We analyze a matching system for the scanning aperture antenna radiating through a layer with unpredictably changing parameters. Improved matching has been achieved by adaptive motion of a dielectric plate in the gap between the aperture and the radome. The system is described within the framework of an infinite layered structure. The validity of the model has been confirmed by numerical simulation using CST Microwave Studio software and by an experiment. It is shown that the reflection coefficient at the input of some types of a matching device, which is due to the deviation of the load impedance from the nominal value, is determined by a compact and versatile formula. The potential efficiency of the proposed matching system is shown by a specific example, and its dependence on the choice of the starting position of the dielectric plate is demonstrated.