Баландин Д.В., Коган М.М. Синтез законов управления на основе линейных матричных неравенств
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H
∞
˙x = Ax + Bu ,
x ∈ R
n
x
u ∈ R
n
u
u = Θx ,
x = 0
Θ
u = (Θ + ∆Θ)x , ∆Θ = F Ω(t)E
∆Θ
Ω(t)
Ω
T
(t)Ω(t) ≤ ρ
2
I ,
F E ρ > 0
˙x = (A + BΘ + BF Ω(t)E)x .
˙x = (A + BΘ)x + BF v
∆
z
∆
= Ex
v
∆
z
∆
|v
∆
|
2
≤ ρ
2
|z
∆
|
2
.
v
∆
= Ω(t)z
∆
V (x) = x
T
Xx
˙
V < 0
x v
∆
S
V (x) = x
T
Xx X = X
T
> 0
x v
∆
˙
V + ρ
2
|z
∆
|
2
− |v
∆
|
2
< 0 , |x|
2
+ |v
∆
|
2
6= 0 .
2x
T
X[(A + BΘ)x + BF v
∆
] + ρ
2
|Ex|
2
− |v
∆
|
2
< 0 ,
(A + BΘ)
T
X + X(A + BΘ) + ρ
2
E
T
E XBF
F
T
B
T
X −I
< 0 ,
(A + BΘ)
T
X + X(A + BΘ) E
T
XBF
E −ηI 0
F
T
B
T
X 0 −I
< 0 ,
η = ρ
−2
> 0
X Θ
Ψ + P
T
Θ
T
Q + Q
T
ΘP < 0 ,
Ψ =
A
T
X + XA E
T
XBF
E −ηI 0
F
T
B
T
X 0 −I
,
P = (I 0 0) , Q = (B
T
X 0 0) ,
X Θ
W
P
=
0 0
I 0
0 I
, W
Q
=
X
−1
W
B
T
0 0
0 I 0
0 0 I
,
−ηI 0
0 −I
< 0 ,
W
T
B
T
(X
−1
A
T
+ AX
−1
)W
B
T
W
T
B
T
X
−1
E
T
0
EX
−1
W
B
T
−ηI 0
0 0 −I
< 0 .
L(Y, η) =
W
B
T
0
0 I
T
Y A
T
+ AY Y E
T
EY −ηI
W
B
T
0
0 I
< 0
Y = Y
T
> 0 η > 0
ρ = η
−1/2
Θ
X = Y
−1
W
T
B
T
(Y A
T
+ AY )W
B
T
< 0 , Y > 0 ,
η
L(Y, η) < 0
X = Y
−1
η
Θ
η > 0
Θ
˙x = Ax + Bu ,
y = Cx ,
x ∈ R
n
x
u ∈ R
n
u
y ∈ R
n
y
Θ =
A
r
B
r
C
r
D
r
k
˙x
r
= A
r
x
r
+ B
r
y ,
u = C
r
x
r
+ D
r
y ,
x
r
∈ R
k
˙x
r
= (A
r
+ ∆A
r
)x
r
+ (B
r
+ ∆B
r
)y ,
u = (C
r
+ ∆C
r
)x
r
+ (D
r
+ ∆D
r
)y
∆Θ =
∆A
r
∆B
r
∆C
r
∆D
r
= F Ω(t)E ,
Ω(t) l × m
Ω
T
(t)Ω(t) ≤ ρ
2
I ,
F E ρ > 0
˙x
c
= (A
c
+ F
c
F Ω(t)EE
c
)x
c
,
x
c
= (x, x
r
)
A
c
=
A + BD
r
C BC
r
B
r
C A
r
!
, F
c
=
0
n
x
×k
B
I
k×k
0
k×n
u
!
,
E
c
=
0
k×n
x
I
k×k
C 0
n
y
×k
!
.
˙x
c
= A
c
x
c
+ F
c
F v
∆
z
∆
= EE
c
x
c
v
∆
z
∆
|v
∆
|
2
≤ ρ
2
|z
∆
|
2
.
v
∆
= Ω(t)z
∆
V (x
c
) = x
T
c
Xx
c
˙
V + ρ
2
|z
∆
|
2
− |v
∆
|
2
< 0 , |x
c
|
2
+ |v
∆
|
2
6= 0 .
A
c
T
X + XA
c
+ ρ
2
E
T
c
E
T
EE
c
XF
c
F
F
T
F
T
c
X −I
< 0
A
T
c
X + XA
c
E
T
c
E
T
XF
c
F
EE
c
−ηI
m×m
0
F
T
F
T
c
X 0 −I
l×l
< 0 ,
η = ρ
−2
A
c
A
c
= A
0
+ F
c
ΘE
c
, A
0
=
A 0
n
x
×k
0
k×n
x
0
k×k
!
,
Ψ + P
T
Θ
T
Q + Q
T
ΘP < 0 ,
Ψ =
A
T
0
X + XA
0
E
T
c
E
T
XF
c
F
EE
c
−ηI
m×m
0
F
T
F
T
c
X 0 −I
l×l
,
P = (E
c
0
(n
y
+k)×m
0
(n
y
+k)×l
) ,
Q = (F
T
c
X 0
(n
u
+k)×m
0
(n
u
+k)×l
) .
W
P
=
W
E
c
0 0
0 I
m×m
0
0 0 I
l×l
, W
Q
=
X
−1
W
F
T
c
0 0
0 I
m×m
0
0 0 I
l×l
,
W
T
E
c
(A
T
0
X + XA
0
)W
E
c
0 W
T
E
c
XF
c
F
0 −ηI 0
F
T
F
T
c
XW
E
c
0 −I
< 0 ,
W
T
F
T
c
(X
−1
A
T
0
+ A
0
X
−1
)W
F
T
c
W
T
F
T
c
X
−1
E
T
c
E
T
0
EE
c
X
−1
W
F
T
c
−ηI 0
0 0 −I
< 0 .
η
L
1
(X) =
W
E
c
0
0 I
T
A
T
0
X + XA
0
XF
c
F
F
T
F
T
c
X −I
W
E
c
0
0 I
< 0 ,
L
2
(Y, η) =
W
F
T
c
0
0 I
T
Y A
T
0
+ A
0
Y Y E
T
c
E
T
EE
c
Y −ηI
W
F
T
c
0
0 I
< 0
X = X
T
> 0
Y = Y
T
> 0 XY = I
ρ = η
−1/2
Θ
˙x
1
= x
2
,
˙x
2
= x
1
+ u ,
y = x
1
.
Θ =
−0.1590 −0.2757
−0.5136 −2.3414
.
˙x
r
= −0.1590x
r
− 0.2757y ,
u = −0.5136x
r
− 2.3414y ,
λ
1,2
= −0.0527 ± 1.1545i , λ
3
= −0.0537 .
˙x
r
= (−0.1590 + ∆A
r
)x
r
+ (−0.2757 + ∆B
r
)y ,
u = (−0.5136 + ∆C
r
)x
r
+ (−2.3414 + ∆ D
r
)y ,
∆A
r
(t) ∆B
r
(t) ∆C
r
(t) ∆D
r
(t)
|∆A
r
(t)| ≤ ρ, |∆B
r
(t)| ≤ ρ, |∆C
r
(t)| ≤ ρ, |∆D
r
(t)| ≤ ρ,
ρ > 0
∆A
r
∆B
r
∆C
r
∆D
r
=
1 1 0 0
0 0 1 1
Ω
1
(t) 0 0 0
0 Ω
2
(t) 0 0
0 0 Ω
3
(t) 0
0 0 0 Ω
4
(t)
1 0
0 1
1 0
0 1
η
A
c
Θ
η = 1682
ρ = η
−1/2
= 0.0244
∆Θ =
0.08 0.08
−0.08 −0.08
,
0.08
ρ
1
Θ =
−6.1618 8.4953
8.5164 −24.2208
.
˙x
r
= −6.1618x
r
+ 8.4953y ,
u = 8.5164x
r
− 24.2208y ,
λ
1,2
= −0.8388 ± 3.8779i , λ
3
= −4.4962 .
H
∞
