PIDcontroller

A PIDController is a type of feedback controller widely used in industrial and engineering applications to regulate a process variable such as temperature, speed, or position. It combines proportional, integral, and derivative actions to achieve a fast, stable response with minimal steady-state error.

The control law of a PID controller is commonly written as u(t) = Kp e(t) + Ki ∫ e(t) dt + Kd de(t)/dt, where e(t) is the error between the desired setpoint and the measured process variable. The proportional term reacts to the current error, the integral term accumulates past errors to eliminate steady-state offset, and the derivative term anticipates future error to dampen oscillations. In discrete time, sampled at intervals Δt, the form becomes u[k] = Kp e[k] + Ki Σ e[i]Δt + Kd (e[k] − e[k-1])/Δt.

Implementation is common in hardware controllers and software running on programmable logic controllers or microcontrollers. Practical considerations include anti-windup strategies to prevent excessive integral action when actuators saturate, filtering or smoothing of the derivative term to reduce noise sensitivity, and appropriate sampling rates to maintain stability and responsiveness.

Tuning is a key part of PID design and can be manual or automatic. Classical methods include Ziegler–Nichols and relay-based auto-tuning, while more modern approaches use model-based optimization or adaptive schemes. Tuning aims to achieve a desired balance of fast rise time, low overshoot, and minimal steady-state error.

Applications span process control, robotics, motor control, and temperature regulation. Variants such as PI, PD, and PID with anti-windup or filtered derivative are common, and gain scheduling is used for nonlinear or time-varying processes. See also control theory and feedback control.