Consider a characteristic equation (denominator of the closed-loop
transfer function) of the form
For example, compare the root-locus plots in Figures 3
and 4 for
In Figure 3, since the high-frequency pole is
moving left along the real axis, the complex-conjugate poles have to
move to the right. In Figure 4, since the high-frequency
pole is moving right along the real axis, the complex-conjugate poles
have to move to the left. We could have also found this result by
considering the centroid of the asymptotes. For Figure 3,
the centroid is
I have heard this fact called ``Grant's Rule.'' However, it is not listed as such in any of the primary root locus books by Walter Evans , John Truxal , or even Chestnut and Mayer .
The only reference that I've been able to find that names this fact ``Grant's Rule'' is Linear Control System Analysis and Design by D'Azzo and Houpis . They reference an unpublished paper (from North American Aviation) by A. J. Grant entitled ``The conservation of the sum of the system roots as applied to the root locus method,'' dated April 10, 1953 . I have tried in vain to find this paper. I have written Professors D'Azzo and Houpis, and even the company historian at Boeing (Boeing now owns North American Aviation) without success.
If you have a copy of this paper that you can send me, or knowledge of other texts or references that label this fact ``Grant's Rule,'' I would greatly appreciate hearing from you.