root locus

The root locus is a map that shows how the poles of a closed-loop system travel across the complex plane as you turn a single knob — usually the controller gain K — from zero up to infinity. Remember that the closed-loop poles set stability and response, but they move when you change the gain. Rather than re-solving the characteristic equation for every K, the root locus draws all their paths at once, so you can see at a glance which gain keeps every pole safely in the left half-plane and which gain pushes one across into instability.

The plot follows elegant rules discovered by Walter Evans in 1948: the branches start at the open-loop poles (K = 0) and end at the open-loop zeros (K = ∞), travel along the real axis to the left of an odd count of poles-and-zeros, and break away to chase asymptotes at predictable angles. The payoff is design intuition: you can see that raising the gain speeds the response but eventually drags a pair of poles across the imaginary axis into oscillation, and you can place a controller zero to bend the locus and reshape the trade-off. It's the classic bridge between 'where are my poles?' and 'what gain should I pick?'