Understanding the Root Locus Using Gain Plots

An alternate graphic representation of the Evans root locus plot, called gain plots, which exposes the relationship between the gain and the pole locations in polar coordinates, is presented. Gain plots portray the magnitude and angle on each closed-loop system eigenvalue in the complex plane as a function of gain. The gain plots highlight several important stability and performance features of the system. Inspection of the gain plots provides information about the closed-loop eigenvalue sensitivity to changes in gain. Gain plots provide a unified approach for single-input, single-output and multivariable systems where compensation dynamics are governed by a single scalar gain amplifying all plant inputs. An example demonstrates the utility of the gain plots for multivariable systems that possess complex conjugate root loci. The state space representation for this example is a linearized model of the vertical plane dynamics of an aircraft with three inputs, three outputs, and five state variables