On spectral mappings, higher order circle criteria, and periodically varying systems

For feedback equations of the form , in which is a linear operator, stability boundaries obtained by the circle criterion method can frequently be improved by finding a function which conformally maps the spectrum of from its initial region in the complex plane into a more suitable region. This idea, which is based on the spectral mapping theorem of Banach algebras, is applied here to a class of equations consisting of a time-invariant part and a periodically time-varying gain. A stability bound is derived which has a minimum whenever the frequency response of the time-invariant part peaks at a frequency near one half that of the time-varying gain.