PID Controller Tuner

Parameters

Transfer Function G(s)

ωₙ (rad/s) 1.5

ζ (Damping) 0.50

Plant Gain K 1.0

Kp — Proportional 4.0

Ki — Integral 1.50

Kd — Derivative 0.50

Setpoint r(t) 1.0

Rise Time — s

Overshoot — %

Settling — s

Steady Error —

Output Setpoint

A PID controller computes a correction signal u(t) from the error between setpoint and output:

u(t) = Kp·e + Ki·∫e dt + Kd·ė

Kp — ProportionalReacts to present error. Higher Kp → faster response but more overshoot and potential instability.
Ki — IntegralEliminates steady-state error by accumulating past error. Too high → integral windup & oscillation.
Kd — DerivativeAnticipates future error via rate of change. Adds damping; sensitive to noise.

Ziegler–Nichols Rule

Increase Kp until sustained oscillations appear (K_u, period T_u). Then set Kp = 0.6K_u, Ti = T_u/2, Td = T_u/8.

Closed-Loop TF

C(s)G(s) / (1 + C(s)G(s)) where C(s) = Kp + Ki/s + Kd·s