Abstract

Scaled relative graphs (SRGs) were recently introduced by Ryu, Hannah and Yin to analyze the convergence of optimization algorithms using two dimensional Euclidean geometry. In this seminar, I’ll show how the SRG can be used to study incremental input/output properties of feedback systems. The SRG of an LTI transfer function is the convex hull of its Nyquist diagram under a particular change of coordinates. The SRG may be plotted or approximated for arbitrary nonlinear operators, and allows classical Nyquist techniques to be applied to nonlinear systems. System interconnection corresponds to graphical manipulations of the systems’ SRGs. I will give a generalization of the Nyquist criterion for stable systems, where the Nyquist diagram and the point −1 are replaced by SRGs of two feedback-interconnected nonlinear operators. This theorem encompasses the incremental small gain and passivity theorems, and mixed small gain/passivity theorems. The distance between the two SRGs is a nonlinear stability margin, and is the reciprocal of the incremental gain of the feedback system. SRGs can be over-approximated based on system properties such as passivity and incremental gain, and can be plotted precisely for some static nonlinearities. I will provide an analytical calculation of the saturation function, and show how SRG analysis of Lur’e systems gives advantages over classical tools ranging from the circle criterion to IQC analysis.

Zoom link: https://zoom.us/j/96507147724