Discrete-Time H∞-Optimal Dynamic Output Feedback Control
A discrete time system operates on a discrete time signal input and produces a discrete time signal output. They are used in digital signal processing, such as digital filters for images or sound. The class of discrete time systems that are both linear and time invariant, known as discrete time LTI systems.
A Dynamic Output feedback controller is designed for a Discrete Time system, to minimize the H∞ norm of the closed loop system with exogenous input
w
k
{\displaystyle w_{k}}
z
k
{\displaystyle z_{k}}
Discrete-Time LTI System with state space realization
(
A
d
,
B
d
,
C
d
,
D
d
)
{\displaystyle (A_{d},B_{d},C_{d},D_{d})}
x
k
+
1
=
A
d
x
k
+
B
d
1
w
k
+
B
d
2
u
k
z
k
=
C
d
1
x
k
+
D
d
11
w
k
+
D
d
12
u
k
y
k
=
C
d
2
x
k
+
D
d
21
w
k
+
D
d
22
u
k
{\displaystyle {\begin{aligned}x_{k+1}&=A_{d}x_{k}+B_{d1}w_{k}+B_{d2}u_{k}\\z_{k}&=C_{d1}x_{k}+D_{d11}w_{k}+D_{d12}u_{k}\\y_{k}&=C_{d2}x_{k}+D_{d21}w_{k}+D_{d22}u_{k}\\\end{aligned}}}
The matrices: System
(
A
d
,
B
d
1
,
B
d
2
,
C
d
1
,
C
d
1
,
D
d
11
,
D
d
12
,
D
d
21
,
D
d
22
)
,
X
1
,
Y
1
,
Z
,
,
X
2
,
Y
2
{\displaystyle (A_{d},B_{d1},B_{d2},C_{d1},C_{d1},D_{d11},D_{d12},D_{d21},D_{d22}),X_{1},Y_{1},Z,,X_{2},Y_{2}}
Controller
(
A
d
c
,
B
d
c
,
C
d
c
,
D
d
c
)
{\displaystyle (A_{dc},B_{dc},C_{dc},D_{dc})}
The Optimization Problem
edit
The following feasibility problem should be optimized:
γ
{\displaystyle \gamma }
Discrete-Time H∞-Optimal Dynamic Output Feedback Control
The LMI formulation
H∞ norm <
γ
{\displaystyle \gamma }
X
1
,
Y
1
∈
S
n
x
;
Z
∈
S
n
z
;
γ
∈
R
>
0
A
d
n
∈
R
n
x
∗
n
x
;
B
d
n
∈
R
n
x
∗
n
y
;
C
d
n
∈
R
n
u
∗
n
x
;
D
d
n
∈
R
n
u
∗
n
y
;
X
1
>
0
Y
1
>
0
Z
>
0
[
X
1
1
X
1
A
d
+
B
d
n
C
d
2
A
d
n
X
1
B
d
1
+
B
d
n
D
d
21
0
∗
Y
1
A
d
+
B
d
2
D
d
n
C
d
2
A
d
Y
1
+
B
d
2
C
d
n
B
d
1
+
B
d
2
D
d
n
D
d
21
0
∗
∗
X
1
1
0
C
d
1
T
+
C
d
2
T
D
d
n
T
D
d
12
T
∗
∗
∗
Y
1
0
Y
1
C
d
1
T
+
C
d
n
T
D
d
12
T
∗
∗
∗
∗
−
γ
I
D
d
11
T
+
D
d
21
T
D
d
n
T
D
d
12
T
∗
∗
∗
∗
∗
−
γ
I
]
>
0
,
[
X
1
1
∗
Y
1
]
>
0
,
{\displaystyle {\begin{aligned}X_{1},Y_{1}\in {S^{n_{x}}};Z\in {S^{n_{z}}};\gamma \in {R_{>0}}\;\\A_{dn}\in {R^{n_{x}*n_{x}}};B_{dn}\in {R^{n_{x}*n_{y}}};C_{dn}\in {R^{n_{u}*n_{x}}};D_{dn}\in {R^{n_{u}*n_{y}}};\\&X_{1}>0\\&Y_{1}>0\\&Z>0\\{\begin{bmatrix}X_{1}&1&X_{1}A_{d}+B_{dn}C_{d2}&A_{dn}&X_{1}B_{d1}+B_{dn}D_{d21}&0\\*&Y_{1}&A_{d}+B_{d2}D_{dn}C_{d2}&A_{d}Y_{1}+B_{d2}C_{dn}&B_{d1}+B_{d2}D_{dn}D_{d21}&0\\*&*&X_{1}&1&0&C_{d1}^{T}+C_{d2}^{T}D_{dn}^{T}D_{d12}^{T}\\*&*&*&Y_{1}&0&Y_{1}C_{d1}^{T}+C_{dn}^{T}D_{d12}^{T}\\*&*&*&*&-\gamma I&D_{d11}^{T}+D_{d21}^{T}D_{dn}^{T}D_{d12}^{T}\\*&*&*&*&*&-\gamma I\end{bmatrix}}&>0,\\{\begin{bmatrix}X_{1}&1\\*&Y_{1}\end{bmatrix}}&>0,\\\end{aligned}}}
The controller is recovered by
A
d
c
=
A
d
k
−
B
d
c
(
1
−
D
d
22
D
d
c
)
−
1
D
d
22
C
d
c
B
d
c
=
B
d
k
(
1
−
D
d
c
D
d
22
)
C
d
c
=
(
1
−
D
d
c
D
d
22
)
C
d
k
D
d
c
=
1
+
D
d
k
D
d
22
)
−
1
D
d
k
{\displaystyle {\begin{aligned}&A_{dc}=A_{dk}-B_{dc}(1-D_{d22}D_{dc})^{-1}D_{d22}C_{dc}\\&B_{dc}=B_{dk}(1-D_{dc}D_{d22})\\&C_{dc}=(1-D_{dc}D_{d22})C_{dk}\\&D_{dc}=1+D_{dk}D_{d22})^{-1}D_{dk}\\\end{aligned}}}
where,
[
A
d
k
B
d
k
C
d
k
D
d
k
]
=
[
X
2
X
1
B
d
2
0
1
]
−
1
(
[
A
d
n
B
d
n
C
d
n
D
d
n
]
−
[
X
1
A
d
Y
1
0
0
0
]
)
[
Y
2
T
0
C
d
2
Y
1
1
]
−
1
{\displaystyle {\begin{aligned}{\begin{bmatrix}A_{dk}&B_{dk}\\C_{dk}&D_{dk}\end{bmatrix}}&={\begin{bmatrix}X_{2}&X_{1}B_{d2}\\0&1\end{bmatrix}}^{-1}({\begin{bmatrix}A_{dn}&B_{dn}\\C_{dn}&D_{dn}\end{bmatrix}}-{\begin{bmatrix}X_{1}A_{d}Y_{1}&0\\0&0\end{bmatrix}}){\begin{bmatrix}Y_{2}^{T}&0\\C_{d2}Y_{1}&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}}
Given
X
1
{\displaystyle X_{1}}
Y
1
{\displaystyle Y_{1}}
X
2
{\displaystyle X_{2}}
Y
2
{\displaystyle Y_{2}}
If
D
d
11
=
0
{\displaystyle D_{d11}=0}
D
d
12
{\displaystyle D_{d12}}
D
d
21
{\displaystyle D_{d21}}
D
d
n
=
0
{\displaystyle D_{dn}=0}
The Discrete-Time H∞-Optimal Dynamic Output Feedback Controller is the system
(
A
d
c
,
B
d
c
,
C
d
c
,
D
d
c
)
{\displaystyle (A_{dc},B_{dc},C_{dc},D_{dc})}
A link to CodeOcean or other online implementation of the LMI
[1] - Continuous Time H∞ Optimal Dynamic Output Feedback Control
A list of references documenting and validating the LMI.
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