Koopman operators and predictive control
Turning a nonlinear system into a linear one on the space of observables — and controlling it there.
The Koopman operator recasts a nonlinear dynamical system as a linear operator acting on functions of the state — the observables. The state evolves nonlinearly, but observables evolve linearly under the operator, at the cost of an infinite-dimensional space. This reframing, first written down by Koopman in 1931, is the basis of a large body of modern data-driven dynamics.
In practice one approximates the operator on a finite dictionary of observables. Extended Dynamic Mode Decomposition estimates it directly from trajectory data, yielding a linear predictor for a nonlinear system. When that linear predictor is placed inside a model-predictive-control loop, nonlinear control becomes a convex problem in the lifted coordinates.
The phase-orchestration studio works in exactly this operator-theoretic setting, treating coherence and phase dynamics through their spectral structure. The related publication documents the domain-agnostic coherence approach that this line of work supports.
The vocabulary of the topic.
- Koopman operator
- The linear operator that advances observables of a nonlinear system in time.
- Observable
- A function of the state; the Koopman operator acts on these rather than on the state itself.
- EDMD
- Extended Dynamic Mode Decomposition — a data-driven finite approximation of the operator.
- Koopman MPC
- Model-predictive control using a Koopman linear predictor, making nonlinear control a convex problem.
Studios working on this.
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The runnable lessons and sealed measurements this site ships for the topic.
Read it at source, in order.
Canonical references for the topic, ordered from the foundations to current work. Every one was verified at source — a DOI resolves through doi.org, a standard through its issuer — so each link goes to the real record.
Start with the foundations
- 1
The original operator view: a nonlinear system as a linear operator on observables.
- 2
Textbook coverage of DMD, Koopman, and data-driven control.
How it works
- 3
The modern spectral revival of Koopman theory.
- 4
EDMD — the estimator that makes Koopman approximation practical from data.
Put to work
- 5 Paper Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control
Koopman lifting turned into a linear MPC problem.
Current work
- 6
A current survey tying operator theory, EDMD, and control together.
Related topics.
Where this topic connects to the rest of the federation's work.