A time-delay approach to vibrational control with square wave dithers

This paper studies stabilization of second-order systems by fast-varying square wave dithers depending on a small parameter ε > 0, which is inverse of the dither frequency. We first employ the known in vibrational control coordinate transformation that allows to cancel ¹_ε multiplying the square wave dithers, and then present a time-delay approach to periodic averaging of the system in new coordinates. The time-delay approach leads to a model where the delay length is equal to ε. The resulting time-delay system is a perturbation of the averaged system in new coordinates which is assumed to be exponentially stable. The stability of the time-delay system guarantees the stability of the original system. We construct an appropriate Lyapunov functional for finding sufficient stability conditions in the form of linear matrix inequalities (LMIs). The upper bound on ε that preserves the exponential stability is found from LMIs. Two numerical examples illustrate the efficiency of the method.