ProportionalIntegralDerivative

Proportional-Integral-Derivative control, commonly abbreviated as PID, is a feedback control strategy used to regulate a wide range of process variables by adjusting a control input to minimize the difference between a desired setpoint and a measured process variable. It is valued for its simplicity, versatility, and ability to provide fast response with low steady-state error in many practical systems.

In continuous time, the PID controller output u(t) is given by

u(t) = Kp e(t) + Ki ∫ e(τ) dτ + Kd de(t)/dt,

where e(t) = setpoint − process variable. The terms are proportional, integral, and derivative actions with gains Kp, Ki, and Kd. The integral term eliminates steady-state error, while the derivative term anticipates future error to improve stability and response. In discrete-time implementations, commonly

u[k] = Kp e[k] + Ki ∑ e[i] Δt + Kd (e[k] − e[k−1])/Δt,

often with filters on the derivative to reduce noise.

Tuning PID parameters involves balancing speed of response, overshoot, and stability. Methods include empirical rules such as Ziegler–Nichols and Cohen–Coon, as well as model-based approaches like Internal Model Control (IMC). Practical considerations include anti-windup measures to prevent integral term buildup when actuators saturate, derivative action sensitivity to measurement noise, and discretization effects. Derivative action is frequently implemented with filtering or modulo techniques to maintain robustness.

Applications span process control, automation, and engineering systems such as temperature, speed, position, and flow regulation. Variants include PI and PD controllers, as well as more advanced forms like fractional-order or robust PID schemes, which adapt the basic structure to improve performance under nonlinearities or uncertainties.