LMIs in Control/pages/quadratic polytopic h2 optimal state feedback control
Quadratic Polytopic Full State Feedback Optimal
H
2
{\displaystyle H_{2}}
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For a system having polytopic uncertainties,
Full State Feedback is a control technique that attempts to place the system's closed-loop system poles in specified locations based on performance specifications given, such as requiring stability or bounding the overshoot of the output. By minimizing the
H
2
{\displaystyle H_{2}}
The System
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Consider System with following state-space representation.
x
˙
(
t
)
=
A
x
(
t
)
+
B
1
q
(
t
)
+
B
2
w
(
t
)
p
(
t
)
=
C
1
x
(
t
)
+
D
11
q
(
t
)
+
D
12
w
(
t
)
z
(
t
)
=
C
2
x
(
t
)
+
D
21
q
(
t
)
+
D
22
w
(
t
)
{\displaystyle {\begin{aligned}{\dot {x}}(t)&=Ax(t)+B_{1}q(t)+B_{2}w(t)\\p(t)&=C_{1}x(t)+D_{11}q(t)+D_{12}w(t)\\z(t)&=C_{2}x(t)+D_{21}q(t)+D_{22}w(t)\\\end{aligned}}}
where
x
∈
R
m
{\displaystyle x\in \mathbb {R} ^{m}}
q
∈
R
n
{\displaystyle q\in \mathbb {R} ^{n}}
w
∈
R
g
{\displaystyle w\in \mathbb {R} ^{g}}
A
∈
R
m
x
m
{\displaystyle A\in \mathbb {R} ^{mxm}}
B
1
∈
R
m
x
n
{\displaystyle B_{1}\in \mathbb {R} ^{mxn}}
B
2
∈
R
m
x
g
{\displaystyle B_{2}\in \mathbb {R} ^{mxg}}
p
∈
R
p
{\displaystyle p\in \mathbb {R} ^{p}}
C
1
∈
R
p
x
m
{\displaystyle C_{1}\in \mathbb {R} ^{pxm}}
D
11
∈
R
p
x
n
{\displaystyle D_{11}\in \mathbb {R} ^{pxn}}
D
12
∈
R
p
x
g
{\displaystyle D_{12}\in \mathbb {R} ^{pxg}}
z
∈
R
s
{\displaystyle z\in \mathbb {R} ^{s}}
C
2
∈
R
s
x
m
{\displaystyle C_{2}\in \mathbb {R} ^{sxm}}
D
21
∈
R
s
x
n
{\displaystyle D_{21}\in \mathbb {R} ^{sxn}}
D
22
∈
R
s
x
g
{\displaystyle D_{22}\in \mathbb {R} ^{sxg}}
t
∈
R
{\displaystyle t\in \mathbb {R} }
Add uncertainty to system matrices
A
,
B
1
,
B
2
,
C
1
,
C
2
,
D
11
,
D
12
{\displaystyle A,B_{1},B_{2},C_{1},C_{2},D_{11},D_{12}}
New state-space representation
x
˙
(
t
)
=
(
A
+
A
i
)
x
(
t
)
+
(
B
1
+
B
i
)
q
(
t
)
+
(
B
2
+
B
i
)
w
(
t
)
p
(
t
)
=
(
C
1
+
C
i
)
x
(
t
)
+
(
D
11
+
D
i
)
q
(
t
)
+
(
D
12
+
D
i
)
w
(
t
)
z
(
t
)
=
C
2
x
(
t
)
+
D
21
q
(
t
)
+
D
22
w
(
t
)
{\displaystyle {\begin{aligned}{\dot {x}}(t)&=(A+A_{i})x(t)+(B_{1}+B_{i})q(t)+(B_{2}+B_{i})w(t)\\p(t)&=(C_{1}+C_{i})x(t)+(D_{11}+D_{i})q(t)+(D_{12}+D_{i})w(t)\\z(t)&=C_{2}x(t)+D_{21}q(t)+D_{22}w(t)\\\end{aligned}}}
The Data
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The matrices necessary for this LMI are
The Optimization Problem:
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Recall the closed-loop in state feedback is:
S
(
P
,
K
)
=
{\displaystyle S(P,K)=}
[
A
+
B
22
F
B
1
C
1
+
D
12
F
D
11
]
{\displaystyle {\begin{aligned}{\begin{bmatrix}A+B_{22}F&&B_{1}\\C_{1}+D_{12}F&&D_{11}\end{bmatrix}}\\\end{aligned}}}
This problem can be formulated as
H
2
{\displaystyle H_{2}}
The LMI: An LMI for Quadratic Polytopic
H
2
{\displaystyle H_{2}}
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State-Feedback Control
|
|
S
(
P
(
Δ
)
,
K
(
0
,
0
,
0
,
F
)
)
|
|
H
2
≤
γ
{\displaystyle ||S(P(\Delta ),K(0,0,0,F))||_{H_{2}}\leq \gamma }
X
>
0
{\displaystyle X>0}
[
A
X
+
B
2
Z
+
X
A
T
+
Z
T
B
2
T
B
1
B
1
T
−
I
]
+
[
A
i
X
+
B
2
,
i
Z
+
X
A
i
T
+
Z
T
B
2
,
I
T
B
1
,
i
B
1
,
i
T
0
]
<
0
i
=
1
,
.
.
.
.
.
.
,
k
{\displaystyle {\begin{aligned}{\begin{bmatrix}AX+B_{2}Z+XA^{T}+Z^{T}B_{2}^{T}&&B_{1}\\B_{1}^{T}&&-I\end{bmatrix}}+{\begin{bmatrix}A_{i}X+B_{2,i}Z+XA_{i}^{T}+Z^{T}B_{2,I}^{T}&&B_{1,i}\\B_{1,i}^{T}&&0\end{bmatrix}}<0\quad i=1,......,k\end{aligned}}}
[
X
(
C
1
X
+
D
12
Z
)
T
C
1
X
+
D
12
Z
W
]
+
[
0
(
C
1
,
i
X
+
D
12
,
i
Z
)
T
C
1
,
i
X
+
D
12
,
i
Z
0
]
>
0
i
=
1
,
.
.
.
.
.
.
,
k
{\displaystyle {\begin{aligned}{\begin{bmatrix}X&&(C_{1}X+D_{12}Z)^{T}\\C_{1}X+D_{12}Z&&W\end{bmatrix}}+{\begin{bmatrix}0&&(C_{1,i}X+D_{12,i}Z)^{T}\\C_{1,i}X+D_{12,i}Z&&0\end{bmatrix}}>0\quad i=1,......,k\end{aligned}}}
T
r
a
c
e
W
<
γ
{\displaystyle {\begin{aligned}\\TraceW<\gamma \end{aligned}}}
Conclusion:
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The
H
2
{\displaystyle H_{2}}
F
=
Z
X
−
1
{\displaystyle F=ZX^{-1}}
Implementation:
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Related LMIs
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External Links
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