Lead and lag compensation
Shape the loop across frequency instead of increasing one gain everywhere.
First, picture it
A lead compensator is like reacting earlier near the frequencies where speed and stability compete. A lag compensator boosts low-frequency accuracy relative to the crossover region, often at the cost of a slower response.
What the model says
A lead network places its zero closer to the origin than its pole (both left-half-plane in a stable minimum-phase design), adding positive phase over a band. A lag network puts its pole closer to the origin and increases relative low-frequency gain. Both change the root locus and Bode curves. A controller must still respect noise, actuator limits and the exact plant.
C_lead(s)=K(s+z)/(s+p), 0<z<p; C_lag(s)=K(s+z)/(s+p), 0<p<zThese are illustrative stable first-order forms; scaling K matters for the final crossover.
Lead adds positive phase over a band; lag adds negative phase. Exact curves and crossover depend on pole, zero and gain choices.
Make it concrete
If a speed loop needs more phase near its gain crossover, consider lead. If it needs lower steady error without lifting crossover as much, consider lag.
Lead is not free speed: it can amplify high-frequency sensor noise. Lag is not a universal cure for drift or saturation.
Check your understanding+
Which network usually adds positive phase near crossover?
A lead compensator, with its zero nearer the origin than its pole.