Колесников А.А. Синергетические методы управления сложными системами: теория системного синтеза
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ϕ
s
(ψ
s
)
ψ
s
ϕ
s
(0)
ϕ
s
(ψ)ψ
s
> 0 ψ
s
=0
ϕ
s
(ψ
s
)
ψ
s
ψ
s
=0
dψ
s
dt
=
n
k=1
∂ψ
s
(x
1
,...,x
n
)
∂x
k
˙x
k
(t)
˙x
k
(t)
m =1
˙x
1
(t)=f
i
(x
1
,...,x
n
); i =1, 2,...,n− 1;
˙x
n
(t)=f
n
(x
1
,...,x
n
)+u,
dψ
dt
=
n
k=1
∂ψ(x
1
,...,x
n
)
∂x
k
f
k
(x
1
,...,x
n
)+
∂ψ
∂x
n
u.
J
Σ
=
∞
0
⎡
⎣
ϕ
2
(ψ)+T
2
n
k=1
∂ψ
∂x
k
f
k
+
∂ψ
∂x
n
u
2
⎤
⎦
dt.
ϕ
s
(ψ) ψ
s
(x
1
,...,x
n
)
ψ
s
ψ
s
ϕ
s
(ψ
s
)
ϕ
s
(ψ
s
)
ψ
s
ψ
3
s
,ψ
5
s
,...,ψ
r
s
ψ
s
(x
1
,...,x
n
)
ψ
3
= ···= ψ
r
∼
=
0
ψ
2
s
ψ
s
ψ
s
ϕ
2
s
(ψ)
˙ϕ
s
(ψ
s
)
T
s
L
e
(u =0)
J =
t
k
t
0
L
e
(x
1
,...,x
n
,t)dt,
u(x
1
,...,x
n
)
L
Σ
= L
e
+ L
u
,
u(x
1
,...,x
n
)
u(x
1
,...,x
n
) ≡ L
u
.
u(x
1
,...,x
n
)
m ψ
s
=0(s =1,...,m)
ϕ
s
(ψ
s
)
ψ
s
|x
k
| A
k
ψ
s
= x
k
+ A
k
th F
k
(x
1
,...,x
n
,...),
ψ
s
=
n−1
k=1
β
k
|x
k
| + β
n
|s|
ψ
s
=
n−1
k=1
β
k
x
2
k
+ β
n
|s|,
s = x
n
+ ϕ(x
1
,...,x
n−1
) ϕ
s
(ψ
s
) ψ
s
(x
1
,...,x
n
)
ψ
s
ϕ
s
(ψ
s
)=ψ
s
ψ
s
=0
ψ
s sup
= ±A
k
+ x
k
J
sup
∼
t
k sup
t
0sup
± A
k
+ x
k
2
+ T
2
k
˙x
2
k
(t)
dt.
f
n
=0
J
sup
∼
t
k sup
t
0sup
± A
k
+ x
k
2
+ T
2
k
U
2
k
(t)
dt.
|x
k
| A
k
ϕ
s
(ψ
s
)=thψ
s
ψ
s
ψ
s
=0 ψ
s sup
∼
=
±1
J
sup
∼
t
k sup
t
0sup
1+T
2
s
˙
ψ
2
sup
(t)
dt.
f
n
=0
J
sup
∼
t
k sup
t
0sup
1+T
2
s
U
2
k
dt,
ψ
s
ϕ
s
(ψ
s
)
m =2
˙
ψ
1
(t)=ψ
2
;
˙
ψ
21
(t)=−U
ψ max
sign µ(ψ
1
,ψ
2
),
µ(ψ
1
,ψ
2
)=ψ
1
+
0, 5
U
ψ max
ψ
2
|ψ
2
|
m =3
˙
ψ
1
(t)=ψ
2
,
˙
ψ
2
(t)=ψ
32
˙
ψ
3
(t)=−U
ψ max
sign µ(ψ
1
,ψ
2
,ψ
3
),
µ(ψ
1
,ψ
2
,ψ
3
)=ψ
1
+
1
3
|ψ
3
|(2ψ
2
+ψ
3
|ψ
3
|)+(ψ
2
+0, 5ψ
3
|ψ
3
|)
ψ
2
+0, 5ψ
3
|ψ
3
|
U
ψ max
µ =0
U
ψ max
ψ
s
=0
ϕ
s
(ψ
s
)
T
s
˙
ψ
s
(t)+ψ
1
2ρ+1
s
=0,ρ=1, 2,...
ρ →∞
T
s
˙
ψ
s
(t)+signψ
s
=0.
J
Σ
=
∞
0
ψ
r
s
dt,
ψ
s
=0
ψ
s
=0
T
2
s
¨
ψ
s
(t)+ϕ
s
(
˙
ψ
s
)+f
s
(ψ
s
)=0,s=1, 2,...,m,
m
ϕ
s
(
˙
ψ
s
) T
2
s
v
s
=0, 5
˙
ψ
2
s
+
1
T
2
s
ψ
s
0
f(ψ
s
)dψ
s
.
ϕ
s
(
˙
ψ
s
)=0
v
s
= const
v
s
v
s
˙v
s
(t)=
¨
ψ
s
(t)+
1
T
2
s
f
s
(ψ
s
)
˙
ψ
s
(t),
˙v
s
(t)=−
1
T
2
s
ϕ
s
(
˙
ψ
s
)
˙
ψ
s
(t).
ϕ
s
(
˙
ψ
s
)
˙
ψ
s
(t) > 0 ˙v
s
(t) 0
v
s
f
s
(ψ
s
)ψ
s
> 0 ϕ
s
(
˙
ψ
s
)=0 ˙v
s
(t)=0
v
s
= const
ψ
s
=
˙
ψ
s
(t)=0
f(ψ
s
)ψ
s
> 0 ψ
s
=0;
ϕ
s
(
˙
ψ
s
)
˙
ψ
s
(t) > 0
˙
ψ
s
(t) =0;
ψ
s
0
f
s
(ψ
s
)dψ
s
→∞ |ψ
s
|→∞.
ψ
s
(x
1
,...,x
n
)=0
˙
ψ
s
(x
1
,...,x
n
, ˙x
1
,..., ˙x
n
)=0
J
Σ
=
∞
0
λ
2s
¨
ψ
2
s
(t)+λ
1s
ϕ
2
s
(
˙
ψ
s
)+f
2
s
(ψ
s
)
dt,
λ
ks
¨
ψ
s
(t)+α
1s
˙
ψ
s
(t)+α
0s
=0
J
Σ
=
∞
0
λ
2s
¨
ψ
2
s
(t)+λ
1s
˙
ψ
2
s
(t)+ψ
2
s
dt,
α
ks
λ
ks
λ
1s
= α
2
1s
− 2α
0s
; λ
2s
= α
2
2s
.
ψ
s
(t) → 0
˙
ψ
s
(t) → 0 t →
∞
J
Σ
=
∞
0
¨
ψ
s
(t)+λ
1s
˙
ψ
s
+ λ
0s
ψ
s
2
dt + c
s
,
c
s
J
Σ
J
Σ
=
∞
0
T
s
˙
ψ
s
(t)+ψ
s
2
dt + c
s
.
J
Σ
J
Σ
ψ
s
(x
1
,...,x
n
)=0
u(ψ)=u(x, z)
ψ
s
(x)=0
˙x
k
(t)=f
k
(x
1
,...,x
n
)+M
k
(t); k =1, 2,...,m− 1; m n,
˙x
k+1
(t)=f
k+1
(x
1
,...,x
n
)+u
k+1
+ M
k+1
(t);
.............................................
˙x
n
(t)=f
n
(x
1
,...,x
n
)+u
n
+ M
r
(t),
x
1
,...,x
n
u
k+1
,...,u
k
M
1
(t),...,M
r
(t)
r
˙z
j
(t)=g
i
(z
1
,...,z
r
,x
1
,...,x
n
),j=1,...,r.
M
1
(t),...,M
r
(t)
˙z
j
(t)=g
i
(z
1
,...,z
r
,x
1
,...,x
n
),j=1,...,r;
˙x
i
(t)=f
i
(x
1
,...,x
n
)+z
j
,i= r +1,...,m− 1;
˙x
i+1
(t)=f
i+1
(x
1
,...,x
n
)+u
i+1
+ z
j+1
;
..........................................
˙x
n
(t)=f
n
(x
1
,...,x
n
)+u
n
+ z
r
.
u(u
1
,...,u
m
)
ψ
s
(x
1
,...,x
n
,z
1
,...,z
r
)=0,s=1, 2,...,m,
ψ
s
=0
M
1
(t),...,M
r
(t)
J
Σ
=
∞
0
m
s=1
ϕ
2
s
(ψ
s
)+
m
s=1
T
2
s
˙
ψ
2
s
(t)
dt.
T
s
˙
ψ
s
(t)+ϕ
s
(ψ
s
)=0,s=1, 2,...,m.
ϕ
s
(ψ
s
)
ψ
s
=0
ϕ
s
(ψ
s
)
ψ
s
ψ
s
=0
˙z
jψ
(t)=g
i
(z
1ψ
,...,z
rψ
,v
i+1
,...,v
m
,x
1ψ
,...,x
m−1ψ
),
˙x
iψ
(t)=f
i
(x
1ψ
,...,x
m−1ψ
,v
i+1
,...,v
m
),
j =1,...,r; i = r +1,...,m− 1,
v
i+1
,...,v
m
v
i+1
,...,v
m
ψ
s
=0
v
k
dim v
k
= m
v
k
ψ
s
= γ
s1
(x
i+1
− v
1
)+···+ γ
sm
(x
n
− v
n
),s=1,...,m.
ψ
s