Scaled Graph Containment for Feedback Stability: Soft-Hard Equivalence and Conic Regions
This work addresses feedback stability analysis for control systems, offering computational improvements and tighter bounds, but it is incremental as it builds on existing scaled graph frameworks.
The paper tackled the problem of feedback stability analysis using scaled graphs, showing that soft and hard containment are equivalent for circular regions with positive-negative multipliers, enabling hard stability certification from soft computations and saving 15-44% in computational time for systems with up to 300 states. It also characterized hyperbolically convex conic regions that provide tighter bounds for nonsymmetric operator scaled graphs.
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.