Editing Proportional–integral–derivative controller (section)

===Derivative term===
[[File:Change with Kd.png|right|thumb|320px|Response of PV to step change of SP vs time, for three values of ''K''<sub>d</sub> (''K''<sub>p</sub> and ''K''<sub>i</sub> held constant)]]

The derivative of the process error is calculated by determining the slope of the error over time and multiplying this rate of change by the derivative gain ''K''<sub>d</sub>. The magnitude of the contribution of the derivative term to the overall control action is termed the derivative gain, ''K''<sub>d</sub>.

The derivative term is given by
:<math>D_\text{out} = K_\text{d} \frac{de(t)}{dt}.</math>

Derivative action predicts system behavior and thus improves settling time and stability of the system.<ref>{{cite web |url=http://ctms.engin.umich.edu/CTMS/index.php?example=Introduction&section=ControlPID |title=Introduction: PID Controller Design |publisher=University of Michigan}}</ref><ref>{{cite journal |title=PID without a PhD |url=http://igor.chudov.com/manuals/Servo-Tuning/PID-without-a-PhD.pdf |publisher=EE Times-India |date=October 2000 |author=Tim Wescott}}</ref> An ideal derivative is not [[causal system|causal]], so that implementations of PID controllers include an additional low-pass filtering for the derivative term to limit the high-frequency gain and noise. Derivative action is seldom used in practice though – by one estimate in only 25% of deployed controllers {{Citation needed|date=November 2017}} – because of its variable impact on system stability in real-world applications.