CV () t = K I ∫ 0 Edt + CV ( t = 0 )

CV () t = K I ∫ 0 Edt + CV ( t = 0 )

where the term CV (t=0) is the value of the output at t = 0. When an integral controller is used in a closed-loop system, it calculates CV (t) for every change in error. So, if the value of the error changes after the controller has calculated

a previous value CV (t) , then it will use this previous value of CV (t) as the CV (t=0) value and calculate a new CV (t) output based on the new error.

The integral gain K I (see Figure 15-38) indicates the sensitivity of the output’s rate of change to the percentage of error that occurs over time. A large value of K I means that a small error will produce a large rate of change in the controller output. Conversely, a small value of K I means that a small error produces a small rate of change in the controller output. In Figure 15-

38, the rate of change of K I1 is greater than that of K I2 .

Figure 15-39 illustrates the reaction of an integral controller’s output to a change in the process variable due to a load disturbance. Note that, at the moment the error occurs, the controller starts the integration of the error

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E=0 Figure 15-38. Integral gain.

value, meaning that the control variable begins to increase as a function of the magnitude of the error. The error in Figure 15-39 is constant, creating a ramp integration of the controller’s output. That is, the amount of error remains constant over time, so the control variable increases at a steady rate.

Figure 15-39. Integral controller’s response to a step change in the process variable.

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To further illustrate the effect of the measured error on the control variable output, let’s examine Figure 15-40, which shows the graph of a direct-acting integral controller’s output response to a change in error. If the error makes

a large jump (1), the controller will respond with a steep increase in output. As the error begins to decrease (2), the rate of increase of the output variable will also decrease (less ramping). When the error becomes zero (3), then the controller will keep its output at its previous level. As the error increases again, but in the opposite direction (4), the output will begin to decrease. As the error decreases, but still remains negative (5), the control variable will continue to decrease but at a less rapid rate. Furthermore, if the error increases positively (6), then the output will increase again. Finally, as the error goes to zero and remains there (7), the controller will level out the control variable and make no more changes to its output level. Thus, an integral controller can adjust its output level to bring the error to zero. An integral controller does not exhibit the limitations of the linear relationship of a proportional controller; thus, it is able to keep a zero error at an output value other than 50% of the controller output.

Figure 15-40. Output response to changes in error.

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The gain of an integral controller (K I ) is defined by the equation: