control//feedback control//PID controller//proportional action

Proportional action sets the actuator in proportion to the current error, \(u=K_pe\), and it is the first thing almost every loop needs: a heater that gives more power the colder the room, a steering correction that grows with the distance from the centre of the lane. It is the term that does most of the work in the first moments after a setpoint change or a disturbance.

Proportional action sets the actuator in proportion to the current error, u=Kpeu=K_peu=Kp​e, and it is the first thing almost every loop needs: a heater that gives more power the colder the room, a steering correction that grows with the distance from the centre of the lane. It is the term that does most of the work in the first moments after a setpoint change or a disturbance.

Its gain trades speed against calm. A small KpK_pKp​ corrects slowly and gently; a large one corrects fast but overshoots, and past a critical value the loop oscillates by itself, because every correction reaches the output late, through the plant's lag, and pushes it past the target. The gain at which the oscillation neither grows nor dies is the ultimate gain, the starting point of the classic Ziegler and Nichols tuning (PID tuning).

Proportional action alone leaves an offset. To hold a steady load (heat lost through the walls, a slope under a car) the actuator needs a non-zero output, and with u=Kpeu=K_peu=Kp​e that takes a non-zero error. For a plant of steady-state gain KKK and a unit setpoint step, the error left is 1/(1+KKp)1/(1+KK_p)1/(1+KKp​): it shrinks as the gain grows and never reaches zero (steady-state error).

Old controllers removed the offset by hand with a manual reset, a constant added to the output, u=Kpe+u0u=K_pe+u_0u=Kp​e+u0​, set so that the offset vanished at one operating point. The integral term does the same automatically at every operating point, which is why it replaced the manual reset and why integral action is still called reset.

Process engineers write the gain as a proportional band: the change of the measurement, in percent of its range, that moves the output across its whole range, PB=100/KpPB=100/K_pPB=100/Kp​ with both signals in percent. A band of 20 % is a gain of 5, and widening the band means a gentler controller.