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. It is said that an arc is proper if it does not pass through the origin and if the net phase change of p(jω) as ω increases over Ω is no more than 180 degrees. The convexity of all proper frequency response arcs associated with a Hurwitz polynomial is established. The ramifications and extensions of arc convexity are discussed. Of particular interest is the fact that the so-called inner frequency response set is convex. This set consists of all points which can be connected to the origin via a continuous path which does not intersect the plot of p(jω) for ω ∈ R. Convexity of the inner frequency response set is shown to lead to an extreme point result for robust stability of a class of feedback systems having a structured unmodeled dynamic in the feedback path. An extension of the arc convexity result for an arbitrary convex root location region D is included