control//feedback control//PID controller//derivative action

Derivative action is the part of a controller that reacts to how fast the error is changing, so that it brakes an approach before the output overshoots, the way a driver eases off the throttle before reaching the speed limit rather than at it. It lets the other two terms use higher gains for the same overshoot, which makes the loop faster.

Derivative action is the part of a controller that reacts to how fast the error is changing, so that it brakes an approach before the output overshoots, the way a driver eases off the throttle before reaching the speed limit rather than at it. It lets the other two terms use higher gains for the same overshoot, which makes the loop faster.

uD(t)=Kd dedt≈Kd ek−ek−1Δtu_D(t)=K_d\,\frac{de}{dt}\approx K_d\,\frac{e_k-e_{k-1}}{\Delta t}uD​(t)=Kd​dtde​≈Kd​Δtek​−ek−1​​

Industry quotes it as a rate time TdT_dTd​, with Kd=KpTdK_d=K_pT_dKd​=Kp​Td​. Read that way, proportional plus derivative action is Kp(e+Td e˙)K_p\bigl(e+T_d,\dot e\bigr)Kp​(e+Td​e˙), the proportional action applied to the error TdT_dTd​ seconds ahead, extrapolated along its current slope.

It amplifies noise. The difference of two noisy readings divided by a short sample time is mostly noise, and the derivative turns sensor jitter into actuator chatter that wears valves and motors. Every practical derivative is filtered, usually by a first-order low-pass with a time constant a fraction of TdT_dTd​; Simulink's PID block sets that filter through a coefficient NNN, 100 by default.

Taken on the error, a setpoint step makes the derivative huge for one sample, a spike called the derivative kick. Taken on the measurement instead, −Kd dy/dt-K_d,dy/dt−Kd​dy/dt, it brakes against disturbances just the same and ignores setpoint changes; this is what the Arduino PID library does, and a common option in industrial controllers.

It is often left out. In noisy, fast loops (flow, pressure) it does more harm than good, so most process loops run as PI. It earns its place on slow plants with a clear lag, such as temperature, and in mechanical systems where it is the damping: the velocity term of the drone's controller is a derivative action on position (vertical drone).