Stability of singularly perturbed functional-differential systems: Spectrum analysis and LMI approaches

A singularly perturbed linear functional-differential system is considered. The delay is assumed to be small of the order of a small parameter multiplying a part of derivatives in the system. It is 'not assumed that the fast subsystem is asymptotically stable'. Two approaches to the study of the exponential stability of the singularly perturbed system are suggested. The first one treats systems with constant delays via the analysis of asymptotic behaviour of the roots of their characteristic equation. The second approach develops a direct Lyapunov-Krasovskii method for systems with time-varying delays leading to stability conditions in terms of linear matrix inequalities. Numerical examples illustrate the efficiency of both approaches.