WIP, Description in progress
This part shows how to design dynamic outpur feedback control in mixed
H
2
{\displaystyle {\mathcal {H}}_{2}}
and
H
∞
{\displaystyle {\mathcal {H}}_{\infty }}
sense for the discrete time .
Consider the discrete-time generalized LTI plant
P
{\displaystyle {\mathcal {P}}}
with minimal state-space realization
x
k
+
1
=
A
d
x
k
+
[
B
d
1
,
1
B
d
1
,
2
]
[
w
1
,
k
w
2
,
k
]
+
B
d
,
2
u
k
,
{\displaystyle x_{k+1}=A_{d}x_{k}+{\begin{bmatrix}B_{d1,1}&B_{d1,2}\end{bmatrix}}{\begin{bmatrix}w_{1,k}\\w_{2,k}\end{bmatrix}}+B_{d,2}u_{k},}
[
z
1
,
k
z
2
,
k
]
=
[
C
d
1
,
1
D
d
1
,
2
]
x
k
+
[
D
d
11
,
11
D
d
11
,
12
D
d
11
,
21
D
d
11
,
22
]
[
w
1
,
k
w
2
,
k
]
+
[
D
12
,
1
D
12
,
2
]
u
k
,
{\displaystyle {\begin{bmatrix}z_{1,k}\\z_{2,k}\end{bmatrix}}={\begin{bmatrix}C_{d1,1}\\D_{d1,2}\end{bmatrix}}x_{k}+{\begin{bmatrix}D_{d11,11}&D_{d11,12}\\D_{d11,21}&D_{d11,22}\end{bmatrix}}{\begin{bmatrix}w_{1,k}\\w_{2,k}\end{bmatrix}}+{\begin{bmatrix}D_{12,1}\\D_{12,2}\end{bmatrix}}u_{k},}
y
k
=
C
d
2
x
k
+
[
D
21
,
1
D
21
,
2
]
[
w
1
,
k
w
2
,
k
]
+
D
d
22
u
k
{\displaystyle y_{k}=C_{d2}x_{k}+{\begin{bmatrix}D_{21,1}&D_{21,2}\end{bmatrix}}{\begin{bmatrix}w_{1,k}\\w_{2,k}\end{bmatrix}}+D_{d22}u_{k}}
A discrete-time dynamic output feedback LTI controller with state-space realization
(
A
d
c
,
B
d
c
,
C
d
c
,
D
d
c
)
{\displaystyle (A_{dc},B_{dc},C_{dc},D_{dc})}
is to be designed to minimize the
H
2
{\displaystyle {\mathcal {H}}_{2}}
norm of the closed loop transfer matrix
T
11
(
z
)
{\displaystyle T_{11}(z)}
from the exogenous
input
w
1
,
k
{\displaystyle w_{1},k}
to the performance output
z
1
,
k
{\displaystyle z_{1},k}
while ensuring the
H
∞
{\displaystyle {\mathcal {H}}_{\infty }}
norm of the closed-loop transfer matrix
T
22
(
z
)
{\displaystyle T_{22}(z)}
from the exogenous input
w
2
,
k
{\displaystyle w_{2},k}
to the performance output
z
2
,
k
{\displaystyle z_{2},k}
is less than
γ
d
{\displaystyle \gamma _{d}}
, where
T
11
(
z
)
=
C
d
C
L
1
,
1
(
z
I
−
A
d
C
L
)
−
1
B
d
C
L
1
,
1
,
{\displaystyle T_{11}(z)=C_{d_{CL}1,1}(zI-A_{d_{CL}})^{-1}B_{d_{CL}1,1},}
T
22
(
z
)
=
C
d
C
L
1
,
2
(
z
I
−
A
d
C
L
)
−
1
B
d
C
L
1
,
2
+
D
d
C
L
11
,
22
,
{\displaystyle T_{22}(z)=C_{d_{CL}1,2}(zI-A_{d_{CL}})^{-1}B_{d_{CL}1,2}+D_{d_{CL}11,22},}
A
d
C
L
=
[
A
d
+
B
d
2
D
d
c
D
~
d
−
1
C
d
2
B
d
2
(
I
+
D
d
c
D
~
d
−
1
D
d
22
)
C
d
c
B
d
c
D
~
d
−
1
C
d
2
A
d
c
+
B
d
c
D
~
d
−
1
D
d
22
C
d
c
]
{\displaystyle A_{d_{CL}}={\begin{bmatrix}A_{d}+B_{d2}D_{dc}{\tilde {D}}_{d}^{-1}C_{d2}&B_{d2}(I+D_{dc}{\tilde {D}}_{d}^{-1}D_{d22})C_{dc}\\B_{dc}{\tilde {D}}_{d}^{-1}C_{d2}&A_{dc}+B_{dc}{\tilde {D}}_{d}^{-1}D_{d22}C_{dc}\end{bmatrix}}}
,
B
d
C
L
1
,
1
=
[
B
d
1
,
1
+
B
d
2
D
d
c
D
~
d
−
1
D
d
21
,
1
B
d
c
D
~
d
−
1
D
d
21
,
1
]
{\displaystyle B_{d_{CL}1,1}={\begin{bmatrix}B_{d1,1}+B_{d2}D_{dc}{\tilde {D}}_{d}^{-1}D_{d21,1}\\B_{dc}{\tilde {D}}_{d}^{-1}D_{d21,1}\end{bmatrix}}}
,
B
d
C
L
1
,
2
=
[
B
d
1
,
2
+
B
d
2
D
d
c
D
~
d
−
1
D
d
21
,
2
B
d
c
D
~
d
−
1
D
d
21
,
2
]
{\displaystyle B_{d_{CL}1,2}={\begin{bmatrix}B_{d1,2}+B_{d2}D_{dc}{\tilde {D}}_{d}^{-1}D_{d21,2}\\B_{dc}{\tilde {D}}_{d}^{-1}D_{d21,2}\end{bmatrix}}}
,
C
d
C
L
1
,
1
=
[
C
d
1
,
1
+
D
d
12
,
1
D
d
c
D
~
d
−
1
C
d
2
,
1
D
d
12
,
1
(
I
+
D
d
c
D
~
d
−
1
D
d
22
)
C
d
c
]
{\displaystyle C_{d_{CL}1,1}={\begin{bmatrix}C_{d1,1}+D_{d12,1}D_{dc}{\tilde {D}}_{d}^{-1}C_{d2,1}&D_{d12,1}(I+D_{dc}{\tilde {D}}_{d}^{-1}D_{d22})C_{dc}\end{bmatrix}}}
,
C
d
C
L
1
,
2
=
[
C
d
1
,
2
+
D
d
12
,
2
D
d
c
D
~
d
−
1
C
d
2
,
2
D
d
12
,
2
(
I
+
D
d
c
D
~
d
−
1
D
d
22
)
C
d
c
]
{\displaystyle C_{d_{CL}1,2}={\begin{bmatrix}C_{d1,2}+D_{d12,2}D_{dc}{\tilde {D}}_{d}^{-1}C_{d2,2}&D_{d12,2}(I+D_{dc}{\tilde {D}}_{d}^{-1}D_{d22})C_{dc}\end{bmatrix}}}
,
D
d
C
L
11
,
22
=
D
d
11
,
22
+
D
d
12
,
2
D
d
c
D
~
d
−
1
D
d
21
,
2
{\displaystyle D_{d_{CL}11,22}=D_{d11,22}+D_{d12,2}D_{dc}{\tilde {D}}_{d}^{-1}D_{d21,2}}
,
and
D
~
d
=
I
−
D
d
22
D
d
c
{\displaystyle {\tilde {D}}_{d}=I-D_{d22}D_{dc}}
.
Solve for
A
d
n
∈
R
n
x
×
n
x
,
B
d
n
∈
R
n
x
×
n
x
,
C
d
n
∈
R
n
u
×
n
x
,
D
d
n
∈
R
n
u
×
n
y
,
X
1
,
Y
1
∈
S
n
x
,
Z
∈
S
n
Z
1
,
{\displaystyle A_{dn}\in \mathbb {R} ^{n_{x}\times n_{x}},B_{dn}\in \mathbb {R} ^{n_{x}\times n_{x}},C_{dn}\in \mathbb {R} ^{n_{u}\times n_{x}},D_{dn}\in \mathbb {R} ^{n_{u}\times n_{y}},X_{1},Y_{1}\in \mathbb {S} ^{n_{x}},Z\in \mathbb {S} ^{n_{Z_{1}}},}
and
μ
∈
R
>
0
{\displaystyle \mu \in \mathbb {R} _{>0}}
that minimizes
J
(
μ
)
=
μ
{\displaystyle {\mathcal {J}}(\mu )=\mu }
subjects to
X
1
>
0
,
Y
1
>
0
Z
>
0
,
{\displaystyle X_{1}>0,\ Y_{1}>0\ Z>0,}
[
X
1
I
X
1
A
d
+
B
d
n
C
d
2
A
d
n
X
1
B
d
1
,
1
+
B
d
n
D
d
21
,
1
∗
Y
1
A
d
+
B
d
2
C
d
n
D
d
2
A
d
Y
1
+
B
d
2
C
d
n
B
d
1
,
1
+
B
d
2
D
d
n
D
d
21
,
1
∗
∗
X
1
I
0
∗
∗
∗
Y
1
0
∗
∗
∗
∗
I
]
>
0
,
{\displaystyle {\begin{bmatrix}X_{1}&I&X_{1}A_{d}+B_{d_{n}}C_{d_{2}}&A_{d_{n}}&X_{1}B_{d1,1}+B{d_{n}}D_{d21,1}\\*&Y_{1}&A_{d}+B_{d2}C_{dn}D_{d2}&A_{d}Y_{1}+B_{d2}C_{dn}&B_{d1,1}+B_{d2}D_{dn}D_{d21,1}\\*&*&X_{1}&I&0\\*&*&*&Y_{1}&0\\*&*&*&*&I\end{bmatrix}}>0,}
[
X
1
I
X
1
A
d
+
B
d
n
C
d
2
A
d
n
X
1
B
d
1
,
2
+
B
d
n
D
d
21
,
2
0
∗
Y
1
A
d
+
B
d
2
C
d
n
D
d
2
A
d
Y
1
+
B
d
2
C
d
n
B
d
1
,
2
+
B
d
2
D
d
n
D
d
21
,
2
0
∗
∗
X
1
I
0
C
d
1
,
2
T
+
C
d
2
T
D
d
n
T
D
d
12
,
2
T
∗
∗
∗
Y
1
0
Y
1
C
d
1
,
2
T
+
C
d
n
T
D
d
12
,
2
T
∗
∗
∗
∗
−
γ
d
I
D
d
11
,
22
T
+
D
d
21
,
2
T
D
d
n
T
D
d
12
,
2
T
∗
∗
∗
∗
∗
−
γ
d
I
]
>
0
{\displaystyle {\begin{bmatrix}X_{1}&I&X_{1}A_{d}+B_{d_{n}}C_{d_{2}}&A_{d_{n}}&X_{1}B_{d1,2}+B{d_{n}}D_{d21,2}&0\\*&Y_{1}&A_{d}+B_{d2}C_{dn}D_{d2}&A_{d}Y_{1}+B_{d2}C_{dn}&B_{d1,2}+B_{d2}D_{dn}D_{d21,2}&0\\*&*&X_{1}&I&0&C_{d1,2}^{T}+C_{d2}^{T}D_{dn}^{T}D_{d12,2}^{T}\\*&*&*&Y_{1}&0&Y_{1}C_{d1,2}^{T}+C_{dn}^{T}D_{d12,2}^{T}\\*&*&*&*&-\gamma _{d}I&D_{d11,22}^{T}+D_{d21,2}^{T}D_{dn}^{T}D{d12,2}^{T}\\*&*&*&*&*&-\gamma _{d}I\end{bmatrix}}>0}
[
Z
C
d
1
,
1
+
D
d
12
,
1
D
d
n
C
d
2
C
d
1
,
1
,
Y
1
T
+
D
d
12
,
1
C
d
n
∗
X
1
I
∗
∗
Y
1
]
>
0
,
{\displaystyle {\begin{bmatrix}Z&C_{d1,1}+D_{d12,1}D_{dn}C_{d2}&C_{d1,1,}Y_{1}^{T}+D_{d12,1}C_{dn}\\*&X_{1}&I\\*&*&Y_{1}\end{bmatrix}}>0,}
D
d
11
,
11
+
D
d
12
,
1
D
d
n
D
d
21
,
1
=
0
,
{\displaystyle D_{d11,11}+D_{d12,1}D_{dn}D_{d21,1}=0,}
[
X
1
I
∗
Y
1
]
>
0
,
{\displaystyle {\begin{bmatrix}X_{1}&I\\*&Y_{1}\end{bmatrix}}>0,}
tr
Z
<
μ
.
{\displaystyle Z<\mu .}
The controller is recovered by
A
d
c
=
A
d
K
−
B
d
c
(
I
−
D
d
22
D
d
c
)
−
1
D
d
22
C
d
c
,
{\displaystyle A_{dc}=A_{dK}-B{dc}(I-D_{d22}D_{dc})^{-1}D_{d22}C_{dc},}
B
d
c
=
B
d
K
(
I
−
D
d
c
D
d
22
)
,
{\displaystyle B_{dc}=B_{dK}(I-D_{dc}D_{d22}),}
C
d
c
=
(
I
−
D
d
c
D
d
22
)
C
d
K
,
{\displaystyle C_{dc}=(I-D_{dc}D_{d22})C_{dK},}
D
d
c
=
(
I
+
D
d
K
D
d
22
)
−
1
D
d
K
,
w
h
e
r
e
<
m
a
t
h
>
[
A
d
K
B
d
K
C
d
K
D
d
K
]
=
[
X
2
X
1
B
d
2
0
I
]
−
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
I
]
−
1
{\displaystyle D_{dc}=(I+D_{dK}D{d22})^{-1}D_{dK},where<math>{\begin{bmatrix}A_{dK}&B_{dK}\\C_{dK}&D_{dK}\end{bmatrix}}={\begin{bmatrix}X_{2}&X_{1}B_{d2}\\0&I\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}&I\end{bmatrix}}^{-1}}
, and the matrices
X
2
{\displaystyle X_{2}}
and
Y
2
{\displaystyle Y_{2}}
satisfy
X
2
Y
2
T
=
I
−
X
1
Y
1
{\displaystyle X_{2}Y_{2}^{T}=I-X_{1}Y_{1}}
. If
D
22
=
0
{\displaystyle D_{22}=0}
, then
A
d
c
=
A
d
K
,
B
d
c
=
B
d
K
,
C
d
c
=
C
d
K
{\displaystyle A_{dc}=A_{dK},B_{dc}=B{dK},C_{dc}=C_{dK}}
and
D
d
c
=
D
d
K
{\displaystyle D_{dc}=D_{dK}}
.
Given
X
1
{\displaystyle X_{1}}
and
Y
1
{\displaystyle Y_{1}}
, the matrices
X
2
{\displaystyle X_{2}}
and
Y
2
{\displaystyle Y_{2}}
can be found using a matrix decomposition, such as
a LU decomposition or a Cholesky decomposition.
If
D
d
11
,
11
=
0
,
D
d
12
,
1
≠
0
,
and
D
d
21
,
1
≠=
0
,
{\displaystyle D_{d11,11}=0,D_{d12,1}\neq 0,{\text{ and }}D{d21,1}\neq =0,}
then it is often simplest to choose
D
d
n
=
0
{\displaystyle D_{dn}=0}
in order
to satisfy the equality constraint
D
d
11
,
11
+
D
d
12
,
1
D
d
n
D
d
21
,
1
=
0
,
{\displaystyle D_{d11,11}+D_{d12,1}D_{dn}D_{d21,1}=0,}
.
WIP, additional references to be added
A list of references documenting and validating the LMI.
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