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Thus, as t increases, the area increases and so the controller output increases. Since, in this example,
the area is proportional to t then the controller output is proportional to t and so increases at a constant
rate. Note that this gives an alternative way of describing integral control as

rate of change of controller output α error
A constant error gives a constant rate of change of controller output.

EXAMPLE 4.6
An integral controller has a value of Ki of 0.10 s-1. What will be the output after times of (a) 1 s, (b)
2 s, if there is a sudden change to a constant error of 20%, as illustrated in Figure 4.17?

FIGURE 4.17 Example.
Answer:
We can use the equation:

controller output = Ki x integral of error with time
a. The area under the graph between a time of 0 and 1 s is 20% s. Thus, the controller output is 0.10

x 20 = 2%.
b. The area under the graph between a time of 0 and 2 s is 40% s. Thus, the controller output is 0.10

x 40 = 4%.
4.6.1 PI Control
The integral mode I of control is not usually used alone but generally in conjunction with the
proportional mode P. When integral action is added to a proportional control system the controller
output is given by

PI controller output = Kp x error + Ki x integral of error with time
where Kp is the proportional control constant and Ki the integral control constant.
Figure 4.18 shows how a system with PI control reacts when there is an abrupt change to a constant
error. The error gives rise to a proportional controller output which remains constant since the error
does not change. There is then superimposed on this a steadily increasing controller output due to the
integral action.

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