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Chebyshev centers and radii for sets induced by quadratic matrix inequalities

Amir Shakouri, Henk J. van Waarde, M. Kanat Camlibel

TL;DR

This paper derives closed-form expressions for the Chebyshev center, Chebyshev radius, and diameter of sets induced by quadratic matrix inequalities (QMIs) with respect to arbitrary unitarily invariant norms, and identifies a common center that remains optimal across all such norms. It provides a parametric description of the QMI-induced sets via Schur complements and establishes how the radius can be computed from the associated symmetric gauge function of the singular values. The results extend to the radius of the largest inscribed ball and to inner-ball approximations, and are applied to data-driven modeling and control by formulating data-consistent system identification as a QMI problem, enabling worst-case estimation analysis and GLS-type solutions. These analytic tools yield exact, norm-agnostic guidance for robust estimation, experiment design, and control synthesis in uncertain linear time-invariant systems. The work advances the practical deployment of QMI-based set-membership methods by providing explicit, computable characterizations that align with common norms used in control and regression tasks.

Abstract

This paper studies sets of matrices induced by quadratic inequalities. In particular, the center and radius of a smallest ball containing the set, called a Chebyshev center and the Chebyshev radius, are studied. In addition, this work studies the diameter of the set, which is the farthest distance between any two elements of the set. Closed-form solutions are provided for a Chebyshev center, the Chebyshev radius, and the diameter of sets induced by quadratic matrix inequalities (QMIs) with respect to arbitrary unitarily invariant norms. Examples of these norms include the Frobenius norm, spectral norm, nuclear norm, Schatten p-norms, and Ky Fan k-norms. In addition, closed-form solutions are presented for the radius of the largest ball within a QMI-induced set. Finally, the paper discusses applications of the presented results in data-driven modeling and control.

Chebyshev centers and radii for sets induced by quadratic matrix inequalities

TL;DR

This paper derives closed-form expressions for the Chebyshev center, Chebyshev radius, and diameter of sets induced by quadratic matrix inequalities (QMIs) with respect to arbitrary unitarily invariant norms, and identifies a common center that remains optimal across all such norms. It provides a parametric description of the QMI-induced sets via Schur complements and establishes how the radius can be computed from the associated symmetric gauge function of the singular values. The results extend to the radius of the largest inscribed ball and to inner-ball approximations, and are applied to data-driven modeling and control by formulating data-consistent system identification as a QMI problem, enabling worst-case estimation analysis and GLS-type solutions. These analytic tools yield exact, norm-agnostic guidance for robust estimation, experiment design, and control synthesis in uncertain linear time-invariant systems. The work advances the practical deployment of QMI-based set-membership methods by providing explicit, computable characterizations that align with common norms used in control and regression tasks.

Abstract

This paper studies sets of matrices induced by quadratic inequalities. In particular, the center and radius of a smallest ball containing the set, called a Chebyshev center and the Chebyshev radius, are studied. In addition, this work studies the diameter of the set, which is the farthest distance between any two elements of the set. Closed-form solutions are provided for a Chebyshev center, the Chebyshev radius, and the diameter of sets induced by quadratic matrix inequalities (QMIs) with respect to arbitrary unitarily invariant norms. Examples of these norms include the Frobenius norm, spectral norm, nuclear norm, Schatten p-norms, and Ky Fan k-norms. In addition, closed-form solutions are presented for the radius of the largest ball within a QMI-induced set. Finally, the paper discusses applications of the presented results in data-driven modeling and control.
Paper Structure (17 sections, 16 theorems, 87 equations, 10 figures)

This paper contains 17 sections, 16 theorems, 87 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Center, radius and diameter of sets
  5. Sets induced by quadratic matrix inequalities
  6. Unitarily invariant norms
  7. Main results
  8. Applications to data-driven modeling and control
  9. Best worst-case estimation
  10. Least-squares solution as a Chebyshev center
  11. Prior knowledge
  12. The choice of norm
  13. Experiment design
  14. Examples
  15. Mass-spring-damper
  16. ...and 2 more sections

Key Result

Proposition 1

Let $\mathcal{X}\subset\mathbb{R}^{p\times q}$ be a compact set. Then:

Figures (10)

  • Figure 1: The Chebyshev radius and the set of Chebyshev centers for the closed area defined by an equilateral triangle in terms of Euclidean and infinity norms.
  • Figure 2: The closed shaded area defined by an ellipsoid is an example of a QMI-induced set.
  • Figure 3: Two ellipses, shown by the dark and light gray areas, with the same Chebyshev radius but different volumes.
  • Figure 4: Examples of the inner balls for QMI-induced sets.
  • Figure 5: Mass-spring-damper system for Example 1.
  • ...and 5 more figures

Theorems & Definitions (30)

  • Proposition 1
  • proof
  • Proposition 2: van2023quadratic
  • Definition 3: bernstein2018scalar
  • Definition 4: mirsky1960symmetric
  • Proposition 5: bernstein2018scalar
  • Proposition 6: ziketak1988characterization
  • Theorem 7
  • Lemma 8
  • proof
  • ...and 20 more