We consider the following state-space representation for a linear system:
x
˙
=
A
x
+
B
u
y
=
C
x
+
D
u
{\displaystyle {\begin{aligned}{\dot {x}}&=Ax+Bu\\y&=Cx+Du\end{aligned}}}
where
A
{\displaystyle A}
,
B
{\displaystyle B}
,
C
{\displaystyle C}
, and
D
{\displaystyle D}
are the state matrix, input matrix, output matrix, and feedforward matrix, respectively.
These are the system (plant) matrices that can be shown as
P
=
(
A
,
B
,
C
,
D
)
{\displaystyle P=(A,B,C,D)}
.
We assume that all the four matrices of the plant,
A
,
B
,
C
,
D
{\displaystyle A,B,C,D}
, are given.
The Optimization Problem
edit
In this problem, we use an LMI to formulate and solve the optimal output-feedback problem to minimize both the <> and <> norms. Giving equal weights to each of the norms, we will have the optimization problem in the following form:
min
|
|
S
(
P
,
K
)
|
|
H
2
2
+
|
|
S
(
P
,
K
)
|
|
H
∞
2
{\displaystyle {\begin{aligned}{\text{min}}\quad ||S(P,K)||_{H_{2}}^{2}+||S(P,K)||_{H_{\infty }}^{2}\end{aligned}}}
The LMI: LMI for mixed
H
2
{\displaystyle H_{2}}
/
H
∞
{\displaystyle H_{\infty }}
edit
Mathematical description of the LMI formulation for a mixed
H
2
{\displaystyle H_{2}}
/
H
∞
{\displaystyle H_{\infty }}
optimal output-feedback problem can be written as follows:
min
γ
1
2
+
γ
2
2
s.t.
[
X
1
I
I
Y
1
]
>
0
[
A
Y
1
+
Y
1
A
T
+
B
2
C
n
+
C
n
B
2
T
∗
T
∗
T
∗
T
A
T
+
A
n
+
(
B
2
D
n
C
2
)
T
X
1
A
+
A
T
+
B
n
C
2
+
C
2
T
B
n
T
∗
T
∗
T
(
B
1
+
B
2
D
n
D
21
)
T
(
X
1
B
1
+
B
n
D
21
)
T
−
γ
I
∗
T
C
1
Y
1
+
D
12
C
n
C
1
+
D
12
D
n
C
2
D
11
+
D
12
D
n
D
21
−
γ
I
]
<
0
[
Y
1
I
(
C
1
Y
1
+
D
12
C
n
)
T
I
X
1
(
C
1
+
D
12
D
n
C
2
)
T
(
C
1
Y
1
+
D
12
C
n
)
(
C
1
+
D
12
D
n
D
21
Z
C
1
Y
1
+
D
12
C
n
C
1
+
D
12
D
n
C
2
D
11
+
D
12
D
n
D
21
−
γ
I
]
>
0
[
A
Y
1
+
Y
1
A
T
+
B
2
C
n
+
C
n
T
B
2
T
∗
T
∗
T
∗
T
(
A
T
+
A
n
+
(
B
2
∗
D
n
∗
C
2
)
T
)
X
1
A
+
A
T
X
1
+
B
n
C
2
+
C
2
T
B
n
T
∗
T
∗
T
(
B
1
+
B
2
D
n
D
21
)
T
(
X
1
B
1
+
B
n
D
21
)
T
−
γ
2
2
I
∗
T
(
C
1
Y
1
+
D
12
C
n
)
(
C
1
+
D
12
D
n
C
2
)
(
D
11
+
D
12
D
n
∗
D
21
)
−
I
]
<
0
trace
(
Z
)
<
γ
1
2
D
11
+
D
12
D
n
D
21
=
0
{\displaystyle {\begin{aligned}&{\text{min}}\quad \gamma _{1}^{2}+\gamma _{2}^{2}\\&{\text{s.t.}}\\&{\begin{bmatrix}X_{1}&I\\I&Y_{1}\end{bmatrix}}>0\\&{\begin{bmatrix}AY_{1}+Y_{1}A^{\text{T}}+B_{2}C_{n}+C_{n}B_{2}^{\text{T}}&*^{\text{T}}&*^{\text{T}}&*^{\text{T}}\\A^{\text{T}}+A_{n}+(B_{2}D_{n}C_{2})^{\text{T}}&X_{1}A+A^{\text{T}}+B_{n}C_{2}+C_{2}^{\text{T}}B_{n}^{\text{T}}&*^{\text{T}}&*^{\text{T}}\\(B_{1}+B_{2}D_{n}D_{21})^{\text{T}}&(X_{1}B_{1}+B_{n}D_{21})^{\text{T}}&-\gamma I&*^{\text{T}}\\C_{1}Y_{1}+D_{12}C_{n}&C_{1}+D_{12}D_{n}C_{2}&D_{11}+D_{12}D_{n}D_{21}&-\gamma I\\\end{bmatrix}}<0\\&{\begin{bmatrix}Y_{1}&I&(C_{1}Y_{1}+D_{12}C_{n})^{\text{T}}\\I&X_{1}&(C_{1}+D_{12}D_{n}C_{2})^{\text{T}}\\(C_{1}Y_{1}+D_{12}C_{n})&(C_{1}+D_{12}D_{n}D_{21}&Z\\C_{1}Y_{1}+D_{12}C_{n}&C_{1}+D_{12}D_{n}C_{2}&D_{11}+D_{12}D_{n}D_{21}&-\gamma I\\\end{bmatrix}}>0\\&{\begin{bmatrix}AY_{1}+Y_{1}A^{\text{T}}+B_{2}C_{n}+C_{n}{\text{T}}B_{2}{\text{T}}&*^{\text{T}}&*^{\text{T}}&*^{\text{T}}\\(A^{\text{T}}+An+(B_{2}*D_{n}*C_{2})^{\text{T}})&X_{1}A+A^{\text{T}}X_{1}+B_{n}C_{2}+C_{2}^{\text{T}}B_{n}^{\text{T}}&*^{\text{T}}&*^{\text{T}}\\(B_{1}+B_{2}D_{n}D_{21})^{\text{T}}&(X_{1}B_{1}+B_{n}D_{21})^{\text{T}}&-\gamma _{2}^{2}I&*^{\text{T}}\\(C_{1}Y_{1}+D_{12}C_{n})&(C_{1}+D_{12}D_{n}C_{2})&(D_{11}+D_{12}D_{n}*D_{21})&-I\\\end{bmatrix}}<0\\&{\text{trace}}(Z)<\gamma _{1}^{2}\\&D_{11}+D_{12}D_{n}D_{21}=0\end{aligned}}}
where
γ
1
2
{\displaystyle \gamma _{1}^{2}}
and
γ
1
2
{\displaystyle \gamma _{1}^{2}}
are defined as the
H
2
{\displaystyle H_{2}}
and
H
∞
{\displaystyle H_{\infty }}
norm of the system:
|
|
S
(
P
,
K
)
|
|
H
2
2
=
γ
1
2
|
|
S
(
P
,
K
)
|
|
H
∞
2
=
γ
2
2
{\displaystyle {\begin{aligned}&||S(P,K)||_{H_{2}}^{2}=\gamma _{1}^{2}\\&||S(P,K)||_{H_{\infty }}^{2}=\gamma _{2}^{2}\end{aligned}}}
Moreover,
X
1
{\displaystyle X_{1}}
,
Y
1
{\displaystyle Y_{1}}
,
A
n
{\displaystyle A_{n}}
,
B
n
{\displaystyle B_{n}}
,
C
n
{\displaystyle C_{n}}
, and
D
n
{\displaystyle D_{n}}
are variable matrices with appropriate dimensions that are found after solving the LMIs.
[1] - LMI in Control Systems Analysis, Design and Applications