A note on relationship between some bounding inequalities in stability analysis of time-delay systems

TL;DR

This paper introduces an extended generalized free matrix inequality for better estimation of integrals and sums in stability analysis of time-delay systems, showing equivalences and relationships with existing inequalities.

Contribution

It presents a unified form of the free matrix based inequality, explores its equivalences, and investigates its relationship with Bessel-based inequalities in stability analysis.

Findings

Unified form of free matrix inequality introduced

Established equivalences with known inequalities

Analyzed relationship with Bessel-based inequalities

Abstract

In this paper, an extension of the generalized free matrix based inequality is introduced in a unified form suitable for the estimation of integrals and sums of quadratic functions. The equivalences of several known variants are shown, including the free matrix based inequalities and its simplified form. Secondly, the relationship between the (simplified) free matrix based estimation and the combination of the Bessel-based inequality with different bounding inequalities affine in the length of the intervals are investigated.