Stability analysis of linear systems with time-varying delays: Delay uncertainty and quenching

This paper addresses the problem of stability analysis of a class of linear systems with time-varying delays. We develop conditions for robust stability that can be tested using semidefinite programming using the sum of squares decomposition of multivariate polynomials and the Lyapunov-Krasovskii theorem. We show how appropriate Lyapunov-Krasovskii functionals can be constructed algorithmically to prove stability of linear systems with a variation in delay, by using bounds on the size and rate of change of the delay. We also explore the quenching phenomenon, a term used to describe the difference in behaviour between a system with fixed delay and one whose delay varies with time. Numerical examples illustrate changes in the stability window as a function of the bound on the rate of change of delay.