Grant's(?) Rule

Consider a characteristic equation (denominator of the closed-loop
transfer function) of the form

If we multiply out this characteristic equation, we get

Clearly, the coefficient

Therefore, if we have a characteristic equation written in terms of the loop transfer function as

and we multiply out the left side, and the order of

then the coefficient of

**Rule 7**- If there are two or more excess poles than zeros (
), then
for any gain
*K*, the sum of the real parts of the closed-loop poles (or the average distance from the -axis) is constant.

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

and for Figure 4 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 [1], John Truxal [2], or even Chestnut and Mayer [3].

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 [4]. They reference an unpublished paper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 [5]. 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.