What derivative controller is and why useful

A derivative controller gives an output proportional to the rate of change of the error:

$$u(t)=K_d\frac{de(t)}{dt},\qquad G_c(s)=K_ds$$

In practice it is used with proportional action as a PD controller, $G_c(s)=K_p+K_ds$.

How it works

 error e(t)        D output
   ^  /\              ^ +
   | /  \             |--+  (rising error)
   |/    \______      |  |   +-- (falling)
   +----------->      +--+---+--->

• When the error is rising, D output is positive and pushes harder.
• When the error is falling quickly (output approaching the set point fast), D output is negative and reduces the drive before overshoot occurs.
• For a constant error, D output is zero.

So it acts on the future trend of the error: it is an anticipatory controller.

Why it is useful

1. Adds damping: for a second-order plant with PD control, $\zeta'=\zeta+\frac{K_d\omega_n}{2}$ (with $K_p=1$), so overshoot falls.
2. Faster settling: $t_s=4/(\zeta'\omega_n)$ decreases.
3. Improves stability: the added zero gives phase lead and raises the phase margin, like a lead compensator.
4. Quick response to sudden disturbances, because it reacts to the rate of change rather than waiting for the error to grow.

Limitations

• Has no effect on steady-state error (zero output for constant error), so it is never used alone.
• Amplifies high-frequency noise ($|G_c|=K_d\omega$).

Example: in a DC motor position servo, PD control lets the shaft reach the target angle quickly with little overshoot; P alone with high gain would oscillate.