1
T =1
λ
1
> 0 λ
2
> 0 m =2 η =1
λ
1
= −1 λ
2
=1 λ
1
= λ
2
=1
x
1s
=
λ
1
λ
2
; x
3s
=
bx
1s
c − x
1s
; x
2s
= λ
2
x
2
1s
− λ
1
x
1s
− ηx
3s
c>x
1s
λ
1
= λ
2
=1
m =2 η =1 T =1
m =3 λ
1
> 0 λ
2
> 0 η =1
x
1s
= ±
λ
1
λ
2
; x
2s
= λ
2
x
2
1s
− λ
1
x
1s
− ηx
3s
;
x
3s
=
bx
1s
c − x
1s
x
1s
< 0 c>|x
1s
|.
λ
1
λ
2
m η =1 x
1s
=
x
2s
= x
3s
=0
u
2
ψ
3
= x
2
+ V
1
(x
1
,x
3
),
T
3
˙
ψ
3
(t)+ψ
3
=0,
u
2
= −x
1
− ax
2
−
∂V
1
∂x
1
˙x
1
(t) −
∂V
1
∂x
3
˙x
3
(t) −
1
T
3
ψ
3
.
ψ
3
=0
˙x
1ψ
(t)=V
1
(x
2ψ
,x
3ψ
) − x
3ψ
;
˙x
3ψ
(t)=bx
1ψ
+ x
1ψ
x
3ψ
− cx
3ψ
,
V
1
(x
1ψ
,x
3ψ
)
V
1
(x
1ψ
,x
3ψ
)
ψ
4
= x
1
+ γx
3
.
ψ
4
T
4
˙
ψ
4
(t)+ψ
4
=0,
V
1
(x
1ψ
,x
3ψ
)=−γbx
1
+ x
3
− γx
1
x
3
+ cx
3
−
1
T
4
ψ
4
.
ψ
4
=0
˙x
3ψ
(t)=−(γb + c)x
3
− γx
2
3ψ
.
γ>0
x
3ψ
=0 0 <γ<c
x
1ψ
=0 V (0, 0) = 0 x
2ψ
=0
u
2
x
1s
= x
2s
= x
3s
=0
u
2
˙x
1
(t) ˙x
2
(t)
V
1
(x
1ψ
,x
3ψ
)
ψ
3
ψ
4
˙x
1
(t)=−x
2
− x
3
;
˙x
2
(t)=x
1
+ ax
2
;
˙x
3
(t)=bx
1
+ x
1
x
3
− cx
3
+ u
3
.
x
1
¨x
2
(t) − a ˙x
2
(t)+x
2
= −x
3
;
˙x
3
(t)=bx
1
+ x
1
x
3
− cx
3
+ u
3
.
ψ
5
= x
3
+ γ
1
˙x
2
(t) − γ
2
˙x
2
2
(t).
T
5
˙
ψ
5
(t)+ψ
5
=0,
u
3
= − bx
1
+ cx
3
− x
1
x
3
+
γ
1
+3γ
2
(x
1
+ ax
2
)
2
×
×
ax
1
+(a
2
− 1)x
2
− x
3
−
1
T
5
ψ
5
.
u
3
ψ
3
=0
¨x
2ψ
(t)+
γ
1
− a + γ
2
˙x
2
2ψ
(t)
˙x
2ψ
(t)+x
2ψ
=0.
γ
1
>a γ
2
0 x
2ψ
=˙x
2ψ
(t)=
0
x
1s
= x
2s
= x
3s
=0
γ
1
=
γ
2
=1
γ
1
−a = −µ γ
1
<a γ
2
= µb
¨x
2ψ
(t) − µ
1 − b ˙x
2ψ
(t)
˙x
2ψ
(t)+x
2ψ
=0,
µ b ψ
3
=0
ψ
3
=0
¨x
2ψ
(t)+γ
1
x
2
+ γ
2
x
3
2
=
γ
3
+ γ
4
x
2
2
˙x
2
(t).
γ
1
,...,γ
4
ψ
6
= x
3
+
γ
3
− a + γ
4
x
2
2
(x
1
+ ax
2
)+(1− γ
1
)x
2
− γ
2
x
3
2
.
ψ
6
T
6
˙
ψ
6
(t)+ψ
6
=0,
u
3
= − bx
1
− x
1
x
3
+ cx
3
+(x
1
+ ax
2
)
γ
1
− 1+(3γ
2
− aγ
3
γ
4
)x
2
2
−
− 2γ
3
γ
4
(x
1
+ ax
2
)x
2
−
a − γ
3
− γ
3
γ
4
x
2
2
(x
2
+ x
3
) −
1
T
6
ψ
6
.
ψ
6
=0
γ
1
=1 γ
2
=1 γ
3
=1 γ
4
= −1
ψ
5
=0
γ
1
= −1 γ
2
=1 γ
3
= −1
γ
4
=1 ψ
6
=0
γ
1
= γ
2
= γ
3
=1 γ
4
= −1 γ
1
= γ
3
= −1 γ
2
= γ
4
=1
ψ
7
= x
3
+ V (x
1
,x
2
).
ψ
7
T
7
˙
ψ
7
(t)+ψ
7
=0,
u
3
= −bx
1
− x
1
x
3
−
1
T
7
+ c
x
3
+
∂V
∂x
1
(x
1
+ x
3
) −
∂V
∂x
2
(x
1
+ ax
2
) −
1
T
7
V (x
1
,x
2
).
u
3
ψ
7
=0
˙x
1ψ
(t)=−x
2ψ
+ V (x
1ψ
,x
2ψ
);
˙x
2ψ
(t)=x
1ψ
+ ax
2ψ
.
ψ
8
= x
1ψ
− γx
2ψ
+ λx
m
2ψ
T
8
˙
ψ
8
(t)+ψ
8
=0,
V (x
1ψ
,x
2ψ
)=x
2ψ
+
γ − λmx
m−1
2ψ
(x
1ψ
+ ax
2ψ
) −
1
T
8
ψ
8
.
ψ
8
=
0
˙x
2ψ
(t)=(γ + a)x
2ψ
− λx
m
2ψ
.
γ<0 |γ| >a λ>0 m =2, 3 x
2ψ
→ 0
x
1ψ
→ 0 V (x
1ψ
,x
2ψ
) → 0
x
1s
= x
2s
= x
3s
=0
γ>0 λ>0 m =2
x
2s
=
γ + a
λ
x
1s
= γx
2s
− λx
2
2s
; x
2s
=
γ + a
λ
;
x
3s
= −x
2s
+(2λx
2s
− γ)(x
1s
+ x
2s
).
γ>0 λ>0 m =3
x
1s
= γx
2s
− λx
3
2s
; x
2s
= ±
γ + a
λ
;
x
3s
= −x
2s
+(3λx
2
2s
− γ)(x
1s
+ ax
2s
).
V
u
3
f(t)=B sin ω
0
t
˙x
1
(t)=−x
2
− x
3
;
˙x
2
(t)=x
1
+ ax
2
;
˙x
3
(t)=bx
1
+ x
1
x
3
− cx
3
+ u
3
+ f(t).
˙w
1
(t)=w
2
;˙w
2
(t)=−ω
2
0
w
1
; f = Bw
1
.
˙x
1
(t)=−x
2
− x
3
;
˙x
2
(t)=x
1
+ ax
2
;
˙x
3
(t)=bx
1
+ x
1
x
3
− cx
3
+ u
3
+ ω
1
;
˙ω
1
(t)=ω
2
+ ϕ
1
;
˙ω
2
(t)=−ω
2
0
ω
1
+ ϕ
2
,
ω
1
ω
2
w
1
w
2
ϕ
1
ϕ
2
ϕ
1
= ϕ
2
=0
u
3
ψ
9
= x
3
− γ
1
x
1
− γ
2
x
2
T
9
˙
ψ
9
(t)+ψ
9
=0,
u
3
= −(b + γ
1
)x
1
− (γ
2
a − γ
1
)x
2
− (c + γ
1
)x
3
− x
1
x
3
− ω
1
−
1
T
9
ψ
9
.
ψ
9
=0
˙x
1ψ
(t)=−γ
1
x
1ψ
− (γ
2
+1)x
2ψ
;
˙x
2ψ
(t)=x
1ψ
+ ax
2ψ
.
γ
1
>a γ
2
>γ
1
a − 1 x
1ψ
=
x
2ψ
=0
x
1ψ
= x
2ψ
= x
3ψ
=0
ϕ
1
= α
1
ψ
9
ϕ
2
= α
2
ψ
9
˙ω
1
(t)=ω
2
+ α
1
ψ
9
;
˙ω
2
(t)=−ω
2
0
ω
1
+ α
2
ψ
9
.
ω
2
¨ω
1
(t)+ω
2
0
ω
1
= α
1
˙
ψ
9
(t)+α
2
ψ
9
.
ψ
9
˙
ψ
9
(t)
x
1
x
2
x
3
u
3
ψ
9
→ 0
˙
ψ
9
(t) → 0
γ
1
=2 γ
2
=0, 38 α
1
= −103 α
2
= −206 T
9
=0, 2
¨x(t)+F
1
(x, z, µ)˙x(t)+F
2
(x, z, µ)=0;
˙z(t)=f
3
(x, z, µ),
F
i
f
3
(x, z, µ)
u
i
=0 u
i
=0
u
1
u
3
r
u
1
u
3
r
u
1
u
3
