LMIs in Control/Stability Analysis/D-Stability

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 $$z$$.

Conic Sector Region Stability via the Dilation Lemma
Consider $$A\in\mathbb{R}^{n \times n}$$ and $$k\in\mathbb{R}_{>0}$$.

The matrix $$A$$ satisfies $$\lambda(A)\subset\mathcal{P}(k)$$, where $$\mathcal{P}(k):=\{\lambda\in\mathbb{C}:\left\vert Im( \lambda) \right\vert0}$$, where $$X>0$$, such that

$$\begin{bmatrix} k(AX+XA^{T}) & AX-XA^{T} \\ * & k(AX+XA^{T}) \end{bmatrix}<0$$.

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

$$\begin{bmatrix} 0 & -kX & X & 0 \\ * & 0 & 0 & -X \\ * & * & 0 & -kX \\ +He\begin{Bmatrix} \begin{bmatrix} A &0 \\ 1 & 0 \\ 0 & 1 \\ 0 & A \end{bmatrix} \begin{bmatrix} F &0 \\ 0 & F \end{bmatrix} \begin{bmatrix} k1 & -\epsilon k1 & \epsilon1 & 1 \\ -1 & -\epsilon1 & \epsilon k1 & k 1 \end{bmatrix}\end{Bmatrix}<0$$.
 * & * & * & 0\end{bmatrix}

Moreover, for every $$X$$ that satisfies

$$\begin{bmatrix} k(AX+XA^{T}) & AX-XA^{T} \\ * & k(AX+XA^{T}) \end{bmatrix}<0$$,

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

$$\begin{bmatrix} 0 & -kX & X & 0 \\ * & 0 & 0 & -X \\ * & * & 0 & -kX \\ +He\begin{Bmatrix} \begin{bmatrix} A &0 \\ 1 & 0 \\ 0 & 1 \\ 0 & A \end{bmatrix} \begin{bmatrix} F &0 \\ 0 & F \end{bmatrix} \begin{bmatrix} k1 & -\epsilon k1 & \epsilon1 & 1 \\ -1 & -\epsilon1 & \epsilon k1 & k 1 \end{bmatrix}\end{Bmatrix}<0$$
 * & * & * & 0\end{bmatrix}

α-Region Stability via the Dilation Lemma
Consider $$A\in\mathbb{R}^{n \times n}$$ and $$\alpha\in\mathbb{R}_{>0}$$. The matrix $$A$$ satisfies $$\lambda(A)\subset\mathcal{H}(\alpha)$$, where $$\mathcal{H}(\alpha):=\{\lambda\in\mathbb{C}:Re(\lambda)<-\alpha\}$$ if and only if there exist $$X\in\mathbb{S}^{n}$$ and $$\epsilon\in\mathbb{R}_{>0}$$, where $$X>0$$, such that

$$AX+XA^{T}+2\alpha X<0$$.

Equivalently, the matrix $$A$$ satisfies $$\lambda(A)\subset\mathcal{H}(\alpha)$$ 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

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

Moreover, for every $$X$$ that satisfies

$$AX+XA^{T}+2\alpha X<0$$

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

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

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}