A Region-Dependent Gain Condition for Asymptotic Stability

TL;DR

This paper introduces a region-dependent small gain condition that ensures local and global asymptotic stability in interconnected dynamical systems, especially when traditional gain conditions fail due to discontinuities or bounds exceeding the identity.

Contribution

It presents an alternative stability criterion based on region-dependent gains, extending the small gain theorem to cases with discontinuities or larger bounds.

Findings

Ensures local asymptotic stability of the origin.

Guarantees global attractivity of a compact set.

Achieves almost global asymptotic stability under certain conditions.

Abstract

A sufficient condition for the stability of a system resulting from the interconnection of dynamical systems is given by the small gain theorem. Roughly speaking, to apply this theorem, it is required that the gains composition is continuous, increasing and upper bounded by the identity function. In this work, an alternative sufficient condition is presented for the case in which this criterion fails due to either lack of continuity or the bound of the composed gain is larger than the identity function. More precisely, the local (resp. non-local) asymptotic stability of the origin (resp. global attractivity of a compact set) is ensured by a region-dependent small gain condition. Under an additional condition that implies convergence of solutions for almost all initial conditions in a suitable domain, the almost global asymptotic stability of the origin is ensured. Two examples illustrate and motivate this approach.