On periodic solutions of adaptive systems in the presence of periodic forcing terms (Q749492)

This paper studies solutions of adaptive feedback loops in the presence of unmodeled dynamics if the discrete-time linear plant. It assumes that the reference output is periodic with period K. The aim is to find K- periodic solutions of the adaptive system and study their stability. Sufficient conditions to have K-periodic solutions are derived. Also, the stability of these solutions is studied assuming that adaptation is slow. A wide class of direct adaptive controllers is used for checking the sufficient conditions for the existence of a periodic solution. It is proved that ``almost always'' the periodic solution exists, but no tools to study its stability, except in the exact modeling case, are available. The present study would be a first step toward a complete description of the behavior of an adaptive controller in feedback with a high-order linear plant.