# Interpreting Root Locus

Last night, I was designing a controller for a feedback system and tinkered alot with root locus. It was a hard time remembering what I learnt 3 years ago but here is a way, how we can interpret a root locus.

Let’s start from root locus definition:

Root locus is a plot of the roots of the characteristic equation of the closed-loop system for all values of a system parameter.

Most commonly used parameter is the gain; however, any other variable of the open-loop transfer function may be used. First thing we need to keep in mind is what complex roots of characteristic equations represent. The roots of an under-damped system are in the form

The real part of root is the decay rate of the system and imaginary part is the frequency of oscillation present in system. Secondly, as gain “k” increases, we move from poles to zeros and lastly, poles closer to origin are dominant and thus contribute greatly to system response.

So lets take a deep dive with an example. Consider a system with root locus

So at k = 0, we have our poles at -ve real axis and as k approches to infinity, poles will be on RHS of s-plane and the system will become unstable. Following observations can be made from this root locus:

- The system is not stable for all values of gain “k”
- Initially the system doesn’t possesses oscillations (poles are real) but as k increases, there are oscillations (inclusion of imaginary part). In short increasing k, increases oscillations.
- As long as the system possesses oscillations and is stable, increasing k, decreases the decay rate of oscillation (real part of poles decreases with increasing k).

To confirm our claim, here is plot showing the step response of this system with three different values of gain:

As we can see, it is clear that our step responses are in accordance with our observations.

That’s all for now. For practicing, try to interpret the root loci of three systems, presented in header image.

**Hint:** MATLAB comes in handy for plotting root loci and step responses of control systems.