The Control Trick That ERASES Disturbances (Integral Action)

Added: 2026-05-16

[00:00:00]A drone hovering at 1 m. A sudden gust of wind pushes it down by 10 cm. The drone's controller is just a proportional gain. Multiply error by a number and apply that as thrust.

[00:00:12]Question, >> [music] >> will the drone return to 1 m? The answer is no. It will hover at 90 cm forever, even though the wind is constant, even though the controller can see the error.

[00:00:24]It just can't fix it. Why?

[00:00:26]This is called steady state [music] error. With a pure proportional controller, a constant disturbance produces a constant offset forever.

[00:00:34][music] Increase the gain, error gets smaller, but never zero. The math says you cannot reach exactly the reference with proportional alone when there's a constant disturbance.

[00:00:46]Now add one thing, [music] an integrator, a box that sums up all past errors. Whatever the error is, the integrator stores it, multiplies it by a small gain, and adds that to the control signal. Watch what happens.

[00:00:59]>> [music] >> The drone's error slowly accumulates in the integrator. The accumulated value pushes the thrust higher, higher, higher, until the error is exactly zero.

[00:01:10]The wind has not changed, but the drone is back at 1 m.

[00:01:15]Why does it work? Because the integrator can only stop growing when the error is exactly zero. As long as there is any non-zero error, the integrator keeps changing. Equilibrium requires zero error. That's it. That's the magic. The integrator forces the system to find the input that perfectly cancels the disturbance.

[00:01:36]And it's not just for hovering. Want to track a constant reference, say, a temperature, an altitude, a speed? The integrator nails it perfectly with zero offset, even if the system has unknown losses. Want to reject any unknown but constant [music] disturbance, wind, friction, gravity offset? Same trick.

[00:01:57]Add an integrator. Done.

[00:01:59]There's a catch. Integrators have memory. If the actuator is saturated, the motor maxes out, the valve fully opens, the integrator keeps integrating, building up windup. When the system finally responds, it overshoots wildly.

[00:02:14]That's why every real PID controller has anti-windup logic to clamp the integrator when the actuator is saturated.

[00:02:22]One integrator, three lines of code, and steady-state error vanishes. This is why every drone, every cruise control, every thermostat, every industrial process loop has integral action somewhere. The simplest, most elegant trick in control theory, and it feels like cheating.

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