Convexity of frequency response arcs associated with a stable polynomial

Associated with a polynomial p(s) and an interval Omega contained in R is a frequency response arc. This arc is obtained by sweeping the frequency omega over Omega and plotting p(j omega ) 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(j omega ) as omega increases over Omega 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(j omega ) for omega in 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.<
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