Checking whether root locus passes through point P

The root locus passes through a point $P$ (the desired dominant closed-loop pole) only if $P$ satisfies the angle criterion. The gain needed there is then found from the magnitude criterion.

Test steps

1. Mark all open-loop poles ($p_i$) and zeros ($z_j$) of $G(s)H(s)$ and the point $P = -\sigma_d + j\omega_d$.
2. Draw vectors from every pole and zero to $P$ and measure their angles from the positive real axis.
3. Compute

$$\phi = \sum \angle(P - z_j) - \sum \angle(P - p_i)$$

4. If $\phi = \pm 180^\circ(2q+1)$, then P lies on the root locus. If not, P cannot be a closed-loop pole for any $K$. The difference $\phi_c = \pm 180^\circ - \phi$ is the angle a compensator must supply (the starting point of lead/lag design).
5. If P is on the locus, find the gain:

$$K = \frac{\prod |P - p_i|}{\prod |P - z_j|}$$

6. Check that the other closed-loop poles at this $K$ lie well to the left (about 5 times farther from the $j\omega$-axis) or are nearly cancelled by zeros. Only then is P truly dominant.

Example

$G(s) = \dfrac{K}{s(s+2)}$, $P = -1 + j\sqrt3$ (ζ = 0.5, $\omega_n = 2$):

• $\angle P = 120^\circ$ and $\angle(P + 2) = 60^\circ$.
• $\phi = -(120^\circ + 60^\circ) = -180^\circ$, so P is on the locus.
• $K = 2 \times 2 = 4$.