Computable Characterisations of Scaled Relative Graphs of Closed Operators

TL;DR

This paper develops methods for exact and computable construction of Scaled Relative Graphs for closed linear operators, enhancing stability analysis tools for linear dynamical systems.

Contribution

It introduces new tools for constructing SRGs for bounded and unbounded operators, including state-space models, using gain computations and the Bounded Real Lemma.

Findings

Provides exact SRG construction methods for closed operators.

Extends SRG analysis to unbounded operators and typical LTI system models.

Utilizes the Bounded Real Lemma for state-space SRG construction.

Abstract

The Scaled Relative Graph (SRG) is a promising tool for stability and robustness analysis of multi-input multi-output systems. In this paper, we provide tools for exact and computable constructions of the SRG for closed linear operators, based on maximum and minimum gain computations. The results are suitable for bounded and unbounded operators, and we specify how they can be used to draw SRGs for the typical operators that are used to model linear-time-invariant dynamical systems. Furthermore, for the special case of state-space models, we show how the Bounded Real Lemma can be used to construct the SRG.