Hinf-Optimal Filter
The goal of optimal filtering is to design a filter that acts on the output
z
{\displaystyle {\mathbf {z}}}
w
{\displaystyle {\mathbf {w}}}
Consider the continuous-time generalized LTI plant, with minimal state-space representation
x
˙
=
A
x
+
B
1
w
,
{\displaystyle {\mathbf {\dot {x}}}={\mathbf {Ax}}+{\mathbf {B}}_{1}{\mathbf {w}},}
z
=
C
1
x
+
D
11
w
,
{\displaystyle {\mathbf {z}}={\mathbf {C}}_{1}{\mathbf {x}}+{\mathbf {D}}_{11}{\mathbf {w}},}
y
=
C
2
x
+
D
21
w
,
{\displaystyle {\mathbf {y}}={\mathbf {C}}_{2}{\mathbf {x}}+{\mathbf {D}}_{21}{\mathbf {w}},}
where it is assumed that
A
{\displaystyle {\mathbf {A}}}
x
˙
f
=
A
f
x
f
+
B
f
y
,
{\displaystyle {\mathbf {\dot {x}}}_{f}={\mathbf {A}}_{f}{\mathbf {x}}_{f}+{\mathbf {B}}_{f}{\mathbf {y}},}
z
^
=
C
f
x
f
+
D
f
y
,
{\displaystyle {\mathbf {\hat {z}}}={\mathbf {C}}_{f}{\mathbf {x}}_{f}+{\mathbf {D}}_{f}{\mathbf {y}},}
is designed to optimize the transfer function from
w
{\displaystyle {\mathbf {w}}}
z
~
=
z
−
z
^
{\displaystyle {\mathbf {\tilde {z}}}={\mathbf {z}}-{\mathbf {\hat {z}}}}
P
~
(
s
)
=
C
~
1
(
s
I
−
A
~
)
−
1
B
~
1
+
D
~
11
,
{\displaystyle {\tilde {\mathbf {P}}}(s)={\tilde {\mathbf {C}}}_{1}(s{\mathbf {I}}-{\tilde {\mathbf {A}}})^{-1}{\tilde {\mathbf {B}}}_{1}+{\tilde {\mathbf {D}}}_{11},}
where
A
~
=
[
A
0
B
f
C
2
A
f
]
,
{\displaystyle {\tilde {\mathbf {A}}}={\begin{bmatrix}{\mathbf {A}}&{\mathbf {0}}\\{\mathbf {B}}_{f}{\mathbf {C}}_{2}&{\mathbf {A}}_{f}\end{bmatrix}},}
B
~
1
=
[
B
1
B
f
D
21
]
,
{\displaystyle {\tilde {\mathbf {B}}}_{1}={\begin{bmatrix}{\mathbf {B}}_{1}\\{\mathbf {B}}_{f}{\mathbf {D}}_{21}\end{bmatrix}},}
C
~
1
=
[
C
1
−
D
f
C
2
−
C
f
]
,
{\displaystyle {\tilde {\mathbf {C}}}_{1}={\begin{bmatrix}{\mathbf {C}}_{1}-{\mathbf {D}}_{f}{\mathbf {C}}_{2}&-{\mathbf {C}}_{f}\end{bmatrix}},}
D
~
11
=
D
11
−
D
f
D
21
.
{\displaystyle {\tilde {\mathbf {D}}}_{11}={\mathbf {D}}_{11}-{\mathbf {D}}_{f}{\mathbf {D}}_{21}.}
Optimal Filtering seeks to minimize the given norm of the transfer function
P
~
(
s
)
.
{\displaystyle {\tilde {\mathbf {P}}}(s).}
Solve for
A
n
∈
R
n
x
×
n
x
,
B
n
∈
R
n
x
×
n
y
,
C
f
∈
R
n
z
×
n
x
,
D
f
∈
R
n
z
×
n
y
,
X
,
Y
∈
§
n
x
,
{\displaystyle {\mathbf {A}}_{n}\in \mathbb {R} ^{n_{x}\times n_{x}},{\mathbf {B}}_{n}\in \mathbb {R} ^{n_{x}\times n_{y}},{\mathbf {C}}_{f}\in \mathbb {R} ^{n_{z}\times n_{x}},{\mathbf {D}}_{f}\in \mathbb {R} ^{n_{z}\times n_{y}},{\mathbf {X,Y}}\in \S ^{n_{x}},}
γ
∈
R
>
0
{\displaystyle \gamma \in \mathbb {R} _{>0}}
J
(
γ
)
=
γ
{\displaystyle J(\gamma )=\gamma }
X
,
Y
>
0
,
{\displaystyle {\mathbf {X,Y}}>0,}
Y
−
X
>
0
,
{\displaystyle {\mathbf {Y}}-{\mathbf {X}}>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
]
<
0.
{\displaystyle {\begin{bmatrix}{\mathbf {YA}}+{\mathbf {A}}^{T}{\mathbf {Y}}+{\mathbf {B}}_{n}{\mathbf {C}}_{2}+{\mathbf {C}}_{2}^{T}{\mathbf {B}}_{n}^{T}&{\mathbf {A}}_{n}+{\mathbf {C}}_{2}^{T}{\mathbf {B}}_{n}^{T}+{\mathbf {A}}^{T}{\mathbf {X}}&{\mathbf {YB}}_{1}+{\mathbf {B}}_{n}{\mathbf {D}}_{21}\\*&{\mathbf {A}}_{n}+{\mathbf {A}}_{n}^{T}&{\mathbf {XB}}_{1}+{\mathbf {B}}_{n}{\mathbf {D}}_{21}\\*&*&-{\mathbf {I}}\end{bmatrix}}<0.}
The optimal Hinf filter is recovered by the state-space matrices
A
f
=
X
−
1
A
n
{\displaystyle {\mathbf {A}}_{f}={\mathbf {X}}^{-1}{\mathbf {A}}_{n}}
B
f
=
X
−
1
B
n
.
{\displaystyle {\mathbf {B}}_{f}={\mathbf {X}}^{-1}{\mathbf {B}}_{n}.}
The problem of optimal filtering can alternatively be formulated as a special case of synthesizing a dynamic output "feedback" controller for the generalized plant given by
x
˙
=
A
x
+
B
1
w
,
{\displaystyle {\mathbf {\dot {x}}}={\mathbf {Ax}}+{\mathbf {B}}_{1}{\mathbf {w}},}
z
=
C
1
x
+
D
11
w
−
u
,
{\displaystyle {\mathbf {z}}={\mathbf {C}}_{1}{\mathbf {x}}+{\mathbf {D}}_{11}{\mathbf {w}}-{\mathbf {u}},}
y
=
C
2
x
+
D
21
w
.
{\displaystyle {\mathbf {y}}={\mathbf {C}}_{2}{\mathbf {x}}+{\mathbf {D}}_{21}{\mathbf {w}}.}
The synthesis method presented in this page takes advantage of the fact that the controller in this case is not a true feedback controller, as it only appears as a feedthrough term in the performance channel.
A list of references documenting and validating the LMI.
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