Complementary root locus revisited

In this paper, a new finding related to the well-known root locus method that is covered in the introductory control systems books is presented, It is shown that some of the complementary root locus rules and properties are not valid for systems with loop transfer functions that are not strictly proper. New definitions for root locus branches have been presented which divide them into branches passing through infinity and branches ending at or starting from infinity. New formulations for calculating the number of branches passing through the infinity point of intersection of the asymptotes on the real axis, and angles of these asymptotes with the real axis have been introduced. It has been shown this type of system with the order of n will have at least one and at most n branches which will pass through infinity. The realization and stability of these systems have been investigated, and their gain plots have been presented. The new finding can be used by educators to complement their lecture materials of the root locus method. By using problems similar to examples presented in the paper, analytical understanding of the students in a classical control systems course can be tested