Баландин Д.В., Коган М.М. Использование LMI toolbox пакета Matlab в синтезе законов управления
Подождите немного. Документ загружается.
A
T
X + XA + C
T
C = 0 .
A
X = X
T
≥ 0
(i) (A, C)
X = X
T
> 0
(A, C)
A
(i) (ii) A
X =
∞
Z
0
e
A
T
t
C
T
Ce
At
dt
¯
X
A
T
(X −
¯
X) + (X −
¯
X)A = 0 .
A
¯
X = X
(A, C)
x
0
6= 0 Xx
0
= 0
x
T
0
Xx
0
=
∞
Z
0
x
T
0
e
A
T
t
C
T
Ce
At
x
0
dt =
∞
Z
0
y
T
y dt ,
y
˙x = Ax ,
y = Cx
x(0) = x
0
y(t) ≡ 0
(iii) X
0
= X
T
0
≥ 0
A Ax
0
= λx
0
x
0
6= 0
λ ≥ 0
0 = x
∗
0
A
T
X
0
x
0
+ x
∗
0
X
0
Ax
0
+ x
∗
0
C
T
Cx
0
= 2( λ)x
∗
0
X
0
x
0
+ x
∗
0
C
T
Cx
0
.
Cx
0
= 0
Ax
0
= λx
0
x
0
6= 0 λ ≥ 0
(A, C)
A
T
XA − X + C
T
C = 0 .
A
X = X
T
≥ 0
(i) (A, C)
X = X
T
> 0
(A, C)
A
(i) (ii) A
X =
∞
X
t=0
(A
t
)
T
C
T
CA
t
¯
X
A
T
(X −
¯
X)A − (X −
¯
X) = 0 ,
A
¯
X = X
(A, C)
x
0
6= 0 Xx
0
= 0
x
T
0
Xx
0
=
∞
X
t=0
x
T
0
(A
t
)
T
C
T
CA
t
x
0
=
∞
X
t=0
y
T
t
y
t
,
y
t
x
t+1
= Ax
t
,
y
t
= Cx
t
x
0
y
t
≡ 0
(iii) X
0
= X
T
0
≥ 0
A Ax
0
= λx
0
x
0
6= 0
|λ| ≥ 1
0 = x
∗
0
A
T
X
0
Ax
0
− x
∗
0
X
0
x
0
+ x
∗
0
C
T
Cx
0
= (|λ|
2
− 1)x
∗
0
X
0
x
0
+ x
∗
0
C
T
Cx
0
.
Cx
0
= 0
Ax
0
= λx
0
x
0
6= 0 |λ| ≥ 1
(A, C)
(m ×n) A (p ×q)
B (mp ×nq)
A ⊗ B =
a
11
B a
12
B ··· a
1n
B
a
21
B a
22
B ··· a
2n
B
··· ··· ··· ···
a
m1
B a
m2
B ··· a
mn
B
.
AC BD
(A ⊗ B)(C ⊗ D) = AC ⊗ BD ;
(A ⊗ B)
T
= A
T
⊗ B
T
A B
A ⊗ B =
¯
A ⊗
¯
B , (A ⊗ B)
∗
= A
∗
⊗ B
∗
;
A B A ⊗B
(A ⊗ B)
−1
= A
−1
⊗ B
−1
.
S
F (x) < 0 , x 6= 0
x ∈ R
n
x
G
i
(x) ≤ 0 , i = 1, . . . , m ,
F (x) G
i
(x)
S(x) = F (x) − τ
1
G
1
(x) − . . . − τ
m
G
m
(x)
S(x) < 0 , x 6= 0
τ
i
≥ 0
S
m = 1
x
0
G
1
(x
0
) < 0
τ
1
> 0
F (x) − τ
1
G
1
(x) < 0 , x 6= 0 .
S
τ
1
= 1
τ
−1
1
F (x) < 0
τ
−1
1
F (x) < 0
m ≥ 2 S
τ
i
≥ 0, i = 1, . . . , m
(A, B)
X
2 x
∗
X(Ax + Bv) − L(x, v) < 0 , ∀x, v, |x| + |v| 6= 0
L(x, v) x, v
v 6= 0
L[(j ωI − A)
−1
Bv, v] > 0 , ∀ω ∈ (−∞, ∞) .
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 .
(A, B)
X
(Ax + Bv)
∗
X(Ax + Bv) − x
T
Xx − L(x, v) < 0 , ∀x, v, |x| + |v| 6= 0
L(x, v) x, v
v 6= 0
L[(e
j ϕ
I − A)
−1
Bv, v] > 0 , ∀ϕ ∈ [0, 2π) .
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 .