Overview of recent advances in stability of linear systems with time-varying delays

• Published in: IET control theory & applications 2019-01, Vol.13 (1), p.1-16
• Main Authors: Zhang, Xian-Ming, Han, Qing-Long, Seuret, Alexandre, Gouaisbaut, Frédéric, He, Yong
• Format: Article
• Language: English
• Subjects: augmented Lyapunov‐Krasovskii functionals; Automatic; canonical Bessel–Legendre inequalities; delay convex analysis approach; delay systems; delays; Engineering Sciences; linear systems; Lyapunov methods; reciprocally convex approach; Review Article; stability criteria; time‐varying delays; time‐varying systems
• ISSN: 1751-8644; 1751-8652; 1751-8652
• DOI: 10.1049/iet-cta.2018.5188

This study provides an overview and in-depth analysis of recent advances in stability of linear systems with time-varying delays. First, recent developments of a delay convex analysis approach, a reciprocally convex approach and the construction of Lyapunov–Krasovskii functionals are reviewed insightfully. Second, in-depth analysis of the Bessel–Legendre inequality and some affine integral inequalities is made, and recent stability results are also summarised, including stability criteria for three cases of a time-varying delay, where information on the bounds of the time-varying delay and its derivative is totally known, partly known and completely unknown, respectively. Third, a number of stability criteria are developed for the above three cases of the time-varying delay by employing canonical Bessel–Legendre inequalities, together with augmented Lyapunov–Krasovskii functionals. It is shown through numerical examples that these stability criteria outperform some existing results. Finally, several challenging issues are pointed out to direct the near future research.