Root loci design using Dicksonʹs technique

Traditional techniques for determining the root loci were developed by Evans. Given the characteristic equation {1 + ( ) ( ) = 0}, Evans root locus method requires the magnitude of ( ) ( ) to be equal to minus one and is well established for determining the pole locations as the system gains are changed. In this paper, nontraditional techniques for developing the root loci of control systems are introduced. Illustrations are developed by using a method for plotting the root loci that involves setting the real and the imaginary portions of the characteristic equation equal to zero. Setting the real and imaginary parts of the characteristic equation equal to zero arises from work in the theory of equations by Dickson. Application of this technique allows adjustment of the gain, to establish the system requirements, and introduces collateral equations subject to unique geometric interpretation.