Reciprocal relationship between detectability and observability in a non-uniform setting

TL;DR

This paper explores the relationship between detectability and observability in non-uniform settings of control systems, establishing conditions under which detectability implies observability, thus extending classical equivalence results.

Contribution

It proves that non-uniform exponential detectability implies non-uniform complete observability under certain conditions, extending classical uniform results.

Findings

Detectability guarantees observability in non-uniform settings.

Preservation of observability properties under output feedback transformations.

Extension of classical equivalence between observability and detectability.

Abstract

Building on the recent notion of non-uniform complete observability, and on the fact that this property ensures non-uniform exponential detectability, this paper establishes the converse implication under suitable additional assumptions. Specifically, we investigate conditions under which non-uniform exponential detectability guarantees non-uniform complete observability. Our approach is based on a refined analysis of the associated output feedback systems and on the preservation of non-uniform observability properties under such feedback transformations. These results extend the classical equivalence between observability and detectability beyond the uniform framework and provide new tools for the qualitative analysis of time-varying control systems.