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

The System

Consider $A,E\in \mathbb {R} ^{n\times n},a,b\in \mathbb {R}$ and $d\in \mathbb {R} _{>0},$ where $b\neq 0.$ The pair $(E,A)$ is D-admissible with $D={z\in \mathbb {C} :a+2bRe(z)+d\vert z\vert ^{2}<0}$ if and only if there exist $X\in \mathbb {R} ^{n\times n}$ and $\alpha \in \mathbb {R}$ such that $E^{T}X=X^{T}E\geq 0$ and

The LMI (Constraint)

                                ${\begin{bmatrix}-aE^{T}X-b(X^{T}A+A^{T}X)&A^{T}X\\*&d^{-1}E^{T}X+\alpha (1-E^{\dagger }E)\end{bmatrix}}>0,$

where $E^{\dagger }$ is the pseudoinverse of $E$. The region D describes a circular region of the complex plane with radius $r={\sqrt {-a/d+b^{2}/d^{2}}}$ centered at $(c,0),$ where $c=-b/d.$

Reference

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