Баландин Д.В., Коган М.М. Использование LMI toolbox пакета Matlab в синтезе законов управления
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X
12
n × k X
22
= X
T
22
> 0 k × k
X
11
− Y
−1
11
= −(I − X
11
Y
11
)Y
−1
11
r
(X
11
− Y
−1
11
) = r .
X
11
−Y
−1
11
X
11
− Y
−1
11
= (U
1
U
2
)
Σ 0
0 0
U
T
1
U
T
2
,
U
1
∈ R
n×r
U
2
∈ R
n×(n−r)
Σ = (λ
1
, ···, λ
r
) > 0
S
Σ 0
r×(k−r)
0
(k−r)×r
I
k−r
S
T
, S = (U
1
U
2
)
I
r
0
r×(k−r)
0
(n−r)×r
0
(n−r)×(k−r)
,
X
12
= S , X
22
= (λ
−1
1
, ···, λ
−1
r
, 1, ···, 1) .
Y
11
Y
12
Y
22
X
22
> 0
X
11
− X
12
X
−1
22
X
T
12
= Y
−1
11
> 0 ,
X > 0
Q = Q
T
∈ R
n×n
A ∈ R
n×m
A <
n
x
T
Qx < 0 , ∀x ∈ {x|A
T
x = 0} ;
∃µ > 0 Q − µAA
T
< 0 .
(C1 −→ C2) R
n
R
n
= N(A
T
) ⊕ R(A) ,
N(A
T
) A
T
R(A) A
Q AA
T
Q =
Q
11
Q
12
Q
T
12
Q
22
, Q
11
< 0 , AA
T
=
0 0
0 DD
T
, DD
T
> 0 .
x = (x
1
, x
2
)
x
T
(Q − µAA
T
)x = (x
T
1
x
T
2
)
Q
11
Q
12
Q
12
Q
22
− µDD
T
x
1
x
2
=
= x
T
1
Q
11
x
1
+ 2x
T
1
Q
12
x
2
+ x
T
2
(Q
22
− µDD
T
)x
2
=
= (x
1
+ Q
−1
11
Q
12
x
2
)
T
Q
11
(x
1
+ Q
−1
11
Q
12
x
2
)+
+x
T
2
(Q
22
− Q
T
12
Q
−1
11
Q
12
− µDD
T
)x
2
< 0 ,
µ Q
22
− Q
T
12
Q
−1
11
Q
12
− µDD
T
< 0 µ > λ
max
[D
−1
(Q
22
−
Q
T
12
Q
−1
11
Q
12
)D
−T
]
(C2 −→ C1) x
T
(Q − µAA
T
)x = x
T
Qx < 0 x
A
T
x = 0
AX = C ,
A C (m × n) (m × q)
X (n ×q)
X
0
X
0
= C = r
C
X
0
= V C ,
V
r
C
C
C
C = (C
1
C
2
) , C
2
= C
1
D
D X = (X
1
X
2
) X
1
∈ R
n×r
C
X
1
AX
1
=
C
1
C
1
¯
X
2
= X
1
D A
¯
X
2
= C
2
X
0
= (X
1
¯
X
2
)
AX
0
= C C
X
0
C = X
0
= r
C
X
0
C X
0
= V C
V
AXB = C
X
AY = C , ZB = C
Y Z
X Y =
XB Z = AX
Y, Z Y
0
r
C
Y
0
= V C
C = AY
0
= AV C = AV ZB
X = V Z
Ax = b ,
A (m ×n) b ∈ R
m
x ∈ R
n
m = n = A N(A) = N(A
T
) = {0}
x = A
−1
b
A N(A) N(A
T
)
m > n A = n A b ∈
R(A)
A
T
A
T
A
x = (A
T
A)
−1
A
T
b .
b 6∈ R(A)
ˆx = min kAx − bk
2
.
b = b
R(A)
+ b
N (A
T
)
b
R(A)
∈ R(A) b
N (A
T
)
∈ N(A
T
)
b
N (A
T
)
⊥R(A)
kAx − b
R(A)
− b
N (A
T
)
k
2
= kAx − b
R(A)
k
2
+ kb
N (A
T
)
k
2
.
min kAx − bk
2
= kb
N (A
T
)
k
2
,
Ax = b
R(A)
x = (A
T
A)
−1
A
T
b
R(A)
(A
T
A)
−1
A
T
b
R(A)
= (A
T
A)
−1
A
T
b
ˆx = (A
T
A)
−1
A
T
b ,
m < n A = m A
¯x ¯x + x
N (A)
x ∈ N(A)
˘x = min
{Ax=b}
kxk .
R
n
R
n
= R(A
T
) ⊕ N(A) ,
N(A) x = x
R(A
T
)
+ x
N (A)
Ax
R(A
T
)
= b x
R(A
T
)
∈ R(A
T
) ξ ∈ R
m
x
R(A
T
)
= A
T
ξ AA
T
ξ = b ξ
AA
T
ξ = (AA
T
)
−1
b
x
R(A
T
)
= A
T
(AA
T
)
−1
b
x = A
T
(AA
T
)
−1
b .
A = r < min(m, n)
A
A = (U
1
U
2
)
Σ 0
0 0
V
T
1
V
T
2
= U
1
ΣV
T
1
,
Σ = (σ
1
, ···, σ
r
) , σ
1
≥ ··· ≥ σ
r
> 0 ,
R(A) = (U
1
) , N(A
T
) = (U
2
) ,
R(A
T
) = (V
1
) , N(A) = (V
2
) ,
U
T
1
U
T
2
(U
1
U
2
) = I ,
V
T
1
V
T
2
(V
1
V
2
) = I .
R
n
R
m
R
n
= R(A
T
) ⊕ N(A) , R
m
= R(A) ⊕ N(A
T
)
x = x
R(A
T
)
+ x
N (A)
, b = b
R(A)
+ b
N (A
T
)
.
kAx − bk
2
= kAx − b
R(A)
k
2
+ kb
N (A
T
)
k
2
=
= kAx
N (A)
+ Ax
R(A
T
)
− b
R(A)
k
2
+ kb
N (A
T
)
k
2
=
= kAx
R(A
T
)
− b
R(A)
k
2
+ kb
N (A
T
)
k
2
,
Ax = b
R(A)
,
˜x R(A
T
)
b
R(A)
= U
1
η
1
η
1
∈ R
r
b
N (A
T
)
= U
2
η
2
η
2
∈ R
m−r
b = U
1
η
1
+ U
2
η
2
U
T
1
U
1
= I U
T
1
U
2
= 0 η
1
= U
T
1
b
b
R(A)
= U
1
U
T
1
b ˜x ∈ R(A
T
) ˜x = V
1
ξ ξ ∈ R
r
AV
1
ξ = U
1
U
T
1
b .
A = U
1
ΣV
T
1
U
T
1
ξ =
Σ
−1
U
T
1
b
˜x = V
1
Σ
−1
U
T
1
b .
A
+
= V
1
Σ
−1
U
T
1
A
A A
+
AA
+
A = A , A
+
AA
+
= A
+
, (AA
+
)
T
= AA
+
, (A
+
A)
T
= A
+
A .
m = n = A −→ A
+
= A
−1
;
m > n, A = n −→ A
+
= (A
T
A)
−1
A
T
;
m < n, A = m −→ A
+
= A
T
(AA
T
)
−1
.
x = A
+
b .
˙x = Ax + Bu , x ∈ R
n
x
, u ∈ R
n
u
(A, B)
t
Z
0
e
Aτ
BB
T
e
A
T
τ
dτ
t > 0
(B AB . . . A
n
x
−1
B) = n
x
;
(sI − A B) = n
x
, ∀s ∈ C;
n
x
Θ
A + BΘ
A e ∈
C
n
x
e
∗
A = λe
∗
λ ∈ C e
∗
B 6= 0
(A, B)
(sI − A B) = n
x
, s ≥ 0;
Θ A + BΘ
e
∗
A = λe
∗
e
∗
B = 0 λ < 0
˙x = Ax ,
y = Cx , x ∈ R
n
x
, y ∈ R
n
y
(A, C)
t
Z
0
e
A
T
τ
C
T
Ce
Aτ
dτ
t > 0
(C
T
A
T
C
T
. . . (A
T
)
n
x
−1
C
T
) = n
x
;
(sI − A
T
C
T
) = n
x
, ∀s ∈ C;
n
x
Θ A + ΘC
A e ∈
C
n
x
Ae = λe λ ∈ C Ce 6= 0
(A, C)
(sI − A
T
C
T
) = n
x
, s ≥ 0;
Θ A + ΘC
Ae = λe Ce = 0 λ < 0