The root locus of an (open-loop) transfer function is a plot of the locations (locus) of all possible closed loop
poles with proportional gain K and unity feedback.
Independently from K, the closed-loop system must always have n poles, where n is the number of poles of the
open loop transfer function. The root locus must have n branches, each branch starts at a pole of the Open Loop
Transfer Function (OLTF) and goes to zero of OLTF. If OLTF has more poles than zeros (as is often the case), m <
n and we say that the OLTF has zeros at infinity. In this case, the limit of OLTF as s → infinity is zero. The number
of zeros at infinity is n-m, the number of poles minus the number of zeros, and is the number of branches of
the root locus that go to infinity (asymptotes). Since the root locus is actually the locations of all possible closed
loop poles, from the root locus we can select a gain such that our closed-loop system will perform the way we
want. If any of the selected poles are on the right half plane, the closed-loop system will be unstable. The poles
that are closest to the imaginary axis have the greatest influence on the closed-loop response, so even though
the system has three or four poles, it may still act like a second or even first order system depending on the
location(s) of the dominant pole(s).
