On the pole placement of scalar linear delay systems with two delays

TL;DR

This paper investigates how two delays in scalar linear delay systems influence spectral properties, root multiplicities, and pole placement, highlighting the delays' role as control parameters for stability and decay rate optimization.

Contribution

It provides new insights into the spectral behavior of systems with two delays and revisits pole placement strategies emphasizing delays as control parameters.

Findings

Delays affect the maximal multiplicity of characteristic roots.

Delays influence the spectrum localization and root dominancy.

Delays can be used as control parameters for partial pole placement.

Abstract

This paper concerns some spectral properties of the scalar dynamical system defined by a linear delay-differential equation with two positive delays. More precisely, the existing links between the delays and the maximal multiplicity of the characteristic roots are explored, as well as the dominancy of such roots compared with the spectrum localization. As a by-product of the analysis, the pole placement issue is revisited with more emphasis on the role of the delays as control parameters in defining a partial pole placement guaranteeing the closed-loop stability with an appropriate decay rate of the corresponding dynamical system.