Constructive method for averaging-based stability via a delay free transformation

We treat input-to-state stability-like (ISS-like) estimates for perturbed linear continuous-time systems with multiple time-scales, under the assumption that the averaged, unperturbed, system is exponentially stable. Such systems contain rapidly-varying, piecewise continuous and almost periodic coefficients with small parameters (time-scales). Our method relies on a novel delay-free system transformation in conjunction with a new system presentation, where the rapidly-varying coefficients are scalars that have zero average. We employ time-varying Lyapunov functions for ISS-like analysis. The analysis yields LMI conditions, leading to explicit bounds on the small parameters, decay rate and ISS-like gains. The novel system presentation plays a crucial role in the ISS-like analysis by allowing to derive essentially less conservative upper bounds on terms containing the small parameters. The obtained LMIs are accompanied by suitable feasibility guarantees. We further extend our approach to rapidly-varying systems subject to either discrete (constant/fast-varying) or distributed delays, where our approach decouples the effects of the delay and small parameters on the stability of the system, and leads to LMI conditions for stability of systems with non-small delays. Extensive numerical examples show that, compared to the existing results, our approach essentially enlarges the small parameter and delay bounds for which the ISS-like/stability property of the original system is preserved.