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

Use the ultimate gain K_u (sustained oscillation gain) and ultimate period T_u to set K_p, K_i, K_d directly.

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`, and conversions

Units: K_i has units of 1/time, K_d has units of time. These formulas correspond to the common PID form u(t)=K_pe(t)+K_i∫e(t)dt+K_d de/dt.

Ultimate gain, K_u

Ultimate period, T_u

P — Kp: —, Ki: 0, Kd: 0

PI — Kp: —, Ki: —, Kd: 0

PID — Kp: —, Ki: —, Kd: —

Tip: Start from these values, then fine-tune for setpoint tracking vs. disturbance rejection, actuator limits, and noise sensitivity.