Open and Closed-Loop Feedback Systems:
Using Gain for Control

To make it easier to understand the function of gain, a microprocessor controller will be used in the coming examples. Later we will be able to apply the term to op amp and pneumatic controllers.

The simplest type of control to understand is called gain, where the controller uses gain as a multiplying factor. The controller uses a mathematical calculation called the algorithm to compute the amount of output for each change of error. This means that the processor is continually checking (sampling) the value from its sensor, which is the PV and it compares this value to the SP. This comparison takes place at the summing junction and the result is called error. It should be pointed out at this time that the function of the summing junction is performed by a calculation and in older controllers this function was performed by op amps. The formula for error is:

Error = Setpoint - Process Variable

or:

E = SP - PV

Note that some applications will use the formula: E = PV - SP

The error can be a positive number or a negative number, depending on the amount of the values. E.g., if the SP for a laboratory furnace is 450°F and the sensor indicates the actual temperature (PV) is 445°F, the error would be a positive 5°F. If the PV temperature is 455°F and the SP is 450°F, the error would be a negative 5°F. In this application the controller is designed to change the output so that heat is added to the furnace if the error is positive, and if the error is negative the controller is designed to turn off the heat source.

The controller takes the error from the summing junction and uses the algorithm to calculate the amount of output. The algorithm can be a complex formula but it can be simplified as shown.

Output = Gain x Error

M_o = K_c x E

In the formula, the output is referred to as M_o the gain is referred to as K_c, which stands for controller (c) constant (K), and E is error. These terms have become standards for the Instrument Society of America (ISA). ISA was formed to standardize and promote the process control and instrumentation industry. Some companies that design process control and motion control equipment may use slightly different designations for the variables in these formulas.

The following example will help explain how the error and gain are used by the controller to change the output to respond to the changes to the system. At the start of this example, the temperature (PV) inside a laboratory furnace is room temperature (75°F) and the SP is 75°F. The amount of error at this point is zero, and the controller sets the output to zero. The engineer starts the oven by entering a new SP at 100°F and sets the gain to a value of 2 in the controller. When the controller sees the new SP of 100°F, the error is calculated (100 - 75 = 25). The controller uses the error of 25 and multiplies it times the gain of 2 and the output is set to 50%. Remember that the output can only be set to values between 0% and 100%.

When the output is set to 50%, the heating element begins to add heat to the furnace and the sensor continually sends the PV signal to the controller to indicate the increasing temperature. The controller is constantly recalculating the changing error and setting the new output. The following table shows a progression of these calculations as the controller samples the change in the PV and recalculates the output. The controller uses a sample time to determine how quickly to recalculate the output signal. The sample time may be fixed, or it may be adjustable in some controllers. Table 1 (below) shows the calculations the controller makes.

It should be noted that error is actually a percentage of the SP-PV over the total number of degrees (span) the system can control. For this example, the span is set for 100°F so the number of degrees of error is also the percentage of error.

From the calculations in Table 1 notice that the controller continually reduces the percentage of output as the PV temperature inside the furnace increases and the temperature gets closer to the SP. The controller will need approximately 30% output to sustain the 85° temperature, and if the output is lowered below 30% the temperature will begin to decrease. This means that the controller will "level out" at some point and hold the output steady.

PREV: Understanding Gain, Reset, and Rate

NEXT: Problems with Using Only Gain for Control

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