Introduction
edit
The goal of optimal filtering is to design a filter that acts on the output z of the generalized plant and optimizes the transfer matrix from w
to the filtered output. An H2-optimal filter is designed to minimize the
H
2
{\displaystyle H_{2}}
P
¯
(
s
)
{\displaystyle {\overline {P}}(s)}
System Dynamics
edit
Consider the discrete-time generalized LTI plant with minimal states space
realization
{
X
k
+
1
=
A
d
X
k
+
B
d
1
w
k
,
Z
k
=
C
d
1
X
k
+
D
d
11
w
k
,
Y
k
=
C
d
2
X
k
+
D
d
21
w
k
,
{\displaystyle {\begin{aligned}&\quad \quad {\begin{cases}X_{k+1}=A_{d}X_{k}+B_{d1}w_{k},\\Z_{k}=C_{d1}X_{k}+D_{d11}w_{k},\\Y_{k}=C_{d2}X_{k}+D_{d21}w_{k},\\\end{cases}}\end{aligned}}}
where it is assumed that A_{d} is Schur. A discrete-time dynamics LTI filter with state-space realization
{
X
f
,
k
+
1
=
A
f
X
f
,
k
+
B
f
Y
k
,
Z
^
k
=
C
f
X
f
,
k
+
D
f
y
k
,
{\displaystyle {\begin{aligned}&\quad \quad {\begin{cases}X_{f,k+1}=A_{f}X_{f,k}+B_{f}Y_{k},\\{\hat {Z}}_{k}=C_{f}X_{f,k}+D_{f}y_{k},\\\end{cases}}\end{aligned}}}
is to be designed to optimize the transfer function from w{k} to
z
¯
k
=
z
k
−
z
^
k
{\displaystyle {\overline {z}}_{k}=z_{k}-{\hat {z}}_{k}}
P
¯
(
z
)
=
C
¯
(
d
1
)
(
z
I
−
A
¯
d
)
−
1
B
¯
d
1
+
D
¯
d
11
{\displaystyle {\overline {P}}(z)={\overline {C}}(d1)(zI-{\overline {A}}_{d})^{-1}{\overline {B}}_{d1}+{\overline {D}}_{d11}}
where
A
¯
d
=
[
A
d
0
B
f
C
d
2
A
f
]
,
B
¯
d
1
=
[
B
d
1
B
f
D
d
21
]
,
C
¯
d
1
=
[
C
d
1
−
D
f
C
d
2
−
C
f
]
,
d
¯
d
11
=
D
d
11
−
D
f
D
d
21
.
{\displaystyle {\overline {A}}_{d}={\begin{bmatrix}A_{d}&0\\B_{f}C_{d2}&A_{f}\end{bmatrix}},{\overline {B}}_{d1}={\begin{bmatrix}B_{d1}\\B_{f}D_{d21}\end{bmatrix}},{\overline {C}}_{d1}={\begin{bmatrix}C_{d1}-D_{f}C_{d2}-C_{f}\\\end{bmatrix}},{\overline {d}}_{d11}=D_{d11}-D_{f}D_{d21}.}
The Optimization Problem
edit
Solve for
A
n
∈
R
n
x
X
n
x
,
B
n
∈
R
n
x
X
n
y
,
C
f
∈
R
n
z
X
n
x
,
D
f
∈
R
n
z
X
n
y
,
X
,
Y
∈
S
n
z
,
a
n
d
{\displaystyle A_{n}\in \mathbb {R} ^{n_{x}Xn_{x}},B_{n}\in \mathbb {R} ^{n_{x}Xn_{y}},C_{f}\in \mathbb {R} ^{n_{z}Xn_{x}},D_{f}\in \mathbb {R} ^{n_{z}Xn_{y}},X,Y\in \mathbb {S} ^{n_{z}},and}
v
∈
R
>
0
{\displaystyle v\in \mathbb {R} >0}
J
(
v
)
=
v
{\displaystyle J(v)=v}
X
>
0
,
Y
>
0
,
Z
>
0.
{\displaystyle X>0,Y>0,Z>0.}
[
Y
A
+
A
T
Y
+
B
n
C
2
+
C
2
T
B
n
T
A
n
+
C
2
T
B
n
T
+
A
T
X
Y
B
1
+
B
n
D
21
∗
A
n
+
A
n
T
X
B
1
+
B
n
D
21
∗
∗
−
I
]
{\displaystyle {\begin{bmatrix}YA+A^{T}Y+B_{n}C_{2}+C_{2}^{T}B_{n}^{T}&A_{n}+C_{2}^{T}B_{n}^{T}+A^{T}X&YB_{1}+B_{n}D_{21}\\*&A_{n}+A_{n}^{T}&XB_{1}+B_{n}D_{21}\\*&*&-I\end{bmatrix}}}
[
−
Z
C
1
−
D
f
C
2
−
C
f
∗
−
Y
−
X
∗
∗
−
X
]
{\displaystyle {\begin{bmatrix}-Z&C_{1}-D_{f}C_{2}&-C_{f}\\*&-Y&-X\\*&*&-X\end{bmatrix}}}
D
11
−
D
f
D
21
=
0
,
{\displaystyle D_{11}-D_{f}D_{21}=0,}
Y
−
X
>
0
,
{\displaystyle Y-X>0,}
t
r
(
Z
)
<
v
.
{\displaystyle tr(Z)<v.}
Conclusion
edit
The filter is recovered by the state-space matrices
A
f
=
X
−
1
A
n
,
B
f
=
X
−
1
B
n
,
C
f
,
a
n
d
D
f
.
{\displaystyle A_{f}=X^{-1}A_{n},Bf=X^{-1}B_{n},C_{f},andD_{f}.}
