control//feedback control//PID controller//integral action
Integral action is the part of a controller that removes the error a loop would otherwise keep for ever: it adds up the error over time and pushes the actuator with that sum, so it keeps growing for as long as any error remains and stops only when the error is zero. It is why an oven reaches exactly 180 degrees instead of settling at 176, and why a cruise control holds its speed up a hill.
Integral action is the part of a controller that removes the error a loop would otherwise keep for ever: it adds up the error over time and pushes the actuator with that sum, so it keeps growing for as long as any error remains and stops only when the error is zero. It is why an oven reaches exactly 180 degrees instead of settling at 176, and why a cruise control holds its speed up a hill.
uI(t)=Ki∫0te(s) dsu_I(t)=K_i\int_0^t e(s)\,dsuI(t)=Ki∫0te(s)ds
At steady state the error is zero and the integral term carries the whole command the load requires: it has learned how much heat the walls lose, or how much throttle the slope needs. Right after a setpoint step the proportional term does most of the pushing; as the error fades the integral part rises to take its place, and at rest it is all that is left.
The price is lag. An integrator answers to the past, so it adds phase lag to the loop (90 degrees at every frequency for a pure integrator) and makes it more oscillatory. A loop that was well damped with P alone overshoots more once integral is added, and too much KiK_iKi makes it swing slowly around the setpoint.
Industry tunes it as a reset time TiT_iTi, with Ki=Kp/TiK_i=K_p/T_iKi=Kp/Ti: the time the integral takes, under a constant error, to add as much again as the proportional term gives (hence minutes per repeat). A shorter reset time means more integral action.
It needs protection. When the actuator saturates, the integral goes on summing an error it cannot act on and later overshoots by what it stored (integral windup).
It is needed only when something steady has to be compensated. A plant that already integrates, a motor driven to a position for instance, reaches zero error under P control for a constant setpoint, and still needs integral action to hold against a constant load torque (steady-state error).