Ziegler–Nichols Tuning (Closed-Loop / Ultimate Gain Method)

Enter K_u and T_u to compute K_p, K_i, K_d, then run the live simulation below.

Formulas (direct in `K_p, K_i, K_d`)

Controller	`K_p`	`K_i`	`K_d`	Notes
P	`0.50 K_u`	`0`	`0`	Proportional only
PI	`0.45 K_u`	`0.54 K_u / T_u`	`0`	From `T_i=T_u/1.2` and `K_i=K_p/T_i`
PID	`0.60 K_u`	`1.20 K_u / T_u`	`0.075 K_u T_u`	From `T_i=T_u/2`, `T_d=T_u/8`

Ultimate gain, K_u

Ultimate period, T_u (s)

P — Kp: —, Ki: 0, Kd: 0

PI — Kp: —, Ki: —, Kd: 0

PID — Kp: —, Ki: —, Kd: —

Units: K_i is 1/time, K_d is time. Controller form: u(t)=K_pe + K_i∫e\,dt + K_d de/dt.

Controller type

Ki (1/s)

Kd (s)

Setpoint

Plant time constant τ (s)

Plant gain K (y/u)

Sample time Δt (s)

Actuator limit |u| ≤

Meas. noise (std)

Window (s)

Plant model: first-order dy/dt = ( -y + K·u ) / τ. Derivative on measurement; simple anti-windup via integral clamping when saturated.

setpoint measurement y control u