Active boundary and interior absorbers for one-dimensional wave propagation: Application to transmission-line metamaterials

We address the problem of generating active absorbers for wave propagation in a class of one-dimensional continuous media. In particular, we consider the emerging application of metamaterials, which are engineered structures supporting unconventional wave propagation phenomena. We design two kinds of absorbers. The first absorber achieves a complete elimination of wave reflections from the metamaterial boundaries independently of the working frequency regime. The associated controller is implemented in a feedback loop and is given in general terms of the metamaterial effective constitutive parameters. The second absorber blocks wave propagation beyond a prescribed location at the metamaterial interior with minimized back-scattering, thus generating a sink without any physical boundary present. It is implemented in a feed-forward loop via a unique near unidirectional control wave method, using two concentrated actuators. Both the interior and boundary absorbers are based on an exact fractional order transfer function model that we derive for the metamaterial. The model explicitly exhibits essential wave characteristics, including delays, dispersion, impedance, boundary reflections etc. The resulting controllers are of fractional order as well and are realized via a dedicated approximation technique. We verify the results by numerical simulations of a representative mechanical metamaterial.