P, PI, PD and PID

Present, past and future — which is why there are three terms and not some other number. Over 90% of industrial loops are PID, and most of those are PI.

1. 1P alone — fast, and settles short of the target, because zero error would mean zero drive.
2. 2PI — the integrator accumulates while any error remains, driving it to exactly zero. Its −90° of phase produces overshoot and ringing.
3. 3PID — derivative acts on the rate, reducing drive before arrival. It brakes rather than reacts, and returns the phase the integrator spent.

When the actuator saturates the plant cannot respond further, but the integrator keeps accumulating — storing a large value that commands nothing extra. When the error finally reverses, the loop stays saturated until that store unwinds, producing a large overshoot long after the disturbance has passed.

Anti-windup — clamping the integrator, or freezing it while saturated — is standard on any real implementation. A textbook PID with no anti-windup will misbehave on the first hard disturbance.

A step change in setpoint has infinite slope, so the D term produces a violent output spike. It also amplifies measurement noise, since differentiating noise makes it worse.

The standard fix is to differentiate the measurement rather than the error, which removes the kick while keeping the damping, and to filter the derivative. Even so, D is the term most often left switched off.

Ziegler-Nichols raises the proportional gain until sustained oscillation, then reads the gain and period — which is exactly the stability boundary Routh-Hurwitz computes from a row of zeros and Bode shows as the axis crossing. The classical rules are an empirical shortcut to the quantities the theory predicts.

They are known to be aggressive, giving roughly quarter-amplitude decay, so most engineers detune the result. Modern controllers run a relay auto-tune that measures the same two numbers automatically.