Root Locus Concept

Topics: Feedback Control Systems

The root locus is a graphical method used to examine how the closed-loop poles of a system move in the s-plane as a gain $k$ in the loop is increased from $0$ to $\infty$. Since closed-loop poles determine the stability of the system, root-locus allows the study of stability as $k$ varies in the feedback loop. The gain $k$ might represent several things like a controller design parameter or a plant parameter.

Considering the open-loop transfer function $L(s)=PKH=\frac{n(s)}{d(s)}$, then the closed-loop transfer function with a gain $k$ added to the loop is:
$T=\frac{kL}{1+kL} = \frac{d\,k\,L}{d+k\,n}$
Then, the characteristic equation of the closed-loop function is

$$q(s) = d(s)+k\,n(s)$$

So, the root locus consists of finding and plotting the roots of $q(s)$ as $k$ goes from 0 to $\infty$.

Note:

• when $k$ is small, $q(s) \approx d(s)$. So, the roots of $q(s)$ are the open-loop poles
• and when $k \rightarrow \infty$, $q(s) \approx k\,n(s)$. So, the roots of $q(s)$ are the open-loop zeros
  Therefore, as $k$ goes from $0$ to $\infty$, the closed-loop poles move from the open-loop poles towards the open-loop zeros.