Discrete-Time H2-Optimal Dynamic Output Feedback Control
A Dynamic Output feedback controller is designed for a Continuous Time system, to minimize the H2 norm of the closed loop system with exogenous input
w
{\displaystyle w}
z
{\displaystyle z}
Continuous-Time LTI System with state space realization
(
A
,
B
,
C
,
D
)
{\displaystyle (A,B,C,D)}
x
˙
=
A
x
+
B
1
w
+
B
2
u
z
=
C
1
x
+
D
11
w
k
+
D
12
u
y
=
C
2
x
+
D
21
w
+
D
22
u
{\displaystyle {\begin{aligned}{\dot {x}}&=A_{x}+B_{1}w+B_{2}u\\z&=C_{1}x+D_{11}w_{k}+D_{12}u\\y&=C_{2}x+D_{21}w+D_{22}u\\\end{aligned}}}
The matrices: System
(
A
,
B
1
,
B
2
,
C
1
,
C
1
,
D
11
,
D
12
,
D
21
,
D
22
)
,
X
1
,
Y
1
,
Z
,
,
X
2
,
Y
2
{\displaystyle (A,B_{1},B_{2},C_{1},C_{1},D_{11},D_{12},D_{21},D_{22}),X_{1},Y_{1},Z,,X_{2},Y_{2}}
Controller
(
A
c
,
B
c
,
C
c
,
D
c
)
{\displaystyle (A_{c},B_{c},C_{c},D_{c})}
The Optimization Problem
edit
The following feasibility problem should be optimized:
ν
{\displaystyle \nu }
Solve for
A
n
∈
R
n
x
×
n
x
,
B
n
∈
R
n
x
×
n
y
,
C
n
∈
R
n
u
×
n
x
,
D
n
∈
R
n
u
×
n
y
,
X
1
,
Y
1
∈
S
n
x
,
Z
∈
S
n
z
,
{\displaystyle A_{n}\in {R^{n_{x}\times n_{x}}},B_{n}\in {R^{n_{x}\times n_{y}}},C_{n}\in {R^{n_{u}\times n_{x}}},D_{n}\in {R^{n_{u}\times n_{y}}},X_{1},Y_{1}\in {S^{n_{x}}},Z\in {S^{n_{z}}},}
ν
∈
R
>
0
{\displaystyle \nu \in {R_{>0}}}
J
(
ν
)
<
ν
{\displaystyle {\mathcal {J}}(\nu )<\nu }
X
1
>
0
,
Y
1
>
0
,
Z
>
0
,
{\displaystyle X_{1}>0,Y_{1}>0,Z>0,}
[
A
Y
1
+
Y
1
A
T
+
B
2
C
n
+
C
n
T
B
2
T
A
+
A
n
T
+
B
2
D
n
C
2
B
1
+
B
2
D
n
D
21
∗
X
1
A
+
A
T
X
1
+
B
n
C
2
+
C
2
T
B
n
T
X
1
B
1
+
B
n
D
21
∗
∗
−
1
]
>
0
,
[
X
1
1
Y
1
C
1
T
+
C
n
T
D
12
T
∗
Y
1
C
1
T
+
C
2
T
D
n
T
D
12
T
∗
∗
Z
]
>
0
,
D
11
+
D
12
D
n
D
21
=
0
[
X
1
1
∗
Y
1
]
>
0
,
t
r
Z
<
ν
{\displaystyle {\begin{aligned}{\begin{bmatrix}AY_{1}+Y_{1}A^{T}+B_{2}C_{n}+C_{n}^{T}B_{2}^{T}&A+A_{n}^{T}+B_{2}D_{n}C_{2}&B_{1}+B_{2}D_{n}D_{21}\\*&X_{1}A+A^{T}X_{1}+B_{n}C_{2}+C_{2}^{T}B_{n}^{T}&X_{1}B_{1}+B_{n}D_{21}\\*&*&-\mathbf {1} \end{bmatrix}}&>0,\\{\begin{bmatrix}X_{1}&\mathbf {1} &Y_{1}C_{1}^{T}+C_{n}^{T}D_{12}^{T}\\*&Y_{1}&C_{1}^{T}+C_{2}^{T}D_{n}^{T}D_{12}^{T}\\*&*&Z\end{bmatrix}}&>0,\\D_{11}+D_{12}D_{n}D_{21}=0\\{\begin{bmatrix}X_{1}&\mathbf {1} \\*&Y_{1}\end{bmatrix}}&>0,\\trZ<\nu \end{aligned}}}
The controller is recovered by
A
c
=
A
k
−
B
c
(
1
−
D
22
D
c
)
−
1
D
22
C
c
B
c
=
B
k
(
1
−
D
c
D
22
)
C
c
=
(
1
−
D
c
D
22
)
C
k
D
c
=
(
1
+
D
k
D
22
)
−
1
D
k
{\displaystyle {\begin{aligned}&A_{c}=A_{k}-B_{c}(1-D_{22}D_{c})^{-1}D_{22}C_{c}\\&B_{c}=B_{k}(1-D_{c}D_{22})\\&C_{c}=(1-D_{c}D_{22})C_{k}\\&D_{c}=(1+D_{k}D_{22})^{-1}D_{k}\\\end{aligned}}}
where,
[
A
k
B
k
C
k
D
k
]
=
[
X
2
X
1
B
2
0
1
]
−
1
(
[
A
n
B
n
C
n
D
n
]
−
[
X
1
A
Y
1
0
0
0
]
)
[
Y
2
T
0
C
Y
1
1
]
−
1
{\displaystyle {\begin{aligned}{\begin{bmatrix}A_{k}&B_{k}\\C_{k}&D_{k}\end{bmatrix}}&={\begin{bmatrix}X_{2}&X_{1}B_{2}\\\mathbf {0} &1\end{bmatrix}}^{-1}({\begin{bmatrix}A_{n}&B_{n}\\C_{n}&D_{n}\end{bmatrix}}-{\begin{bmatrix}X_{1}AY_{1}&\mathbf {0} \\\mathbf {0} &\mathbf {0} \end{bmatrix}}){\begin{bmatrix}Y_{2}^{T}&\mathbf {0} \\CY_{1}&\mathbf {1} \end{bmatrix}}^{-1}\end{aligned}}}
X
2
{\displaystyle X_{2}}
Y
2
{\displaystyle Y_{2}}
X
2
Y
2
T
=
1
−
X
1
Y
1
{\displaystyle X_{2}Y_{2}^{T}=1-X_{1}Y_{1}}
D
22
=
0
,
{\displaystyle D_{22}=0,}
A
c
=
A
k
,
B
c
=
B
k
,
C
c
=
C
k
{\displaystyle A_{c}=A_{k},B_{c}=B_{k},C_{c}=C_{k}}
D
c
=
D
k
.
{\displaystyle D_{c}=D_{k}.}
Given
X
1
{\displaystyle X_{1}}
Y
1
{\displaystyle Y_{1}}
X
2
{\displaystyle X_{2}}
Y
2
{\displaystyle Y_{2}}
If
D
11
=
0
{\displaystyle D_{11}=0}
D
12
≠
0
,
{\displaystyle D_{12}\neq 0,}
D
21
≠
0
{\displaystyle D_{21}\neq 0}
D
n
=
0
{\displaystyle D_{n}=0}
The Continuous-Time H2-Optimal Dynamic Output feedback controller is the system
(
A
c
,
B
c
,
C
c
,
D
c
)
{\displaystyle (A_{c},B_{c},C_{c},D_{c})}
The LMI given above can be implemented and solved using a tool such as YALMIP, along with an LMI solver such as MOSEK.
A list of references documenting and validating the LMI.
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