Колесников А.А. Синергетические методы управления сложными системами: теория системного синтеза
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x = −r cos θ, y = r sin θ, r =
!
x
2
+ y
2
.
x = x
1
y = y
1
˙x
1
(t)=x
2
;˙x
2
(t)=U
x
;
˙y
1
(t)=y
2
;˙y
2
(t)=U
y
.
U
x
U
y
•
ω
1
= r − ex
1
− p =0,
•
ω
2
= x
1
y
2
− x
2
y
1
− h =0,
U
x
= U
x
(x
1
,x
2
,y
1
,y
2
) U
y
= U
y
(x
1
,x
2
,y
1
,y
2
),
ψ
1
=˙ω
1
(t)= ˙r(t) − ex
2
=0; ψ
2
= ω
2
= x
1
y
2
− x
2
y
1
− h =0.
˙ω
1
(t)=0 ω
2
=0
¨ω
1
(t)+ϕ(x
1
,x
2
,y
1
,y
2
) · ˙ω
1
(t)=0
˙ω
2
(t)+ϕ(x
1
,x
2
,y
1
,y
2
) · ω
2
=0,
ϕ(x
1
,x
2
,y
1
,y
2
)
U
x
= U
xN
+ U
xs
= −
(ω
2
+ h)
2
(ω
1
+ p)r
3
x
1
−
rx
1
˙ω
1
(t) − y
1
ω
2
(ω
1
+ p)r
ϕ
U
y
= U
yN
+ U
ys
= −
(ω
2
+ h)
2
(ω
1
+ p)r
3
y
1
−
ry
1
˙ω
1
(t)+(x
1
− er)ω
2
(ω
1
+ p)r
ϕ,
U
xN
U
yN
U
xs
U
ys
U
x
U
y
U
xs
U
ys
ω
1
=˙ω
1
(t)=0 ω
2
=0
a
x
=¨x(t)=−
(ω
2
+ h)
2
(ω
1
+ p)r
3
x −
rx ˙ω
1
(t) − yω
2
(ω
1
+ p)r
ϕ
a
y
=¨y(t)=−
(ω
2
+ h)
2
(ω
1
+ p)r
3
y −
ry ˙ω
1
(t)+(x − er)ω
2
(ω
1
+ p)r
ϕ.
˙ω
1
(t)= ˙r(t) − e ˙x(t) ω
2
= x ˙y(t) − y ˙x(t) − h,
ϕ
¨x
ω
(t)=−
h
2
pr
3
x
ω
, ¨y
ω
(t)=−
h
2
pr
3
y
ω
¨x
ω
(t)=−
GM
r
3
x
ω
, ¨y
ω
(t)=−
GM
r
3
y
ω
.
ω
1
= ω
2
=˙ω
1
(t)=0.
¨r(t)=
(ω
2
+ h)
2
r
3
+
1
r
(xU
x
+ yU
y
).
U
x
U
y
U
rΣ
= U
rN
+ U
rs
= −
(ω
2
+ h)
2
(ω
1
+ p)r
2
−
r ˙ω
1
(t) − eω
2
sin θ
(ω
1
+ p)
ϕ.
˙ω
1
(t) ω
2
ϕ
F
rω
= mU
rω
= −
mh
2
pr
2
= −
GmM
r
2
U
rΣ
U
rN
= −
(ω
2
+ h)
2
(ω
1
+ p)r
2
U
rs
= −
r
2
˙ω
1
(t) − eyω
2
(ω
1
+ p)
ϕ.
U
rN
U
rs
U
rN
ω
1
= ω
2
=0
U
rs
U
rΣ
U
rs
˙ω
1
(t)
ϕ
U
rs
ϕ
ϕ
ϕ
ϕ =
2˙r
2
(t)
h
=
T ˙r
2
(t)
πab
=
T ˙r
2
(t)
πa
√
1 − e
2
,
T h = const a
V
c
=0, 5h r
πab
ϕ
U
rΣ
U
rΣ
= −
(ω
2
+ h)
2
(ω
1
+ p)r
2
−
r ˙ω
1
(t) − eω
2
sin θ
(ω
1
+ p)h
2˙r
2
(t).
U
rΣ
ϕ
U
rΣ
U
rΣ
ω
1
ω
2
˙ω
1
(t)
ϕ
ω
1
ω
2
ω
2
=0
ω
2
(t)
U
rΣ
ω
2
≡ 0 U
rΣ
U
rΣ
U
rΣ
ω
1
=˙ω
1
(t)=0
ω
2
=0 U
rΣ
U
rΣ
U
rΣ
˙r(t) ˙x(t) ˙y(t)
U
rΣ
U
rΣ
U
rΣ
U
rs
U
rΣ
U
rs
ω
1
=˙ω
1
(t)=0 ω
2
=0
U
rΣ
→ →
→
→
→ →
U
rΣ
