I am curious to know the difference between a lead compensator (lead filter) and a PD controller with a low pass filter. From the theory I can see that the transfer function of both these controllers are same. By choosing appropriate gains and placing zeros and poles at right frequencies, we can obtain same bodeplot (magnitude and phase plot). So, what is the real difference between the two? Why use different names for the same controllers?
A lead filter implies that the zero has a lower frequency than the pole. While a PD controller with a low-pass-filter does not necessarily imply that order. Also a lead filter (usually) does not have the zero and pole to far apart from each other, meaning that the bode diagram does not get very close to the asymptotes of +1 slope for magnitude and 90° for phase between the zero and pole. While the PD and lowpass in general can be spaced very far apart.
But you are right that there is theoretically not really a difference, besides that the PD and lowpass has an additional degree of freedom, due to the P action. Namely for both you have to place a zero and a pole.
However when implementing both you might get different results due to the way you might implement each filter when discretizing them. But usually there is only a significant difference when you get close to your Nyquist frequency.