Scaled Relative Graph Analysis of General Interconnections of SISO Nonlinear Systems

TL;DR

This paper introduces a new SRG-based method combined with the Nyquist criterion to analyze stability and performance of interconnected nonlinear systems, extending classical criteria like the circle criterion.

Contribution

It presents a reformulation of SRGs for LTI operators and a theorem for stability and $L_2$-gain analysis of nonlinear interconnections, broadening applicability.

Findings

Provides a generalized circle criterion for nonlinearities.

Derives $L_2$-gain bounds for nonlinear system interconnections.

Demonstrates effectiveness on multiple example systems.

Abstract

Scaled Relative Graphs (SRGs) provide a novel graphical frequency-domain method for the analysis of nonlinear systems. However, we show that the current SRG analysis suffers from a pitfall that limits its applicability in analyzing practical nonlinear systems. We overcome this pitfall by introducing a novel reformulation of the SRG of a linear time-invariant operator and combining the SRG with the Nyquist criterion. The result is a theorem that can be used to assess stability and $L_2$-gain performance for general interconnections of nonlinear dynamic systems. We provide practical calculation results for canonical interconnections and apply our result to Lur'e systems to obtain a generalization of the celebrated circle criterion, which deals with broader class of nonlinearities, and we derive (incremental) $L_2$-gain performance bounds. We illustrate the power of the new approach on the analysis of several examples.