LMI Characterisation of Robust Stability for Time-Delay Systems: Singular Perturbation Approach

We study here robust stability of linear systems with several uncertain incommensurate delays, more precisely delay-dependent stability. The main result of this paper consists in establishing that this property is equivalent to the feasibility of some linear matrix inequality (LMI), a convex optimization problem. The method is based on two main ideas: use of Fade approximation to transform the system into some singularly perturbed finite-dimensional system, for which robust dichotomy has to be checked; and recursive applications of generalized Kalman-Yakubovich-Popov (KYP) lemma to characterize by an LMI the previous property