Koopman operator

Move nonlinearity into observable functions#

Ordinary state dynamics update the state itself. The Koopman operator instead advances an observable, a function of that state, by composition with the dynamics. This operator is linear even when the underlying state transition is nonlinear, but the exact collection of observables is generally infinite-dimensional.

Brunton and colleagues study finite Koopman-invariant subspaces for control. They show why finding a finite set that includes the original state and exactly closes under complex nonlinear dynamics is restrictive, rather than an automatic linearisation of every system.

Learn a lifted predictor for control#

A practical method chooses or learns observables, lifts measured state into that feature space, and fits a finite linear transition. Korda and Mezić construct data-driven controlled predictors by a nonlinear lift followed by linear least squares, then use the fitted model with model predictive control.

For robotics, the lifted state can model locomotion, soft-body motion, fluid interaction, or another nonlinear process for which a local linear model is too narrow. This is related to system identification, but “Koopman” identifies the observable-space operator structure, not merely any learned dynamics model.

A finite lift is an approximation, not hidden exact physics#

Predictions can fail when the selected observables do not close under the dynamics, training data misses a regime, contacts switch mode, or control inputs affect the system differently from the fitted model. A large feature space may improve fit while increasing estimation and optimisation cost.

Linear evolution in lifted coordinates does not make state constraints, uncertainty, or safety automatically linear or correct. Validation must cover multi-step prediction and closed-loop behaviour, not only one-step training error. Physical structure and fallback control remain important outside the data distribution.