Баландин Д.В., Коган М.М. Синтез законов управления на основе линейных матричных неравенств
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µ
Q = {Q ∈ ∆ : Q
∗
Q = I
n
}
L = { (L
1
, . . . , L
r
, l
1
I
m
1
, . . . , l
f
I
m
f
) :
L
i
∈ C
k
i
×k
i
, L
i
= L
∗
i
> 0, l
j
> 0} .
∆ ∈ ∆ Q ∈ Q L ∈ L
Q
∗
∈ Q , Q∆ ∈ ∆ , ∆Q ∈ ∆ ,
σ
max
(Q∆) = σ
max
(∆Q) = σ
max
(∆) , L∆ = ∆L .
(I − M∆) = (I − ML
−1
L∆) =
= (I − ML
−1
∆L) = (I − LML
−1
∆) ,
(I − MR) =
I R
M I
= (I − RM) .
µ
∆
(MQ) = µ
∆
(QM) = µ
∆
(M) = µ
∆
(LML
−1
) ,
max
Q∈Q
ρ(QM) ≤ µ
∆
(M) ≤ inf
L∈L
σ
max
(LML
−1
) .
σ
max
(LML
−1
) < γ
LML
−2
M
∗
L − γ
2
I < 0 ,
ML
−2
M
∗
− γ
2
L
−2
< 0 .
−γ
2
L
−2
M
M
∗
−L
2
< 0 ,
M
∗
L
2
M − γ
2
L
2
< 0 .
µ
∆
(M)
γ
2
γ
2
S − M
∗
SM > 0 , S ∈ L ,
˙x = Ax + Bv
∆
,
z
∆
= Cx + Dv
∆
,
v
∆
= ∆z
∆
,
H(s) = C(sI −A)
−1
B + D
(n×n) ∆(s)
A
∆(s)
k∆(s)k
∞
< η .
sup
ω∈(−∞,∞)
µ
∆(jω)
(H(jω)) ≤
1
η
.
ω γ(ω)
γ
∗
= max
ω
γ(ω)
η ≤ γ
−1
∗
µ
H
T
(−j ω)SH(j ω) < η
−2
S , ∀ω .
S
−1/2
S
−1/2
H
T
(−j ω)SH(j ω)S
−1/2
< η
−2
I , ∀ω
kS
1/2
H(s)S
−1/2
k
∞
< η
−1
.
ω
ζ X > 0
S ∈ L
A
T
X + XA XB C
T
S
B
T
X −ζS D
T
S
SC SD −ζS
< 0 ,
η = ζ
−1
˙x = Ax + B
∆
v
∆
+ B
2
u ,
z
∆
= C
∆
x + D
∆∆
v
∆
+ D
∆2
u ,
y = C
2
x + D
2∆
v
∆
,
v
∆
= ∆(t)z
∆
,
x ∈ R
n
x
v
∆
∈ R
n
v∆
u ∈ R
n
u
z
∆
∈ R
n
z∆
y ∈ R
n
y
n
v∆
= n
z∆
= n
∆
∆(t)
∆(t) = (δ
1
(t)I
k
1
, . . . , δ
r
(t)I
k
r
, ∆
1
(t), . . . , ∆
f
(t)) ,
r f
m
1
, . . . , m
f
∆
T
(t)∆(t) ≤ η
2
I , ∀t ≥ 0 .
(I − ∆(t)D
∆∆
) 6= 0
k
˙x
r
= A
r
x
r
+ B
r
y ,
u = C
r
x
r
+ D
r
y ,
x
r
∈ R
k
∆(t)
˙x
c
= A
c
x
c
+ B
c
v
∆
,
z
∆
= C
c
x
c
+ D
c
v
∆
,
v
∆
= ∆(t)z
∆
A
c
=
A + B
2
D
r
C
2
B
2
C
r
B
r
C
2
A
r
, B
c
=
B
∆
+ B
2
D
r
D
2∆
B
r
D
2∆
,
C
c
= (C
∆
+ D
∆2
D
r
C
2
D
∆2
C
r
) , D
c
= D
∆∆
+ D
∆2
D
r
D
2∆
.
ζ = η
−1
A
T
c
X + XA
c
XB
c
C
T
c
S
B
T
c
X −ζS D
T
c
S
SC
c
SD
c
−ζS
< 0
X = X
T
> 0 S = S
T
> 0
S = (S
1
, . . . , S
r
, s
1
I
m
1
, . . . , s
f
I
m
f
) ,
r
k
1
, . . . , k
r
k
1
+ ··· + k
r
+ m
1
+ ··· + m
f
= n
∆
A
T
c
X + XA
c
XB
c
C
T
c
B
T
c
X −ζS D
T
c
C
c
D
c
−ζS
−1
< 0 .
Θ =
A
r
B
r
C
r
D
r
,
A
c
= A
0
+ BΘC , B
c
= B
0
+ BΘD
1
,
C
c
= C
0
+ D
2
ΘC , D
c
= D
∆∆
+ D
2
ΘD
1
,
A
0
=
A 0
n
x
×k
0
k×n
x
0
k×k
,
B =
0
n
x
×k
B
2
I
k
0
k×n
u
, C =
0
k×n
x
I
k
C
2
0
n
y
×k
,
B
0
=
B
∆
0
k×n
∆
, C
0
= (C
∆
0
n
∆
×k
) ,
D
1
=
0
k×n
∆
D
2∆
, D
2
= (0
n
∆
×k
D
∆2
) .
Θ
Ψ + P
T
Θ
T
Q + Q
T
ΘP < 0 ,
Ψ =
A
T
0
X + XA
0
XB
0
C
T
0
B
T
0
X −ζS D
T
∆∆
C
0
D
∆∆
−ζS
−1
,
P = (C D
1
0
(n
y
+k)×n
∆
) , Q = ( B
T
X 0
(n
u
+k)×n
∆
D
T
2
) .
W
T
P
A
T
0
X + XA
0
XB
0
C
T
0
B
T
0
X −ζS D
T
∆∆
C
0
D
∆∆
−ζS
−1
W
P
< 0 ,
W
T
Q
A
T
0
X + XA
0
XB
0
C
T
0
B
T
0
X −ζS D
T
∆∆
C
0
D
∆∆
−ζS
−1
W
Q
< 0 ,
W
P
P
W
Q
Q
Q = R
X 0 0
0 I 0
0 0 I
, R = (B
T
0 D
T
2
) ,
W
Q
=
X
−1
0 0
0 I 0
0 0 I
W
R
.
ζ > 0
X = X
T
> 0 (n
x
+ k) × (n
x
+ k) S = S
T
> 0
n
∆
× n
∆
W
T
P
A
T
0
X + XA
0
XB
0
C
T
0
B
T
0
X −ζS D
T
∆∆
C
0
D
∆∆
−ζS
−1
W
P
< 0 ,
W
T
R
X
−1
A
T
0
+ A
0
X
−1
B
0
X
−1
C
T
0
B
T
0
−ζS D
T
∆∆
C
0
X
−1
D
∆∆
−ζS
−1
W
R
< 0 ,
k
η = ζ
−1
X S Θ
Ψ
P Q
Y = X
−1
Σ = S
−1
X Y S
Σ
W
T
P
A
T
0
X + XA
0
XB
0
C
T
0
B
T
0
X −ζS D
T
∆∆
C
0
D
∆∆
−ζΣ
W
P
< 0 ,
W
T
R
Y A
T
0
+ A
0
Y B
0
Y C
T
0
B
T
0
−ζS D
T
∆∆
C
0
Y D
∆∆
−ζΣ
W
R
< 0 .
ˆ
X = (X, S)
ˆ
Y = (Y, Σ)
ζ
∗
ˆ
X
ˆ
Y = I
η
∗
≥ ζ
−1
∗
m
¨
ξ + b
˙
ξ + cξ = u ,
y = ξ ,
m = m
0
(1 + w
m
δ
m
) b = b
0
(1 + w
b
δ
b
) c = c
0
(1 + w
c
δ
c
) m
0
b
0
c
0
|δ
m
| ≤ η |δ
b
| ≤ η |δ
c
| ≤ η
x
1
= ξ x
2
=
˙
ξ
˙x
1
= x
2
˙x
2
= −
c
0
m
0
x
1
−
b
0
m
0
x
2
− w
m
δ
m
˙x
2
− w
c
δ
c
c
0
m
0
x
1
− w
b
δ
b
b
0
m
0
x
2
+
1
m
0
u .
v
∆
= (v
m
, v
c
, v
b
) z
∆
= (z
m
, z
c
, z
b
)
v
m
= −δ
m
˙x
2
, v
c
= −δ
c
c
0
m
0
x
1
, v
b
= −δ
b
b
0
m
0
x
2
z
m
= −˙x
2
=
c
0
m
0
x
1
+
b
0
m
0
x
2
− w
m
v
m
− w
c
v
c
− w
b
v
b
−
1
m
0
u,
z
c
= −
c
0
m
0
x
1
, z
b
= −
b
0
m
0
x
2
,
A =
0 1
−
c
0
m
0
−
b
0
m
0
, B
∆
=
0 0 0
w
m
w
c
w
b
, B
2
=
0
1
m
0
,
C
∆
=
c
0
m
0
b
0
m
0
−
c
0
m
0
0
0 −
b
0
m
0
, D
∆∆
=
−w
m
−w
c
−w
b
0 0 0
0 0 0
, D
∆2
=
−
1
m
0
0
0
,
C
2
= (1 0) , D
2∆
= (0 0 0) ,
∆(t) =
δ
m
(t) 0 0
0 δ
c
(t) 0
0 0 δ
b
(t)
.
(k = 0)
A
0
= A , B
0
= B
∆
, C
0
= C
∆
,
P = (1 0 0 0 0 0 0 0) , R = (0
1
m
0
0 0 0 −
1
m
0
0 0) .
m
0
= 1 b
0
= 1 c
0
= 100 w
m
= w
b
= w
c
= 0.1 η = 0.46
X =
29.575 0.1504
0.1504 0.3587
, S =
0.0029 0 0
0 0.0034 0
0 0 0.035