[Paper Review] Geometric properties of LMI regions

Olga Y. Kushel|arXiv (Cornell University)|Oct 23, 2019

Stability and Control of Uncertain Systems56 references4 citations

TL;DR

This paper provides a comprehensive geometric analysis of Linear Matrix Inequality (LMI) regions in the complex plane, establishing connections between the matrix properties of the generating functions and the topological and convex geometric features of the regions. It derives explicit conditions for inscribing a disk within an LMI region, offering a constructive method using spectral properties of the defining matrices L and M, with key results on recession cones, lineality spaces, and canonical forms for region decomposition.

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 $D(x_0, r) \subseteq {\mathfrak D}$?

Motivation & Objective

• To establish a rigorous link between the matrix characteristics of L and M and the geometric properties of LMI regions in the complex plane.
• To solve the problem of finding the largest inscribed disk within a given LMI region, particularly for pole placement and robust stability analysis.
• To characterize topological features such as boundaries, closures, recession cones, and lineality spaces of LMI regions using matrix-theoretic tools.
• To decompose LMI regions into intersections of elementary regions (halfplanes, stripes, conic sectors, hyperbolas) under specific matrix conditions.
• To provide constructive criteria for inclusion relations, transformations (shifts, reflections), and angle estimates of recession cones.

Proposed method

• The paper uses matrix analysis, particularly the spectral decomposition of symmetric and skew-symmetric parts of M, to characterize the geometry of LMI regions.
• It applies Cholesky decomposition to the matrix function L(x) = L + Sym(M)·2x to analyze the positivity domain and derive bounds on the imaginary part of z.
• The method involves transforming the region via real shifts to center it at the origin, enabling analysis of the skew-symmetric part of M under the transformed metric.
• It employs canonical forms for commuting normal matrices L and M to decompose the LMI region into intersections of elementary polynomial regions.
• The derivation of the inscribed disk radius r(D,x) relies on computing eigenvalues of the transformed skew-symmetric matrix (B^T)^{-1} Skew(M) B^{-1} and relating them to the imaginary axis bounds.
• Theoretical results are validated through explicit examples, including the left half-plane, unit disk, and stability parabola, with closed-form expressions for r(D,x).

Experimental results

Research questions

• RQ1 Under what conditions on matrices L and M does an LMI region coincide with the intersection of elementary regions such as halfplanes, stripes, or conic sectors?
• RQ2 How can the maximal inscribed disk D(x₀,r) within a given LMI region D be computed using only the spectral properties of L and M?
• RQ3 What are the precise geometric characteristics (e.g., recession cone, lineality space) of an LMI region, and how are they determined by the structure of L and M?
• RQ4 How do transformations such as shifts, reflections, or contractions affect the shape and inclusion properties of LMI regions?
• RQ5 What is the analytical expression for the radius r(D,x) of the largest disk centered at a real point x within a given LMI region D?

Key findings

• The recession cone of an LMI region D is characterized by the null space of Sym(M), with D being a cone if and only if Sym(M) is positive semidefinite and L is negative definite.
• An LMI region is bounded if and only if the symmetric part of M is positive definite, ensuring the region does not extend to infinity along any direction.
• For any x in the real projection of D, the maximal inscribed disk radius r(D,x) is given by r(D,x) = |x|√(−ε²x)/√(x² − ε²x) for the stability parabola region, with explicit dependence on ε and x.
• When M is normal and commutes with L, the LMI region D decomposes into an intersection of elementary regions, including halfplanes, horizontal stripes, and hyperbolic sectors.
• The lineality space of D is trivial (i.e., {0}) if and only if the symmetric part of M is positive definite, indicating no nontrivial line segments through the origin are contained in D.
• The radius of the largest inscribed disk centered at x₀ ∈ ℝ is determined by the eigenvalues of the transformed skew-symmetric matrix (B^T)^{-1} Skew(M) B^{-1}, with the imaginary eigenvalues directly yielding the upper bound on the imaginary part of z.