Period-Aware Asymptotic Gain with Application to a Periodically Forced Synchronization Circuit

The classical asymptotic gain (AG) is a concept known from the input-to-state stability theory. Given a uniform input bound, AG estimates the asymptotic bound of the output. Sometimes, however, more information is known about the input than just a bound. In this paper we consider the case of a periodic input. Under the assumption that the system converges to a periodic solution, we introduce a new gain, called period-aware asymptotic gain (PAG), which employs periodicity to enable a sharper asymptotic estimation of the output. Since the PAG can distinguish between short-period ("high-frequency") and long-period ("low-frequency") signals, it is able to rigorously quantify such properties as bandwidth, resonant behavior, and high-frequency damping. We discuss how the PAG can be computed and illustrate it with a numerical example from the field of power electronics.