Amplitude Dependent Bode Diagrams via Scaled Relative Graphs

Julius P. J. Krebbekx; Roland TÃ³th; Roland Tóth; Amritam Das; Thomas Chaffey

paper

Amplitude Dependent Bode Diagrams via Scaled Relative Graphs

arXiv:2603.09382

Abstract

Scaled Relative Graphs (SRGs) provide an intuitive graphical frequency-domain method for the analysis of Nonlinear (NL) systems, generalizing the Nyquist diagram. In this paper, we develop a method for computing $L_{2}$ -gain bounds for Lur'e systems over bounded frequency and amplitude ranges. We do this by restricting the input space of the SRG both in frequency and energy content, and using methods from Sobolev theory. The resulting gain bounds over restricted sets of inputs are less conservative than bounds computed over the entire $L_{2}$ , and yield three-dimensional NL generalization of the Bode diagram, plotting $L_{2}$ -gain as function of both input frequency and energy content. In the zero-energy limit, the Linear Time-Invariant (LTI) Bode diagram is recovered, and at the infinite-energy zero-frequency limit, we recover the $L_{2}$ -gain. The effectiveness of our method is demonstrated on an example that resembles Phase-Locked Loop dynamics.

Accepted for publication in IEEE Control Systems Letters (L-CSS). This is the author's version, see the copyright notice in the PDF/source file