A rolled-off passivity theorem

TL;DR

This paper generalizes passivity and small gain theorems for interconnected nonlinear systems, providing a graphical method to determine finite incremental gain and broadening the applicability of passivity-based analysis.

Contribution

It introduces a new condition for the negative feedback interconnection of systems with small incremental gains, extending existing theorems and using SRGs for graphical analysis.

Findings

Provides a generalized passivity theorem for nonlinear systems

Uses Scaled Relative Graphs for graphical verification

Enables analysis of systems with small incremental gains

Abstract

Given two nonlinear systems which only violate incremental passivity when their incremental gains are sufficiently small, we give a condition for their negative feedback interconnection to have finite incremental gain, which generalizes the incremental small gain and incremental passivity theorems. The property may be determined graphically by plotting the Scaled Relative Graphs (SRGs) of the systems, which provides engineering significance to the mathematical result.