PIcontroller

PIController is a feedback control element that implements proportional-integral (PI) control. It computes a control signal u(t) = Kp e(t) + Ki ∫ e(τ) dτ, where e(t) is the difference between the desired setpoint and the measured process variable. The proportional term provides immediate correction based on current error, while the integral term accumulates past errors to reduce or eliminate steady-state offset, improving accuracy against constant disturbances. The integral action can introduce lag and, if gains are too high, may destabilize the system, so proper tuning is essential. In continuous-time form, the PI controller has a transfer function U(s)/E(s) = Kp + Ki/s.

In digital or discrete-time implementations with sampling period Ts, a common form is u[k] = Kp e[k] + Ki ∑ e[i] Ts, which can be implemented efficiently with an accumulator z[k+1] = z[k] + e[k] Ts and u[k] = Kp e[k] + Ki z[k]. This discretization avoids continuous integral calculations and aligns with typical microcontroller or PLC deployments.

Practical considerations include anti-windup strategies to prevent integrator buildup when the actuator saturates, bumpless transfer during setpoint changes, and appropriate anti-noise measures since integral action can amplify low-frequency disturbances. Tuning methods range from manual adjustment to established rules such as Ziegler-Nichols or Cohen-Coon, with digital implementations sometimes using model-based or auto-tuning approaches.

PI controllers are widely used in industrial and embedded systems where steady-state accuracy is important and where derivative action is either unnecessary or undesirable due to noise sensitivity.