A small-gain theorem for 2-contraction of nonlinear interconnected systems

TL;DR

This paper develops small-gain conditions for 2-contraction in interconnected nonlinear systems, enabling stability analysis even when subsystems are not individually contractive, with practical examples illustrating the theory.

Contribution

It introduces novel small-gain criteria for 2-contraction in interconnected systems based on the second additive compound matrix, expanding stability analysis tools.

Findings

The conditions apply even if individual subsystems are not contractive.

Examples demonstrate the effectiveness and scope of the proposed criteria.

The approach broadens the applicability of contraction-based stability analysis.

Abstract

This paper introduces small-gain sufficient conditions for $2$-contraction of feedback interconnected systems, on the basis of individual gains of suitable subsystems arising from a modular decomposition of the second additive compound equation. The condition applies even to cases when individual subsystems might fail to be contractive (due to the extra margin of contraction afforded by the second additive compound matrix). Examples of application are provided to illustrate the theory and show its degree of conservatism and scope of applicability.