Short note: Lead-lag compensator

A lag–lead compensator is a single network that combines a lag section and a lead section, used when both the transient response and the steady-state accuracy must be improved.

$$G_c(s) = K_c \underbrace{\left(\frac{s + 1/T_1}{s + \beta/T_1}\right)}_{\text{lead}} \underbrace{\left(\frac{s + 1/T_2}{s + 1/(\beta T_2)}\right)}_{\text{lag}}, \qquad \beta > 1$$

• The lag section (pole and zero very near the origin) acts at low frequency. It raises the low-frequency gain, so $K_v$ or $K_a$ increases and steady-state error falls.
• The lead section acts near the gain crossover frequency. It adds positive phase, raising the phase margin, lowering overshoot and increasing bandwidth.
• Frequencies are ordered $\dfrac{1}{\beta T_2} < \dfrac{1}{T_2} < \dfrac{1}{T_1} < \dfrac{\beta}{T_1}$.

Electrical network: two resistors and two capacitors ($R_1 \parallel C_1$ in series, $R_2$ with $C_2$ in shunt) give this transfer function.

Design idea: design the lead part to meet the phase margin or the dominant-pole location, then the lag part to meet the error constant without disturbing the transient.

Its effect is similar to a PID controller: the lag part acts like PI and the lead part like PD.