LMIs in Control/Click here to continue/LMIs in system and stability Theory/General LMI Region D-Admissibility

The System

Consider $A,E\in \mathbb {R} ^{n\times n}$. The pair ($E$,$A$) is D-admissible if it is regular and causal, and the eigenvalues of ($E$,$A$) lie within the LMI region D of the complex plane, which is defined as

$D=\{z\in \mathbb {C} :f_{D}(z)<0}\$, where

$f_{D}(z):=\Lambda +z\Phi +{\bar {z}}\Phi ^{T}=[\lambda _{kl}+\Phi _{kl}z+\Phi _{lk}{\bar {z}}]_{1\leq k,l\leq m},$

$\Lambda \in \mathbb {S} ^{m},\Phi \in \mathbb {R} ^{m\times m}$, and ${\bar {z}}$ is the complex complex conjugate of $z$.

Conditions

The pair ($E$,$A$) is D-admissible if and only if any of the following equivalent conditions are satisfied.

1. There exist $P\in \mathbb {S} ^{n},S\in \mathbb {R} ^{(n-n_{e})\times (n-n_{e})},$ $U,V\in \mathbb {R} ^{n\times (n-n_{e})},$ where $n_{e}=$ rank$(E),{\mathcal {R}}(U)={\mathcal {N}}(E^{T}),{\mathcal {R}}(V)={\mathcal {N}}(E),$and $P>0,$ satisfying $[\lambda _{kl}EPE^{T}+\phi _{kl}E^{T}PA^{T}+AVSU^{T}+US^{T}V^{T}A^{T}]_{1\leq k,l\leq m}<0,$
2. There exist $P,Q\in \mathbb {S} ^{n},$ where $P>0,$ satisfying $E^{T}QE\geq 0$ and $[\lambda _{kl}EPE^{T}+\phi _{kl}APE+\phi _{lk}E^{T}PA^{T}+A^{T}QA]_{1\leq k,l\leq m}<0,$
3. There exist $P\in \mathbb {S} ^{n},S\in \mathbb {R} ^{(n-n_{e})\times (n-n_{e})},U\in \mathbb {R} ^{n\times (n-n_{e})},$ where $n_{e}=$ rank$(E),UE=0,$ and $P>0,$ satisfying $[\lambda _{kl}EPE^{T}+\phi _{kl}APE+\phi _{lk}E^{T}PA^{T}+A^{T}U^{T}SUA]_{1\leq k,l\leq m}<0,$
4. There exist $P\in \mathbb {S} ^{n},S\in \mathbb {R} ^{(n-n_{e})\times (n-n_{e})},$$U,V\in \mathbb {R} ^{n\times (n-n_{e})},$ where $n_{e}=$ rank$(E),{\mathcal {R}}(U)={\mathcal {N}}(E^{T}),{\mathcal {R}}(V)={\mathcal {N}}(E),$ and $P>0,$ satisfying $\Lambda \otimes EPE^{T}+\Phi \otimes (APE)+\Phi ^{T}\otimes (EPA^{T})+1_{mm}\otimes (AVSU^{T}+US^{T}V^{T}A^{T})<0,$ where $\otimes$ is the Kroenecker product and $1_{mm}$ is an $m\times m$ matrix filled with ones.
5. There exist $P,Q\in \mathbb {S} ^{n},$ where $P>0,$ satisfying $E^{T}QE\geq 0$ and $\Lambda \otimes EPE^{T}+\Phi \otimes (APE)+\Phi ^{T}\otimes (EPA^{T})+1_{mm}\otimes (A^{T}QA)<0,$ where $\otimes$ is the Kroenecker product and $1_{mm}$ is an $m\times m$ matrix filled with ones.
6. There exist $P\in \mathbb {S} ^{n},S\in \mathbb {R} ^{(n-n_{e})\times (n-n_{e})},U\in \mathbb {R} ^{n\times (n-n_{e})},$ where $n_{e}=$ rank$(E),UE=0,$ and $P>0,$ satisfying $\Lambda \otimes EPE^{T}+\Phi \otimes (APE)+\Phi ^{T}\otimes (EPA^{T})+1_{mm}\otimes (A^{T}U^{T}SUA)<0,$ where $\otimes$ is the Kroenecker product and $1_{mm}$ is an $m\times m$ matrix filled with ones.

Reference

Caverly, Ryan; Forbes, James (2021). LMI Properties and Applications in Systems, Stability, and Control Theory.