Stability, Delays and Multiple Characteristic Roots in Dynamical Systems: A Guided Tour

TL;DR

This paper reviews stability analysis techniques for linear dynamical systems with delays, focusing on multiple roots of characteristic functions, their parameter dependence, and control methods like partial pole placement, with practical examples.

Contribution

It introduces a comprehensive overview of methods for analyzing multiple characteristic roots in delayed systems, emphasizing the multiplicity-induced-dominancy property and practical control strategies.

Findings

Methods for characterizing multiple roots and their parameter effects.

Introduction of frequency-sweeping and multiplicity-induced-dominancy criteria.

Practical implementation via software toolboxes and illustrative examples.

Abstract

This paper presents a guided tour of some specific problems encountered in the stability analysis of linear dynamical systems including delays in their systems' representation. More precisely, we will address the characterization of multiple roots of the corresponding characteristic function with a particular emphasis on the way these roots are affected by the system's parameters and the way that they can be used to control. The paper covers several approaches (perturbation techniques, hypergeometric functions) leading to some methods and criteria (frequency-sweeping, multiplicity-induced-dominancy) that can be implemented (software toolboxes) for analyzing the qualitative and quantitative properties induced by the delays and other parameters on the system's dynamics. A particular attention will be paid to the so-called partial pole placement method based on the multiplicity-induced-dominancy property. The presentation is as simple as possible, focusing more on the main intuitive ideas and appropriate mathematical reasoning by analogy in the presentation of the theoretical results as well as their potential use in practical applications. Illustrative examples complete the paper.