How does a cruise control hold your speed on a hill? A PID controller watches the ERROR — how far off target you are — and pushes back in proportion to it (P), to its build-up over time (I), and to how fast it’s changing (D). P alone leaves a stubborn offset; the I term erases it; D damps the overshoot. It’s the workhorse of every self-regulating machine. Slide the integral gain and watch the offset close.
A PID controller computes u = Kp·e + Ki·∫e dt + Kd·de/dt from the error e = target − measurement: the PROPORTIONAL term reacts to the present error, the INTEGRAL to the accumulated past (it removes steady-state offset), and the DERIVATIVE anticipates the future (it damps overshoot). With a constant load, a P-only controller settles SHORT of the target (a residual offset), because it needs a nonzero error to produce the force that fights the load; adding integral action drives that offset to zero by ramping up until the error vanishes. A fail-loud self-check throws unless the PID reaches the target more closely than P-only. ◆ real control theory, node-verified.
A first-order plant with a fixed load (the exact P-offset and integral-removes-it behaviour); real tuning trades speed vs overshoot vs stability (Ziegler–Nichols) and too much I or D destabilises — the roles of the three terms are exact.