PID — integral term

Summary¶

The integral term (I) accumulates error over time. If a small error persists — the drone consistently drifts right because one motor is slightly weaker, or the centre of mass is off-centre — I builds up a stored correction that grows until it exactly eliminates the residual. I is the term that allows the drone to hold a precise position against steady wind or compensate for hardware imbalance without constant stick input. Its risk is windup: if the error never reduces (commanded attitude is physically impossible), I accumulates without bound and delivers an unexpected large correction when conditions change.

Concept¶

Why P alone is insufficient¶

The proportional term can only correct what it can currently see. If the drone needs a constant small correction — say, a 1% motor speed difference to fly level — P provides that correction, but as an equilibrium condition where the error is exactly the size needed to produce the balancing correction. The drone is slightly off the setpoint forever, because reducing the error would reduce the correction, which would let the disturbance push the drone further off.

This is steady-state error. P cannot eliminate it; I can.

How I works¶

The integral term accumulates error over time:

I_output = K_I × sum(error × dt)

Every PID cycle, the current error is multiplied by the cycle time and added to a running total. As long as any error exists, this total grows. The I output grows with it. Eventually the I term alone provides enough correction to balance the steady disturbance — at which point the error drops to zero, the integrator stops accumulating, and the total stabilises. The drone holds the setpoint exactly.

This is the mechanism that allows a drone to maintain a perfectly level hover when one arm is slightly heavier, or to hold a fixed altitude despite slow battery voltage sag changing the effective motor power.

I windup — the failure mode¶

If the commanded setpoint is physically impossible to achieve — the drone is commanded to hover but is being held down by a hand — the error never reduces to zero. The integrator accumulates without limit. When the constraint is removed (the hand lets go), the accumulated integral delivers a large, sudden correction. The drone shoots upward.

Betaflight limits I accumulation to prevent runaway windup. The I term is also anti-wound during throttle saturation (when motors are at maximum) and during arming/disarming.

I and long-duration hover¶

For libdrone's air quality mapping missions, which may involve hovering at a fixed position for 30–120 seconds, the I term is critical. Battery voltage sags continuously during the flight, reducing motor output per unit throttle command. Without I, the drone would gradually descend as the battery empties, requiring the operator to continuously apply upward throttle. With I, the integrator detects the altitude error and automatically increases throttle to compensate. The operator sees a stable hover; the integrator silently compensates for 5–8V of battery sag over the flight.

Reference¶

I effects on flight behaviour¶

I and GPS position hold¶

In GPS position hold, the outer GPS loop generates position error and feeds it as a setpoint to the attitude loop. The I term in the position loop compensates for sustained wind — if a 10 km/h wind consistently pushes the drone north, the position I accumulates a southward pitch bias that counters the wind without requiring continuous GPS correction. This is the mechanism behind position hold that "leans into" sustained wind rather than constantly chasing the setpoint.

Procedure¶

I tuning sequence (Betaflight)¶

1. Tune P first. Then tune D. Only adjust I after both P and D are set.
2. In calm air, hover and observe whether the drone maintains attitude without stick input. If it drifts consistently in one direction, I is too low.
3. After a large pitch or roll manoeuvre, observe whether the drone returns cleanly to level. If it overshoots and oscillates slowly: I is too high.
4. Adjust I in 10% steps. I changes are slow-acting — evaluate over a 10–30 second hover, not a brief hover.
5. The ideal: drone holds a fixed hover position in calm air without stick input, returns cleanly to level after manoeuvres, no slow oscillation.

Rationale¶

Why I is tuned last¶

I operates on accumulated error over time. If P or D are wrong, the error accumulation seen by I is also wrong — I would compensate for oscillations caused by incorrect P or D rather than for real steady-state error. Tuning I after P and D ensures it only compensates for genuine steady disturbances, not for tuning artefacts.

Connections¶

requires: - closed-loop-control - pid-proportional-term related: - pid-derivative-term - feed-forward-control leads_to: - pid-derivative-term - feed-forward-control