A lead or lag compensator adds a pole and a zero to reshape a feedback loop without replacing the plant model.

A common first-order compensator has the form

$$C(s)=K\frac{1+s/z}{1+s/p}.$$

For a lead compensator, the zero is encountered before the pole as frequency increases. It contributes positive phase over a frequency range and can increase phase margin or move dominant closed-loop poles toward a faster, better-damped region.

For a lag compensator, the pole is encountered before the zero. It increases low-frequency loop gain relative to high-frequency gain and can improve steady-state tracking while changing high-frequency behavior less strongly.

Lead and lag compensation expose a general design idea: poles and zeros are deliberately placed to reshape loop gain and phase where performance requires it.

The compensator is judged through the resulting closed loop, not through its isolated transfer function.