Scaled relative graphs for system analysis

TL;DR

This paper links scaled relative graphs to classical input/output system theory, enabling visualization and analysis of system properties and interconnections through graphical methods, including a proof of the incremental passivity theorem.

Contribution

It establishes a connection between scaled relative graphs and the Nyquist diagram, offering new graphical tools for system analysis and verification.

Findings

Nyquist diagram is the convex hull of the scaled relative graph

SRGs can visualize static nonlinearities like describing functions

Graphical manipulations of SRGs verify system interconnection properties

Abstract

Scaled relative graphs were recently introduced to analyze the convergence of optimization algorithms using two dimensional Euclidean geometry. In this paper, we connect scaled relative graphs to the classical theory of input/output systems. It is shown that the Nyquist diagram of an LTI system on $L_2$ is the convex hull of its scaled relative graph under a particular change of coordinates. The SRG may be used to visualize approximations of static nonlinearities such as the describing function and quadratic constraints, allowing system properties to be verified or disproved. Interconnections of systems correspond to graphical manipulations of their SRGs. This is used to provide a simple, graphical proof of the classical incremental passivity theorem.