Root-locus invariance exploiting alternative arrival and departure points

The same complete root locus is generated by all loop transfer functions whose numerator and denominator are linear combinations of the polynomials n(s) and d(s). Therefore, a complete root locus can be constructed starting from different sets of arrival and departure points. This property can be used to find the asymptotes of the negative locus for an exactly proper loop transfer function and to determine the configuration of the locus branches around the breakaway points. The root-locus invariance property can also be exploited to characterize a stabilization procedure leading to an exactly proper loop transfer function whose Nyquist diagram travels along a circle centered at -1 + j0