Derivatife term

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_d. The magnitude of the contribution of the derivative term to the overall control action is termed the derivative gain, K_d.

The derivative term is given by:

Derivative action predicts system behavior and thus improves settling time and stability of the system.^[11][12] An ideal derivative is not 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.^[13] Derivative action is seldom used in practice though - by one estimate in only 25% of deployed controllers^[13] - because of its variable impact on system stability in real-world applications.