PD controller principle and working

A PD (proportional–derivative) controller produces a control signal proportional to the error plus the rate of change of the error:

$$u(t)=K_pe(t)+K_d\frac{de(t)}{dt},\qquad G_c(s)=K_p+K_ds=K_p(1+T_ds),\ T_d=\frac{K_d}{K_p}$$

          +-->[ Kp ]-----+
 e(t) ----|              (+)---> u(t)
          +-->[ Kd d/dt ]+

Principle

• The P part acts on the present error.
• The D part acts on the trend of the error. When the error is falling fast (output rushing toward the set point), $de/dt$ is negative and the D term reduces the drive before the output overshoots. So D gives anticipatory action and extra damping.
• It adds a zero at $s=-K_p/K_d$ to the open-loop transfer function, i.e. phase lead.

Working performance (second-order system)

For $G(s)=\dfrac{\omega_n^2}{s(s+2\zeta\omega_n)}$ with PD control (taking $K_p=1$):

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

Example: with $\omega_n=2$ rad/s and $\zeta=0.25$, choosing $T_d=0.3$ s gives $\zeta'=0.25+0.3=0.55$, cutting the step overshoot from about 44% to about 16% (the added zero keeps it a little above the 12.6% of a plain ζ = 0.55 system).