Averaging of systems with fast-varying coefficients and non-small delays with application to stabilization of affine systems via time-dependent switching. (May 2023) - Caiazzo, Bianca; Fridman, Emilia; Yang, Xuefei
Abstract: This paper investigates the stability of systems with fast-varying piecewise-continuous coefficients and non-small delays. Starting from a recent constructive time-delay approach to periodic averaging, that allowed finding upper bound on small parameter ϵ > 0 preserving the stability of the original delay-free systems, here we extend the method to systems with non-small delays and provide their input-to-state stability (ISS) analysis. The original time-delay system is transformed into a neutral type one embedding both initial non-small delay, whose upper bound is essentially larger than ϵ and does not vanish for ϵ → 0, and an additional induced delay due to transformation, whose length is proportional to ϵ . By exploiting Lyapunov–Krasovskii theory, we derive ISS conditions expressed as Linear Matrix Inequalities (LMIs), whose solution allows evaluating upper bounds both on small parameter ϵ and non-small delays preserving the ISS of the original time-delay system, as well as the resulting ultimate bound of its solutions. We further apply our results to stabilization of delayed affine systems by time-dependent switching. Three numerical examples illustrate the effectiveness of the approach.