How it works
The PID controller output is u(t) = K_p[e(t) + (1/T_i)∫e(t)dt + T_d·de/dt]. Proportional gain K_p multiplies current error directly; increasing K_p reduces steady-state error but increases overshoot. Integral time T_i sets how fast the integral term accumulates — smaller T_i means faster integration and stronger integral action but more overshoot. Derivative time T_d amplifies the rate of error change, providing predictive correction but amplifying measurement noise. In the s-domain, PID transfer function is K_p(1 + 1/T_i·s + T_d·s).
Key points to remember
Ziegler-Nichols tuning method 1 (process reaction): apply a step, measure dead time L and time constant T from the open-loop step response, set K_p = 1.2T/(K·L), T_i = 2L, T_d = 0.5L. Tuning method 2 (ultimate gain): increase K_p until sustained oscillation at period P_u, then K_p = 0.6K_u, T_i = 0.5P_u, T_d = 0.125P_u. Integral windup occurs when the output saturates and the integrator keeps accumulating error; anti-windup limits the integrator during saturation. P-only control has steady-state error; PI eliminates it at cost of slower response; PD improves speed without eliminating offset; PID combines all benefits.
Exam tip
The examiner always asks you to compare P, PI, PD, and PID controllers in terms of steady-state error, speed of response, and stability — write a clean comparison table and explain physically why the integral term eliminates offset.