A pendulum’s equation of motion is nonlinear in the angle. A motor’s torque equation is nonlinear once you include saturation, cogging, or back-EMF. A fluid is famously nonlinear. Yet most of the analysis and control tools we actually trust — root locus, frequency response, the LQR, Kalman filtering, linear MPC — were built for linear systems. The standard answer is to linearize around a trim point, with full awareness that the resulting model degrades the moment the system moves too far away from that point.
In 1931, Bernard Koopman proposed a much more radical answer: don’t linearize the system, change what you’re looking at. Instead of tracking the state itself, track scalar-valued functions of the state — what Koopman called observables. The infinite-dimensional vector space of these observables evolves under a linear operator, no matter how nonlinear the underlying system is. The trade is an honest one — you have given up the finite state dimension, but in exchange you have global, exact linearity.
For sixty years that was a beautiful mathematical fact that nobody could compute with. Then in the 2000s a data-driven approximation called Dynamic Mode Decomposition arrived, and shortly after that EDMD, deep Koopman, and the family of algorithms that lets engineers actually build a finite-dimensional linear surrogate from data. Today the operator sits underneath a substantial slice of modern model-predictive control and a growing fraction of model-based reinforcement learning.
This primer is the working tour I wish I’d had when I first encountered the idea.
What it covers
Seven sections, about twenty-five minutes if you read the prose. Longer if you stop to play with the figures, which is the point.
§ i — The big idea. Change what you look at, and nonlinearity disappears. The Duffing oscillator drawn in the original state space (crooked, looping trajectories) and then in a higher-dimensional space of observables (straight lines). The geometric intuition, before any operator notation.
§ ii — The operator. The Koopman operator $\mathcal{K}$ as the linear object that pushes any observable forward by one time step: $\mathcal{K} \varphi = \varphi \circ F$. Why this is exactly linear, what it costs (infinite dimensionality), and the eigendecomposition that organizes everything that follows.
§ iii — Lifting. What an “observable” actually is, how the choice of dictionary determines the quality of your linear model, and the standard families — monomials, RBFs, Fourier, neural-network features. The geometric picture: a curved manifold of trajectories in state space becomes a flat invariant subspace in observable space.
§ iv — DMD. Dynamic Mode Decomposition — the algorithm that made Koopman computable. The SVD-based least-squares fit, the eigenmodes and their physical meaning (frequencies and growth rates), and the connection between DMD’s spectrum and Koopman’s. A live “fit a DMD model to a noisy nonlinear trajectory” demo where you can pick the rank.
§ v — EDMD and beyond. Extended DMD with explicit dictionaries; kernel DMD when you can’t write down good features by hand; deep Koopman, where the dictionary itself is learned end-to-end by a neural network.
§ vi — Koopman MPC. The headline practical application: build a linear surrogate of a nonlinear plant from data, then run linear MPC — fast QPs, predictable convergence, real-time guarantees — over the lifted state. Korda & Mezić (2018) showed this works on real systems. The trade-offs versus nonlinear MPC and feedback linearization.
§ vii — Koopman + RL. The newer frontier: linear dynamics in observable space turn the value function into something with structure you can exploit, give you a model-based RL setup whose model is exact (in the limit) rather than approximate, and yield interpretable failure modes when the dictionary is too small.
Playground. A final interactive panel where you can pick a nonlinear vector field, choose a dictionary, fit a Koopman model, and watch the linear-prediction error grow or shrink with dictionary size.
Read it
The primer lives at its own URL with a light editorial layout — amber and copper accents, serif body, MathJax-typeset equations, and live canvases for every figure. It is meant to be read once cover to cover and then reached for again whenever a specific term (“EDMD?”, “the lifted-state trick?”) needs a quick refresher.
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