A time-delay approach to Lie-brackets-based averaging of affine systems

In this paper, we study stability of affine systems with a small parameter ϵ > 0. We present a time-delay approach to Lie-brackets-based averaging, where we transform the system to a time-delay one. The latter has a form of perturbed Lie-brackets system. The practical stability of the time-delay system guarantees the same for the original system. We present the direct Lyapunov-Krasovskii method for the time-delay system and provide sufficient conditions for the local input-to-state stability of the original one. We further apply the results to stabilization under unknown control directions. In contrast to the existing results that are all qualitative, we derive constructive linear matrix inequalities for finding quantitative upper bounds on ϵ that ensures the practical stability and on the resulting ultimate bound. We also give new qualitative conditions for semiglobal stabilization. Numerical examples illustrate the efficiency of the results.