Convexity of Frequency Response Arcs Associated with a Stable Polynomial

Associated with a polynomial p(s) and an interval Ω ⊆ R is a frequency response arc. This arc is obtained by sweeping the frequency ω over Ω and plotting p(jω) in the complex plane. We say that an arc is proper if it does not pass through the origin and the net phase change of p(jω) for all ω ϵ Ω is no more than 180 degrees. In this paper, we establish convexity of all proper frequency response arcs associated with a Hurwitz polynomial. The latter part of the paper deals with ramifications and extensions. Of particular interest is the fact that the so-called inner frequency response set is convex.