New Free-Matrix-Based Integral Inequality: Application to Stability Analysis of Systems With Additive Time-Varying Delays

This paper is concerned with the problem of stability analysis for systems with additive time-varying delays (ATDs). This paper proposes a new free-matrix-based integral inequality that provides estimate of the energy of the vector that contains the state and its derivative at the same time. Consequently, the proposed inequality enables the Lyapunov-Krasovskii functional (LKF) to take into account not only the respective energies of the state and its derivatives but also the correlated effect of them. Then, based on the proposed inequality, this paper derives a new stability criterion of systems with ATDs. This paper constructs LKF by proposing new sets of multiple subintervals of the ATDs, and makes use of the proposed inequality when estimating the derivatives of the LKF. This allows the resulting stability criterion to take full advantage of the information of the multiple subintervals of the ATDs. This paper also successfully applies the proposed free-matrix-based integral inequality to the system with a single time-varying delay. Three numerical examples demonstrate the effectiveness of the proposed methods.