A note on the stability of linear dynamical systems with time delay

Abstract

This paper presents a stability study of linear time-invariant and periodic systems with time delay. The methods of semi-discretization, continuous time approximation and Lyapunov stability theory are used to study the stability of two benchmark systems. It is found that for linear time-invariant systems, the Lyapunov method is usually conservative leading to a much smaller domain of stability in a parameter space than the true solution, with the exception of the complete Lyapunov functional due to Gu, which gives highly accurate predictions with little conservatism. For periodic systems, it is difficult to find appropriate Lyapunov–Krasovskii functionals. Numerical methods such as semi-discretization and continuous time approximation are more appealing, and can compute geometrically complex stability boundaries in the parameter space with high accuracy.