Purpose of lead and lag compensators

A compensator is an additional network placed in a control loop to change the system's response so that it meets the specifications that adjusting gain alone cannot meet. Lead and lag compensators are the most common.

Lead compensator

$$G_c(s)=\frac{1+Ts}{1+\alpha Ts},\quad\alpha<1\ \text{(zero closer to origin than pole)}$$

Purpose:

• Supplies positive phase (lead), up to $\phi_m=\sin^{-1}\frac{1-\alpha}{1+\alpha}$ at $\omega_m=1/(T\sqrt\alpha)$.
• Increases phase margin and damping, so overshoot falls.
• Increases gain crossover frequency and bandwidth, so the response is faster (smaller rise and settling time).
• Reshapes the root locus to the left (improves transient response and stability).

Lag compensator

$$G_c(s)=\frac{1+Ts}{1+\beta Ts},\quad\beta>1\ \text{(pole closer to origin)}$$

Purpose:

• Raises low-frequency gain by β, improving steady-state accuracy ($K_p$, $K_v$, $K_a$ larger) with little change to the transient.
• Attenuates high frequencies by $20\log\beta$, which lowers the gain crossover so that the PM increases (when the plant has enough phase at low frequency).
• Reduces bandwidth, so it filters noise but slows the response.

Summary

A lag–lead compensator combines both when both transient and steady-state improvements are needed.