control//adaptive control

2024-09-14

Adaptive control is a family of controllers with adjustable parameters plus a slower mechanism that tunes them online from measurements, and it is used when the plant changes under the controller in ways a fixed design cannot absorb: a delivery drone that drops its parcel, a robot arm picking parts of different weight, a heat exchanger fouling over months. Real plants drift. A battery falling from 4.2 to 3.6 V per cell costs about 25 % of thrust for the same command; a nicked propeller weakens one motor and adds a parasitic torque; a worn valve lowers the process gain and adds friction.

The first defence is already in every loop: the integrator is a one-parameter adapter that learns how much thrust is missing and supplies it, so a constant offset such as a payload's weight needs nothing more. It fails when the change touches the dynamics. With 50 % more mass the same force gives two thirds of the acceleration, so the loop crosses over at a lower frequency, responds more slowly and changes its damping. An adaptive controller then runs two loops at two speeds, the fast control loop and the slow adaptation loop, and the whole is nonlinear even for a linear plant, because adjusted parameters multiply signals; that is why its analysis is delicate.

Adapt only what you cannot measure.

If the variable that drives the change can be measured and varies slowly, schedule the gains on it: that is deterministic, cheap and certifiable point by point. Adaptation earns its risk only for changes that leave no measurable trace, such as a damaged propeller.

Gain scheduling is the workhorse and the flyswatter: it does not learn, it looks up a table indexed by a measured variable. Industry everywhere, from airliners to wind turbines.

Direct adaptation adjusts the controller gains themselves; MRAC makes the plant imitate a reference model, with a Lyapunov argument for stability. L1 adaptive control adapts very fast and filters the result before the actuator.

Indirect adaptation identifies the plant online and redesigns the controller as if the estimate were true (self-tuning regulator); PX4's hover-thrust estimator is its humblest form in production.

Everything that adapts needs persistent excitation and fails without its protections (robust adaptation). Lyapunov proves that the tracking error goes to zero, not that the gains reach their ideal values: tracking can be perfect with wrong parameters if the signals never showed the difference. The ideal adapter would also probe on purpose (dual control).

The cautionary tale is 1967: an X-15 carrying the MH-96 adaptive flight control system broke up on re-entry and its pilot, Michael Adams, died, with the adaptive system's behaviour part of the accident chain. An adapter not analysed against the dynamics it ignores is a gamble, which is why practice puts a robust baseline underneath that flies alone if adaptation is switched off or saturates.