Optimal filtering is a means of adaptive extraction of a weak desired signal in the presence of noise and interfering signals. The goal of optimal filtering is to design a filter that acts on the output
z
{\displaystyle z}
Consider the continuous-time generalized LTI plant with minimal states-space realization
x
˙
=
A
x
+
B
1
w
z
=
C
1
x
+
D
11
w
,
y
=
C
2
x
+
D
21
w
,
{\displaystyle {\begin{aligned}{\dot {x}}&=Ax+B_{1}w\\z&=C_{1}x+D_{11}w,\\y&=C_{2}x+D_{21}w,\end{aligned}}}
where it is assumed that
A
{\displaystyle A}
The matrices needed as inputs are
A
,
B
1
,
C
2
,
C
1
,
,
D
11
,
D
21
{\displaystyle A,B_{1},C_{2},C_{1},,D_{11},D_{21}}
An
H
∞
{\displaystyle H\infty }
H
∞
{\displaystyle H_{\infty }}
P
~
(
s
)
{\displaystyle {\tilde {P}}(s)}
P
~
(
s
)
=
C
~
1
(
s
I
−
A
~
)
−
1
B
~
1
+
D
~
11
,
where
A
~
=
[
A
0
B
f
C
2
A
f
]
<
0
B
~
1
=
[
B
1
B
f
D
21
]
<
0
C
~
1
=
[
C
1
−
D
f
C
2
−
C
f
]
<
0
D
~
11
=
D
11
−
D
f
D
21
{\displaystyle {\begin{aligned}{\tilde {P}}(s)={\tilde {C}}_{1}(sI-{\tilde {A}})^{-}1{\tilde {B}}_{1}+{\tilde {D}}_{11},\\{\text{where}}\\{\tilde {A}}={\begin{bmatrix}A&&0\\B_{f}C_{2}&&A_{f}\end{bmatrix}}&<0\\{\tilde {B}}_{1}={\begin{bmatrix}B_{1}\\B_{f}D_{21}\end{bmatrix}}&<0\\{\tilde {C}}_{1}={\begin{bmatrix}C_{1}-D_{f}C_{2}-C_{f}\end{bmatrix}}&<0\\{\tilde {D}}_{11}=D_{11}-D_{f}D_{21}\\\end{aligned}}}
The LMI:
H
∞
{\displaystyle H_{\infty }}
[ edit | edit source ] Solve for
A
n
∈
R
n
x
×
n
x
,
B
n
∈
R
n
x
×
n
y
,
C
f
∈
R
n
x
×
n
x
{\displaystyle A_{n}\in \mathbb {R} ^{n_{x}\times n_{x}},B_{n}\in \mathbb {R} ^{n_{x}\times n_{y}},C_{f}\in \mathbb {R} ^{n_{x}\times n_{x}}}
X
,
Y
∈
S
n
x
{\displaystyle X,Y\in \mathbb {S} ^{n_{x}}}
ν
∈
R
>
0
{\displaystyle \nu \in \mathbb {R} _{>0}}
ζ
(
ν
)
=
ν
{\displaystyle \zeta (\nu )=\nu }
X
>
0
,
Y
>
0
{\displaystyle X>0,Y>0}
[
Y
A
+
A
T
Y
+
B
n
C
2
A
n
+
C
2
T
B
n
T
+
A
T
X
Y
B
1
+
B
n
D
21
C
1
T
−
C
2
T
D
f
T
⋆
A
n
+
A
n
T
X
B
1
+
B
n
D
21
−
C
f
T
⋆
⋆
−
γ
I
D
1
1
T
−
D
2
1
T
D
f
T
⋆
⋆
⋆
−
γ
I
]
<
0
Y
−
X
>
0
{\displaystyle {\begin{aligned}{\begin{bmatrix}YA+A^{T}Y+B_{n}C_{2}&&A_{n}+C_{2}^{T}B_{n}^{T}+A^{T}X&&YB_{1}+B_{n}D_{21}&&{C_{1}}^{T}-{C_{2}}^{T}{D_{f}}^{T}\\\star &&A_{n}+A_{n}^{T}&&XB_{1}+B_{n}D_{21}&&-{C_{f}}^{T}\\\star &&\star &&-\gamma I&&{D_{1}1}^{T}-{D_{2}1}^{T}{D_{f}}^{T}\\\star &&\star &&\star &&-\gamma I\end{bmatrix}}&<0\\\\Y-X>0\\\end{aligned}}}
The filter is recovered by
A
f
=
X
−
1
A
n
{\displaystyle A_{f}=X^{-1}A_{n}}
B
f
=
X
−
1
B
n
{\displaystyle B_{f}=X^{-1}B_{n}}
