On the Stability of Three-Time-Scale Linear Time-Invariant Singularly Perturbed Systems with State Delay

The purpose of this study is to obtain the conditions for the exponential stability of three-time-scale singularly perturbed linear time-invariant systems with multiple commensurate delays (TSPLTISD) in the slow state variables, independent of the small parameters and valid for all of their sufficiently small values. The considered approach is the asymptotic decomposition of the system, depending on the two small parameters of singular perturbation in the system, based on the non-degenerate decoupling transformation of the Chang-type. This splits the original system into the three subsystems of lower dimensions without compromising the qualitative behaviour of the original system. The obtained approximated subsystems are simpler than the original system and do not depend on the small parameters. Based on the First Lyapunov method, it is proved that the exponential stability of the asymptotically approximated subsystems guarantees the exponential stability of the original three-time-scale system with delay and the established conditions are robust with respect to the small parameters.