Hierarchical stability conditions and iterative reciprocally high-order polynomial inequalities for two types of time-varying delay systems

• Published in: Automatica (Oxford) 2024-05, Vol.163, p.111526, Article 111526
• Main Authors: Zhai, Zhengliang, Yan, Huaicheng, Chen, Shiming, Li, Zhichen, Xu, Chengjie
• Format: Article
• Language: English
• Subjects: Hierarchical Lyapunov–Krasovskii functional; Iterative reciprocally high-order polynomial; Negative definite conditions; Time-varying delay
• ISSN: 0005-1098; 1873-2836
• DOI: 10.1016/j.automatica.2024.111526

This paper investigates the stability problem of linear systems with a time-varying delay that the delay’s derivative has an upper bound or no constraint. Based on the state vectors and multiple integral state vectors involved in the delay related Bessel–Legendre inequalities (BLIs), the hierarchical Lyapunov–Krasovskii functionals (LKFs) are proposed. In order to deal with the fractions of the delay introduced by the BLIs, a novel iterative reciprocally high-order polynomial (IRHP) combination lemma is developed, which encompasses the existing reciprocally convex inequalities as special cases. Then, by introducing the specific matrix valued negative definite conditions (NDCs) of the odd number high degree polynomials, the hierarchical stability conditions are expressed in the form of linear matrix inequalities (LMIs). Eventually, the validity and advantages of the proposed conditions are illustrated through some classical numerical examples.