Short note: Phase-lead/phase-lag compensator

A compensator is an extra block placed in the loop to change the root locus or the frequency response so that the specifications are met. Its digital form is

$$G_D(z) = K\,\frac{z - z_c}{z - p_c}$$

Phase-lead compensator

• The zero is nearer to $z = 1$ than the pole ($p_c < z_c$), so it gives positive phase, at most $\phi_m$ where $\sin\phi_m = \frac{1-\alpha}{1+\alpha}$ (w-plane form $\frac{1+\tau w}{1+\alpha\tau w}$, $\alpha<1$).
• It moves the root locus to the left (towards the origin of the z-plane), so it increases phase margin, damping and bandwidth.
• Result: faster response and less overshoot, but more high-frequency noise.

Phase-lag compensator

• The pole is nearer to $z = 1$ than the zero ($p_c > z_c$), both close to 1. It gives negative phase but high low-frequency gain.
• It raises the static error constants ($K_p$, $K_v$) by the factor $\frac{1-z_c}{1-p_c} > 1$ (its DC gain at $z = 1$ for $K = 1$) without much change to the transient response.
• Result: lower steady-state error and less noise, but a slower response.

A lag–lead compensator combines the two when both transient and steady-state performance must be improved.