Баландин Д.В., Коган М.М. Использование LMI toolbox пакета Matlab в синтезе законов управления
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x
t+1
= Ax
t
+ Bu
t
,
y
t
= Cx
t
,
x
t
∈ R
n
x
u
t
∈ R
n
u
y
t
∈ R
n
y
k
x
(r)
t+1
= A
r
x
(r)
t
+ B
r
y
t
,
u
t
= C
r
x
(r)
t
+ D
r
y
t
,
x
(r)
t
∈ R
k
¯x
t+1
= A
c
¯x
t
, A
c
=
A + BD
r
C BC
r
B
r
C A
r
,
¯x
t
= (x
t
, x
(r)
t
)
V
t
(¯x
t
) = ¯x
T
t
X ¯x
t
X
T
= X > 0
V
t+1
− V
t
= ¯x
T
t+1
X ¯x
t+1
− ¯x
T
t
X ¯x
t
= ¯x
T
t
(A
T
c
XA
c
− X)¯x
t
< 0 .
A
T
c
XA
c
− X < 0 ,
−X
−1
A
c
A
T
c
−X
< 0 .
Θ =
A
r
B
r
C
r
D
r
A
c
= A
0
+ B
0
ΘC
0
,
A
0
=
A 0
0 0
, B
0
=
0 B
I 0
, C
0
=
0 I
C 0
,
Θ
Ψ + P
T
Θ
T
Q + Q
T
ΘP < 0 ,
Ψ =
−X
−1
A
0
A
T
0
−X
, P = (0 C
0
) , Q =
B
T
0
0
.
W
P
=
0 I
W
C
0
0
, W
Q
=
W
B
T
0
0
0 I
,
(n
x
+k)×
(n
x
+ k) X = X
T
> 0
0 I
W
C
0
0
T
−X
−1
A
0
A
T
0
−X
0 I
W
C
0
0
< 0 ,
W
B
T
0
0
0 I
T
−X
−1
A
0
A
T
0
−X
W
B
T
0
0
0 I
< 0 .
X
Y = X
−1
X Y
0 I
W
C
0
0
T
−Y A
0
A
T
0
−X
0 I
W
C
0
0
< 0 ,
W
B
T
0
0
0 I
T
−Y A
0
A
T
0
−X
W
B
T
0
0
0 I
< 0 .
X Y
XY = I
B
0
=
0
n
x
×k
B
I
k
0
k×n
u
, C
0
=
0
k×n
x
I
k
C 0
n
y
×k
W
B
T
0
W
C
0
B
T
0
W
B
T
0
=
0 C
0
W
C
0
= 0
W
B
T
0
=
W
B
T
0
, W
C
0
=
W
C
0
.
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
W
T
C
(A
T
X
11
A − X
11
)W
C
< 0
W
T
B
T
(AY
11
A
T
− Y
11
)W
B
T
< 0 .
(n
x
×n
x
)
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 ,
A
r
=
0 B
r
= 0 C
r
= 0 Θ = D
r
Ψ =
−X
−1
A
A
T
−X
, P = (0 C) , Q =
B
T
0
.
W
T
C
(A
T
XA − X)W
C
< 0 ,
W
T
B
T
(AX
−1
A
T
− X
−1
)W
B
T
< 0 ,
X (n
x
× n
x
)
C = I (0 C) =
n
x
Ψ =
−X
−1
A
A
T
−X
, P = (0 I) , Q =
B
T
0
.
I
0
T
−X
−1
A
A
T
−X
I
0
< 0 ,
W
B
T
0
0 I
T
−X
−1
A
A
T
−X
W
B
T
0
0 I
< 0 ,
X
−1
> 0
W
T
B
T
(AX
−1
A
T
− X
−1
)W
B
T
< 0 .
W
T
B
T
(AY A
T
− Y )W
B
T
< 0
Y = Y
T
LMI
D
˙x = Ax D
A D A
D D
D
A
X
AX + XA
T
< 0 , X > 0 .
(m × m)
f(z) = α + zβ + ¯zβ
T
,
α = α
T
∈ R
m×m
β ∈ R
m×m
D = {z ∈ C : f(z) < 0}
LMI f(z)
LMI
x = (z) y = (z) LMI
z ∈ D
f(¯z) = f(z) < 0 ,
LMI
LMI
f(z) (m × m)
M(A, X) = α ⊗ X + β ⊗ (AX) + β
T
⊗ (AX)
T
,
⊗
M
ij
(A, X) = α
ij
X + β
ij
AX + β
ji
XA
T
, i, j = 1, ···, m .
M(A, X) f(z)
(X, AX, XA
T
) ↔ (1, z, ¯z) .
D LMI A D
X = X
T
M(A, X) < 0 , X > 0 .
λ v
A v
∗
A = λv
∗
(I ⊗ v)
∗
M(A, X)(I ⊗ v) = (v
∗
Xv)f(λ) .
X > 0 f(λ) < 0 λ ∈ D
A = (λ
1
, ···, λ
n
) D
M(A, X)
A X = X
∗
M(A, X) = α ⊗ X + β ⊗ (AX) + β
T
⊗ (AX)
∗
.
M(A, I) = U
T
(f(λ
1
), ···, f(λ
n
))U ,
U f(λ
i
) < 0, i = 1, ···, n
M(A, X) < 0 X = I A
LMI
D
1
= {z : z < −µ}
µ
f
1
(z) = z + ¯z + 2µ A
µ
X = X
T
AX + XA
T
+ 2µX < 0 , X > 0 .
LMI D
2
= {z : |z + q| < r}
r (−q, 0)
f
2
(z) =
−r q + z
q + ¯z −r
< 0 ,
−rX qX + AX
qX + XA
T
−rX
< 0 , X > 0 .
D
3
= {z : −µ
2
< z < −µ
1
}
f
3
(z) =
(z + ¯z) + 2µ
1
0
0 −(z + ¯z) − 2µ
2
AX + XA
T
+ 2µ
1
X 0
0 −AX − XA
T
− 2µ
2
X
< 0 , X > 0 ,
D
4
= {z : z < 0 , −ν < z < ν}
f
4
(z) =
−2ν z − ¯z
−(z − ¯z) −2ν
−2νX AX − XA
T
−AX + XA
T
−2νX
< 0 , X > 0 .
D
5
= {z : z ϕ < −| z|}
f
5
=
(z + ¯z) sin ϕ (z − ¯z) cos ϕ
−(z − ¯z) cos ϕ (z + ¯z) sin ϕ
(AX + XA
T
) sin ϕ (AX − XA
T
) cos ϕ
(XA
T
− AX) cos ϕ (AX + XA
T
) sin ϕ
< 0 , X > 0 .
LMI
LMI LMI
D
1
T
D
2
LMI D
1
D
2
f
1
f
2
f = (f
1
, f
2
)
M(A, X) = (M
1
(A, X), M
2
(A, X)) .
A
LMI D
1
D
2
X
A D
1
D
2
X = X
T
> 0 M
1
(A, X) < 0 M
2
(A, X) < 0
LMI
˙x = Ax + Bu ,
x ∈ R
n
x
u ∈ R
n
u
u = Θx ,
Θ
D
LMI
X = X
T
> 0
Θ M(A + BΘ, X) < 0 Z = ΘX
M(A, X) + β ⊗ BZ + β
T
⊗ Z
T
B
T
< 0
X Z
Θ = ZX
−1
˙x
1
= x
3
,
˙x
2
= x
4
,
˙x
3
= 2x
1
− x
2
+ u ,
˙x
4
= −2x
1
+ 2x
2
.
z = −1
0.1
LMI D
2
−0.1X X + AX + BZ
X + XA
T
+ Z
T
B
T
−0.1X
< 0 , X > 0 ,
A =
0 0 1 0
0 0 0 1
2 −1 0 0
−2 2 0 0
, B =
0
0
1
0
.
Θ = (−10.0067, 9.5078, −4.0023, 6.0056) ,
λ
1,2
= −0.9890 ± 0.0111i , λ
3
= −1.0104 , λ
4
= −1.0140 .
