Geometric properties of LMI regions

Abstract: LMI (Linear Matrix Inequalities) regions is an important class of convex subsets of $\mathbb C$ arising in control theory. An LMI region $\mathfrak D$ is defined by its matrix-valued characteristic function $f_{\mathfrak D}(z) = {\mathbf L} + z{\mathbf M}+\bar{z}{\mathbf M}^T$ as follows: ${\mathfrak D} := {z \in {\mathbb C}: f_{\mathfrak D}(z)\prec 0}$. In this paper, we study LMI regions from the point of view of convex geometry, describing their boundaries, recession cones, lineality spaces and other characteristic in terms of the properties of matrices $\mathbf M$ and $\mathbf L$. Conversely, we study the link between the properties of matrices $\mathbf M$ and $\mathbf L$, e.g. normality, positive and negative definiteness, and the corresponding properties of an LMI region $\mathfrak D$. We provide the conditions, when an LMI region coincides with the intersection of elementary regions such as halfplanes, stripes, conic sectors and sides of hyperbolas. We also analyze the following problem, connected to pole placement: for a given LMI region $\mathfrak D$, defined by $f_{\mathfrak D}$, how to find a closed disk $D(x_0, r)$ centered at the real axis, such that $D(x_0, r) \subseteq {\mathfrak D}$?