Approximation of distributed delays

TL;DR

This paper presents a new methodology for approximating distributed delays, which are convolution operators with bounded support, crucial in control systems with time delays, using rational functions within a Banach algebra framework.

Contribution

It introduces a general approximation approach for distributed delays using graph topology and convolution algebra, with constructive methods and analysis in time and frequency domains.

Findings

The methodology effectively approximates distributed delays.

Simulations demonstrate improved stabilization control.

The approach is applicable within a Banach algebra setting.

Abstract

We address in this paper the approximation problem of distributed delays. Such elements are convolution operators with kernel having bounded support, and appear in the control of time-delay systems. From the rich literature on this topic, we propose a general methodology to achieve such an approximation. For this, we enclose the approximation problem in the graph topology, and work with the norm defined over the convolution Banach algebra. The class of rational approximates is described, and a constructive approximation is proposed. Analysis in time and frequency domains is provided. This methodology is illustrated on the stabilization control problem, for which simulations results show the effectiveness of the proposed methodology.