Controller to change transient properties

To change (improve) the transient properties of a system (overshoot, settling time, rise time), a controller with derivative action is recommended: a PD controller (or derivative/rate feedback), or in compensator terms a lead compensator.

How PD control changes the transient

$$G_c(s)=K_p+K_ds=K_p(1+T_ds)$$

For the standard plant $G(s)=\dfrac{\omega_n^2}{s(s+2\zeta\omega_n)}$ (with $K_p=1$):

$$\frac{C(s)}{R(s)}=\frac{\omega_n^2(1+T_ds)}{s^2+(2\zeta\omega_n+T_d\omega_n^2)s+\omega_n^2},\qquad\zeta'=\zeta+\frac{T_d\omega_n}{2}$$

1. Damping increases (ζ′ > ζ), so peak overshoot falls.
2. Settling time falls ($t_s=4/\zeta'\omega_n$).
3. Anticipation: D action responds to the rate of change of error, applying correction before the error becomes large, like braking a car before the stop line.
4. The added zero gives phase lead, which raises the phase margin and bandwidth, so the response is faster and more stable.
5. Steady-state error is not changed (the system type is the same); if needed, it is handled separately by I action.

Caution: derivative action amplifies high-frequency noise, so $K_d$ is kept moderate or a filtered derivative is used.

Example: $\omega_n=4$ rad/s, $\zeta=0.25$ ($M_p=44\%$, $t_s=4$ s). With $T_d=0.15$ s: $\zeta'=0.25+0.3=0.55$, $t_s\approx4/(\zeta'\omega_n)=1.8$ s, and the overshoot drops from 44% to about 16% (a little above the 12.6% of a pure ζ = 0.55 system, because of the added zero).