PID Controller (Three-Term Feedback Loop)

Closed-loop feedback controller computing corrective actuator output from proportional, integral, and derivative error terms.

Specifications & Metrics

type: Control & Sensing Mechanism
governing equation: u(t) = K_p e(t) + K_i ∫₀ᵗ e(τ) dτ + K_d (de(t)/dt), where e(t) = r(t) − y(t)
primary function: Continuous setpoint tracking and disturbance rejection in automated systems
key parameters: Proportional gain (K_p), integral reset time (T_i = K_p/K_i), derivative time (T_d = K_d/K_p)
originator: Elmer Sperry (1911) & Nicolas Minorsky (1922 formal control law)

Russian-American engineer Nicolas Minorsky formulated the mathematical three-term PID control law in 1922 after analyzing how US Navy helmsmen steered the battleship USS New Mexico using current course error, accumulated past drift, and rate of yaw.
The Proportional term (K_p·e) provides immediate response proportional to current error; the Integral term (K_i·∫e dt) eliminates residual steady-state offset; and the Derivative term (K_d·de/dt) anticipates overshoot by damping rapid rates of change.
First commercialized as Foxboro's pneumatic 'Stabilog' nozzle-and-flapper controller in 1930 and standardized by Ziegler–Nichols loop tuning rules (1942), PID loops account for over 90% of industrial process control loops in chemical plants, robotics, and flight autopilots.