Баландин Д.В., Коган М.М. Синтез законов управления на основе линейных матричных неравенств
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W
T
Q
−Y
11
−Y
12
A 0 B
∆
0
? −Y
22
0 0 0 0
? ? −X
11
−X
12
0 C
T
∆
? ? ? −X
22
0 0
? ? ? ? −ζS D
T
∆∆
? ? ? ? ? −ζΣ
W
Q
< 0 .
−W
(1)
P
T
X
11
W
(1)
P
− ζW
(2)
P
T
SW
(2)
P
? ?
AW
(1)
P
+ B
∆
W
(2)
P
0
!
−Y ?
C
∆
W
(1)
P
+ D
∆∆
W
(2)
P
0 −ζΣ
< 0,
Y > 0
−W
(1)
P
T
X
11
W
(1)
P
− ζW
(2)
P
T
SW
(2)
P
W
(1)
P
T
C
T
∆
+ W
(2)
P
T
D
T
∆∆
C
∆
W
(1)
P
+ D
∆∆
W
(2)
P
−ζΣ
+
+
W
(1)
P
T
A
T
+ W
(2)
P
T
B
T
∆
0
0 0
Y
−1
AW
(1)
P
+ B
∆
W
(2)
P
0
0 0
< 0 .
Y
−1
= X
X
11
X
11
ˆ
W
T
P
A
T
X
11
A − X
11
A
T
X
11
B
∆
| C
T
∆
? −ζS + B
T
∆
X
11
B
∆
| D
T
∆∆
− − | −
? ? | −ζΣ
ˆ
W
P
< 0 ,
ˆ
W
P
=
N
1
| 0
− | −
0 | I
,
N
1
= (W
(1)
P
, W
(2)
P
)
(C
2
D
2∆
)
ˆ
W
T
Q
AY
11
A
T
− Y
11
AY
11
C
T
∆
| B
∆
? −ζS + C
∆
Y
11
C
T
∆
| D
∆∆
− − | −
? ? | −ζΣ
ˆ
W
Q
< 0 ,
ˆ
W
Q
=
N
2
| 0
− | −
0 | I
,
N
2
= (W
(1)
Q
, W
(2)
Q
)
(B
T
2
D
T
∆2
) H
∞
k
(n
x
× n
x
) X
11
= X
T
11
> 0
Y
11
= Y
T
11
> 0
X
11
I
I Y
11
≥ 0 ,
(I − X
11
Y
11
) ≤ k ,
˙x = Ax + B
1
ϕ(σ, t) ,
x ∈ R
n
x
σ = Cx σ ∈ R
n
σ
ϕ(σ, t) ∈ R
n
σ
t ≥ 0 σ
Γ
T
= Γ > 0
ϕ
T
(σ, t)Γ[ϕ(σ, t) − σ] ≤ 0
ϕ(0, t) ≡ 0 n
σ
= 1
0 ≤
ϕ(σ, t)
σ
≤ 1 , ∀σ 6= 0 ,
ϕ(σ, t)
ϕ(y, t)
˙x = Ax + B
1
v , σ = Cx ,
v σ
v
T
Γ(v − σ) ≤ 0 .
V (x) = x
T
Xx
x v
˙
V < 0 .
S
V (x) = x
T
Xx X = X
T
> 0
x v
2x
T
X(Ax + B
1
v) − v
T
Γ(v − Cx ) < 0 , |x|
2
+ |v|
2
6= 0 .
V (x)
A
T
X + XA XB
1
+
1
2
C
T
Γ
B
T
1
X +
1
2
ΓC −Γ
< 0 .
X = X
T
> 0
˙x = Ax + B
1
ϕ(σ, t) + B
2
u ,
σ = Cx ϕ(σ, t)
u = Θx ,
˙x = (A + B
2
Θ)x + B
1
ϕ(σ, t)
(A + B
2
Θ)
T
X + X(A + B
2
Θ) XB
1
+
1
2
C
T
Γ
B
T
1
X +
1
2
ΓC −Γ
< 0 .
X
−1
0
0 I
X
−1
= Y Z = ΘY
Y A
T
+ AY + B
2
Z + Z
T
B
T
2
B
1
+
1
2
Y C
T
Γ
B
T
1
+
1
2
ΓCY −Γ
< 0
Y = Y
T
> 0 Z
Θ = ZY
−1
Y A
T
+ AY + B
2
Z + Z
T
B
T
2
< 0 ,
(A, B
2
)
H
∞
˙x = Ax + 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
∆
x ∈ R
n
x
v
∆
∈ R
n
v∆
v ∈
R
n
v
u ∈ R
n
u
z
∆
∈ R
n
z∆
y ∈ R
n
y
n
v∆
= n
z∆
= n
∆
∆(t)
∆(t) = (δ
1
(t)I
k
1
, . . . , δ
r
(t)I
k
r
, ∆
1
(t), . . . , ∆
f
(t))
∆
T
(t)∆(t) ≤ η
2
I , ∀t ≥ 0 .
(I − ∆(t)D
∆∆
) 6= 0
H
∞
k
˙x
r
= A
r
x
r
+ B
r
y ,
u = C
r
x
r
+ D
r
y ,
x
r
∈ R
k
∆(t) γ
sup
v6≡0
kzk
kvk
< γ ,
H
∞
˙x = Ax +
ˆ
B
1
ˆv + B
2
u
ˆz =
ˆ
C
1
x +
ˆ
D
11
ˆv +
ˆ
D
12
u
y = C
2
x +
ˆ
D
21
ˆv
ˆv ˆz
ˆ
B
1
= (B
∆
B
1
)L
1
,
ˆ
C
1
= L
2
C
∆
C
1
,
ˆ
D
11
= L
2
D
∆∆
D
∆1
D
1∆
D
11
L
1
,
ˆ
D
12
= L
2
D
∆2
D
12
,
ˆ
D
21
= (D
2∆
D
21
)L
1
,
L
1
=
γηS
−1/2
0
0 I
, L
2
=
S
1/2
0
0 I
,
S = S
T
> 0
S = (S
1
, . . . , S
r
, s
1
I
m
1
, . . . , s
f
I
m
f
) ,
r
k
1
, . . . , k
r
k
1
+ ··· + k
r
+ m
1
+ ··· + m
f
= n
∆
H
∞
kˆzk < γkˆvk , ∀ˆv 6≡ 0 ,
k · k L
2
ˆv = L
−1
1
v
∆
v
!
, ˆz = L
2
z
∆
z
!
kzk
2
− γ
2
kvk
2
< −(kS
1/2
z
∆
k
2
− η
−2
kS
1/2
v
∆
k
2
) .
v
∆
= ∆(t)z
∆
∆(t)
kS
1/2
z
∆
k
2
− η
−2
kS
1/2
v
∆
k
2
≥ 0
kzk
2
− γ
2
kvk
2
< 0 .
H
∞
H
∞
˙x
c
= A
c
x
c
+
ˆ
B
c
ˆv ,
ˆz =
ˆ
C
c
x
c
+
ˆ
D
c
ˆv ,
A
c
=
A + B
2
D
r
C
2
B
2
C
r
B
r
C
2
A
r
,
ˆ
B
c
=
ˆ
B
1
+ B
2
D
r
ˆ
D
21
B
r
ˆ
D
21
= B
c
L
1
, B
c
=
B
∆
+ B
2
D
r
D
2∆
B
1
+ B
2
D
r
D
21
B
r
D
2∆
B
r
D
21
,
ˆ
C
c
=
C
1
+
ˆ
D
12
D
r
C
2
ˆ
D
12
C
r
= L
2
C
c
, C
c
=
C
∆
+ D
∆2
D
r
C
2
D
∆2
C
r
C
1
+ D
12
D
r
C
2
D
12
C
r
,
ˆ
D
c
=
ˆ
D
11
+
ˆ
D
12
D
r
ˆ
D
21
= L
2
D
c
L
1
,
D
c
=
D
∆∆
+ D
∆2
D
r
D
2∆
D
∆1
+ D
∆2
D
r
D
21
D
1∆
+ D
12
D
r
D
2∆
D
11
+ D
12
D
r
D
21
.
γ
X = X
T
> 0
A
T
c
X + XA
c
X
ˆ
B
c
ˆ
C
T
c
ˆ
B
T
c
X −γI
ˆ
D
T
c
ˆ
C
c
ˆ
D
c
−γI
=
A
T
c
X + XA
c
XB
c
L
1
C
T
c
L
2
L
1
B
T
c
X −γI L
1
D
T
c
L
2
L
2
C
c
L
2
D
c
L
1
−γI
< 0 .
H
∞
L
1
L
2
A
T
c
X + XA
c
XB
c
C
T
c
B
T
c
X −
γ
−1
η
−2
S 0
0 γI
!
D
T
c
C
c
D
c
−
γS
−1
0
0 γI
!
< 0 .
ˆ
S = γ
−1
η
−1
S ζ = η
−1
A
T
c
X + XA
c
XB
c
C
T
c
B
T
c
X −
ζ
ˆ
S 0
0 γI
!
D
T
c
C
c
D
c
−
ζ
ˆ
S
−1
0
0 γI
!
< 0 .
A
c
B
c
D
c
C
c
A
c
= A
0
+ BΘC , B
c
= B
0
+ BΘD
1
,
C
c
= C
0
+ D
2
ΘC , D
c
= D
0
+ D
2
ΘD
1
,
A
0
=
A 0
n
x
×k
0
k×n
x
0
k×k
,
B =
0
n
x
×k
B
2
I
k
0
k×n
u
, C =
0
k×n
x
I
k
C
2
0
n
y
×k
,
B
0
=
B
∆
B
1
0
k×n
∆
0
k×n
v
, C
0
=
C
∆
0
n
∆
×k
C
1
0
n
z
×k
,
D
1
=
0
k×n
∆
0
k×n
v
D
2∆
D
21
, D
2
=
0
n
∆
×k
D
∆2
0
n
z
×k
D
12
,
D
0
=
D
∆∆
D
∆1
D
1∆
D
11
, Θ =
A
r
B
r
C
r
D
r
.