11.1.1 The Proportional Term

The proportional term is merely a scaled error $e(t)$. This term makes perfect sense. When $e(t)$ is large, it means our set point is far away from the feedback. As a result, the output needs to be ``very high'' in order to bring the feedback closer to the set point. As our feedback gets closer, $e(t)$ becomes smaller, and it makes sense to reduce the output so we don't overshoot.

In a lossless system, just relying on $K_pe(t)$ should get us to the set point, eventually. However, because all practical systems are not lossless, $K_pe(t)$ never gets us to the set point. This is because at some point, the term $K_pe(t)$ is merely enough to counter the loss, and the system enters a steady state in which the input (from the heater or motor) cancels the output (heat loss or friction).

Despite the fact that $K_pe(t)$ is not sufficient by itself, it is the main component during the first phase of approaching the set point. Physically, $K_pe(t)$ is the bulk of the response to changes.