Why can't the root locus travel along the vertical imaginary axis ?
You would know if you read the 10 rules of drawing a Root Locus. Actually, by the time you get to the 8th rule, you already have answered your question.
The root locus of an (open-loop) transfer function is a plot of the locations (locus) of all possible closed loop poles" ? Could anyone give further maths explanation on this statement ?
That is a common question. In all major control systems books, they call the Open Loop transfer function to what is actually called in electronics the
Loop Gain. Loop Gain = Open Loop gain * Feedback factor.
Math is simple: Root locus simply plots the roots of A(s)*H(s)=-1 or equivalently, 1+A(s)*H(s)=0, which are the closed loop poles i.e. the zeros of the characteristic polynomial.
I repeat again. Open Loop transfer function is
NOT A(s), but it is what is called in electronics the loop gain i.e. A(s)*H(s).
A simple conclusion one could draw from this is that the Root Locus can be applied
ONLY FOR NEGATIVE FEEDBACK SYSTEMS.
"For negative feedback systems, the closed-loop poles move along the root-locus from the open-loop poles to the open-loop zeroes as the gain is increased" ?
Poles tend to repel the branches of the Root Locus and zeros tend to attract them... It is a conclusion from the rules.