Баландин Д.В., Коган М.М. Синтез законов управления на основе линейных матричных неравенств
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m
¨
ξ
1
= −2b
˙
ξ
1
− 2cξ
1
+ b
˙
ξ
2
+ cξ
2
− m
¨
ξ
0
(t)
.................................................
m
¨
ξ
s
= −2b
˙
ξ
s
− 2cξ
s
+ b
˙
ξ
s−1
+ cξ
s−1
+ b
˙
ξ
s+1
+ cξ
s+1
+ U − m
¨
ξ
0
(t)
.................................................
m
¨
ξ
n
= −b
˙
ξ
n
− cξ
n
+ b
˙
ξ
n−1
+ cξ
n−1
− m
¨
ξ
0
(t) ,
ξ = (ξ
1
, . . . , ξ
n
) ξ
i
i
U s
ξ
0
m b c
β =
b
m
, ω
2
=
c
m
, u = m
−1
U , v
1
= −
¨
ξ
0
,
¨
ξ = −βK
˙
ξ − ω
2
Kξ + qu + pv
1
,
ξ = (ξ
1
, ξ
2
, ..., ξ
n
)
T
K =
2 −1 0 0 ... 0
−1 2 −1 0 ... 0
0 −1 2 −1 ... 0
... ... ... ... ... ...
... ... 0 −1 2 −1
0 0 0 ... −1 1
, q =
0
0
.
1
.
0
, p =
1
1
.
.
.
1
.
s q s
1
0
˙x = Ax + Bv
1
+ B
2
u ,
x = (ξ,
˙
ξ) A, B, B
2
A =
0 I
−ω
2
K −βK
, B =
0
p
, B
2
=
0
q
.
y
1
= x
1
+ αv
21
y
2
= x
2
− x
1
+ αv
22
............................
y
n
= x
n
− x
n−1
+ αv
2n
,
αv
2
= α (v
21
, v
22
, . . . , v
2n
) α
y =
(y
1
, y
2
, . . . , y
n
)
y = C
2
x + αv
2
,
C
2
=
1 0 0 0 ... 0 0 ... 0
−1 1 0 0 ... 0 0 ... 0
0 −1 1 0 ... 0 0 ... 0
... ... ... ... ... ... ... ... ...
0 0 0 ... −1 1 0 ... 0
.
y
˙x
r
= A
r
x
r
+ B
r
y ,
u = C
r
x
r
+ D
r
y , x
r
(0) = 0 ,
x
r
v
1
(t) v
2
(t)
v(t) = (v
1
(t), v
2
(t))
[0, ∞)
J(u, v) =
∞
Z
0
(x
T
Qx + ρ
2
u
2
)d t ,
Q ρ
Q
Q =
ω
2
K 0
0 I
γ
∞
Z
0
(x
T
Qx + ρ
2
u
2
)d t
∞
Z
0
|v|
2
d t
< γ
2
γ
z =
√
Q
0
x +
0
ρ
u ,
∞
Z
0
|z|
2
d t
∞
Z
0
|v|
2
d t
< γ
2
.
H
∞
kzk
kvk
< γ , ∀v 6≡ 0
˙x = Ax + B
1
v + B
2
u ,
z = C
1
x + D
12
u
y = C
2
x + D
21
v ,
kzk L
2
A B
2
C
2
B
1
= (B 0) , C
1
=
√
Q
0
, D
12
=
0
ρ
, D
21
= (0 αI) .
H
∞
ω
2
β
ω
2
β
ω
2
= ω
2
∗
[1 + f
1
Ω
1
(t)] , β = β
∗
[1 + f
2
Ω
2
(t)] ,
β
∗
ω
2
∗
f
1
f
2
Ω
1
(t) Ω
2
(t)
|Ω
1
(t)| ≤ 1 , |Ω
2
(t)| ≤ 1 .
A
A = A
∗
+ F Ω(t)E ,
Ω(t) = (Ω
1
(t)I
n×n
, Ω
2
(t)I
n×n
)
A
∗
=
0 I
−ω
2
∗
K −β
∗
K
!
, F = (F
1
F
2
) , E =
E
1
E
2
!
,
F
1
=
0
f
1
I
!
, F
2
=
0
f
2
I
!
, E
1
= (−ω
2
∗
K 0), E
2
= (0 − β
∗
K).
˙x = A
∗
x + B
∆
v
∆
+ B
1
v + B
2
u ,
z
∆
= C
∆
x + D
∆∆
v
∆
+ D
∆1
v + D
∆2
u ,
z = C
1
x + D
1∆
v
∆
+ D
11
v + D
12
u ,
y = C
2
x + D
2∆
v
∆
+ D
21
v ,
v
∆
= ∆(t)z
∆
,
B
∆
= F C
∆
= E D
∆∆
= 0 D
∆1
= 0 D
∆2
= 0 D
1∆
= 0 D
11
= 0
D
2∆
= 0
∆(t)
∆(t) = (Ω
1
(t)I
n×n
, Ω
2
(t)I
n×n
)
∆
T
(t)∆(t) ≤ I , ∀t ≥ 0 .
H
∞
γ
ω
2
β
n = 10 n
x
= 20
β
∗
= 1
−1
ω
2
∗
= 100
−2
ρ = 1
184.4
k = 20
α = 1
2
α = 0.01
2
α
s
γ (α = 1)
γ (α = 0.01)
α = 0.01
2
s
γ
H
∞
(s = 7) α = 1 c
2
ω
2
[99, 101]
β [0.8, 1.2] f
1
= 0.01 f
2
= 0.2
γ = 18
γ = 16.1