Grant's(?) Rule

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

P(s) = (s+a)(s+b)(s+c)(s+d) = 0.

If we multiply out this characteristic equation, we get

P(s) = s⁴ + ws³ + xs² + ys + z = 0.

Clearly, the coefficient w is the sum of all the pole locations.

w = a+b+c+d

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

$$P(s) = 1+L(s) = 1+K\frac{n(s)}{d(s)} = 0$$

P(s) = d(s)+Kn(s) = 0

and we multiply out the left side, and the order of d(s) is two or more greater than the order of n(s), if

$$P-Z \geq 2$$

then the coefficient of s^P-1 is independent of K. In other words, the sum of all the roots (closed-loop poles) is constant and equal to the sum of the open-loop poles. This result gives us our seventh root-locus rule:

Rule 7

If there are two or more excess poles than zeros ( $P-Z \geq 2$), then for any gain K, the sum of the real parts of the closed-loop poles (or the average distance from the $j\omega$-axis) is constant.

For example, compare the root-locus plots in Figures 3 and 4 for

$$L_a(s) = \frac{K(s+4)}{s(s+2)(s+3)} \quad \mbox{and} \quad L_b(s) = \frac{K(s+3)}{s(s+2)(s+4)}$$

  

Figure 3: Root locus plot showing Grant's Rule moving poles to the right

  

Figure 4: Root locus plot showing Grant's Rule moving poles to the left

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

$$\sigma_a = \frac{\sum p_i - \sum z_j}{P-Z} = \frac{-2-3+4}{2} = -0.5$$

and for Figure 4 the centroid is

$$\sigma_a = \frac{\sum p_i - \sum z_j}{P-Z} = \frac{-2-4+3}{2} = -1.5$$

The Question

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 They reference an unpublished paper (from North American Aviation) by A. J. Grant  entitled ``The conservation of the sum of the system roots as applied to the root locus method,'' dated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.

 

• The Question