# P D Control Systems

**Proportional-Derivative**

Proportional plus rate describes a control mode in which a derivative section is added to a proportional controller. This derivative section responds to the rate of change of the error signal, not the amplitude; 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 signal 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 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. Figure 25 shows the input versus output relationship of 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 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 shown in Figure 26. As long as the input rate of change is constant, the magnitude of the output is constant. A new input rate of change would give a new output magnitude.

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 were used as a controller, the input signal it would receive is the error signal. As just described, a steady-state error signal corresponds to any number of necessary output signals for the positioning of the final control element. Therefore, derivative action is combined with proportional action in a manner such that the proportional section output serves as the derivative section input.

Proportional plus rate controllers take advantage of both proportional and rate control modes.

As seen in Figure 27, 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 action, when added, provides quick response to the error.

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