Scaled Relative Graphs and Dynamic Integral Quadratic Constraints: Connections and Computations for Nonlinear Systems

TL;DR

This paper introduces a systematic method for computing scaled relative graphs of nonlinear systems using dynamic IQCs, providing new graphical tools for stability analysis.

Contribution

It presents a novel approach combining SRGs and dynamic IQCs, including computational techniques and stability results for Lur'e-type systems.

Findings

Computed tractable SRG overbounds for nonlinear systems.

Derived an SRG-based feedback stability criterion.

Provided a graphical interpretation of IQC stability results.

Abstract

Scaled relative graphs (SRGs) enable graphical analysis and design of nonlinear systems. In this paper, we present a systematic approach for computing both soft and hard SRGs of nonlinear systems using dynamic integral quadratic constraints (IQCs). These constraints are exploited via application of the S-procedure to compute tractable SRG overbounds. In particular, we show that the multipliers associated with the IQCs define regions in the complex plane. Soft SRG computations are formulated through frequency-domain conditions, while hard SRGs are obtained via hard factorizations of multipliers and linear matrix inequalities. The overbounds are used to derive an SRG-based feedback stability result for Lur'e-type systems, providing a new graphical interpretation of classical IQC stability results with dynamic multipliers.