Баландин Д.В., Коган М.М. Использование LMI toolbox пакета Matlab в синтезе законов управления
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γ
2
I − H
T
(e
−j ϕ
)H(e
j ϕ
) > 0 , ∀ϕ ∈ [0, 2π) .
(Ax
t
+ Bv
t
)
∗
X(Ax
t
+ Bv
t
) − x
T
t
Xx
t
+ (Cx
t
+ Dv
t
)
∗
(Cx
t
+ Dv
t
) − γ
2
v
∗
t
v
t
< 0 .
X γX γ
−1
x
∗
t
(A
T
XA − X + γ
−1
C
T
C)x
t
+ 2 x
∗
t
(A
T
XB + γ
−1
C
T
D)v
t
+
+v
∗
t
(γ
−1
D
T
D − γI + B
T
XB)v
t
< 0 .
x
t
v
t
A
T
XA − X + γ
−1
C
T
C A
T
XB + γ
−1
C
T
D
(A
T
XB + γ
−1
C
T
D)
T
−γI + B
T
XB + γ
−1
D
T
D
< 0
F
d
(X, γ) =
A
T
XA − X A
T
XB C
T
B
T
XA −γI + B
T
XB D
T
C D −γI
< 0 .
A
T
XA − X < 0 ,
A
X > 0
A
γ
(n
x
× n
x
) X > 0
X γ
γ
∗
= inf
F
d
(X,γ)<0
γ ,
F
d
(X, γ) < 0
H
∞
x
t+1
= Ax
t
+ B
1
v
t
+ B
2
u
t
,
z
t
= C
1
x
t
+ D
11
v
t
+ D
12
u
t
,
y
t
= C
2
x
t
+ D
21
v
t
,
x
t
∈ R
n
x
v
t
∈ R
n
v
u
t
∈ R
n
u
z
t
∈ R
n
z
y
t
∈ R
n
y
k
x
(r)
t+1
= A
r
x
(r)
t
+ B
r
y
t
,
u
t
= C
r
x
(r)
t
+ D
r
y
t
,
x
(r)
t
∈ R
k
kzk
kvk
< γ , ∀v, kvk 6= 0
γ
x
(c)
t+1
= A
c
x
(c)
t
+ B
c
v
t
,
z
t
= C
c
x
(c)
t
+ D
c
v
t
,
A
c
=
A + B
2
D
r
C
2
B
2
C
r
B
r
C
2
A
r
!
, B
c
=
B
1
+ B
2
D
r
D
21
B
r
D
21
!
,
C
c
= (C
1
+ D
12
D
r
C
2
D
12
C
r
) , D
c
= D
11
+ D
12
D
r
D
21
.
Θ =
A
r
B
r
C
r
D
r
!
A
c
= A
0
+ BΘC , B
c
= B
0
+ BΘD
21
,
C
c
= C
0
+ D
12
ΘC , D
c
= D
11
+ D
12
ΘD
21
,
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
1
0
k×n
v
, C
0
= (C
1
0
n
z
×k
) ,
D
12
= (0
n
z
×k
D
12
) , D
21
=
0
k×n
v
D
21
.
H
∞
v
t
z
t
H
c
(q) = D
c
+ C
c
(qI − A
c
)
−1
B
c
kH
c
k
∞
< γ .
A
T
c
XA
c
− X A
T
c
XB
c
C
T
c
B
T
c
XA
c
−γI + B
T
c
XB
c
D
T
c
C
c
D
c
−γI
< 0 .
X = X
T
> 0 H
∞
Θ
Θ
−X
−1
A
c
B
c
0
A
T
c
−X 0 C
T
c
B
T
c
0 −γI D
T
c
0 C
c
D
c
−γI
< 0 ,
Θ
X > 0 ,
−X 0 C
T
c
0 −γI D
T
c
C
c
D
c
−γI
+
A
T
c
B
T
c
0
X(A
c
B
c
0) < 0 .
A
c
A
T
c
XA
c
− X A
T
c
XB
c
C
T
c
B
T
c
XA
c
−γI + B
T
c
XB
c
D
T
c
C
c
D
c
−γI
< 0 ,
A
T
c
XA
c
− X A
T
c
XB
c
B
T
c
XA
c
−γI + B
T
c
XB
c
+ γ
−1
C
T
c
D
T
c
(C
c
D
c
) < 0 .
Θ
Ψ + P
T
Θ
T
Q + Q
T
ΘP < 0 ,
Ψ =
−X
−1
A
0
B
0
0
A
T
0
−X 0 C
T
0
B
T
0
0 −γI D
T
11
0 C
0
D
11
−γI
,
P = (0
(n
y
+k)×(n
x
+k)
C D
21
0
(n
y
+k)×n
z
) ,
Q = (B
T
0
(n
u
+k)×(n
x
+k)
0
(n
u
+k)×n
v
D
T
12
)
H
∞
k γ
(n
x
+ k) × (n
x
+ k) X =
X
T
> 0
W
T
P
−X
−1
A
0
B
0
0
A
T
0
−X 0 C
T
0
B
T
0
0 −γI D
T
11
0 C
0
D
11
−γI
W
P
< 0 ,
W
T
Q
−X
−1
A
0
B
0
0
A
T
0
−X 0 C
T
0
B
T
0
0 −γI D
T
11
0 C
0
D
11
−γI
W
Q
< 0 .
X Θ
Y = X
−1
H
∞
X Y
W
T
P
−Y A
0
B
0
0
A
T
0
−X 0 C
T
0
B
T
0
0 −γI D
T
11
0 C
0
D
11
−γI
W
P
< 0 ,
W
T
Q
−Y A
0
B
0
0
A
T
0
−X 0 C
T
0
B
T
0
0 −γI D
T
11
0 C
0
D
11
−γI
W
Q
< 0 .
X Y XY = I
A
0
B
0
C
0
X Y
X =
X
11
X
12
X
T
12
X
22
, Y =
Y
11
Y
12
Y
T
12
Y
22
.
P =
0
k×n
x
0
k×k
0
k×n
x
I
k
0
k×n
v
0
k×n
z
0
n
y
×n
x
0
n
y
×k
C
2
0
n
y
×k
D
21
0
n
y
×n
z
,
Q =
0
k×n
x
I
k
0
k×n
x
0
k×k
0
k×n
v
0
k×n
z
B
T
2
0
n
u
×k
0
n
u
×n
x
0
n
u
×k
0
n
u
×n
v
D
T
12
,
W
P
=
0 I 0 0
0 0 I 0
W
(1)
P
0 0 0
0 0 0 0
W
(2)
P
0 0 0
0 0 0 I
, W
Q
=
W
(1)
Q
0 0 0
0 0 0 0
0 I 0 0
0 0 I 0
0 0 0 I
W
(2)
Q
0 0 0
,
W
(1)
P
W
(2)
P
W
(1)
Q
W
(2)
Q
C
2
W
(1)
P
+ D
21
W
(2)
P
= 0 , B
T
2
W
(1)
Q
+ D
T
12
W
(2)
Q
= 0 .
W
T
P
−Y
11
−Y
12
A 0 B
1
0
? −Y
22
0 0 0 0
? ? −X
11
−X
12
0 C
T
1
? ? ? −X
22
0 0
? ? ? ? −γI D
T
11
? ? ? ? ? −γI
W
P
< 0 ,
W
T
Q
−Y
11
−Y
12
A 0 B
1
0
? −Y
22
0 0 0 0
? ? −X
11
−X
12
0 C
T
1
? ? ? −X
22
0 0
? ? ? ? −γI D
T
11
? ? ? ? ? −γI
W
Q
< 0 .
−W
(1)
P
T
X
11
W
(1)
P
− γW
(2)
P
T
W
(2)
P
? ?
AW
(1)
P
+ B
1
W
(2)
P
0
!
−Y ?
C
1
W
(1)
P
+ D
11
W
(2)
P
0 −γI
< 0,
Y > 0
−W
(1)
P
T
X
11
W
(1)
P
− γW
(2)
P
T
W
(2)
P
W
(1)
P
T
C
T
1
+ W
(2)
P
T
D
T
11
C
1
W
(1)
P
+ D
11
W
(2)
P
−γI
+
+
W
(1)
P
T
A
T
+ W
(2)
P
T
B
T
1
0
0 0
Y
−1
AW
(1)
P
+ B
1
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
1
| C
T
1
? −γI + B
T
1
X
11
B
1
| D
T
11
− − | −
? ? | −γI
ˆ
W
P
< 0 ,
H
∞
ˆ
W
P
=
N
1
| 0
− | −
0 | I
,
N
1
= (W
(1)
P
, W
(2)
P
) (C
2
D
21
)
ˆ
W
T
Q
AY
11
A
T
− Y
11
AY
11
C
T
1
| B
1
? −γI + C
1
Y
11
C
T
1
| D
11
− − | −
? ? | −γI
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
12
)
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
1
= x
2
,
˙x
2
= x
1
+ u ,
y = x
1
u = Θy
X = X
T
> 0 Y = Y
T
> 0
W
T
C
(A
T
X + XA)W
C
< 0 ,
W
T
B
T
(Y A
T
+ AY )W
B
T
< 0 .
X =
x
11
x
12
x
12
x
22
, Y =
y
11
y
12
y
12
y
22
,
A =
0 1
1 0
, B =
0
1
, C = (1 0)
W
B
T
=
1
0
, W
C
=
0
1
,
x
12
< 0 , y
12
< 0 .
X =
1 + δ −δ
2
−δ
2
1 + δ
, Y =
1 + δ −δ
2
−δ
2
1 + δ
,
|δ| 1
Θ
XY 6= I
X Y XY = I
L
i
(X, Y ) < 0,
i = 1, 2 X Y
λ
min
= min{λ : X − Y
−1
< λI, X > 0, Y > 0, L
i
(X, Y ) < 0, i = 1, 2, 3} ,
L
3
(X, Y ) =
−X I
I −Y
!
.
L
3
(X, Y ) < 0
X > Y
−1
A1 λ
min
= 0
X Y A
A1
X − Y
−1
< λI
A1
A1