Averaging-Based ISS Analysis of Systems With Rapidly Varying Periodic Delays

We study the stability of a linear system with a periodic delay that is rapidly varying around a nominal value. We are inspired by previous results on rapidly varying periodic delays, where the stabilizing effect of the rapidly-varying periodic delay on the system stability was studied via a trajectory-based comparison to a suitable comparison (averaged) system with distributed delays. The stability of the original system was proved to be asymptotic, provided the delay modulation was rapid enough, but quantitative bounds on the delay period which preserve the stability of the original system were not provided. In this work we aim to provide such quantitative bounds for the exponential stability of the original system. We introduce a novel state transformation that explicitly maps the original system into a perturbed version of the comparison system. Then, assuming knowledge of an exponential bound on the norm of the fundamental solution of the comparison system, and augmenting the state of the original system with that of the transformed system, we employ a recently developed trajectory-based inequality to derive constructive conditions on the delay period and on the exponential decay rate of the original system. In the case of perturbed linear systems, we prove that the same delay periods guarantee input-to-state stability (ISS) of the original system. Numerical examples demonstrate the efficacy of the proposed approach.