A small-gain criterion for 2-contraction of large scale interconnected systems

TL;DR

This paper develops small-gain like conditions to ensure 2-contraction in large-scale interconnected systems, enabling convergence analysis even with multiple equilibria, using L2 gain bounds derived from individual channels.

Contribution

It introduces a novel small-gain criterion for 2-contraction based on L2 gain bounds from individual channels, applicable to systems with multiple equilibria.

Findings

Conditions guarantee exponential contraction of the system's Jacobian

L2 gains are computed via Linear Matrix Inequalities

Application to interconnected Thomas' systems demonstrates effectiveness

Abstract

Despite modular conditions to guarantee stability for large-scale systems have been widely studied, few methods are available to tackle the case of networks with multiple equilibria. This paper introduces small-gain like sufficient conditions for 2-contraction of large-scale interconnected systems on the basis of a family of upper-bounds to the $L_2$ gains that arise from the gains computed on individual channels of the second additive variational equation. Such a condition guarantee the 2-additive compound of the system's Jacobian to be exponentially contractive, thus implying convergence towards equilibria of the system's solutions. The gains are obtained by solving suitable Linear Matrix Inequalities. Three interconnected Thomas' systems are considered in order to illustrate the application of the theory and the degree of conservatism.