Real-Axis Rule

Topics: Feedback Control Systems

The real-axis rule allows to determine which parts of the real axis belong to the root locus.
Starting from the open-loop transfer function $L$ in the following form

$$L = \beta \frac{\prod_{i=1}^{d_n}(s-z_i)}{\prod_{i=1}^{d_d}(s-p_i)}$$

Then, the real-axis rule includes the following two parts.

Odd Rule: $\beta k > 0$

A segment of the real axis belongs to the root locus if the number of finite open-loop poles and finite open-loop zeros to its right on the real axis is odd.

Even Rule: $\beta k > 0$

A segment of the real axis belongs to the root locus if the number of finite open-loop poles and finite open-loop zeros to its right on the real axis is even.