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

Silviu-Iulian Niculescu (1, 2) , Islam Boussaada (1, 2, 3) , Xu-Guang Li (4) , Guilherme Mazanti (1, 2) , César-Fernando Méndez-Barrios (5)

1 L2S - Laboratoire des signaux et systèmes
2 DISCO - Dynamical Interconnected Systems in COmplex Environments
3 IPSA - Institut Polytechnique des Sciences Avancées
4 School of Information Science and Engineering [Shenyang]
5 UASLP - Facultad de Ingeniería

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.

Mots clés: delay; dynamical systems; parameter-based methods; frequency-sweeping; Weierstrass polynomial; multiplicity-induced-dominancy; asymptotic behavior

Domaines: Optimisation et contrôle [math.OC]; Systèmes dynamiques [math.DS]

Silviu-Iulian Niculescu, Islam Boussaada, Xu-Guang Li, Guilherme Mazanti, César-Fernando Méndez-Barrios. Stability, Delays and Multiple Characteristic Roots in Dynamical Systems: A Guided Tour. TDS 2021 - 16th IFAC Workshop on Time Delay Systems, Sep 2021, Guangzhou, China. pp.222-239, ⟨10.1016/j.ifacol.2021.11.143⟩. ⟨hal-03430358⟩