Баландин Д.В., Коган М.М. Синтез законов управления на основе линейных матричных неравенств
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Θ = 17.534 η > η
∗
= 0.46
u = 17.534y
|δ
m
(t)| ≤ η
∗
|δ
b
(t)| ≤ η
∗
|δ
c
(t)| ≤ η
∗
0.4013
˙x
r
= −31.123x
r
− 275.727y ,
u = −29.943x
r
− 232.505y
2.5
k
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
.
P =
0
k×n
x
I
k
0
k×n
∆
0
k×n
∆
C
2
0
n
y
×k
D
2∆
0
n
y
×n
∆
,
R =
0
k×n
x
I
k
0
k×n
v∆
0
k×n
∆
B
T
2
0
n
u
×k
0
n
u
×n
∆
D
T
∆2
,
W
P
W
R
W
P
=
W
(1)
P
0
0 0
W
(2)
P
0
0 I
, W
R
=
W
(1)
R
0
0 0
0 I
W
(2)
R
0
,
W
(1)
P
W
(2)
P
W
(1)
R
W
(2)
R
C
2
W
(1)
P
+ D
2∆
W
(2)
P
= 0 , B
T
2
W
(1)
R
+ D
T
∆2
W
(2)
R
= 0 .
W
(1)
P
0
0 0
W
(2)
P
0
0 I
T
A
T
X
11
+ X
11
A A
T
X
12
X
11
B
∆
C
T
∆
? 0 X
T
12
B
∆
0
? ? −ζS D
T
∆∆
? ? ? −ζΣ
W
(1)
P
0
0 0
W
(2)
P
0
0 I
,
W
(1)
R
0
0 0
0 I
W
(2)
R
0
T
Y
11
A
T
+ AY
11
AY
12
B
∆
Y
11
C
T
∆
? 0 0 Y
T
12
C
T
∆
? ? −ζS D
T
∆∆
? ? ? −ζΣ
W
(1)
R
0
0 0
0 I
W
(2)
R
0
,
N
1
| 0
− − −
0 | I
T
A
T
X
11
+ X
11
A X
11
B
∆
| C
T
∆
B
T
∆
X
11
−ζS | D
T
∆∆
− − − −
C
∆
D
∆∆
| −ζΣ
N
1
| 0
− − −
0 | I
< 0 ,
N
2
| 0
− − −
0 | I
T
Y
11
A
T
+ AY
11
Y
11
C
T
∆
| B
∆
C
∆
Y
11
−ζΣ | D
∆∆
− − − −
B
T
∆
D
T
∆∆
| −ζS
N
2
| 0
− − −
0 | I
< 0 ,
N
1
= (W
(1)
P
, W
(2)
P
) N
2
= (W
(1)
R
, W
(2)
R
)
(C
2
D
2∆
) (B
T
2
D
T
∆2
)
k
S = S
T
> 0 Σ = Σ
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 ,
C
2
= I , D
2∆
= 0 , A
r
= 0 , B
r
= 0 , C
r
= 0 .
N
1
= (0 I)
−ζS D
T
∆∆
D
∆∆
−ζΣ
< 0 ,
−ζΣ ΣD
T
∆∆
D
∆∆
Σ −ζΣ
< 0
Σ
Y
11
Σ
N
2
| 0
− − −
0 | I
T
Y
11
A
T
+ AY
11
Y
11
C
T
∆
| B
∆
Σ
C
∆
Y
11
−ζΣ | D
∆∆
Σ
− − − −
ΣB
T
∆
ΣD
T
∆∆
| −ζΣ
N
2
| 0
− − −
0 | I
< 0 ,
N
2
(B
T
2
D
T
∆2
)
ζ > 0
Y
11
= Y
T
11
> 0 n
x
×n
x
Σ = Σ
T
> 0 n
∆
×n
∆
u = Θx
η = ζ
−1
Y
11
Σ Θ
Ψ =
A
T
Y
−1
11
+ Y
−1
11
A Y
−1
11
B
∆
C
T
∆
B
T
∆
Y
−1
11
−ζΣ
−1
D
T
∆∆
C
∆
D
∆∆
−ζΣ
,
P = (I 0 0) , Q = (B
T
2
Y
−1
11
0 D
T
∆2
) .
ˆ
A = A + F Ω(t)E
Ω(t)
D
∆∆
= 0 , D
∆2
= 0 , S = Σ = I , B
∆
= F , C
∆
= E .
ζ > 0
F
c
(Y
11
, ζ) < 0 F
c
(Y
11
, ζ)
W
B
T
2
0 0
0 I 0
0 0 I
T
Y
11
A
T
+ AY
11
Y
11
E
T
F
EY
11
−ζI 0
F
T
0 −ζI
W
B
T
2
0 0
0 I 0
0 0 I
.
η
max
˙x = (A + F Ω(t)E)x + B
2
u ,
Ω
T
(t)Ω(t) ≤ η
2
I
u = Θx
η
max
≥ η
∗
, η
∗
= ζ
−1
∗
, ζ
∗
= inf
F
c
(Y
11
,ζ)<0
ζ .
m
0
¨
ξ + b
0
(1 + f
1
Ω
1
(t))
˙
ξ + c
0
(1 + f
2
Ω
2
(t))ξ = u ,
u = Θ
1
ξ + Θ
2
˙
ξ
A =
0 1
−
c
0
m
0
−
b
0
m
0
, F =
0 0
f
1
f
2
,
E =
0 −
b
0
m
0
−
c
0
m
0
0
, B
2
=
0
1
m
0
.
m
0
= 1 b
0
= 1 c
0
= 100 f
1
= f
2
= 0, 1
η
∗
∼ 10
11
0.9074
x
t+1
= Ax
t
+ B
∆
v
∆t
+ B
2
u
t
,
z
∆t
= C
∆
x
t
+ D
∆∆
v
∆t
+ D
∆2
u
t
,
y
t
= C
2
x
t
+ D
2∆
v
∆t
,
v
∆t
= ∆
t
z
∆t
,
x
t
∈ R
n
x
v
∆t
∈ R
n
v∆
u
t
∈ R
n
u
z
∆t
∈ R
n
z∆
y
t
∈ 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)
t+1
= A
r
x
(r)
t+1
+ B
r
y
t
,
u
t
= C
r
x
(r)
t
+ D
r
y
t
,
x
(r)
t
∈ R
k
∆
t
¯x
t
= A
c
¯x
t
+ B
c
v
∆t
,
z
∆t
= C
c
¯x
t
+ D
c
v
∆t
,
v
∆
= ∆
t
z
∆t
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
XA
c
− X A
T
c
XB
c
C
T
c
S
B
T
c
XA
c
−ζS + B
T
c
XB
c
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
XA
c
− X A
T
c
XB
c
C
T
c
B
T
c
XA
c
−ζS + B
T
c
XB
c
D
T
c
C
c
D
c
−ζS
−1
< 0 ,
−X
−1
A
c
B
c
0
A
T
c
−X 0 C
T
c
B
T
c
0 −ζS D
T
c
0 C
c
D
c
−ζS
−1
< 0 .
X > 0
−X 0 C
T
c
0 −ζS D
T
c
C
c
D
c
−ζS
−1
+
A
T
c
B
T
c
0
X(A
c
B
c
0) < 0 ,
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 =
A
r
B
r
C
r
D
r
.
Θ
Ψ + P
T
Θ
T
Q + Q
T
ΘP < 0 ,
Ψ =
−X
−1
A
0
B
0
0
A
T
0
−X 0 C
T
0
B
T
0
0 −ζS D
T
∆∆
0 C
0
D
∆∆
−ζS
−1
,
P = (0
(n
y
+k)×(n
x
+k)
C D
1
0
(n
y
+k)×n
∆
) ,
Q = (B
T
0
(n
u
+k)×(n
x
+k)
0
(n
u
+k)×n
∆
D
T
2
) .
ζ > 0
X = X
T
> 0 (n
x
+ k) × (n
x
+ k) S = S
T
> 0
n
∆
×n
∆
W
T
P
−X
−1
A
0
B
0
0
A
T
0
−X 0 C
T
0
B
T
0
0 −ζS D
T
∆∆
0 C
0
D
∆∆
−ζS
−1
W
P
< 0 ,
W
T
Q
−X
−1
A
0
B
0
0
A
T
0
−X 0 C
T
0
B
T
0
0 −ζS D
T
∆∆
0 C
0
D
∆∆
−ζS
−1
W
Q
< 0 ,
k
η = ζ
−1
X S Θ
Ψ P Q
Y = X
−1
Σ = S
−1
X
P
W
T
P
−Y A
0
B
0
0
A
T
0
−X 0 C
T
0
B
T
0
0 −ζS D
T
∆∆
0 C
0
D
∆∆
−ζΣ
W
P
< 0 ,
W
T
Q
−Y A
0
B
0
0
A
T
0
−X 0 C
T
0
B
T
0
0 −ζS D
T
∆∆
0 C
0
D
∆∆
−ζΣ
W
Q
< 0 .
ˆ
X = (X, S)
ˆ
Y = (Y, Σ)
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
.
P =
0
k×n
x
0
k×k
0
k×n
x
I
k
0
k×n
∆
0
k×n
∆
0
n
y
×n
x
0
n
y
×k
C
2
0
n
y
×k
D
2∆
0
n
y
×n
∆
,
Q =
0
k×n
x
I
k
0
k×n
x
0
k×k
0
k×n
∆
0
k×n
∆
B
T
2
0
n
u
×k
0
n
u
×n
x
0
n
u
×k
0
n
u
×n
∆
D
T
∆2
,
W
P
=
0 I 0 0
0 0 I 0
W
(1)
P
0 0 0
0 0 0 0
W
(2)
P
0 0 0
0 0 0 I
, W
Q
=
W
(1)
Q
0 0 0
0 0 0 0
0 I 0 0
0 0 I 0
0 0 0 I
W
(2)
Q
0 0 0
,
W
(1)
P
W
(2)
P
W
(1)
Q
W
(2)
Q
C
2
W
(1)
P
+ D
2∆
W
(2)
P
= 0 , B
T
2
W
(1)
Q
+ D
T
∆2
W
(2)
Q
= 0 .
W
T
P
−Y
11
−Y
12
A 0 B
∆
0
? −Y
22
0 0 0 0
? ? −X
11
−X
12
0 C
T
∆
? ? ? −X
22
0 0
? ? ? ? −ζS D
T
∆∆
? ? ? ? ? −ζΣ
W
P
< 0 ,