Scaled Graph Containment for Feedback Stability: Soft-Hard Equivalence and Conic Regions

TL;DR

This paper introduces a geometric framework for feedback stability analysis using scaled graphs, establishing conditions for containment within multiplier-defined regions and demonstrating computational efficiency and tighter bounds.

Contribution

It develops new containment conditions for scaled graphs within multiplier-defined regions, enabling more efficient and tighter stability certificates, especially for nonsymmetric operators.

Findings

Soft and hard SG containment are equivalent for positive-negative multipliers.

Numerical experiments show 15-44% computational savings.

Hyperbolically convex conic regions provide tighter bounds than circles.

Abstract

Scaled graphs (SGs) offer a geometric framework for feedback stability analysis. This paper develops containment conditions for SGs within multiplier-defined regions, addressing both circular and conic geometries. For circular regions, we show that soft and hard SG containment are equivalent whenever the associated multiplier is positive-negative. This enables hard stability certification from soft computations alone, bypassing both the positive semidefinite storage constraint and the homotopy condition of existing methods. Numerical experiments on systems with up to 300 states demonstrate computational savings of 15-44 % for the circular containment framework. We further characterize which conic regions are hyperbolically convex, a condition our frequency-domain certificate requires, and demonstrate that such regions provide tighter SG bounds than circles whenever the operator SG is nonsymmetric.