LMIs in Control/pages/H-2 filtering
For systems that have disturbances, filtering can be used to reduce the effects of these disturbances. Described on this page is a method of attaining a filter that will reduce the effects of the disturbances as completely as possible. To do this, we look to find a set of new coefficient matrices that describe the filtered system. The process to achieve such a new system is described below. The H2-filter tries to minimize the average magnitude of error.
For the application of this LMI, we will look at linear systems that can be represented in state space as
x
˙
=
A
x
+
B
w
,
x
(
0
)
=
x
0
y
=
C
x
+
D
w
z
=
L
x
{\displaystyle {\begin{aligned}{\dot {x}}&=Ax+Bw,x(0)=x_{0}\\y&=Cx+Dw\\z&=Lx\end{aligned}}}
where
x
∈
R
n
,
y
∈
R
l
,
z
∈
R
m
{\displaystyle x\in R^{n},y\in R^{l},z\in R^{m}}
represent the state vector, the measured output vector, and the output vector of interest, respectively,
w
∈
R
p
{\displaystyle w\in R^{p}}
is the disturbance vector, and
A
,
B
,
C
,
D
{\displaystyle A,B,C,D}
and
L
{\displaystyle L}
are the system matrices of appropriate dimension.
To further define:
x
{\displaystyle x}
is
∈
R
n
{\displaystyle \in R^{n}}
and is the state vector,
A
{\displaystyle A}
is
∈
R
n
∗
n
{\displaystyle \in R^{n*n}}
and is the state matrix,
B
{\displaystyle B}
is
∈
R
n
∗
r
{\displaystyle \in R^{n*r}}
and is the input matrix,
w
{\displaystyle w}
is
∈
R
r
{\displaystyle \in R^{r}}
and is the exogenous input,
C
{\displaystyle C}
is
∈
R
m
∗
n
{\displaystyle \in R^{m*n}}
and is the output matrix,
D
{\displaystyle D}
and
L
{\displaystyle L}
are
∈
R
m
∗
r
{\displaystyle \in R^{m*r}}
and are feedthrough matrices, and
y
{\displaystyle y}
and
z
{\displaystyle z}
are
∈
R
m
{\displaystyle \in R^{m}}
and are the output and the output of interest, respectively.
The Optimization Problem
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We need to design a filter that will eliminate the effects of the disturbances as best we can. For this, we take a filter of the following form:
σ
˙
=
A
f
σ
+
B
f
y
,
σ
(
0
)
=
σ
0
z
^
=
C
f
σ
,
{\displaystyle {\begin{aligned}{\dot {\sigma }}&=A_{f}\sigma +B_{f}y,\sigma (0)=\sigma _{0}\\{\hat {z}}&=C_{f}\sigma ,\end{aligned}}}
where
σ
∈
R
n
{\displaystyle \sigma \in R^{n}}
is the state vector,
z
^
∈
R
m
{\displaystyle {\hat {z}}\in R^{m}}
is the estimation vector, and
A
f
,
B
f
,
C
f
{\displaystyle A_{f},B_{f},C_{f}}
are the coefficient matrices of appropriate dimensions.
Note that the combined complete system can be represented as
x
˙
e
=
A
~
x
e
+
B
~
w
,
x
e
(
0
)
=
x
e
0
z
~
=
C
~
x
e
,
{\displaystyle {\begin{aligned}{\dot {x}}_{e}&={\tilde {A}}x_{e}+{\tilde {B}}w,x_{e}(0)=x_{e0}\\{\tilde {z}}&={\tilde {C}}x_{e},\end{aligned}}}
where
z
~
=
z
−
z
^
∈
R
m
{\displaystyle {\tilde {z}}=z-{\hat {z}}\in R^{m}}
is the estimation error,
x
e
=
[
x
σ
]
{\displaystyle {\begin{aligned}x_{e}={\begin{bmatrix}x\\\sigma \end{bmatrix}}\\\end{aligned}}}
is the state vector of the system, and
A
~
,
B
~
,
C
~
{\displaystyle {\tilde {A}},{\tilde {B}},{\tilde {C}}}
are the coefficient matrices, defined as:
A
~
=
[
A
0
B
f
C
A
f
]
,
B
~
=
[
B
B
f
D
]
,
C
~
=
[
L
−
C
f
]
{\displaystyle {\begin{aligned}{\tilde {A}}={\begin{bmatrix}A&0\\B_{f}C&A_{f}\end{bmatrix}},{\tilde {B}}={\begin{bmatrix}B\\B_{f}D\end{bmatrix}},\\{\tilde {C}}={\begin{bmatrix}L&-C_{f}\end{bmatrix}}\end{aligned}}}
In other words, for the system defined above we need to find
A
f
,
B
f
,
C
f
{\displaystyle A_{f},B_{f},C_{f}}
such that
|
|
G
z
~
w
(
s
)
|
|
2
<
γ
,
{\displaystyle {\begin{aligned}||G_{{\tilde {z}}w}(s)||_{2}<\gamma ,\end{aligned}}}
where
γ
{\displaystyle \gamma }
is a positive constant, and
G
z
~
w
(
s
)
=
C
~
(
s
I
−
A
~
)
−
1
B
~
{\displaystyle {\begin{aligned}G_{{\tilde {z}}w}(s)={\tilde {C}}(sI-{\tilde {A}})^{-1}{\tilde {B}}\end{aligned}}}
The LMI: H-2 Filtering
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For this LMI, the solution exists if one of the following sets of LMIs hold:
Matrices
R
,
X
,
M
,
N
,
Z
,
Q
{\displaystyle R,X,M,N,Z,Q}
exist that obey the following LMIs:
R
−
X
>
0
,
t
r
a
c
e
(
Q
)
<
γ
2
,
[
−
Q
∗
∗
L
T
−
R
∗
−
N
T
−
X
−
X
]
<
0
,
[
R
A
+
A
T
R
+
Z
C
+
C
T
Z
T
∗
∗
M
T
+
Z
C
+
X
A
M
T
+
M
∗
B
T
R
+
D
T
Z
T
B
T
X
+
D
T
Z
T
−
I
]
<
0.
{\displaystyle {\begin{aligned}R-X&>0,\\trace(Q)&<\gamma ^{2},\\{\begin{bmatrix}-Q&*&*\\L^{T}&-R&*\\-N^{T}&-X&-X\end{bmatrix}}&<0,\\{\begin{bmatrix}RA+A^{T}R+ZC+C^{T}Z^{T}&*&*\\M^{T}+ZC+XA&M^{T}+M&*\\B^{T}R+D^{T}Z^{T}&B^{T}X+D^{T}Z^{T}&-I\end{bmatrix}}&<0.\\\end{aligned}}}
or
Matrices
R
¯
,
X
¯
,
M
¯
,
N
¯
,
Z
¯
,
Q
¯
{\displaystyle {\bar {R}},{\bar {X}},{\bar {M}},{\bar {N}},{\bar {Z}},{\bar {Q}}}
exist that obey the following LMIs:
R
¯
−
X
¯
>
0
,
t
r
a
c
e
(
Q
¯
)
<
γ
2
,
[
−
Q
¯
∗
∗
R
¯
B
+
Z
¯
D
−
R
¯
∗
X
¯
B
+
Z
¯
D
−
X
¯
−
I
]
<
0
,
[
R
¯
A
+
A
T
R
¯
+
Z
¯
C
+
C
T
Z
¯
T
∗
∗
M
¯
T
+
Z
¯
C
+
X
¯
A
M
¯
T
+
M
¯
∗
L
−
N
¯
−
I
]
<
0.
{\displaystyle {\begin{aligned}{\bar {R}}-{\bar {X}}&>0,\\trace({\bar {Q}})&<\gamma ^{2},\\{\begin{bmatrix}-{\bar {Q}}&*&*\\{\bar {R}}B+{\bar {Z}}D&-{\bar {R}}&*\\{\bar {X}}B+{\bar {Z}}D&-{\bar {X}}&-I\end{bmatrix}}&<0,\\{\begin{bmatrix}{\bar {R}}A+A^{T}{\bar {R}}+{\bar {Z}}C+C^{T}{\bar {Z}}^{T}&*&*\\{\bar {M}}^{T}+{\bar {Z}}C+{\bar {X}}A&{\bar {M}}^{T}+{\bar {M}}&*\\L&-{\bar {N}}&-I\end{bmatrix}}&<0.\\\end{aligned}}}
To find the corresponding filter, use the optimized matrices from the first solution to find:
A
f
=
X
−
1
M
,
B
f
=
X
−
1
Z
,
C
f
=
N
{\displaystyle A_{f}=X^{-1}M,B_{f}=X^{-1}Z,C_{f}=N}
Or the second solution to find:
A
f
=
X
¯
−
1
M
¯
,
B
f
=
X
¯
−
1
Z
¯
,
C
f
=
N
¯
{\displaystyle A_{f}={\bar {X}}^{-1}{\bar {M}},B_{f}={\bar {X}}^{-1}{\bar {Z}},C_{f}={\bar {N}}}
These matrices can then be used to produce
A
~
,
B
~
,
C
~
{\displaystyle {\tilde {A}},{\tilde {B}},{\tilde {C}}}
to construct the final filter below, that will best eliminate the disturbances of the system.
x
˙
e
=
A
~
x
e
+
B
~
w
,
x
e
(
0
)
=
x
e
0
z
~
=
C
~
x
e
,
{\displaystyle {\begin{aligned}{\dot {x}}_{e}&={\tilde {A}}x_{e}+{\tilde {B}}w,x_{e}(0)=x_{e0}\\{\tilde {z}}&={\tilde {C}}x_{e},\end{aligned}}}
This LMI comes from
[1] - "LMIs in Control Systems: Analysis, Design and Applications" by Guang-Ren Duan and Hai-Hua Yu
Other resources:
Duan, G. (2013). LMIs in control systems: analysis, design and applications. Boca Raton: CRC Press, Taylor & Francis Group.
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