Convergence and Equivalence results for the Jensen's inequality - Application to time-delay and sampled-data systems

TL;DR

This paper investigates the conservatism of Jensen's inequality in time-delay and sampled-data systems, demonstrating how fragmentation schemes and bounds can reduce the Jensen's gap and improve analysis accuracy.

Contribution

It proves that the Jensen's gap can be made arbitrarily small with sufficiently large uniform fragmentation and characterizes a family of bounds with improved properties.

Findings

Jensen's gap can be minimized with large uniform fragmentation

Non-uniform fragmentation accelerates convergence in some cases

Other bounds are equivalent to Jensen's and have advantageous properties

Abstract

The Jensen's inequality plays a crucial role in the analysis of time-delay and sampled-data systems. Its conservatism is studied through the use of the Gr\"{u}ss Inequality. It has been reported in the literature that fragmentation (or partitioning) schemes allow to empirically improve the results. We prove here that the Jensen's gap can be made arbitrarily small provided that the order of uniform fragmentation is chosen sufficiently large. Non-uniform fragmentation schemes are also shown to speed up the convergence in certain cases. Finally, a family of bounds is characterized and a comparison with other bounds of the literature is provided. It is shown that the other bounds are equivalent to Jensen's and that they exhibit interesting well-posedness and linearity properties which can be exploited to obtain better numerical results.