Control engineering: lead and lag compensators

iiJDSii

Not sure if this is the right forum, but I'm an electrical engineer studying controls so here goes.

Here's my background on what I know. Lead or lag compensators generally have the form C(s) = K*(s+z)/(s+p), i.e. the controller has a DC gain and introduces a pole-zero pair. In the case of a lag compensator, the pole occurs before the zero, and the opposite for the lead compensator.

From what I understand, these compensators can help meet system specifications for steady-state error and phase margin (probably other stuff, but my course tends to focus on those two).

A lead compensator can help add phase around the critical point (where magnitude of the loop gain, |C(s)P(s)| = 1) to help with the phase margin; while the gain K of the controller can help with the steady-state error.

Alright, so what does a lag compensator do that's better/different that a lead compensator? As far as I can tell the phase added by the lag controller doesn't really help/is unwanted, and really you're just interested in the gain for steady-state error, while designing the pole-zero pair placement such that you fix the phase margin after adding your desired DC gain. That's my interpretation thus far.

Summaries of my confusion:
- When would one opt for a lead, vs a lag, vs a lead-lag dual compensator?
- How can a lag controller be better (more suitable, whatever) than a lead?

Thanks for any help, much appreciated

DragonPetter

Are you familiar with PID controllers? A lag (integrator) pole will allow you to have 0 steady state error but also increase your chance for instability with faster response, while a lead (differentiator) zero will decrease your system's response speed making it respond more slowly to changes in the command signal or to disturbances. The combination with the right gain is the design task to compromise between speed, accuracy, and stability.