Баландин Д.В., Коган М.М. Использование LMI toolbox пакета Matlab в синтезе законов управления
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γ
Y
I
0
(I 0) γI
> 0 .
Θ =
A
r
B
r
C
r
D
r
A
c
= A
0
+ BΘC , C
c
= C
0
+ DΘC ,
A
0
=
A 0
n
x
×k
0
k×n
x
0
k×k
, B =
0
n
x
×k
B
I
k
0
k×n
u
,
C =
0
k×n
x
I
k
C
2
0
n
y
×k
, C
0
= (C
1
0
n
z
×k
) ,
D = (0
n
z
×k
D) .
Θ
Ψ + P
T
Θ
T
Q + Q
T
ΘP < 0 ,
Ψ =
A
0
Y + Y A
T
0
Y C
T
0
C
0
Y −γI
,
P = (CY 0) , Q = (B
T
D
T
) .
Θ
W
T
P
A
0
Y + Y A
T
0
Y C
T
0
C
0
Y −γI
W
P
< 0 ,
W
T
Q
A
0
Y + Y A
T
0
Y C
T
0
C
0
Y −γI
W
Q
< 0 ,
W
P
P W
Q
Q
P = (CY 0) = G
Y 0
0 I
, G = (C 0) ,
W
P
=
Y
−1
0
0 I
W
G
.
γ k
(n
x
+ k) × (n
x
+ k)
X = X
T
> 0 Y = Y
T
> 0
W
T
G
A
T
0
X + XA
0
C
T
0
C
0
−γI
W
G
< 0 ,
W
T
Q
Y A
0
+ A
T
0
Y Y C
T
0
C
0
Y −γI
W
Q
< 0 ,
Y
I
0
(I 0) γI
> 0 .
Y Θ
X Y XY = I
A
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
.
G =
0
k×n
x
I
k
0
k×n
z
C
2
0
n
y
×k
0
n
y
×n
z
, Q =
0
k×n
x
I
k
0
k×n
z
B
T
0
n
u
×k
D
T
,
W
G
W
Q
W
G
=
W
C
2
0
0 0
0 I
, W
Q
=
W
(1)
Q
0
W
(2)
Q
,
W
(1)
Q
W
(2)
Q
B
T
W
(1)
Q
+ D
T
W
(2)
Q
= 0 .
γ
W
C
2
0
0 0
0 I
T
A
T
X
11
+ X
11
A A
T
X
12
C
T
1
? 0 0
? ? −γI
W
C
2
0
0 0
0 I
,
W
(1)
Q
0
W
(2)
Q
T
Y
11
A
T
+ AY
11
AY
12
Y
11
C
T
1
? 0 Y
T
12
C
T
1
? ? −γI
W
(1)
Q
0
W
(2)
Q
,
X
11
Y
11
X
Y
W
C
2
0
0 I
T
A
T
X
11
+ X
11
A C
T
1
C
1
−γI
W
C
2
0
0 I
< 0 ,
N
T
Y
11
A
T
+ AY
11
Y
11
C
T
1
C
1
Y
11
−γI
N < 0 ,
W
C
2
N = (W
(1)
Q
, W
(2)
Q
) C
2
(B
T
D
T
)
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
X
11
< γI .
Y
11
Y
12
I
Y
T
12
Y
22
0
I 0 γI
> 0 ,
Y
22
> 0 ,
Y
11
− Y
12
Y
−1
22
Y
T
12
I
I γI
> 0 .
Y > 0
X
11
Y
11
X
−1
11
I
I γI
> 0 ,
X
11
< γI
γ k
(n
x
× n
x
)
X
11
= X
T
11
> 0 Y
11
= Y
T
11
> 0
γ (k = n
x
)
γ
(n
x
×n
x
) X
11
= X
T
11
> 0
Y
11
= Y
T
11
> 0
W
C
2
0
0 I
T
A
T
X
11
+ X
11
A C
T
1
C
1
−γI
W
C
2
0
0 I
< 0 ,
N
T
Y
11
A
T
+ AY
11
Y
11
C
T
1
C
1
Y
11
−γI
N < 0 ,
X
11
< γI ,
X
11
I
I Y
11
≥ 0 ,
W
C
2
N C
2
(B
T
D
T
)
Y
11
Y
Θ Y
12
Y
22
Y
11
− X
−1
11
= Y
12
Y
−1
22
Y
T
12
,
Y
12
= Y
22
= V > 0
V = Y
11
− X
−1
11
γ
˙x
1
= x
2
,
˙x
2
= −ω
2
0
x
1
− δx
2
+ u ,
z
1
= x
1
,
z
2
= u ,
y = x
1
γ
A, B, D
C
1
=
1 0
0 0
, C
2
= (1 0) .
γ ω
0
= 10 δ = 0.1
γ = 2.0354
X
11
=
2.0354 0.0011
0.0011 1.5211
, Y
11
=
0.4914 −0.0593
−0.0593 49.1470
,
γ
˙x
r
=
−57.8622 −0.0702
−0.0671 −0.1424
x
r
+
0.1201
−98.6673
y ,
u = (−0.0255 − 0.0420)x
r
− 0.0050y .
˙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
∗
= 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
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
,
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 .