On the necessity of looped-functionals arising in the analysis of pseudo-periodic, sampled-data and hybrid systems

TL;DR

This paper proves that looped-functional conditions for analyzing impulsive pseudo-periodic systems are both sufficient and necessary, broadening their applicability to various hybrid and sampled-data systems.

Contribution

It establishes the necessity of looped-functional conditions for a wide class of linear systems, extending their relevance beyond previous sufficiency-only results.

Findings

Looped-functional conditions are necessary and sufficient for impulsive pseudo-periodic systems.

The results unify analysis methods for sampled-data, impulsive, and switched systems.

Broader class of systems analyzed with these conditions.

Abstract

Looped-functionals have been shown to be relevant for the analysis of a wide variety of systems. However, the conditions obtained in previous works on the analysis of sampled-data, impulsive and switched systems have only been shown to be sufficient for the characterization of their associated discrete-time stability conditions. We prove here that these conditions are also \necessary. This result is derived for a wider class of linear systems, referred to as impulsive pseudo-periodic systems, that encompass periodic, impulsive, sampled-data and switched systems as special cases.