Баландин Д.В., Коган М.М. Синтез законов управления на основе линейных матричных неравенств
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η
max
η
max
≥ η
∗
, η
∗
= ζ
−1
∗
, ζ
∗
= inf
F
c
(X,ζ)<0
ζ ,
F
c
(X, ζ) < 0
˙x = Ax + Bv
∆
z
∆
= Cx + Dv
∆
v
∆
= ∆(t)z
∆
,
∆(t)
∆
T
(t)∆(t) ≤ η
2
I .
A
(I − ∆(t)D) 6= 0
∆(t)
η
η
max
η
˙x = Ax + Bv
∆
z
∆
= Cx + Dv
∆
,
v
∆
z
∆
|v
∆
|
2
≤ η
2
|z
∆
|
2
.
I − η
2
D
T
D > 0 .
x = 0
v
T
∆
(I − η
2
D
T
D)v
∆
≤ 0 ,
v
∆
= 0 x = 0
V (x) = x
T
Xx
x v
∆
˙
V < 0 .
S
τ > 0 V (x) = x
T
Xx
X = X
T
> 0 x v
∆
˙
V + τ(|z
∆
|
2
− η
−2
|v
∆
|
2
) < 0 , |x|
2
+ |v
∆
|
2
6= 0 .
τ = 1
X τX
x v
∆
˙
V + |z
∆
|
2
− η
−2
|v
∆
|
2
< −ε(|x|
2
+ |v
∆
|
2
) < −ε|x|
2
,
ε > 0 V (x)
A
(A, B)
V (x)
H
T
(−j ω)H(j ω) − η
−2
I < 0 , ∀ω ∈ (−∞, ∞)
kH(s)k
∞
< η
−1
, H(s) = D + C(sI − A)
−1
B .
η
−1
ζ = η
−1
X = X
T
> 0
F
c
(X, ζ) =
A
T
X + XA XB C
T
B
T
X −ζI D
T
C D −ζI
< 0 ,
ζ
η
max
η
∗
= ζ
−1
∗
, ζ
∗
= inf
F
c
(X,ζ)<0
ζ ,
F
c
(X, ζ) < 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
= 1 b
0
= 1 c
0
= 100
f
1
= f
2
= 0.1
η
∗
= 0.7027
m
0
(1 + w
m
δ
m
)
¨
ξ + b
0
(1 + w
b
δ
b
)
˙
ξ + c
0
(1 + w
c
δ
c
)ξ = 0 ,
m
0
= 1 b
0
= 1 c
0
= 100
w
m
= w
b
= w
c
= 0.1 η
∗
= 0.4013
˙x = Ax + Bv
∆
z
∆
= Cx + Dv
∆
v
∆
= ∆(t)z
∆
,
∆(t)
∆
T
(t)∆(t) ≤ η
2
I .
∆(t) n
∆
×n
∆
∆(t) = (δ
1
(t)I
k
1
, . . . , δ
r
(t)I
k
r
, ∆
1
(t), . . . , ∆
f
(t)) ,
r f
m
1
, . . . , m
f
S
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
∆
v
∆
z
∆
v
T
∆
Sv
∆
≤ η
2
z
T
∆
Sz
∆
S
v
T
∆
Sv
∆
= z
T
∆
∆
T
(t)S∆(t)z
∆
,
S
S∆ = ∆S
˙x = Ax + BS
−1/2
˜v
∆
˜z
∆
= S
1/2
Cx + S
1/2
DS
−1/2
˜v
∆
,
˜v
∆
˜z
∆
|˜v
∆
|
2
≤ η
2
|˜z
∆
|
2
.
v
∆
= S
−1/2
˜v
∆
z
∆
= S
−1/2
˜z
∆
˙x = Ax +
˜
B˜v
∆
˜z
∆
=
˜
Cx +
˜
D˜v
∆
,
˜
B = BS
−1/2
˜
C = S
1/2
C
˜
D = S
1/2
DS
−1/2
S − η
2
D
T
SD > 0 ,
A
T
X + XA X
˜
B
˜
C
T
˜
B
T
X −ζI
˜
D
T
˜
C
˜
D −ζI
< 0 .
x y z
x
T
(A
T
X + XA)x + 2x
T
XBS
−1/2
y + 2x
T
C
T
S
1/2
z−
−ζy
T
y + 2y
T
S
−1/2
D
T
S
1/2
z − ζz
T
z < 0 .
¯y = S
−1/2
y ¯z = S
−1/2
z
ζ = η
−1
X = X
T
> 0
S = S
T
> 0
A
T
X + XA XB C
T
S
B
T
X −ζS D
T
S
SC SD −ζS
< 0 ,
˜
B
˜
C
˜
D X
kS
1/2
H(s)S
−1/2
k
∞
< η
−1
, H(s) = D + C(sI − A)
−1
B .
X S ζ
ζ
ζ
X S
ζ
X
S = I
η
∗
= 0.9074
x
t+1
= Ax
t
+ Bv
∆t
z
∆t
= Cx
t
+ Dv
∆t
v
∆t
= ∆
t
z
∆t
,
∆
t
∆
T
t
∆
t
≤ η
2
I .
A
(I − ∆
t
D) 6= 0
∆
t
η
η
max
η
x
t+1
= Ax
t
+ Bv
∆t
z
∆t
= Cx
t
+ Dv
∆t
,
v
∆t
z
∆t
|v
∆t
|
2
≤ η
2
|z
∆t
|
2
.
I − η
2
D
T
D > 0 .
x
t
≡ 0
v
T
∆t
(I − η
2
D
T
D)v
∆t
≤ 0 ,
v
∆t
≡ 0 x = 0
V (x) = x
T
Xx
x
t
v
∆t
V
t+1
− V
t
< 0 .
S
V (x) = x
T
Xx X = X
T
> 0
x
t
v
∆t
V
t+1
− V
t
+ |z
∆t
|
2
− η
−2
|v
∆t
|
2
< 0 , |x
t
|
2
+ |v
∆t
|
2
6= 0 .
x
t
v
∆t
V
t+1
− V
t
+ |z
∆t
|
2
− η
−2
|v
∆t
|
2
< −ε(|x
t
|
2
+ |v
∆t
|
2
) < −ε|x
t
|
2
,
ε > 0 V (x)
A
(A, B)
V (x)
kH(z)k
∞
< η
−1
, H(z) = D + C(zI − A)
−1
B .
η
−1
ζ = η
−1
X = X
T
> 0
F
d
(X, γ) =
A
T
XA − X A
T
XB C
T
B
T
XA −ζI + B
T
XB D
T
C D −ζI
< 0 ,
η
max
η
max
≥ η
∗
, η
∗
= ζ
−1
∗
, ζ
∗
= inf
F
d
(X,ζ)<0
ζ .
∆(t)
∆
t
= (δ
1
(t)I
k
1
, . . . , δ
r
(t)I
k
r
, ∆
1
(t), . . . , ∆
f
(t)) ,
r f
m
1
, . . . , m
f
µ
x
t+1
= Ax
t
+ BS
−1/2
˜v
∆t
˜z
∆t
= S
1/2
Cx
t
+ S
1/2
DS
−1/2
˜v
∆t
,
S
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
∆
ζ = η
−1
X = X
T
> 0
S = S
T
> 0
A
T
XA − X A
T
XB C
T
S
B
T
XA −ζS + B
T
XB D
T
S
SC SD −ζS
< 0 ,
µ
µ
µ
M ∈ C
n×n
∆ ∈ C
n×n
∆ = (δ
1
I
k
1
, . . . , δ
r
I
k
r
, ∆
1
, . . . , ∆
f
) ,
r f
m
1
, . . . , m
f
∆
B∆ = {∆ ∈ ∆ : σ
max
(∆) ≤ 1} .
µ
∆
(M) M ∈
C
n×n
∆
µ
∆
(M) =
1
min{σ
max
(∆) : ∆ ∈ ∆, (I − M∆) = 0}
,
∆ ∈ ∆ (I −
M∆) = 0 µ
∆
(M) = 0
µ
∆
(M) = max
∆∈B∆
ρ(M∆) ,
ρ
µ
∆
(M)
y = Mv v = ∆y
(I − M∆)
(I − M∆) 6= 0
y = 0 v = 0 (I − M∆) = 0
µ
∆
(M)
∆ y v
∆ = {δI
n
} ,
µ
∆
(M) = ρ(M)
∆ = {∆ ∈ C
n×n
}
µ
∆
(M) = σ
max
(M)
∆
i
, i = 1, . . . , f
(n × n)
ρ(M) ≤ µ
∆
(M) ≤ σ
max
(M) .
µ
∆
(M) ρ(M) σ
max
(M)