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 $$E^TX=X^TE \geq 0$$ and

The LMI (Constraint)
$$\begin{bmatrix} -aE^TX-b(X^TA+A^TX) & A^TX \\ * & d^{-1}E^TX+\alpha(1-E^\dagger E) \end{bmatrix} >0,$$ 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.