Loop Stability (FOPDT)

Dead time is what makes loops unstable — it adds phase lag without attenuating anything. This finds the frequency where the open loop hits −180°, the proportional gain that would put the loop exactly on the stability limit (K_u), and how much margin you have below it.

Aim for a gain margin of at least 2× (6 dB). Ku and Pu here feed straight into the closed-loop Ziegler-Nichols table, which is a safer way to get them than driving real plant into oscillation.