Баландин Д.В., Коган М.М. Использование LMI toolbox пакета Matlab в синтезе законов управления
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H
∞
˙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
∞
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 ,
H
∞
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 ,
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
) .
H
∞
Θ
W
T
P
A
T
0
X + XA
0
XB
0
C
T
0
B
T
0
X −γI D
T
11
C
0
D
11
−γI
W
P
< 0 ,
W
T
Q
A
T
0
X + XA
0
XB
0
C
T
0
B
T
0
X −γI D
T
11
C
0
D
11
−γI
W
Q
< 0 ,
W
P
N(P ) P
W
Q
N(Q) Q
Q = (B
T
X 0 D
T
12
) = R
X 0 0
0 I 0
0 0 I
, R = (B
T
0
(n
u
+k)×n
v
D
T
12
) ,
W
Q
=
X
−1
0 0
0 I 0
0 0 I
W
R
.
H
∞
k γ
(n
x
+ k) × (n
x
+ k) X =
X
T
> 0
W
T
P
A
T
0
X + XA
0
XB
0
C
T
0
B
T
0
X −γI D
T
11
C
0
D
11
−γI
W
P
< 0 ,
W
T
R
X
−1
A
T
0
+ A
0
X
−1
B
0
X
−1
C
T
0
B
T
0
−γI D
T
11
C
0
X
−1
D
11
−γI
W
R
< 0 .
X Θ
Y = X
−1
X Y
W
T
P
A
T
0
X + XA
0
XB
0
C
T
0
B
T
0
X −γI D
T
11
C
0
D
11
−γI
W
P
< 0 ,
W
T
R
Y A
T
0
+ A
0
Y B
0
Y C
T
0
B
T
0
−γI D
T
11
C
0
Y D
11
−γI
W
R
< 0 .
X Y XY = I
H
∞
˙x
1
= x
2
,
˙x
2
= −ω
2
0
x
1
− δx
2
+ v + u ,
z
1
= x
1
,
z
2
= u ,
y = x
1
A, B
1
, B
2
, C
1
, D
11
, D
12
C
2
= (1 0) , D
21
= 0 .
(k = 0)
u = Θy .
Θ
Ψ =
A
T
X + XA XB
1
C
T
1
B
T
1
X −γI 0
C
1
0 −γI
,
P = (C
2
D
21
0) , Q = (B
T
2
X 0 D
T
12
) ,
X
ω
0
= 10 δ = 0.1 γ
∗
=
1.001 Θ = 0.0390 γ
γ
∗
= 1
u = 0
˙x
r
= −0.9572x
r
− 103.25y ,
u = −0.092x
r
− 0.0945y
H
∞
γ
∗
= 0.708
γ
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
I
k
0
k×n
v
0
k×n
z
C
2
0
n
y
×k
D
21
0
n
y
×n
z
,
R =
0
k×n
x
I
k
0
k×n
v
0
k×n
z
B
T
2
0
n
u
×k
0
n
u
×n
v
D
T
12
,
W
P
W
R
W
P
=
W
(1)
P
0
0 0
W
(2)
P
0
0 I
, W
R
=
W
(1)
R
0
0 0
0 I
W
(2)
R
0
,
W
(1)
P
W
(2)
P
W
(1)
R
W
(2)
R
C
2
W
(1)
P
+ D
21
W
(2)
P
= 0 , B
T
2
W
(1)
R
+ D
T
12
W
(2)
R
= 0 .
W
(1)
P
0
0 0
W
(2)
P
0
0 I
T
A
T
X
11
+ X
11
A A
T
X
12
X
11
B
1
C
T
1
? 0 X
T
12
B
1
0
? ? −γI D
T
11
? ? ? −γI
W
(1)
P
0
0 0
W
(2)
P
0
0 I
,
W
(1)
R
0
0 0
0 I
W
(2)
R
0
T
Y
11
A
T
+ AY
11
AY
12
B
1
Y
11
C
T
1
? 0 0 Y
T
12
C
T
1
? ? −γI D
T
11
? ? ? −γI
W
(1)
R
0
0 0
0 I
W
(2)
R
0
,
X
11
Y
11
X Y
N
1
| 0
− − −
0 | I
T
A
T
X
11
+ X
11
A X
11
B
1
| C
T
1
B
T
1
X
11
−γI | D
T
11
− − − −
C
1
D
11
| −γI
N
1
| 0
− − −
0 | I
< 0 ,
N
2
| 0
− − −
0 | I
T
Y
11
A
T
+ AY
11
Y
11
C
T
1
| B
1
C
1
Y
11
−γI | D
11
− − − −
B
T
1
D
T
11
| −γI
N
2
| 0
− − −
0 | I
< 0 ,
N
1
= (W
(1)
P
, W
(2)
P
) N
2
= (W
(1)
R
, W
(2)
R
)
(C
2
D
21
) (B
T
2
D
T
12
)
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 > 0 Y > 0 X
11
Y
11
H
∞
k
(n
x
× n
x
) X
11
= X
T
11
> 0 Y
11
= Y
T
11
> 0
D
11
= 0
v
γ > 0
H
∞
(k = n
x
)
γ
H
∞
˙x
(1)
r
= 0.0592x
(1)
r
− 0.8358x
(2)
r
− 24.76y ,
˙x
(2)
r
= 1.0667x
(1)
r
− 0.8604x
(2)
r
+ 20.24y ,
u = −0.2586x
(1)
r
+ 0.1836x
(2)
r
− 0.0389y
γ
∗
= 0.7072
γ
x
t+1
= Ax
t
+ Bv
t
,
z
t
= Cx
t
+ Dv
t
,
x
t
∈ R
n
x
v
t
∈ R
n
v
z
t
∈ R
n
z
A
l
2
v
t
kvk = (
∞
X
t=0
|v
t
|
2
)
1/2
< ∞ .
γ
∗
= sup
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
ϕ∈[0,2π)
kH(e
j ϕ
)k = kHk
∞
,
H(q) = D + C(qI − A)
−1
B v
z j =
√
−1 k · k
kHk = max
i
σ
i
(H) ,
σ
i
i H k · k
∞
∞ H(q)
sup
|q|≥1
kH(q)k < ∞
kHk
∞
< γ
H
T
(e
−j ϕ
)H(e
jϕ
) < γ
2
I , ∀ϕ ∈ [0, 2π) .
γ
H(e
j ϕ
), ϕ ∈ [0, 2π)
γ
V
t+1
− V
t
− L(x
t
, v
t
) < 0 , ∀x
t
, v
t
, |x
t
| + |v
t
| 6= 0
L(x, v) x ∈ C
n
x
v ∈ C
n
v
V
t
=
V (x
t
) = x
T
t
Xx
t
X
X = X
∗
(Ax
t
+ Bv
t
)
∗
X(Ax
t
+ Bv
t
) − x
T
t
Xx
t
− L(x
t
, v
t
) < 0 , ∀x
t
, v
t
, |x
t
| + |v
t
| 6= 0
L[(e
j ϕ
I − A)
−1
Bv
t
, v
t
] > 0 , ∀ϕ ∈ [0, 2π) , ∀|v
t
| 6= 0 .
L(x, v) = (x
T
, v
T
) L
x
v
, L =
L
11
L
12
L
T
12
L
22
,
A
T
XA − X − L
11
A
T
XB − L
12
B
T
XA − L
T
12
B
T
XB − L
22
< 0 ,
(e
−j ϕ
I − A)
−1
B
I
T
L
11
L
12
L
T
12
L
22
(e
j ϕ
I − A)
−1
B
I
> 0 .
L(x, v) = γ
2
v
∗
v − (Cx + Dv)
∗
(Cx + Dv)