Stability of LTI Delayed Systems: Lyapunov Based Delay-Independent Methods

The paper presents the Lyapunov - Krasovskii and the Lyapunov-Razumikhin criteria to analyse the stability of Linear Time Invariant Delayed Systems (LTI-DS). These approaches are an adaptation to Time Delay Systems (TDS) of the seminal Lyapunov theorem on stability, primarily designed for systems without delay. A reminder of how delay may enter a system is addressed. The two main approaches are then investigated. Due to the infinite dimensional characteristics of TDS, a functional is considered in the Lyapunov - Krasovskii approach. However, in contrast, an ordinary Lyapunov function is chosen in the Lyapunov- Razumikhin method. These are delay independent methods and result in Linear Matrix Inequalities (LMIs).