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Proportional -Only Control

Here is where math starts to enter the algorithm: a proportional controller calculates the difference between the process variable signal and the setpoint signal, and calls it the error. This is a measure of how far off the process is deviating from its setpoint, and may be calculated as SPPV or as PVSP, depending on whether or not the controller has to produce an increasing output signal to cause an increase in the process variable, or output a decreasing signal to do the same thing. This choice in how we subtract determines whether the controller will be reverse-acting or direct-acting. The direction of action required of the controller is determined by the nature of the process, transmitter, and final control element. In this case, we are assuming that an increasing output signal sent to the valve results in increased steam flow, and consequently higher temperature, so our algorithm will need to be reverse-acting (i.e. an increase in measured temperature results in a decrease in output signal; error calculated as SPPV). This error signal is then multiplied by a constant value called the gain, which is programmed into the controller. The resulting figure, plus a “bias” quantity, becomes the output signal sent to the valve to proportion it:


m = Kpe + b


  m = Controller output

  e = Error (difference between PV and SP)

  Kp = Proportional gain

  b = Bias


If this equation appears to resemble the standard slope-intercept form of linear equation (y = mx + b), it is more than coincidence. Often, the response of a proportional controller is shown graphically as a line, the slope of the line representing gain and the y-intercept of the line representing the output bias point, or what value the output signal will be when there is zero error (PV precisely equals SP):


In this graph the bias value is 50% and the gain of the controller is 1.

Proportional controllers give us a choice as to how “sensitive” we want the controller to be to changes in process variable (PV) and setpoint (SP). With the simple on/off (“bang-bang”) approach, there was no adjustment. Here, though, we get to program the controller for any desired level of aggressiveness. The gain value (Kp) of a controller is something which may be altered by a technician or engineer. In pneumatic controllers, this takes the form of a lever or valve adjustment; in analog electronic controllers, a potentiometer adjustment; in digital control systems, a programmable parameter.

If the controller could be configured for infinite gain, its response would duplicate on/off control. That is, any amount of error will result in the output signal becoming “saturated” at either 0% or 100%, and the final control element will simply turn on fully when the process variable drops below setpoint and turn off fully when the process variable rises above setpoint. Conversely, if the controller is set for zero gain, it will become completely unresponsive to changes in either process variable or setpoint: the valve will hold its position at the bias point no matter what happens to the process.

Obviously, then, we must set the gain somewhere between infinity and zero in order for this algorithm to function any better than on/off control. Just how much gain a controller needs to have depends on the process and all the other instruments in the control loop.

If the gain is set too high, there will be oscillations as the PV converges on a new setpoint value:


If the gain is set too low, the process response will be stable under steady-state conditions, but “sluggish” to changes in setpoint because the controller does not take aggressive enough action to cause quick changes in the process:

With proportional-only control, the only way to obtain fast-acting response to setpoint changes or “upsets” in the process is to set the gain constant high enough that some “overshoot” results:


As with on/off control, instances of overshoot (the process variable rising above setpoint) and undershoot (drifting below setpoint) are generally undesirable, and for the same reasons. Ideally, the controller will be able to respond in such a way that the process variable is made equal to setpoint as quickly as the process dynamics will allow, yet with no substantial overshoot or undershoot. With plain proportional control, however, this ideal goal is nearly impossible.

An unnecessarily confusing aspect of proportional control is the existence of two completely different ways to express the “aggressiveness” of proportional action. In the proportional-only equation shown earlier, the degree of proportional action was specified by the constant Kp, called gain. However, there is another way to express the sensitivity of proportional action, and that is to state the percentage of error change necessary to make the output (m) change by 100%.

Mathematically, this is the inverse of gain, and it is called proportional band (PB):


Gain is always specified as a unitless value1, whereas proportional band is always specified as a percentage. For example, a gain value of 2.5 is equivalent to a proportional band value of 40%.

Due to the existence of these two completely opposite conventions for specifying proportional action, you may see the proportional term of the control equation written differently depending on whether the author assumes the use of gain or the use of proportional band:

Many modern digital electronic controllers allow the user to conveniently select the unit they wish to use for proportional action. However, even with this ability, anyone tasked with adjusting a controller’s “tuning” values may be required to translate between gain and proportional band, especially if certain values are documented in a way that does not match the unit configured for the controller.

When you communicate the proportional action setting of a process controller, you should always be careful to specify either “gain” or “proportional band” to avoid ambiguity. Never simply say something like, “The proportional setting is twenty,” for this could mean either:

  • Gain = 20; Proportional band = 5% . . . or . . .

  • Proportional band = 20%; Gain = 5


1In electronics, the unit of decibels is commonly used to express gains. Thankfully, the world of process control was spared the introduction of decibels as a unit of measurement for controller gain. The last thing we need is a third way to express the degree of proportional action in a controller!

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