Schur Detectability
Schur detectability is a dual concept of Schur stabilizability and is defined as follows, the matrix pair
(
A
,
C
)
,
{\displaystyle (A,C),}
L
{\displaystyle L}
A
+
L
C
{\displaystyle A+LC}
We consider the following system:
x
(
k
+
1
)
=
A
x
(
k
)
+
B
u
(
k
)
y
(
k
)
=
C
x
(
k
)
+
D
u
(
k
)
{\displaystyle {\begin{aligned}x(k+1)=Ax(k)+Bu(k)\\y(k)=Cx(k)+Du(k)\\\end{aligned}}}
where the matrices
A
∈
R
n
×
n
{\displaystyle A\in \mathbb {R} ^{n\times n}}
B
∈
R
n
×
r
{\displaystyle B\in \mathbb {R} ^{n\times r}}
C
∈
R
m
×
n
{\displaystyle C\in \mathbb {R} ^{m\times n}}
D
∈
R
m
×
r
{\displaystyle D\in \mathbb {R} ^{m\times r}}
x
∈
R
n
{\displaystyle x\in \mathbb {R} ^{n}}
y
∈
R
m
{\displaystyle y\in \mathbb {R} ^{m}}
u
∈
R
r
{\displaystyle u\in \mathbb {R} ^{r}}
Moreover,
k
{\displaystyle k}
k
+
1
{\displaystyle k+1}
The state feedback control law is defined as follows:
u
(
k
)
=
K
x
(
k
)
{\displaystyle {\begin{aligned}u(k)=Kx(k)\end{aligned}}}
where
K
∈
R
n
×
r
{\displaystyle K\in \mathbb {R} ^{n\times r}}
x
(
k
+
1
)
=
(
A
+
B
K
)
x
(
k
)
{\displaystyle {\begin{aligned}x(k+1)=(A+BK)x(k)\end{aligned}}}
The matrices
A
,
B
,
C
,
D
{\displaystyle A,B,C,D}
There exist a symmetric matrix
P
{\displaystyle P}
[
−
P
A
T
P
+
C
T
W
T
P
A
+
W
C
P
]
<
0
{\displaystyle {\begin{aligned}{\begin{bmatrix}-P&A^{T}P+C^{T}W^{T}\\PA+WC&P\end{bmatrix}}<0\\\end{aligned}}}
P
{\displaystyle P}
[
−
N
c
T
P
N
c
N
c
T
A
T
P
P
A
N
c
−
P
]
<
0
{\displaystyle {\begin{aligned}{\begin{bmatrix}-N_{c}^{T}PN_{c}&N_{c}^{T}A^{T}P\\PAN_{c}&-P\end{bmatrix}}<0\\\end{aligned}}}
N
c
{\displaystyle N_{c}}
C
{\displaystyle C}
[
−
P
P
A
A
T
P
−
P
−
γ
C
T
C
]
<
0
{\displaystyle {\begin{aligned}{\begin{bmatrix}-P&PA\\A^{T}P&-P-\gamma C^{T}C\end{bmatrix}}<0\\\end{aligned}}}
γ
>
1
{\displaystyle \gamma >1}
The LMI for Schur detecability can be written as minimization of the scalar,
γ
{\displaystyle \gamma }
min
γ
s.t.
{\displaystyle {\begin{aligned}&{\text{min}}\quad \gamma \\&{\text{s.t.}}\end{aligned}}}
[
−
P
A
T
P
+
C
T
W
T
P
A
+
W
C
P
]
<
0
{\displaystyle {\begin{aligned}{\begin{bmatrix}-P&A^{T}P+C^{T}W^{T}\\PA+WC&P\end{bmatrix}}<0\\\end{aligned}}}
[
−
N
c
T
P
N
c
N
c
T
A
T
P
P
A
N
c
−
P
]
<
0
{\displaystyle {\begin{aligned}{\begin{bmatrix}-N_{c}^{T}PN_{c}&N_{c}^{T}A^{T}P\\PAN_{c}&-P\end{bmatrix}}<0\\\end{aligned}}}
[
−
P
P
A
A
T
P
−
P
−
γ
C
T
C
]
<
0
{\displaystyle {\begin{aligned}{\begin{bmatrix}-P&PA\\A^{T}P&-P-\gamma C^{T}C\end{bmatrix}}<0\\\end{aligned}}}
Thus by proving the above conditions we prove that the matrix pair
(
A
,
C
)
{\displaystyle (A,C)}
A link to Matlab codes for this problem in the Github repository:
Schur Detectability
LMI for Hurwitz stability LMI for Schur stability Hurwitz Detectability
[1] - LMI in Control Systems Analysis, Design and ApplicationsLMIs in Control/Tools
