Synthesis of positive real feedback systems: a simple derivation via Parrott's theorem

Abstract

A simple derivation of the formulas for controller rendering a closed-loop system strongly positive real is presented. The derivation is based on Parrott's theorem and the positive real lemma. When the positive real lemma is applied to the closed loop system, a bilinear matrix inequality results. The significance of Parrott's theorem in the context of a strongly positive real synthesis problem is that it allows one to convert this bilinear matrix inequality into two linear matrix inequalities.