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On the Existence of Koopman Linear Embeddings for Controlled Nonlinear Systems

Xu Shang, Masih Haseli, Jorge Cortés, Yang Zheng

arXiv:2602.14537v1math.OCeess.SY

TL;DR

The paper addresses the lack of a complete characterization of when controlled nonlinear systems admit exact finite-dimensional Koopman linear embeddings. It characterizes the required CAP structure and autonomous Koopman closure, and relates embedding existence to symbolic CAP certification.

  • Problem

    The precise system-level conditions under which controlled nonlinear systems admit finite-dimensional Koopman linear embeddings remain incompletely characterized, limiting understanding of Koopman linear modeling's scope.

  • Method

    The paper decomposes embedding characterization through N-step linear predictors, proves their equivalence to a CAP structure under coordinate transformation, and uses symbolic procedures to identify that structure.

  • Results

    Finite-dimensional Koopman linear embeddings require a CAP structure with an input-driven affine subsystem, an autonomous nonlinear subsystem, and a finite-dimensional Koopman-invariant subspace capturing the system's nonlinearities.

  • Takeaways & Limitations

    Koopman linear embeddings bridge LTI systems and autonomous systems admitting finite-dimensional Koopman representations, while clarifying when exact controlled-system representations can exist.

Abstract

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Koopman linear representations have become a popular tool for control design of nonlinear systems, yet it remains unclear when such representations are exact. In this paper, we establish sufficient and necessary conditions under which a controlled nonlinear system admits an exact finite-dimensional Koopman linear representation, which we term Koopman linear embedding. We show that such a system must be transformable into a special control-affine preserved (CAP) structure, which enforces affine dependence of the state on the control input and isolates all nonlinearities into an autonomous subsystem. We further prove that this autonomous subsystem must itself admit a finite-dimensional Koopman linear model with a sufficiently-rich Koopman invariant subspace. Finally, we introduce a symbolic procedure to determine whether a given controlled nonlinear system admits the CAP structure, thereby elucidating whether Koopman approximation errors arise from intrinsic system dynamics or from the choice of lifting functions.

I. INTRODUCTION

Koopman methods seek tractable linear representations of nonlinear systems, but exact finite-dimensional embeddings for controlled systems remain difficult to characterize. This paper identifies structural conditions for exactness and introduces a symbolic procedure to assess whether approximation error is intrinsic or due to lifting choices.

  • Motivation: Koopman methods lift nonlinear dynamics into linear representations, supporting control design while avoiding some computational demands of direct nonlinear control.The paper motivates linear and approximately linear representations as alternatives to computationally demanding nonlinear controllers with difficult performance guarantees.
  • Open challenges: Controlled nonlinear systems may fail to admit an exact Koopman linear representation, so finite-dimensional models can incur modeling error.The difficulty is distinct from the autonomous case, where an infinite-dimensional Koopman representation is guaranteed.
  • Open challenges: The paper addresses whether a controlled nonlinear system admits an exact finite-dimensional Koopman model and whether suitable lifting functions can be synthesized.Determining exact embeddability helps distinguish intrinsic dynamical error from error caused by inadequate lifting functions.
  • Approach: N-step linear predictability is introduced as an intermediate concept for characterizing Koopman linear embeddings.The paper uses recursive affine-composition arguments and coordinate transformations to connect prediction structure with CAP structure.
  • Algorithmic certification: A symbolic procedure verifies whether a system can be transformed into CAP structure, informing lifting design and the source of approximation error.If CAP structure is absent, the paper characterizes the approximation error as intrinsic; if it exists, the identified nonlinear components guide possible lifting functions.

B. Koopman Control Family and Koopman linear embedding

Koopman control families extend Koopman analysis to controlled systems, while finite-dimensional Koopman embeddings impose stronger structural requirements. The paper frames the open problem as an exact system-level characterization of controlled nonlinear systems that admit such embeddings.

  • Koopman control family: Koopman control families represent controlled dynamics through operators indexed by constant input values.Each fixed input produces an autonomous system, and the resulting operators act on observables closed under composition with those dynamics.
  • Koopman control family: A common finite-dimensional invariant subspace yields an input-state separable lifted representation with input-dependent linear dynamics.The lifted state evolves through a matrix-valued function of the input, but coupling between input and lifted state remains more structured than the original nonlinear model.
  • Koopman linear embedding: A Koopman linear embedding uses lifting functions whose lifted state evolves as z+ = Az + Bu while the original state is recovered as x = Cz.The matrices A, B, and C are constant, and the state must lie in the image of the lifting.
  • Approximation limits: Controlled Koopman models do not necessarily become exact by increasing lifting functions and data because controlled systems may lack an exact embedding even in infinite dimensions.This contrasts with the autonomous setting, where increasing the lifting subspace and data can asymptotically reduce finite-horizon EDMD prediction error.
  • Problem statement: The paper seeks necessary and sufficient structural conditions for all discrete-time controlled nonlinear systems admitting a finite-dimensional Koopman embedding.Its stated objective is to clarify the scope and limitations of exact Koopman linear modeling.

A. Motivating example and informal characterizations

The slow-manifold example illustrates how exact finite-dimensional Koopman embeddings arise when input-affine dynamics are coupled to an autonomous nonlinear subsystem with finite-dimensional Koopman closure. The paper formalizes these requirements through multi-step linear predictors and the CAP structure.

  • Motivating example: The slow-manifold system admits a finite-dimensional Koopman linear embedding whose exact lifting reproduces the original trajectories.Inexact lifted initial conditions can generate Koopman-system trajectories that do not correspond to trajectories of the original nonlinear system.
  • Motivating example: The example separates a state component with affine input propagation from an autonomous nonlinear component.The nonlinear term enters through the autonomous component and appears in the evolution of the input-affected state.
  • Informal characterization: A finite-dimensional Koopman embedding requires input-affine cascaded dynamics, an autonomous nonlinear generator, and finite-dimensional Koopman closure.The closure must contain the autonomous subsystem state and all nonlinear terms appearing in the controlled dynamics.
  • N-step linear predictors: An N-step linear predictor expresses finite-horizon evolution through a nonlinear function of the initial state and a linear function of the input sequence.The concept connects finite-horizon prediction with Koopman linear embeddings.
  • N-step linear predictors: A finite-dimensional Koopman linear model implies ∞-step linear prediction, making the latter a necessary condition for a Koopman linear embedding.The characterization separates the problem into establishing ∞-step prediction and adding requirements for a Koopman embedding.
  • N-step linear predictors: One-step input-affine prediction need not preserve affine dependence under recursion, as shown by x+ = x^2 + u producing a u^2 term after two steps.Additional structural conditions are needed for affine dependence to persist over multiple horizons.
  • N-step linear predictors: Arbitrarily accurate finite-horizon linear approximation requires an exact N-step linear predictor under the stated compact-input assumptions.Proposition 1 establishes this implication for every approximation tolerance, and Proposition 1 notes that exact prediction is therefore necessary.
  • CAP structure: Theorem 1 characterizes ∞-step linear prediction by a coordinate transformation into CAP form, where nonlinearities are isolated from the directly controlled subsystem.The CAP structure preserves affine input dependence because the input does not enter the nonlinear component.

C. Koopman linear embeddings

An exact finite-dimensional Koopman linear embedding is equivalent to a CAP structure, up to similarity transformation, together with a finite-dimensional Koopman-closed representation of the autonomous nonlinear component. The section also connects this characterization to multi-step prediction and constructive lifting design.

  • Characterization: A finite-dimensional Koopman linear embedding necessarily yields N-step linear predictors for every N≥1, although the converse generally fails.Thus, exact embeddings impose stronger requirements than one-step or finite-horizon linear prediction alone.
  • Characterization: Theorem 2 establishes equivalence between an exact Koopman linear embedding and a CAP structure with Koopman closure under stated domain and surjectivity assumptions.The CAP form is obtained up to a similarity transformation and includes compatible lifting and constant system matrices.
  • Structural conditions: The CAP structure separates an autonomous nonlinear subsystem from a control-driven subsystem whose state evolves linearly with the input.All nonlinear terms are concentrated in the autonomous component, whose Koopman-invariant subspace contains its state and nonlinear generator terms.
  • Structural conditions: The Koopman-closure condition supplies a finite-dimensional invariant subspace for the autonomous generator, enabling construction of the full embedding by concatenating the control-driven state with lifted autonomous coordinates.The lifting is constructed as Ψ(x)=ˆΨ(Tx), with ˆΨ combining the transformed control-driven state and autonomous lifting functions.
  • Special cases: Affine systems arise as a special CAP case when the autonomous nonlinear generator vanishes, with lifting Ψ(x)=col(x,1).The section also notes that the framework reduces to classical LTI systems without the autonomous generator and to autonomous Koopman representations when control is removed.
  • Multi-step prediction: One-step linear prediction does not guarantee infinite-step prediction because recursive composition can destroy affine dependence on the input.The scalar example x+ = x^2 + u has an immediate one-step predictor, but its two-step evolution need not preserve the required affine form.

A. Affine functions on general sets and their composition

This section develops affine-composition tools used to prove when recursive system dynamics preserve affine dependence on inputs. Full-row-rank input directions force global affinity, while rank-deficient inputs constrain only the input-driven coordinates.

  • Definitions: An affine function on X is the restriction of a map Φ(x)=Cx+v, without requiring X to be convex or connected.Connectedness becomes relevant when proving that a locally constant affine offset is shared across the entire domain.
  • Technical conditions: Openness and connectedness support the affine conclusion by ensuring gradients exist and the offset is constant across the domain.On disconnected sets, different connected components may have different affine offsets.
  • Composition: Affine maps are closed under composition: Φ2∘Φ1 remains affine with linear part A2A1 and offset A2b1+b2.This closure property motivates the recursive-composition analysis used later for multi-step prediction.
  • Full-row-rank recursion: Under surjectivity, open-domain assumptions, and full-row-rank B, affine dependence of Φ(Φ(x)+Bu) on u forces Φ itself to be affine on X.The proof perturbs the input using a right inverse B†, derives a constant increment relation, and then obtains a constant gradient.
  • Full-row-rank recursion: The full-row-rank condition is essential because input perturbations must generate variations in every state-space direction.Without it, the lemma fails: a rank-deficient example satisfies the composition assumptions while Φ is not affine on X.
  • Rank-deficient recursion: For rank-deficient B, the dynamics can be partitioned into input-driven and complementary coordinates, yielding affine dependence on the former while allowing functions of the latter.Lemma 3 formalizes this structure through constant matrices C1 and C2 and functions g1 and g2 of the complementary state coordinates.

B. Linear predictors for 1D and 2D systems

Low-dimensional systems admitting multi-step linear predictors are forced into the CAP structure, with nonlinearities either isolated in an autonomous subsystem or eliminated entirely. The proof uses rank cases and affine-composition constraints.

  • n = 1: For n = 1, an autonomous system is already CAP, while full-row-rank input forces the state dynamics to be affine.The latter follows because a 2-step linear predictor forces Φ(x) = Cx+v.
  • n = 2: For n = 2 with rank-one input, a coordinate change yields an input-driven state and an additional state whose dynamics depend on functions g1 and g2.Two-step predictability makes both state-update components affine in the input-driven coordinate.
  • n = 2: When the autonomous coordinate is genuinely nonlinear, the transformed dynamics have CAP form with the input-driven state separated from the autonomous nonlinear generator.The autonomous component is ˜x2, while ˜x1 carries the input dependence.
  • n = 2: If c2 ≠ 0, the predictor constraints force both g1 and g2 to be affine, reducing the system to an affine CAP special case without a nonlinear generator.The same conclusion is obtained by comparing multi-step linear predictors with the composed dynamics.

C. N-step linear predictors for general nonlinear systems

For general finite-dimensional systems, recursive affine-composition arguments progressively concentrate nonlinearities into lower-dimensional substates. Rank-based reductions terminate in a CAP representation after at most n − 1 reductions.

  • Recursive reduction: A recursive affine-composition strategy progressively constrains the dynamics and terminates after at most n − 1 steps because the nonlinear generator dimension strictly decreases.The construction generalizes the low-dimensional proof to arbitrary state dimension.
  • Rank-based characterization: When the intermediate coefficient has full row rank, multi-step predictability forces the remaining nonlinear functions to be affine; otherwise, further recursive reduction is required.This distinguishes immediate affine collapse from continued concentration of nonlinearities.
  • Recursive reduction: The general system is decomposed into state blocks whose nonlinearities originate only in the final substate.This block structure admits linear predictions through the number of blocks and supports recursive analysis.
  • Rank-based characterization: The rank-based characterization gives CAP structure directly when the final coefficient matrix is zero or full row rank.For intermediate rank, an invertible coordinate change produces a smaller nonlinear component for further reduction.
  • Proof completion: Applying the rank-based proposition repeatedly yields a transformed system with a zero or full-row-rank terminal coefficient, after which CAP structure follows.Each application strictly reduces the nonlinear-generator dimension, ensuring termination.

V. EXISTENCE OF KOOPMAN LINEAR EMBEDDINGS

The paper derives structural properties of Koopman linear embeddings that make the lifted model observable and separate linear from nonlinear components. These properties support the equivalence between embeddings and CAP-based representations.

  • Observable embeddings: Any Koopman linear embedding can be transformed into an observable embedding by removing the unobservable lifted component.The resulting reduced model still reconstructs the original state.
  • Observable embeddings: The observable component reconstructs the state, whereas the unobservable component can be discarded without invalidating the Koopman embedding.The reduced lifted dynamics use only the observable block.
  • Separable lifting: An invertible change of lifted coordinates can strictly separate the observable linear component from nonlinear lifting functions containing no linear terms in the state.The transformed lifting induces another Koopman linear model, and observability is preserved.

B. Proof of Theorem 2

Theorem 2 proves that Koopman linear embeddings are equivalent to CAP dynamics together with the required invariant lifted representation. The proof uses observability, separability, and multi-step prediction to show that nonlinear lifting terms depend only on the autonomous subsystem.

  • Direction 1: (16) ⇒ (15): CAP structure directly produces a Koopman embedding by lifting the input-driven state together with a Koopman lifting of the autonomous subsystem.This establishes the constructive reverse implication.
  • Direction 2: (15) ⇒ (16): A Koopman embedding implies CAP structure after a suitable coordinate transformation because it supplies N-step linear predictors for every positive N.Theorem 1 then yields the CAP decomposition into input-driven and autonomous components.
  • Direction 2: (15) ⇒ (16): The nonlinear component of the lifting is independent of the input-driven state, so it can be written solely as a function of the autonomous subsystem.Observable lifted dynamics and CAP state evolution together force this independence.
  • Algorithmic motivation: The paper identifies constructing or certifying the CAP coordinate transformation as necessary for verifying whether an exact Koopman embedding exists.This motivates the symbolic certification procedure introduced afterward.

A. Iterative Algorithm for CAP Structures Certification

Algorithms 1 and 2 iteratively test whether coordinate transformations can expose a CAP structure. The procedure terminates within n iterations and certifies existence or nonexistence of an ∞-step linear predictor.

  • Algorithmic procedure: Algorithm 1 repeatedly invokes the Subsystem Certification and Decomposition routine to transform subsystems and concentrate nonlinearities.The routine examines subsystem actuation and affine dependence, then updates the coordinate transformation when certification fails.
  • Termination and guarantee: Algorithm 1 terminates in at most n iterations and returns a similarity transformation yielding CAP structure when an ∞-step linear predictor exists.If the structure cannot be certified, the algorithm verifies that no ∞-step linear predictor exists.
  • Illustrative example: In the illustrative three-state system, the CAP structure is identified successfully after three iterations.The final transformed system has an autonomous nonlinear state, while the remaining states evolve affinely with the input and transformed coordinates.
  • Algorithmic procedure: The procedure separates fully controlled states from uncontrolled states using invertible coordinate transformations.At each iteration, the algorithm decomposes the input matrix and transforms the residual nonlinear subsystem.
  • Certification conditions: If transformed dynamics are affine in the controlled coordinates, the algorithm separates the remaining nonlinearity into an autonomous component; otherwise, it reports that no ∞-step predictor exists.The decomposition records the affine terms and residual nonlinear generator before continuing or terminating.

VII. CONCLUSION

The paper establishes necessary and sufficient conditions for finite-dimensional Koopman linear embeddings through N-step predictors and CAP structure. It also identifies structural criteria for deciding when lifting refinement can help or when intrinsic limitations require another modeling strategy.

  • Main conclusions: The paper proves sufficient and necessary conditions for a discrete-time controlled nonlinear system to admit a finite-dimensional Koopman linear embedding.The result is organized around the equivalence between N-step linear predictors and CAP structure.
  • Main conclusions: The CAP structure isolates nonlinear dynamics in a self-evolving autonomous subsystem.The controlled portion remains affine, while the autonomous subsystem contains the nonlinear dynamics relevant to the embedding.
  • Future directions: Future work includes data-driven CAP verification, approximate CAP structures, and quantifying how structural deviations affect approximate-embedding prediction.These directions define the current boundary of the paper’s exact structural analysis.

APPENDIX

The appendix develops technical results showing that limits of affine approximations remain affine and that affine composition forces a separated form. These results support the structural decomposition used in the paper’s main proofs.

  • Affine-limit argument: Uniform limits of affine functions of the input remain affine, yielding a representation with state-dependent offset and input coefficient.The appendix then establishes that the input coefficient is independent of the initial state.
  • Affine-limit argument: The input coefficient is shown to be a constant matrix by evaluating affine relationships on affinely independent input sequences.Invertibility of the resulting matrix allows convergence of the coefficient differences to imply state independence.
  • Affine composition: The principle of affine composition states that if a composed system remains affine in an input entering one argument, the outer function must be affine in that argument.The resulting representation separates the affine argument from a function of the remaining variables.
  • Assumptions: The appendix’s assumptions include open convex domains and full-row-rank input matrices.These conditions support the perturbation and slice arguments used to derive affine dependence.
  • Affine composition: The proof establishes differentiability on open convex slices and uses a constant derivative to obtain the affine decomposition.For each fixed second component, the function is affine in the first component.

C. Proof of Proposition 2

The proof constructs auxiliary states and coordinate decompositions to show that repeated linear predictability concentrates nonlinearity into autonomous variables. Full-row-rank decompositions and affine-composition arguments then yield the CAP structure.

  • State decomposition: The proof begins by separating the state into components whose propagated coefficient matrices have full row rank.Lemma 7 supplies a similarity transformation that isolates a fully actuated component.
  • Recursive prediction: Recursive substitution expresses future states as affine functions of prior states and input sequences under the assumed linear predictors.The construction tracks residual terms while preserving the affine input relationship across prediction horizons.
  • Affine composition: Applying the affine-composition principle shows that the relevant transition functions are affine in the decomposed state variables.The resulting relation has the form of an affine term in the current state plus a function of the next-state component.
  • CAP conclusion: When the coefficient matrix is zero, the system is already in CAP structure; when it has full row rank, the proof shows the system is affine.These cases establish the decomposition needed for the induction.
  • CAP conclusion: An auxiliary state is propagated autonomously, so the remaining system nonlinearity comes only from that auxiliary state.This identifies the autonomous nonlinear generator required by the CAP structure.
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