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arxiv: 2605.12244 · v2 · submitted 2026-05-12 · 📡 eess.SY · cs.SY· math.RA

Estimation Problems and the Modulating Function Method: The Algebra of Modulating Functions

Davi G. Accioli , Jerome Jouffroy This is my paper

Pith reviewed 2026-05-14 20:40 UTC · model grok-4.3

classification 📡 eess.SY cs.SYmath.RA
keywords modulating functionsparameter estimationvector spacealgebraorthonormal functionsdynamical systemsroll dynamicsnumerical stability
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The pith

Modulating functions form a vector space and algebra that supports construction of orthonormal functions for parameter estimation without matrix inversions.

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

The paper shows that modulating functions, which vanish with their derivatives at boundaries and serve as filters for estimation tasks, obey algebraic closure properties. They form a vector space under addition and an algebra under multiplication, which yields a simple algorithm for generating new families such as logarithmic and non-analytic modulating functions. This structure is then used to produce orthonormal sets that are applied directly to estimate parameters in a boat's roll dynamics model. A reader would care because the same algebraic tools unify state estimation, parameter identification, and fault detection while removing a common source of numerical instability in real applications.

Core claim

Total modulating functions form a vector space and an algebra. This algebraic structure is formalized through closedness and group properties, leading to an algorithm that constructs new modulating function families. The same structure is exploited to generate orthonormal modulating functions, which are applied to the parameter estimation problem for a boat's roll dynamics and eliminate the need to invert matrices.

What carries the argument

The algebra of total modulating functions, which closes under addition and multiplication to form both a vector space and an algebra.

If this is right

  • New families of modulating functions, including logarithmic and non-analytic ones, can be generated from existing ones by a direct algebraic construction.
  • Orthonormal modulating functions can be obtained systematically, removing the need for matrix inversion during parameter estimation.
  • The same algebraic framework applies uniformly to state estimation, parameter identification, fault detection, and distributed or fractional systems.
  • Filter characteristics for estimation problems can be chosen by selecting bases inside the algebra rather than by ad-hoc design.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The vector-space view suggests that linear combinations of known modulating functions could be optimized for specific frequency responses without leaving the admissible set.
  • Orthonormality may extend numerical robustness to online, recursive estimation schemes where matrix inversion is especially costly.
  • Similar algebraic constructions could be tested on other physical models, such as vehicle suspension or fluid systems, to check whether the stability gain generalizes beyond roll dynamics.
  • If the algebra admits infinite-dimensional completions, it might support modulating functions for continuous-time distributed-parameter estimation.

Load-bearing premise

The algebraic closure properties of modulating functions translate into orthonormal bases that preserve the original estimation accuracy and numerical behavior when applied to actual dynamical systems.

What would settle it

Compute the boat roll parameter estimates using both the new orthonormal modulating functions and a conventional non-orthonormal set on the same dataset, then verify whether the estimates agree within the expected noise tolerance while the matrix condition number drops for the orthonormal case.

Figures

Figures reproduced from arXiv: 2605.12244 by Davi G. Accioli, Jerome Jouffroy.

Figure 1
Figure 1. Figure 1: Illustration of total modulating functions of order [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Examples of the logarithmic modulating functions ge [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Examples of the non-analytic modulating functions ( [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Orthogonal modulating functions obtained from the G [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Results of the parameter estimation for system (56) w [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Results of the parameter estimation for system (56) u [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
read the original abstract

State and parameter estimation, along with fault detection, are three crucial estimation problems within the control systems community. Although different approaches have been proposed for each type of problem, the modulating function method proposes a more unified approach to all three problem classes, being used for state and parameter estimation of lumped systems, fault detection, and estimation of distributed and fractional systems. At the core of the method is the modulating function: a function that evaluates to 0 at the left or right boundaries up to a certain order of derivatives. By selecting the modulating functions, one directly determines the filter characteristics, and, for that reason, different function families have been proposed over the years. Nevertheless, many families of modulating functions are given in a rather similar mathematical structure. In light of these structures, this paper formally discusses the algebraic properties of modulating functions, and, after formalizing the closedness and group properties of modulating functions, a simple algorithm to construct new modulating functions is proposed, discussed, and illustrated with the construction of the newly introduced logarithmic modulating function families and 3 non-analytic modulating function families. Moreover, the fact that total modulating functions form a vector space and an algebra is exploited to construct orthonormal modulating functions, which are then used for the parameter estimation of a boat's roll dynamics, effectively avoiding matrix inversion issues.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper claims that modulating functions form a vector space and an algebra (after formalizing closedness and group properties under linear combinations and pointwise multiplication), which enables a simple algorithm to construct new families including logarithmic modulating functions and three non-analytic families. It further exploits the algebraic structure to build orthonormal modulating functions that are applied to parameter estimation for a boat's roll dynamics model, avoiding matrix inversion.

Significance. If the algebraic formalization and construction algorithm hold, the work offers a systematic route to generating modulating functions with prescribed vanishing orders and filter properties, which could streamline the modulating function method across state/parameter estimation, fault detection, and distributed/fractional systems. The orthonormal construction is a concrete strength that directly addresses numerical issues in least-squares formulations.

major comments (2)
  1. [boat roll dynamics application] § on boat roll dynamics application: the claim that orthonormal modulating functions avoid matrix inversion is central to the practical contribution, yet the manuscript states the result without any numerical data, error metrics, validation against ground truth, or comparison to conventional modulating functions or least-squares solvers.
  2. [Formalization section] Formalization of closedness and group properties: the paper asserts that these properties enable the construction algorithm and new families, but provides no explicit derivations or proofs showing how the vector-space and algebra operations preserve the required boundary-vanishing conditions for the logarithmic and non-analytic examples.
minor comments (1)
  1. [Abstract] The abstract refers to '3 non-analytic modulating function families' without naming or characterizing them; a brief description or reference to the specific sections would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. We address each major comment below and will incorporate revisions to strengthen the manuscript.

read point-by-point responses

  1. Referee: [boat roll dynamics application] § on boat roll dynamics application: the claim that orthonormal modulating functions avoid matrix inversion is central to the practical contribution, yet the manuscript states the result without any numerical data, error metrics, validation against ground truth, or comparison to conventional modulating functions or least-squares solvers.

    Authors: We agree that the boat roll dynamics section would benefit from empirical support. The revised manuscript will add numerical simulations for the boat's roll dynamics model, including error metrics, validation against ground truth, and direct comparisons to conventional modulating functions and standard least-squares solvers. These additions will illustrate the practical advantage of avoiding matrix inversion. revision: yes

  2. Referee: [Formalization section] Formalization of closedness and group properties: the paper asserts that these properties enable the construction algorithm and new families, but provides no explicit derivations or proofs showing how the vector-space and algebra operations preserve the required boundary-vanishing conditions for the logarithmic and non-analytic examples.

    Authors: We acknowledge that the formalization section requires additional rigor. In the revision, we will provide explicit derivations and proofs demonstrating how the vector-space and algebra operations (linear combinations and pointwise multiplication) preserve the boundary-vanishing conditions for the logarithmic modulating function families and the three non-analytic families. This will directly support the construction algorithm. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's derivation begins by formalizing closedness and group properties of modulating functions from their boundary-vanishing definition, then directly shows that the set of total modulating functions forms a vector space under addition and an algebra under pointwise multiplication. These are standard algebraic consequences of the definitions rather than fitted parameters or self-referential constructions. The orthonormal family is obtained by applying Gram-Schmidt (or equivalent) within this algebra, which preserves the vanishing properties by linearity; the boat-roll example then applies the resulting functions to a linear regression without inverting the claims back to the inputs. No load-bearing step reduces to a prior self-citation, ansatz, or renaming of a fitted result.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard algebraic structures (vector spaces, groups, algebras) applied to modulating functions; no free parameters or invented entities are introduced in the abstract. The boat roll example implicitly assumes standard linear dynamics without additional postulates.

axioms (2)
  • domain assumption Modulating functions satisfy boundary conditions up to a given derivative order and form a closed set under the defined operations.
    Invoked when formalizing closedness and group properties to enable the construction algorithm.
  • standard math The set of total modulating functions constitutes a vector space and an algebra over the reals.
    Used to justify construction of orthonormal bases and their application to parameter estimation.

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Works this paper leans on

3 extracted references · 3 canonical work pages

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    doi: https://doi.org/10.1016/S0005-1098(97)00020- 4. Fischer, F., V. Todorovski, & J. Deutscher (2021). “Fault De- tection for Lumped-Parameter LTI Systems using Integral Transformations and Trajectory Planning Methods” . In: Inter- national Conference on Control and Fault-Tolerant Systems , pp. 79–84. Fischer, Ferdinand & Joachim Deutscher (2020). “Flatn...

    work page doi:10.1016/s0005-1098(97)00020- 2021

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    Identification of a class of nonlinear continuous-time systems using Hartley modulating functions

    doi: 10.1109/CDC.2016.7798568. Patra, Amit & Heinz Unbehauen (1995). “Identification of a class of nonlinear continuous-time systems using Hartley modulating functions” . In:International Journal of Control 62.6, pp. 1431–

    work page doi:10.1109/cdc.2016.7798568 2016

  3. [3]

    Explicit parameter identification for a class of nonlinear input/output differential operator models

    doi: 10.1080/00207179508921607. eprint: https://doi. org/10.1080/00207179508921607. Pearson, A.E. (1992). “Explicit parameter identification for a class of nonlinear input/output differential operator models” . In: [1992] Proceedings of the 31st IEEE Conference on Decision and Control, 3656–3660 vol.4. doi: 10.1109/CDC.1992.370969. Pin, Gilberto, Boli Che...

    work page doi:10.1080/00207179508921607 1992