branches passing through infinity in the root locus method

in this paper, construction rules of the root locus of a positive feedback system whose loop transfer function is not strictly proper have been investigated. the root locus method is being used since 1950. it is demonstrated that some of the steps in this technique are not general and for a positive feedback system whose loop transfer function is not strictly proper are not valid. to overcome this problem, new definitions for root locus branches have been presented which divide them into branches passing through infinity and branches ending/starting at/from infinity. we have also considered the realization and stability of these systems.