User:Margav06/sandbox/Click here to continue/LMIs in system and stability Theory/Circular Region Stability via the Dilation Lemma

Definition
Consider $$A\in\mathbb{R}^{n \times n}$$. The matrix $$A$$ is $$\mathcal{D}$$-stable if and only if there exists $$P\in\mathbb{S}^{n}$$, where $$P>0$$, such that

$$[\lambda_{kl}P+\phi_{kl}AP+\phi_{lk}PA^{T}]_{1<k,l<m}<0$$,

or equivalent

$$\Lambda\otimes P+\Phi\otimes (AP)+\Phi^{T}\otimes (PA^{T})<0$$,

where $$\otimes$$ is the Kroenecker product,

The eigenvalues of a $$\mathcal{D}$$-stable matrix lie within the LMI region $$\mathcal{D}$$, which is defined as

$$\mathcal {D}=\{z\in\mathbb{C}:f_{D}(z)<0\}$$, 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 conjugate of $$z$$.

Circular Region Stability via the Dilation Lemma
Consider $$A\in\mathbb{R}^{n \times n}$$, $$r\in\mathbb{R}_{>0}$$, and $$c\in\mathbb{R}_{<0}$$, where $$c<-r$$. The matrix $$A$$ satisfies $$\lambda(A)\subset\mathcal{G}(c,r)$$, where $$\mathcal{G}(c,r):=\{\lambda\in\mathbb{C}:\left\vert \lambda-c \right\vert0}$$, where $$X>0$$, such that

$$AX+XA^{T}-\frac{c^{2}-r^{2}}{c}X-\frac{1}{c}AXA^{T}<0$$.

Equivalently, the matrix $$A$$ satisfies $$\lambda(A)\subset\mathcal{G}(c,r)$$ if and only if there exist $$X\in\mathbb{S}^{n}$$, $$\epsilon\in\mathbb{R}_{>0}$$, and $$F\in\mathbb{R}^{n \times n}$$, where $$X>0$$, such that

$$\begin{bmatrix} 0 & -X & X & 0 \\ * & 0 & 0 & -X \\ * & * & \frac{c}{c^{2}-r^{2}}X & 0 \\ +He\begin{Bmatrix} \begin{bmatrix} A \\ 1 \\ 0 \\ 0 \end{bmatrix}F \begin{bmatrix} 1 & -\epsilon1 & \epsilon1 & 1 \end{bmatrix} \end{Bmatrix}<0$$
 * & * & * & cX \end{bmatrix}

Moreover, for every $$X$$ that satisfies

$$AX+XA^{T}-\frac{c^{2}-r^{2}}{c}X-\frac{1}{c}AXA^{T}<0$$

$$X$$ and $$F=-\epsilon^{ -1}(A-\epsilon^{ -1}1)^{ -1}X$$ are solutions to

$$\begin{bmatrix} 0 & -X & X & 0 \\ * & 0 & 0 & -X \\ * & * & \frac{c}{c^{2}-r^{2}}X & 0 \\ +He\begin{Bmatrix} \begin{bmatrix} A \\ 1 \\ 0 \\ 0 \end{bmatrix}F \begin{bmatrix} 1 & -\epsilon1 & \epsilon1 & 1 \end{bmatrix} \end{Bmatrix}<0$$
 * & * & * & cX \end{bmatrix}