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Exact Finite Koopman Embedding of Block-Oriented Polynomial Systems

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Pith's one-line read Every nonlinear system built from a series/parallel chain of LTI and polynomial blocks has an exact finite-dimensional Koopman embedding.

desk verdict The iterative embedding construction is solid, but the main theorem overclaims: the decoupling bound it cites is false, so the universality result fails as stated. read the letter →

arxiv 2507.15093 v1 pith:3NGG47MT submitted 2025-07-20 eess.SY cs.SY

Lucian Cristian Iacob , Roland Tóth , Maarten Schoukens This is my paper · ORCID
classification eess.SYcs.SY MSC 47-0847A1547A6747B3393B28
keywords Koopmanoperatorexactfinite-dimensionalembeddingbilinearsystemsblock-orientedmodelspolynomialWiener-HammersteinKroneckerproductsconstructivealgorithm
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper claims that every nonlinear system built by connecting linear time-invariant (LTI) blocks and static polynomial blocks in series and parallel, with no feedback, has an exact finite-dimensional Koopman embedding. The embedding takes a polynomial-input time-invariant (PITI) form in which state and output matrices are constant and the input influence enters polynomially; when the LTI blocks have no feedthrough and the chain does not start with a static nonlinearity, the embedding reduces to a bilinear time-invariant system. The proof is constructive and doubles as an algorithm that processes the blocks one by one, producing the model without data or approximation. If true, this removes a central obstacle to using Koopman models for analysis and control design: the model structure and dimension are dictated by the system, not chosen ad hoc.

What carries the argument

The machinery is an inductive block-by-block embedding built from three components: the decomposition of each multivariate polynomial static block as $f(u)=Wg(V^\top u)$ with univariate polynomials $g$, imported from the cited tensor-decomposition literature; Kronecker-power state augmentation $z=[1,z^\top,(z^{(2)})^\top,\dots,(z^{(p)})^\top]^\top$, which makes the time derivative of each power a linear function of the augmented state plus a structured input term; and the Kronecker gradient identities $\partial x^{(i)}/\partial x\,Ax={}_iA\,x^{(i)}$ and $\partial x^{(i)}/\partial x\,B={}_iB\,x^{(i-1)}$, which convert the derivatives into explicit matrices. Lemmas 3.6, 3.8, 3.10, and 3.12 state how PITI forms compose through series LTI blocks, series nonlinear blocks, input junctions, and output junctions respectively, and together they carry the induction.

What would settle it

Run the paper's algorithm on the MIMO Wiener-Hammerstein example with a third-order polynomial feedthrough term added to the final LTI block, simulate the original system and the PITI model under the same random input, and check whether the output difference stays at rounding level; any deviation above roughly $10^{-12}$ would refute the claimed exactness of Theorem 3.1.

Watch

Extended reading notes

Core claim

The central discovery is Theorem 3.1: given a nonlinear system represented as a block chain of LTI blocks and static polynomial blocks joined by input and output junctions, the system admits an exact finite-dimensional Koopman representation in the PITI form $\dot{z}=Az+L(z)R(u)u$, $y=Cz+\bar{L}(z)\bar{R}(u)u$, where $L$ and $\bar{L}$ are linear in the lifted state and $R$ and $\bar{R}$ are polynomial in the input. Corollary 3.2 adds that if every LTI block has zero feedthrough and the first block is not a static nonlinearity, the representation reduces to the bilinear form $\dot{z}=Az+\sum_j\bar{B}_j z_j u$, $y=Cz$. The proof is inductive over the block chain: each elementary block or junction operation is replaced by a PITI or bilinear building block whose state is augmented with Kronecker powers of the previous lifted state, and these building blocks compose exactly. Simulations of a MIMO Wiener-Hammerstein system and a SISO block chain confirm that the embedded outputs match the original dynamics to roughly $10^{-13}$, i.e. at machine precision.

Load-bearing premise

The construction is exact only if every static polynomial block admits an exact decomposition $f(u)=Wg(V^\top u)$ into univariate polynomials of linear forms, a fact imported from the cited literature rather than proved here; if that decomposition is approximate or its required order is not attainable, the resulting model is not an exact embedding.

Editorial extensions

If this is right

  • Every Wiener-Hammerstein system, and any series/parallel network of LTI and polynomial blocks without feedback, has a finite-dimensional Koopman model that is exact, with no data and no truncation error involved.
  • When feedthrough is absent in the LTI blocks and the chain starts with a linear dynamics block, the exact model is bilinear, so existing bilinear analysis and controller-design tools apply without the usual approximation gap.
  • The lifted state may contain many duplicate entries from Kronecker products, but a linear projection removes them, reducing 17 to 12 and 931 to 103 in the paper's examples while preserving exactness.
  • If the chain starts with a static nonlinearity or has feedthrough, the input term is polynomial rather than bilinear; the paper notes this can be circumvented by treating the nonlinear block's output as a virtual input, at the cost of complicating control use.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The construction suggests a dimension trade-off: the lifted state size is governed by products of the polynomial degrees encountered along each path, so reordering blocks could change the model dimension; the paper does not discuss this optimization.
  • Because the construction is exact and parametric, it could serve as a benchmark generator for data-driven Koopman learning, providing ground-truth finite-dimensional embeddings against which learned models can be measured.
  • The no-feedback restriction is a natural boundary for this approach; systems with algebraic loops or state feedback would need an implicit or fixed-point PITI form, which the present proof does not cover.
  • If the decoupling order $r$ can be computed a priori from the polynomial degree, a question left to the cited tensor-decomposition literature, the algorithm becomes fully automatic and its dimension and exactness follow from the system data alone.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

1 major / 5 minor

Summary. The paper proposes a constructive method for obtaining exact finite-dimensional Koopman embeddings of nonlinear systems that can be represented as block-oriented interconnections of LTI blocks and static polynomial nonlinearities. The main theorem (Theorem 3.1) states that every such block-chain polynomial system admits an exact finite-dimensional PITI Koopman representation of the form (3.1), and Corollary 3.2 gives conditions, namely no feedthrough in the LTI blocks and no static nonlinearity at the start of the chain, under which the representation becomes bilinear. The proof proceeds inductively over the elementary block operations (LD, SN, IJ, OJ), and it yields an iterative algorithm for constructing the embedding. Two numerical examples, a MIMO Wiener-Hammerstein system and a more complex SISO block chain, validate the construction to machine precision. The key external ingredient is the decoupling f_i(u)=W_i g_i(V_i^T u) of each multivariate polynomial block, assumed to exist with r bounded by n_y n_u.

Significance. The constructive, data-free nature of the algorithm is a real strength: the derivations in Appendix A are detailed, the proof gives an explicit state construction at each block operation, and the numerical examples reproduce the original system outputs to about 1e-13. If the scope of the polynomial blocks is stated correctly, the result would be a useful addition to the exact Koopman-embedding literature, going beyond Carleman truncation and giving simple structural conditions for bilinear Koopman models. The paper's main quantitative claim, however, is currently broader than what the decoupling premise supports.

major comments (1)
  1. [Section 2.3.2 and Lemma 3.8] The assertion that every matrix polynomial f admits an exact decoupling (2.29) with 'sufficiently high r <= n_y n_u' is dimensionally impossible for generic blocks, and this premise is load-bearing for Theorem 3.1. Concretely, for n_y=1, n_u=3 and degree p=4, the space of scalar polynomials of degree at most 4 in 3 variables has dimension 35, while the family (2.29) with r<=3 and univariate degree at most 4 has at most 27 parameters (9 in V, 3 in W, and 15 in the gamma coefficients), so a generic ternary quartic cannot be represented in that form. Lemma 3.8 starts from (3.7c), which assumes (2.29) for every Sigma_NL block; since that assumption fails for generic polynomial blocks, Theorem 3.1's claim that every block-chain polynomial system admits the PITI embedding is not supported, and the algorithm as stated cannot be exact for the full stated class. Please either supply a correct finite bound for r (allowing r to grow with the degree and number of variables) together with an exact computation, or restrict the main theorem, corollary, and algorithm to polynomial blocks that explicitly admit an exact decomposition (2.29). The reference to the 'approximate polynomial decoupling' toolbox [15] should also be reconciled with the paper's no-approximation claim.
minor comments (5)
  1. [Lemma 3.8, Eq. (3.9b)] The displayed block vector for \bar{R}_i(\bar{u}_i) in Eq. (3.9b) appears to have duplicated or misplaced entries (for example, two consecutive \bar{R}_{i-1} terms and a \bar{R}_{i-1}^{(2)}(I \otimes \bar{u}_{i-1}) \bar{R}_{i-1} term); please restate it as a clean block vector indexed by k, consistent with Eq. (A.28) and with the block structure of \bar{L}_{i,e} in Eq. (3.11).
  2. [Section 3.1 and Lemmas 3.3, 3.5] The PITI definition in Eq. (3.1) says L(z) and \bar{L}(z) are linear in z, but Lemmas 3.3 and 3.5 set L_1(z_1) \equiv B_1 and \bar{L}_1(z_1) \equiv I, which are constant functions; if 'linear' is meant to include affine terms or relies on a constant coordinate in z, this should be stated explicitly so the linearity claims in Lemma 3.8 and its proof are well-defined.
  3. [Section 4.1.2 and Figure 11] In Section 4.1.2 and the caption of Figure 11, setting 'D1 = D2 = 0' is inconsistent with the notation of Section 4.1.1, where the two LTI blocks are G1 and G3 and their feedthrough matrices are D1 and D3; the text and caption should refer to D1 and D3.
  4. [Section 4.2.9] The sentence 'This results in Figure 13b, completing the process' should refer to Figure 13e, since this is the final absorption of the nonlinear block f6.
  5. [Throughout] There are several typographical slips to correct: 'the the' in Corollary 3.7, 'feedtrough' for 'feedthrough' throughout, 'defiend' in Section 2.3.2, 'appropirate' in Appendix A.1, and 'N > iand' in the proof of Theorem 3.1.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Koopman embedding is obtained by explicit constructive lifting, with no fitted parameters; the only self-citation is a non-load-bearing elementary identity, and the decoupling premise is an external correctness caveat rather than a circular reduction.

full rationale

The central derivation is constructive and self-contained in the sense relevant to circularity. Theorem 3.1 is proved by induction over the block-chain operations: each lemma (3.3, 3.5, 3.6, 3.8, 3.10, 3.12) gives explicit matrices and functions in terms of A_i, B_i, C_i, D_i, V_i, W_i and the gamma coefficients, and the lifted state is built from stacked Kronecker powers, so no constants are fitted and no output is statistically forced. The static-nonlinearity base case (Lemma 3.5) uses the exact factorization f(u) = f(0) + tilde f(u) u, cited as Lemma 1 of the authors' [20]; this is an elementary, parameter-free line-integral identity and does not import the target Koopman result, so it does not make the embedding circular. The PITI-to-BLTI simplifications (Corollaries 3.4, 3.7, 3.9, 3.11, 3.13) only track feedthrough terms algebraically. The main caveat is an external correctness premise, not a circular reduction: Lemma 3.8 and Appendix A.4 assume the decomposition f_i(bar y_{i-1}) = W_i g_i(V_i^T bar y_{i-1}) for every polynomial sigma_NL block, citing [10,51] with the bound r <= n_y n_u. If that theorem or bound fails for generic multivariate polynomials, Theorem 3.1 overclaims in scope, but that is an unsupported or false premise, not an equivalence of the conclusion to its own input. There is no self-citation chain that forces the stated result.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The construction introduces no fitted parameters, new entities, or ad hoc constants. The PITI form is a model class, not an invented physical quantity. All parameters (W, V, polynomial coefficients) come from exact polynomial decoupling, not from fitting.

assumptions (4)
  • domain assumption Every multivariate polynomial static block admits an exact decoupling f(u)=W g(V^T u) with univariate polynomials g_e.
    Invoked in Section 2.3.2 and used in Lemma 3.8 and Appendix A.4. The paper cites [10,51] rather than proving it; if only approximate, the embedding is not exact.
  • domain assumption The nonlinear system is exactly representable as a block-chain of LTI and static polynomial blocks with series/parallel operations and no feedback.
    Section 2.3.3 restricts scope; feedback interconnections are excluded for well-posedness and for the conversion theory.
  • domain assumption The state space X is open and forward invariant under the flow; f is Lipschitz; the Koopman semigroup is strongly continuous.
    Section 2.1, used to define the Koopman generator and the lifted dynamics.
  • standard math Kronecker product properties (mixed product, (AB)^(k)=A^(k)B^(k), x⊗y=(x⊗I)y) hold.
    Appendix A.1, used throughout the lifting derivations.

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Cite this review

Pith. "Pith review of Exact Finite Koopman Embedding of Block-Oriented Polynomial Systems." pith.science (2026). https://pith.science/paper/3NGG47MT

@misc{pith2026250715093,
  author       = {Pith},
  title        = {Pith review of: Exact Finite Koopman Embedding of Block-Oriented Polynomial Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3NGG47MT}},
  note         = {Machine review of arXiv:2507.15093}
}
read the original abstract

The challenge of finding exact and finite-dimensional Koopman embeddings of nonlinear systems has been largely circumvented by employing data-driven techniques to learn models of different complexities (e.g., linear, bilinear, input affine). Although these models may provide good accuracy, selecting the model structure and dimension is still ad-hoc and it is difficult to quantify the error that is introduced. In contrast to the general trend of data-driven learning, in this paper, we develop a systematic technique for nonlinear systems that produces a finite-dimensional and exact embedding. If the nonlinear system is represented as a network of series and parallel linear and nonlinear (polynomial) blocks, one can derive an associated Koopman model that has constant state and output matrices and the input influence is polynomial. Furthermore, if the linear blocks do not have feedthrough, the Koopman representation simplifies to a bilinear model.

Figures

Figures reproduced from arXiv: 2507.15093 by the authors.

Figure 1
Figure 1. Example of a block-chain interconnection of LTI blocks Σ [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Block oriented description of Wiener (left) and Hammerstein (right) systems. [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Embedding an LTI block in a PITI Koopman representation. [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figures from the paper (11 more)
Figure 4
Figure 4. Figure 4: Embedding a static nonlinear block in a PITI Koopman representation. [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]
Figure 5
Figure 5. Figure 5: Embedding the series interconnection of a PITI block and a linear dynamic block [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]
Figure 6
Figure 6. Figure 6: Embedding the series interconnection of a PITI block and a static nonlinear block [PITH_FULL_IMAGE:figures/full_fig_p014_6.png]
Figure 7
Figure 7. Figure 7: Embedding a PITI block followed by a junction into a parallel connection of PITI [PITH_FULL_IMAGE:figures/full_fig_p016_7.png]
Figure 8
Figure 8. Figure 8: Embedding of m paralel branches of PITI blocks via an output junction into a single PITI Koopman representation. and with Ai =     Ai−1,1 . . . Ai−1,m     , Ci = h Ci−1,1 · · · Ci−1,mi (3.12a) , Li(zi) ≡     Li−1,1(zi−1,1) . . . Li−1,m(zi−1,m)     , Ri(…
Figure 9
Figure 9. Figure 9: Block-chain form of the MIMO Wiener-Hammerstein system. [PITH_FULL_IMAGE:figures/full_fig_p019_9.png]
Figure 10
Figure 10. Figure 10: Embedding steps of the Wiener-Hammerstein system into a PITI Koopman model. [PITH_FULL_IMAGE:figures/full_fig_p020_10.png]
Figure 11
Figure 11. Figure 11: Simulated output responses of the MIMO Wiener-Hammerstein block chain system [PITH_FULL_IMAGE:figures/full_fig_p024_11.png]
Figure 12
Figure 12. Figure 12: Interconnection structure of the SISO nonlinear block chain system. [PITH_FULL_IMAGE:figures/full_fig_p025_12.png]
Figure 13
Figure 13. Figure 13: Visual representation of the steps taken to embed the SISO nonlinear blockchain [PITH_FULL_IMAGE:figures/full_fig_p027_13.png]
Figure 14
Figure 14. Figure 14: Simulated output response (y x ) of the SISO NL block chain system depicted in [PITH_FULL_IMAGE:figures/full_fig_p030_14.png]

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    14th IF AC Conference on Methods and Models in Automation and Robotics

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.