The closed-loop transfer function $M(s)$ is: $$M(s) = \fracL(s)1 + L(s)$$

Machine learning classifiers are being trained to identify ( T_\sigma / T_1 ) ratios from step response shapes and then recommend optimal MO parameters—even for highly nonlinear or time-varying processes.

While revolutionary for its time, the ZN method has a fundamental flaw: it is designed to provide a "quarter amplitude decay" ratio. This means that after a setpoint change or a disturbance, the process variable oscillates such that each peak is a quarter of the height of the previous one. In many modern applications, particularly in motion control and high-speed manufacturing, this level of oscillation is unacceptable. It results in:

Given ( G_p(s) = \fracK_p e^-sT_d1 + sT_1 ), we approximate the dead time as an additional small time constant (e.g., using the Padé or Taylor expansion). Let ( T_\sigma = T_d + \text(other small lags) ).