Root locus properties and sensitivity relations in control systems

The differential properties of root loci including pole sensitivity, angle of slope, and curvature at ordinary and irregular points are investigated in a unified manner. A relation between the sensitivity function and pole sensitivity is established. The sensitivity is shown to determine variations in the transfer function due to large (not only infinitesimal) variations in K. Additional properties of loci which are developed include loci of a variable pole position and the existence of asymptotes for open-loop transfer functions with the poles or zeros at infinity. The locus is treated as a transformation of a line (the real axis) in the K plane to the s plane, and properties of analytic functions are used to simplify calculations and results. It is shown that the properties obtained can be extended to the general root locus of a nonreal K.