Баландин Д.В., Коган М.М. Синтез законов управления на основе линейных матричных неравенств
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H
2
H
2
γ
γ
2
γ
γ
γ
γ
γ
H
2
RL
2
H(s)
∞
Z
−∞
[H
T
(−jω)H(jω)] dω < ∞ .
H
2
kHk
2
= {
1
2π
∞
Z
−∞
[H
T
(−jω)H(jω)] dω}
1/2
.
H(s) = C(sI − A)
−1
B .
H
2
˙x = Ax + Bv , x(0) = 0
z = Cx ,
x ∈ R
n
x
v ∈ R
n
v
z ∈ R
n
z
H
2
v
(i)
(t) = δ(t) e
i
δ(t) e
i
i (n
v
×n
v
)
i = 1, 2, . . . , n
v
z
(i)
Z
(i)
(jω)
Z
(i)
(jω) = H(jω)e
i
kHk
2
2
=
1
2π
∞
Z
−∞
[H
T
(−jω)H(jω)] dω =
=
1
2π
∞
Z
−∞
n
v
X
i=1
e
T
i
H
T
(−jω)H(jω)e
i
dω =
1
2π
∞
Z
−∞
n
v
X
i=1
Z
(i)T
(−jω)Z
(i)
(jω) dω =
=
n
v
X
i=1
∞
Z
0
z
(i)T
z
(i)
dt .
H
2
z
(i)
(t) = Ce
At
Be
i
, t > 0 (K
T
SK) =
(SKK
T
) K
S
kHk
2
2
=
n
v
X
i=1
∞
Z
0
e
T
i
B
T
e
A
T
t
C
T
Ce
At
Be
i
dt =
=
n
v
X
i=1
∞
Z
0
(e
A
T
t
C
T
Ce
At
Be
i
e
T
i
B
T
) dt =
=
∞
Z
0
(e
A
T
t
C
T
Ce
At
n
v
X
i=1
Be
i
e
T
i
B
T
) dt =
=
∞
Z
0
(e
A
T
t
C
T
Ce
At
BB
T
) dt =
= (B
T
∞
Z
0
e
A
T
t
C
T
Ce
At
dtB) = (C
∞
Z
0
e
At
BB
T
e
A
T
t
dtC
T
) .
A
P = P
o
=
∞
Z
0
e
A
T
t
C
T
Ce
At
dt
A
T
P + P A + C
T
C = 0 ,
P = P
r
=
∞
Z
0
e
At
BB
T
e
A
T
t
dt
AP + P A
T
+ BB
T
= 0 .
H
2
kHk
2
2
= (B
T
P
o
B) ,
P
o
kHk
2
2
= (CP
r
C
T
) ,
H
2
P
r
v
E[v(t)v
T
(t + τ)] = δ(τ)I
J = lim
t→∞
E |z(t)|
2
,
E
J = lim
t→∞
E (zz
T
) =
= lim
t→∞
E [
t
Z
0
Φ(t, τ)v(τ) dτ
t
Z
0
v
T
(σ)Φ
T
(t, σ) dσ] =
= lim
t→∞
[
t
Z
0
Φ(t, τ)
t
Z
0
E [v(τ)v
T
(σ)]Φ
T
(t, σ) dσ dτ =
= lim
t→∞
[
t
Z
0
Φ(t, τ)Φ
T
(t, τ) dτ] ,
Φ(ξ, ν) = Ce
A(ξ−ν)
B µ = t − τ
J = lim
t→∞
[C
t
Z
0
e
Aµ
BB
T
e
A
T
µ
dµC
T
] = [C
∞
Z
0
e
Aµ
BB
T
e
A
T
µ
dµC
T
] = kHk
2
2
.
H
2
H
2
A
kHk
2
< γ
X = X
T
> 0 S = S
T
A
T
X + XA XB
B
T
X −γI
< 0 ,
X C
T
C S
> 0 , (S) < γ;
Y = Y
T
> 0 R = R
T
AY + Y A
T
Y C
T
CY −γI
< 0 ,
Y B
B
T
R
> 0 , (R) < γ .
(B
T
P
o
B) < γ
2
P
o
X = X
T
> 0
A
T
X + XA + C
T
C < 0 , (B
T
XB) < γ
2
.
A
T
X + XA + C
T
C + ε
2
I = 0 ,
X > P
o
(B
T
P
o
B) ≤
(B
T
XB) < γ
2
(B
T
P
o
B) < γ
2
ε (B
T
XB) < γ
2
X = Y
−1
Y R
Y A
T
+ AY + Y C
T
CY < 0 , B
T
Y
−1
B < R , (R) < γ
2
.
Y γ
−1
Y R γR
kHk
2
= kH
T
k
2
γ
γ
˙x = Ax + Bu , x(0) = x
0
,
z = Cx + Du
u = Θx ,
γ
γ > 0
J =
∞
Z
0
|z(t)|
2
dt < γ
2
|x
0
|
2
∀x
0
6= 0 .
γ
J
γ γ
γ
C
T
D = 0 D
T
D > 0 (A, B)
(A, C)
u = −(D
T
D)
−1
B
T
P x ,
P = P
T
≥ 0
A
T
P + P A − P B(D
T
D)
−1
B
T
P + C
T
C = 0 ,
min J = x
T
0
P x
0
γ
2
γ
P
γ
˙x = Ax + x
0
δ(t) + Bu , x(0) = 0 ,
z = Cx + Du ,
δ(t)
˙x = (A + BΘ)x + x
0
δ(t) , x(0) = 0 ,
z = (C + DΘ)x .
H
2
J H
2
k(C + DΘ)[sI − (A + BΘ)]
−1
x
0
k
2
< γ|x
0
| ∀x
0
6= 0 .
Y = Y
T
> 0 R = R
T
(A + BΘ)Y + Y (A + BΘ)
T
Y (C + DΘ)
T
(C + DΘ)Y −γ|x
0
|I
< 0 ,
Y x
0
x
T
0
R
> 0 , R < γ|x
0
| .
R
Y |x
0
|Y R |x
0
|R
(A + BΘ)Y + Y (A + BΘ)
T
Y (C + DΘ)
T
(C + DΘ)Y −γI
< 0 ,
Y |x
0
| x
0
x
T
0
R|x
0
|
> 0 , R < γ .
Z = ΘY
x
T
0
Y
−1
x
0
|x
0
|
2
< γ , ∀x
0
6= 0 ,
Y I
I γI
> 0 ,
Θ γ
Θ = ZY
−1
Y = Y
T
> 0 Z
AY + Y A
T
+ BZ + Z
T
B
T
Y C
T
+ Z
T
D
T
CY + DZ −γI
< 0 ,
Y I
I γI
> 0 .
γ
γ
γ
γ
Ψ + P
T
Θ
T
Q + Q
T
ΘP < 0 ,
Ψ =
Y A
T
+ AY Y C
T
CY −γI
,
P = (Y 0) , Q = ( B
T
D
T
) .
Θ
W
T
P
Y A
T
+ AY Y C
T
CY −γI
W
P
< 0 ,
W
T
Q
Y A
T
+ AY Y C
T
CY −γI
W
Q
< 0 ,
W
P
=
0
I
, W
Q
=
W
(1)
Q
W
(2)
Q
,
W
(1)
Q
W
(2)
Q
B
T
W
(1)
Q
+ D
T
W
(2)
Q
= 0 .
Θ γ
Y = Y
T
> 0
W
T
Q
Y A
T
+ AY Y C
T
CY −γI
W
Q
< 0 ,
Y I
I γI
> 0 .
γ
˙x
1
= x
2
,
˙x
2
= −ω
2
0
x
1
− δx
2
+ u ,
z
1
= x
1
,
z
2
= u ,
A =
0 1
−ω
2
0
−δ
, B =
0
1
,
C =
1 0
0 0
, D =
0
1
.
ω
0
= 10 δ = 0.1 γ
Y =
0.4914 −0.0593
−0.0593 49.1460
, Z = (0 − 2.0354) ,
Θ = (−0.005 −0.0414)
γ = 2.0354
P =
4.1428 0.0050
0.0050 0.0414
, Θ = −(D
T
D)
−1
B
T
P = (−0.005 − 0.0414).
γ