proportional, integral, and derivative terms

These are the three building blocks inside a PID controller, and each one answers a different question about the error — the gap between where you want to be and where you actually are. The Proportional term asks 'how big is the error right now?', the Integral term asks 'how much error has piled up over time?', and the Derivative term asks 'how fast is the error changing?'. Each term has its own tuning knob, called a gain, that sets how loudly its answer is allowed to speak in the final correction.

The Proportional (P) term is the muscle of the moment: it produces a push in direct proportion to the present error, so a big gap means a big shove and a small gap a gentle one. On its own, though, P often stalls just short of the goal — it relaxes as the error shrinks, and a small steady error can linger forever, like a spring that pulls hard at first but settles a little below the mark. The Integral (I) term fixes exactly that: it keeps a running total of every bit of error left over, second after second, and that growing sum eventually becomes large enough to wipe out even a tiny stubborn gap — patient where P is impatient. Its weakness is being slow and prone to overshoot, since it has to build up before it bites.

The Derivative (D) term looks not at the error itself but at its speed and direction of change — it anticipates. If the error is closing fast, D senses the rush and pushes back early, acting like a shock absorber that smooths out overshoot and damps oscillation before it starts. The catch is that measuring rate of change magnifies any jitter in the sensor, so too much D makes a system twitchy and nervous. In short: P reacts to the present, I remembers the past, and D predicts the near future — and tuning is the art of balancing the three so they cooperate instead of fighting.