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Optimal construction of Koopman eigenfunctions for prediction and control
Milan Korda, Igor Mezić
TL;DR
The paper addresses how to construct Koopman eigenfunctions rich enough for accurate linear prediction of nonlinear systems, particularly in non-recurrent, off-attractor regions. It proposes an optimization-based, data-driven construction, extends it to controlled prediction and Koopman MPC, and proves that the resulting family can be dense in the continuous-function space. Numerical examples include prediction errors around one percent with 28 eigenfunctions in both controlled and uncontrolled Duffing settings.
Problem
Existing Koopman prediction and control require embeddings that span invariant subspaces and contain the observable, motivating systematic eigenfunction construction for nonlinear dynamics.
Method
The method constructs eigenfunctions from non-recurrent trajectories and optimized boundary functions and eigenvalues, then identifies control inputs by minimizing multi-step prediction error through linear least squares.
Results
The eigenfunction family is proved dense in C(XT), and with 28 eigenfunctions the reported Duffing prediction error is around one percent in both controlled and uncontrolled scenarios.
Takeaways & Limitations
The framework provides data-driven linear predictors usable for nonlinear prediction, feedback control, and state estimation through established linear-system tools.
Takeaways & Limitations
The convex Koopman MPC problem approximates the nonlinear MPC problem because the predicted output is only an approximation of the true output.
Abstract
from arXiv · showhide
This work presents a novel data-driven framework for constructing eigenfunctions of the Koopman operator geared toward prediction and control. The method leverages the richness of the spectrum of the Koopman operator away from attractors to construct a rich set of eigenfunctions such that the state (or any other observable quantity of interest) is in the span of these eigenfunctions and hence predictable in a linear fashion. The eigenfunction construction is optimization-based with no dictionary selection required. Once a predictor for the uncontrolled part of the system is obtained in this way, the incorporation of control is done through a multi-step prediction error minimization, carried out by a simple linear least-squares regression. The predictor so obtained is in the form of a linear controlled dynamical system and can be readily applied within the Koopman model predictive control framework of [12] to control nonlinear dynamical systems using linear model predictive control tools. The method is entirely data-driven and based purely on convex optimization, with no reliance on neural networks or other non-convex machine learning tools. The novel eigenfunction construction method is also analyzed theoretically, proving rigorously that the family of eigenfunctions obtained is rich enough to span the space of all continuous functions. In addition, the method is extended to construct generalized eigenfunctions that also give rise Koopman invariant subspaces and hence can be used for linear prediction. Detailed numerical examples with code available online demonstrate the approach, both for prediction and feedback control.
1 Introduction
The paper introduces a data-driven, optimization-based construction of Koopman eigenfunctions for linear prediction and control, exploiting the rich off-attractor spectrum without dictionary selection. It extends this construction to controlled prediction and Koopman MPC using convex or least-squares procedures.
- Motivation and contribution: The method targets transient, off-attractor dynamics, where non-recurrent surfaces support continuous eigenfunctions for arbitrary complex eigenvalues.The associated eigenspaces are infinite-dimensional and parameterized by boundary functions.
- Motivation and contribution: The constructed eigenfunctions are selected so observables of interest lie in their span, yielding invariant embeddings and linear prediction.The observable may include the state or another quantity of interest.
- Eigenfunction construction: Given trajectory initial conditions, eigenvalues, and continuous boundary functions, the algorithm propagates boundary values along non-periodic trajectories and extends them across the state space.The extension is obtained by interpolation or approximation, with continuous extensions established theoretically.
- Optimization: Boundary functions are chosen through a convex reformulation into linear least squares, while eigenvalue selection is low-dimensional but intrinsically non-convex.These optimization steps suffice to construct linear predictors in the uncontrolled setting.
- Controlled prediction and MPC: For control, an uncontrolled predictor is built first, then a controlled multi-step prediction error identifies the input matrix through linear least squares.This is possible because the dynamics and output matrices are already identified, unlike methods that convexly minimize only one-step error.
- Controlled prediction and MPC: The resulting predictors are applied within Koopman MPC, whose multi-step horizon motivates the multi-step error objective; the construction also applies to state estimation.The eigenfunction and linear-predictor methods are described as general beyond control.
2 Koopman operator
The Koopman framework represents nonlinear dynamics through linear evolution of observables, with eigenfunctions providing coordinates that evolve according to linear ODEs. Their span yields linear predictors whose output error is determined by observable projection.
- Koopman operator: The Koopman semigroup maps continuous observables through composition with the system flow and is linear on continuous functions.The flow is assumed well defined for all states and nonnegative times.
- Eigenfunctions: A Koopman eigenfunction is a nonzero continuous complex-valued function associated with an eigenvalue and satisfying the eigenfunction relation.Its value evolves as a linear coordinate along system trajectories.
- Linear predictors: Eigenfunctions with associated eigenvalues define a predictor with linear dynamics ż = Az, initialization z0 = φ(x0), and output ŷ = Cz.The matrix C projects the observable onto the eigenfunction span.
- Prediction error: Because eigenfunction evolution is error-free, prediction error arises solely from projecting the observable onto the span of the selected eigenfunctions.If the observable belongs to that span, the prediction is exact.
- Section goals: The paper’s primary goal is to construct eigenfunctions that minimize projection error, then use them for controlled prediction and Koopman model predictive control.The framework is intended to support linear tools for nonlinear-system control and estimation.
3 Non-recurrent sets and eigenfunctions
Non-recurrent sets enable construction of Koopman eigenfunctions by propagating boundary-function values along trajectories. With suitable eigenvalues and boundary functions, these eigenfunctions are theoretically rich enough to approximate every continuous observable on the flow image.
- 3.1 Non-recurrent sets and eigenfunctions: The eigenfunction is constructed by solving the defining ODE separately from each initial condition in Γ and propagating g forward along the flow.For non-periodic trajectories, this consistently and uniquely defines eigenfunction values across the data-generated flow image.
- 3.1 Non-recurrent sets and eigenfunctions: For any non-recurrent set Γ, continuous boundary function g, and complex λ, the construction defines a Koopman eigenfunction on the flow image X_T.If g is Lipschitz continuous, the resulting eigenfunction inherits Lipschitz regularity.
- 3.1 Non-recurrent set vs Non-recurrent surface: The non-recurrent surface need not be known explicitly: randomly selected discrete initial conditions are sufficient under non-periodic dynamics, although the resulting surface may be irregular.This allows the construction to use sampled trajectories rather than an explicitly identified geometric surface.
- 3.1 Non-recurrent set vs Non-recurrent surface: A finite set of initial conditions on distinct trajectories can be embedded in a non-recurrent surface when the flow is rectifiable on a suitable compact domain.The surface may be constructed as a closed, n−1 dimensional graph containing the finite set.
- 3.2 Span of the eigenfunctions: Using arbitrary complex eigenvalues and a dense family of boundary functions, linear combinations of the constructed eigenfunctions are dense in C(X_T).The associated projection error can therefore be made arbitrarily small by enlarging the eigenfunction family, while the theorem is asymptotic in the number of eigenfunctions.
- 3.3 Generalized eigenfunctions: Generalized eigenfunctions extend the construction to Koopman-invariant subspaces that can be used for linear prediction.Jordan blocks of size one recover the ordinary eigenfunction case, while arbitrary block sizes and continuous boundary functions provide additional construction freedom.
4 Learning eigenfunctions from data
The data-driven learning procedure selects eigenvalues and boundary-function values so that learned eigenfunctions minimize observable projection error. Boundary-function selection admits a convex least-squares formulation, followed by interpolation or approximation over the state space.
- 4.2 Optimal selection: Eigenvalues and boundary functions are optimized to minimize projection error of the observables onto the eigenfunction span.Boundary-function selection has a convex reformulation as a linear least-squares problem; eigenvalue selection remains low-dimensional but generally non-convex.
- 4.1 Data-driven construction: The method learns eigenfunctions from distinct, equidistantly sampled trajectories using chosen complex eigenvalues and continuous boundary functions.The initial conditions form the finite non-recurrent set Γ, and the boundary functions specify eigenfunction values there.
- 4.1 Data-driven construction: Eigenfunction values on the full data set are computed by applying the flow-based construction to each trajectory and each selected eigenvalue-boundary-function pair.The existence of a non-recurrent surface through the initial conditions implies these samples come from continuous eigenfunctions under Lemma 1 assumptions.
- 4.1 Data-driven construction: The learned eigenfunctions are extended from data points to the state space through interpolation or approximation using selected basis functions.Interpolation is exact on the data set, while accuracy beyond it depends on learnability by the chosen basis and the regularity of the underlying eigenfunctions.
- 4.3 Generalized eigenfunctions from data: Generalized-eigenfunction learning permits arbitrary Jordan-block sizes and pairings with continuous boundary functions, enriching the invariant subspaces available from data.A Jordan block of size one reduces the procedure to ordinary eigenfunction learning.
5 Linear predictors for controlled systems
The paper builds controlled linear predictors in two stages: eigenfunctions and the uncontrolled linear model are identified first, then control dynamics are fitted by convex multi-step regression. The resulting predictor supports convex Koopman MPC, while its accuracy determines how closely the controller approximates nonlinear MPC.
- Controlled predictor: The controlled predictor has the linear form ż = Az + Bu for a nonlinear system with state x and control input u.The formulation can represent general control systems through state inflation, including constraints on the derivative of the original control input.
- Two-step construction: The procedure first constructs approximate eigenfunctions and matrices A and C for the uncontrolled system.This uses the eigenfunction-construction procedure applied with the control set to zero.
- Two-step construction: The second step estimates B from controlled data by minimizing a multi-step prediction error.The method assumes separate uncontrolled and controlled datasets, whose trajectory counts and lengths may differ.
- Two-step construction: Because A and C are fixed, the multi-step prediction error depends affinely on B and can be minimized by convex optimization.This contrasts with EDMD-type approaches, which minimize only one-step-ahead prediction error in the corresponding procedure.
- Koopman model predictive control: The resulting predictor can be embedded in Koopman MPC, where nonlinearities are represented through the output mapping and the optimization becomes a convex quadratic program.After replacing the true output with its predictor, the objective is convex quadratic and the stage and terminal constraints are affine in the optimization variables.
- Koopman model predictive control: The convex MPC problem is only an approximation of nonlinear MPC because the predicted output is an approximation of the true output.When prediction is accurate, the paper expects the linear MPC solution to be close to the nonlinear optimum and to yield near-optimal closed-loop performance.
- Koopman model predictive control: The MPC problem can be reduced so its solve cost is independent of the number of eigenfunctions once a fixed product is evaluated.This supports using many eigenfunctions to reduce prediction error without increasing the reduced optimization cost.
6 Numerical examples
Numerical examples on the Van der Pol and damped Duffing oscillators evaluate prediction under controlled and uncontrolled settings, showing strong performance away from attractors and singularities. Eigenvalue optimization improves accuracy, while the damped Duffing predictor remains robust to reduced training data and supports feedback control through Koopman MPC.
- Experimental setup: The experiments evaluate state prediction for the Van der Pol and damped Duffing oscillators using varying numbers of Koopman eigenfunctions.Both controlled and uncontrolled settings are considered, with trajectories generated from system-specific initial-condition distributions.
- Prediction accuracy: Eigenvalue optimization significantly improves prediction performance, especially when only a small number of eigenfunctions is used.This pattern is reported for both oscillator examples.
- Spatial error distribution: Sampling Van der Pol trajectories from the smaller radius-0.05 circle produces broader state-space coverage and lower mean and standard-deviation prediction errors than radius 0.2 sampling.The comparison uses controlled square-wave predictions and independently generated test initial conditions.
- Damped Duffing oscillator: With 28 eigenfunctions, the damped Duffing prediction error is around 1% in both controlled and uncontrolled scenarios.The controlled tests use square-wave and sinusoidal signals distinct from the identification forcing.
- Data robustness: Using only 10 trajectories for the damped Duffing construction increases mean prediction error to 5.4%, indicating limited sensitivity to training-data quantity.The comparison of 100 and 25 trajectories showed only a small deterioration, and optimized eigenvalue locations were also robust.
7 Conclusion
The work develops a systematic, data-driven framework for learning Koopman eigenfunctions in non-recurrent state-space regions for prediction and control. It uses these eigenfunctions to build linear predictors for nonlinear systems with limited user input and convex-optimization-based computation.
- The framework learns Koopman eigenfunctions in non-recurrent regions to support prediction, feedback control, and state estimation for nonlinear systems.
- The method exploits the rich Koopman spectrum away from attractors to reduce projection error by constructing many eigenfunctions spanning the state or another observable.
- The approach relies on linear algebra and convex optimization, has complexity comparable to DMD-type methods, and requires selecting only an interpolation method.
- Future work will consider extending the method to settings involving recurrences.
Appendix
The appendix expresses the predictive-control optimization matrices directly through the data-derived system matrices and block-structured weighting matrices.
- The matrices in (57) are constructed from the data of (54).
- H1 combines R with the quadratic input term B⊤QB, while h combines q⊤B with r⊤.
- H2 is formed from the state-transition matrix A, the weighting matrix Q, and the input matrix B.
- Q and R are block-diagonal matrices assembled from stage-wise matrices Q0 through QNp and R0 through RNp−1.
- E and q are assembled from stage-wise components, while r collects the corresponding stage terms through RNp−1.