The System
edit
Consider
A
,
E
∈
R
n
×
n
{\displaystyle A,E\in \mathbb {R} ^{n\times n}}
E
{\displaystyle E}
A
{\displaystyle A}
D -admissible if it is regular and causal, and the eigenvalues of (
E
{\displaystyle E}
A
{\displaystyle A}
D of the complex plane, which is defined as
D
=
{
z
∈
C
:
f
D
(
z
)
<
0
}
{\displaystyle D=\{z\in \mathbb {C} :f_{D}(z)<0}\}
f
D
(
z
)
:=
Λ
+
z
Φ
+
z
¯
Φ
T
=
[
λ
k
l
+
Φ
k
l
z
+
Φ
l
k
z
¯
]
1
≤
k
,
l
≤
m
,
{\displaystyle f_{D}(z):=\Lambda +z\Phi +{\bar {z}}\Phi ^{T}=[\lambda _{kl}+\Phi _{kl}z+\Phi _{lk}{\bar {z}}]_{1\leq k,l\leq m},}
Λ
∈
S
m
,
Φ
∈
R
m
×
m
{\displaystyle \Lambda \in \mathbb {S} ^{m},\Phi \in \mathbb {R} ^{m\times m}}
z
¯
{\displaystyle {\bar {z}}}
z
{\displaystyle z}
Conditions
edit
The pair (
E
{\displaystyle E}
A
{\displaystyle A}
D -admissible if and only if any of the following equivalent conditions are satisfied.
There exist
P
∈
S
n
,
S
∈
R
(
n
−
n
e
)
×
(
n
−
n
e
)
,
{\displaystyle P\in \mathbb {S} ^{n},S\in \mathbb {R} ^{(n-n_{e})\times (n-n_{e})},}
U
,
V
∈
R
n
×
(
n
−
n
e
)
,
{\displaystyle U,V\in \mathbb {R} ^{n\times (n-n_{e})},}
n
e
=
{\displaystyle n_{e}=}
(
E
)
,
R
(
U
)
=
N
(
E
T
)
,
R
(
V
)
=
N
(
E
)
,
{\displaystyle (E),{\mathcal {R}}(U)={\mathcal {N}}(E^{T}),{\mathcal {R}}(V)={\mathcal {N}}(E),}
P
>
0
,
{\displaystyle P>0,}
[
λ
k
l
E
P
E
T
+
ϕ
k
l
E
T
P
A
T
+
A
V
S
U
T
+
U
S
T
V
T
A
T
]
1
≤
k
,
l
≤
m
<
0
,
{\displaystyle [\lambda _{kl}EPE^{T}+\phi _{kl}E^{T}PA^{T}+AVSU^{T}+US^{T}V^{T}A^{T}]_{1\leq k,l\leq m}<0,}
There exist
P
,
Q
∈
S
n
,
{\displaystyle P,Q\in \mathbb {S} ^{n},}
P
>
0
,
{\displaystyle P>0,}
E
T
Q
E
≥
0
{\displaystyle E^{T}QE\geq 0}
[
λ
k
l
E
P
E
T
+
ϕ
k
l
A
P
E
+
ϕ
l
k
E
T
P
A
T
+
A
T
Q
A
]
1
≤
k
,
l
≤
m
<
0
,
{\displaystyle [\lambda _{kl}EPE^{T}+\phi _{kl}APE+\phi _{lk}E^{T}PA^{T}+A^{T}QA]_{1\leq k,l\leq m}<0,}
There exist
P
∈
S
n
,
S
∈
R
(
n
−
n
e
)
×
(
n
−
n
e
)
,
U
∈
R
n
×
(
n
−
n
e
)
,
{\displaystyle P\in \mathbb {S} ^{n},S\in \mathbb {R} ^{(n-n_{e})\times (n-n_{e})},U\in \mathbb {R} ^{n\times (n-n_{e})},}
n
e
=
{\displaystyle n_{e}=}
(
E
)
,
U
E
=
0
,
{\displaystyle (E),UE=0,}
P
>
0
,
{\displaystyle P>0,}
[
λ
k
l
E
P
E
T
+
ϕ
k
l
A
P
E
+
ϕ
l
k
E
T
P
A
T
+
A
T
U
T
S
U
A
]
1
≤
k
,
l
≤
m
<
0
,
{\displaystyle [\lambda _{kl}EPE^{T}+\phi _{kl}APE+\phi _{lk}E^{T}PA^{T}+A^{T}U^{T}SUA]_{1\leq k,l\leq m}<0,}
There exist
P
∈
S
n
,
S
∈
R
(
n
−
n
e
)
×
(
n
−
n
e
)
,
{\displaystyle P\in \mathbb {S} ^{n},S\in \mathbb {R} ^{(n-n_{e})\times (n-n_{e})},}
U
,
V
∈
R
n
×
(
n
−
n
e
)
,
{\displaystyle U,V\in \mathbb {R} ^{n\times (n-n_{e})},}
n
e
=
{\displaystyle n_{e}=}
(
E
)
,
R
(
U
)
=
N
(
E
T
)
,
R
(
V
)
=
N
(
E
)
,
{\displaystyle (E),{\mathcal {R}}(U)={\mathcal {N}}(E^{T}),{\mathcal {R}}(V)={\mathcal {N}}(E),}
P
>
0
,
{\displaystyle P>0,}
Λ
⊗
E
P
E
T
+
Φ
⊗
(
A
P
E
)
+
Φ
T
⊗
(
E
P
A
T
)
+
1
m
m
⊗
(
A
V
S
U
T
+
U
S
T
V
T
A
T
)
<
0
,
{\displaystyle \Lambda \otimes EPE^{T}+\Phi \otimes (APE)+\Phi ^{T}\otimes (EPA^{T})+1_{mm}\otimes (AVSU^{T}+US^{T}V^{T}A^{T})<0,}
⊗
{\displaystyle \otimes }
1
m
m
{\displaystyle 1_{mm}}
m
×
m
{\displaystyle m\times m}
There exist
P
,
Q
∈
S
n
,
{\displaystyle P,Q\in \mathbb {S} ^{n},}
P
>
0
,
{\displaystyle P>0,}
E
T
Q
E
≥
0
{\displaystyle E^{T}QE\geq 0}
Λ
⊗
E
P
E
T
+
Φ
⊗
(
A
P
E
)
+
Φ
T
⊗
(
E
P
A
T
)
+
1
m
m
⊗
(
A
T
Q
A
)
<
0
,
{\displaystyle \Lambda \otimes EPE^{T}+\Phi \otimes (APE)+\Phi ^{T}\otimes (EPA^{T})+1_{mm}\otimes (A^{T}QA)<0,}
⊗
{\displaystyle \otimes }
1
m
m
{\displaystyle 1_{mm}}
m
×
m
{\displaystyle m\times m}
There exist
P
∈
S
n
,
S
∈
R
(
n
−
n
e
)
×
(
n
−
n
e
)
,
U
∈
R
n
×
(
n
−
n
e
)
,
{\displaystyle P\in \mathbb {S} ^{n},S\in \mathbb {R} ^{(n-n_{e})\times (n-n_{e})},U\in \mathbb {R} ^{n\times (n-n_{e})},}
n
e
=
{\displaystyle n_{e}=}
(
E
)
,
U
E
=
0
,
{\displaystyle (E),UE=0,}
P
>
0
,
{\displaystyle P>0,}
Λ
⊗
E
P
E
T
+
Φ
⊗
(
A
P
E
)
+
Φ
T
⊗
(
E
P
A
T
)
+
1
m
m
⊗
(
A
T
U
T
S
U
A
)
<
0
,
{\displaystyle \Lambda \otimes EPE^{T}+\Phi \otimes (APE)+\Phi ^{T}\otimes (EPA^{T})+1_{mm}\otimes (A^{T}U^{T}SUA)<0,}
⊗
{\displaystyle \otimes }
1
m
m
{\displaystyle 1_{mm}}
m
×
m
{\displaystyle m\times m}
Reference
edit
Caverly, Ryan; Forbes, James (2021). LMI Properties and Applications in Systems, Stability, and Control Theory .
![{\displaystyle f_{D}(z):=\Lambda +z\Phi +{\bar {z}}\Phi ^{T}=[\lambda _{kl}+\Phi _{kl}z+\Phi _{lk}{\bar {z}}]_{1\leq k,l\leq m},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e1965b7abad2942a381470e91b7092a79fd6b51c)
![{\displaystyle [\lambda _{kl}EPE^{T}+\phi _{kl}E^{T}PA^{T}+AVSU^{T}+US^{T}V^{T}A^{T}]_{1\leq k,l\leq m}<0,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b26caa6fcefeb946f471e499b824c3c7e0eaca66)
![{\displaystyle [\lambda _{kl}EPE^{T}+\phi _{kl}APE+\phi _{lk}E^{T}PA^{T}+A^{T}QA]_{1\leq k,l\leq m}<0,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7a935042a445cf9e0cd40a12403ff70b09e50b41)
![{\displaystyle [\lambda _{kl}EPE^{T}+\phi _{kl}APE+\phi _{lk}E^{T}PA^{T}+A^{T}U^{T}SUA]_{1\leq k,l\leq m}<0,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/65a18c945f316c12ee632b574d04d74a4dbf7300)
