Reversals in stability of linear time-delay systems: A finer characterization

• Published in: Automatica (Oxford) 2019-10, Vol.108, p.108479, Article 108479
• Main Authors: Li, Xu, Liu, Jian-Chang, Li, Xu-Guang, Niculescu, Silviu-Iulian, Çela, Arben
• Format: Article
• Language: English
• Subjects: Automatic Control Engineering; Computer Science; Critical imaginary roots; Multiple critical imaginary roots; Stability; Stability reversal; Time-delay systems
• ISSN: 0005-1098; 1873-2836
• DOI: 10.1016/j.automatica.2019.06.031

In most of the numerical examples of time-delay systems proposed in the literature, the number of unstable characteristic roots remains positive before and after a multiple critical imaginary root (CIR) appears (as the delay, seen as a parameter, increases). This fact may lead to some misunderstandings: (i) A multiple CIR may at most affect the instability degree; (ii) It cannot cause any stability reversals (stability transitions from instability to stability). As far as we know, whether the appearance of a multiple CIR can induce stability is still unclear (in fact, when a CIR generates a stability reversal has not been specifically investigated). In this paper, we provide a finer analysis of stability reversals and some new insights into the classification: the link between the multiplicity of a CIR and the asymptotic behavior with the stabilizing effect. Based on these results, we present an example illustrating that a multiple CIR’s asymptotic behavior is able to cause a stability reversal. To the best of the authors’ knowledge, such an example is a novelty in the literature on time-delay systems.