Home / Curated / Model Families / Wiener–Hammerstein model

Wiener–Hammerstein model

Model Families AIMech:WienerHammersteinModel DRAFT

A real-valued, single-input single-output, discrete-time deterministic dynamical model family formed by a causal linear time-invariant block, a memoryless static nonlinear function, and a second causal linear time-invariant block in that order.

Identity Notes & Concept Scope Compared all 442 pinned AIO IDs, labels and scoped synonyms and 536 native IDs, names and aliases against both cascade names, punctuation variants, expanded descriptions and IDs: no normalized identity match. Ignored- and hidden-inclusive searches of the accessible recovery and previous worktrees returned 746 and zero lines. All 22 distinct matched texts were inspected; they concern existing two-block records and their review/report paths, or the pending combined-cascade cohort. Original Documents files were inaccessible during the initial search. After access returned, a full ignored- and hidden-inclusive search there at 2026-10-12T14:36:27Z found no candidate-name, expanded-description or reference hits; all 11 user files matched their preserved hashes. Read all 41 semantic AIO rows, 65 native name/definition neighbors, nine complete annotated CLASS nodes with incident edges, nine generated records and eight complete native neighbors. The existing Wiener/Hammerstein families have two blocks, their correlation records specify estimators, and NARX/NARMAX specify observed-history recurrences. A linear state-space layer is a neural component; GPR conditions a process prior. None supplies the proposed three-block identity. AIO GLM specifies a statistical link/variance structure, ARDL lag structure and ETS smoothing/trend/seasonality; neural layer/network classes require different scope. Shared linear special cases do not make the families equivalent. This is bounded to the inspected pin and accessible records.

Claims & Grounded Evidence

4 Claims

Mechanism #linear-static-linear

For zero-state linear operators R and S and a memoryless function f, the response is y=S(f(Ru)). The first block mixes input history, f acts pointwise on the resulting internal signal, and S filters that nonlinear signal. With causal finite filters R=(1,2), S=(1,3), inputs (1,2,-1), zero prehistory and f(v)=v², the first three outputs are (1,19,57). This is an exact curator example of the specified composition, not a fitted benchmark.

supports Source: Initial estimates for Wiener-Hammerstein models using phase-coupled multisines · Locator: Section 2.2, Figure 1 and Equations 3–4, PDF/printed pp. 2–3; curator calculation in research/discovery/three-block-cascade-boundaries.py

Scope #two-block-special-cases

An identity input block R gives the Hammerstein order, and an identity output block S gives the Wiener order. If f is the identity, the cascade reduces to the linear composition SR. The families therefore overlap through degenerate or linear special cases; their different ordered structures are not interchangeable names, and a three-block representation need not be minimal.

supports Source: Initial estimates for Wiener-Hammerstein models using phase-coupled multisines · Locator: Introduction and Figure 1, PDF/printed pp. 1–2; direct algebraic specializations of Equation 3 on p. 3, checked in research/discovery/three-block-cascade-boundaries.py

Limitation #internal-gain-ambiguity

For nonzero a and b in the zero-state deterministic formulation, replacing R by aR, f(v) by b*f(v/a), and S by S/b leaves the external response unchanged on the corresponding signal domains. The finite example with a=2,b=3 still gives (1,19,57). External data alone therefore cannot fix these two internal scale choices without normalization or other information; this is not a complete classification of all ambiguities.

supports Source: Initial estimates for Wiener-Hammerstein models using phase-coupled multisines · Locator: Direct curator reparameterization of Section 2.2, Equation 3, PDF/printed p. 3; exact check in research/discovery/three-block-cascade-boundaries.py

Limitation #linear-approximation-boundary

The source’s Gaussian best-linear approximation exposes a scalar multiple of the product RS rather than uniquely assigning dynamics to R and S. An informative nonzero gain is an additional requirement: non-evenness of f alone does not suffice. For iid standard Gaussian input, R=(1,2) gives internal variance 5, and f(q)=q³-15q has zero first input/output correlations despite being a nonconstant odd function. The curator counterexample qualifies the source text after Theorem 1. Estimating or separating blocks is a distinct identification problem.

supports Source: Initial estimates for Wiener-Hammerstein models using phase-coupled multisines · Locator: Section 3.2, Equations 8–10 and text after Theorem 1, PDF/printed p. 4; Section 4.3, p. 7; curator exact Gaussian moments in research/discovery/three-block-cascade-boundaries.py

⚠️ Curated Limitations & Negative Results

This maintained record selects real-valued scalar discrete-time deterministic cascades. Noise models, multivariable, parallel, feedback, time-varying and neural variants require explicit additional scope. Finite examples use causal FIR blocks and zero prehistory; general linear blocks may have infinite impulse-response memory or specified nonzero initial states.
A memoryless function is a single-valued map on its stated domain. Forward composition does not in itself require inverse functions, monotonicity, Gaussian inputs, polynomial parameterization or a GP prior. Any use of inverse-coordinate predictors must state the valid domain and stronger assumptions.
Representation, minimality, internal identifiability, stable simulation and estimation accuracy are separate questions. Internal gain normalizations alone do not settle all ambiguities; overlapping special cases do not establish family equivalence.
The source’s phase-coupled-multisine initialization method has separate excitation, odd-term, steady-state and noise assumptions. Its higher-degree combinatorial formulas need a separate audit; they are not used as model-definition evidence. The reported electronics experiment was not replicated.
All pages of the cited version were visually read. No model was trained, no empirical result was replicated and no paid provider was run. This record remains DRAFT; author self-review is not independent scientific approval.

Artificial Intelligence Ontology (AIO) Grounding

Mapped AIO Classes

AIO Target Class	Mapping Relation	Mapping Rationale
`AIO:TimeSeriesAnalysis` Time Series Analysis	related_match	This ordered dynamical model family represents temporal input-output relationships for system identification and analysis. The inspected AIO target is an analysis task, so the link is a model/task relationship rather than exact identity or an asserted subclass.

Evidence Sources & Primary Literature

1 Source

Source ID	Publication / Document	Type / Access	Reference & Checksum
`tiels-schoukens2015`	Initial estimates for Wiener-Hammerstein models using phase-coupled multisines Koen Tiels, Maarten Schoukens, Johan Schoukens	primary paper full_text	ID: https://doi.org/10.1016/j.automatica.2015.07.020 sha256:3368b292c40e…

Curation Provenance

Curator: Codex

Method: ai_assisted

Model: GPT-6

Created: 2026-10-12T14:34:20Z

Updated: 2026-10-12T14:42:40Z

Source File: data/curated/WienerHammersteinModel.yaml