PIDcontrollers

A PID controller is a feedback controller that uses three terms—proportional, integral, and derivative—to compute a control signal. It is widely used in engineering to regulate variables such as temperature, speed, or position.

The continuous-time form is u(t) = Kp e(t) + Ki ∫ e(τ) dτ + Kd de/dt, where e(t) is the difference between the desired setpoint and the process variable. Kp, Ki, and Kd are tunable gains that determine the influence of each term. The proportional term provides a response proportional to the current error, the integral term eliminates steady-state error by accumulating past errors, and the derivative term dampens rapid changes by predicting future error.

In digital control systems, the PID is implemented in discrete time with a sampling period T. The discrete form uses e[k], u[k], and finite-difference approximations for the integral and derivative. Derivative filtering is commonly applied to reduce measurement noise, and integral windup prevention is essential to avoid excessive actuator saturation or sluggish response.

Tuning aims to achieve desired stability and performance. Common methods include Ziegler–Nichols, Cohen–Coon, and model-based IMC. Practical tuning often combines heuristic rules with simulation or experimental sweeps, and may involve tailoring the controller to a specific plant model or implementing adaptive elements.

Applications span process control, robotics, CNC machines, and automotive systems. Limitations include sensitivity to sensor noise, model inaccuracy, and actuator saturation. Variants such as PI, PD, or anti-windup configurations extend the basic PID to address particular dynamics and reliability requirements.