Root Locus - 2
Note a couple of special cases. What will the characteristic equation look like for K = 0?
This is just the denominator of G times the denominator of H. So the roots of this equation are the open-loop poles.
As K increases in the characteristic equation, terms with K in them become more important. In the extreme, as K approaches infinity, the terms without K in them become insignificant. So the characteristic equation becomes
But the roots of this equation are just the open-loop zeros.
In between 0 and infinity, the roots describe some path. But we now know the starting and ending points for the roots: they start at the open-loop poles and end at the open-loop zeros. We can describe the root locus in this way: the root locus is the path taken by the closed-loop poles as they migrate from the open-loop poles to the open-loop zeros as K varies from 0 to infinity.
It is often the case that there are not as many open-loop zeros as there are open-loop poles. Often there are not open-loop zeros. Actually the zeros that you see in the open-loop transfer function are finite zeros. There are infinte zeros also, i.e. zeros at infinity. In fact (# open-loop poles) - (# finite open-loop zeros) = (#infinite open-loop zeros). So, for example
the open-loop system above has 4 poles, two finite zeros (at 0 and -3), and two infinite zeros.