ProportionalIntegralDifferential

ProportionalIntegralDifferential, commonly known as a PID controller, is a feedback control loop mechanism widely used in industrial and engineering applications to regulate a process variable by adjusting a control input based on the error between a desired setpoint and the measured process variable.

The standard continuous-time form combines three terms: proportional, integral, and derivative. The control signal is u(t) = Kp e(t) + Ki ∫ e(τ) dτ + Kd de/dt, where e(t) = SP − PV, and Kp, Ki, Kd are tunable gains. In discrete time, often used in digital implementations, a typical update is u[k] = Kp e[k] + Ki Σ e[i] Δt + Kd (e[k] − e[k−1])/Δt, with Δt as the sampling interval. The proportional term reacts to current error, the integral term addresses accumulated past error to eliminate steady-state error, and the derivative term anticipates future error to dampen rapid changes.

Tuning aims to achieve a fast, stable response with minimal overshoot and steady-state error. Common methods include Ziegler–Nichols and Cohen–Coon rules, manual tuning, relay feedback, and autotuning. Practical challenges include integral windup when actuators saturate, and sensitivity of the derivative term to measurement noise, often mitigated by anti-windup strategies and filtering or by using a filtered derivative.

Applications span process control, temperature regulation, motor speed and position control, robotics, and aerospace. Variants such as PI or PD controllers, and PID implementations with derivative filtering or anti-windup, are widely used to suit specific system dynamics. The PID concept dates to early 20th-century control theory, with tuning rules popularized in the 1940s by Ziegler and Nichols.