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Data-driven discovery of Koopman eigenfunctions for control

Eurika Kaiser, J. Nathan Kutz, Steven L. Brunton

arXiv:1707.01146v4math.OCmath.DS

TL;DR

The paper addresses the challenge of controlling strongly nonlinear systems when finite-dimensional Koopman approximations may fail to close. It develops KRONIC by identifying and validating Koopman eigenfunctions, formulating control in the resulting intrinsic coordinates, and demonstrates the approach across nonlinear, Hamiltonian, and double-gyre systems.

  • Problem

    Finite-dimensional Koopman approximations can suffer closure problems and produce spurious eigenfunctions, while data-driven discovery of eigenfunctions remains a central challenge for representing strongly nonlinear systems.

  • Method

    KRONIC identifies Koopman eigenfunctions from data using regression, sparse extensions to EDMD, and an implicit formulation, then formulates nonlinear control in the resulting eigenfunction coordinates.

  • Results

    KRONIC accurately approximates lightly damped or undamped eigenfunctions from data and demonstrates data-driven control on nonlinear, Hamiltonian, and double-gyre systems.

  • Takeaways & Limitations

    Validated eigenfunctions yield reduced-order models that are closed by design, while lightly damped eigenfunctions capture persistent phenomena such as conserved quantities relevant to long-time control.

  • Takeaways & Limitations

    The eigenfunction-coordinate cost is exactly equivalent to the original state-space cost only when the state itself is a Koopman eigenfunction; eigenfunctions may also lack an exact inverse mapping.

Abstract

from arXiv · show

Data-driven transformations that reformulate nonlinear systems in a linear framework have the potential to enable the prediction, estimation, and control of strongly nonlinear dynamics using linear systems theory. The Koopman operator has emerged as a principled linear embedding of nonlinear dynamics, and its eigenfunctions establish intrinsic coordinates along which the dynamics behave linearly. Previous studies have used finite-dimensional approximations of the Koopman operator for model-predictive control approaches. In this work, we illustrate a fundamental closure issue of this approach and argue that it is beneficial to first validate eigenfunctions and then construct reduced-order models in these validated eigenfunctions. These coordinates form a Koopman-invariant subspace by design and, thus, have improved predictive power. We show then how the control can be formulated directly in these intrinsic coordinates and discuss potential benefits and caveats of this perspective. The resulting control architecture is termed Koopman Reduced Order Nonlinear Identification and Control (KRONIC). It is demonstrated that these eigenfunctions can be approximated with data-driven regression and power series expansions, based on the partial differential equation governing the infinitesimal generator of the Koopman operator. Validating discovered eigenfunctions is crucial and we show that lightly damped eigenfunctions may be faithfully extracted from EDMD or an implicit formulation. These lightly damped eigenfunctions are particularly relevant for control, as they correspond to nearly conserved quantities that are associated with persistent dynamics, such as the Hamiltonian. KRONIC is then demonstrated on a number of relevant examples, including 1) a nonlinear system with a known linear embedding, 2) a variety of Hamiltonian systems, and 3) a high-dimensional double-gyre model for ocean mixing.

1 Introduction

The paper addresses the challenge of controlling strongly nonlinear systems by using Koopman eigenfunctions as intrinsic coordinates for linear representations. It proposes validating these eigenfunctions before constructing reduced-order control models, forming the KRONIC architecture.

  • Nonlinear control remains difficult because many complex systems lack simple models suitable for control design.
  • Koopman theory embeds nonlinear dynamics in a linear framework, supporting prediction, estimation, and control through measurement functions and eigenfunctions.
  • KRONIC reformulates Koopman-based control in eigenfunction coordinates using models built from validated eigenfunctions.
  • Koopman eigenfunctions provide intrinsic coordinates closed under the Koopman operator, and control can be formulated directly in these coordinates.The paper connects eigenfunctions to geometric properties and coherent structures, including Hamiltonian energy.
  • Sparse regression is used to discover smooth Koopman eigenfunctions and to regularize EDMD or identify eigenfunctions implicitly.
  • Validated lightly damped eigenfunctions can support reduced-order models because they correspond to persistent phenomena such as conserved quantities.The framework is demonstrated on nonlinear, Hamiltonian, and double-gyre systems.

2 Motivation

The motivation is a closure problem in finite-dimensional Koopman approximations: incomplete bases can produce spurious eigenfunctions and inaccurate predictions. The paper shows that validation, sparsification, and reduced closed representations address this issue.

  • Finite-dimensional EDMD projections may fail to close on the chosen basis, causing spurious eigenfunctions.
  • In the slow-manifold example, third-degree monomials generate higher-degree terms, so the fitted EDMD model aliases omitted dynamics.
  • Some EDMD eigenvectors are spurious because their observed evolution disagrees with the linear prediction from their associated eigenvalues.
  • Selecting accurate eigenfunctions yields a five-dimensional linear system that remains closed under the Koopman operator.
  • Noisy data cause least-squares EDMD to overfit with many small nonzero coefficients and poor prediction performance.
  • Sparsifying EDMD or constructing models from validated eigenfunctions improves prediction accuracy and robustness to noise.

3 Background

The background introduces Koopman operator theory, finite-dimensional approximations, DMD, and optimal control as foundations for control in Koopman eigenfunction coordinates. It emphasizes the tension between linear operator structure and infinite-dimensional or non-invariant representations.

  • Koopman spectral theory: The Koopman operator advances measurement functions linearly, even when the underlying dynamical system is nonlinear.
  • Koopman spectral theory: The Koopman operator is infinite dimensional, so applied analysis approximates its evolution on finite-dimensional subspaces or invariant subspaces.
  • Koopman spectral theory: Koopman eigenfunctions provide intrinsic observables whose discovery enables globally linear representations of strongly nonlinear systems.
  • Dynamic mode decomposition: DMD computes a best-fit finite-dimensional linear operator from time-shifted measurements, typically using a pseudoinverse and singular value decomposition.
  • Dynamic mode decomposition: Because linear measurements generally do not span a Koopman-invariant subspace, DMD can capture limit-cycle dynamics while missing nonlinear transients off the attractor.
  • Optimal control: The control background defines an infinite-horizon quadratic cost over state and input and seeks full-state feedback driving the system to the origin.

4 Koopman operator control in eigenfunctions

KRONIC formulates control directly in validated Koopman eigenfunction coordinates, where nonlinear unforced dynamics become linear and control enters through structured additional terms. The approach supports optimal control objectives in intrinsic coordinates while retaining problem-specific selection and approximation caveats.

  • KRONIC constructs a closed, linear-by-design control model from validated Koopman eigenfunctions.
  • Control in eigenfunction coordinates can target fixed points, periodic orbits, and coherent structures through eigenfunction-related quantities such as the Hamiltonian.
  • For control-affine nonlinear systems, eigenfunction dynamics split into a linear unforced component and a control-associated bilinear component.
  • The optimal-control objective is quadratic in a reduced vector of r eigenfunctions, with each eigenfunction associated with an eigenvalue.
  • Matching eigenfunction-space costs to state-space costs may require modified weights or an approximate inverse mapping because eigenfunctions may not be exactly invertible.
  • The selected eigenfunctions and cost formulation are problem specific, while eigenfunction identification may be accurate only within a limited domain and training regime.

5 Identifying Koopman eigenfunctions from data

The paper addresses EDMD’s closure problem by identifying and validating Koopman eigenfunctions before constructing reduced-order models. It develops sparse data-driven formulations based on the Koopman PDE and emphasizes trajectory-based validation to select reliable coordinates.

  • The proposed strategy identifies dominant lightly damped eigenfunctions and uses them to construct low-dimensional, closed models.
  • Sparsity-promoting techniques identify eigenfunctions directly from the partial differential equation governing Koopman-generator evolution.
  • The continuous-time formulation assumes continuous, differentiable dynamics and smooth eigenfunctions.
  • Implicit-SINDy searches for sparse vectors in a null space while alternately updating the eigenvalue and selecting candidates by test-data validation.
  • EDMD can produce spurious eigenfunctions, whereas lightly damped eigenfunctions may be well approximated and used for reduced-order Koopman models.
  • Candidate eigenfunctions must be tested for linear trajectory behavior, ranked by error, and retained below a threshold for reduced-order modeling.
  • Validation yields a model that is closed and linear by design, with increased predictive power claimed by the authors.

6 Example: System with a slow manifold

The slow-manifold example shows that validating Koopman eigenfunctions before constructing reduced-order models improves robustness, especially under measurement noise. The validated intrinsic coordinates also support control design that can outperform standard linearization and feedback linearization while handling uncontrollable directions.

  • 6.1 Identification of eigenfunctions and models: Increasing noise makes EDMD transition matrices denser through overfitting, whereas implicit and L1-regularized EDMD produce sparse matrices.The sparse structure is more apparent at the higher noise magnitude η = 0.9.
  • 6.1 Identification of eigenfunctions and models: Eigenpairs are validated by comparing eigenvalue-predicted evolution with eigenfunction evaluation on a test trajectory, then using accurate pairs to construct linear ROMs.This validation distinguishes accurate eigenfunctions from spurious ones and preserves Koopman-invariant coordinates by construction.
  • 6.1 Identification of eigenfunctions and models: At low noise, the EDMD-based ROM fails because the eigenfunction associated with β = −0.19 is rejected despite its eigenvalue being correct.Including that additional eigenfunction restores performance comparable to the full-state EDMD model.
  • 6.1 Identification of eigenfunctions and models: At high noise, LARS and its ROM significantly outperform EDMD and the EDMD-based ROM, while the implicit method recovers only one accurate eigenfunction and predicts steady-state convergence but not exact transients.Adding one or two eigenfunctions improves the EDMD-ROM prediction, but it remains unable to predict steady-state behavior.
  • 6.2 Control design: KRONIC applies linear optimal control directly in Koopman intrinsic coordinates, yielding a nonlinear controller in the original state coordinates.For the tested unstable x2 direction, intrinsic-coordinate control matches observable-coordinate control and outperforms linearized LQR and feedback linearization.
  • 6.4 Stabilization with unstable slow dynamics: When an uncontrollable unstable eigenfunction direction appears, intrinsic-coordinate control can discard that direction and design the controller in the controllable subspace.This contrasts with standard control tools that fail for the corresponding uncontrollable, unstable representation.

7 Example: Hamiltonian energy control

Hamiltonian energy is a zero-eigenvalue Koopman eigenfunction, enabling direct energy-based control in intrinsic coordinates. The examples show accurate data-driven recovery and control of pendulum and Duffing dynamics, while highlighting eigenfunction-coordinate ambiguity.

  • Hamiltonian eigenfunctions: Conserved Hamiltonian energy is a Koopman eigenfunction with eigenvalue λ = 0, and trajectories remain on constant-energy hypersurfaces.Energy-level stabilization corresponds to controlling oscillatory modes and invariant manifolds in phase space.
  • Control formulation: For control-affine Hamiltonian systems, the Hamiltonian eigenfunction evolves through the control term ∇xϕ · Bu.With ϕ = H, the resulting cost penalizes energy deviations and control expenditure.
  • Control formulation: The energy feedback law is linear in Hamiltonian energy but nonlinear in the state, with Q/R setting the speed-cost trade-off.For Q = R, the law reduces to u = −sign(BH)H(x).
  • Pendulum example: The pendulum controller steers the system toward selected energy levels, and Koopman control outperforms an LQR controller based on linearized dynamics near the center.The target energies are E ∈ {−1, 0, 1, 2}.
  • Duffing example: The discovered Duffing Hamiltonian predicts energy over the full state space from a single trajectory with error O(10^-17).The eigenfunction evaluated along a trajectory oscillates with tiny amplitude around the true energy level.
  • Duffing example: With 1792 measurements, the identified zero-eigenvalue eigenfunction recovers the Hamiltonian up to scaling with error O(10^-8); 56 measurements suffice at a larger time step with error O(10^-6).The measurement dependence also includes identification time and control performance.

8 Example: Basin hopping in a double well

The double-well example uses Hamiltonian level control together with switching around the homoclinic orbit to move a trajectory between basins. A single energy objective cannot distinguish symmetric fixed points, motivating basin-aware switching.

  • Basin-hopping challenge: Controlling only toward the right-well fixed point can trap a trajectory in the left well because it cannot cross the saddle energy.The controller must first raise energy to the saddle transition, then lower it after basin crossing.
  • Switching strategy: The proposed switching strategy uses Koopman eigenfunctions and the homoclinic orbit to transport particles from the left basin to the right basin.Control is turned off during natural transit between basins and reapplied after crossing the saddle.
  • Switching strategy: The switching law targets the saddle-associated energy below transition, applies zero control at the transition under a location condition, and then targets the right-well energy.The three regimes correspond to increasing energy, natural transit, and final stabilization.
  • Data-driven construction: The controller can be derived from data by identifying zero-eigenvalue eigenfunctions, locating equilibria and saddles from roots and extrema, and using connecting orbits as transit paths.Within each basin, eigenfunction control drives the system toward the desired state.

9 Example: Double Gyre flow

The double-gyre example extends Koopman eigenfunction control to a high-dimensional, time-dependent flow. Controllers steer drifter ensembles toward a stream-function level set in both autonomous and periodically forced cases.

  • Flow setting: The periodically driven double gyre models transport between convection cells and uses time-dependent Koopman eigenfunctions.The system represents a model relevant to ocean-front transport.
  • Flow setting: For the autonomous flow, the stream function is a zero-eigenvalue Koopman eigenfunction and is conserved along unforced trajectories.Particles follow streamlines, which are stream-function isolines.
  • Control formulation: In the non-integrable case, the stream function evolves as dΨ/dt = AΨΨ + BΨu, with a time-dependent contribution from ∂Ψ/∂t.This provides the bilinear control representation for the forced flow.
  • Control results: Controllers steer autonomous and forced drifter ensembles toward the level set Ψ = 0.2 using a quadratic cost and state-dependent Riccati equations.For the forced system, the Riccati equation is solved at each point in space and time.
  • Control results: In the non-autonomous flow, particles follow the moving reference isocurve and exhibit small oscillations around the desired stream-function value.The reference curve periodically shifts and compresses or expands in the x-direction.
  • Interpretation and caveat: For a constrained spatial domain, the distributed control gain may be precomputed when controlling the external flow field.The framework can also be interpreted as local control of particles or swimmers in an external field.

10 Discussion and conclusions

The paper argues for validating Koopman eigenfunctions before constructing closed reduced-order models and demonstrates control directly in those coordinates. It concludes that lightly damped eigenfunctions are useful but that representation and control-objective choices remain limiting issues.

  • Contributions: The framework extends Koopman theory to actuation, formulates nonlinear control as bilinear control in eigenfunction coordinates, and identifies lightly damped or undamped eigenfunctions from data.These eigenfunctions correspond to conserved or nearly conserved quantities.
  • Contributions: KRONIC constructs a reduced-order model from validated Koopman eigenfunctions so the model is closed by design, unlike basis-projected finite-dimensional approximations that can produce spurious eigenfunctions.The paper contrasts this with EDMDc, which uses a high-dimensional observable span and an approximated affine control term.
  • Open problems: Accurate eigenfunction identification is essential, and some eigenfunctions may themselves be irrepresentable, imposing a fundamental uncertainty on their data-driven representation.Future work is directed toward improved extraction algorithms and machine-learning advances.
  • Representation caveat: Koopman eigenfunctions are global representations, so a Hamiltonian eigenfunction retains energy altitude while folding fixed-point location into a single scalar.This loss of spatial information explains ambiguity between states sharing the same energy.
  • Open problems: Selecting eigenfunctions for particular control goals and interpreting controllability and observability in eigenfunction coordinates remain open problems.The paper notes that state-space formulations or learned inverse mappings may sometimes be preferable for model-predictive control.
  • Implications: Undamped or lightly damped eigenfunctions correspond to conserved or nearly conserved quantities with potential applications including simultaneous energy and angular-momentum control.The paper also identifies possible applications in space mission design and other nonlinear-control domains.
  • Optimal-control context: Solving the nonlinear HJB equation yields a globally optimal state-feedback law but is computationally feasible only for low-dimensional problems.This motivates reduced or alternative control formulations for larger systems.
  • Optimal-control context: Pontryagin-based optimal control can address higher-dimensional problems than HJB, but practical solutions generally find only local minima and may require expensive simulations.The limitation is especially important for systems with many degrees of freedom, such as fluid flows.

A.3 Linear or factorized systems

The section contrasts conventional linear-quadratic control with state-dependent extensions for nonlinear systems, then motivates eigenfunction coordinates as a way to recover simpler control formulations.

  • Linear systems: Linear systems admit an algebraic Riccati equation whose solution produces a globally precomputed state-feedback law.This is the linear quadratic regulator formulation.
  • Factorized nonlinear systems: Control-affine nonlinear systems can be factored into a linear-like structure with state-dependent transition and actuation matrices.
  • Factorized nonlinear systems: The state-dependent Riccati equation extends LQR to nonlinear systems by solving the algebraic Riccati equation point-wise in x.The weight matrices Q and R may also depend on x.
  • Factorized nonlinear systems: Unlike an offline algebraic Riccati solution, SDRE requires online computation at every x, which may be infeasible for high-dimensional systems.Despite this cost, SDRE retains simple implementation and often yields near-optimal controllers.

B Effect of misrepresentation of eigenfunctions

The analysis examines how eigenfunction approximation errors affect closed-loop control and derives an upper error bound under explicit control-affine and measurement assumptions.

  • Setup: The controlled eigenfunction problem assumes a control-affine system and seeks to regulate one Koopman eigenfunction using positive weights Q and R.
  • Setup: With full-state measurements, known actuation effects B(x), and a known eigenfunction, the control term C(x) can be evaluated directly rather than represented in eigenfunction coordinates.
  • Control formulation: The feedback control and resulting closed-loop dynamics follow from solving the scalar Riccati equation for the eigenfunction coordinate.
  • Approximation error: Eigenfunction misrepresentation is modeled as ˆϕ(x)=ϕ(x)+εψ(x), with small ε, and its dynamics include both autonomous and control-induced derivative terms.
  • Approximation error: An upper bound is provided for the error in the closed-loop quantity µ caused by misrepresenting the eigenfunction.
  • Eigenfunction construction: Koopman eigenfunctions can also be computed from the governing PDE using standard Taylor or Laurent series techniques.

C.1 Linear dynamics

For linear dynamics, substituting a Taylor-series eigenfunction into the governing PDE constrains its coefficients and yields monomial eigenfunctions for positive integer eigenvalues.

  • Taylor expansion: A Taylor expansion represents the eigenfunction as ϕ(x)=c0+c1x+c2x^2+c3x^3+⋯.
  • Taylor expansion: The gradient and directional derivative are obtained term-by-term from the Taylor coefficients.
  • Eigenfunction solutions: For positive integer λ=k, only one Taylor coefficient may be nonzero, giving ϕ(x)=cx^k as an eigenfunction.
  • Eigenfunction solutions: When λ=1, the resulting eigenfunction is ϕ(x)=x.

C.2 Quadratic nonlinear dynamics

For quadratic nonlinear dynamics, the Taylor-series approach can fail completely, so Laurent expansions are used to represent nontrivial eigenfunctions through nonpositive powers.

  • Quadratic nonlinear dynamics: The nonlinear system considered in this section is introduced as the setting for testing the eigenfunction PDE.
  • Failure of Taylor series: No Taylor series satisfies the eigenfunction PDE except the trivial solution ϕ=0 for λ=0.
  • Laurent expansion: A Laurent series extends the representation to negative powers alongside the constant and positive-power terms.
  • Laurent expansion: The gradient and directional derivatives of the Laurent expansion are likewise computed term-by-term.
  • Eigenfunction solutions: All positive-index coefficients vanish, while the remaining coefficients obey the recursion λc_{k+1}=kc_k for k≤−1.
  • Eigenfunction solutions: The section notes that other Koopman eigenfunctions may also be identified from the series.
  • Generalization: The subsequent general nonlinear system extends the analysis beyond the specific quadratic example.
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