H
∞
{\displaystyle H_{\infty }}
Optimal Output Controllability for Systems With Transients
This LMI provides an
H
∞
{\displaystyle H_{\infty }}
optimal output controllability problem to check if such controllers for systems with unknown exogenous disturbances and initial conditions can exist or not.
x
˙
=
A
x
+
B
1
v
+
B
2
u
,
x
(
0
)
=
x
0
,
z
=
C
1
x
+
D
11
v
+
D
12
u
,
y
=
C
2
x
+
D
21
v
,
{\displaystyle {\begin{aligned}{\dot {x}}&=Ax+B_{1}v+B_{2}u,x(0)=x_{0},\\z&=C_{1}x+D_{11}v+D_{12}u,\\y&=C_{2}x+D_{21}v,\end{aligned}}}
where
x
∈
R
n
{\displaystyle x\in \mathbb {R} ^{n}}
is the state,
v
∈
R
r
{\displaystyle v\in \mathbb {R} ^{r}}
is the exogenous input,
u
∈
R
m
{\displaystyle u\in \mathbb {R} ^{m}}
is the control input,
y
∈
R
p
{\displaystyle y\in \mathbb {R} ^{p}}
is the measured output and
z
∈
R
s
{\displaystyle z\in \mathbb {R} ^{s}}
is the regulated output.
System matrices
(
A
,
B
1
,
B
2
,
C
1
,
C
2
,
D
11
,
D
12
,
D
21
,
D
22
)
{\displaystyle (A,B_{1},B_{2},C_{1},C_{2},D_{11},D_{12},D_{21},D_{22})}
need to be known. It is assumed that
v
∈
L
2
[
0
,
∞
)
{\displaystyle v\in L_{2}[0,\infty )}
.
N
1
,
N
2
{\displaystyle N_{1},N_{2}}
are matrices with their columns forming the bais of kernels of
C
2
D
21
{\displaystyle C_{2}D_{21}}
and
C
2
D
12
{\displaystyle C_{2}D_{12}}
respectively.
The Optimization Problem
edit
For a given
γ
{\displaystyle \gamma }
, the following
H
∞
{\displaystyle H_{\infty }}
condition needs to be fulfilled:
γ
w
=
s
u
p
‖
v
‖
∞
2
+
x
0
⊤
R
x
0
≠
0
‖
z
‖
∞
(
‖
v
‖
∞
2
+
x
0
⊤
R
x
0
)
1
/
2
<
γ
w
,
{\displaystyle \gamma _{w}=sup_{\|v\|_{\infty }^{2}+x_{0}^{\top }Rx_{0}\neq 0}{\frac {\|z\|_{\infty }}{(\|v\|_{\infty }^{2}+x_{0}^{\top }Rx_{0})^{1/2}}}<\gamma _{w},}
The LMI:
H
∞
{\displaystyle H_{\infty }}
Output Feedback Controller for Systems With Transients
edit
min
γ
,
X
11
,
Y
11
:
γ
subj. to:
X
11
>
0
,
Y
11
>
0
,
[
N
1
0
0
I
]
⊤
[
A
⊤
X
11
+
X
11
A
X
11
B
1
C
1
⊤
∗
−
γ
2
I
D
11
⊤
∗
∗
−
I
]
[
N
1
0
0
I
]
<
0
,
[
N
2
0
0
I
]
⊤
[
A
Y
11
+
Y
11
A
⊤
Y
11
C
1
⊤
B
1
∗
−
I
D
11
∗
∗
−
γ
2
I
]
[
N
2
0
0
I
]
<
0
,
[
X
11
I
I
Y
11
]
≥
0
,
X
11
<
γ
2
R
,
{\displaystyle {\begin{aligned}&{\text{min}}_{\gamma ,X_{11},Y_{11}}:\gamma \\&{\text{subj. to: }}X_{11}>0,Y_{11}>0,\\&\quad {\begin{bmatrix}N_{1}&0\\0&I\end{bmatrix}}^{\top }{\begin{bmatrix}A^{\top }X_{11}+X_{11}A&X_{11}B_{1}&C_{1}^{\top }\\*&-\gamma ^{2}I&D_{11}^{\top }\\*&*&-I\end{bmatrix}}{\begin{bmatrix}N_{1}&0\\0&I\end{bmatrix}}<0,\\&\quad {\begin{bmatrix}N_{2}&0\\0&I\end{bmatrix}}^{\top }{\begin{bmatrix}AY_{11}+Y_{11}A^{\top }&Y_{11}C_{1}^{\top }&B_{1}\\*&-I&D_{11}\\*&*&-\gamma ^{2}I\end{bmatrix}}{\begin{bmatrix}N_{2}&0\\0&I\end{bmatrix}}<0,\\&\quad {\begin{bmatrix}X_{11}&I\\I&Y_{11}\end{bmatrix}}\geq 0,X_{11}<\gamma ^{2}R,\end{aligned}}}
Solution of the above LMI gives a check to see if an
H
∞
{\displaystyle H_{\infty }}
optimal output controller for systems with transients can exist or not.
A link to CodeOcean or other online implementation of the LMI
Links to other closely-related LMIs
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