# Proportional Control Action

Proportional action is the simplest and most commonly encountered of all continuous control modes. In this type of action, the controller produces an output signal which is proportional to the error. Hence, the greater the magnitude of the error, the larger is the corrective action applied.

• ## Mathematical Description

Mathematically, proportional control could be expressed as:

Where

• V is the adjustment or signal for the adjustment from the controller.
• is the error.
• = S - L
• L is the measured value of the controlled variable.
• S is the setpoint.
• K is the proportional constant, named as the gain which shows the sensitivity of the control.
• Vo is the signal output when no error exists.The gain is often replaced with another parameter, called the proportional band, PB . This quantity is defined as the error required to move the final control element over its whole range and is expressed as a percentage of the total range of the measured variable. What is the relationship between K and PB .According to this definition we can see that the whole range of the final control element adjustment should be Vmin to Vmax .At point Vmin At point Vmax The error required to move from Vmin to Vmax will be

Therefore

Recall that the proportional band, PB , is defined as the error required to move the final control element over it’s whole range expressed as a %. So for the controlled variable, L , with its total range Lmin to Lmax the definition for the proportional band is

or

Therefore we have the relationship between gain K and proportional band PB as

With proportional band, the relationship between the adjustment and the error can be expressed as

It can be seen both from the expression above and by running the experiments in the Virtual Laboratory that the larger the gain K , or equivalently the smaller the proportional band PB , the higher the sensitivity of the controller’s actuating signal to deviations will be.

• ## Dynamic Response

Now let’s examine the dynamic response of the proportional control. Assume the process is at steady state and the level is at the setpoint. At time = 0, an increase in the inlet flowrate, regarded as a disturbance, enters into the process. If no control action is taken, i.e. the outlet flowrate is not altered, the level (controlled variable) will increase.

With proportional control, the level is brought back and maintained in a certain range near the setpoint. The history curve could typically be like that shown below. Different responses are obtained depending on the proportional band, B , of the controller.

As can be seen the smaller the proportional band the closer to the setpoint the controlled variable becomes but the more oscillatory the response.