Compensator Design:
System Type :
input = A/s^q ; System type = G/s^N;
If N = q-1 ; finite amount of steady state error.
N > q-1; zero steady state error.
N < q-1 steady state error is infinite.
'N' should be satisfied before getting into design details.

Open loop poles tend to repel the branches of the root locus and open loop zeros tend to attract the root locus. So a lead compensator [------x-----o---] will try to pull the root locus (poles sitting at origin)towards left half plane. Stability is improved.
Same case with lag compensator- [-----o-----x----] will push the root locus away from the left half plane.