A survey of inequality techniques for stability analysis of time‐delay systems

• Published in: International journal of robust and nonlinear control Vol. 32; no. 11; pp. 6412 - 6440
• Main Authors: Chen, Jun, Park, Ju H., Xu, Shengyuan, Zhang, Baoyong
• Format: Journal Article
• Language: English
• Published: Bognor Regis Wiley Subscription Services, Inc 25.07.2022
• Subjects: inequality technique; Linear matrix inequalities; Lyapunov–Krasovskii functional; Mathematical analysis; reciprocally convex combination lemma; stability; Stability analysis; time‐delay system
• ISSN: 1049-8923; 1099-1239
• DOI: 10.1002/rnc.6151

During the past decades, much attention has been paid to the stability problem of linear time‐delay systems. In order to obtain tractable stability conditions shown in the linear matrix inequality form, a great number of remarkable results have been reported in the literature. This article first gives a survey of inequality techniques recently developed to estimate integral quadratic terms and reciprocally convex combination terms arising in the estimation of the time derivative of a Lyapunov–Krasovskii functional candidate. Emphases are specially placed on the evolution of various integral and matrix inequalities and the relationships among them. Second, several stability conditions are obtained and carefully compared to illustrate the effectiveness of different inequality techniques on reducing the conservatism. Finally, the quadratic negative‐definiteness lemmas are insightfully reviewed and meanwhile, a simple method to calculate matrix coefficients of a quadratic matrix‐valued function is presented.