Колесников А.А. Синергетические методы управления сложными системами: механические и электромеханические системы
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m
z j
τ
j
β
j
f
j
pr
f
j
tr
m
0
¨x(t)=
m
j=1
(f
j
pr
cos β
j
− f
j
tr
sin β
j
)cosα −
m
j=1
(f
j
pr
sin β
j
+ f
j
tr
cos β
j
)sinα;
m
0
¨y(t)=
m
j=1
(f
j
pr
cos β
j
− f
j
tr
sin β
j
)sinα +
m
j=1
(f
j
pr
sin β
j
+ f
j
tr
cos β
j
)cosα;
J
0
¨α(t)=
m
j=1
(z
j
1
sin β
j
− z
j
2
cos β
j
)f
j
pr
+(z
j
1
cos β
j
+ z
j
2
sin β
j
)f
j
tr
,
m τ
j
V
j
1
=cosβ
j
V
x
cos α + V
y
sin α − z
j
2
ω
+sinβ
j
V
y
cos α − V
x
sin α + z
j
1
ω
;
V
j
2
= − sin β
j
V
x
cos α + V
y
sin α − z
j
2
ω
+cosβ
j
V
y
cos α − V
x
sin α + z
j
1
ω
,
V
x
V
y
f
j
tr
= −k
j
V
j
2
,
k
j
•
•
•
•
•
•
•
˙x
1
(t)=x
2
;
m
0
˙x
2
(t)=F
1
cos x
5
− F
2
sin x
5
;
˙x
3
(t)=x
4
;
m
0
˙x
4
(t)=F
1
sin x
5
+ F
2
cos x
5
;
˙x
5
(t)=x
6
;
J
0
˙x
6
(t)=M,
x
1
=
x
x
2
=˙x(t) x
3
= y x
4
=˙y(t) x
5
= α x
6
=˙α(t)
f
j
tr
= −k
j
−sin β
j
x
2
cos x
5
+x
4
sin x
5
−z
j
2
x
6
+cosβ
j
x
4
cos x
5
−x
2
sin x
5
+z
j
1
x
6
F
1
=
m
j=1
(f
j
pr
cos β
j
− f
j
tr
sin β
j
);
F
2
=
m
j=1
(f
j
pr
sin β
j
+ f
j
tr
cos β
j
);
M =
m
j=1
(z
j
1
sin β
j
− z
j
2
cos β
j
)f
j
pr
+(z
j
1
cos β
j
+ z
j
2
sin β
j
)f
j
tr
.
f
j
pr
β
j
2m
F
1
F
2
M
2m
X
ϑ
P
F
1
(x),F
2
(x),M(x)
f
pr
β
β
f
pr
f
pr
β
f
pr
M
d
= f
pr
r r
β
(M = M
0
=const)
(θ
r
= θ
r0
=const)
ψ
r
= ψ
r0
=const
θ
r
= α
0
+
α
1
t + α
2
sin ω
0
t ψ
r
= ψ
r0
α
0
,α
1
,α
2
,ω
0
ψ
r0
α
0
=const,α
1
=0,α
2
=0,ω
0
=0.
M
c
ω
rj
=
cos β
j
(x
1
cos x
6
+x
3
sin x
6
− z
j
2
x
5
)+ sin β
j
(x
3
cos x
6
−x
1
sin x
6
+z
j
1
x
5
)
r
,
r j
ψ
1
=0 ψ
2
=0
dθ
r
dt
= ω
r
;
J
dω
r
dt
=
m
3
pk
r
ψ
r
ϕ
2
− M
c
;
dψ
r
dt
= r
r
k
r
ϕ
1
−
1
T
r
ψ
r
.
ψ
3
= ψ
r
− ψ
r0
.
ψ
r0
T
3
˙
ψ
3
(t)+ψ
3
=0,
T
3
> 0 ψ
3
=0
J
d
2
θ
r
dt
2
=
m
3
pk
r
ψ
const
ϕ
2
− M
c
.
ϕ
1
ϕ
1
=
ψ
r
T
r
r
r
k
r
−
ψ
r
− ψ
const
T
3
r
r
k
r
.
u
sx
u
sy
u
sx
=
L
∗
s
i
sx
(T
∗
s
− T
3
)
T
3
+
c
1
L
∗
s
(ψ
r0
− ψ
r
)
T
1
T
2
T
3
r
r
k
r
β
0
+
ψ
r
L
∗
s
(T
3
− T
∗
r
)
T
2
r
T
3
r
r
k
r
−
L
∗
s
c
1
c
3
T
r
r
r
k
r
T
1
T
2
β
0
+
+
β
12
β
22
L
∗
s
(ϕ
2
− i
sy
)(T
1
− T
2
)
T
1
T
2
β
0
+ i
sy
ω
r
L
∗
s
+
r
r
k
r
i
2
sy
L
∗
s
ψ
r
+
(k
r
ψ
r
− L
∗
s
i
sx
)
T
r
;
u
sy
=
c
2
L
∗
s
(ϕ
2
− i
sy
)
T
1
T
2
β
0
− ω
r
(L
∗
s
i
sx
+ k
r
ψ
r
)+
i
sy
L
∗
s
T
∗
s
+
r
r
k
r
i
sx
i
sy
L
∗
s
ψ
r
−
−
β
21
β
11
L
∗
s
c
3
(T
1
− T
2
)
T
1
T
2
T
∗
r
r
r
k
r
β
0
−
β
21
β
11
L
∗
s
(ψ
r
− ψ
r0
)(T
1
+ T
2
)
T
1
T
2
T
3
r
r
k
r
β
0
,
β
0
= β
11
β
22
−β
12
β
21
c
1
= T
1
β
12
β
21
−T
2
β
11
β
22
c
2
= T
2
β
12
β
21
−T
1
β
11
β
22
c
3
= ψ
r
−i
sx
T
∗
r
r
r
k
r
F
1
F
2
M
f
j
pr
β
j
F
1
F
2
M
f
j
pr
β
j
I
1
=
ν
j=1
q
j
f
j
pr
2
−→ min,
q
j
ν
f
j
pr
=
M
j
d
r
j
=
m
(j)
3
p
(j)
L
(j)
m
r
j
L
(j)
r
ψ
(j)
const
ϕ
(j)
2
I
1
=
ν
j=1
q
j
m
(j)
3
p
(j)
L
(j)
m
r
j
L
(j)
r
ψ
(j)
const
ϕ
(j)
2
2
−→ min,
ϕ
(j)
β
j
m =3
z
1
1
=5,z
1
2
=0;z
2
1
= −3,z
2
2
= −3; z
3
1
=5,z
3
2
=3
k
1
= k
2
= k
3
=0
β
j
{−π, π} b
j
= arctan(β
j
),j=1...3
F
1
=
m
(1)
3
1
1+b
2
1
p
(1)
L
(1)
m
r
1
L
(1)
r
ψ
(1)
const
ϕ
(1)
2
+
+
1
1+b
2
2
p
(2)
L
(2)
m
r
2
L
(2)
r
ψ
(2)
const
ϕ
(2)
2
+
1
1+b
2
3
p
(3)
L
(3)
m
r
3
L
(3)
r
ψ
(3)
const
ϕ
(3)
2
;
F
2
=
m
(2)
2
b
1
1+b
2
1
p
(1)
L
(1)
m
r
1
L
(1)
r
ψ
(1)
const
ϕ
(1)
2
+
+
b
2
1+b
2
2
p
(2)
L
(2)
m
r
2
L
(2)
r
ψ
(2)
const
ϕ
(2)
2
+
b
3
1+b
2
3
p
(3)
L
(3)
m
r
3
L
(3)
r
ψ
(3)
const
ϕ
(3)
2
;
M =
m
(3)
2
5b
1
1+b
2
1
p
(1)
L
(1)
m
r
1
L
(1)
r
ψ
(1)
const
ϕ
(1)
2
+
+
3 − 3b
2
1+b
2
2
p
(2)
L
(2)
m
r
2
L
(2)
r
ψ
(2)
const
ϕ
(2)
2
−
3+3b
3
1+b
2
3
p
(3)
L
(3)
m
r
3
L
(3)
r
ψ
(3)
const
ϕ
(3)
2
.
ϕ
(1)
2
= ϕ
(1)
2
(F
1
,F
2
,M,b
1
,b
2
,b
3
);
ϕ
(2)
2
= ϕ
(2)
2
(F
1
,F
2
,M,b
1
,b
2
,b
3
);
ϕ
(3)
2
= ϕ
(3)
2
(F
1
,F
2
,M,b
1
,b
2
,b
3
).
b
j
b
1
= b
2
= b
3
=0
ϕ
(1)
2
,ϕ
(2)
2
,b
1
ϕ
(1)
2
=
3
m
(1)
r
1
L
(1)
r
p
(1)
L
(1)
m
ψ
(1)
const
/
0
0
0
0
1
A
1
3r
1
L
(3)
r
2
C
1
3r
3
L
(3)
r
+ C
2
2
×
×
(1+b
2
3
) ((8b
2
−3) F
1
+M−5F
2
)
3r
3
L
(3)
r
+
1+b
2
3
(8b
3
−8b
2
+6) 3p
(3)
L
(3)
m
ψ
(3)
const
(8b
2
− 3) (1 + b
2
3
)
3r
3
L
(3)
r
;
ϕ
(2)
2
=
3
m
(2)
·
r
2
L
(2)
r
p
(2)
L
(2)
m
ψ
(2)
const
·
1+b
2
2
(1 + b
2
3
)(5F
2
− M) − f
3
pr
1+b
2
3
(3 + 8b
3
)
(8b
2
− 3) (1 + b
2
3
)
;
b
1
=
1+b
2
3
3F
2
(b
2
− 1) + Mb
2
3r
3
L
(3)
r
1+b
2
3
(F
1
(8b
2
− 3) − 5F
2
+ M)3r
3
L
(3)
r
+ m
(3)
p
(3)
L
(3)
m
+
+
m
(3)
p
(3)
L
(3)
m
K
(3)
ψ
(3)
const
(b
3
+ b
2
)
1+b
2
3
F
1
(8b
2
− 3) − 5F
2
+ M
3r
3
L
(3)
r
+ m
(3)
p
(3)
L
(3)
m
ψ
(3)
const
(8b
3
− 8b
2
+6),
C
1
=
1+b
2
3
(8b
2
−3)F
1
−5F
2
+M
C
2
=
6+8b
3
−8b
2
m
(3)
p
(3)
L
(3)
m
ψ
(3)
const
A
1
=
⎛
⎝
9−48b
2
+64b
2
2
F
2
1
+
34−18b
2
+9b
2
2
F
2
2
+
b
2
2
+1
M
2
−
−
6b
2
−6b
2
2
+10
MF
2
+ ((30−80b
2
) F
2
+(16b
2
−6) M) F
1
⎞
⎠
1+b
2
3
+
+
⎛
⎜
⎜
⎜
⎝
−128b
2
2
+ (128b
3
+ 144) b
2
− 36 − 48b
3
F
1
+
+
18b
2
2
+(18b
3
+ 62) b
2
− 60 − 98b
3
F
2
+
+ ((6b
3
− 16) b
2
+12+16b
3
) M
⎞
⎟
⎟
⎟
⎠
3
m
(j)
p
(3)
L
(3)
m
r
3
L
(3)
r
ψ
(3)
const
ϕ
(3)
2
1+b
2
3
+
+
73b
2
2
− (96 + 110b
3
)b
2
+36+96b
3
+73b
2
3
3
m
(j)
p
(3)
L
(3)
m
r
3
L
(3)
r
ψ
(3)
const
ϕ
(3)
2
2
.
ϕ
(1)
2
ϕ
(2)
2
m =3
ϕ
(3)
2
,b
2
b
3
I
1
= q
1
ϕ
(1)
2
ϕ
(3)
2
,b
2
,b
3
2
+ q
2
ϕ
(2)
2
ϕ
(3)
2
,b
2
,b
3
2
+ q
3
ϕ
(3)
2
2
−→ min .
∂I(ϕ
(3)
2
,b
2
,b
3
)
∂ϕ
(3)
2
=0;
∂I(ϕ
(3)
2
,b
2
,b
3
)
∂b
2
=0;
∂I(ϕ
(3)
2
,b
2
,b
3
)
∂b
3
=0.
ϕ
(3)
2
,b
2
b
3
∂I(ϕ
(3)
2
,b
2
,b
3
)
∂ϕ
(3)
2
=0
ϕ
(3)
2
=
3
m
(3)
r
3
L
(3)
r
p
(3)
L
(3)
m
ψ
(3)
const
(ν
8
b
2
b
3
−F
1
ν
1
−F
2
ν
3
−Mν
3
−ν
7
b
3
) q
1
+(8b
3
+3) ν
4
(M−5F
2
)q
2
ν
−
1
2
5
(ν
6
q
1
+(9+64b
2
3
+48b
3
) ν
4
q
2
+(64b
2
2
− 48b
2
+9)ν
5
q
3
)
,
ν
1
=18− 72b
2
+64b
2
2
ν
2
=30− 9b
2
2
− 31b
2
ν
3
=8b
2
−6 − 3b
2
2
ν
4
=1+b
2
2
ν
5
=1+b
2
3
ν
6
=
73(ν
4
+ν
5
)−110(1+b
1
b
2
)+96(b
3
−b
2
) ν
7
=24F
1
+49F
2
−8M ν
8
=64F
1
+9F
2
+3M
ϕ
(3)
2
∂
2
I
∂
ϕ
(3)
2
2
=
3
m
(3)
r
3
L
(3)
r
p
(3)
L
(3)
m
ψ
(3)
const
2
×
×
2(73b
2
2
−(96+110b
3
)b
2
+36+96b
3
+73b
2
3
) q
1
+2 (3+8b
3
)
2
q
2
(1+b
2
3
)(8b
2
−3)
2
+2q
3
.
I(ϕ
(3)
2
,b
2
,b
3
)
ϕ
(3)
2
b
3
∂I(b
2
,b
3
)
∂b
3
=0
b
3
b
{1}
3
=
(ν
1
F
1
+ ν
2
F
2
+ ν
3
M) q
1
+ ν
4
(15F
2
− 3M) q
2
(ν
8
− ν
7
) q
1
+ ν
4
(8M −40F
2
) q
2
;
b
{2}
3
= −
A
2
A
3
,
A
2
=(3b
2
F
1
+(3b
2
− 6) F
2
+ Mb
2
)6q
2
1
+ ν
4
(8M −40F
2
) q
3
q
2
+(ν
4
ν
7
q
2
+(ν
8
− ν
7
) q
3
) q
1
,
A
3
=(3F
1
− 3F
2
− M)6q
2
1
+3ν
4
(5F
2
− M) q
3
q
2
+
ν
4
(64F
1
− 9F
2
− 3M) q
2
+
+(ν
1
F
1
+ ν
2
F
2
+ ν
3
M) q
3
q
1
+3ν
4
(5F
2
− M) q
3
q
2
.
F
1
=const,F
2
=0,M =0
b
{1}
3
(b
2
=0,F
1
=const,F
2
=0,M =0)=−
1
24
,
b
{2}
3
(b
2
=0,F
1
=const,F
2
=0,M =0)=0.
b
{2}
3
=0
b
2
β
2
∂I(b
2
,b
{2}
3
)
∂b
2
=0
b
{1}
2
=
(21F
2
−82F
1
−9M) q
3
q
1
−(3F
1
+M+3F
2
)6q
2
1
−(M−5F
2
)3q
2
3
q
3
((49F
2
−24F
1
−8M) q
1
−(M−5F
2
)8q
3
)
;
b
{2}
2
=
q
3
(36q
2
1
F
2
− (q
2
+ q
3
)ν
7
q
1
− (M − 5F
2
)8q
3
q
2
)
ν
11
(q
2
+ q
3
)6q
2
1
+(ν
9
q
2
+ ν
10
q
3
) q
1
q
3
− (5F
2
− M)3q
2
3
q
2
,
ν
9
=82F
1
+9M −21F
2
ν
10
=3M +64F
1
+9F
2
ν
11
=3F
2
+3F
1
+ M
F
1
,F
2
,M
β
j
b
j
b
j
b
j
β
j
= arctan
x
3
cos x
6
− x
1
sin x
6
+ z
1
1
x
5
x
1
cos x
6
+ x
3
sin x
6
− z
1
2
x
5
,j =1..m,
b
j
=
x
3
cos x
6
− x
1
sin x
6
+ z
j
1
x
5
x
1
cos x
6
+ x
3
sin x
6
− z
j
2
x
5
,j =1..m,