Generalized Jensen Inequalities with Application to Stability Analysis of Systems with Distributed Delays over Infinite Time-Horizons

TL;DR

This paper introduces generalized Jensen inequalities for continuous and discrete systems, improving stability analysis of systems with distributed delays over infinite horizons by reducing conservativeness in stability conditions.

Contribution

It presents new generalized integral and summation inequalities that extend Jensen inequalities, enhancing stability analysis of systems with gamma- and poisson-distributed delays.

Findings

Improved stability conditions for continuous-time systems with gamma-distributed delays.

Less conservative stability criteria for discrete-time systems with poisson-distributed delays.

Demonstrated effectiveness through illustrative examples.

Abstract

The Jensen inequality has been recognized as a powerful tool to deal with the stability of time-delay systems. Recently, a new inequality that encompasses the Jensen inequality was proposed for the stability analysis of systems with finite delays. In this paper, we first present a generalized integral inequality and its double integral extension. It is shown how these inequalities can be applied to improve the stability result for linear continuous-time systems with gamma-distributed delays. Then, for the discrete-time counterpart we provide an extended Jensen summation inequality with infinite sequences, which leads to less conservative stability conditions for linear discrete-time systems with poisson-distributed delays. The improvements obtained thanks to the introduced generalized inequalities are demonstrated by examples.