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Hammerstein model

Model Families AIMech:HammersteinModel DRAFT

A real-valued, single-input single-output, discrete-time deterministic nonlinear dynamical model family formed by a memoryless static nonlinearity followed by a causal linear time-invariant system.

Identity Notes & Concept Scope Compared all 442 pinned AIO IDs, labels and all scoped synonyms and 532 native IDs, names and aliases with both candidate names, expanded cascade descriptions and IDs: no normalized identity match. Ignored- and hidden-inclusive full-text searches of the accessible recovery worktree and readable previous worktree returned four and zero lines; all four were read and concern prior discovery leads or a source example. Original Documents untracked files remain inaccessible and are not claimed re-audited. Read all 55 semantic AIO rows and 62 native name/definition neighbors, nine complete annotated AIO CLASS nodes with incident edges, all nine contextual generated YAMLs, and the complete IteratedWienerFilter, LinearStateSpaceLayer, NARX and NARMAX records. The existing iterated Wiener filter is a fixed-prior image-denoising algorithm. Neural linear state-space layers and NARX/NARMAX recurrences can overlap in representation without defining these ordered two-block model families. GLM specifies a statistical link/variance model; ARDL specifies lag structure; neither specifies this cascade identity. AIO convolutional layers and recurrent networks require neural-component/network scope, and ETS adds smoothing/trend/seasonality structure. These are not exact matches. This assessment is bounded to the inspected pin and accessible records.

Claims & Grounded Evidence

4 Claims

Mechanism #static-then-linear

Set z_t=g(u_t) and pass z through the linear dynamics, giving the zero-state response y_t=sum_{j=0}^{t-1} h_j*g(u_(t-j)). The nonlinearity acts on each input value before temporal mixing. Reversing these blocks generally changes the response; a Wiener model instead applies g to the mixed linear output. The family specifies the composition, not a particular nonlinear basis, fitting algorithm or neural implementation.

supports Source: Input-Output Parametric Models for Nonlinear Systems. Part II Stochastic Nonlinear Systems · Locator: Section 3, Figure 3.2 and Equations 3.11–3.14, PDF pp. 15–16 / printed pp. 12–13

Mechanism #recurrence-without-inversion

When the finite-dimensional linear block has coefficients a_i and b_j for its difference equation, substitution gives y_(t+n)=sum_{i=1}^n a_i*y_(t+n-i)+sum_{j=0}^n b_j*g(u_(t+n-j)). This construction uses the input nonlinearity directly and does not require g to be invertible. Its validity on the specified input domain does not imply stable trajectories, a correct fitted structure or uniquely recovered component parameters.

supports Source: Input-Output Parametric Models for Nonlinear Systems. Part II Stochastic Nonlinear Systems · Locator: Section 3, Equations 3.11–3.15 and following discussion, PDF pp. 15–16 / printed pp. 12–13

Scope #sufficient-rank-condition-boundary

A general local-realization theorem can impose stronger sufficient conditions than this special structure needs. The source example x_next=x+u², y=x has the exact recurrence y_next=y+u² even though the derivative of the input nonlinearity at zero vanishes. For inputs (1,2,-1) from zero the outputs after each transition are (1,5,6). Failure of that equilibrium derivative condition is therefore not proof that an explicit Hammerstein recurrence is unavailable.

supports Source: Input-Output Parametric Models for Nonlinear Systems. Part II Stochastic Nonlinear Systems · Locator: Section 2, Equations 2.31–2.32, PDF p. 13 / printed p. 10; Section 3, Equations 3.15–3.18 and discussion, PDF p. 16 / printed p. 13; curator exact calculations in research/discovery/cascade-model-boundaries.py

Limitation #internal-scale-ambiguity

Without a normalization, multiplying g by a nonzero constant c and dividing the linear operator L by c preserves L(g(u)) for every admissible input in the zero-state response. A c=3 finite example gives the same outputs. This gain exchange alone prevents unique recovery of both block scales from external responses; it is an algebraic boundary, not a claim that all other parameters are identifiable or that a particular estimator converges.

supports Source: Input-Output Parametric Models for Nonlinear Systems. Part II Stochastic Nonlinear Systems · Locator: Section 3, Equations 3.11–3.14, PDF pp. 15–16 / printed pp. 12–13; direct curator consequence of the cascade equations; curator exact calculations in research/discovery/cascade-model-boundaries.py

⚠️ Curated Limitations & Negative Results

The scoped evidence covers real-valued, single-input single-output, discrete-time deterministic cascades with a causal time-invariant linear block and a memoryless single-valued nonlinearity. Direct feedthrough is permitted. Continuous-time, multivariable, parallel and stochastic variants need explicit additional assumptions.
A model family is distinct from its parameter estimator, internal-state realization, trained instance and physical mechanism. Noise placement, initial state, stability, excitation, identifiability and generalization are separate choices; no universal identification or forecasting guarantee is asserted.
Invertibility assumptions used to derive a particular output-history recurrence are not definition-level requirements for a cascade model. Nonlinear and linear special cases can overlap across model families without making their ordered structures equivalent.
Evidence is the fully read 1984 technical report. The final 1985 journal version and earlier correlation-based identification reports remain follow-up sources. Several displayed equations in the report have scope or transcription problems; the cited cascade definitions and directly checked algebra are used here, not its unchecked global claims.
No model was trained and no empirical result was replicated. Exact finite examples are curator derivations, not author-reported experiments. Both records remain DRAFT; author self-review does not constitute 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	The source develops this ordered input-output dynamical representation for analyzing and identifying nonlinear systems from temporal observations. The pinned target denotes the analysis task; a related model/task mapping preserves that distinction and does not assert that the cascade is a task, neural layer or particular estimator.

Evidence Sources & Primary Literature

1 Source

Source ID	Publication / Document	Type / Access	Reference & Checksum
`leontaritis-billings1984-part2`	Input-Output Parametric Models for Nonlinear Systems. Part II Stochastic Nonlinear Systems I. J. Leontaritis, S. A. Billings	author technical report full_text	ID: https://eprints.whiterose.ac.uk/id/eprint/76841/ sha256:6ef50a90a753…

Curation Provenance

Curator: Codex

Method: ai_assisted

Model: GPT-6

Created: 2026-10-12T13:46:47Z

Updated: 2026-10-12T13:55:47Z

Source File: data/curated/HammersteinModel.yaml