proportionalintegralderivat

Proportionalintegralderivat, often called a PID controller, is a feedback control mechanism used to regulate dynamic systems. It combines three terms—proportional, integral, and derivative—each acting on the error e(t) = setpoint minus process variable. The controller output is u(t) = Kp e(t) + Ki ∫ e(τ) dτ + Kd de/dt.

The proportional term provides immediate correction proportional to present error, the integral term accumulates past errors to eliminate steady-state error, and the derivative term dampens rapid error changes to improve stability and transient response.

In discrete-time implementations with sampling interval Δt, the equation becomes u[k] = Kp e[k] + Ki Σ e[i] Δt + Kd (e[k] − e[k−1])/Δt.

PID control is versatile and widely used in process control, robotics, motor control, and temperature regulation. Tuning Kp, Ki, and Kd trades off speed, stability, overshoot, and robustness. Common methods include heuristic approaches such as Ziegler–Nichols and Cohen–Cuunliffe, as well as model-based tuning like Internal Model Control and optimization-based techniques.

Practical concerns include anti-windup to prevent integral saturation when actuators saturate, derivative filtering to reduce noise sensitivity, and careful handling of actuator limits. In some implementations the derivative term is applied to the filtered process variable to avoid derivative kick caused by abrupt setpoint changes. While the term proportionalintegralderivat may appear in instructional or concatenated forms, PID remains the standard abbreviation in professional writing.