Averaging of systems with fast-varying coefficients and non-small delays with application to stabilization of affine systems via time-dependent switching

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.