This paper presents a way to learn compact, dynamical models of disturbances that interfere with a robot or machine and to use those models inside an adaptive controller. By “disturbance” the authors mean time-varying forces or motions that affect the main system, for example fluid sloshing in a tank or a swinging pendulum attached to a vehicle. The learned model runs a small hidden state (a “latent” state) forward in time and decodes that state into a prediction of the disturbance acting on the plant. The controller uses those predictions to compensate for the disturbance and keep the system on track.

To build the disturbance model the authors propose a statistically principled training loop. They alternate between a hard expectation–maximization (hard-EM) step that infers the most likely latent trajectory for each data window and an M-step that updates the model parameters. The hard E-step uses a Kalman smoother, a standard method that computes a best estimate of a time series of hidden states given noisy observations. The learned representation is constrained to be “contractive,” meaning its latent dynamics shrink differences over time. That property makes the latent state insensitive to unknown initial conditions and is compatible with closed-form Bayesian filters used online.

At run time the paper pairs the learned model with a Bayesian filter (a Kalman–Bucy style filter) to estimate the latent state from measurements and then adds a disturbance-rejection term to a nominal tracking controller. This results in a composite adaptive controller that can predict disturbances ahead of time and adjust control inputs accordingly. The authors prove a stability result: the controller drives tracking error exponentially fast to a bounded neighborhood. In plain terms, the method gives provable, fast convergence to a small error band rather than guaranteed perfect tracking.