# Derivative Control

With derivative action (also called rate action), the controller output is proportional to the rate of change of the error. This means the faster the change in level, the faster the change in controller output and control valve settings. By the same token, if the level remains constant, even with a large error, the controller output would be zero. This makes the use of derivative action by itself impractical.

Derivative action is normally combined with proportional action or proportionalplus-integral action. Derivative action, being proportional to the rate of change of the measured variable, introduces a “lead” (anticipation) element into the controller. This increases the speed of response of the controller and compensates for the lags introduced by proportional and integral actions. Figure 300-14 illustrates derivative action.

The output from a proportional-plus-derivative controller may be expressed as follows:

where:
O = controller output
Kc = controller gain
En = error at time n
TD = derivative time, minutes
Mn = measurement at time n
Mn-1 = measurement at previous sampling time
S = Time between measurements (sampling time)

The derivative action is greatest when integral and proportional action are just beginning to respond. Derivative action also responds to the change in sign of the measured variable. This opposes the tendency of integral and proportional action to overshoot the setpoint and enables the controlled variable to settle out faster than with either proportional or proportional-plus-integral action.

In Figure 300-14, area A represents the proportional component of controller output. Note that the proportional response is a function of the difference between the setpoint and the measured variable. Areas B and C represent the component added or subtracted by derivative action. As the measured variable stops decreasing and starts increasing, the sign of the derivative function changes. The integral action (area D) eliminates offset by not returning to zero when the proportional and derivative actions return to zero output. Areas E and F represent the corrections that result from all three actions taken together.

Derivative action, being sensitive to the rate of change of the measured variable, cannot be used in processes that require fast response, or that have rapid fluctuations or high noise levels. These conditions cause instability through large increases in the derivative gain, and rapidly change direction (sign). Although derivative action is difficult to tune because of its extreme sensitivity to measurement noise and other high frequency disturbances, it does have some applications. Most importantly, it is used with proportional and integral action in temperature processes that have large time lags.

Derivative action can be very helpful in controlling processes that have significant deadtime, but using it can be difficult. Sometimes adding derivative action can make the control loop appear slow and inactive with some types of process disturbances. This sluggishness might lead one to increase the amount of derivative and perhaps also increase the controller gain. However, these new tunings might make the controller unstable when a different disturbance occurs in the plant.