1. The Function of the Integral Term: The integral action calculates the sum of the error over time. As long as the controlled variable is not at the setpoint (error $\neq 0$), the integral term will continue to increase or decrease the controller's output.
2. The "Reset" Terminology: In the early days of industrial control, operators had to manually adjust the bias ($P_0$) of proportional controllers to eliminate the steady-state offset. This was called "manual reset." Because integral action performs this task automatically by shifting (resetting) the proportional band until the error is zero, it became known as the
Automatic Reset or simply the
RESET control mode.
3. Mathematical Representation: $$u(t) = \frac{1}{T_i} \int e(t) \, dt$$
Where $T_i$ is the integral time or "reset time." A smaller $T_i$ results in a faster "reset" of the offset, though it may lead to instability if too aggressive.