Колесников А.А. Синергетические методы управления сложными системами: теория системного синтеза
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ψ
s
=0
ψ =0
u(x
1
,...,x
n
)
˙x
i
(t)=R
i
(x
1
,...,x
n
),i= 1,n,
R
k
=f
k
(x
1
,...,x
n
) k = 1,n− 1 R
n
=f
n
(x
1
,...,x
n
)+u(x
1
,...,x
n
)
x
1
,...,x
n
ψ(x
1
,...,x
n
)
ψ =0
t ψ =0
ψ =0
n − 1
ψ = const
ψ =0
ψ =0
ψ = const
ψ(x
1
,...,x
n
)
ψ(x
1
,...,x
n
)
ψ =0
ψ(x
1
,...,x
n
) t
ψ t
dψ
dt
=
n
i=1
∂ψ(x
1
,...,x
n
)
∂x
i
˙x
i
,
dψ
dt
=
n
i=1
∂ψ(x
1
,...,x
n
)
∂x
i
R
i
(x
1
,...,x
n
),
ψ(x
1
,...,x
n
) n x
1
,...,x
n
˙
ψ(t)=ωψ,
˙
ψ(t)
ω x
1
,...,x
n
˙
ψ(t)
n
i=1
∂ψ(x
1
,...,x
n
)
∂x
i
R
i
(x
1
,...,x
n
)=ω(x
1
,...,x
n
)ψ(x
1
,...,x
n
).
ψ =0
t
ψ =0 ωψ
Φ(ψ, x
1
,...,x
n
)
dψ
dt
=
n
i=1
∂ψ(x
1
,...,x
n
)
∂x
i
R
i
(x
1
,...,x
n
)=Φ(ψ, x
1
,...,x
n
),
Φ(0,x
1
,...,x
n
)=0
ψ =0
m
˙x
i
(t)=f
i
(x
1
,...,x
n
)+u
j
(x
1
,...,x
n
),i= 1,n, j = 1,m, m<n,
u
j
˙x
i
(t)=R
i
(x
1
,...,x
n
),
R
i
=f
i
+u
j
ψ
s
(x
1
,...,x
n
)=0,s=1, 2,...m n,
T
s
˙
ψ
s
(t)+Φ
s
(ψ
1
,...,ψ
m
)=0,
J
Σ
=
∞
0
m
s=1
Φ
2
s
(ψ
s
)+
m
s=1
T
2
s
˙
ψ
2
s
(t)
dt,
Φ
s
(ψ
s
) · ψ
s
> 0 T
s
> 0 ψ
s
(t)
m
x
1
,...,x
n
ψ
s
x
i
m
rangD =rang
∂ψ
1
∂x
1
∂ψ
2
∂x
1
...
∂ψ
m
∂x
1
∂ψ
1
∂x
2
∂ψ
2
∂x
2
...
∂ψ
m
∂x
2
... ... ... ...
∂ψ
1
∂x
m
∂ψ
2
∂x
m
...
∂ψ
m
∂x
m
= m.
x
1
,...,x
n
n − m
ψ
s
∂ψ
s
∂x
1
,
∂ψ
s
∂x
2
,...,
∂ψ
s
∂x
n
,s=1, 2,...,m,
lim
x
i
→a
i
ψ
s
(x
1
,...,x
n
)=ψ
s
(a
1
,...,a
n
).
ψ
s
(x
1
,...,x
n
)
dψ
s
dt
=
n
i=1
∂ψ
s
(x
1
,...,x
n
)
∂x
i
R
i
(x
1
,...,x
n
)=
m
k=1
ω
sk
ψ
s
,s=1, 2,...,m<n.
ψ
s
(x
1
,...,x
n
) t ˙x
i
(t) m
m
n − m
m
x
1
,x
2
,...,x
m
x
m+1
,...,x
n
x
1
,x
2
,...,x
m
n − m
n − m
˙x
m+µ
(t)=R
∗
m+µ
(x
m+1
,...,x
n
),µ=1, 2,...,n− m,
R
∗
m+µ
R
i
(x
1
,...,x
m
)
x
m+1
,...,x
n
x
m+µ
(t)
m
x
1
,x
2
,...,x
m
x
m+µ
µ =1, 2,...,n
x
1
,x
2
,...,x
m
t
ω
sk
ψ
s
Φ
s
(ψ
1
,...,ψ
m
,x
1
,...,x
m
)
dψ
s
dt
=
n
i=1
∂ψ
s
(x
1
,...,x
n
)
∂x
i
R
i
(x
1
,...,x
n
)=Φ
s
(ψ
1
,...,ψ
m
,x
1
,...,x
m
),
Φ
s
(ψ
1
,...,ψ
m
,x
1
,...,x
m
)
Φ
s
(0,x
1
,...,x
m
)=0
R
i
(x
1
,...,x
n
)
u
j
x
1
,...,x
n
u
j
m
ψ
s
ψ
s0
=(x
10
,...,x
no
) =0,s=1, 2,...,m n.
˙
ψ
s
(t)=Φ
s
Φ
s
(ψ
1
,...,ψ
m
,x
1
,...,x
m
)
ψ
1
=0,...,ψ
m
=0
Φ
s
ψ
1
,...,ψ
m
Φ
s
(ψ
1
,...,ψ
m
)
ψ
s
=0
x
1k
=0,...,x
nk
=0
n − m
u
j
ψ =0
ψ =0
ψ(x
1
,...,x
n
, ˙x
1
,..., ˙x
n
,t)=0,
t
ψ =0 t
ψ(x
1
,...,x
n
,t)=0
˙x
1
(t),..., ˙x
n
(t)
ψ(x
1
,...,x
n
,t)=0
t ψ(x
1
,...,x
n
)=0
ψ
s
=0 s =1, 2,...,m
ψ
s
=0
ψ
s
=0 s =1, 2,...,m s
x
i
n−s s ψ
s
=0
ψ
s
=0
ψ
s
=0
