Proportional plus Derivative (PD) Control System

Proportional plus derivative (rate) control is a control mode in which a derivative section is added to the proportional controller. This derivative section responds to the rate of change of the error signal, not the amplitude as with the integral control; this derivative action responds to the rate of change the instant it starts. This causes the controller output to be initially larger in direct relation with the error signal rate of change. The higher the error rate of change, the sooner the final control element is positioned to the desired value. The added derivative action reduces initial overshoot of the measured variable, and therefore aids in stabilizing the process sooner.

This control mode is called proportional plus derivative/rate (PD) control because the derivative section responds to the rate of change of the error signal.

Definition of Derivative Control

A device that produces a derivative signal is called a differentiator.

The differentiator provides an output that is directly related to the rate of change of the input and a constant that specifies the function of the differentiation. The derivative constant is expressed in units of seconds and defines the differential controller output

The differentiator acts to transform a changing signal to a constant magnitude signal as illustrated in the figure below:

Derivative cannot be used alone as a control mode. This is because a steady-state input produces a zero output in a differentiator. If the differentiator was used as a controller, the input signal it would receive is the error signal. A steady-state error corresponds to any number of necessary output signals for the positioning of the final control. Hence a derivative action is combined with proportion action in such a way such that the proportion section output serves as the derivative section output. Proportional plus derivative (rate) controllers take advantage of both proportional and derivative control modes.

As illustrated in Figure (c), the proportional action provides an output proportional to the error. If the error is not a step change, but is slowly changing, the proportional action is slow. Rate or derivative action, when added, provides quick response to the error.

Example of PD Control

Consider the industrial application below using a temperature controller (TC) as a PD controller.

The proportional only control mode responds to the decrease in demand, but because of the inherent characteristics of proportional control, a residual offset error remains. Adding the derivative action affects the response by allowing only one small overshot and rapid stabilization to the new control point. Thus, derivative action provides stability to the system, but does not eliminate the offset error.

Industrial applications of Proportional plus Derivative (PD) Control

Proportional plus derivative control is usually used with large capacity or slow-responding processes such as temperature control. The leading action of the controller output compensates for the lagging characteristics of large capacity slow processes.

The derivative/rate action is not usually used with fast responding processes such as flow control or noisy processes because derivative action responds to any rate or change with error signal, including the noise.

PD controllers are useful with processes which are frequently started up and shut down because they are not susceptible to reset windup which is the main problem with PI controllers.