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Koopman invariant subspaces and finite linear representations of nonlinear dynamical systems for control

Steven L. Brunton, Bingni W. Brunton, Joshua L. Proctor, J. Nathan Kutz

arXiv:1510.03007v2math.DS

TL;DR

The paper addresses how to choose observables that yield finite-dimensional Koopman models of nonlinear systems, particularly for control. It combines Koopman-invariant subspaces with SINDy-based observable identification and shows that state-inclusive models are broadly restricted, while optimal-control laws can be constructed for supported nonlinear systems.

  • Problem

    Choosing nonlinear observable functions that form useful Koopman-invariant subspaces, especially for control, remains an open challenge, while linear DMD observables can be too restrictive for nonlinear systems.

  • Method

    The paper uses SINDy to identify nonlinear dynamics, selects observables spanning those dynamics, and restricts the Koopman operator to the resulting invariant subspace.

  • Results

    For a large class of nonlinear systems with a single isolated fixed point, the paper obtains state-inclusive finite-dimensional Koopman-invariant subspaces and demonstrates optimal control using linear-control techniques.

  • Takeaways & Limitations

    Finite-dimensional Koopman models can support nonlinear optimal control when appropriate invariant observables are available, but including the original state is highly restrictive.

  • Takeaways & Limitations

    State-inclusive finite-dimensional Koopman representations are impossible for systems with multiple fixed points or more general attractors, and uncontrollable unstable observables can prevent standard LQR solvers from returning a controller.

Abstract

from arXiv · show

In this work, we explore finite-dimensional linear representations of nonlinear dynamical systems by restricting the Koopman operator to an invariant subspace. The Koopman operator is an infinite-dimensional linear operator that evolves observable functions of the state-space of a dynamical system [Koopman 1931, PNAS]. Dominant terms in the Koopman expansion are typically computed using dynamic mode decomposition (DMD). DMD uses linear measurements of the state variables, and it has recently been shown that this may be too restrictive for nonlinear systems [Williams et al. 2015, JNLS]. Choosing nonlinear observable functions to form an invariant subspace where it is possible to obtain linear models, especially those that are useful for control, is an open challenge. Here, we investigate the choice of observable functions for Koopman analysis that enable the use of optimal linear control techniques on nonlinear problems. First, to include a cost on the state of the system, as in linear quadratic regulator (LQR) control, it is helpful to include these states in the observable subspace, as in DMD. However, we find that this is only possible when there is a single isolated fixed point, as systems with multiple fixed points or more complicated attractors are not globally topologically conjugate to a finite-dimensional linear system, and cannot be represented by a finite-dimensional linear Koopman subspace that includes the state. We then present a data-driven strategy to identify relevant observable functions for Koopman analysis using a new algorithm to determine terms in a dynamical system by sparse regression of the data in a nonlinear function space [Brunton et al. 2015, arxiv]; we show how this algorithm is related to DMD. Finally, we demonstrate how to design optimal control laws for nonlinear systems using techniques from linear optimal control on Koopman invariant subspaces.

1 Introduction

Koopman analysis represents nonlinear dynamics through an infinite-dimensional linear operator, but useful finite-dimensional control models require carefully chosen observable functions. The paper develops invariant-subspace and data-driven strategies for making such representations practical.

  • Koopman analysis provides an operator-theoretic perspective by evolving observable functions with an infinite-dimensional linear operator.
  • Finite-rank Koopman approximations are promising for nonlinear-system control but trade nonlinear dynamics for infinite-dimensional linear dynamics.
  • DMD uses linear state measurements, which are too limited to describe rich dynamics in nonlinear systems.
  • Selecting nonlinear observables for extended DMD remains an open problem, despite kernel methods that reduce its computational cost.
  • The paper identifies Koopman-invariant observables, uses SINDy to identify nonlinear dynamics, and develops finite-dimensional models suitable for nonlinear optimal control.

2 Background on Koopman analysis

Koopman analysis lifts nonlinear state dynamics into linear evolution of observable functions in a Hilbert space. The Koopman operator acts by composition with the system flow and can be represented in discrete or continuous time.

  • A continuous-time system evolves an n-dimensional state x on a smooth manifold M under a vector field f.
  • The flow map F_t advances x(t_0) to x(t_0+t), inducing a discrete-time system by sampling the continuous trajectory.
  • Observable functions g:M→R belong to an infinite-dimensional Hilbert space, commonly a space of square-integrable functions.
  • The Koopman operator is an infinite-dimensional linear operator that advances observables through composition with the state dynamics.
  • For Hamiltonian fluids, the Koopman operator is unitary, while its continuous-time evolution can also be described by an infinitesimal generator.
  • The schematic emphasizes recovering the original state x from evolved observables y_k.

3 Koopman invariant subspaces and exact finite-dimensional models

A finite-dimensional Koopman-invariant subspace yields linear dynamics for selected observables, including the original state only under restrictive conditions. The paper connects observable selection to nonlinear-term identification using SINDy and DMD.

  • A Koopman-invariant subspace remains closed under the Koopman operator, allowing its restriction to produce a finite-dimensional linear system.
  • Including the state variables requires the nonlinear right-hand-side dynamics f to belong to the same observable subspace.
  • Finite-dimensional state-inclusive Koopman models are impossible for systems with multiple fixed points or more general attractors because finite-dimensional linear systems have a single fixed point.
  • For an isolated fixed point or periodic orbit, Koopman coordinates may represent the basin through an associated linear system, although recovering states can be challenging.
  • SINDy uses sparse regression over nonlinear candidate functions to identify active terms in the dynamics and select corresponding observables.
  • After active observables are selected, additional regressions can determine their linear evolution and the procedure can be iterated until the subspace converges.
  • When the candidate-function library equals the state measurements, the SINDy formulation reduces to standard DMD, whose solution minimizes squared error.

4 Systems with Koopman-invariant subspaces containing the state

The paper constructs nonlinear systems whose Koopman-invariant subspaces include the original state, yielding finite-dimensional linear representations despite nonlinear governing dynamics. Polynomial slow manifolds provide explicit examples, including continuous- and discrete-time formulations and intrinsic eigen-observable coordinates.

  • Continuous-time formulation: A finite-dimensional Koopman-invariant subspace containing the state exists for the constructed family, which necessarily has a single isolated fixed point.More general attractors and multiple fixed points cannot be represented by finite-dimensional linear systems in this way.
  • Continuous-time formulation: For polynomial P(x1), augmenting the state with the active polynomial terms closes the Koopman dynamics in a finite-dimensional linear system.For λ ≪ |µ| < 0, the manifold x2 = P(x1) is asymptotically attracting, and the closure advances the original state exactly.
  • Continuous-time examples: The quadratic and quartic slow-manifold examples illustrate how nonlinear dynamics embed into higher-dimensional linear observable coordinates.The quadratic example uses x2 = x1^2, while the quartic example uses x2 = x1^4 − 2x1^2.
  • Continuous-time examples: In the quadratic example, trajectories remain on the constraint y3 = y1^2, rapidly approach the slow subspace, and then slowly approach the fixed point.The slow subspace is associated with eigenvalues µ and 2µ, while the original attracting manifold appears as a separate parabolic surface.
  • Intrinsic coordinates defined by eigen-observables of the Koopman operator: Koopman eigenfunctions obtained from left eigenvectors define intrinsic coordinates and preserve invariant observable subspaces under coordinate transformations.The paper demonstrates this invariance after rotating the coordinate system by 45°.
  • Discrete-time formulation: The discrete-time formulation similarly converges to a slow manifold when |λ| ≪ |µ| and |λ| < 1, with the continuous-time scaling replaced by µ^N.The resulting finite-dimensional update is represented by a structured linear matrix for polynomial observables.

5 Koopman operator optimal control

The paper designs nonlinear controllers by applying linear optimal control to truncated Koopman representations, achieving substantially lower cost than standard LQR in the motivating example. It also identifies structural limitations, including loss of invariance and uncontrollable unstable modes in some systems.

  • Control design: Koopman operator optimal controllers apply linear control theory to a truncated Koopman operator and induce nonlinear control laws on the original state-space.The controller is linear in the Koopman coordinates but nonlinear when interpreted on the original state.
  • Control design: The LQR cost penalizes state deviations and control expenditure, with the motivating example weighting all state deviations and control expenditures equally.The quadratic cost uses Q for state deviations and R for control expenditure.
  • 5.1 Simple motivating example: Standard LQR is optimal only near the fixed point under valid linearization; outside that vicinity, nonlinear terms remove its optimality guarantees.The limitation arises when nonlinear terms become large away from the fixed point.
  • 5.1 Simple motivating example: Approximately 1/3 the cost of standard LQR is achieved by the KOOC in the motivating nonlinear fixed-point example.The comparison is between standard LQR based on linearization and the truncated-Koopman controller.
  • 5.2 Limitations of Koopman operator optimal control: In a limitation example, the Koopman subspace is no longer invariant because the derivative of y3 contains an additional nonlinear term involving x1u.The altered actuation and instability produce a more complicated derivative expression.
  • 5.2 Limitations of Koopman operator optimal control: The same limitation example contains an uncontrollable state y3 with positive eigenvalue 2µ, causing standard LQR software to fail for such systems.The paper notes that many off-the-shelf packages will not return an LQR controller in this setting.

6 Discussion

The paper studies finite-dimensional Koopman-invariant subspaces that include the state, identifies their restrictive existence conditions, and applies linear control methods on them. It shows that data-driven eigen-observable identification and Koopman optimal control are promising but remain limited in scope.

  • The paper seeks finite-dimensional Koopman-invariant subspaces that contain the original state variables and remain invariant under Koopman evolution.Such subspaces yield finite-dimensional linear systems that directly advance the state.
  • Identifying relevant eigenfunctions, inverting them to recover states, and classifying systems admitting state-containing subspaces remain open problems.The paper highlights eigenfunction-based control as a future direction for broader nonlinear-control applications.
  • For a large class of nonlinear systems with a single isolated fixed point, relevant state-containing subspaces can be identified using sparse regression and left eigenvectors.SINDy identifies the nonlinear dynamics, while the restricted Koopman operator provides the eigen-observables.
  • KOOC applies LQR techniques to the finite-dimensional Koopman system while retaining a cost defined on the original state.The resulting controller is nonlinear when expressed in the original state variables.
  • State-containing finite-dimensional Koopman subspaces are rare because systems with multiple fixed points, periodic orbits, or complex attractors cannot generally be represented this way.The paper links this restriction to the inability of such systems to be topologically conjugate to a finite-dimensional linear system with a single fixed point.

Appendix: Systems without Koopman-invariant subspaces that explicitly span the state

The appendix illustrates why polynomial observables often fail to produce finite state-spanning Koopman representations. Logistic-map and finite-time-blow-up examples show infinite expansions or unstable truncations, while eigenfunction coordinates can sometimes restore a usable linear model.

  • Systems with multiple fixed points, periodic orbits, or attracting or repelling structures cannot have finite-dimensional Koopman subspaces that explicitly include the state.The stated reason is that these systems cannot be topologically conjugate to a finite-dimensional linear system with a single fixed point.
  • Appendix: Systems without Koopman-invariant subspaces that explicitly span the state: For the logistic map, advancing x^2 requires cubic and quartic terms, which recursively require higher polynomial orders without closure.The construction therefore produces an infinite Koopman expansion.
  • Appendix: Systems without Koopman-invariant subspaces that explicitly span the state: Polynomial-basis truncations for the logistic map can agree with the true dynamics for only a small number of iterations.The passage attributes this poor behavior to the lack of closure and large determinants of finite-rank truncations for r > 1.
  • Appendix: Systems without Koopman-invariant subspaces that explicitly span the state: For the finite-time-blow-up example, derivatives of polynomial observables generate successively higher powers, yielding an infinite Koopman expansion.The stated sequence includes dty3 = 3x2 ˙x = 3x4 and continues indefinitely.
  • Appendix: Systems without Koopman-invariant subspaces that explicitly span the state: An eigenfunction coordinate ϕ(x) = e−1/x evolves linearly under the example dynamics and may be inverted to recover the state.The appendix presents this as an alternative to finite polynomial truncation.
  • Appendix: Systems without Koopman-invariant subspaces that explicitly span the state: The appendix identifies data-driven eigenfunction discovery and control using eigenfunction-based linear models as a high-priority future direction.
  • Appendix: Systems without Koopman-invariant subspaces that explicitly span the state: Figure 5 depicts convergence of the Koopman linear system toward the true solution as truncation rank r increases.
  • Appendix: Systems without Koopman-invariant subspaces that explicitly span the state: In the control example, the Koopman controller uses an augmented linear system and induces a controller nonlinear in the original state.The implementation retains quadratic state terms in the controlled dynamics and compares cumulative costs against LQR.
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