11.1.3 The Derivative Term

The derivative term, $K_d\frac{de(t)}{dt}$ only contributes when there is an abrupt change of $e(t)$. This change can be due to a change of set point, or it can be due to a change of feedback. A sudden change of feedback can be due to external factors such as a bump on the road.

This term is also called the dampening term because it dampens the effect of the other two terms. As $e(t)$ gets smaller, $K_d\frac{de(t)}{dt}$ is negative. In fact, for motor control, $K_d\frac{de(t)}{dt}$ is very negative to begin with because the response of a motor is stronger at lower speeds.

So why do we need the derivative term?

One use of the derivative term is so that we can crank up $K_p$. With a dampening term, we can adjust the other two terms so that the output does not ``fizzle'' out as the feedback gets closer to the set point. Note that $K_pe(t)$ fizzles out by itself anyway, but $K_i\int_0^t{e(x)dx}$ contributes more in time. Also, the response of a motor decreases as speed increases.