PID Tuning

Two ways to get starting gains. Closed loop: raise proportional gain until the loop oscillates steadily — that gain is K_u and the oscillation period is P_u. Open loop: bump the output and fit a first-order-plus-dead-time curve (gain K, time constant τ, dead time θ). Both give a starting point to detune from, not a final answer.

Ziegler-Nichols targets roughly quarter-amplitude decay, which is deliberately aggressive — expect overshoot and halve Kp if the loop rings. Cohen-Coon handles dead-time-dominant processes better. Always detune on real plant.