Complementary root locus revisited

Abstract

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 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 will have at least one and at most 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.

Extension of the root-locus method to a certain class of fractional-order systems

“…Final results can easily be extended for non-proper systems but such an extension to bi-proper transfer functions is complicated. This is similar to the difficulty that occurs when the root-locus plot for an integer-order system with bi-proper transfer function is involved [6]. • N(s) and D(s) have no common roots.…”

A revision of root locus method with applications

“…Note also the singularity ink for k = − c −1 . This will cause some branches to reach infinity even for finite gains, see [8].…”

“…General algebraic equations for RL in polar and Cartesian coordinates have been considered in [7]. Some recent results in this regard are presented in [8,9]. Several numerical procedures for RL plotting have also been proposed.…”

Simple answers to usual questions about unusual forms of the Evans' root locus plot

“…By taking G(s) ≡ b(s)/a(s), where a(s) and b(s) are real coprime polynomials of degrees n and m, respectively, then (1) can be written as: Evans derived rules that allow to sketch the loci (the branches) of the roots of equation (2) as k is varied from zero to infinity, without solving such an equation (the case k < 0 is studied, for instance, by Eydgahi and Ghavamzadeh, 2001;Teixeira et alii, 2004). This sketch is based on the values of the poles p j of G(s) (j = 1, ..., n), which are the n roots of the polynomial a(s); and on the zeros z j of G(s) (j = 1, ..., m), which are the m roots of the polynomial b(s).…”