Баландин Д.В., Коган М.М. Использование LMI toolbox пакета Matlab в синтезе законов управления
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x = (x
1
, ···, x
m
)
F (x) = F
0
+ x
1
F
1
+ ··· + x
m
F
m
> 0 ,
F
0
, F
1
, ···, F
m
n ×n
F
i
= F
T
i
∈ R
n×n
, i = 0, 1, ···, m > 0
u
T
F (x)u > 0 ∀u ∈ R
n
, u 6= 0 .
F (x)
λ
min
(F (x)) > 0 λ(·)
F (x) > 0 F (x)
F (x) > 0
V
S
n
= {M| M = M
T
∈ R
n×n
}
F (x) = F
0
+ L(x) ,
F
0
∈ S
n
L(x) V −→ S
n
x ∈ V
{e
1
, ···, e
m
} V
L(x) =
m
X
j=1
x
j
F
j
,
x =
m
X
j=1
x
j
e
j
, F
j
= L(e
j
) , j = 1, ···, m ,
F (x) > 0
A
T
X + XA + Q > 0 ,
A, Q ∈ R
n×n
X ∈ R
n×n
Q V
S
n
(n ×n)
R
m
m = n(n+1)/2 {E
1
, ···, E
m
}
X =
P
m
j=1
x
j
E
j
Q +
m
X
j=1
x
j
(A
T
E
j
+ E
j
A) > 0 ,
A =
0 1
2 3
, Q =
0 0
0 0
,
X = x
1
1 0
0 0
+ x
2
0 1
1 0
+ x
3
0 0
0 1
=
x
1
x
2
x
2
x
3
x
1
0 1
1 0
+ x
2
4 3
3 2
+ x
3
0 2
2 6
> 0 .
> ≥
F (x) < 0 F (x) > G(x) F (x) G(x)
−F (x) > 0 F (x) − G(x) > 0
x F = {x| F (x) > 0}
x
1
, x
2
∈ F α ∈ [0, 1] F (x)
F (αx
1
+ (1 − α)x
2
) = αF (x
1
) + (1 − α)F (x
2
) > 0 .
F
1
(x) > 0, ···, F
k
(x) > 0 .
F (x) =
F
1
(x) 0 ··· 0
0 F
2
(x) ··· 0
0 0 ··· F
k
(x)
> 0 .
F (x)
F
1
(x), ···, F
k
(x)
F (x)
F
i
(x), i = 1, ···, k x
F (x) > 0
(
F (x) > 0
Ax = b
(
F (x) > 0
x = Ay + b
y
(
F (x) > 0
x ∈ M ,
M R
n
M = x
0
+ M
0
= {x
0
+ m| m ∈ M
0
} ,
x
0
∈ R
n
M
0
R
n
{e
1
, ···, e
k
} M
0
F (x) = F
0
+ L(x)
0 < F (x) = F
0
+ L(x
0
+
k
P
j=1
x
j
e
j
) = F
0
+ L(x
0
) +
k
P
j=1
x
j
L(e
j
) =
=
¯
F
0
+ x
1
¯
F
1
+ ··· + x
k
¯
F
k
=
¯
F (¯x) ,
¯
F
0
= F
0
+ L(x
0
)
¯
F
j
= L(e
j
) ¯x = (x
1
, ···, x
k
) x ∈ R
n
¯
F (¯x) > 0 x ¯x x = x
0
+
P
k
j=1
x
j
e
j
M
M =
M
11
M
12
M
T
12
M
22
M
11
= M
T
11
x
T
Mx = x
T
1
M
11
x
1
+ 2x
T
1
M
12
x
2
+ x
T
2
M
22
x
2
=
= (x
1
+ M
−1
11
M
12
x
2
)
T
M
11
(x
1
+ M
−1
11
M
12
x
2
) + x
T
2
(M
22
− M
T
12
M
−1
11
M
12
)x
2
,
x = (x
1
, x
2
) M
M > 0 M
11
> 0 S = M
22
− M
T
12
M
−1
11
M
12
> 0
S M
11
M
F : V −→ S
n
F (x) =
F
11
(x) F
12
(x)
F
T
12
(x) F
22
(x)
,
F
22
(x) F (x) > 0
F
22
(x) > 0 , F
11
(x) − F
12
(x)F
−1
22
(x)F
T
12
(x) > 0 .
x
x
H
∞
A
T
X + XA + XBR
−1
B
T
X + Q < 0 .
−A
T
X − XA − Q XB
B
T
X R
!
> 0 .
X
X > Y
−1
> 0
X I
I Y
> 0
X Y
x
F (x) > 0
f :
V −→ R
µ
opt
= inf
F ( x)>0
f(x) .
ε x F (x) > 0
f(x) ≤ µ
opt
+ ε ε
λ ∈ R
λF (x) − G(x) > 0 ,
F (x) > 0 ,
H(x) > 0 ,
F (x) G(x) H(x) ∈ S
n
˙x = Ax ,
A ∈ R
n×n
A
X = X
T
> 0
A
T
X + XA < 0 .
X 0
0 −A
T
X − XA
> 0 .
x
t+1
= Ax
t
A
X = X
T
> 0
A
T
XA − X < 0 .
X 0
0 X − A
T
XA
> 0 .
˙x = Ax + Bv , x(0) = 0 ,
z = Cx + Dv ,
L
2
v(t)
kvk = (
∞
Z
0
|v(t)|
2
dt)
1/2
< ∞ ,
γ
∗
= sup
kvk6=0
kzk
kvk
.
γ
A
T
X + XA XB C
T
B
T
X −γI D
T
C D −γI
< 0 .
F (x) = F
T
(x) > 0
f(x) = λ
max
(F (x))
λ
max
(F (x)) = λ
1/2
max
(F
T
(x)F (x))
λ
max
(F
T
(x)F (x)) < γ ⇐⇒ γI − F
T
(x)F (x) > 0 ⇐⇒
γI F
T
(x)
F (x) I
> 0.
¯x =
x
γ
,
¯
F (¯x) =
γI F
T
(x)
F (x) I
,
¯
f(¯x) = γ ,
¯
F (¯x)
F (x)
¯
f(¯x)
¯
F (¯x) > 0
˙x = Ax .
λ
∗
< 0
A
T
X + XA − λX < 0
X = X
T
> 0
|λ
∗
|/2
˙x = Ax + F v , x(0) = 0
v = γ
−2
Hx ,
H γ > 0
J(v) =
∞
Z
0
(x
T
Qx − γ
2
v
T
v)d t
Q = Q
T
≥ 0
Q = Q
T
≥ 0
v
∗
= γ
−2
F
T
Xx ,
X = X
T
≥ 0
A
T
X + XA + γ
−2
XF F
T
X + Q = 0 ,
A + γ
−2
F F
T
X
A
T
X + XA + γ
−2
H
T
H ≤ 0 ,
(A + γ
−2
F H)
T
X
1
+ X
1
(A + γ
−2
F H) < 0
X
1
= X
T
1
> 0 X = X
T
≥ 0
F
T
X = H ,
Ψ + P
T
Θ
T
Q + Q
T
ΘP < 0
Ψ + P
T
Θ
T
Q + Q
T
ΘP < 0 ,
Ψ (n ×n) P Q
(l × n) (k × n)
Θ (k ×l)
P Q n
Q
T
ΘP = K Θ K
K < −(1/2)Ψ
P Q n n P
Q n
Ψ = Ψ
T
∈ R
n×n
P ∈ R
l×n
Q ∈ R
k×n
P = n Q = r
Q
< n
Θ ∈ R
k×l
W
T
Q
ΨW
Q
< 0 ,
W
Q
Q
W
T
Q
W
Q
R
n
R
n
= R(Q
T
) ⊕ N(Q) ,
R(Q
T
) Q
T
N(Q) Q
Q
Q = (Q
1
0) ,
Ψ + P
T
Θ
T
Q + Q
T
ΘP < 0
Q
1
k × r
Q
k × (n − r
Q
) P Ψ
P = (P
1
P
2
) , Ψ =
Ψ
11
Ψ
12
Ψ
T
12
Ψ
22
.
W
Q
QW
Q
= 0
Q W
T
Q
= (0 I)
W
T
Q
ΨW
Q
< 0 Ψ
22
< 0
Ψ
11
+ Q
T
1
ΘP
1
+ P
T
1
Θ
T
Q
1
Ψ
12
+ Q
T
1
ΘP
2
Ψ
T
12
+ P
T
2
Θ
T
Q
1
Ψ
22
< 0 .
K = (K
1
K
2
)
Θ
Q
T
1
Θ(P
1
P
2
) = (K
1
K
2
) .
Y Z
Q
T
1
Y = K, Z(P
1
P
2
) = K .
(r
Q
× k) Q
T
1
r
Q
≤ k (l × n) (P
1
P
2
)
n K
i
, i = 1, 2
Θ
Ψ
22
Θ
Ψ = Ψ
T
∈ R
n×n
P ∈ R
l×n
Q ∈ R
k×n
P = r
P
< n Q = r
Q
< n
Θ ∈ R
k×l
W
T
P
ΨW
P
< 0 , W
T
Q
ΨW
Q
< 0 ,
W
P
P W
Q
Q
W
T
P
W
P
W
T
Q
W
Q
R
n
R
n
= {N(P )\[N(P )
T
N(Q)]} ⊕ [N(P )
T
N(Q)]⊕
⊕{N(Q)\[N(P )
T
N(Q)]} ⊕ M ,
N(P ) N(Q) P Q M N(P )⊕N(Q) R
n
P Q
P = (0 0 P
1
P
2
) , Q = (Q
1
0 0 Q
2
) .