Баландин Д.В., Коган М.М. Синтез законов управления на основе линейных матричных неравенств
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˙x = Ax + Bv
z = Cx + Dv , x(0) = 0 ,
x ∈ R
n
x
v ∈ R
n
v
z ∈ R
n
z
L
2
v(t)
kvk = (
∞
Z
0
|v(t)|
2
dt)
1/2
< ∞ .
γ
∗
= s up
kvk6=0
kzk
kvk
.
γ
∗
= inf
γ
{γ :
kzk
kvk
< γ, ∀v, kvk 6= 0} ,
γ > 0
kzk
kvk
< γ , ∀v, kvk 6= 0 .
sup
kvk6=0
kzk
kvk
= sup
ω∈(−∞,∞)
kH(j ω)k = kHk
∞
,
H(s) = D + C(sI − A)
−1
B
v z j =
√
−1 k · k
kHk = max
i
σ
i
(H) ,
σ
i
(H) = λ
1/2
i
(HH
∗
) i H ∗
k·k
∞
∞ H(s)
sup
Re s≥0
kH(s)k < ∞
kHk
∞
< γ
H(j ω)H
T
(−jω) < γ
2
I , ∀ω ∈ (−∞, ∞) .
−I H
T
(−j ω)
H(j ω) −γ
2
I
< 0 ,
H
T
(−j ω)H(jω) < γ
2
I , ∀ω ∈ (−∞, ∞) .
γ
H(j ω), ω ∈ (−∞, ∞)
γ
L[(j ωI − A)
−1
Bv, v] > 0 , ∀ω ∈ (−∞, ∞) , ∀|v| 6= 0
L(x, v) x ∈ C
n
x
v ∈ C
n
v
(A, B)
V (x) = x
T
Xx X = X
∗
˙
V − L(x, v) < 0 , ∀x, v, |x| + |v| 6= 0
2 x
∗
X(Ax + Bv) − L(x, v) < 0 , ∀x, v, |x| + |v| 6= 0 .
L(x, v) = (x
T
, v
T
) L
x
v
, L =
L
11
L
12
L
T
12
L
22
,
A
T
X + XA − L
11
XB − L
12
B
T
X − L
T
12
−L
22
< 0 ,
(−j ωI − A)
−1
B
I
T
L
11
L
12
L
T
12
L
22
(j ωI − A)
−1
B
I
> 0 .
L(x, v) = γ
2
v
∗
v − (Cx + Dv)
∗
(Cx + Dv) ,
γ
2
I − H
T
(−j ω)H(j ω) > 0 , ∀ω ∈ (−∞, ∞) ,
2 x
∗
X(Ax + Bv) + (Cx + Dv)
∗
(Cx + Dv) − γ
2
v
∗
v < 0 .
X γX γ
−1
x
∗
(A
T
X+XA+γ
−1
C
T
C)x+2 x
∗
(XB+γ
−1
C
T
D)v+v
∗
(γ
−1
D
T
D−γI)v < 0 .
x v
A
T
X + XA + γ
−1
C
T
C XB + γ
−1
C
T
D
(XB + γ
−1
C
T
D)
T
−γI + γ
−1
D
T
D
< 0 ,
F
c
(X, γ) =
A
T
X + XA XB C
T
B
T
X −γI D
T
C D −γI
< 0 .
A
T
X + XA < 0 ,
A
X > 0
A
γ
(n
x
× n
x
) X > 0
X γ
γ
∗
= inf
F
c
(X,γ)<0
γ ,
F
c
(X, γ) < 0
˙x
1
= x
2
,
˙x
2
= −ω
2
0
x
1
− δx
2
+ v ,
z = x
1
,
H
∞
A =
0 1
−ω
2
0
−δ
, B =
0
1
, C = (1 0) , D = 0 .
γ
ω
0
= 10 δ = 1 γ
∗
= 0.1001 ω
0
= 10 δ = 0.1
γ
∗
= 1.000
H
∞
˙x = Ax + B
1
v + B
2
u ,
z = C
1
x + D
11
v + D
12
u ,
y = C
2
x + D
21
v ,
x ∈ R
n
x
v ∈ R
n
v
u ∈ R
n
u
z ∈ R
n
z
y ∈ R
n
y
H
∞
γ
sup
v6≡0
kzk
kvk
< γ .
H
∞
γ H
∞
H
∞
γ
C
2
= I D
21
= 0
u = Θx .
˙x
c
= A
c
x
c
+ B
c
v ,
z = C
c
x
c
+ D
c
v ,
A
c
= A + B
2
Θ , B
c
= B
1
, C
c
= C
1
+ D
12
Θ , D
c
= D
11
.
v z
H
c
(s) = D
c
+ C
c
(sI − A
c
)
−1
B
c
kH
c
k
∞
< γ .
A
T
c
X + XA
c
XB
c
C
T
c
B
T
c
X −γI D
T
c
C
c
D
c
−γI
< 0
X = X
T
> 0
X
−1
0 0
0 I 0
0 0 I
Y (A + B
2
Θ)
T
+ (A + B
2
Θ)Y B
1
Y (C
1
+ D
12
Θ)
T
B
T
1
−γI D
T
11
(C
1
+ D
12
Θ)Y D
11
−γI
< 0 ,
Y = X
−1
Y Z = ΘY γ
F (Y, Z, γ) =
Y A
T
+ AY + Z
T
B
T
2
+ B
2
Z B
1
Y C
T
1
+ Z
T
D
T
12
B
T
1
−γI D
T
11
C
1
Y + D
12
Z D
11
−γI
< 0 .
H
∞
H
∞
γ n
x
× n
x
Y = Y
T
> 0 n
u
×n
x
Z
Y Z
Θ
Θ = ZY
−1
γ
γ
∗
= inf
F ( Y,Z,γ)<0, Y >0
γ .
H
∞
˙x
1
= x
2
,
˙x
2
= −ω
2
0
x
1
− δx
2
+ v + u ,
z
1
= x
1
,
z
2
= u ,
A =
0 1
−ω
2
0
−δ
, B
1
= B
2
=
0
1
,
C
1
=
1 0
0 0
, D
11
=
0
0
, D
12
=
0
1
.
γ
ω
0
= 10 δ = 0.1 γ
∗
= 0.7071
Θ = (−0.005, −0.1) H
∞
u = −0.005x
1
− 0.1x
2
γ
∗
= 0.7071
H
∞
Ψ + P
T
Θ
T
Q + Q
T
ΘP < 0 ,
Ψ =
Y A
T
+ AY B
1
Y C
T
1
B
T
1
−γI D
T
11
C
1
Y D
11
−γI
,
P = (Y 0 0) , Q = (B
T
2
0 D
T
12
) .
Θ
W
T
P
Y A
T
+ AY B
1
Y C
T
1
B
T
1
−γI D
T
11
C
1
Y D
11
−γI
W
P
< 0 ,
W
T
Q
Y A
T
+ AY B
1
Y C
T
1
B
T
1
−γI D
T
11
C
1
Y D
11
−γI
W
Q
< 0 ,
W
P
=
0 0
I 0
0 I
, W
Q
=
W
(1)
Q
0
0 I
W
(2)
Q
0
,
W
(1)
Q
W
(2)
Q
B
T
2
W
(1)
Q
+ D
T
12
W
(2)
Q
= 0 .
−γI D
T
11
D
11
−γI
< 0 ,
γ
2
> λ
max
(D
T
11
D
11
)
N
2
| 0
− − −
0 | I
T
Y A
T
+ AY Y C
T
1
| B
1
C
1
Y −γI | D
11
− − − −
B
T
1
D
T
11
| −γI
N
2
| 0
− − −
0 | I
< 0 ,
H
∞
N
2
= (W
(1)
Q
, W
(2)
Q
)
(B
T
2
D
T
12
)
H
∞
γ
n
x
× n
x
Y = Y
T
> 0
Y
Θ
γ
H
∞
k
˙x
r
= A
r
x
r
+ B
r
y ,
u = C
r
x
r
+ D
r
y ,
x
r
∈ R
k
k = 0
u = D
r
y
˙x
c
= A
c
x
c
+ B
c
v ,
z = C
c
x
c
+ D
c
v ,
x
c
= (x, x
r
)
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
.
v z
H
c
(s) = D
c
+ C
c
(sI − A
c
)
−1
B
c
kH
c
k
∞
< γ .
A
T
c
X + XA
c
XB
c
C
T
c
B
T
c
X −γI D
T
c
C
c
D
c
−γI
< 0
X = X
T
> 0
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
.
Θ
Ψ + P
T
Θ
T
Q + Q
T
ΘP < 0 ,
Ψ =
A
T
0
X + XA
0
XB
0
C
T
0
B
T
0
X −γI D
T
11
C
0
D
11
−γI
,
P = (C D
21
0
(n
y
+k)×n
z
) , Q = ( B
T
X 0
(n
u
+k)×n
v
D
T
12
) .