11.1.2 The Integration Term

The integration term, $K_i\int_0^t{e(x)dx}$ is the area between the set point and the feedback in a plot against time. As the proportional term $K_pe(t)$ ``runs out of gas'' as the feedback gets closer to the set point, the integration term $K_i\int_0^t{e(x)dx}$ continues to increase. As a result, the summation of $K_pe(t)+K_i\int_0^t{e(x)dx}$ should eventually become large enough so that any loss of the system is balanced by the contribution of $K_i\int_0^t{e(x)dx}$.

Note that that the integration term is not entirely independent from the proportional term. As the integration term helps to bring feedback closer to the set point, the proportional term gets even smaller. Eventually, in a tuned system, as the feedback becomes the same as the set point, the entire output comes from the integration term.

Physically, $K_i\int_0^t{e(x)dx}$ is the amount of output necessary to balance loss of the system. This term is very dependent on how the other two terms are set up, although it will, eventually, get the feedback to match the set point.