Баландин Д.В., Коган М.М. Синтез законов управления на основе линейных матричных неравенств
Подождите немного. Документ загружается.
˙x =
ˆ
Ax ,
ˆ
A = A + F Ω(t)E ,
A
F E
Ω(t)
t
Ω
T
(t)Ω(t) ≤ η
2
I
η 6= 0
A =
a
11
a
12
a
13
a
21
a
22
a
23
a
31
a
32
a
33
,
Ω
1
(t)
Ω
2
(t)
Ω
2
1
(t) + Ω
2
2
(t) ≤ η
2
.
ˆ
A =
a
11
a
12
+ Ω
1
(t) a
13
a
21
a
22
a
23
a
31
a
32
+ Ω
2
(t) a
33
.
ˆ
A = A +
1 0
0 0
0 1
Ω
1
(t)
Ω
2
(t)
(0 1 0) .
ˆ
A =
a
11
a
12
+ Ω
1
(t) a
13
+ Ω
2
(t)
a
21
a
22
a
23
a
31
a
32
a
33
.
ˆ
A = A +
1
0
0
(Ω
1
(t) Ω
2
(t))
0 1 0
0 0 1
,
Ω
2
1
(t) Ω
1
(t)Ω
2
(t)
Ω
1
(t)Ω
2
(t) Ω
2
2
(t)
≤ η
2
I ,
ˆ
A =
a
11
a
12
+ Ω
1
(t) a
13
a
21
a
22
a
23
a
31
a
32
a
33
+ Ω
2
(t)
.
ˆ
A = A +
1 0
0 0
0 1
Ω
1
(t) 0
0 Ω
2
(t)
0 1 0
0 0 1
,
Ω
2
1
(t) 0
0 Ω
2
2
(t)
≤ η
2
I .
|Ω
1
(t)| ≤ η , |Ω
2
(t)| ≤ η ,
˙x =
ˆ
Ax ,
ˆ
A = A +
n
c
X
i=1
F
i
Ω
i
(t)E
i
,
F
i
E
i
Ω
i
(t)
|Ω
i
(t)| ≤ η , i = 1, . . . , n
c
η 6= 0
m
0
¨
ξ + b
0
(1 + f
1
Ω
1
(t))
˙
ξ + c
0
(1 + f
2
Ω
2
(t))ξ = 0 ,
m
0
b
0
c
0
f
1
f
2
Ω
1
(t) Ω
2
(t)
|Ω
1
(t)| ≤ 1 , |Ω
2
(t)| ≤ 1 .
x
1
= ξ x
2
=
˙
ξ
˙x
1
= x
2
˙x
2
= −
c
0
m
0
(1 + f
2
Ω
2
(t))x
1
−
b
0
m
0
(1 + f
1
Ω
1
(t))x
2
,
A =
0 1
−
c
0
m
0
−
b
0
m
0
, F
1
=
0
f
1
, F
2
=
0
f
2
,
E
1
= (0 −
b
0
m
0
) , E
2
= (−
c
0
m
0
0) .
˙x = Ax + Bv
∆
z
∆
= Cx + Dv
∆
v
∆
= ∆(t)z
∆
.
v
∆
z
∆
∆(t)
∆
T
(t)∆(t) ≤ η
2
I .
∆(t) = (δ
1
(t)I
k
1
, . . . , δ
r
(t)I
k
r
, ∆
1
(t), . . . , ∆
f
(t)) ,
r f
m
1
, . . . , m
f
v
∆
˙x = (A + B∆(t)(I − D∆(t))
−1
C)x
(I − D∆(t)) 6= 0
D = 0
˙x = (A + B∆(t)C)x ,
B = (F
1
, F
2
, ···, F
n
c
), ∆(t) =
Ω
1
(t) 0 ··· 0
0 Ω
2
(t) ··· 0
··· ··· ··· ···
0 0 ··· Ω
n
c
(t)
, C =
E
1
E
2
···
E
n
c
.
m
¨
ξ + b
˙
ξ + cξ = 0 ,
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
.
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
,
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
,
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
,
∆(t) =
δ
m
(t) 0 0
0 δ
c
(t) 0
0 0 δ
b
(t)
.
˙x = Ax + Bv
∆
z
∆
= Cx + Dv
∆
v
∆
= ∆z
∆
A B C D
∆ L
2
k∆k = sup
z
∆
6≡0
kv
∆
(t)k
2
kz
∆
(t)k
2
≤ η ,
η
m
1
¨
ξ
1
= −b
1
˙
ξ
1
− c
1
ξ
1
− b
2
(
˙
ξ
1
−
˙
ξ
2
) − c
2
(ξ
1
− ξ
2
) ,
m
2
¨
ξ
2
= b
2
(
˙
ξ
1
−
˙
ξ
2
) + c
2
(ξ
1
− ξ
2
) ,
m
1
m
2
b
1
b
2
c
1
c
2
m
2
b
2
c
2
x
1
= ξ
1
x
2
=
˙
ξ
1
v
∆
= −m
2
¨
ξ
2
˙x
1
= x
2
˙x
2
= −
c
1
m
1
x
1
−
b
1
m
1
x
2
+
1
m
1
v
∆
.
sy(t) =
dy
dt
(m
2
s
2
+ b
2
s + c
2
)
¨
ξ
2
= (b
2
s + c
2
)
¨
ξ
1
z
∆
= −
c
1
m
1
x
1
−
b
1
m
1
x
2
+
1
m
1
v
∆
,
(m
2
s
2
+ b
2
s + c
2
)v
∆
= −m
2
(b
2
s + c
2
)z
∆
.
v
∆
(0) = 0 ˙v
∆
(0) = 0
∆ : L
2
→ L
2
z
∆
(t)
v
∆
(t)
k∆k = sup
ω∈(−∞,∞)
kH(j ω)k = kHk
∞
≤ η ,
H(s) = −
m
2
(b
2
s + c
2
)
m
2
s
2
+ b
2
s + c
2
.
˙x = Ax + ∆(x) ,
∆(x)
|∆(x)| ≤ α|x|
α > 0 |·|
x ε ≤ α |∆(x)| = ε|x| ∆(x)
x Ω(x)
x Ω
T
(x)Ω(x) = I
∆(x)
|∆(x)|
= Ω(x)
x
|x|
, ∆(x) 6= 0 .
∆(x)
∆(x) = εΩ(x)x .
˙x = [A + αΩ(x)]x , Ω
T
(x)Ω(x) ≤ I .
˙x = (A + B∆C)x ,
A B C ∆
∆
T
∆ ≤ η
2
I .
V (x) = x
T
Xx X = X
T
> 0
∆
˙
V = x
T
[(A + B∆C)
T
X + X(A + B∆C)]x < 0 ∀x 6= 0 .
η
max
η
x
T
(A
T
X + XA)x + 2x
T
XB∆Cx < 0 ∀x 6= 0
max
{∆: ∆
T
∆≤η
2
I}
2(B
T
Xx)
T
∆(Cx) < −x
T
(A
T
X + XA)x ∀x 6= 0 .
max
{∆: ∆
T
∆≤η
2
I}
a
T
∆ b = η|a||b| ,
x
T
(A
T
X + XA)x + 2η|B
T
Xx||Cx| < 0 ∀x 6= 0 .
x
T
(A
T
X + XA)x + 2ηv
T
B
T
Xx < 0
x v
v
T
v ≤ x
T
C
T
Cx .
v
X = X
T
> 0
S
X = X
T
> 0 τ > 0 (x, v)
x
T
(A
T
X + XA)x + 2ηv
T
B
T
Xx − τ(v
T
v − x
T
C
T
Cx) < 0 .
τ
−1
η τ
−1
ηX
X
A
T
X + XA + ηC
T
C ηXB
ηB
T
X −ηI
< 0 .
X = X
T
> 0 ζ = η
−1
F
c
(X, ζ) =
A
T
X + XA XB C
T
B
T
X −ζI 0
C 0 −ζI
< 0 .