Scaled Relative Graph Analysis of Lur'e Systems and the Generalized Circle Criterion

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 limit its applicability in analyzing practical nonlinear systems. We overcome this pitfall by modifying the SRG of a linear time invariant operator, combining the SRG with the Nyquist criterion, and apply our result to Lur’e systems. We thereby obtain a generalization of the celebrated circle criterion, which deals with a broader class of nonlinearities, and provides (incremental) $L_2$-gain performance bounds.