proportionalintegralderiv

ProportionalIntegralDeriv, commonly abbreviated PID, is a widely used feedback control loop in engineering, designed to regulate a process variable toward a desired setpoint. It combines three concurrent actions—proportional, integral, and derivative—to respond to current error, accumulated past error, and predicted future error. The approach is standard in process control, automotive systems, and robotics, and exists in both analog and digital versions.

In continuous time, the controller output u(t) is given by u(t) = Kp e(t) + Ki ∫ e(τ) dτ + Kd de/dt, where e(t) = setpoint minus process variable. The proportional term provides immediate correction proportional to the error; the integral term eliminates steady-state error by integrating the error; the derivative term damps changes and reduces overshoot by reacting to the rate of error change. The constants Kp, Ki, Kd (sometimes expressed as Kp, Ti, Td) tune the influence of each term.

Discrete implementation applies to digital controllers: u[k] = Kp e[k] + Ki T ∑ e[i] + Kd (e[k]-e[k-1]) / T, with sampling interval T. Tuning methods such as Ziegler-Nichols and Cohen-Coon provide initial settings; further refinement uses model-based or optimization methods. Anti-windup measures prevent integral term from accumulating when actuator saturates; filtering on the derivative term reduces sensitivity to measurement noise.

Applications and limitations: PID is versatile and common for regulating temperature, speed, position, and pressure. It is robust and adaptable but sensitive to sensor noise, especially on the derivative term; integral windup can cause overshoot or sluggish response; proper anti-windup, filtering, and gain scheduling are often required for nonlinear or time-varying processes.